Tarleton State University.The MTR TOT for 9-12 is designed to be a two-day training. It includes an...

Table of Contents

MathematicsTEKSRefinementProject (9-12)

1 Refinement in theMathematics TEKS

2 Texas Essential Knowledgeand Skills Mathematics

3 Algebra I

4 Algebra II

5 Geometry

6 Assessment

7 Math for EnglishLanguage Learners (MELL)

Mathematics TEKS Refinement 2006 – 9-12 Tarleton State University

Introductory Materials i

© 2006 Texas Education Agency and Tarleton State University Copyright © Notice: The materials to which this notice is affixed (Materials) are copyrighted as the property of the Texas Education Agency (TEA) and Tarleton State University and may not be reproduced without the express written permission of Tarleton State University, except under the following conditions:

1. Texas public school districts, charter schools, and Education Service Centers may reproduce and use copies of the Materials for the districts’ and schools’ own educational use, without obtaining permission from Tarleton State University.

2. Residents of the State of Texas may reproduce and use copies of the Materials for individual and personal use only without obtaining written permission from Tarleton State University.

3. Any portion reproduced must be reproduced in its entirety and remain unedited, unaltered, and unchanged in any way. Distribution beyond the professional development intent of the original grant is prohibited.

4. This copyright notice may not be removed. 5. No monetary charge can be made for reproduced materials or any document

containing them; however, a reasonable charge to cover only the cost of reproduction and distribution may be charged.

Private entities or persons located in Texas that are not Texas public school districts, Texas Education Service Centers, or Texas charter school or any entity, whether public or private, educational or non-educational, located outside the State of Texas, MUST obtain written approval from Tarleton State University and will be required to enter into a license agreement that may involve payment of a licensing fee or royalty. For information contact: Pam Littleton Professor of Mathematics Tarleton State University Box T-0470 Stephenville, TX 76402 [email protected] 254.968.9705 254.968.9534 FAX


Introductory Materials ii

The Mathematics TEKS Refinement (MTR) project was developed under the direction and assistance of the following: Project Director Pam Littleton Tarleton State University 9-12 Lead Writer Bowen Brawner Tarleton State University Writers Joe Ashcraft Fort Worth ISD Pamela Harris Consultant Pam Littleton Tarleton State University Betsey Norris Fort Worth ISD Mary Peters Tarleton State University Kay Teague Tarleton State University Bryant Wyatt Tarleton State University Editors Veronica Meeks Fort Worth ISD Lynn Rambo Fort Worth ISD, Retired Doug Rensi Math Editor, Austin, TX Kay Teague Tarleton State University Advisory Committee Beverly Anderson Region XVII ESC Jamie Ashby Texarkana College Cathy Banks Texas Womans University Jennie Bennett Houston ISD Sana Brennan Region IV ESC Eduardo Cancino Hidalgo ISD Anita Carter Fort Worth ISD Shirl Chapman Region VII ESC Dr. James Epperson UT-Arlington Linda Gann Northside ISD (San Antonio) Amy Gaskins Delmar College Gaye Glenn Region II ESC Lynn Granzin Region XV ESC Kathy Hale Region XIV ESC Basia Hall Houston ISD Donna Harris Region XI ESC Diane McGowan Consultant Lois Moseley Consultant Becky Ontiveros Region XIX ESC


Introductory Materials iii

Jessica Pussman Socorro ISD (El Paso) Tamara Ramsey Region XIII ESC Cindy Schimek Katy ISD Kristy Smith Conroe ISD Pam Walker Dana Center Jim Wohlgehogen Plano ISD A special thanks to Fort Worth ISD and Texas Rural Systemic Initiative (TRSI) for allowing us to work with their teachers to field test content. In addition, a heartfelt thank you to all dedicated individuals that participated in this project at the Dallas, Arlington, and Fort Worth meetings. This work was made possible by your thoughtful suggestions, insights, and contributions. Project Evaluation Evaluation services are provided by the West Texas Office of Evaluation and Research (WTER) at West Texas A&M University in Canyon, Texas. WTER Associates involved in the MTR evaluation include: Amy Andersen Jarvis Hampton Gary Kelley Judy Kelley Russell Lowery-Hart Jessica Mallard Jan Spears


Introductory Materials iv

Mathematics TEKS Refinement (MTR) Training of Trainers MTR Philosophy Teachers at all levels need to be aware of the state curriculum and the refinements that have occurred. Teachers need to be aware of the knowledge level of the students arriving in the classrooms. Teachers also need to understand how the knowledge the students acquire within the classroom will affect the students’ learning in the future. As life-long learners, it is important for teachers to increase their mathematical content knowledge. This includes knowledge regarding children’s mathematical understandings and the teacher’s conceptual knowledge of mathematical content. Assessment is an on-going process that provides teachers information about their students’ levels of learning in order to guide instructional practices. Teachers who understand the foundations of mathematics are able to identify student weaknesses and to provide necessary instruction to improve student understanding. Assessment strategies within the MTR training elicit deep levels of mathematical understanding and proficiency. MTR training is designed to balance intense thinking with hands-on experiences. Throughout MTR training, mathematical ideas will be presented using multiple representations (concrete, pictorial, tabular, verbal, and symbolic). The use of multiple representations helps both teachers and students understand mathematical relationships in different ways. The emphasis of MTR Training of Trainers (TOT) is on mathematics, not on learning about particular manipulative materials. However, such tools are used in a variety of ways throughout the training. MTR Overview The MTR TOT for 9-12 is designed to be a two-day training. It includes an overview of the refined Texas Essential Knowledge and Skills (TEKS), specific information to address the refinements regarding Algebra 1, Algebra 2 and Geometry, and how to use assessment to improve instruction. The following is a suggested schedule in delivering the material: Day 1: TEKS Refinements Tabs 1-2 (3 hours)

Algebra and Geometry Tab 3-5 (3 hours) Day 2: Algebra and Geometry continued Tab 3-5 (3 hours) Assessment Tab 6 (3 hours) Within the MTR training, participants will investigate the refinements to the TEKS and consider implications on instruction. In addition, they will examine the refinements vertically to understand how instruction at each grade level impacts and complements


Introductory Materials v

other grades. Special attention will be given to the implementation of refined TEKS and the modifications that will need to take place to make sure all children receive instruction in the new concepts.

The second topic is selected content in Algebra 1, Algebra 2, and Geometry. The content modules explore concepts in-depth, providing the participants with an understanding beyond that in ordinary secondary texts. The idea is to provide the background knowledge needed to understand the underlying nature of the mathematics and the use of language.

During the assessment module, participants will develop definitions for formative and summative assessments to inform classroom instruction. Examples of student work will be shown to demonstrate various types of assessment. Even though this training of trainers is a two day (12 hour) session, as a trainer you should use your professional judgment in deciding scheduling. New teachers and teachers that have not been involved in alignment activities would benefit greatly by doing a complete alignment rather than just focusing on the refinements. Trainers should keep in mind that the activities in this training use a hands-on approach which facilitates learning for all students and is especially appropriate for ELL and other students with special needs. Building a strong conceptual foundation for mathematics is important for all students’ success. Some materials are not available within the tabs of the notebook, but they can be accessed by clicking on links provided in materials lists. Since the TOT materials are intended for use by trainers only, they should not be shared with individuals who have not completed the TOT and signed the MTR trainer agreement. The MTR writers have developed student lessons that address significant changes in the TEKS. These lessons should be shared only with teachers that have completed the MTR training. The student lessons can be accessed through links on this CD. It is the hope of the writing team that these materials will be used as a springboard for continual professional development. The state of Texas has many resources and several in development. Don’t forget that TEXTEAMS is still a wonderful resource. Encourage teachers to continue to use what is working, but they must apply the lens of the refinements to make sure the content is on target with what the state expects. Suggested Scope and Sequence for MTR TOT This schedule is flexible and is provided as a guideline for trainers. The delivery of materials will depend upon the needs of the participants. The ordering of selection


Introductory Materials vi

content maybe rearranged. Several of the activities are layered in parts allowing the trainer to control the depth to which the concept is explored. Day 1 Day 2 Morning

TEKS Refinements o What are the

Changes?

Selected Content o Understanding

Correlation with a Visual Model

o Picture This! o Texas “T”

Afternoon Selected Content o Understanding

Functions o Spaghetti Regression

Assessment o Assessment Pyramid o What’s Your

Problem? o The Power of

Creating o Assessment Should

Drive Instruction

Selected Content Understanding Functions Function is the single most important concept in all of mathematics. Functions enable us to mathematically model and describe the world around us. The intent of this module is to increase the understanding of Algebra through functions. We use the process definition of function which is best suited for teaching and understanding algebra. Additional emphasis is placed on the function-based perspective of an equation. Spaghetti Regression Fitting the graph of an equation to a data set is covered in all mathematics courses from Algebra I to Calculus and beyond. This module explores the concept in-depth providing the background knowledge needed to understand the process of modeling. This activity is layered in three parts plus an appendix. Understanding Correlation with a Visual Model This module encourages participants to use interactive applets to explore the meaning of the correlation coefficient. The relationship between correlation and causation is also studied. This activity is layered in three parts.


Introductory Materials vii

Picture This! This activity encourages participants to explore and draw orthographic and isometric views. It explores some non-technical aspects of orthographic drawings and the relationship between isometric and orthographic drawings. Texas “T” This activity encourages participants to explore patterns in perimeter, surface area, and volume of similar figures when the dimensions are changed.


Tab 1: Refinement in the Mathematics TEKS: Table of Contents 1-i

Tab 1: Refinement in the Mathematics TEKS Table of Contents

Master Materials List 1-ii What Are the Changes 9-12? 1-1 Transparency 1 1-5 Handout 1 1-6 Transparency 2 1-7

Algebra I, Algebra II, and Geometry TEKS 1-8

6-8 Mathematics TEKS 1-19

K-12 Mathematics TEKS 1-30

Significant Changes Guide 9-12 1-71

Significant Changes Chart 9-12 1--85


Tab 1: Refinement in the Mathematics TEKS: Master Materials List 1-ii

Tab 1: Refinement in the Mathematics TEKS Master Materials List

Chart paper Highlighters Markers Algebra I, Algebra II, and Geometry TEKS 6-8 Mathematics TEKS K-12 Mathematics TEKS What Are the Changes 9-12? Transparencies and Handouts The following materials are not in the notebook. They can be accessed on the CD through the links below. Mathematics TEKS for Grades 6-8, Algebra 1, Geometry, and Algebra 2 on mailing

labels Algebra 1, Algebra 2, and Geometry TEKS with a blank column for notes PowerPoint


What Are the Changes 9-12 1-1

Activity: What Are the Changes 9-12? TEKS: Mathematics 6th- 8th grade TEKS and TEKS for Algebra I, Algebra II, and

Geometry Materials: Chart paper TEKS for Algebra 1, Algebra 2, and Geometry (page 1-8 – 1-18)

Mathematics TEKS for Grades 6-8, Algebra 1, Geometry, and Algebra 2 on mailing labels, may be used instead of cutting the copies of the TEKS

6-8 Mathematics TEKS (page 1-19 – 1-29) K-12 Mathematics TEKS, one per group (page 1-30 - 72)

Algebra I, Algebra II, and Geometry TEKS with a blank column for notes

Transparencies 1 and 2 (pages 1-5, 1-7) Markers Highlighters PowerPoint slides Handout 1 (page 1-6) Significant Changes Guide 9-12 (page 1-73 – 1-86) Significant Changes Chart 9-12 (page 1-87 – 1-92) Overview: Participants will investigate the refinements to the Texas Essential

Knowledge and Skills and consider implications to instruction. They will also look at the refinements vertically to understand how instruction in each course and at preceding grade levels impact and complements each other. Special attention will be given to the implementation of new TEKS and the modifications that will need to take place to make sure all students receive instruction in the new concepts.

Grouping: Large group and small group Time: 3 hours Lesson:

Procedures Notes 1. Share PowerPoint slides that are relevant

to changes in the 6-12 TEKS. (See the materials list for link to the PowerPoint.)

Point out that this is just a sample of what will happen to the TEKS and the TAKS over the next two years and what needs to be taught for student success on TAKS.

This gives teachers a reason to want to do the alignment activity because it is just a sample of the critical changes to the TEKS and TAKS that will impact instruction in the classroom. What type of instructional strategies and questioning techniques must



Procedures Notes teachers use to address the spirit of the TEKS? What should the culture of the mathematics classroom look like and sound like?

2. Sort participants by course: Algebra I, Algebra II, and Geometry. It is okay to have several groups working on the same course because one group may report back differently from the other.

If a teacher teaches more than one course, then he/she must choose one course for this activity. Make sure that you have at least one group for each course.

3. Focus of each course group is:

Algebra I: Consider the refinements in Algebra I TEKS and look back for connections in grades 6-8 TEKS and forward to connections in Algebra II. Geometry: Consider the refinements in Geometry TEKS and look back for connections in Algebra I and Grades 6-8 Geometry and Measurement Strands and forward to connections in Algebra II. Algebra II: Consider the refinements in Algebra II TEKS and look back for connections in Geometry, Algebra I, and 8th Grade (if time permits) and forward to connections in PreCalculus.

Stress that it is really important that the participants not ignore the Basic Understandings for the assigned course.

4. Determine where each group will work and have teachers move.

Have someone from each group pick up the TEKS that the group will need.

5. Direct each group to study the assigned content and starting with the refinements to the assigned course, look “forward” and “back” to the assigned grade level and course TEKS.

Each group will make a chart showing: • the “new” or refined TEKS • where the concept was introduced • where the concept is mastered • any gaps. (These are places in the

TEKS where a concept is addressed

Use Transparency 1 (page 1-5) and make sure instructions are understood. Each group will have TEKS for the assigned course (pages 1-8 – 1-18) and the grade 6-8 TEKS (pages 1-19 – 1-30). Handout 1 (page 1-6) will help groups organize. Encourage participants to follow a concept back to where it is first



Procedures Notes before the course and after the course but not in the course being studied.)

Because of the nature of the activity, some groups will finish before others. For those that finish early, have them look through the materials for English Language Learners in Tab 7. (Looking at these materials will be a working lunch assignment for everyone.)

introduced. Suggest that groups cut apart TEKS one course at a time. Also, since the K&S statement may be cut apart from the SE, suggest that they label each SE with the K&S number. (See the materials list for a link for the TEKS on peel and stick labels.) A set of the K-12 TEKS (page 1-30 – 1-72) should also be provided so that participants can look to see if something deleted was actually moved to a different course or strand. **If you are working with a group of novice teachers or teachers that have not been part of an alignment activity, you have the materials to do a complete alignment instead of just focusing on the refinements. This will take more time but will be very beneficial in the long run. Working with high school teachers, the complete alignment should be grades 6-12 at a minimum.

5. While in the course groups, participants should look at the changes in each course and identify the “most significant” changes by course.

Each person should record biggest changes for his/her particular course on the recording sheet provided.

6. Use Transparency 2 (page 1-7) to give instructions for the next part of the activity.

Post the most significant changes as well as the vertical alignment of the changes around the room. Make sure all algebra 1 groups post their findings together. Do the same for Geometry and Algebra II groups.

The debriefing of this part of the activity should first involve a sharing among the course groups to compare their findings. Groups may need to make modifications to their findings and may even decide to make a combined chart from the work that they have done individually. A gallery walk will follow so each group needs to make sure that they are clearly presenting to other groups what they have learned about the changes in the



Procedures Notes TEKS for their course. Note: If something is deleted from the TEKS, the group needs to determine whether it has moved to a different course or if it is really deleted. Also, if something is added, is it really new or moved from another course? When debriefing this part of the activity, ask participants to stick to the changes instead of giving an overview of the course.

7. Conduct a gallery walk so that all participants see the work completed for all three courses.

8. Return to the large group and conduct a debriefing of the significant changes in the three courses as well as connections that have been discovered during the participants work.

A guide (pages 1-73 – 1-86) that highlights the significant changes and a chart (pages 1-87 – 1-92) with information about the type of changes and notes about those changes are provided for the Trainers to use. Copies of the TEKS on wide paper with the blank column provide a good place for participants to make notes about significant changes. (See the materials list for link to this version of the TEKS.)


Transparency 1 What Are the Changes 9-12? 1-5

Study your assigned course piece. Identify the “most significant” changes within the

course.

On chart paper develop a vertical alignment for the changes in the assigned course part. Note any gaps or overlaps in the concepts. Trace each concept back to

where it was first introduced. Be sure to note when a concept should be mastered.

If a concept has been deleted, is it no

longer a part of the TEKS, or has it been moved to another course?

If a concept has been added to your strand,

is it really new or has it been moved from another course?

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Transparency 2 What Are the Changes 9-12? 1-7

Record “most significant” changes for your course on chart paper.

Post



Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter C. High School

§111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12.

The provisions of this subchapter shall be implemented beginning September 1, 1998, and at that time, shall supersede ¤75.63(e)-(g) of this title (relating to Mathematics).

§111.32. Algebra I (One Credit). (a) Basic understandings.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities.

(3) Function concepts. [Functions represent the systematic dependence of one quantity on another.] A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations , and to analyze and interpret relationships.

(4) Relationship between equations and functions. Equations and inequalities arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and inequalities and use a variety of methods to solve them [these equations] .

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical , and verbal ), tools, and technology [,] ( including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers ) to [with graphing capabilities and] model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, and reasoning (justification and proof) to make connections within and outside mathematics . Students also use [, and reasoning, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts [justification and proof].

(b) Knowledge and skills (A.1) Foundations for functions. The

student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways.

The student is expected to: (A) describe[s] independent and dependent quantities in

functional relationships ; [.] (B) gather[s] and record[s] data [,] and use [or uses] data

sets [,] to determine functional [(systematic)] relationships between quantities ; [.]

(C) describe[s] functional relationships for given problem situations and write [writes] equations or inequalities to answer questions arising from the situations ; [.]

(D) represent[s] relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities ; and [.]

(E) interpret and make decisions, predictions, and critical judgments [interprets and makes inferences] from functional relationships.



(A.2) Foundations for functions. The student uses the properties and attributes of functions.

The student is expected to: (A) identify[s] and sketch[s] the general forms of linear (y =

x) and quadratic (y = x2) parent functions ; [.]

(B) identify[s] the mathematical domains and ranges and determine [determines] reasonable domain and range values for given situations, both continuous and discrete; [.]

(C) interpret[s] situations in terms of given graphs or creates situations that fit given graphs ; and [.]

(D) collect[s] and organize[s] data, make and interpret [makes and interprets] scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations) , and model, predict, and make [models, predicts, and makes] decisions and critical judgments in problem situations .

(A.3) Foundations for functions. The

student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations.

The student is expected to: (A) use[s] symbols to represent unknowns and variables;

and [.] (B) look[s] for patterns and represent [represents]

generalizations algebraically.

(A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

The student is expected to: (A) find[s] specific function values, simplify [simplifies]

polynomial expressions, transform and solve [transforms and solves] equations, and factor [factors] as necessary in problem situations ; [.]

(B) use[s] the commutative, associative, and distributive properties to simplify algebraic expressions ; and [.]

(C)connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

(A.5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations.

The student is expected to: (A) determine[s] whether or not given situations can be

represented by linear functions ; [.] (B) determine[s] the domain and range [values] for [which]

linear functions in [make sense for] given situations ; and [.]

(C) use[s], translate[s], and make connections [The student translates] among [and uses] algebraic, tabular, graphical, or verbal descriptions of linear functions.

(A.6) Linear functions. The student

understands the meaning of the slope and intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations.

The student is expected to: (A) develop[s] the concept of slope as rate of change and

determine [determines] slopes from graphs, tables, and algebraic representations ; [.]

(B) interpret[s] the meaning of slope and intercepts in situations using data, symbolic representations, or graphs ; [.]

(C) investigate[s], describe[s], and predict[s] the effects of changes in m and b on the graph of y = mx + b ; [.]

(D) graph[s] and write[s] equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept ; [.]



(E) determine[s] the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations ; [.]

(F) interpret[s] and predict[s] the effects of changing slope and y-intercept in applied situations ; and [.]

(G) relate[s] direct variation to linear functions and solve [solves] problems involving proportional change.


formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) analyze[s] situations involving linear functions and

formulate [formulates] linear equations or inequalities to solve problems ; [.]

(B) investigate[s] methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select [selects] a method, and solve [solves] the equations and inequalities ; and [.]

(C) interpret[s] and determine[s] the reasonableness of solutions to linear equations and inequalities.


formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) analyze[s] [The student analyzes] situations and

formulate [formulates] systems of linear equations in two unknowns to solve problems ; [.]

(B) solve[s] [The student solves] systems of linear equations using concrete models, graphs, tables, and algebraic methods ; and [.]

(C) interpret[s] and determine[s] [For given contexts, the student interprets and determines] the reasonableness of solutions to systems of linear equations.

(A.9) Quadratic and other nonlinear

functions. The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions.

The student is expected to: (A) determine[s] the domain and range [values] for [which]

quadratic functions in [make sense for] given situations ; [.]

(B) investigate[s], describe[s], and predict[s] the effects of changes in a on the graph of y = ax2 + c; [y = ax2 .]

(C) investigate[s], describe[s], and predict[s] the effects of changes in c on the graph of y = ax2 + c; and [y = x2 + c.]

(D) analyze[s] graphs of quadratic functions and draw [draws] conclusions.


functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

The student is expected to: (A) solve[s] quadratic equations using concrete models,

tables, graphs, and algebraic methods ; and [.] (B) make connections among the solutions (roots) [The

student relates the solutions] of quadratic equations , the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph [to the roots] of the function [their functions].


functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and

The student is expected to: (A) use[s] patterns to generate the laws of exponents and

apply [applies] them in problem-solving situations ; [.] (B) analyze[s] data and represent [represents] situations

involving inverse variation using concrete models,



models the situations. tables, graphs, or algebraic methods ; and [.] (C) analyze[s] data and represent [represents] situations

involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.




§111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12.


§111.33. Algebra II (One-Half to One Credit). (a) Basic understandings.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra.

(3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.

(4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical , and verbal ), tools, and technology [,] ( including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers ) to [with graphing capabilities and] model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, and reasoning (justification and proof) to make connections within and outside mathematics . Students also use [, and reasoning, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts [justification and proof] .

(b) Knowledge and skills. (2A.1) Foundations for functions. The

student uses properties and attributes of functions and applies functions to problem situations.

The student is expected to (A) identify[s] [For a variety of situations, the student

identifies] the mathematical domains and ranges of functions and determine [determines] reasonable domain and range values for continuous and discrete [given] situations ; and [.]

(B) collect[s] and organize[s] [In solving problems, the student collects data and records results, organizes the] data, make and interpret [makes] scatterplots, fit [fits] the graph of a [curves to the appropriate parent] function to the data , interpret [interprets] the results, and proceed [proceeds] to model, predict, and make decisions and critical judgments.



(2A.2) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

The student is expected to: (A) use[s] tools including [matrices,] factoring [,] and

properties of exponents to simplify expressions and to transform and solve equations ; and [.]

(B) use[s] complex numbers to describe the solutions of quadratic equations.

[(C) The student connects the function notation of y = and f(x) =.]

(2A.3) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations.

The student is expected to: (A) analyze[s] situations and formulate [formulates] systems

of equations in two or more unknowns or inequalities in two [or more] unknowns to solve problems ; [.]

(B) use[s] algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities ; and [.]

(C) interpret[s] and determine[s] the reasonableness of solutions to systems of equations or inequalities for given contexts.

(2A.4) Algebra and geometry. The

student connects algebraic and geometric representations of functions.

The student is expected to: (A) identify[s] and sketch[s] graphs of parent functions,

including linear (f(x) = x) [(y = x)] , quadratic (f(x) = x2) [(y = x2)] , [square root (y = √x), inverse (y = 1/x),] exponential (f(x) = ax) [(y = ax)] , and logarithmic (f(x) = logax) [(y = logax)] functions , absolute value of x (f(x) = |x|), square root of x (f(x) = √x), and reciprocal of x (f(x) = 1/x); [.]

(B) extend[s] parent functions with parameters such as a in f(x) = a/x [m in y = mx] and describe the effects of the [describes] parameter changes on the graph of parent functions ; and [.]

(C) describe and analyze the relationship between a function and its inverse [The student recognizes inverse relationships between various functions] .


student knows the relationship between the geometric and algebraic descriptions of conic sections.

The student is expected to: (A) describe[s] a conic section as the intersection of a plane

and a cone ; [.] (B) [In order to] sketch graphs of conic sections to relate[s] [,

the student relates] simple parameter changes in the equation to corresponding changes in the graph ; [.]

(C) identify[s] symmetries from graphs of conic sections ; [.] (D) identify[s] the conic section from a given equation ; and

[.] (E) use[s] the method of completing the square.

(2A.6) Quadratic and square root functions. The student understands that quadratic functions can be represented in different ways and translates among their various representations.

The student is expected to: (A) determine[s] the reasonable domain and range values of

quadratic functions, as well as interpret[s] and determine[s] [interprets and determines] the reasonableness of solutions to quadratic equations and inequalities ; [.]

(B) relate[s] representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions ;



and [.] (C) determine[s] a quadratic function from its roots or a

graph.

(2A.7) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations.

The student is expected to: (A) use[s] characteristics of the quadratic parent function to

sketch the related graphs and connect [connects] between the y = ax

2 + bx + c and the y = a(x - h)

2 + k

symbolic representations of quadratic functions ; and (B) use[s] the parent function to investigate, describe, and

predict the effects of changes in a, h, and k on the graphs of y = a(x - h)

2 + k form of a function in applied

and purely mathematical situations.

(2A.8) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation

The student is expected to: (A) analyze[s] situations involving quadratic functions and

formulate [formulates] quadratic equations or inequalities to solve problems ; [.]

(B) analyze[s] and interpret[s] the solutions of quadratic equations using discriminants and solve [solves] quadratic equations using the quadratic formula ; [.]

(C) compare[s] and translate[s] between algebraic and graphical solutions of quadratic equations ; and [.]

(D) solve[s] quadratic equations and inequalities using graphs, tables, and algebraic methods .

(2A.9) Quadratic and square root

functions. The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) use[s] the parent function to investigate, describe, and

predict the effects of parameter changes on the graphs of square root functions and describe [describes] limitations on the domains and ranges ; [.]

(B) relate[s] representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions ; [.]

(C) determine[s] the reasonable domain and range values of square root functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to square root equations and inequalities ; [.]

(D) determine solutions of [The student solves] square root equations [and inequalities] using graphs, tables, and algebraic methods ; [.]

(E) determine solutions of square root inequalities using graphs and tables;

(F) analyze[s] situations modeled by square root functions, formulate [formulates] equations or inequalities, select [selects] a method, and solve [solves] problems ; and [.]

(G) connect [The student expresses] inverses of square root functions with quadratic functions [using square root functions].

(2A.10) Rational functions. The student

formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and

The student is expected to: (A) use[s] quotients of polynomials to describe the graphs of

rational functions, predict the effects of parameter changes, describe [describes] limitations on the domains and ranges, and examine [examines] asymptotic



analyzes the solutions in terms of the situation.

behavior ; [.] (B) analyze[s] various representations of rational functions

with respect to problem situations ; [.] (C) determine[s] the reasonable domain and range values of

rational functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to rational equations and inequalities ; [.]

(D) determine the solutions of [The student solves] rational equations [and inequalities] using graphs, tables, and algebraic methods ; [.]

(E) determine solutions of rational inequalities using graphs and tables;

(F) analyze[s] [The student analyzes] a situation modeled by a rational function, formulate [formulates] an equation or inequality composed of a linear or quadratic function, and solve [solves] the problem ; and [.]

(G) use[s] [The student uses direct and inverse variation] functions to model and [as models to] make predictions in problem situations involving direct and inverse variation .

(2A.11) Exponential and logarithmic

functions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) develop[s] [the definition of logarithms by exploring and

describing the relationship between exponential functions and their inverses ; [.]

(B) use[s] the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe [describes] limitations on the domains and ranges, and examine [examines] asymptotic behavior ; [.]

(C) determine[s] the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to exponential and logarithmic equations and inequalities ; [.]

(D) determine solutions of [The student solves] exponential and logarithmic equations [and inequalities] using graphs, tables, and algebraic methods ; [.]

(E) determine solutions of exponential and logarithmic inequalities using graphs and tables; and

(F) analyze[s] a situation modeled by an exponential function, formulate [formulates] an equation or inequality, and solve [solves] the problem.




§111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12. The provisions of this subchapter shall be implemented beginning September 1, 1998, and at that time, shall supersede ¤75.63(e)-(g) of this title (relating to Mathematics).

§111.34. Geometry (One Credit). (a) Basic understandings.


(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric [shapes and] figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them.

(3) Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures.

(4) The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems.

(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal [algebraic, and coordinate] ), tools, and technology [,] ( including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers ) [with graphing capabilities] to solve meaningful problems by representing and transforming figures [, transforming figures,] and analyzing relationships [, and proving things about them] .

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use [, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts [justification and proof].

(b) Knowledge and skills. (G.1) Geometric structure. The student

understands the structure of, and relationships within, an axiomatic system.

The student is expected to: (A) develop[s] an awareness of the structure of a

mathematical system, connecting definitions, postulates, logical reasoning, and theorems ;

(B) recognize[s] [Through] the historical development of geometric systems[, the student recognizes that] and know mathematics is developed for a variety of purposes

(C) compare[s] and contrast[s] the structures and implications of Euclidean and non-Euclidean geometries.

(G.2) Geometric structure. The student analyzes geometric relationships in order to make and verify conjectures

The student is expected to: (A) use[s] constructions to explore attributes of geometric

figures and to make conjectures about geometric relationships ; and [.]

(B) make[s] [The student makes and verifies] conjectures about angles, lines, polygons, circles, and three-



dimensional figures and determine the validity of the conjectures , choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

(G.3) Geometric structure. The student applies [understands the importance of] logical reasoning to justify and prove mathematical statements [, justification, and proof in mathematics] .

The student is expected to: (A) determine[s] the validity [The student determines if the

converse] of a conditional statement , its converse, inverse, and contrapositive; [is true or false.]

(B) construct[s] and justify[s] statements about geometric figures and their properties ; [.]

(C) use logical reasoning [The student demonstrates what it means] to prove statements are true and find counter examples to disprove [mathematically that] statements that are false; [ true.]

(D) use[s] inductive reasoning to formulate a conjecture; and [.]

(E) use[s] deductive reasoning to prove a statement.

(G.4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems

The student is expected to select[s] an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

(G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems [identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures] .

The student is expected to: (A) use[s] numeric and geometric patterns to develop

algebraic expressions representing geometric properties; [to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.]

(B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles;

(C) use[s] properties of transformations and their compositions to make connections between mathematics and the real world , [in applications] such as tessellations ; and [or fractals.]

(D) identify[s] and apply[s] [The student identifies and applies] patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

(G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures [objects] and related two-dimensional representations and uses these representations to solve problems.

The student is expected to; (A) describe and draw the intersection of a given plane with

various [The student describes, and draws cross sections and other slices of] three-dimensional geometric figures; [objects.]

(B) use[s] [The student uses] nets to represent and construct three- dimensional geometric figures; and [objects.]

(C) use orthographic and isometric views [The student uses top, front, side, and corner views] of three-dimensional geometric figures [objects] to represent and construct three-dimensional geometric figures [create accurate and complete representations] and solve problems.



(G.7) Dimensionality and the geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

The student is expected to: (A) use[s] one- and two-dimensional coordinate systems to

represent points, lines, rays, line segments, and figures ; [.]

(B) use[s] slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons ; and [.]

(C) derive and use[s] [The student develops and uses] formulas involving length, slope, [including distance] and midpoint.

(G.8) Congruence and the geometry of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, [perimeter,] and volume in problem situations.

The student is expected to: (A) find[s] areas of regular polygons, circles, and composite

figures ; (B) find[s] areas of sectors and arc lengths of circles using

proportional reasoning ; [.] (C) derive[s], extend[s], and use[s] [The student develops,

extends, and uses] the Pythagorean Theorem ; and [.] (D) find[s] surface areas and volumes of prisms, pyramids,

spheres, cones, [and] cylinders , and composites of these figures in problem situations.

(G.9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures.

The student is expected to: (A) formulate[s] and test[s] conjectures about the properties

of parallel and perpendicular lines based on explorations and concrete models; [.]

(B) formulate[s] and test[s] conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models; [.]

(C) formulate[s] and test[s] conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and [.]

(D) analyze[s] the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models .

(G.10) Congruence and the geometry of size. The student applies the concept of congruence to justify properties of figures and solve problems

The student is expected to: (A) use[s] congruence transformations to make conjectures

and justify properties of geometric figures including figures represented on a coordinate plane; and [.]

(B) justify[s] and apply[s] triangle congruence relationships.

(G.11) Similarity and the geometry of shape. The student applies the concepts of similarity to justify properties of figures and solve problems.

The student is expected to: (A) use[s] and extend [The student uses] similarity

properties and transformations to explore and justify conjectures about geometric figures ; [.]

(B) use[s] ratios to solve problems involving similar figures ; [.]

(C) develop[s], apply[s], and justify[s] triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and [.]

(D) describe[s] the effect on perimeter, area, and volume when one or more dimensions [length, width, or height] of a figure are [three-dimensional solid is] changed and apply [applies] this idea in solving problems.



Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter A. Elementary (6-8)

§111.11. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades K-5. The provisions of this subchapter shall be implemented by school districts beginning with the 2006-2007 school year. [September 1, 1998, and at that time shall supersede §75.27(a)-(f) of this title (relating to Mathematics) §111.22. Mathematics, Grade 6. (a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 6 are using ratios to describe direct proportional relationships involving number, geometry, measurement, [and] probability , and adding and subtracting decimals and fractions.

(2) Throughout mathematics in Grades 6-8, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other; and they connect verbal, numeric, graphic, and symbolic representations of relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures [objects] or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, reasoning, and concepts of probability to draw conclusions, evaluate arguments, and make recommendations.

(3) Problem solving in meaningful contexts , language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology [(at least four-function calculators for whole numbers, decimals, and fractions)] and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.

(b) Knowledge and skills. (6.1)

Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order non-negative rational numbers; (B) generate equivalent forms of rational numbers including

whole numbers, fractions, and decimals; (C) use integers to represent real-life situations; (D) write prime factorizations using exponents; [and] (E) identify factors of a positive integer, [and multiples

including] common factors , and the greatest common factor of a set of positive integers; and [common multiples.]

(F) identify multiples of a positive integer and common multiples and the least common multiple of a set of positive integers.

(6.2) Number, operation, and

quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions.

The student is expected to: (A) model addition and subtraction situations involving

fractions with objects, pictures, words, and numbers; (B) use addition and subtraction to solve problems

involving fractions and decimals; (C) use multiplication and division of whole numbers to

solve problems including situations involving equivalent ratios and rates; [and]



(D) estimate and round to approximate reasonable results and to solve problems where exact answers are not required ; and [.]

(E) use order of operations to simplify whole number expressions (without exponents) in problem solving situations.

(6.3) Patterns, relationships, and

algebraic thinking. The student solves problems involving direct proportional relationships.

The student is expected to: (A) use ratios to describe proportional situations; (B) represent ratios and percents with concrete models,

fractions, and decimals; and (C)use ratios to make predictions in proportional situations

(6.4) Patterns, relationships, and algebraic thinking. The student uses letters as variables in mathematical expressions to describe how one quantity changes when a related quantity changes.

The student is expected to: (A) use tables and symbols to represent and describe

proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change) , perimeter and [,] area [, etc.] ; and

(B) use tables of data to generate formulas representing [to represent] relationships involving perimeter, area, volume of a rectangular prism, etc. [, from a table of data.]


algebraic thinking. The student uses letters to represent an unknown in an equation.

The student is expected to formulate equations [an equation] from [a] problem situations described by linear relationships [situation].

(6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.

The student is expected to: (A) use angle measurements to classify angles as acute,

obtuse, or right; (B) identify relationships involving angles in triangles and

quadrilaterals; and (C) describe the relationship between radius, diameter, and

circumference of a circle.

(6.7) Geometry and spatial reasoning. The student uses coordinate geometry to identify location in two dimensions.

The student is expected to locate and name points on a coordinate plane using ordered pairs of non-negative rational numbers.

(6.8 Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume [capacity] , weight, and angles.

The student is expected to: (A) estimate measurements (including circumference) and

evaluate reasonableness of results; (B) select and use appropriate units, tools, or formulas to

measure and to solve problems involving length (including perimeter [and circumference] ), area, time, temperature, volume [capacity] , and weight;

(C) measure angles; and (D) convert measures within the same measurement

system (customary and metric) based on relationships between units.



(6.9) Probability and statistics. The student uses experimental and theoretical probability to make predictions.

The student is expected to: (A) construct sample spaces using lists and [,] tree

diagrams [, and combinations] ; and (B) find the probabilities of a simple event and its

complement and describe the relationship between the two.

(6.10) Probability and statistics. The

student uses statistical representations to analyze data.

The student is expected to: (A) select and use an appropriate representation for

presenting and displaying [draw and compare] different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot ;

(B) identify mean (using concrete objects and pictorial models), [use] median, mode, and range of a set of [to describe] data;

(C) sketch circle graphs to display data; and (D) solve problems by collecting, organizing, displaying,

and interpreting data.

(6.11) Underlying processes and mathematical tools. The student applies Grade 6 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.

The student is expected to: (A) identify and apply mathematics to everyday

experiences, to activities in and outside of school, with other disciplines, and with other mathematical topics;

(B) use a problem-solving model that incorporates understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and

(D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.

(6.12) Underlying processes and

mathematical tools. The student communicates about Grade 6 mathematics through informal and mathematical language, representations, and models.

The student is expected to: (A) communicate mathematical ideas using language,

efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and

(B) evaluate the effectiveness of different representations to communicate ideas.


mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions

The student is expected to: (A) make conjectures from patterns or sets of examples

and non-examples; and (B) validate his/her conclusions using mathematical

properties and relationships.



§111.23. Mathematics, Grade 7. (a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 7 are using direct proportional relationships in number, geometry, measurement, and probability; applying addition, subtraction, multiplication, and division of decimals, fractions, and integers; and using statistical measures to describe data.



(b) Knowledge and skills. (7.1) Number, operation, and

quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order integers and positive rational

numbers; (B) convert between fractions, decimals, whole numbers,

and percents mentally, on paper, or with a calculator; and

(C) represent squares and square roots using geometric models.


quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

The student is expected to: (A) represent multiplication and division situations involving

fractions and decimals with [concrete] models , including concrete objects , pictures, words, and numbers;

(B) use addition, subtraction, multiplication, and division to

solve problems involving fractions and decimals; (C) use models , such as concrete objects, pictorial models,

and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms;

(D) use division to find unit rates and ratios in proportional relationships such as speed, density, price, recipes, and student-teacher ratio;

(E) simplify numerical expressions involving order of operations and exponents;

(F) select and use appropriate operations to solve problems and justify the selections; and

(G) determine the reasonableness of a solution to a problem



(7.3) Patterns, relationships, and algebraic thinking. The student solves problems involving direct proportional relationships.

The student is expected to: (A) estimate and find solutions to application problems

involving percent; and (B) estimate and find solutions to application problems

involving proportional relationships such as similarity, scaling, unit costs, and related measurement units.


algebraic thinking. The student represents a relationship in numerical, geometric, verbal, and symbolic form.

The student is expected to: (A) generate formulas involving unit conversions,

perimeter, area, circumference, volume, and scaling; (B) graph data to demonstrate relationships in familiar

concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

(C) use words and symbols to describe the relationship between the terms in an arithmetic [a] sequence (with a constant rate of change) and their positions in the sequence


algebraic thinking. The student uses equations to solve problems.

The student is expected to: (A) use concrete and pictorial models to solve equations

and use symbols to record the actions; and (B) formulate [a possible] problem situations [situation]

when given a simple equation and formulate an equation when given a problem situation

(7.6) Geometry and spatial reasoning.

The student compares and classifies two- and three-dimensional figures [shapes and solids] using geometric vocabulary and properties.

The student is expected to: (A) use angle measurements to classify pairs of angles as

complementary or supplementary; (B) use properties to classify [shapes including] triangles

and [,] quadrilaterals [, pentagons, and circles] ; (C) use properties to classify three-dimensional figures

[solids] , including pyramids, cones, prisms, and cylinders; and

(D) use critical attributes to define similarity

(7.7) Geometry and spatial reasoning. The student uses coordinate geometry to describe location on a plane.

The student is expected to: (A) locate and name points on a coordinate plane using

ordered pairs of integers; and (B) graph reflections across the horizontal or vertical axis

and graph translations on a coordinate plane

(7.8) Geometry and spatial reasoning. The student uses geometry to model and describe the physical world.

The student is expected to: (A) sketch three-dimensional figures [a solid] when given

the top, side, and front views; (B) make a net (two-dimensional model) of the surface area

of a three-dimensional figure [solid] ; and (C) use geometric concepts and properties to solve

problems in fields such as art and architecture.

(7.9) Measurement. The student solves application problems involving estimation and measurement.

The student is expected to : [estimate measurements and solve application problems involving length (including perimeter and circumference), area, and volume.]

(A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes;



(B) connect models for volume of prisms (triangular and rectangular) and cylinders to formulas of prisms (triangular and rectangular) and cylinders; and

(C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders


student recognizes that a physical or mathematical model can be used to describe the experimental and theoretical probability of real-life events.

The student is expected to: (A) construct sample spaces for simple or composite

experiments [compound events (dependent and independent)] ; and

(B) find the [approximate] probability of independent events [a compound event through experimentation] .


student understands that the way a set of data is displayed influences its interpretation.


presenting and displaying relationships among collected data , including line plot, line graph, bar graph, stem and leaf plot, circle graph, and Venn diagrams, and justify the selection; and

(B) make inferences and convincing arguments based on an analysis of given or collected data.


student uses measures of central tendency and range to describe a set of data.

The student is expected to: (A) describe a set of data using mean, median, mode, and

range; and (B) choose among mean, median, mode, or range to

describe a set of data and justify the choice for a particular situation.


mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines, and activities in and outside of school.





(D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems





(B) evaluate the effectiveness of different representations to communicate ideas




mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions.







(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 8 are using basic principles of algebra to analyze and represent both proportional and non-proportional linear relationships and using probability to describe data and make predictions.




quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations.

The student is expected to: (A) compare and order rational numbers in various forms

including integers, percents, and positive and negative fractions and decimals;

(B) select and use appropriate forms of rational numbers to solve real-life problems including those involving proportional relationships;

(C) approximate (mentally and with calculators) the value of irrational numbers as they arise from problem situations ( such as π, √2); and

(D) express numbers in scientific notation, including negative exponents, in appropriate problem situations [using a calculator] .


quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions.

The student is expected to: (A) select [and use] appropriate operations to solve

problems involving rational numbers and justify the selections;

(B) use appropriate operations to solve problems involving [add, subtract, multiply, and divide] rational numbers in problem situations;

(C) evaluate a solution for reasonableness; and (D) use multiplication by a constant factor (unit rate) to

represent proportional relationships [; for example, the arm span of a gibbon is about 1.4 times its height, a = 1.4h] .


algebraic thinking. The student identifies proportional or non-

The student is expected to: (A) compare and contrast proportional and non-proportional

linear relationships; and



proportional linear relationships in problem situations and solves problems.

(B) estimate and find solutions to application problems involving percents and other proportional relationships such as similarity and rates.

(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship.

The student is expected to generate a different representation of data given another [one] representation of data ( such as a table, graph, equation, or verbal description ) .

(8.5) Patterns, relationships, and algebraic thinking. The student uses graphs, tables, and algebraic representations to make predictions and solve problems.

The student is expected to: (A) predict, [estimate,] find, and justify solutions to

application problems using appropriate tables, graphs, and algebraic equations; and

(B) find and evaluate [use] an algebraic expression to determine [find] any term in an arithmetic [a] sequence (with a constant rate of change) .

(8.6) Geometry and spatial

reasoning. The student uses transformational geometry to develop spatial sense.

The student is expected to: (A) generate similar figures [shapes] using dilations

including enlargements and reductions; and (B) graph dilations, reflections, and translations on a

coordinate plane.


The student is expected to: (A) draw three-dimensional figures [solids] from different

perspectives; (B) use geometric concepts and properties to solve

problems in fields such as art and architecture; (C) use pictures or models to demonstrate the Pythagorean

Theorem; and (D) locate and name points on a coordinate plane using

ordered pairs of rational numbers.

(8.8) Measurement. The student uses procedures to determine measures of three-dimensional figures [solids].

The student is expected to: (A) find lateral and total surface area of prisms , pyramids,

and cylinders using concrete models and nets (two-dimensional models);

(B) connect models [to formulas for volume] of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects ; and

(C) estimate measurements [answers] and use formulas to solve application problems involving lateral and total surface area and volume.

(8.9) Measurement. The student uses indirect measurement to solve problems.

The student is expected to: (A) use the Pythagorean Theorem to solve real-life

problems; and (B) use proportional relationships in similar two-dimensional

figures or similar three-dimensional figures [shapes] to find missing measurements.



(8.10) Measurement. The student describes how changes in dimensions affect linear, area, and volume measures.

The student is expected to: (A) describe the resulting effects on perimeter and area

when dimensions of a shape are changed proportionally; and

(B) describe the resulting effect on volume when dimensions of a solid are changed proportionally.


student applies concepts of theoretical and experimental probability to make predictions.

The student is expected to: (A) find the probabilities of dependent and independent

[compound] events [(dependent and independent)] ; (B) use theoretical probabilities and experimental results to

make predictions and decisions; and (C) select and use different models to simulate an event.

(8.12) Probability and statistics. The student uses statistical procedures to describe data.

The student is expected to: (A) select the appropriate measure of central tendency or

range to describe a set of data and justify the choice for a particular situation [purpose] ;

(B) draw conclusions and make predictions by analyzing trends in scatterplots; and

(C) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, [construct] circle graphs, bar graphs, box and whisker plots, [and] histograms, and Venn diagrams, with and without the use of technology.

(8.13) Probability and statistics. The student evaluates predictions and conclusions based on statistical data.

The student is expected to: (A) evaluate methods of sampling to determine validity of an

inference made from a set of data; and (B) recognize misuses of graphical or numerical information

and evaluate predictions and conclusions based on data analysis.



The student is expected to: (A) identify and apply mathematics to everyday experiences,

to activities in and outside of school, with other disciplines, and with other mathematical topics;



(D) select tools such as real objects, manipulatives,

paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems.



(8.15) Underlying processes and mathematical tools. The student communicates about Grade 8 mathematics through informal and mathematical language, representations, and models.






The student is expected to: (A) make conjectures from patterns or sets of examples and

nonexamples; and (B) validate his/her conclusions using mathematical



K-12 Mathematics TEKS – Marked Up Version 1-30

Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter A. Elementary

§111.11. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades K-5. The provisions of this subchapter shall be implemented by school districts beginning with the 2006-2007 school year. [September 1, 1998, and at that time shall supersede §75.27(a)-(f) of this title (relating to Mathematics) §111.12. Mathematics, Kindergarten. (a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Kindergarten are developing whole-number concepts and using patterns and sorting to explore number, data, and shape.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [patterns to describe] objects to create and identify patterns and use those patterns to [,] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from informal to formal [use informal] language [and observation of geometric properties] to describe two- and three-dimensional geometric figures [shapes, solids,] and likenesses [locations] in the physical world. Students [and] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.

(3) Throughout mathematics in Kindergarten-Grade 2, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Kindergarten-Grade 2 use basic number sense to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 2, students know basic addition and subtraction facts and are using them to work flexibly, efficiently, and accurately with numbers during addition and subtraction computation.

(4) [(3)] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Kindergarten-Grade 2, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills. (K.1) Number, operation, and

quantitative reasoning. The student uses numbers to name quantities.

The student is expected to: (A) use one-to-one correspondence and language such as

more than, same number as, or two less than to describe relative sizes of sets of concrete objects;

(B) use sets of concrete objects to represent quantities given in verbal or written form (through 20 [9]); and

(C) use numbers to describe how many objects are in a set (through 20) using verbal and symbolic descriptions.

(K.2) Number, operation, and

quantitative reasoning. The student describes order of events or objects.

The student is expected to: (A) use language such as before or after to describe relative

position in a sequence of events or objects; and (B) name the ordinal positions in a sequence such as first,

second, third, etc.



(K.3) Number, operation, and quantitative reasoning. The student recognizes that there are quantities less than a whole.

The student is expected to: (A) share a whole by separating it into two equal parts; and (B) explain why a given part is half of the whole.

(K.4) Number, operation, and quantitative reasoning. The student models addition (joining) and subtraction (separating)

.

The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(K.5) Patterns, relationships, and algebraic thinking. The student identifies, extends, and creates patterns.

The student is expected to identify, extend, and create patterns of sounds, physical movement, and concrete objects.

(K.6) Patterns, relationships, and algebraic thinking. The student uses patterns to make predictions.

The student is expected to: (A) use patterns to predict what comes next, including cause-

and-effect relationships; and (B) count by ones to 100.

(K.7) Geometry and spatial reasoning. The student describes the relative positions of objects

The student is expected to: (A) describe one object in relation to another using informal

language such as over, under, above, and below; and (B) place an object in a specified position.

(K.8) Geometry and spatial reasoning. The student uses attributes to determine how objects are alike and different.

The student is expected to: (A) describe and identify an object by its attributes using

informal language; (B) compare two objects based on their attributes; and (C) sort a variety of objects including two- and three-

dimensional geometric figures according to their attributes and describe how the objects are sorted [those groups are formed].

(K.9) Geometry and spatial

reasoning. The student recognizes attributes [characteristics] of two- and three-dimensional geometric figures [shapes and solids]..

The student is expected to: (A) describe and compare the attributes of real-life objects

such as balls, boxes, cans, and cones or models of three-dimensional geometric figures [solids];

(B) recognize shapes in real-life three-dimensional geometric figures [objects] or models of three-dimensional geometric figures [solids]; and

(C) describe, identify, and compare circles, triangles, [and] rectangles , and [including] squares (a special type of rectangle).

(K.10) Measurement. The student

directly compares the [uses] attributes of [such as] length, area, weight/mass, [weight, or] capacity , and/or relative temperature [to compare and order objects]. The student uses comparative language to solve problems and answer questions.

The student is expected to: (A) compare and order two or three concrete objects

according to length (longer/shorter than, or the same); [(shorter or longer), capacity (holds more or holds less), or weight (lighter or heavier); and]

(B) compare the areas of two flat surfaces of two-dimensional figures (covers more, covers less, or covers the same);

[(B) find concrete objects that are about the same as, less than, or greater than a given object according to length, capacity, or weight.]



(C) compare two containers according to capacity (holds

more, holds less, or holds the same); (D) compare two objects according to weight/mass (heavier

than, lighter than or equal to); and (E) compare situations or objects according to relative

temperature (hotter/colder than, or the same as).

(K.11) Measurement. The student uses time [and temperature] to describe, compare , and order events [,] and situations [, and/or objects]

The student is expected to: [(A) compare situations or objects according to temperature

such as hotter or colder;] (A) [(B)] compare events according to duration such as more

time than or less time than; (B) [(C)] sequence events (up to three); and (C) [(D)] read a calendar using days, weeks, and months.

(K.12) Probability and statistics. The student constructs and uses graphs of real objects or pictures to answer questions.

The student is expected to: (A) construct graphs using real objects or pictures in order to

answer questions; and (B) use information from a graph of real objects or pictures in

order to answer questions.

(K.13) Underlying processes and mathematical tools. The student applies Kindergarten mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (A) identify mathematics in everyday situations; (B) solve problems [use a problem-solving model,] with

guidance [,] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and

(D) use tools such as real objects, manipulatives, and technology to solve problems.

(K.14) Underlying processes and

mathematical tools. The student communicates about Kindergarten mathematics using informal language

The student is expected to: (A) communicate mathematical ideas [explain and record

observations] using objects, words, pictures, numbers, and technology; and

(B) relate everyday language to mathematical language and symbols.


mathematical tools. The student uses logical reasoning [to make sense of his or her world]

The student is expected to justify [reason and support] his or her thinking using objects, words, pictures, numbers, and technology.




(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 1 are building number sense through number relationships, adding and subtracting whole numbers, [and] organizing and analyzing data, and working with two- and three-dimensional geometric figures.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [patterns to describe] objects to create and identify patterns and use those patterns to [,] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from informal to formal [use informal] language [and observation of geometric properties] to describe two- and three-dimensional geometric figures [shapes, solids,] and likenesses [locations] in the physical world. Students [and] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.




quantitative reasoning. The student uses whole numbers to describe and compare quantities.

The student is expected to: (A) compare and order whole numbers up to 99 (less than,

greater than, or equal to) using sets of concrete objects and pictorial models;

(B) create sets of tens and ones using concrete objects to describe, compare, and order whole numbers;

(C) identify individual coins by name and value and describe relationships among them; and

[(C) use words and numbers to describe the values of individual coins such as penny, nickel, dime, and quarter and their relationships; and]

(D) read and write numbers to 99 to describe sets of concrete objects.


quantitative reasoning. The student uses pairs of whole numbers to describe fractional parts of whole objects or sets of objects.

The student is expected to: (A) separate [share] a whole [by separating it] into two,

three, or four equal parts and use appropriate language to describe the parts such as three out of four equal parts; and

(B) use appropriate language to describe part of a set such as three out of the eight crayons are red.



(1.3) Number, operation, and quantitative reasoning. The student recognizes and solves problems in addition and subtraction situations.

The student is expected to: (A) model and create addition and subtraction problem

situations with concrete objects and write corresponding number sentences; and

(B) use concrete and pictorial models to [learn and] apply basic addition and subtraction facts (up to 9 + 9 = 18 and 18 – 9 = 9) [(sums to 18) using concrete models].


algebraic thinking. The student uses repeating patterns and additive patterns to make predictions.

The student is expected to identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems. [:]

[(A) identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems; and]

[(B) use patterns to skip count by twos, fives, and tens.].

(1.5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers and operations.

The student is expected to: (A) use patterns to skip count by twos, fives, and tens; (B) [(A)] find patterns in numbers, including odd and even; (C) [(B)] compare and order whole numbers using place

value; [and] (D) use patterns to develop strategies to solve basic addition

and basic subtraction problems; and (E) [(C)] identify patterns in related addition and subtraction

sentences (fact families for sums to 18) such as 2 + 3 = 5, 3 + 2 = 5, 5 – 2 = 3, and 5 – 3 = 2.


The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both [, compare, and contrast shapes and solids]

The student is expected to: [(A) describe and identify objects in order to sort them

according to a given attribute using informal language;] (A) [(B)] describe and identify two-dimensional geometric

figures, including circles, triangles, [and] rectangles, and [including] squares (a special type of rectangle); [, and describe the shape of balls, boxes, cans, and cones; and]

(B) describe and identify three-dimensional geometric figures, including spheres, rectangular prisms (including cubes), cylinders, and cones;

(C) describe and identify two- and three-dimensional geometric figures in order to sort them according to a given attribute using informal and formal language; and

(D) [(C)] use concrete models to combine two-dimensional geometric figures [shapes] to make new geometric figures.

(1.7) Measurement. The student

directly compares the attributes of length, area, weight/mass, capacity, and temperature. The student uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length [, weight, and capacity] .

The student is expected to: (A) estimate and measure length [, capacity, and weight of

objects] using nonstandard units such as paper clips or sides of color tiles; [and]

(B) compare and order two or more concrete objects according to length (from longest to shortest);

(C) [(B)] describe the relationship between the size of the unit and the number of units needed to measure the length of an object; [in a measurement.]



(D) compare and order the area of two or more two-

dimensional surfaces (from covers the most to covers the least);

(E) compare and order two or more containers according to capacity (from holds the most to holds the least);

(F) compare and order two or more objects according to weight/mass (from heaviest to lightest); and

(G) compare and order two or more objects according to relative temperature (from hottest to coldest).


understands that time [and temperature] can be measured. The student uses time to describe and compare situations.

The student is expected to: [(A) recognize temperatures such as a hot day or a cold day;] [(B) describe time on a clock using hours and half hours;

and] (A) [(C)] order three or more events according to duration; and

[by how much time they take.] (B) read time to the hour and half-hour using analog and

digital clocks.

(1.9) Probability and statistics. The student displays data in an organized form.

The student is expected to: (A) collect and sort data; and (B) use organized data to construct real-object graphs,

picture graphs, and bar-type graphs.

(1.10) Probability and statistics. The student uses information from organized data.

The student is expected to: (A) draw conclusions and answer questions using

information organized in real-object graphs, picture graphs, and bar-type graphs; and

(B) identify events as certain or impossible such as drawing a red crayon from a bag of green crayons.


mathematical tools. The student applies Grade 1 mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (A) identify mathematics in everyday situations; (B) solve problems [use a problem-solving model,] with

guidance [as needed,] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and



mathematical tools. The student communicates about Grade 1 mathematics using informal language.

The student is expected to: (A) explain and record observations using objects, words,

pictures, numbers, and technology; and (B) relate informal language to mathematical language and

symbols.

(1.13) Underlying processes and mathematical tools. The student uses logical reasoning [to make sense of his or her world] .





(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 2 are developing an understanding of the base-ten place value system, comparing and ordering whole numbers, applying addition and subtraction, and using measurement processes.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use [patterns to describe] objects to create and identify patterns and use those patterns to [,] express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from [use] informal to formal language [and observation of geometric properties] to describe two- and three-dimensional geometric figures [shapes, solids,] and likenesses [locations] in the physical world. Students [and] begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.




quantitative reasoning. The student understands how place value is used to represent whole numbers.

The student is expected to: [use concrete models to represent, compare, and order whole numbers (through 999), read the numbers, and record the comparisons using numbers and symbols (>, <, =).]

(A) use concrete models of hundreds, tens, and ones to represent a given whole number (up to 999) in various ways;

(B) use place value to read, write, and describe the value of whole numbers to 999; and

(C) use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<, =, >).


quantitative reasoning. The student describes how fractions are used [uses fraction words] to name parts of whole objects or sets of objects.

The student is expected to: (A) use concrete models to represent and name fractional

parts of a whole object (with denominators of 12 or less); [(not to exceed twelfths) when given a concrete representation; and]

(B) use concrete models to represent and name fractional parts of a set of objects (with denominators of 12 or less); and [(not to exceed twelfths) when given a concrete representation.]

(C) use concrete models to determine if a fractional part of a whole is closer to 0, ½, or 1.




quantitative reasoning. The student adds and subtracts whole numbers to solve problems.

The student is expected to: (A) recall and apply basic addition and subtraction facts

([sums] to 18); (B) model addition and subtraction of two-digit numbers with

objects, pictures, words, and numbers; (C) [(B)] select addition or subtraction to [and] solve problems

using two-digit numbers, whether or not regrouping is necessary; [and]

(D) [(C)] determine the value of a collection of coins up to [less than] one dollar; and [.]

(E) describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collection of coins.


quantitative reasoning. The student models multiplication and division.

The student is expected to: (A) model, create, and describe multiplication situations in

which equivalent sets of concrete objects are joined; and (B) model, create, and describe division situations in which a

set of concrete objects is separated into equivalent sets.

(2.5) Patterns, relationships, and algebraic thinking. The student uses patterns in numbers and operations.

The student is expected to: (A) find patterns in numbers such as in a 100s chart; (B) use patterns in place value to compare and order whole

numbers through 999; and (C) use patterns and relationships to develop strategies to

remember basic addition and subtraction facts. Determine patterns in related addition and subtraction number sentences (including fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 – 8 = 9, and 17 – 9 = 8. [; and]

[(D) solve subtraction problems related to addition facts (fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 – 8 = 9, and 17 – 9 = 8.]


algebraic thinking. The student uses patterns to describe relationships and make predictions.

The student is expected to: (A) generate a list of paired numbers based on a real-life

situation such as number of tricycles related to number of wheels;

(B) identify patterns in a list of related number pairs based on a real-life situation and extend the list; and

(C) identify, describe, and extend repeating and additive patterns to make predictions and solve problems.


The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both [, compare, and contrast shapes and solids].

The student is expected to: (A) describe [identify] attributes (the number of vertices,

faces, edges, sides) of two- and three-dimensional geometric figures such as circles, polygons, spheres, cones, cylinders, prisms, and pyramids, etc. [any shape or solid] ;

(B) use attributes to describe how 2 [two] two-dimensional figures or 2 three-dimensional geometric figures [shapes or two solids] are alike or different; and

(C) cut two-dimensional geometric figures [shapes] apart and identify the new geometric figures formed [shapes made] .



(2.8) Geometry and spatial reasoning. The student recognizes that [numbers can be represented by points on] a line can be used to represent a set of numbers and its properties.

The student is expected to use whole numbers to locate and name points on a number line.

(2.9) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length, area, capacity, and weight/mass. The student recognizes and uses models that approximate standard units ( from both SI, also known as metric, and customary systems ) of length, weight/mass [weight], capacity, and time.

The student is expected to: (A) identify concrete models that approximate standard units

of length and use them to measure length [, capacity, and weight];

(B) select a non-standard unit of measure such as square tiles to determine the area of a two-dimensional surface;

[(B) measure length, capacity, and weight using concrete models that approximate standard units; and]

(C) select a non-standard unit of measure such as a bathroom cup or a jar to determine the capacity of a given container; and

[(C) describe activities that take approximately one second, one minute, and one hour.]

(D) select a non-standard unit of measure such as beans or marbles to determine the weight/mass of a given object.

(2.10) Measurement. The student uses

standard tools to estimate and measure time and temperature (in degrees Fahrenheit).

The student is expected to: (A) read a thermometer to gather data; [and] (B) read and write times shown [describe time] on analog

and digital clocks using five-minute increments; and [a clock using hours and minutes.]

(C) describe activities that take approximately one second, one minute, and one hour.


student organizes data to make it useful for interpreting information.

The student is expected to: (A) construct picture graphs and bar-type graphs; (B) draw conclusions and answer questions based on

picture graphs and bar-type graphs; and (C) use data to describe events as more likely or less likely

such as drawing a certain color crayon from a bag of seven red crayons and three green crayons.



The student is expected to: (A) identify the mathematics in everyday situations; (B) solve problems with guidance [use a problem-solving

model] that incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and



mathematical tools. The student communicates about Grade 2 mathematics using informal





language. symbols.

(2.14) Underlying processes and mathematical tools. The student uses logical reasoning [to make sense of his or her world].



(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 3 are multiplying and dividing whole numbers, connecting fraction symbols to fractional quantities, and standardizing language and procedures in geometry and measurement.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [shapes and solids] ; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.

(3) Throughout mathematics in Grades 3-5, students develop numerical fluency with conceptual understanding and computational accuracy. Students in Grades 3-5 use knowledge of the base-ten place value system to compose and decompose numbers in order to solve problems requiring precision, estimation, and reasonableness. By the end of Grade 5, students know basic addition, subtraction, multiplication, and division facts and are using them to work flexibly, efficiently, and accurately with numbers during addition, subtraction, multiplication, and division computation.

(4) [(3)] Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 3-5, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.


quantitative reasoning. The student uses place value to communicate about increasingly large whole numbers in verbal and written form, including money.

The student is expected to: (A) use place value to read, write (in symbols and words),

and describe the value of whole numbers through 999,999;

(B) use place value to compare and order whole numbers through 9,999; and

(C) determine the value of a collection of coins and bills.

(3.2) Number, operation, and quantitative reasoning. The student uses fraction names and symbols (with denominators of 12 or less) to describe fractional parts of whole objects or sets of objects.

The student is expected to: (A) construct concrete models of fractions; (B) compare fractional parts of whole objects or sets of

objects in a problem situation using concrete models; (C) use fraction names and symbols to describe fractional

parts of whole objects or sets of objects [with denominators of 12 or less]; and

(D) construct concrete models of equivalent fractions for fractional parts of whole objects.



(3.3) Number, operation, and quantitative reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers.

The student is expected to: (A) model addition and subtraction using pictures, words,

and numbers; and (B) select addition or subtraction and use the operation to

solve problems involving whole numbers through 999.

(3.4) Number, operation, and quantitative reasoning. The student recognizes and solves problems in multiplication and division situations.

The student is expected to: (A) learn and apply multiplication facts through 12 by 12 [the

tens] using concrete models and objects; (B) solve and record multiplication problems (up to two digits

times one digit) [(one-digit multiplier)]; and (C) use models to solve division problems and use number

sentences to record the solutions.

(3.5) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results.

The student is expected to: (A) round whole numbers [two-digit numbers] to the nearest

ten or hundred to approximate reasonable results in problem situations [and three-digit numbers to the nearest hundred]; and

(B) use strategies including rounding and compatible numbers to estimate solutions to addition and subtraction problems.

[(B) estimate sums and differences beyond basic facts.]

(3.6) Patterns, relationships, and algebraic thinking. The student uses patterns to solve problems.

The student is expected to: (A) identify and extend whole-number and geometric

patterns to make predictions and solve problems; (B) identify patterns in multiplication facts using concrete

objects, pictorial models, or technology; and (C) identify patterns in related multiplication and division

sentences (fact families) such as 2 x 3 = 6, 3 x 2 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2.


algebraic thinking. The student uses lists, tables, and charts to express patterns and relationships.

The student is expected to: (A) generate a table of paired numbers based on a real-life

situation such as insects and legs; and (B) identify and describe patterns in a table of related

number pairs based on a meaningful problem [real-life situation] and extend the table.


The student uses formal geometric vocabulary.

The student is expected to identify, classify, and [name,] describe two- and three-dimensional geometric figures by their attributes. The student compares two- dimensional figures, three-dimensional figures, or both by their attributes [, and compare shapes and solids] using formal geometry [geometric] vocabulary.


The student recognizes congruence and symmetry.

The student is expected to: (A) identify congruent two-dimensional figures [shapes]; (B) create two-dimensional figures [shapes] with lines of

symmetry using concrete models and technology; and (C) identify lines of symmetry in two-dimensional geometric

figures [shapes].

(3.10) Geometry and spatial reasoning. The student recognizes that

The student is expected to locate and name points on a number line using whole numbers and fractions, including



[numbers can be represented by points on] a line can be used to represent numbers and fractions and their properties and relationships .

[such as] halves and fourths.

(3.11) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses standard units to describe length, area, capacity/volume, and weight/mass.

[(3.11) Measurement. The student selects and uses appropriate units and procedures to measure length and area.]

The student is expected to: (A) use linear measurement tools to estimate and measure

lengths using standard units [such as inch, foot, yard, centimeter, decimeter, and meter];

(B) use standard units [linear measure] to find the perimeter of a shape; [and]

(C) use concrete and pictorial models of square units to determine the area of two-dimensional surfaces; [shapes.]

(D) identify concrete models that approximate standard units of weight/mass and use them to measure weight/mass;

(E) identify concrete models that approximate standard units for capacity and use them to measure capacity; and

(F) use concrete models that approximate cubic units to determine the volume of a given container or other three-dimensional geometric figure.

(3.12) Measurement. The student reads

and writes time and measures [time and] temperature in degrees Fahrenheit to solve problems

The student is expected to: (A) use a thermometer to measure temperature; and (B) [(A)] tell and write time shown on analog [traditional] and

digital clocks. [; and] [(B) use a thermometer to measure temperature.]

[(3.13) Measurement. The student applies measurement concepts.]

[The student is expected to measure to solve problems involving length, area, temperature, and time.]

(3.13) [(3.14)] Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.

The student is expected to: (A) collect, organize, record, and display data in pictographs

and bar graphs where each picture or cell might represent more than one piece of data;

(B) interpret information from pictographs and bar graphs; and

(C) use data to describe events as more likely than, less likely than, or equally likely as.

(3.14) [(3.15)] Underlying processes

and mathematical tools. The student applies Grade 3 mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (A) identify the mathematics in everyday situations; (B) solve problems [use a problem-solving model] that

incorporate [incorporates] understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and


(3.15) [(3.16)] Underlying processes

and mathematical tools. The The student is expected to: (A) explain and record observations using objects, words,



student communicates about Grade 3 mathematics using informal language.


symbols.

(3.16) [(3.17)] Underlying processes and mathematical tools. The student uses logical reasoning [to make sense of his or her world] .

The student is expected to: (A) make generalizations from patterns or sets of examples

and nonexamples; and (B) justify why an answer is reasonable and explain the

solution process.


(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 4 are comparing and ordering fractions and decimals, applying multiplication and division, and developing ideas related to congruence and symmetry.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [shapes and solids]; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.




quantitative reasoning. The student uses place value to represent whole numbers and decimals.

The student is expected to: (A) use place value to read, write, compare, and order whole

numbers through 999,999,999 [the millions place]; and (B) use place value to read, write, compare, and order

decimals involving tenths and hundredths, including money, using concrete objects and pictorial models.


quantitative reasoning. The student describes and compares fractional parts of whole objects or sets of objects.

The student is expected to: (A) use concrete objects and pictorial models to generate

equivalent fractions [using concrete and pictorial models];

(B) model fraction quantities greater than one using concrete objects and pictorial models [materials and pictures];

(C) compare and order fractions using concrete objects and



pictorial models; and (D) relate decimals to fractions that name tenths and

hundredths using concrete objects and pictorial models.

(4.3) Number, operation, and quantitative reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers and decimals.

The student is expected to: (A) use addition and subtraction to solve problems involving

whole numbers; and (B) add and subtract decimals to the hundredths place using

concrete objects and pictorial models.

(4.4) Number, operation, and quantitative reasoning. The student multiplies and divides to solve meaningful problems involving whole numbers.

The student is expected to: (A) model factors and products using arrays and area

models; (B) represent multiplication and division situations in picture,

word, and number form; (C) recall and apply multiplication facts through 12 x 12; (D) use multiplication to solve problems (no more than two

digits times two digits without technology) [involving two-digit numbers]; and

(E) use division to solve problems (no more than one-digit divisors and three-digit dividends without technology) [involving one-digit divisors]


quantitative reasoning. The student estimates to determine reasonable results.

The student is expected to: (A) round whole numbers to the nearest ten, hundred, or

thousand to approximate reasonable results in problem situations; and

(B) use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems.

[(B) estimate a product or quotient beyond basic facts.]

(4.6) Patterns, relationships, and algebraic thinking. The student uses patterns in multiplication and division.

The student is expected to: (A) use patterns and relationships to develop strategies to

remember basic multiplication and division facts (such as the patterns in related multiplication and division number sentences (fact families) such as 9 x 9 = 81 and 81 ÷ 9 = 9); and

[(B) solve division problems related to multiplication facts (fact families) such as 9 x 9 = 81 and 81 ÷ 9 = 9; and]

(B) [(C)] use patterns to multiply by 10 and 100.

(4.7) Patterns, relationships, and algebraic thinking. The student uses organizational structures to analyze and describe patterns and relationships.

The student is expected to describe the relationship between two sets of related data such as ordered pairs in a table.

(4.8) Geometry and spatial reasoning. The student identifies and describes attributes of geometric figures [lines, shapes, and solids] using formal geometric language.

The student is expected to: (A) identify and describe right, acute, and obtuse angles; (B) identify and describe [models of] parallel and intersecting

(including perpendicular) lines using concrete objects and pictorial models; and

(C) use essential attributes to define two- and three-dimensional geometric figures.

[(C) describe shapes and solids in terms of vertices, edges,



and faces.]

(4.9) Geometry and spatial reasoning. The student connects transformations to congruence and symmetry.

The student is expected to: (A) demonstrate translations, reflections, and rotations using

concrete models; (B) use translations, reflections, and rotations to verify that

two shapes are congruent; and (C) use reflections to verify that a shape has symmetry.

(4.10) Geometry and spatial reasoning. The student recognizes the connection between numbers and their properties and points on a [number] line.

The student is expected to locate and name points on a number line using whole numbers, fractions such as halves and fourths, and decimals such as tenths.

(4.11) Measurement. The student applies measurement concepts. The student is expected to estimate and measure to solve problems involving length (including perimeter) and area. The student uses measurement tools to measure capacity/volume and weight/mass.

The student is expected to: (A) estimate and use measurement tools to determine

length (including perimeter), area, capacity and weight/mass using standard units SI (metric) and customary;

(B) perform simple conversions between different units of length, between different units of capacity, and between different units of weight within the customary measurement system;

(C) use concrete models of standard cubic units to measure volume;

(D) estimate volume in cubic units; and (E) explain the difference between weight and mass.

[(4.11) Measurement. The student selects and uses appropriate units and procedures to measure weight and capacity.]

[The student is expected to:] [(A) estimate and measure weight using standard units

including ounces, pounds, grams, and kilograms; and] [(B) estimate and measure capacity using standard units

including milliliters, liters, cups, pints, quarts, and gallons.]


applies measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius).

The student is expected to : [measure to solve problems involving length, including perimeter, time, temperature, and area.]

(A) use a thermometer to measure temperature and changes in temperature; and

(B) use tools such as a clock with gears or a stopwatch to solve problems involving elapsed time.


student solves problems by collecting, organizing, displaying, and interpreting sets of data.

The student is expected to: (A) use concrete objects or pictures to make generalizations

about determining all possible combinations of a given set of data or of objects in a problem situation; and

[(A) list all possible outcomes of a probability experiment such as tossing a coin;]

[(B) use a pair of numbers to compare favorable outcomes to all possible outcomes such as four heads out of six tosses of a coin; and]

(B) [(C)] interpret bar graphs.



(4.14) Underlying processes and mathematical tools. The student applies Grade 4 mathematics to solve problems connected to everyday experiences and activities in and outside of school.









symbols.

(4.16) Underlying processes and mathematical tools. The student uses logical reasoning [to make sense of his or her world] .


and non-examples; and (B) justify why an answer is reasonable and explain the

solution process.


(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 5 are comparing and contrasting lengths, areas [area], and volumes [volume] of two- or three-dimensional geometric figures [geometric shapes and solids]; representing and interpreting data in graphs, charts, and tables; and applying whole number operations in a variety of contexts.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures [shapes and solids]; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.







The student is expected to: (A) use place value to read, write, compare, and order whole

numbers through the 999,999,999,999 [billions place]; and

(B) use place value to read, write, compare, and order decimals through the thousandths place.


quantitative reasoning. The student uses fractions in problem-solving situations.

The student is expected to: (A) generate a fraction equivalent to a given fraction such as

1/2 and 3/6 or 4/12 and 1/3; [(A) generate equivalent fractions;] (B) generate a mixed number equivalent to a given improper

fraction or generate an improper fraction equivalent to a given mixed number;

(C) [(B)] compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators; and

(D) [(C)] use models to relate decimals to fractions that name tenths, hundredths, and thousandths.


quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.

The student is expected to: (A) use addition and subtraction to solve problems involving

whole numbers and decimals; (B) use multiplication to solve problems involving whole

numbers (no more than three digits times two digits without technology);

(C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit dividends without technology) , including interpreting the remainder within a given context;

(D) identify [prime factors of a whole number and] common factors of a set of whole numbers; and

(E) model situations using [and record] addition and/or [and] subtraction involving [of] fractions with like denominators using concrete objects, pictures, words, and numbers [in problem-solving situations].



The student is expected to use strategies, including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division problems. [(A) round whole numbers and decimals through tenths to

approximate reasonable results in problem situations; and]

[(B) estimate to solve problems where exact answers are not required.]


algebraic thinking. The student makes generalizations based on observed patterns and relationships.

The student is expected to: [(A) use concrete objects or pictures to make generalizations

about determining all possible combinations;] (A) [(B)] describe the relationship between sets of data in

graphic organizers such as [use] lists, tables, charts, and diagrams [to find patterns and make generalizations such as a procedure for determining equivalent fractions]; and



(B) [(C)] identify prime and composite numbers using concrete objects, pictorial models, and patterns in factor pairs.


algebraic thinking. The student describes relationships mathematically.

The student is expected to select from and use diagrams and equations such as y = 5 + 3 [number sentences] to represent meaningful problem [real-life] situations.

(5.7) Geometry and spatial reasoning. The student generates geometric definitions using critical attributes.

The student is expected to identify essential attributes including parallel, perpendicular, and congruent parts of two- and three-dimensional geometric figures. [:]

[(A) identify critical attributes including parallel, perpendicular, and congruent parts of geometric shapes and solids; and]

[(B) use critical attributes to define geometric shapes or solids.]


The student models transformations.

The student is expected to: (A) sketch the results of translations, rotations, and

reflections on a Quadrant I coordinate grid; and (B) identify [describe] the transformation that generates one

figure from the other when given two congruent figures on a Quadrant I coordinate grid .


The student recognizes the connection between ordered pairs of numbers and locations of points on a plane.

The student is expected to locate and name points on a coordinate grid using ordered pairs of whole numbers.

(5.10) Measurement. The student applies measurement concepts involving length (including perimeter), area, capacity/volume, and weight/mass to solve problems [selects and uses appropriate units and procedures to measure volume] .

The student is expected to: (A) perform simple conversions within the same

measurement system (SI (metric) or customary); [(A) measure volume using concrete models of cubic units;

and] (B) connect models for perimeter, area, and volume with

their respective formulas; and [(B) estimate volume in cubic units.] (C) select and use appropriate units and formulas to

measure length, perimeter, area, and volume.

(5.11) Measurement. The student applies measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius).

The student is expected to: (A) solve problems involving changes in temperature; and [(A) measure to solve problems involving length (including

perimeter), weight, capacity, time, temperature, and area; and]

(B) solve problems involving elapsed time. [(B) describe numerical relationships between units of

measure within the same measurement system such as an inch is one-twelfth of a foot.]


student describes and predicts the results of a probability experiment.

The student is expected to: (A) use fractions to describe the results of an experiment;

[and] (B) use experimental results to make predictions; and [.] (C) list all possible outcomes of a probability experiment such



as tossing a coin.

(5.13) Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.

The student is expected to: (A) use tables of related number pairs to make line graphs; (B) describe characteristics of data presented in tables and

graphs including median, mode, and range [the shape and spread of the data and the middle number]; and

(C) graph a given set of data using an appropriate graphical representation such as a picture or line graph.











symbols.

(5.16) Underlying processes and mathematical tools. The student uses logical reasoning [to make sense of his or her world]


and non-examples; and (B) justify why an answer is reasonable and explain the

solution process. §111.22. Mathematics, Grade 6. (a) Introduction.

(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 6 are using ratios to describe direct proportional relationships involving number, geometry, measurement, [and] probability , and adding and subtracting decimals and fractions.


(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in



mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology [(at least four-function calculators for whole numbers, decimals, and fractions)] and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.

(b) Knowledge and skills.

(6.1) Number, operation, and quantitative reasoning. The student represents and uses rational numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order non-negative rational numbers; (B) generate equivalent forms of rational numbers including

whole numbers, fractions, and decimals; (C) use integers to represent real-life situations; (D) write prime factorizations using exponents; [and] (E) identify factors of a positive integer, [and multiples

including] common factors, and the greatest common factor of a set of positive integers; and [common multiples.]



quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions.

The student is expected to: (A) model addition and subtraction situations involving

fractions with objects, pictures, words, and numbers; (B) use addition and subtraction to solve problems

involving fractions and decimals; (C) use multiplication and division of whole numbers to

solve problems including situations involving equivalent ratios and rates; [and]

(D) estimate and round to approximate reasonable results and to solve problems where exact answers are not required; and [.]




The student is expected to: (A) use ratios to describe proportional situations; (B) represent ratios and percents with concrete models,

fractions, and decimals; and (C) use ratios to make predictions in proportional situations


The student is expected to: (A) use tables and symbols to represent and describe

proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter and [,] area [, etc.]; and

(B) use tables of data to generate formulas representing [to represent] relationships involving perimeter, area, volume of a rectangular prism, etc. [, from a table of data.]



The student is expected to formulate equations [an equation] from [a] problem situations described by linear relationships [situation].




reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.

The student is expected to: (A) use angle measurements to classify angles as acute,

obtuse, or right; (B) identify relationships involving angles in triangles and

quadrilaterals; and (C) describe the relationship between radius, diameter, and

circumference of a circle.



(6.8 Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume [capacity], weight, and angles.

The student is expected to: (A) estimate measurements (including circumference) and

evaluate reasonableness of results; (B) select and use appropriate units, tools, or formulas to

measure and to solve problems involving length (including perimeter [and circumference]), area, time, temperature, volume [capacity], and weight;

(C) measure angles; and (D) convert measures within the same measurement

system (customary and metric) based on relationships between units.


student uses experimental and theoretical probability to make predictions.

The student is expected to: (A) construct sample spaces using lists and [,] tree

diagrams [, and combinations]; and (B) find the probabilities of a simple event and its





presenting and displaying [draw and compare] different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot ;

(B) identify mean (using concrete objects and pictorial models), [use] median, mode, and range of a set of [to describe] data;

(C) sketch circle graphs to display data; and (D) solve problems by collecting, organizing, displaying,

and interpreting data.





(C) select or develop an appropriate problem-solving strategy from a variety of different types, including drawing a picture, looking for a pattern, systematic



guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and








mathematical tools. The student uses logical reasoning to make conjectures and verify conclusions



properties and relationships. §111.23. Mathematics, Grade 7. (a) Introduction.



(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology [(at least four-function calculators for whole numbers, decimals, and fractions)] and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.


quantitative reasoning. The student represents and uses numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order integers and positive rational

numbers; (B) convert between fractions, decimals, whole numbers,

and percents mentally, on paper, or with a calculator; and

(C) represent squares and square roots using geometric



models.

(7.2) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

The student is expected to: (A) represent multiplication and division situations involving

fractions and decimals with [concrete] models, including concrete objects, pictures, words, and numbers;

(B) use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals;

(C) use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms;




(G) determine the reasonableness of a solution to a problem



The student is expected to: (A) estimate and find solutions to application problems

involving percent; and (B) estimate and find solutions to application problems

involving proportional relationships such as similarity, scaling, unit costs, and related measurement units.




perimeter, area, circumference, volume, and scaling; (B) graph data to demonstrate relationships in familiar

concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

(C) use words and symbols to describe the relationship between the terms in an arithmetic [a] sequence (with a constant rate of change) and their positions in the sequence



The student is expected to: (A) use concrete and pictorial models to solve equations

and use symbols to record the actions; and (B) formulate [a possible] problem situations [situation]

when given a simple equation and formulate an equation when given a problem situation


The student compares and classifies two- and three-dimensional figures [shapes and solids] using geometric vocabulary and properties.

The student is expected to: (A) use angle measurements to classify pairs of angles as

complementary or supplementary; (B) use properties to classify [shapes including] triangles

and [,] quadrilaterals [, pentagons, and circles]; (C) use properties to classify three-dimensional figures

[solids], including pyramids, cones, prisms, and cylinders; and

(D) use critical attributes to define similarity




The student is expected to: (A) locate and name points on a coordinate plane using

ordered pairs of integers; and (B) graph reflections across the horizontal or vertical axis

and graph translations on a coordinate plane


The student is expected to: (A) sketch three-dimensional figures [a solid] when given

the top, side, and front views; (B) make a net (two-dimensional model) of the surface area

of a three-dimensional figure [solid]; and (C) use geometric concepts and properties to solve

problems in fields such as art and architecture.

(7.9) Measurement. The student solves application problems involving estimation and measurement.

The student is expected to: [estimate measurements and solve application problems involving length (including perimeter and circumference), area, and volume.]

(A) estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes;


(C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders



The student is expected to: (A) construct sample spaces for simple or composite

experiments [compound events (dependent and independent)]; and

(B) find the [approximate] probability of independent events [a compound event through experimentation] .


student understands that the way a set of data is displayed influences its interpretation.


presenting and displaying relationships among collected data, including line plot, line graph, bar graph, stem and leaf plot, circle graph, and Venn diagrams, and justify the selection; and




The student is expected to: (A) describe a set of data using mean, median, mode, and

range; and (B) choose among mean, median, mode, or range to

describe a set of data and justify the choice for a particular situation.


mathematical tools. The student applies Grade 7 mathematics to solve problems connected to everyday experiences, investigations in other disciplines,



(B) use a problem-solving model that incorporates understanding the problem, making a plan, carrying out



and activities in and outside of school.

the plan, and evaluating the solution for reasonableness;


(D) select tools such as real objects, manipulatives, paper/pencil, and technology or techniques such as mental math, estimation, and number sense to solve problems












(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 8 are using basic principles of algebra to analyze and represent both proportional and non-proportional linear relationships and using probability to describe data and make predictions.


(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology [(at least four-function calculators for whole numbers, decimals, and fractions)] and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.




quantitative reasoning. The student understands that different forms of numbers are appropriate for different situations.

The student is expected to: (A) compare and order rational numbers in various forms

including integers, percents, and positive and negative fractions and decimals;


(C) approximate (mentally and with calculators) the value of irrational numbers as they arise from problem situations (such as π, √2); and

(D) express numbers in scientific notation, including negative exponents, in appropriate problem situations [using a calculator].


quantitative reasoning. The student selects and uses appropriate operations to solve problems and justify solutions.

The student is expected to: (A) select [and use] appropriate operations to solve


(B) use appropriate operations to solve problems involving [add, subtract, multiply, and divide] rational numbers in problem situations;

(C) evaluate a solution for reasonableness; and (D) use multiplication by a constant factor (unit rate) to

represent proportional relationships [; for example, the arm span of a gibbon is about 1.4 times its height, a = 1.4h] .


algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems.

The student is expected to: (A) compare and contrast proportional and non-proportional

linear relationships; and (B) estimate and find solutions to application problems

involving percents and other proportional relationships such as similarity and rates.

(8.4) Patterns, relationships, and algebraic thinking. The student makes connections among various representations of a numerical relationship.

The student is expected to generate a different representation of data given another [one] representation of data (such as a table, graph, equation, or verbal description).


The student is expected to: (A) predict, [estimate,] find, and justify solutions to

application problems using appropriate tables, graphs, and algebraic equations; and

(B) find and evaluate [use] an algebraic expression to determine [find] any term in an arithmetic [a] sequence (with a constant rate of change).


reasoning. The student uses transformational geometry to develop spatial sense.

The student is expected to: (A) generate similar figures [shapes] using dilations

including enlargements and reductions; and (B) graph dilations, reflections, and translations on a

coordinate plane.




The student is expected to: (A) draw three-dimensional figures [solids] from different

perspectives; (B) use geometric concepts and properties to solve

problems in fields such as art and architecture; (C) use pictures or models to demonstrate the Pythagorean

Theorem; and (D) locate and name points on a coordinate plane using

ordered pairs of rational numbers.

(8.8) Measurement. The student uses procedures to determine measures of three-dimensional figures [solids].

The student is expected to: (A) find lateral and total surface area of prisms, pyramids,

and cylinders using concrete models and nets (two-dimensional models);

(B) connect models [to formulas for volume] of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects; and

(C) estimate measurements [answers] and use formulas to solve application problems involving lateral and total surface area and volume.


indirect measurement to solve problems.

The student is expected to: (A) use the Pythagorean Theorem to solve real-life

problems; and (B) use proportional relationships in similar two-dimensional

figures or similar three-dimensional figures [shapes] to find missing measurements.


describes how changes in dimensions affect linear, area, and volume measures.

The student is expected to: (A) describe the resulting effects on perimeter and area

when dimensions of a shape are changed proportionally; and




The student is expected to: (A) find the probabilities of dependent and independent

[compound] events [(dependent and independent)] ; (B) use theoretical probabilities and experimental results to

make predictions and decisions; and (C) select and use different models to simulate an event.


The student is expected to: (A) select the appropriate measure of central tendency or

range to describe a set of data and justify the choice for a particular situation [purpose];


(C) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, [construct] circle graphs, bar graphs, box and whisker plots, [and] histograms, and Venn diagrams, with and without the use of technology.




The student is expected to: (A) evaluate methods of sampling to determine validity of an

inference made from a set of data; and (B) recognize misuses of graphical or numerical information

and evaluate predictions and conclusions based on data analysis.



The student is expected to: (A) identify and apply mathematics to everyday experiences,

to activities in and outside of school, with other disciplines, and with other mathematical topics;











The student is expected to: (A) make conjectures from patterns or sets of examples and

nonexamples; and (B) validate his/her conclusions using mathematical





¤111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12.


§111.32. Algebra I (One Credit). (a) Basic understandings.

(1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their understanding through other mathematical experiences.


(3) Function concepts. [Functions represent the systematic dependence of one quantity on another.] A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations, and to analyze and interpret relationships.

(4) Relationship between equations and functions. Equations and inequalities arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and inequalities and use a variety of methods to solve them [these equations].

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical, and verbal), tools, and technology [,] (including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers) to [with graphing capabilities and] model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use [, and reasoning, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts [justification and proof].

(b) Knowledge and skills (A.1) Foundations for functions. The


The student is expected to: (A) describe independent and dependent quantities in

functional relationships; [.] (B) gather and record data [,] and use [or uses] data sets [,]

to determine functional [(systematic)] relationships between quantities; [.]

(C) describe functional relationships for given problem situations and write [writes] equations or inequalities to answer questions arising from the situations; [.]

(D) represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities; and [.]

(E) interpret and make decisions, predictions, and critical judgments from functional relationships.




The student is expected to: (A) identify and sketch the general forms of linear (y = x)

and quadratic (y = x2) parent functions; [.]

(B) identify the mathematical domains and ranges and determine [determines] reasonable domain and range values for given situations, both continuous and discrete;

(C) interpret situations in terms of given graphs or creates situations that fit given graphs; and [.]

(D) collect and organize data, make and interpret [makes and interprets] scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make [models, predicts, and makes] decisions and critical judgments in problem situations.



The student is expected to: (A) use symbols to represent unknowns and variables; and (B) look for patterns and represent [represents]

generalizations algebraically.


The student is expected to: (A) find specific function values, simplify [simplifies]

polynomial expressions, transform and solve [transforms and solves] equations, and factor [factors] as necessary in problem situations; [.]

(B) use the commutative, associative, and distributive properties to simplify algebraic expressions; and [.]

(C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.

(A.5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations.

The student is expected to: (A) determine whether or not given situations can be

represented by linear functions; [.] (B) determine the domain and range [values] for [which]

linear functions in [make sense for] given situations; and (C) use, translate, and make connections [The student

translates] among [and uses] algebraic, tabular, graphical, or verbal descriptions of linear functions.



The student is expected to: (A) develop the concept of slope as rate of change and

determine [determines] slopes from graphs, tables, and algebraic representations;

(B) interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs; [.]

(C) investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b;

(D) graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept;

(E) determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and



algebraic representations; (F) interpret and predict the effects of changing slope and y-

intercept in applied situations; and (G) relate direct variation to linear functions and solve

[solves] problems involving proportional change.

(A.7) Linear functions. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) analyze situations involving linear functions and

formulate [formulates] linear equations or inequalities to solve problems;

(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select [selects] a method, and solve [solves] the equations and inequalities; and

(C) interpret and determine the reasonableness of solutions to linear equations and inequalities.



The student is expected to: (A) analyze [The student analyzes] situations and formulate

[formulates] systems of linear equations in two unknowns to solve problems;

(B) solve [The student solves] systems of linear equations using concrete models, graphs, tables, and algebraic methods; and [.]

(C) interpret and determine [For given contexts, the student interprets and determines] the reasonableness of solutions to systems of linear equations.



The student is expected to: (A) determine the domain and range [values] for [which]

quadratic functions in [make sense for] given situations; (B) investigate, describe, and predict the effects of changes

in a on the graph of y = ax2 + c; [y = ax2 .] (C) investigate, describe, and predict the effects of changes

in c on the graph of y = ax2 + c; and [y = x2 + c.] (D) analyze graphs of quadratic functions and draw [draws]

conclusions.

(A.10) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

The student is expected to: (A) solve quadratic equations using concrete models, tables,

graphs, and algebraic methods; and [.] (B) make connections among the solutions (roots) [The

student relates the solutions] of quadratic equations , the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph [to the roots] of the function [their functions].


functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.

The student is expected to: (A) use patterns to generate the laws of exponents and

apply [applies] them in problem-solving situations; [.] (B) analyze data and represent [represents] situations

involving inverse variation using concrete models, tables, graphs, or algebraic methods; and [.]

(C) analyze data and represent [represents] situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.



§111.33. Algebra II (One-Half to One Credit). (a) Basic understandings.





(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical, and verbal), tools, and technology [,] (including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers) to [with graphing capabilities and] model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use [, and reasoning, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts [justification and proof].



The student is expected to (A) identify [For a variety of situations, the student identifies]

the mathematical domains and ranges of functions and determine [determines] reasonable domain and range values for continuous and discrete [given] situations; and

(B) collect and organize [In solving problems, the student collects data and records results, organizes the] data, make and interpret [makes] scatterplots, fit [fits] the graph of a [curves to the appropriate parent] function to the data , interpret [interprets] the results, and proceed [proceeds] to model, predict, and make decisions and critical judgments.

(2A.2) Foundations for functions. The

student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem

The student is expected to: (A) use tools including [matrices,] factoring [,] and properties

of exponents to simplify expressions and to transform and solve equations; and [.]

(B) use complex numbers to describe the solutions of quadratic equations.

[(C) The student connects the function notation of y = and f(x) =.]



situations. (2A.3) Foundations for functions. The

student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations.

The student is expected to: (A) analyze situations and formulate [formulates] systems of

equations in two or more unknowns or inequalities in two [or more] unknowns to solve problems; [.]

(B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and [.]

(C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts.


student connects algebraic and geometric representations of functions.

The student is expected to: (A) identify and sketch graphs of parent functions, including

linear (f(x) = x) [(y = x)], quadratic (f(x) = x2) [(y = x2)] , [square root (y = √x), inverse (y = 1/x),] exponential (f(x) = ax) [(y = ax)], and logarithmic (f(x) = logax) [(y = logax)] functions, absolute value of x (f(x) = |x|), square root of x (f(x) = √x), and reciprocal of x (f(x) = 1/x); [.]

(B) extend parent functions with parameters such as a in f(x) = a/x [m in y = mx] and describe the effects of the [describes] parameter changes on the graph of parent functions; and [.]

(C) describe and analyze the relationship between a function and its inverse [The student recognizes inverse relationships between various functions] .


student knows the relationship between the geometric and algebraic descriptions of conic sections.

The student is expected to: (A) describe a conic section as the intersection of a plane

and a cone; [.] (B) [In order to] sketch graphs of conic sections to relate [,

the student relates] simple parameter changes in the equation to corresponding changes in the graph; [.]

(C) identify symmetries from graphs of conic sections; [.] (D) identify the conic section from a given equation; and [.] (E) use the method of completing the square.


The student is expected to: (A) determine the reasonable domain and range values of

quadratic functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to quadratic equations and inequalities; [.]

(B) relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions; and [.]

(C) determine a quadratic function from its roots or a graph.

(2A.7) Quadratic and square root functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations.

The student is expected to: (A) use characteristics of the quadratic parent function to

sketch the related graphs and connect [connects] between the y = ax

2 + bx + c and the y = a(x - h)

2 + k

symbolic representations of quadratic functions; and (B) use the parent function to investigate, describe, and

predict the effects of changes in a, h, and k on the graphs of y = a(x - h)

2 + k form of a function in applied



and purely mathematical situations.

(2A.8) Quadratic and square root functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation

The student is expected to: (A) analyze situations involving quadratic functions and

formulate [formulates] quadratic equations or inequalities to solve problems; [.]

(B) analyze and interpret the solutions of quadratic equations using discriminants and solve [solves] quadratic equations using the quadratic formula; [.]

(C) compare and translate between algebraic and graphical solutions of quadratic equations; and [.]

(D) solve quadratic equations and inequalities using graphs, tables, and algebraic methods .



The student is expected to: (A) use the parent function to investigate, describe, and

predict the effects of parameter changes on the graphs of square root functions and describe [describes] limitations on the domains and ranges; [.]

(B) relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions; [.]

(C) determine the reasonable domain and range values of square root functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to square root equations and inequalities; [.]

(D) determine solutions of [The student solves] square root equations [and inequalities] using graphs, tables, and algebraic methods; [.]


(F) analyze situations modeled by square root functions, formulate [formulates] equations or inequalities, select [selects] a method, and solve [solves] problems; and [.]

(G) connect [The student expresses] inverses of square root functions with quadratic functions [using square root functions].


formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) use quotients of polynomials to describe the graphs of

rational functions, predict the effects of parameter changes, describe [describes] limitations on the domains and ranges, and examine [examines] asymptotic behavior; [.]

(B) analyze various representations of rational functions with respect to problem situations; [.]

(C) determine the reasonable domain and range values of rational functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to rational equations and inequalities; [.]

(D) determine the solutions of [The student solves] rational equations [and inequalities] using graphs, tables, and algebraic methods; [.]

(E) determine solutions of rational inequalities using graphs and tables;

(F) analyze [The student analyzes] a situation modeled by a rational function, formulate [formulates] an equation or



inequality composed of a linear or quadratic function, and solve [solves] the problem; and [.]

(G) use [The student uses direct and inverse variation] functions to model and [as models to] make predictions in problem situations involving direct and inverse variation.



The student is expected to: (A) develop [the definition of logarithms by exploring and

describing the relationship between exponential functions and their inverses; [.]

(B) use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe [describes] limitations on the domains and ranges, and examine [examines] asymptotic behavior; [.]

(C) determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine [interprets and determines] the reasonableness of solutions to exponential and logarithmic equations and inequalities; [.]

(D) determine solutions of [The student solves] exponential and logarithmic equations [and inequalities] using graphs, tables, and algebraic methods; [.]


(F) analyze a situation modeled by an exponential function, formulate [formulates] an equation or inequality, and solve [solves] the problem.

§111.34. Geometry (One Credit). (a) Basic understandings.


(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric [shapes and] figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them.



(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal [algebraic, and coordinate]), tools, and technology [,] (including, but not limited to, [powerful and accessible hand-held] calculators with graphing capabilities, data collection devices, and computers) [with graphing capabilities] to solve meaningful problems by representing and transforming figures [, transforming figures,] and analyzing relationships [, and proving things about them] .



(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, [computation in problem-solving contexts,] language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use [, as well as] multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts [justification and proof].

(b) Knowledge and skills. (G.1) Geometric structure. The

student understands the structure of, and relationships within, an axiomatic system.

The student is expected to: (A) develop an awareness of the structure of a mathematical

system, connecting definitions, postulates, logical reasoning, and theorems;

(B) recognize [Through] the historical development of geometric systems[, the student recognizes that] and know mathematics is developed for a variety of purposes

(C) compare and contrast the structures and implications of Euclidean and non-Euclidean geometries.

(G.2) Geometric structure. The

student analyzes geometric relationships in order to make and verify conjectures

The student is expected to: (A) use constructions to explore attributes of geometric

figures and to make conjectures about geometric relationships; and [.]

(B) make [The student makes and verifies] conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

(G.3) Geometric structure. The

student applies [understands the importance of] logical reasoning to justify and prove mathematical statements [, justification, and proof in mathematics] .

The student is expected to: (A) determine the validity [The student determines if the

converse] of a conditional statement, its converse, inverse, and contrapositive; [is true or false.]

(B) construct and justify statements about geometric figures and their properties; [.]

(C) use logical reasoning [The student demonstrates what it means] to prove statements are true and find counter examples to disprove [mathematically that] statements that are false; [ true.]

(D) use inductive reasoning to formulate a conjecture; and [.] (E) use deductive reasoning to prove a statement.

(G.4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems

The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.

(G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems [identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures] .

The student is expected to: (A) use numeric and geometric patterns to develop algebraic

expressions representing geometric properties; [to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.]

(B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and



solids, and angle relationships in polygons and circles; (C) use properties of transformations and their compositions

to make connections between mathematics and the real world, [in applications] such as tessellations; and [or fractals.]

(D) identify and apply [The student identifies and applies] patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

(G.6) Dimensionality and the

geometry of location. The student analyzes the relationship between three-dimensional geometric figures [objects] and related two-dimensional representations and uses these representations to solve problems.

The student is expected to; (A) describe and draw the intersection of a given plane with

various [The student describes, and draws cross sections and other slices of] three-dimensional geometric figures; [objects.]

(B) use [The student uses] nets to represent and construct three- dimensional geometric figures; and [objects.]

(C) use orthographic and isometric views [The student uses top, front, side, and corner views] of three-dimensional geometric figures [objects] to represent and construct three-dimensional geometric figures [create accurate and complete representations] and solve problems.

(G.7) Dimensionality and the

geometry of location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

The student is expected to: (A) use one- and two-dimensional coordinate systems to

represent points, lines, rays, line segments, and figures; (B) use slopes and equations of lines to investigate

geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and [.]

(C) derive and use [The student develops and uses] formulas involving length, slope, [including distance] and midpoint.

(G.8) Congruence and the geometry

of size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, [perimeter,] and volume in problem situations.

The student is expected to: (A) find areas of regular polygons, circles, and composite

figures; (B) find areas of sectors and arc lengths of circles using

proportional reasoning; [.] (C) derive, extend, and use [The student develops, extends,

and uses] the Pythagorean Theorem; and [.] (D) find surface areas and volumes of prisms, pyramids,

spheres, cones, [and] cylinders, and composites of these figures in problem situations.

(G.9) Congruence and the geometry of size. The student analyzes properties and describes relationships in geometric figures.

The student is expected to: (A) formulate and conjectures about the properties of

parallel and perpendicular lines based on explorations and concrete models; [.]

(B) formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models; [.]

(C) formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and [.]

(D) analyze the characteristics of polyhedra and other three-



dimensional figures and their component parts based on explorations and concrete models .

(G.10) Congruence and the geometry

of size. The student applies the concept of congruence to justify properties of figures and solve problems

The student is expected to: (A) use congruence transformations to make conjectures

and justify properties of geometric figures including figures represented on a coordinate plane; and [.]

(B) justify and apply triangle congruence relationships.


The student is expected to: (A) use and extend [The student uses] similarity properties

and transformations to explore and justify conjectures about geometric figures; [.]

(B) use ratios to solve problems involving similar figures; [.] (C) develop, apply, and justify triangle similarity

relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and [.]

(D) describe the effect on perimeter, area, and volume when one or more dimensions [length, width, or height] of a figure are [three-dimensional solid is] changed and apply [applies] this idea in solving problems.

§111.35. Precalculus (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September 1,

1998, and at that time shall supersede §75.63(bb) of this title (relating to Mathematics). Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisites: Algebra II, Geometry.

(b) Introduction. (1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry

foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model functions and equations and solve real-life problems.

(2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof) . Students also use multiple representations, technology, applications and modeling, [justification and proof,] and numerical fluency [computation] in problem-solving contexts.

(c) Knowledge and skills. (P.1) The student defines functions,

describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric, and piecewise-

The student is expected to: (A) describe parent functions symbolically and graphically,

including f(x) = xn [y = xn], f(x) = 1n x [y = ln x], f(x) = loga x [y = loga x], f(x) = 1/x [y =1/x], f(x) = ex [y = ex], f(x) = |x|, f(x) = ax [y = ax], f(x) = sin x [y = sin x], f(x) = arcsin x, etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(C) describe symmetry of graphs of even and odd functions;



defined functions. (D) recognize and use connections among significant values [points] of a function (zeros, [roots,] maximum values [points], [and] minimum values, etc. [points]), points on the graph of a function, and the symbolic representation of a function; and

(E) investigate the concepts of continuity, end behavior, [vertical and horizontal] asymptotes, and limits and connect these characteristics to functions represented graphically and numerically [the graph of a function]

(P.2) The student interprets the meaning

of the symbolic representations of functions and operations on functions to solve meaningful problems [within a context]

The student is expected to: (A) apply basic transformations, including a • f(x), f(x) + d,

f(x - c), f(b • x), and compositions with absolute value functions, including |f(x)|, and f(|x|), to the parent functions;

(B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and

(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.

(P.3) The student uses functions and

their properties , tools and technology, to model and solve meaningful [real-life] problems.

The student is expected to: (A) investigate properties of trigonometric and polynomial

functions; (B) use functions such as logarithmic, exponential,

trigonometric, polynomial, etc. to model real-life data; (C) use regression to determine the appropriateness of a

linear function to model real-life data (including using technology to determine the correlation coefficient);

(D) use properties of functions to analyze and solve problems and make predictions; and

(E) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed .

(P.4) The student uses sequences and

series as well as tools and technology to represent, analyze, and solve real-life problems.

The student is expected to: (A) represent patterns using arithmetic and geometric

sequences and series; (B) use arithmetic, geometric, and other sequences and

series to solve real-life problems; (C) describe limits of sequences and apply their properties

to investigate convergent and divergent series; and (D) apply sequences and series to solve problems including

sums and binomial expansion.

(P.5) The student uses conic sections, their properties, and parametric representations, as well as tools and technology, to model physical situations.

The student is expected to: (A) use conic sections to model motion, such as the graph of

velocity vs. position of a pendulum and motions of planets;

(B) use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound;

(C) convert between parametric and rectangular forms of functions and equations to graph them; and



(D) use parametric functions to simulate problems involving motion.

(P.6) The student uses vectors to model

physical situations. The student is expected to: (A) use the concept of vectors to model situations defined by

magnitude and direction; and (B) analyze and solve vector problems generated by real-life

situations. §111.36. Mathematical Models with Applications (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September 1,

1998. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisite: Algebra I.

(b) Introduction. (1) In Mathematical Models with Applications, students continue to build on the K-8 and Algebra I

foundations as they expand their understanding through other mathematical experiences. Students use algebraic, graphical, and geometric reasoning to recognize patterns and structure, to model information, and to solve problems from various disciplines. Students use mathematical methods to model and solve real-life applied problems involving money, data, chance, patterns, music, design, and science. Students use mathematical models from algebra, geometry, probability, and statistics and connections among these to solve problems from a wide variety of advanced applications in both mathematical and nonmathematical situations. Students use a variety of representations (concrete, pictorial, numerical, symbolic, [algorithmic,] graphical, and verbal ), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to link modeling techniques and purely mathematical concepts and to solve applied problems.

(2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, [justification and proof,] and numerical fluency [computation] in problem-solving contexts.

(c) Knowledge and skills. (M.1) The student uses a variety of

strategies and approaches to solve both routine and non-routine problems.

The student is expected to: (A) compare and analyze various methods for solving a real-

life problem; (B) use multiple approaches (algebraic, graphical, and

geometric methods) to solve problems from a variety of disciplines; and

(C) select a method to solve a problem, defend the method, and justify the reasonableness of the results.

(M.2) The student uses graphical and numerical techniques to study patterns and analyze data.

The student is expected to: (A) interpret information from various graphs, including line

graphs, bar graphs, circle graphs, histograms, [and] scatterplots, line plots, stem and leaf plots, and box and whisker plots to draw conclusions from the data;

(B) analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences;

(C) analyze graphs from journals, newspapers, and other sources to determine the validity of stated arguments; and

(D) use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information.

(M.3) The student develops and The student is expected to:



implements a plan for collecting and analyzing data in order to make decisions.

(A) formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions;

(B) communicate methods used, analyses [analysis] conducted, and conclusions drawn for a data-analysis project by written report, visual display, oral report, or multi-media presentation; and

(C) determine the appropriateness of a model for making predictions from a given set of data.

(M.4) The student uses probability models to describe everyday situations involving chance.

The student is expected to: (A) compare theoretical and empirical probability; and (B) use experiments to determine the reasonableness of a

theoretical model such as binomial, geometric, etc. (M.5) The student uses functional

relationships to solve problems related to personal income.

The student is expected to: (A) use rates, linear functions, and direct variation to solve

problems involving personal finance and budgeting, including compensations and deductions;

(B) solve problems involving personal taxes; and (C) analyze data to make decisions about banking.

(M.6) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit

The student is expected to: (A) analyze methods of payment available in retail

purchasing and compare relative advantages and disadvantages of each option;

(B) use amortization models to investigate home financing and compare buying and renting a home; and

(C) use amortization models to investigate automobile financing and compare buying and leasing a vehicle.

(M.7) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning.

The student is expected to: (A) analyze types of savings options involving simple and

compound interest and compare relative advantages of these options;

(B) analyze and compare coverage options and rates in insurance; and

(C) investigate and compare investment options including stocks, bonds, annuities, and retirement plans.

(M.8) The student uses algebraic and geometric models to describe situations and solve problems.

The student is expected to: (A) use geometric models available through technology to

model growth and decay in areas such as population, biology, and ecology;

(B) use trigonometric ratios and functions available through technology to calculate distances and model periodic motion; and

(C) use direct and inverse variation to describe physical laws such as Hook's, Newton's, and Boyle's laws.

(M.9) The student uses algebraic and geometric models to represent patterns and structures.

The student is expected to: (A) use geometric transformations, symmetry, and

perspective drawings to describe mathematical patterns and structure in art and architecture; and

(B) use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.



TEKS High School Significant Changes

This document is intended as a resource for the trainer and not necessarily to be copied for participants. However, different trainers have different styles, thus the document is formatted so that if it is copied on a black and white copier, the changes are still apparent. New refinements are in bold red and underlined, deletions are in plain blue text with a strike through.

TEKS Algebra I Significant Changes

(a) Basic understandings.

(3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships.

(3) Function concepts. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations, and to analyze and interpret relationships. The change here is that there is to be greater emphasis on the relationship – developing the ability to find either quantity from the other, rather than a one-directional dependence of one quantity on another. A complete understanding of the concept of a function necessitates an understanding of the interdependence of the two variables.

(4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations.

(4) Relationship between equations and functions. Equations and inequalities arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and inequalities and use a variety of methods to solve them. The significant change is in the addition of inequalities. Inequalities are generalizations of equations, and students need to be able to make connections between equations, inequalities, and the situations that they model.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, and graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with



graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. This rewording more explicitly requires teachers to emphasize and demonstrate concepts in a variety of ways to help students build a more complete understanding of functions and equations. The ultimate purpose is that students not only improve their understanding of these important concepts, but also their communication of their ideas and questions. Problem solving requires verbal and written communication of ideas about what is being learned, not just manipulation of symbols. More emphasis should be placed on conceptual understanding, not just computations.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts. The change here (besides rewording for clarity) is the addition of technology and numerical fluency. The emphasis is on numerical fluency, rather than just computation. From the elementary school level and upwards, students are expected to develop numerical fluency, and that development is continued throughout the high school courses as well. Students need to be able to apply their knowledge in new contexts, not just work problems similar to examples they have seen. (b) Knowledge and skills. (A.1) Foundations for functions. The student understands that a function represents

a dependence of one quantity on another and can be described in a variety of ways.

(E) The student interprets and makes inferences from functional relationships. The student is expected to: (E) interpret and make decisions, predictions, and critical judgments from

functional relationships. The change here is subtle. Inference involves drawing conclusions from given facts or data; making decisions involves acting upon such a conclusion, prediction involves application of such a conclusion to a similar situation, and critical judgments involves the evaluation of such a conclusion. In each case, the student is expected go beyond the inference and build upon it for deeper application and evaluation. (A.2) Foundations for functions. The student uses the properties and attributes of

functions.



(B) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations.

The student is expected to: (B) identify mathematical domains and ranges and determine reasonable

domain and range values for given situations, both continuous and discrete.

The addition of continuous and discrete is designed to emphasize that these are distinctly different situations, and students should be able to distinguish between them, see the relationship between them, and understand the differences in the situations they model. The development of this concept is continued in Algebra II. (See TEK 2A.1(A).) (D) In solving problems, the student collects and organizes data, makes and

interprets scatterplots, and models, predicts, and makes decisions and critical judgments.

The student is expected to: (D) collect and organize data, make and interpret scatterplots (including

recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

To enrich the study of functions, the TEKS call for the inclusion of problem situations which can illustrate how mathematics can model aspects of the” real world.” (In real life, functions arise from data gathered through observations or experiments). A discussion of correlation is also very useful in helping students interpret scatterplots, and in giving them appropriate terminology with which to do so. This is an important foundation on which students will build their understanding in Algebra II. (A.4) Foundations for functions. The student understands the importance of the

skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

The student is expected to: (C) connect equation notation with function notation, such as y = x + 1 and

f(x) = x + 1. Function notation is introduced in Algebra I now instead of Algebra II. This gives students more time to become familiar with the notation for functions, which is a fundamental concept of Algebra I. (A.5) Linear functions. The student understands that linear functions can be

represented in different ways and translates among their various representations.

(C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions.

The student is expected to:



(C) use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.

The addition of making connections among the different representations of linear functions is extremely important. The tendency is to show the different representations without making any clear connections between them; as a result, students have disjointed ideas of a function that have no connection or relationship, and they cannot move easily from one representation to another. Connecting the representations, and showing how they are all different pictures of the same object (a function!), teachers can help students build a much deeper, more complete understanding of functions. (A.6) Linear functions. The student understands the meaning of the slope and

intercepts of the graphs of linear functions and zeros of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations.

(Note that the change here filters through all the objectives, (A) through (G), so that the language of zeros of functions and intercepts of graphs is emphasized throughout.) (E) The student determines the intercepts of linear functions from graphs, tables,

and algebraic representations. The student is expected to: (E) determine the intercepts of the graphs of linear functions and zeros of

linear functions from graphs, tables, and algebraic representations. The change here is similar to that of (A.5) (C). Connecting the zeros and the intercepts also reinforces the connection of a linear equation to a linear function, and the graphical, equation, and tabular representations of that function, and gives students terminology with which to talk about these relationships. (A.8) Linear functions. The student formulates systems of linear equations from

problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(A) The student analyzes situations and formulates systems of linear equations to solve problems.

The student is expected to: (A) analyze situations and formulate systems of linear equations in two

unknowns to solve problems. This is a clarification; in Algebra I, students are expected to learn how to solve systems of two unknowns, while systems of three unknowns are addressed in Algebra II. (See TEK 2A.3 (A).) (A.9) Quadratic and other non-linear functions. The student understands that the

graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions.

(B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax2.




(B) investigate, describe, and predict the effects of changes in a on the graph of y = ax2 + c.

(C) The student investigates, describes, and predicts the effects of changes in c

on the graph of y = x2 + c. The student is expected to: (C) investigate, describe, and predict the effects of changes in c on the

graph of y = ax2 + c. Both of these changes are a generalization of changing parameters of quadratic functions. The tendency for most teachers is to isolate parameter changes and teach them individually without showing the combined effect of more than one change. After demonstrating the individual effects of single parameter changes, teachers should then help students “put it all together.” (A.10) Quadratic and other non-linear functions. The student understands there is

more than one way to solve a quadratic equation and solve them using appropriate methods.

(B) The student relates the solutions of quadratic equations to the roots of their functions.

The student is expected to: (B) make connections among the solutions (roots) of quadratic equations,

the zeros of their related function, and the horizontal intercepts (x-intercepts) of the graph of the function.

This change is another that reinforces making connections between the different representations of a function. Connecting the different representations, and giving the language with which to communicate the connections, is vital to helping students master the concept of a function.



TEKS Algebra II Significant Changes (a) Basic understandings.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, and graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. (See the explanation under Algebra I changes.) (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts. There are several important changes here. Students need to be able to make their own connections, forming for themselves the framework that they will use as the basis of their understanding and application of new concepts. The addition of technology and numerical fluency means that students need to have opportunities to use technology in meaningful ways and in activities that will help them to develop an understanding of what methods are appropriate for use in solving problems with which they have little or no prior experience. (b) Knowledge and skills. (2A.1) Foundations for functions. The student uses properties and attributes of

functions and applies functions to problem situations. (A) For a variety of situations, the student identifies the mathematical domains

and ranges and determines reasonable domain and range values for given situations.




(A) identify the mathematical domains and ranges of functions and determine reasonable domain and range values for continuous and discrete situations.

The change here is the clarification and distinction between continuous and discreet situations. This was first introduced in Algebra I (see TEK A.2 (B)), and further developed in Algebra II. Since this concept is continued here, student mastery of the concept is not expected in Algebra I; however, teachers should not have to introduce the vocabulary of continuous and discrete in Algebra II as there is an expectation that students will already have an introductory level knowledge of this concept. (B) In solving problems, the student collects data and records results, organizes

the data, makes scatterplots, fits the curves to the appropriate parent function, interprets the results, and proceeds to model, predict, and make decisions and critical judgments.

The student is expected to: (B) collect and organize data, make and interpret scatterplots, fit the graph

of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

The additions of interpretation and graph-fitting build upon what was supposed to have been learned in Algebra I. As students collect and organize data, they will begin to see how real-world situations can be modeled as functions, and they will be able to meaningfully interpret their data. The correlation coefficient can be taught in this context, giving students a measurement of the relationship between variables. In addition, graphing calculators can be a tremendous aid in determining lines of best fit; teachers can also use them to demonstrate non-linear curves of best fit that would be difficult or tedious to calculate by hand. (2A.2) Foundations for functions. The student understands the importance of the

skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

(A) Matrices removed (C) Moved to algebra I

(2A.3) Foundations for functions. The student formulates systems of equations and

inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations.

(A) The student analyzes situations and formulates systems of equations or inequalities in two or more unknowns to solve problems.

The student is expected to: (A) analyze situations and formulate systems of equations in two or more

unknowns or inequalities in two unknowns to solve problems; In Algebra I, students are expected to learn how to form and solve systems of equations in two unknowns. This is extended in Algebra II in two different directions: solving systems of three unknowns, and solving a system of inequalities (still with two unknowns). (See TEKS A.8 (A).)



(2A.4) Algebra and geometry. The student connects algebraic and geometric

representations of functions. (A) The student identifies and sketches graphs of parent functions, including

linear (y = x), quadratic (y = x2), square root (y = x ), inverse (y = 1/x), exponential (y = ax), and logarithmic (y = logax) functions.

The student is expected to: (A) identify and sketch graphs of parent functions, including linear (f(x) =

x), quadratic (f(x) = x2), exponential (f(x) = a

x), and logarithmic (f(x) =

logax) functions, absolute value of x (f(x) = |x|), square root of x (f(x) =

√x), and reciprocal of x (f(x) = 1/x). Here, the “inverse function” is appropriately renamed the reciprocal function (as “inverse function” has a much more generalized definition). Also, function notation is used, since it was already introduced in Algebra I, and students would be expected to use and understand it in Algebra II. (B) The student extends parent functions with parameters such as m in y = mx

and describes parameter changes on the graph of parent functions. The student is expected to: (B) extend parent functions with parameters such as a in f(x) = a/x and

describe the effects of the parameter changes on the graph of parent functions.

The change here is intended to emphasize the use of both linear and non-linear functions. Teachers may have been limiting their examples to only linear functions, and they need to show that parameter changes affect non-linear functions and linear functions in the same way. Students also need to be able to describe these changes, both verbally and in written form. (C) The student recognizes inverse relationships between various functions. The student is expected to: (C) describe and analyze the relationship between a function and its

inverse. Students should be able to describe and analyze relationships between functions and their inverses, rather than just recognizing that there is a relationship. This means that teachers need to teach students the language of inverse relationships so that students can describe them in both oral and written forms; teachers must also help students to understand what inverses do, how they work, when they exist, and how to find them. (2A.8) Quadratic and square root functions. The student formulates equations and

inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(D) The student solves quadratic equations and inequalities. The student is expected to: (D) solve quadratic equations and inequalities using graphs, tables, and

algebraic methods.



Problem solving should not be limited to only symbolic manipulation; students should learn how to solve equations and inequalities in other ways that may be faster or less prone to error. They should also be able to interpret their solutions in context, apply them in different contexts, and evaluate when a solution must be excluded. All of these representations can help students to understand what they are doing, rather than just manipulating symbols. (See the multiple representations emphasis in Basic understandings statement (5).) (2A.9), (2A.10), (2A.11) (D) and (E) all separate solving inequalities from equations and

place it in a new objective, eliminating the specification of using algebraic methods to solve inequalities.

Solutions to inequalities often require non-algebraic methods; thus solutions to equations and inequalities have been separated. (2A.9) Quadratic and square root functions. The student formulates equations and

inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(G) The student expresses inverses of quadratic functions using square root functions.

The student is expected to: (G) connect inverses of square root functions with quadratic functions. Since inverses of quadratic functions are not functions, care must be taken in any discussion of inverses of quadratic functions and connecting them to square root functions. The emphasis has been reversed so that students are led to see that while there is an inverse relationship between quadratic and square-root functions, they are not directly inverses of each other. A discussion of domain and range is vital to helping students understand this inverse relationship. (2A.10) Rational functions. The student formulates equations and inequalities based

on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

(A) The student uses quotients to describe the graphs of rational functions, describes limitations on the domains and ranges, and examines asymptotic behavior.

The student is expected to: (A) use quotients of polynomials to describe the graphs of rational

functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior.

The first change here is just a clarification that reinforces the definition of rational functions as quotients of polynomial functions. The second change is somewhat ambiguous in meaning. This group has interpreted this change to mean the effects of

the parameters a, b, and c in the basic rational function ( ) cbx

axf +−

= . By

understanding how these parameter changes affect the graph, students will gain a better understanding of domains, ranges, end behavior, and asymptotic behavior of rational functions.



TEKS Geometry Significant Changes (a) Basic understandings.

(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, algebraic, and coordinate), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities to solve meaningful problems by representing figures, transforming figures, analyzing relationships, and proving things about them.

(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships. The major changes here focus on using multiple representations that include the use of technology. Note that data collection devices are a requirement now, so schools will need to provide classroom sets for teachers and students to use in explorations. Proofs are not explicitly included here; however, they are implied in the analysis of relationships among figures and in the transformations of figures.

(6) Underlying mathematical processes. Many processes underlie all content

areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problem-solving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts. The change here (besides rewording for clarity) is the addition of technology and numerical fluency. The emphasis is on numerical fluency, rather than just computation. From the elementary school level and upwards, students are expected to develop numerical fluency, and that development is continued throughout the high school courses as well. Students need to be able to apply their knowledge in new contexts, not just work problems similar to examples they have seen. (b) Knowledge and skills. (G.3) Geometric structure: The student understands the importance of logical

reasoning, justification, and proof in mathematics. (G.3) Geometric structure. The student applies logical reasoning to justify and

prove mathematical statements.



The major change here is the change from understanding to application; note that this applies to all of the sub-topics, objectives (A) through (E). This requires more of the student. To apply logical reasoning, they must understand its importance and act on it. (A) The student determines if the converse of a conditional statement is true or

false. The student is expected to: (A) determine the validity of a conditional statement, its converse, inverse, and

contrapositive. This change is from only teaching the converse to including also the inverse and the contrapositive. This adds a dimension to the logical reasoning that was not previously present by examining a statement for truth from several different perspectives. (C) The student demonstrates what it means to prove mathematically that

statements are true. The student is expected to: (C) use logical reasoning to prove statements are true and find

counterexamples to disprove statements that are false. This is a clarification of what proofs are designed to do. Not only should students be able to logically show that something is true; they also need to be able to prove that something is NOT true, and develop techniques to do this. i.e. How do you show that a conditional statement is false? This also begins to develop the concept of proof by contradiction. (G.5) Geometric patterns: The student identifies, analyzes, and describes patterns

that emerge from two and three dimensional geometric figures. (G.5) Geometric patterns. The student uses a variety of representations to

describe geometric relationships and solve problems. As in Algebra I and II, the emphasis is on multiple representations. The more ways a student can think about and describe geometric relationships, the deeper his/her understanding of those concepts will be; in addition, he/she will be more likely to be able to apply learned concepts to solving problems in new contexts. (A) The student is expected to use numeric and geometric patterns to

develop algebraic expressions representing geometric properties. This is an addition not in the previous TEKS. (Note that the former (A) is now in (B).) Students need to be able to use the patterns they observe to actually predict and describe the relationships they see in symbolic form, making connections between geometric relationships and their algebraic expression. Students also need to be able to determine when a geometric pattern can be represented algebraically. This continues the idea of multiple representations that is emphasized throughout Algebra I and Algebra II. (D) The student identifies and applies patterns from right triangles to solve

problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.



The student is expected to: (D) identify and apply patterns from right triangles to solve meaningful

problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

The emphasis is on meaningful applications. Students often ask, “When am I going to use this?” By showing them practical applications, they will see the necessity for geometry and be more motivated (we would hope!) to learn it. (G.6) Dimensionality and the geometry of location: The student analyzes the

relationship between three-dimensional objects and related two-dimensional representations and uses these representations to solve problems.

(G.6) Dimensionality and the geometry of location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

This change is better vocabulary. (A) The student describes, and draws cross sections and other slices of three-

dimensional objects; The student is expected to: (A) describe and draw the intersection of a given plane with various three-

dimensional geometric figures; This change is better vocabulary. (C) The student uses top, front, side, and corner views of three-dimensional

objects to create accurate and complete representations and solve problems. The student is expected to: (C) use orthographic and isometric views of three-dimensional geometric

figures to represent and construct three-dimensional geometric figures and solve problems.

This is an update on vocabulary, as well as an emphasis on using special drawings to help to visualize geometric figures. (G.7) Dimensionality and the geometry of location. The student understands that

coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

(C) The student develops and uses formulas including distance and midpoint. The student is expected to: (C) derive and use formulas involving length, slope, and midpoint. This change removes “distance” and replaces it with “length” and adds slope. This is more general; rather than limiting the idea to just distances on the coordinate plane, students are expected to connect distances with lengths (not necessarily on the plane) and be able to use the technique of finding distance on the coordinate plane to find lengths in other contexts. In addition, slope is an important concept of Algebra I, so there is connection to linear functions, their applications, and their graphical



representations, reinforcing again the multiple representations that are emphasized in both Algebra I and Algebra II. (G.8) Congruence and the geometry of size: The student extends measurement

concepts to find area, perimeter, and volume in problem situations. (G.8) Congruence and the geometry of size. The student uses tools to determine

measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations.

The addition of the use of tools of measurement is significant in several ways. First, students should recognize what can be used as a tool of measurement, and how it is used; this includes traditional tools, such as a ruler or a protractor (what is the significance of the various marks on a ruler and on a protractor?), but also non-traditional tools, such as cubes for measuring volume, a piece of string for measuring length, etc. Second, students must understand what attribute is being measured with these tools (we do not measure a desk, but its length and width), and what measurements are involved in calculating perimeter, area, and volume. Third, students need to be very familiar with the different units of measure and the conversions between them. And lastly, students need to understand the limitations of these tools; students need to recognize that all measurements are approximate, and that the accuracy of the measurement depends upon the accuracy of the tool. Note that this skill is first introduced in the elementary and middle school grades; thus mastery is not yet assumed, although students should have considerable experience by this point. (A) The student finds areas of regular polygons and composite figures. The student is expected to: (A) find areas of regular polygons, circles, and composite figures. This refinement adds finding the areas of circles as well as other two-dimensional geometric figures. While this is not an obvious change, it is important in that it emphasizes that students need to be able to understand and use formulas that involve irrational numbers, not just integers or rational numbers. (D) The student finds surface areas and volumes of prisms, pyramids, spheres,

cones, and cylinders in problem situations. The student is expected to: (D) find surface areas and volumes of prisms, pyramids, spheres, cones,

cylinders, and composites of these figures in problem situations. Students need to see how composite figures can be broken apart and different formulas used to find areas or volumes of the parts to make up the whole. (How can you find the equation that can be used to find the volume of a composite figure?) (G.9) Congruence and the geometry of size. The student analyzes properties and

describes relationships in geometric figures. (D) The student analyzes the characteristics of three-dimensional figures and

their component parts. The student is expected to:



(D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.

The addition of polyhedra is an emphasis on new vocabulary. Students need to learn appropriate vocabulary with which to describe and communicate their thoughts and questions about three-dimensional figures both verbally and in written form. The addition of “based on explorations and concrete models” continues from the previous TEKS the pattern of using explorations and concrete models to give meaning and relevance to all of these concepts. (G.10) Congruence and the geometry of size. The student applies the concept of

congruence to justify properties of figures and solve problems. (A) The student uses congruence transformations to make conjectures and

justify properties of geometric figures. The student is expected to: (A) use congruence transformations to make conjectures and justify

properties of geometric figures including figures represented on a coordinate plane.

The addition here gives yet another reinforcement to multiple representations. A figure on a coordinate plane can give another connection between graphs and tables and formulas. (How can coordinate geometry be used to show two figures on a coordinate plane are congruent?)

(G.11) Similarity and the geometry of shape. The student applies the concepts of

similarity to justify properties of figures and solve problems. (A) The student uses similarity properties and transformations to explore and

justify conjectures about geometric figures. The student is expected to: (A) use and extend similarity properties and transformations to explore and

justify conjectures about geometric figures. The addition of “and extend” is important; students are expected to extend the properties and transformations of similarity, not just use them. Teachers must therefore provide opportunities for extension in ways that are meaningful to students. (D) The student describes the effect on perimeter, area, and volume when

length, width, or height of a three-dimensional solid is changed and applies this idea in solving problems.

The student is expected to: (D) describe the effect on perimeter, area, and volume when one or more

dimensions of a figure are changed and apply this idea in solving problems.

This change is a generalization of the effects of dimension changes; instead of focusing only on three-dimensional figures, this change makes this objective apply to all geometric figures. (What is the effect on perimeter, area and volume when dimensions of similar figures are changed?)

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repl

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chan

ged


Texas Essential Knowledge and Skills: Mathematics 2-1

Text of Adopted Amendments to 19 TAC Chapter 111. Texas Essential Knowledge and Skills for Mathematics

Subchapter A. Elementary

§111.11. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades K-5. The provisions of this subchapter shall be implemented by school districts beginning with the 2006-2007school year.

§111.12. Mathematics, Kindergarten.

(a) Introduction. (1) Within a well-balanced mathematics curriculum, the primary focal points at Kindergarten are

developing whole-number concepts and using patterns and sorting to explore number, data, and shape.

(2) Throughout mathematics in Kindergarten-Grade 2, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use numbers in ordering, labeling, and expressing quantities and relationships to solve problems and translate informal language into mathematical language and symbols. Students use objects to create and identify patterns and use those patterns to express relationships, make predictions, and solve problems as they build an understanding of number, operation, shape, and space. Students progress from informal to formal language to describe two- and three-dimensional geometric figures and likenesses in the physical world. Students begin to develop measurement concepts as they identify and compare attributes of objects and situations. Students collect, organize, and display data and use information from graphs to answer questions, make summary statements, and make informal predictions based on their experiences.


(4) Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Kindergarten-Grade 2, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.

(b) Knowledge and skills. (K.1) Number, operation, and quantitative

reasoning. The student uses numbers to name quantities.

The student is expected to: (A) use one-to-one correspondence and language

such as more than, same number as, or two less than to describe relative sizes of sets of concrete objects;

(B) use sets of concrete objects to represent quantities given in verbal or written form (through 20); and

(C) use numbers to describe how many objects are in a set (through 20) using verbal and symbolic descriptions.

(K.2) Number, operation, and quantitative

reasoning. The student describes order of events or objects.

The student is expected to: (A) use language such as before or after to

describe relative position in a sequence of events or objects; and



(B) name the ordinal positions in a sequence such as first, second, third, etc.

(K.3) Number, operation, and quantitative

reasoning. The student recognizes that there are quantities less than a whole.

The student is expected to: (A) share a whole by separating it into two equal

parts; and (B) explain why a given part is half of the whole.

(K.4) Number, operation, and quantitative reasoning. The student models addition (joining) and subtraction (separating).

The student is expected to model and create addition and subtraction problems in real situations with concrete objects.

(K.5) Patterns, relationships, and algebraic thinking. The student identifies, extends, and creates patterns.

The student is expected to identify, extend, and create patterns of sounds, physical movement, and concrete objects.

(K.6) Patterns, relationships, and algebraic thinking. The student uses patterns to make predictions.

The student is expected to: (A) use patterns to predict what comes next,

including cause-and-effect relationships; and (B) count by ones to 100.

(K.7) Geometry and spatial reasoning. The student describes the relative positions of objects.

The student is expected to: (A) describe one object in relation to another

using informal language such as over, under, above, and below; and

(B) place an object in a specified position.

(K.8) Geometry and spatial reasoning. The student uses attributes to determine how objects are alike and different.

The student is expected to: (A) describe and identify an object by its attributes

using informal language; (B) compare two objects based on their attributes;

and (C) sort a variety of objects including two- and

three-dimensional geometric figures according to their attributes and describe how the objects are sorted.

(K.9) Geometry and spatial reasoning.

The student recognizes attributes of two- and three-dimensional geometric figures.

The student is expected to: (A) describe and compare the attributes of real-life

objects such as balls, boxes, cans, and cones or models of three-dimensional geometric figures;

(B) recognize shapes in real-life three-dimensional geometric figures or models of three-dimensional geometric figures; and

(C) describe, identify, and compare circles, triangles, rectangles, and squares (a special type of rectangle).



(K.10) Measurement. The student directly

compares the attributes of length, area, weight/mass, capacity, and/or relative temperature. The student uses comparative language to solve problems and answer questions.

The student is expected to: (A) compare and order two or three concrete

objects according to length (longer/shorter than, or the same);

(B) compare the areas of two flat surfaces of two-dimensional figures (covers more, covers less, or covers the same);

(C) compare two containers according to capacity (holds more, holds less, or holds the same);

(D) compare two objects according to weight/mass (heavier than, lighter than or equal to); and

(E) compare situations or objects according to relative temperature (hotter/colder than, or the same as).

(K.11) Measurement. The student uses time

to describe, compare, and order events and situations.

The student is expected to: (A) compare events according to duration such as

more time than or less time than; (B) sequence events (up to three); and (C) read a calendar using days, weeks, and

months.

(K.12) Probability and statistics. The student constructs and uses graphs of real objects or pictures to answer questions.

The student is expected to: (A) construct graphs using real objects or pictures

in order to answer questions; and (B) use information from a graph of real objects or

pictures in order to answer questions.

(K.13) Underlying processes and mathematical tools. The student applies Kindergarten mathematics to solve problems connected to everyday experiences and activities in and outside of school.

The student is expected to: (A) identify mathematics in everyday situations; (B) solve problems with guidance that

incorporates the processes of understanding the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;

(C) select or develop an appropriate problem solving strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and



mathematical tools. The student communicates about Kindergarten mathematics using informal language.

The student is expected to: (A) communicate mathematical ideas using

objects, words, pictures, numbers, and technology; and

(B) relate everyday language to mathematical language and symbols.


mathematical tools. The student uses logical reasoning.

The student is expected to justify his or her thinking using objects, words, pictures, numbers, and technology.



§111.13. Mathematics, Grade 1. (a) Introduction. (1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 1 are building

number sense through number relationships, adding and subtracting whole numbers, organizing and analyzing data, and working with two- and three-dimensional geometric figures.




(b) Knowledge and skills. (1.1) Number, operation, and quantitative

reasoning. The student uses whole numbers to describe and compare quantities.

The student is expected to: (A) compare and order whole numbers up to 99

(less than, greater than, or equal to) using sets of concrete objects and pictorial models;

(B) create sets of tens and ones using concrete objects to describe, compare, and order whole numbers;

(C) identify individual coins by name and value and describe relationships among them; and

(D) read and write numbers to 99 to describe sets of concrete objects.

(1.2) Number, operation, and quantitative

reasoning. The student uses pairs of whole numbers to describe fractional parts of whole objects or sets of objects.

The student is expected to: (A) separate a whole into two, three, or four equal

parts and use appropriate language to describe the parts such as three out of four equal parts; and

(B) use appropriate language to describe part of a set such as three out of the eight crayons are red.


reasoning. The student recognizes and solves problems in addition and subtraction situations.

The student is expected to: (A) model and create addition and subtraction

problem situations with concrete objects and write corresponding number sentences; and



(B) use concrete and pictorial models to apply basic addition and subtraction facts (up to 9 + 9 = 18 and 18 – 9 = 9).


algebraic thinking. The student uses repeating patterns and additive patterns to make predictions.

The student is expected to identify, describe, and extend concrete and pictorial patterns in order to make predictions and solve problems.

(1.5) Patterns, relationships, and algebraic thinking. The student recognizes patterns in numbers and operations.

The student is expected to: (A) use patterns to skip count by twos, fives, and

tens; (B) find patterns in numbers, including odd and

even; (C) compare and order whole numbers using

place value; (D) use patterns to develop strategies to solve

basic addition and basic subtraction problems; and

(E) identify patterns in related addition and subtraction sentences (fact families for sums to 18) such as 2 + 3 = 5, 3 + 2 = 5, 5 – 2 = 3, and 5 – 3 = 2.


The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both.

The student is expected to: (A) describe and identify two-dimensional

geometric figures, including circles, triangles, rectangles, and squares (a special type of rectangle);

(B) describe and identify three-dimensional geometric figures, including spheres, rectangular prisms (including cubes), cylinders, and cones;

(C) describe and identify two- and three-dimensional geometric figures in order to sort them according to a given attribute using informal and formal language; and

(D) use concrete models to combine two-dimensional geometric figures to make new geometric figures.

(1.7) Measurement. The student directly

compares the attributes of length, area, weight/mass, capacity, and temperature. The student uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length.

The student is expected to: (A) estimate and measure length using

nonstandard units such as paper clips or sides of color tiles;

(B) compare and order two or more concrete objects according to length (from longest to shortest);

(C) describe the relationship between the size of the unit and the number of units needed to measure the length of an object;

(D) compare and order the area of two or more two-dimensional surfaces (from covers the most to covers the least);



(E) compare and order two or more containers according to capacity (from holds the most to holds the least);

(F) compare and order two or more objects according to weight/mass (from heaviest to lightest); and

(G) compare and order two or more objects according to relative temperature (from hottest to coldest).


understands that time can be measured. The student uses time to describe and compare situations.

The student is expected to: (A) order three or more events according to

duration; and (B) read time to the hour and half-hour using

analog and digital clocks.

(1.9) Probability and statistics. The student displays data in an organized form.

The student is expected to: (A) collect and sort data; and (B) use organized data to construct real-object

graphs, picture graphs, and bar-type graphs.

(1.10) Probability and statistics. The student uses information from organized data.

The student is expected to: (A) draw conclusions and answer questions using

information organized in real-object graphs, picture graphs, and bar-type graphs; and

(B) identify events as certain or impossible such as drawing a red crayon from a bag of green crayons.



The student is expected to: (A) identify mathematics in everyday situations; (B) solve problems with guidance that


(C) select or develop an appropriate problem solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and




The student is expected to: (A) explain and record observations using objects,

words, pictures, numbers, and technology; and

(B) relate informal language to mathematical language and symbols.







(1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 2 are developing an understanding of the base-ten place value system, comparing and ordering whole numbers, applying addition and subtraction, and using measurement processes.





reasoning. The student understands how place value is used to represent whole numbers.

The student is expected to: (A) use concrete models of hundreds, tens, and

ones to represent a given whole number (up to 999) in various ways;

(B) use place value to read, write, and describe the value of whole numbers to 999; and

(C) use place value to compare and order whole numbers to 999 and record the comparisons using numbers and symbols (<, =, >).


reasoning. The student describes how fractions are used to name parts of whole objects or sets of objects.

The student is expected to: (A) use concrete models to represent and name

fractional parts of a whole object (with denominators of 12 or less);

(B) use concrete models to represent and name fractional parts of a set of objects (with denominators of 12 or less); and

(C) use concrete models to determine if a fractional part of a whole is closer to 0, 1/2, or 1.


reasoning. The student adds and subtracts whole numbers to solve problems.

The student is expected to: (A) recall and apply basic addition and subtraction

facts (to 18);



(B) model addition and subtraction of two-digit numbers with objects, pictures, words, and numbers;

(C) select addition or subtraction to solve problems using two-digit numbers, whether or not regrouping is necessary;

(D) determine the value of a collection of coins up to one dollar; and

(E) describe how the cent symbol, dollar symbol, and the decimal point are used to name the value of a collection of coins.


reasoning. The student models multiplication and division.

The student is expected to: (A) model, create, and describe multiplication

situations in which equivalent sets of concrete objects are joined; and

(B) model, create, and describe division situations in which a set of concrete objects is separated into equivalent sets.


algebraic thinking. The student uses patterns in numbers and operations.

The student is expected to: (A) find patterns in numbers such as in a 100s

chart; (B) use patterns in place value to compare and

order whole numbers through 999; and (C) use patterns and relationships to develop

strategies to remember basic addition and subtraction facts. Determine patterns in related addition and subtraction number sentences (including fact families) such as 8 + 9 = 17, 9 + 8 = 17, 17 – 8 = 9, and 17 – 9 = 8.


algebraic thinking. The student uses patterns to describe relationships and make predictions.

The student is expected to: (A) generate a list of paired numbers based on a

real-life situation such as number of tricycles related to number of wheels;

(B) identify patterns in a list of related number pairs based on a real-life situation and extend the list; and

(C) identify, describe, and extend repeating and additive patterns to make predictions and solve problems.


The student uses attributes to identify two- and three-dimensional geometric figures. The student compares and contrasts two- and three-dimensional geometric figures or both.

The student is expected to: (A) describe attributes (the number of vertices,

faces, edges, sides) of two- and three-dimensional geometric figures such as circles, polygons, spheres, cones, cylinders, prisms, and pyramids, etc.;

(B) use attributes to describe how 2 two-dimensional figures or 2 three-dimensional geometric figures are alike or different; and

(C) cut two-dimensional geometric figures apart and identify the new geometric figures formed.



(2.8) Geometry and spatial reasoning. The student recognizes that a line can be used to represent a set of numbers and its properties.

The student is expected to use whole numbers to locate and name points on a number line.

(2.9) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses nonstandard units to describe length, area, capacity, and weight/mass. The student recognizes and uses models that approximate standard units (from both SI, also known as metric, and customary systems) of length, weight/mass, capacity, and time.

The student is expected to: (A) identify concrete models that approximate

standard units of length and use them to measure length;

(B) select a non-standard unit of measure such as square tiles to determine the area of a two-dimensional surface;

(C) select a non-standard unit of measure such as a bathroom cup or a jar to determine the capacity of a given container; and

(D) select a non-standard unit of measure such as beans or marbles to determine the weight/mass of a given object.


standard tools to estimate and measure time and temperature (in degrees Fahrenheit).

The student is expected to: (A) read a thermometer to gather data; (B) read and write times shown on analog and

digital clocks using five-minute increments; and

(C) describe activities that take approximately one second, one minute, and one hour.


student organizes data to make it useful for interpreting information.

The student is expected to: (A) construct picture graphs and bar-type graphs; (B) draw conclusions and answer questions based

on picture graphs and bar-type graphs; and (C) use data to describe events as more likely or

less likely such as drawing a certain color crayon from a bag of seven red crayons and three green crayons.



The student is expected to: (A) identify the mathematics in everyday

situations; (B) solve problems with guidance that


(C) select or develop an appropriate problem solving plan or strategy including drawing a picture, looking for a pattern, systematic guessing and checking, or acting it out in order to solve a problem; and












§111.15. Mathematics, Grade 3.

(a) Introduction. (1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 3 are

multiplying and dividing whole numbers, connecting fraction symbols to fractional quantities, and standardizing language and procedures in geometry and measurement.

(2) Throughout mathematics in Grades 3-5, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use algorithms for addition, subtraction, multiplication, and division as generalizations connected to concrete experiences; and they concretely develop basic concepts of fractions and decimals. Students use appropriate language and organizational structures such as tables and charts to represent and communicate relationships, make predictions, and solve problems. Students select and use formal language to describe their reasoning as they identify, compare, and classify two- or three-dimensional geometric figures; and they use numbers, standard units, and measurement tools to describe and compare objects, make estimates, and solve application problems. Students organize data, choose an appropriate method to display the data, and interpret the data to make decisions and predictions and solve problems.


(4) Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 3-5, students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve meaningful problems as they do mathematics.


reasoning. The student uses place value to communicate about increasingly large whole numbers in verbal and written form, including money.

The student is expected to: (A) use place value to read, write (in symbols and

words), and describe the value of whole numbers through 999,999;

(B) use place value to compare and order whole numbers through 9,999; and

(C) determine the value of a collection of coins and bills.


reasoning. The student uses fraction names and symbols (with denominators of 12 or less) to describe fractional parts of whole objects or sets of objects.

The student is expected to: (A) construct concrete models of fractions; (B) compare fractional parts of whole objects or

sets of objects in a problem situation using concrete models;

(C) use fraction names and symbols to describe fractional parts of whole objects or sets of objects; and



(D) construct concrete models of equivalent fractions for fractional parts of whole objects.


reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers.

The student is expected to: (A) model addition and subtraction using pictures,

words, and numbers; and (B) select addition or subtraction and use the

operation to solve problems involving whole numbers through 999.


reasoning. The student recognizes and solves problems in multiplication and division situations.

The student is expected to: (A) learn and apply multiplication facts through 12

by 12 using concrete models and objects; (B) solve and record multiplication problems (up

to two digits times one digit); and (C) use models to solve division problems and use

number sentences to record the solutions.

(3.5) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results.

The student is expected to: (A) round whole numbers to the nearest ten or

hundred to approximate reasonable results in problem situations; and

(B) use strategies including rounding and compatible numbers to estimate solutions to addition and subtraction problems.


algebraic thinking. The student uses patterns to solve problems.

The student is expected to: (A) identify and extend whole-number and

geometric patterns to make predictions and solve problems;

(B) identify patterns in multiplication facts using concrete objects, pictorial models, or technology; and

(C) identify patterns in related multiplication and division sentences (fact families) such as 2 x 3 = 6, 3 x 2 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2.


algebraic thinking. The student uses lists, tables, and charts to express patterns and relationships.

The student is expected to: (A) generate a table of paired numbers based on

a real-life situation such as insects and legs; and

(B) identify and describe patterns in a table of related number pairs based on a meaningful problem and extend the table.


The student uses formal geometric vocabulary.

The student is expected to identify, classify, and describe two- and three-dimensional geometric figures by their attributes. The student compares two- dimensional figures, three-dimensional figures, or both by their attributes using formal geometry vocabulary.



(3.9) Geometry and spatial reasoning. The student recognizes congruence and symmetry.

The student is expected to: (A) identify congruent two-dimensional figures; (B) create two-dimensional figures with lines of

symmetry using concrete models and technology; and

(C) identify lines of symmetry in two-dimensional geometric figures.


The student recognizes that a line can be used to represent numbers and fractions and their properties and relationships.

The student is expected to locate and name points on a number line using whole numbers and fractions, including halves and fourths.

(3.11) Measurement. The student directly compares the attributes of length, area, weight/mass, and capacity, and uses comparative language to solve problems and answer questions. The student selects and uses standard units to describe length, area, capacity/volume, and weight/mass.

The student is expected to: (A) use linear measurement tools to estimate and

measure lengths using standard units; (B) use standard units to find the perimeter of a

shape; (C) use concrete and pictorial models of square

units to determine the area of two-dimensional surfaces;

(D) identify concrete models that approximate standard units of weight/mass and use them to measure weight/mass;

(E) identify concrete models that approximate standard units for capacity and use them to measure capacity; and

(F) use concrete models that approximate cubic units to determine the volume of a given container or other three-dimensional geometric figure.

(3.12) Measurement. The student reads and

writes time and measures temperature in degrees Fahrenheit to solve problems.

The student is expected to: (A) use a thermometer to measure temperature;

and (B) tell and write time shown on analog and digital

clocks.

(3.13) Probability and statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.

The student is expected to: (A) collect, organize, record, and display data in

pictographs and bar graphs where each picture or cell might represent more than one piece of data;

(B) interpret information from pictographs and bar graphs; and

(C) use data to describe events as more likely than, less likely than, or equally likely as.




situations; (B) solve problems that incorporate understanding

the problem, making a plan, carrying out the plan, and evaluating the solution for reasonableness;



(C) select or develop an appropriate problem solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem; and









The student is expected to: (A) make generalizations from patterns or sets of

examples and non-examples; and (B) justify why an answer is reasonable and

explain the solution process. §111.16. Mathematics, Grade 4.


comparing and ordering fractions and decimals, applying multiplication and division, and developing ideas related to congruence and symmetry.




(b) Knowledge and skills



(4.1) Number, operation, and quantitative reasoning. The student uses place value to represent whole numbers and decimals.

The student is expected to: (A) use place value to read, write, compare, and

order whole numbers through 999,999,999; and

(B) use place value to read, write, compare, and order decimals involving tenths and hundredths, including money, using concrete objects and pictorial models.


quantitative reasoning. The student describes and compares fractional parts of whole objects or sets of objects.

The student is expected to: (A) use concrete objects and pictorial models to

generate equivalent fractions; (B) model fraction quantities greater than one

using concrete objects and pictorial models; (C) compare and order fractions using concrete

objects and pictorial models; and (D) relate decimals to fractions that name tenths

and hundredths using concrete objects and pictorial models.


quantitative reasoning. The student adds and subtracts to solve meaningful problems involving whole numbers and decimals.

The student is expected to: (A) use addition and subtraction to solve problems

involving whole numbers; and (B) add and subtract decimals to the hundredths

place using concrete objects and pictorial models.


quantitative reasoning. The student multiplies and divides to solve meaningful problems involving whole numbers.

The student is expected to: (A) model factors and products using arrays and

area models; (B) represent multiplication and division situations

in picture, word, and number form; (C) recall and apply multiplication facts through 12

x 12; (D) use multiplication to solve problems (no more

than two digits times two digits without technology); and

(E) use division to solve problems (no more than one-digit divisors and three-digit dividends without technology).



The student is expected to: (A) round whole numbers to the nearest ten,

hundred, or thousand to approximate reasonable results in problem situations; and

(B) use strategies including rounding and compatible numbers to estimate solutions to multiplication and division problems.


algebraic thinking. The student uses patterns in multiplication and division.

The student is expected to: (A) use patterns and relationships to develop

strategies to remember basic multiplication and division facts (such as the patterns in related multiplication and division number sentences (fact families) such as 9 x 9 = 81 and 81 ÷ 9 = 9); and



(B) use patterns to multiply by 10 and 100.

(4.7) Patterns, relationships, and algebraic thinking. The student uses organizational structures to analyze and describe patterns and relationships.

The student is expected to describe the relationship between two sets of related data such as ordered pairs in a table.

(4.8) Geometry and spatial reasoning. The student identifies and describes attributes of geometric figures using formal geometric language.

The student is expected to: (A) identify and describe right, acute, and obtuse

angles; (B) identify and describe parallel and intersecting

(including perpendicular) lines using concrete objects and pictorial models; and

(C) use essential attributes to define two- and three-dimensional geometric figures.


The student connects transformations to congruence and symmetry.

The student is expected to: (A) demonstrate translations, reflections, and

rotations using concrete models; (B) use translations, reflections, and rotations to

verify that two shapes are congruent; and (C) use reflections to verify that a shape has

symmetry.

(4.10) Geometry and spatial reasoning. The student recognizes the connection between numbers and their properties and points on a line.

The student is expected to locate and name points on a number line using whole numbers, fractions such as halves and fourths, and decimals such as tenths.

(4.11) Measurement. The student applies measurement concepts. The student is expected to estimate and measure to solve problems involving length (including perimeter) and area. The student uses measurement tools to measure capacity/volume and weight/mass.

The student is expected to: (A) estimate and use measurement tools to

determine length (including perimeter), area, capacity and weight/mass using standard units SI (metric) and customary;

(B) perform simple conversions between different units of length, between different units of capacity, and between different units of weight within the customary measurement system;

(C) use concrete models of standard cubic units to measure volume;

(D) estimate volume in cubic units; and (E) explain the difference between weight and

mass.

(4.12) Measurement. The student applies measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius).

The student is expected to: (A) use a thermometer to measure temperature

and changes in temperature; and (B) use tools such as a clock with gears or a

stopwatch to solve problems involving elapsed time.





The student is expected to: (A) use concrete objects or pictures to make

generalizations about determining all possible combinations of a given set of data or of objects in a problem situation; and

(B) interpret bar graphs.

(4.14) Underlying processes and mathematical tools. The student applies Grade 4 mathematics to solve problems connected to everyday experiences and activities in and outside of school.















explain the solution process. §111.17. Mathematics, Grade 5.


comparing and contrasting lengths, areas, and volumes of two- or three-dimensional geometric figures; representing and interpreting data in graphs, charts, and tables; and applying whole number operations in a variety of contexts.








The student is expected to: (A) use place value to read, write, compare, and

order whole numbers through the 999,999,999,999; and

(B) use place value to read, write, compare, and order decimals through the thousandths place.


quantitative reasoning. The student uses fractions in problem-solving situations.

The student is expected to: (A) generate a fraction equivalent to a given

fraction such as 1/2 and 3/6 or 4/12 and 1/3; (B) generate a mixed number equivalent to a

given improper fraction or generate an improper fraction equivalent to a given mixed number;

(C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators; and

(D) use models to relate decimals to fractions that name tenths, hundredths, and thousandths.


quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.

The student is expected to: (A) use addition and subtraction to solve problems

involving whole numbers and decimals; (B) use multiplication to solve problems involving

whole numbers (no more than three digits times two digits without technology);

(C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit dividends without technology), including interpreting the remainder within a given context;

(D) identify common factors of a set of whole numbers; and

(E) model situations using addition and/or subtraction involving fractions with like denominators using concrete objects, pictures, words, and numbers

. (5.4) Number, operation, and


The student is expected to use strategies, including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division problems.



(5.5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships.

The student is expected to: (A) describe the relationship between sets of data

in graphic organizers such as lists, tables, charts, and diagrams; and

(B) identify prime and composite numbers using concrete objects, pictorial models, and patterns in factor pairs.


algebraic thinking. The student describes relationships mathematically.

The student is expected to select from and use diagrams and equations such as y = 5 + 3 to represent meaningful problem situations.

(5.7) Geometry and spatial reasoning. The student generates geometric definitions using critical attributes.

The student is expected to identify essential attributes including parallel, perpendicular, and congruent parts of two- and three-dimensional geometric figures.

(5.8) Geometry and spatial reasoning. The student models transformations.

The student is expected to: (A) sketch the results of translations, rotations,

and reflections on a Quadrant I coordinate grid; and

(B) identify the transformation that generates one figure from the other when given two congruent figures on a Quadrant I coordinate grid.


The student recognizes the connection between ordered pairs of numbers and locations of points on a plane.

The student is expected to locate and name points on a coordinate grid using ordered pairs of whole numbers.

(5.10) Measurement. The student applies measurement concepts involving length (including perimeter), area, capacity/volume, and weight/mass to solve problems.

The student is expected to: (A) perform simple conversions within the same

measurement system (SI (metric) or customary);

(B) connect models for perimeter, area, and volume with their respective formulas; and

(C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.

(5.11) Measurement. The student applies

measurement concepts. The student measures time and temperature (in degrees Fahrenheit and Celsius).

The student is expected to: (A) solve problems involving changes in

temperature; and (B) solve problems involving elapsed time.

(5.12) Probability and statistics. The student describes and predicts the results of a probability experiment.

The student is expected to: (A) use fractions to describe the results of an

experiment; (B) use experimental results to make predictions;

and (C) list all possible outcomes of a probability

experiment such as tossing a coin.





The student is expected to: (A) use tables of related number pairs to make

line graphs; (B) describe characteristics of data presented in

tables and graphs including median, mode, and range; and

(C) graph a given set of data using an appropriate graphical representation such as a picture or line graph.

















explain the solution process.



Chapter 111. Texas Essential Knowledge and Skills for Mathematics

Subchapter B. Middle School §111.22. Mathematics, Grade 6.

(a) Introduction. (1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 6 are using

ratios to describe direct proportional relationships involving number, geometry, measurement, probability, and adding and subtracting decimals and fractions.

(2) Throughout mathematics in Grades 6-8, students build a foundation of basic understandings in number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry and spatial reasoning; measurement; and probability and statistics. Students use concepts, algorithms, and properties of rational numbers to explore mathematical relationships and to describe increasingly complex situations. Students use algebraic thinking to describe how a change in one quantity in a relationship results in a change in the other; and they connect verbal, numeric, graphic, and symbolic representations of relationships. Students use geometric properties and relationships, as well as spatial reasoning, to model and analyze situations and solve problems. Students communicate information about geometric figures or situations by quantifying attributes, generalize procedures from measurement experiences, and use the procedures to solve problems. Students use appropriate statistics, representations of data, reasoning, and concepts of probability to draw conclusions, evaluate arguments, and make recommendations.

(3) Problem solving in meaningful contexts, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Throughout mathematics in Grades 6-8, students use these processes together with graphing technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems as they do mathematics.


reasoning. The student represents and uses rational numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order non-negative rational

numbers; (B) generate equivalent forms of rational numbers

including whole numbers, fractions, and decimals;

(C) use integers to represent real-life situations; (D) write prime factorizations using exponents; (E) identify factors of a positive integer, common

factors, and the greatest common factor of a set of positive integers; and



reasoning. The student adds, subtracts, multiplies, and divides to solve problems and justify solutions.

The student is expected to: (A) model addition and subtraction situations

involving fractions with objects, pictures, words, and numbers;

(B) use addition and subtraction to solve problems involving fractions and decimals;

(C) use multiplication and division of whole numbers to solve problems including situations involving equivalent ratios and rates;



(D) estimate and round to approximate reasonable results and to solve problems where exact answers are not required; and




The student is expected to: (A) use ratios to describe proportional situations; (B) represent ratios and percents with concrete

models, fractions, and decimals; and (C) use ratios to make predictions in proportional

situations.


The student is expected to: (A) use tables and symbols to represent and

describe proportional and other relationships such as those involving conversions, arithmetic sequences (with a constant rate of change), perimeter and area; and

(B) use tables of data to generate formulas representing relationships involving perimeter, area, volume of a rectangular prism, etc.



The student is expected to formulate equations from problem situations described by linear relationships.

(6.6) Geometry and spatial reasoning. The student uses geometric vocabulary to describe angles, polygons, and circles.

The student is expected to: (A) use angle measurements to classify angles as

acute, obtuse, or right; (B) identify relationships involving angles in

triangles and quadrilaterals; and (C) describe the relationship between radius,

diameter, and circumference of a circle.



(6.8) Measurement. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles.

The student is expected to: (A) estimate measurements (including

circumference) and evaluate reasonableness of results;

(B) select and use appropriate units, tools, or formulas to measure and to solve problems involving length (including perimeter), area, time, temperature, volume, and weight;

(C) measure angles; and (D) convert measures within the same

measurement system (customary and metric) based on relationships between units.



(6.9) Probability and statistics. The student uses experimental and theoretical probability to make predictions.

The student is expected to: (A) construct sample spaces using lists and tree

diagrams; and (B) find the probabilities of a simple event and its




The student is expected to: (A) select and use an appropriate representation

for presenting and displaying different graphical representations of the same data including line plot, line graph, bar graph, and stem and leaf plot;

(B) identify mean (using concrete objects and pictorial models), median, mode, and range of a set of data;

(C) sketch circle graphs to display data; and (D) solve problems by collecting, organizing,

displaying, and interpreting data.










language, efficient tools, appropriate units, and graphical, numerical, physical, or algebraic mathematical models; and


. (6.13) Underlying processes and


The student is expected to: (A) make conjectures from patterns or sets of

examples and non-examples; and (B) validate his/her conclusions using

mathematical properties and relationships.








reasoning. The student represents and uses numbers in a variety of equivalent forms.

The student is expected to: (A) compare and order integers and positive

rational numbers; (B) convert between fractions, decimals, whole

numbers, and percents mentally, on paper, or with a calculator; and

(C) represent squares and square roots using geometric models.


reasoning. The student adds, subtracts, multiplies, or divides to solve problems and justify solutions.

The student is expected to: (A) represent multiplication and division situations

involving fractions and decimals with models, including concrete objects, pictures, words, and numbers;

(B) use addition, subtraction, multiplication, and division to solve problems involving fractions and decimals;

(C) use models, such as concrete objects, pictorial models, and number lines, to add, subtract, multiply, and divide integers and connect the actions to algorithms;






(G) determine the reasonableness of a solution to a problem.



The student is expected to: (A) estimate and find solutions to application

problems involving percent; and (B) estimate and find solutions to application

problems involving proportional relationships such as similarity, scaling, unit costs, and related measurement units.




perimeter, area, circumference, volume, and scaling;

(B) graph data to demonstrate relationships in familiar concepts such as conversions, perimeter, area, circumference, volume, and scaling; and

(C) use words and symbols to describe the relationship between the terms in an arithmetic sequence (with a constant rate of change) and their positions in the sequence.



The student is expected to: (A) use concrete and pictorial models to solve

equations and use symbols to record the actions; and

(B) formulate problem situations when given a simple equation and formulate an equation when given a problem situation.


The student compares and classifies two- and three-dimensional figures using geometric vocabulary and properties.

The student is expected to: (A) use angle measurements to classify pairs of

angles as complementary or supplementary; (B) use properties to classify triangles and

quadrilaterals; (C) use properties to classify three-dimensional

figures, including pyramids, cones, prisms, and cylinders; and

(D) use critical attributes to define similarity .


The student is expected to: (A) locate and name points on a coordinate plane

using ordered pairs of integers; and (B) graph reflections across the horizontal or

vertical axis and graph translations on a coordinate plane.


The student uses geometry to model and describe the physical world.

The student is expected to: (A) sketch three-dimensional figures when given

the top, side, and front views; (B) make a net (two-dimensional model) of the

surface area of a three-dimensional figure; and



(C) use geometric concepts and properties to solve problems in fields such as art and architecture.

(7.9) Measurement. The student solves

application problems involving estimation and measurement.

The student is expected to: (A) estimate measurements and solve application

problems involving length (including perimeter and circumference) and area of polygons and other shapes;


(C) estimate measurements and solve application problems involving volume of prisms (rectangular and triangular) and cylinders.



The student is expected to: (A) construct sample spaces for simple or

composite experiments; and (B) find the probability of independent events.

(7.11) Probability and statistics. The student understands that the way a set of data is displayed influences its interpretation.

The student is expected to: (A) select and use an appropriate representation

for presenting and displaying relationships among collected data, including line plot, line graph, bar graph, stem and leaf plot, circle graph, and Venn diagrams, and justify the selection; and




The student is expected to: (A) describe a set of data using mean, median,

mode, and range; and (B) choose among mean, median, mode, or range

to describe a set of data and justify the choice for a particular situation.



















mathematical properties and relationships. §111.24. Mathematics, Grade 8.

(a) Introduction. (1) Within a well-balanced mathematics curriculum, the primary focal points at Grade 8 are using

basic principles of algebra to analyze and represent both proportional and non-proportional linear relationships and using probability to describe data and make predictions.




reasoning. The student understands that different forms of numbers are appropriate for different situations.

The student is expected to: (A) compare and order rational numbers in

various forms including integers, percents, and positive and negative fractions and decimals;




(C) approximate (mentally and with calculators) the value of irrational numbers as they arise from problem situations (such as π, √2); and

(D) express numbers in scientific notation, including negative exponents, in appropriate problem situations.


reasoning. The student selects and uses appropriate operations to solve problems and justify solutions.

The student is expected to: (A) select appropriate operations to solve


(B) use appropriate operations to solve problems involving rational numbers in problem situations;

(C) evaluate a solution for reasonableness; and (D) use multiplication by a constant factor (unit

rate) to represent proportional relationships.

(8.3) Patterns, relationships, and algebraic thinking. The student identifies proportional or non-proportional linear relationships in problem situations and solves problems.

The student is expected to: (A) compare and contrast proportional and non-

proportional linear relationships; and (B) estimate and find solutions to application

problems involving percents and other proportional relationships such as similarity and rates.


algebraic thinking. The student makes connections among various representations of a numerical relationship.

The student is expected to generate a different representation of data given another representation of data (such as a table, graph, equation, or verbal description).


The student is expected to: (A) predict, find, and justify solutions to application

problems using appropriate tables, graphs, and algebraic equations; and

(B) find and evaluate an algebraic expression to determine any term in an arithmetic sequence (with a constant rate of change).


The student uses transformational geometry to develop spatial sense.

The student is expected to: (A) generate similar figures using dilations

including enlargements and reductions; and (B) graph dilations, reflections, and translations on

a coordinate plane.


The student is expected to: (A) draw three-dimensional figures from different

perspectives; (B) use geometric concepts and properties to

solve problems in fields such as art and architecture;



(C) use pictures or models to demonstrate the Pythagorean Theorem; and

(D) locate and name points on a coordinate plane using ordered pairs of rational numbers.


procedures to determine measures of three-dimensional figures.

The student is expected to: (A) find lateral and total surface area of prisms,

pyramids, and cylinders using concrete models and nets (two-dimensional models);

(B) connect models of prisms, cylinders, pyramids, spheres, and cones to formulas for volume of these objects; and

(C) estimate measurements and use formulas to solve application problems involving lateral and total surface area and volume.


indirect measurement to solve problems.

The student is expected to: (A) use the Pythagorean Theorem to solve real-

life problems; and (B) use proportional relationships in similar two-

dimensional figures or similar three-dimensional figures to find missing measurements.

(8.10) Measurement. The student describes

how changes in dimensions affect linear, area, and volume measures.

The student is expected to: (A) describe the resulting effects on perimeter and

area when dimensions of a shape are changed proportionally; and




The student is expected to: (A) find the probabilities of dependent and

independent events; (B) use theoretical probabilities and experimental

results to make predictions and decisions; and (C) select and use different models to simulate an

event.


The student is expected to: (A) select the appropriate measure of central

tendency or range to describe a set of data and justify the choice for a particular situation;


(C) select and use an appropriate representation for presenting and displaying relationships among collected data, including line plots, line graphs, stem and leaf plots, circle graphs, bar graphs, box and whisker plots, histograms, and Venn diagrams, with and without the use of technology.




The student is expected to: (A) evaluate methods of sampling to determine

validity of an inference made from a set of data; and

(B) recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.

















mathematical properties and relationships.




§111.32. Algebra I (One Credit).

(a) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic

understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their understanding through other mathematical experiences.


(3) Function concepts. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations, and to analyze and interpret relationships.

(4) Relationship between equations and functions. Equations and inequalities arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and inequalities and use a variety of methods to solve them.

(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.

(b) Knowledge and skills. (A.1) Foundations for functions. The


The student is expected to: (A) describe independent and dependent

quantities in functional relationships; (B) gather and record data and use data sets to

determine functional relationships between quantities;

(C) describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations;

(D) represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities; and

(E) interpret and make decisions, predictions, and critical judgments from functional relationships.


student uses the properties and attributes of functions.

The student is expected to: (A) identify and sketch the general forms of linear

(y = x) and quadratic (y = x2) parent functions;



(B) identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete;

(C) interpret situations in terms of given graphs or creates situations that fit given graphs; and

(D) collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.



The student is expected to: (A) use symbols to represent unknowns and

variables; and (B) look for patterns and represent generalizations

algebraically.


The student is expected to: (A) find specific function values, simplify

polynomial expressions, transform and solve equations, and factor as necessary in problem situations;

(B) use the commutative, associative, and distributive properties to simplify algebraic expressions; and

(C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1.


understands that linear functions can be represented in different ways and translates among their various representations.

The student is expected to: (A) determine whether or not given situations can

be represented by linear functions; (B) determine the domain and range for linear

functions in given situations; and (C) use, translate, and make connections among

algebraic, tabular, graphical, or verbal descriptions of linear functions.



The student is expected to: (A) develop the concept of slope as rate of

change and determine slopes from graphs, tables, and algebraic representations;

(B) interpret the meaning of slope and intercepts in situations using data, symbolic representations, or graphs;

(C) investigate, describe, and predict the effects of changes in m and b on the graph of y = mx + b;

(D) graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept;

(E) determine the intercepts of the graphs of linear functions and zeros of linear functions from graphs, tables, and algebraic representations;



(F) interpret and predict the effects of changing slope and y-intercept in applied situations; and

(G) relate direct variation to linear functions and solve problems involving proportional change.


formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) analyze situations involving linear functions

and formulate linear equations or inequalities to solve problems;

(B) investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities; and

(C) interpret and determine the reasonableness of solutions to linear equations and inequalities.



The student is expected to: (A) analyze situations and formulate systems of

linear equations in two unknowns to solve problems;

(B) solve systems of linear equations using concrete models, graphs, tables, and algebraic methods; and

(C) interpret and determine the reasonableness of solutions to systems of linear equations.



The student is expected to: (A) determine the domain and range for quadratic

functions in given situations; (B) investigate, describe, and predict the effects of

changes in a on the graph of y = ax2 + c;

(C) investigate, describe, and predict the effects of changes in c on the graph of y = ax2

+ c; and (D) analyze graphs of quadratic functions and

draw conclusions.

(A.10) Quadratic and other nonlinear functions. The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods.

The student is expected to: (A) solve quadratic equations using concrete

models, tables, graphs, and algebraic methods; and

(B) make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function.


functions. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations.

The student is expected to: (A) use patterns to generate the laws of

exponents and apply them in problem-solving situations;

(B) analyze data and represent situations involving inverse variation using concrete models, tables, graphs, or algebraic methods; and



(C) analyze data and represent situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods.

§111.33. Algebra II (One-Half to One Credit).


understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.




(5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, and reasoning (justification and proof) to make connections within and outside mathematics. Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.



The student is expected to: (A) identify the mathematical domains and ranges

of functions and determine reasonable domain and range values for continuous and discrete situations; and

(B) collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

(2A.2) Foundations for functions. The

student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations.

The student is expected to: (A) use tools including factoring and properties of

exponents to simplify expressions and to transform and solve equations; and

(B) use complex numbers to describe the solutions of quadratic equations.



(2A.3) Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations.

The student is expected to: (A) analyze situations and formulate systems of

equations in two or more unknowns or inequalities in two unknowns to solve problems;

(B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and

(C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts.

(2A.4) Algebra and geometry. The student

connects algebraic and geometric representations of functions.

The student is expected to: (A) identify and sketch graphs of parent functions,

including linear (f(x) = x), quadratic (f(x) = x2), exponential (f(x) = ax), and logarithmic (f(x) = logax) functions, absolute value of x (f(x) = |x|), square root of x (f(x) = √x), and reciprocal of x (f(x) = 1/x);

(B) extend parent functions with parameters such as a in f(x) = a/x and describe the effects of the parameter changes on the graph of parent functions; and

(C) describe and analyze the relationship between a function and its inverse.

(2A.5) Algebra and geometry. The student

knows the relationship between the geometric and algebraic descriptions of conic sections.

The student is expected to: (A) describe a conic section as the intersection of

a plane and a cone; (B) sketch graphs of conic sections to relate

simple parameter changes in the equation to corresponding changes in the graph;

(C) identify symmetries from graphs of conic sections;

(D) identify the conic section from a given equation; and

(E) use the method of completing the square.


The student is expected to: (A) determine the reasonable domain and range

values of quadratic functions, as well as interpret and determine the reasonableness of solutions to quadratic equations and inequalities;

(B) relate representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions; and

(C) determine a quadratic function from its roots or a graph.


functions. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations.

The student is expected to: (A) use characteristics of the quadratic parent

function to sketch the related graphs and connect between the y = ax2

+ bx + c and the y = a(x - h)2

+ k symbolic representations of quadratic functions; and



(B) use the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x - h)2

+ k form of a function in applied and purely mathematical situations.


functions. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) analyze situations involving quadratic

functions and formulate quadratic equations or inequalities to solve problems;

(B) analyze and interpret the solutions of quadratic equations using discriminants and solve quadratic equations using the quadratic formula;

(C) compare and translate between algebraic and graphical solutions of quadratic equations; and

(D) solve quadratic equations and inequalities using graphs, tables, and algebraic methods.



The student is expected to: (A) use the parent function to investigate,

describe, and predict the effects of parameter changes on the graphs of square root functions and describe limitations on the domains and ranges;

(B) relate representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions;

(C) determine the reasonable domain and range values of square root functions, as well as interpret and determine the reasonableness of solutions to square root equations and inequalities;

(D) determine solutions of square root equations using graphs, tables, and algebraic methods;


(F) analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems; and

(G) connect inverses of square root functions with quadratic functions.


formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation.

The student is expected to: (A) use quotients of polynomials to describe the

graphs of rational functions, predict the effects of parameter changes, describe limitations on the domains and ranges, and examine asymptotic behavior;

(B) analyze various representations of rational functions with respect to problem situations;

(C) determine the reasonable domain and range values of rational functions, as well as interpret and determine the reasonableness of solutions to rational equations and inequalities;



(D) determine the solutions of rational equations

using graphs, tables, and algebraic methods; (E) determine solutions of rational inequalities

using graphs and tables; (F) analyze a situation modeled by a rational

function, formulate an equation or inequality composed of a linear or quadratic function, and solve the problem; and

(G) use functions to model and make predictions in problem situations involving direct and inverse variation.



The student is expected to: (A) develop the definition of logarithms by

exploring and describing the relationship between exponential functions and their inverses;

(B) use the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describe limitations on the domains and ranges, and examine asymptotic behavior;

(C) determine the reasonable domain and range values of exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities;

(D) determine solutions of exponential and logarithmic equations using graphs, tables, and algebraic methods;


(F) analyze a situation modeled by an exponential function, formulate an equation or inequality, and solve the problem.

§111.34. Geometry (One Credit).


understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences.

(2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; geometric figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them.





(5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to solve meaningful problems by representing and transforming figures and analyzing relationships.

(6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem solving contexts.

(b) Knowledge and skills. (G.1) Geometric structure. The student

understands the structure of, and relationships within, an axiomatic system.

The student is expected to: (A) develop an awareness of the structure of a

mathematical system, connecting definitions, postulates, logical reasoning, and theorems;

(B) recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes; and

(C) compare and contrast the structures and implications of Euclidean and non-Euclidean geometries.

(G.2) Geometric structure. The student

analyzes geometric relationships in order to make and verify conjectures.

The student is expected to: (A) use constructions to explore attributes of

geometric figures and to make conjectures about geometric relationships; and

(B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic.

(G.3) Geometric structure. The student

applies logical reasoning to justify and prove mathematical statements.

The student is expected to: (A) determine the validity of a conditional

statement, its converse, inverse, and contrapositive;

(B) construct and justify statements about geometric figures and their properties;

(C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false;

(D) use inductive reasoning to formulate a conjecture; and

(E) use deductive reasoning to prove a statement.

(G.4) Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems.

The student is expected to select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems.



(G.5) Geometric patterns. The student uses a variety of representations to describe geometric relationships and solve problems.

The student is expected to: (A) use numeric and geometric patterns to

develop algebraic expressions representing geometric properties;

(B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles;

(C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and

(D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples.

(G.6) Dimensionality and the geometry of

location. The student analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

The student is expected to: (A) describe and draw the intersection of a given

plane with various three-dimensional geometric figures;

(B) use nets to represent and construct three-dimensional geometric figures; and

(C) use orthographic and isometric views of three-dimensional geometric figures to represent and construct three-dimensional geometric figures and solve problems.

(G.7) Dimensionality and the geometry of

location. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly.

The student is expected to: (A) use one- and two-dimensional coordinate

systems to represent points, lines, rays, line segments, and figures;

(B) use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons; and

(C) derive and use formulas involving length, slope, and midpoint.

(G.8) Congruence and the geometry of

size. The student uses tools to determine measurements of geometric figures and extends measurement concepts to find perimeter, area, and volume in problem situations.

The student is expected to: (A) find areas of regular polygons, circles, and

composite figures; (B) find areas of sectors and arc lengths of circles

using proportional reasoning; (C) derive, extend, and use the Pythagorean

Theorem; and (D) find surface areas and volumes of prisms,

pyramids, spheres, cones, cylinders, and composites of these figures in problem situations.




size. The student analyzes properties and describes relationships in geometric figures.

The student is expected to: (A) formulate and test conjectures about the

properties of parallel and perpendicular lines based on explorations and concrete models;

(B) formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models;

(C) formulate and test conjectures about the properties and attributes of circles and the lines that intersect them based on explorations and concrete models; and

(D) analyze the characteristics of polyhedra and other three-dimensional figures and their component parts based on explorations and concrete models.


size. The student applies the concept of congruence to justify properties of figures and solve problems.

The student is expected to: (A) use congruence transformations to make

conjectures and justify properties of geometric figures including figures represented on a coordinate plane; and

(B) justify and apply triangle congruence relationships.

(G.11) Similarity and the geometry of

shape. The student applies the concepts of similarity to justify properties of figures and solve problems.

The student is expected to: (A) use and extend similarity properties and

transformations to explore and justify conjectures about geometric figures;

(B) use ratios to solve problems involving similar figures;

(C) develop, apply, and justify triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples using a variety of methods; and

(D) describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed and apply this idea in solving problems.



§111.35. Precalculus (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September 1,

1998, and at that time shall supersede §75.63(bb) of this title (relating to Mathematics). Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisites: Algebra II, Geometry. (b) Introduction.

(1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model functions and equations and solve real-life problems.

(2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.

(c) Knowledge and skills. (P.1) The student defines functions,

describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, power (including radical), exponential, logarithmic, trigonometric, and piecewise-defined functions.

The student is expected to: (A) describe parent functions symbolically and

graphically, including f(x) = xn, f(x) = 1n x, f(x) = loga x, f(x) = 1/x, f(x) = ex, f(x) = |x|, f(x) = ax, f(x) = sin x, f(x) = arcsin x, etc.;

(B) determine the domain and range of functions using graphs, tables, and symbols;

(C) describe symmetry of graphs of even and odd functions;

(D) recognize and use connections among significant values of a function (zeros, maximum values, minimum values, etc.), points on the graph of a function, and the symbolic representation of a function; and

(E) investigate the concepts of continuity, end behavior, asymptotes, and limits and connect these characteristics to functions represented graphically and numerically.

(P.2) The student interprets the meaning of

the symbolic representations of functions and operations on functions to solve meaningful problems.

The student is expected to: (A) apply basic transformations, including a • f(x),

f(x) + d, f(x - c), f(b • x), and compositions with absolute value functions, including |f(x)|, and f(|x|), to the parent functions;

(B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and

(C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties.



(P.3) The student uses functions and their properties, tools and technology, to model and solve meaningful problems.

The student is expected to: (A) investigate properties of trigonometric and

polynomial functions; (B) use functions such as logarithmic, exponential,

trigonometric, polynomial, etc. to model real-life data;

(C) use regression to determine the appropriateness of a linear function to model real-life data (including using technology to determine the correlation coefficient);

(D) use properties of functions to analyze and solve problems and make predictions; and

(E) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas and incorporate radian measure where needed.

(P.4) The student uses sequences and

series as well as tools and technology to represent, analyze, and solve real-life problems.

The student is expected to: (A) represent patterns using arithmetic and

geometric sequences and series; (B) use arithmetic, geometric, and other

sequences and series to solve real-life problems;

(C) describe limits of sequences and apply their properties to investigate convergent and divergent series; and

(D) apply sequences and series to solve problems including sums and binomial expansion.

(P.5) The student uses conic sections, their

properties, and parametric representations, as well as tools and technology, to model physical situations.

The student is expected to: (A) use conic sections to model motion, such as

the graph of velocity vs. position of a pendulum and motions of planets;

(B) use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound;

(C) convert between parametric and rectangular forms of functions and equations to graph them; and

(D) use parametric functions to simulate problems involving motion.

(P.6) The student uses vectors to model

physical situations. The student is expected to: (A) use the concept of vectors to model situations

defined by magnitude and direction; and (B) analyze and solve vector problems generated

by real-life situations.



§111.36. Mathematical Models with Applications (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September

1, 1998. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisite: Algebra I.

(b) Introduction. (1) In Mathematical Models with Applications, students continue to build on the K-8 and Algebra I

foundations as they expand their understanding through other mathematical experiences. Students use algebraic, graphical, and geometric reasoning to recognize patterns and structure, to model information, and to solve problems from various disciplines. Students use mathematical methods to model and solve real-life applied problems involving money, data, chance, patterns, music, design, and science. Students use mathematical models from algebra, geometry, probability, and statistics and connections among these to solve problems from a wide variety of advanced applications in both mathematical and nonmathematical situations. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to link modeling techniques and purely mathematical concepts and to solve applied problems.

(2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning (justification and proof). Students also use multiple representations, technology, applications and modeling, and numerical fluency in problem-solving contexts.

(c) Knowledge and skills. (M.1) The student uses a variety of strategies

and approaches to solve both routine and non-routine problems.

The student is expected to: (A) compare and analyze various methods for

solving a real-life problem; (B) use multiple approaches (algebraic, graphical,

and geometric methods) to solve problems from a variety of disciplines; and

(C) select a method to solve a problem, defend the method, and justify the reasonableness of the results.

(M.2) The student uses graphical and

numerical techniques to study patterns and analyze data.

The student is expected to: (A) interpret information from various graphs,

including line graphs, bar graphs, circle graphs, histograms, scatterplots, line plots, stem and leaf plots, and box and whisker plots to draw conclusions from the data;

(B) analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences;

(C) analyze graphs from journals, newspapers, and other sources to determine the validity of stated arguments; and

(D) use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information.

(M.3) The student develops and implements

a plan for collecting and analyzing data in order to make decisions.

The student is expected to: (A) formulate a meaningful question, determine

the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions;



(B) communicate methods used, analyses conducted, and conclusions drawn for a data-analysis project by written report, visual display, oral report, or multi-media presentation; and

(C) determine the appropriateness of a model for making predictions from a given set of data.

(M.4) The student uses probability models to

describe everyday situations involving chance.

The student is expected to: (A) compare theoretical and empirical probability;

and (B) use experiments to determine the

reasonableness of a theoretical model such as binomial, geometric, etc.

(M.5) The student uses functional

relationships to solve problems related to personal income.

The student is expected to: (A) use rates, linear functions, and direct variation

to solve problems involving personal finance and budgeting, including compensations and deductions;

(B) solve problems involving personal taxes; and (C) analyze data to make decisions about

banking.

(M.6) The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit.

The student is expected to: (A) analyze methods of payment available in retail

purchasing and compare relative advantages and disadvantages of each option;

(B) use amortization models to investigate home financing and compare buying and renting a home; and

(C) use amortization models to investigate automobile financing and compare buying and leasing a vehicle.

(M.7) The student uses algebraic formulas,

numerical techniques, and graphs to solve problems related to financial planning.

The student is expected to: (A) analyze types of savings options involving

simple and compound interest and compare relative advantages of these options;

(B) analyze and compare coverage options and rates in insurance; and

(C) investigate and compare investment options including stocks, bonds, annuities, and retirement plans.

(M.8) The student uses algebraic and

geometric models to describe situations and solve problems.

The student is expected to: (A) use geometric models available through

technology to model growth and decay in areas such as population, biology, and ecology;

(B) use trigonometric ratios and functions available through technology to calculate distances and model periodic motion; and

(C) use direct and inverse variation to describe physical laws such as Hook's, Newton's, and Boyle's laws.



(M.9) The student uses algebraic and

geometric models to represent patterns and structures.

The student is expected to: (A) use geometric transformations, symmetry, and

perspective drawings to describe mathematical patterns and structure in art and architecture; and

(B) use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music.


Tab 3: Algebra I: Table of Contents 3-i

Tab 3: Algebra I Table of Contents

Master Materials List 3-ii

Spaghetti Regression 3-1 Handout 1: Spaghetti Regression 3-6 Transparency 1/Handout 2: Scatterplot 3-7 Handout 3: Activity 1 Goodness-of-Fit 3-8 Transparency 2 3-11 Transparency 3 3-12 Transparency 4 3-13 Transparency 5 3-14 Handout 4: Measuring 3-15 Transparency 6 3-16 Handout 5: Activity 2 3-17 Handout 6: Activity 3 Absolute Value vs. Squaring 3-27 Handout 7: Supplemental Material 3-31

Understanding Correlation Properties with a Visual Model 3-34 Handout 1: Activity 1 3-42 Handout 2: Activity 2 3-53 Handout 3: Activity 3 - Correlation vs. Causation 3-61 Handout 4: Activity 3, Part B – Headlines 3-65 Handout 5: Supplemental Reading 3-66


Tab 3: Algebra I: Master Materials List 3-ii

Tab 3: Algebra I Master Materials List

Graphing calculator Spaghetti or linguine Tape Colored markers Straightedge Computer with internet access and Java 1.4 Yard stick Spaghetti Regression: Transparencies and handouts Correlation: Transparencies and handouts The following materials are not in the notebook. They can be accessed on the CD through the links below. Java Applet PowerPoint presentation: Correlation vs. Causation


Spaghetti Regression 3-1

Activity: Spaghetti Regression Overview: Participants will investigate the concept of the “goodness-of-fit” and its

significance in determining the regression line or best-fit line for the data. TEKS: This activity supports teacher content knowledge underlying the following

TEKS. (A.2) Foundations of functions. The student uses the properties and

attributes of functions. The student is expected to: (D) collect and organize data, make and interpret scatterplots (including

recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

Background: Fitting the graph of an equation to a data set is covered in all mathematics courses from Algebra I to Calculus and beyond. This module explores the concept in-depth, providing the participants with an understanding beyond that in ordinary secondary texts. The idea is to provide the background knowledge needed to understand the process of modeling. To enrich the study of functions, the TEKS call for the inclusion of problem situations which illustrate how mathematics can model aspects of the world. In real life, functions arise from data gathered through observations or experiments. This data rarely falls neatly into a straight line or along a curve. There is variability in real data and it is up to the student to find the function that best 'fits' the data. Regression, in its many facets, is probably the most widely used statistical methodology in existence. It is the basis of almost all modeling. This activity supports teacher knowledge underlying TEKS A.2.D, wherein students create scatterplots to develop an understanding of the relationships of bivariate data. This includes studying correlations and creating models from which they will predict and make critical judgments. As always, it is beneficial for students to generate their own data. This gives them ownership of the data and gives them insight into the process of collecting reliable data. Teachers should naturally encourage the students to discuss important concepts such as goodness-of-fit. Using the graphing calculator facilitates this understanding. Students will be curious about how the linear functions are created, and teachers should help students develop this understanding.



Knuth and Hartmann in Technology-Supported Mathematics Learning Environments discuss the common approach to this topic:

A common instructional practice is to have students plot the data on a coordinate plane, and then ask them to use a piece of spaghetti to represent the line that they will “fit” to the data. Students are typically instructed to position the spaghetti noodle so that it appears to be as close as possible to each point—visually determining the “best” fit. At this point students might determine the equation for their line and then use that equation in making predictions about additional points. Alternatively, the objective for the lesson might be to determine a line of best fit analytically, usually by using the statistical capabilities of a graphing calculator, and then to use the resulting equation in a similar fashion (i.e., to make predictions). In the former situation, the line that students identified as their line of best fit has not been determined mathematically and may or may not be the best fit in reality. In the latter example, the line has been determined mathematically, but students may not have an understanding of “what the calculator did” in determining the equation for the line or why the line is called a least squares line of best fit (the most commonly used line of best fit). Moreover, teachers often may not attempt to explain the underlying ideas, since the focus of the lesson may be on the use of the equation for the line. In either situation, ideas underlying the least squares line of best fit are not beyond the grasp of students and should be a topic of discussion.

Participants will investigate the concept of the “goodness-of-fit” and its significance in determining the regression line or best-fit-line for the data. Development sequence: Activity 1

What is meant by “best”? What are non-analytical methods used by students to determine fit? Develop an analytical measure for fit. Discuss various measures, including residuals.

Activity 2

Develop the least squares regression method via absolute value regression.

Activity 3

Explore the effects of squaring the residuals and contrast it with using the absolute value of the residuals.

Appendix

Deriving the regression formula via algebra and then thru calculus.

Historical notes. Materials: Graphing calculator



Spaghetti or linguine Tape Colored markers Straightedge Computer with internet access (Activity 3) Transparencies: 1-6 (pages 3-7, 3-11 – 3-14, 3-16) Handout 1 (page 3-6) Handout 2 (page 3-7) Handout 3 (pages 3-8 – 3-10) Handout 4 (page 3-15) Handout 5 (pages 3-17 – 3-21) Handout 6 (pages 3-27 – 3-28 Handout 7 (pages 3-31 – 3-33)

Grouping: 4-5 per group Time: 1½ -2 hours Lesson:

Procedures Notes Activity 1 Have participants read and discuss Handout 1, Spaghetti Regression: Overview/Learning Objectives/Background, (page 3-6).

Discuss the importance of modeling and student discussions of concepts such as goodness-of-fit (see the Trainer Notes Background discussion above.)

Give each participant 3-5 pieces of spaghetti, the Transparency 1/Handout 2, Scatterplot (page 3-7) and Handout 3, Activity 1: Goodness of Fit, (page 3-8 ).

Have the participants examine the plot and visually determine a line of best-fit (or trend line) using a piece of spaghetti. They then tape the spaghetti line onto their graph.

This should be done individually so that there is variation in the choice of lines within each group.

Ask: Who has the best line in your group? How can we determine this? Ask: What is meant by best? Ask: What is meant by a close fit? See, How Do You Find the Line of Best-Fit? (page 3-10), to discuss methods students use for placing trend lines. (Do not

This page discusses the general idea behind linear regression. To determine a line of best fit you must have an agreed upon measure of “goodness”. If that measure is “closeness of the points to the line”, the best line is then the line with the least total distance of points to the line. There are many methods for measuring “closeness.” The most common is the method of least



Procedures Notes discuss how to measure yet; see below.)

squares. Intuitively, we think of a close fit as a good fit. We look for a line with little space between the line and the points it is supposed to fit. We would say that the best fitting line is the one that has the least space between itself and the data points which represent actual measurements.

Have the participants use a second piece of spaghetti to measure the distance from each point to the line and break off that length. Each member of a group must measure the same way. Thus, each group must decide their method for measuring before they begin. Groups may measure vertically, horizontally, perpendicularly, etc.

Encourage diversity in measuring methods among the groups to add depth to the following discussions.

Have the participants line up their spaghetti distances to determine who in their group has the closest fit. Then, they replace the segments and tape them to their scatterplot.

This will determine the total error (i.e., total distance from their line to the data).

Have each group present their method and results. A good way to accomplish this is to have the “winner” from each table come up to the front. They can then be grouped by their method of measurement. Have each share, discuss, compare, and contrast.

Discuss the fact that since the groups used different methods of measuring, we cannot determine best-of-fit for the entire class. Discuss accuracy of measurement. Did they measure from the edge of each point or the middle, etc.?

Distribute Handout 4, Measuring, (page 3-15) to discuss three ways (vertically, horizontally, perpendicularly) to measure the space between a point and the line. Discuss the meaning of a residual and why it is used in evaluating the accuracy of a model.

Why measure vertically? The sole purpose in making a regression line is to use it to predict the output for a given input. The vertical distances (residuals) represent how far off the predictions are from the data we actually measured.

Activity 2



Procedures Notes Distribute Handout 5, Activity 2, (pages 3-17 – 3-21). Tell the participants we will now determine who has the best trend line in the class. Tell participants to look for “FYI:” in the activity for calculator help. Have participants stop when they finish #5 and use overhead 2 to cultivate a class discussion of the questions in #5 before proceeding. It is important that participants understand why the residuals must be absolute valued or squared before summing. Transparency 6 reproduces the figure on page 3-19.

In Activity 1 the groups used different measures of goodness-of-fit; thus the best trend line of the class could not be determined. The participants will need a Graphing Calculator. Encourage the calculator-capable participants to help out within their groups.

Activity 3

Distribute Handout 6, Activity 3, (pages 3-27 – 3-28). Participants will need a computer with Java version 1.4. Have participants open the applet Regression and work through handout.

In this activity, an interactive java applet is used to investigate several data sets and contrast geometrically and numerically the effect of using the square of the residuals vs. the absolute value of the residuals. Encourage the participants to test their own conjectures and share/discuss with group.

Supplemental Material

Ask participants to read the Handout 7, Supplemental Material, (pages 3-31 – 3-33).

The Supplemental Material discusses two ways to minimize the sum of the squared residuals which leads to the formula that the calculator uses to find the least squares regression line. Historical notes are included about who originally developed least squares regression and where the term regression comes from.


Handout 1 Spaghetti Regression 3-6

Spaghetti Regression

Overview

Participants will investigate the concept of the “goodness-of-fit” and its significance in determining the regression line or best-fit line for the data.

Learning Objectives This activity supports Teacher Content Knowledge needed for A2D: The student is expected to collect and organize data, make and interpret scatterplots (including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

Background

Fitting the graph of an equation to a data set is covered in all mathematics courses from Algebra I to Calculus and beyond. The objective of this module is to explore the concept in-depth to provide understanding beyond that in ordinary secondary texts.

To enrich the study of functions, the TEKS call for the inclusion of problem situations which illustrate how mathematics can model aspects of the world. In real life, functions arise from data gathered through observations or experiments. This data rarely falls neatly into a straight line or along a curve. There is variability in real data and it is up to the student to find the function that best 'fits' the data. Regression, in its many facets, is probably the most widely use statistical methodology in existence. It is the basis of almost all modeling.

This activity supports teacher knowledge underlying TEKS A.2.D, wherein students create scatterplots to develop an understanding of the relationships of bivariate data; this includes studying correlations and creating models from which they will predict and make critical judgments. As always, it is beneficial for students to generate their own data. This gives them ownership of the data and gives them insight into the process of collecting reliable data. Teachers should naturally encourage the students to discuss important concepts such as goodness-of fit. Using the graphing calculator facilitates this understanding. Students will be curious about how the linear functions are created, and teachers should help students develop this understanding.


Transparency 1/Handout 2 Spaghetti Regression 3-7

Scatterplot



Activity 1 Goodness-of-Fit Objective: To Investigate the concept of goodness of fit and develop an understanding of residuals in determining a line of best-fit. 1. Examine the plot provided and visually determine a line of best-fit (or trend line)

using a piece of spaghetti. Tape your spaghetti line onto your graph. 2. Now let us investigate the “goodness” of the fit. Use a second piece of spaghetti to

measure the distance from the first point to the line. Break off this piece to represent that distance. Each person at the table must measure in the same way, so discuss the method you will use before starting. Repeat this for each point.

3. Line up your spaghetti distances to determine who in your group has the closest fit.

Determine the total error; i.e., total distance from your line to the data. Then replace the segments and tape them to your scatterplot.

Total error = _______



Activity 1 Goodness-of-Fit – Possible Solutions Objective: To Investigate the concept of goodness of fit and develop an understanding of residuals in determining a line of best-fit 1. Examine the plot provided and visually determine a line of best-fit (or trend line)

using a piece of spaghetti. Tape your spaghetti line onto your graph. Trainer notes: Use the page titled How Do You Find the Line of Best-Fit? to discuss methods students use for placing trend lines. This page discusses the general idea behind linear regression. To determine a line-of best fit, you must have an agreed upon measure of “goodness.” If that measure is closeness of the points to the line, the best line is then the line with the least total distance. There are many methods for measuring “closeness.” The most common is the method of least squares. 2. Now let us investigate the “goodness” of the fit. Use a second piece of spaghetti to

measure the distance from the first point to the line. Break off this piece to represent that distance. Each person at the table must measure in the same way, so discuss the method you will use before starting. Repeat this for each point.

Encourage at least one group to use the shortest distance from the point to the line (i.e., the perpendicular distance.) Have each group present their method and results. A good way to accomplish this is to have the “winner” from each table come up to the front. They can then be grouped by their method of measurement. Have each share, discuss, compare, and contrast. Discuss the fact that since that the groups used different methods of measuring, we cannot determine best-of-fit for the entire class. Discuss the accuracy of their measurements. Did they measure from the edge of each point or the middle, etc.? 3. Line up your spaghetti distances to determine who in your group has the closest fit.

Determine the total error. (i.e., total distance from your line to the data.) Then, replace the segments and tape them to your scatterplot.

Total error = _______ Use the page titled Measuring to discuss three ways to measure the space between a point and the line. Discuss the meaning of a residual and why it is used in evaluating the accuracy of a model.



How Do You Find the Line of Best Fit? – Possible Solutions

So you’ve observed some data. You have a set of data points (x,y). You've plotted them, and they seem to be pretty much linear. How do you find the line that best fits those points? "That’s simple," your students say. "Put them into a TI-83 and look at the answer." Okay, but let us ask a deeper question: How does the calculator find the answer?

What is meant by Best?

First, we have to agree on what we mean by the "best fit" of a line to a set of points. Why do we say that the line on the left fits the points better than the line on the right? And can we say that some other line might fit them better still?

Transparency 2 (page 3-11)

Look at the following students’ responses to the task: draw a line of best fit for the data. What reasoning might they have given for their choice of lines?

Passes through the most points, equal number of points above and below, passes through the end points, etc. [Transparencies 3-5 (pages 3-12 – 3-14)]

Usually we think of a close fit as a good fit. But, what do we mean by close? How close are these points?


Transparency 2 Spaghetti Regression 3-11

Discuss criteria that might be used to assess the “closeness” of these points? How many different ways might it be done?



Measuring There are at least three ways to measure the space between a point and the line: vertically in the y direction, horizontally in the x direction, and the shortest distance from a point to the line (on a perpendicular to the line.) In regression, we usually choose to measure the space vertically. These distances are known as residuals. • Why would you want to measure this way? What do the residuals represent in relation to our function? Consider the purpose of the line and the following diagram. The purpose of regression is to find a function that can model a data set. The function is then used to predict the y values (or outputs, f(x) ) for any given input x. So, the vertical distance represents how far off the prediction is from the actual data point (i.e., the “error” in each prediction.) Residuals are calculated by subtracting the model’s predicted values, f(xi), from the observed values, yi.

Residual = )( ii xfy −


Handout 5-1 Spaghetti Regression 3-17

Activity 2 Objective: Investigate various methods of regression. Whose model makes the best predictions? Let us compare everyone’s lines using the residuals. Before we begin, we need to know the equation for your spaghetti function, f(x) = mx + b. Assume the lower left corner of the graph is (0,0). f(x) = __________________ 1. Enter your function at Y1= in the calculator. 2. Enter the actual data into L1 and L2. Put the x-values in L1 and the y-values in L2.

Make certain that the x’s are typed in correspondence to the y’s.

x 2 5 6 10 12 15 16 20 20 y 14 19 9 21 7 21 18 10 22

3. Place the predicted values, f(xi), created by your function, in L3. To do this, place

your cursor on L3 and enter your function, using L1 as the inputs of the function. (See below.)

FYI: Y1 can be found under [vars] → [Y-vars] → [1:function] → 1:Y1



4. Compute the residuals (the distances between the predicted values, f(xi) , and actual y values) and place them in L4. This can be done by entering L4 = L2-L3.

5. On your home screen compute Sum(L4). Record your group’s functions and the

corresponding sums.

FYI: Sum can be found under [2nd][stat] → [math] → 5:sum

Function Sum of the residual errors

• Examine your values in L4. What is the meaning of a negative residual in terms of the graph and in terms of the function’s predictions? What is the meaning of a positive or negative total for the functions in #5?



Examine the following student’s work.

• In L4 what is the meaning of 39.23? What is the corresponding value in your table? Describe its meaning.

• What is the meaning of a low total residual error? Is it a good measure of fit? Why or why not?



There are two possible ways to fix the above problem. One way is to take the absolute value of the residual; the other is to square the residual. Taking the absolute value of the residuals is synonymous with using our spaghetti segments to measure the vertical error. 6. Find Sum(abs(L4)). Record your group’s functions and the corresponding sums.

FYI: abs can be found under [2nd][0]

Function Sum of the errorresidual

• Compare with those in the class to determine who now has the lowest total error.

Note: The calculator’s regression method uses the squared residuals when measuring the goodness-of-fit of a regression line. Let us compare our lines of best-fit, using the squared residuals. 7. Find the total of the squared residuals by Sum((L4)2) . This is often referred to as

the Sum of the Squared Errors, noted SSE.

Function SSE



• Compare with those in the class to determine who has the lowest sum of the squared errors. Did the best line in the group change? Why or why not?

Let us compare our lines against the calculator’s regression line. 8. Use your calculator to compute the linear regression function, f(x) = mx + b.

f(x) = ___________________

9. Enter the function into Y1 and place the function’s predicted values f(xi) in L3, i.e., L3 =

Y1(L1). 10. Quickly, compute the sum of squared errors by using SUM((L2- L3)2). SSE = ________

• How do the functions in the class compare to this one?

11. Create a scatterplot and graph your group’s functions and the calculator’s regression

function. Examine visually the goodness of fit of each in regard to their SSE. At least two methods exist for evaluating goodness of fit: taking the absolute value of the residuals and squaring the residuals. Although taking the absolute value seems most intuitive, relying on squaring does several things. The most desirable one is that it simplifies the mathematics needed to guarantee the “best” line. (See the appendix.) In Activity 3, you can investigate how squaring the residuals when measuring our goodness-of-fit affects the choice of the regression line. Understanding what you are looking for is always the toughest part of any problem, so the hard part is done. You now know how to measure “goodness” of fit. We can also say exactly what the calculator means by the line of best-fit. If we compute the residuals (i.e., the error in the y direction), square each one, and add up the squares, we say the line of best-fit is the line for which that sum is the least. Since it is a sum of squares, the method is called the Method of Least Squares! This is the most commonly used method but, as we have seen, it isn’t the only way!



Activity 2 - Possible solutions Objective: Investigate various methods of regression. Whose model makes the best predictions? Let us compare everyone’s lines using the residuals. Before we begin, we need to know the equation for your spaghetti function, f(x) = mx + b. Assume the lower left corner of the graph is (0,0). f(x) = __1/3 x + 9________________ 1. Enter your function at Y1= in the calculator.

2. Enter the actual data into L1 and L2. Put the x-values in L1 and the y-values in L2. Make

certain that the x’s are typed in correspondence to the y’s.

x 2 5 6 10 12 15 16 20 20 y 14 19 9 21 7 21 18 10 22

3. Place the predicted values, f(xi), created by your function, in L3. To do this, place your cursor on L3 and enter your function, using L1 as the inputs of the function. (See below.)

FYI: Y1 can be found under [vars] → [Y-vars] → [1:function] → 1:Y1



4. Compute the residuals (the distances between the predicted values, f(xi) , and actual y values) and place them in L4. This can be done by entering L4 = L2-L3.

5. On your home screen compute Sum(L4). Record your group’s functions and the

corresponding sums.

FYI: Sum can be found under [2nd][stat] → [math] → 5:sum

Function Sum of the residual errors

Y= 1/3 x + 9 24.66

Y= ¼ x + 11 15.5

Y= 5/4 x 8.5

Y= 2x + 3 -98

• Examine your values in L4. What is the meaning of a negative residual in terms of the graph and in terms of the function’s predictions? What is the meaning of a positive or negative total for the functions in #5? In the graph, a negative residual in L4 means the actual point is below the line. In terms of the function’s predictions a negative residual means the function over predicted value. A positive sum of the residuals means you have more total under predictions than over predictions and vise versa for a negative sum of the residuals.



Examine the following student’s work.

• In L4 what is the meaning of 39.23? What is the corresponding value in your table? Describe its meaning.

It means this person’s function under predicted the value by 39.32.

• What is the meaning of a low total residual error? Is it a good measure of fit? Why or why not?

This is not a good measure of fit because large under predictions could be cancelled by large over predictions hence making the sum small, as in the above example.



There are two possible ways to fix the above problem. One way is to take the absolute value of the residual; the other is to square the residual. Taking the absolute value of the residuals is synonymous with using our spaghetti segments to measure the vertical error. 6. Find Sum(abs(L4)). Record your group’s functions and the corresponding sums.

FYI: abs can be found under [2nd][0]

Function Sum of the errorresidual

Y= 1/3 x + 9 52

Y= ¼ x + 11 48.5

Y= 5/4 x 64.5

Y= 2x + 3 124

• Compare with those in the class to determine who now has the lowest total error. Note: The calculator’s regression method uses the squared residuals when measuring the goodness-of-fit of a regression line. Let us compare our lines of best-fit, using the squared residuals. 7. Find the total of the squared residuals by Sum((L4)2) . This is often referred to as the Sum

of the Squared Errors, noted SSE.

Function SSE

Y= 1/3 x + 9 338

Y= ¼ x + 11 289.375

Y= 5/4 x 676.375

Y= 2x + 3 2488



• Compare with those in the class to determine who has the lowest sum of the squared errors. Did the best line in the group change? Why or why not? The best line could change. In regression using the absolute value and using the square may not agree, because it changes how you define what the best line is.

Let us compare our lines against the calculator’s regression line. 8. Use your calculator to compute the linear regression function, f(x) = mx + b.

f(x) = _.156 x + 13.83__________________

9. Enter the function into Y1 and place the function’s predicted values f(xi) in L3, i.e., L3 =

Y1(L1). 10. Quickly, compute the sum of squared errors by using SUM((L2- L3)2). SSE = _259.67_______

• How do the functions in the class compare to this one? The calculator linear regression function should have a lower SEE than the classes functions.

11. Create a scatterplot and graph your group’s functions and the calculator’s regression

function. Examine visually the goodness of fit of each in regard to their SSE. At least two methods exist for evaluating goodness of fit: taking the absolute value of the residuals and squaring the residuals. Although taking the absolute value seems most intuitive, relying on squaring does several things. The most desirable one is that it simplifies the mathematics needed to guarantee the “best” line. (See the appendix.) In Activity 3, you can investigate how squaring the residuals when measuring our goodness-of-fit affects the choice of the regression line. Understanding what you are looking for is always the toughest part of any problem, so the hard part is done. You now know how to measure “goodness” of fit. We can also say exactly what the calculator means by the line of best-fit. If we compute the residuals (i.e., the error in the y direction), square each one, and add up the squares, we say the line of best-fit is the line for which that sum is the least. Since it is a sum of squares, the method is called the Method of Least Squares! This is the most commonly used method but, as we have seen, it isn’t the only way!



Activity 3 Absolute Value vs. Squaring OBJECTIVE: It is important to understand the effect squaring has on the residuals and the placement of a regression line. In this activity, we will use an interactive java applet to investigate several data sets and contrast geometrically and numerically the effect of using the square of the residuals vs. the absolute value of the residuals. 1. Place three points forming a triangle on the graph. Select “plot line” and place a

trend line on the graph. 2. Select “Draw residuals.” Using the handle points, adjust your line to visually

minimize the length of the residuals.

Select “Show Trend Line Equation.” ____________________ 3. Select “Draw (residuals)2.” Using the handle points, adjust your line to visually

minimize the area of the squares.

Equation of the line: ____________________ 4. Now select “Sum of the residuals” and adjust your line to numerically minimize the

|residuals|. Record the equation and total: ___________________ 5. Now select “Sum of the (residuals)2” and adjust your line to numerically minimize the

(residuals)2. Record the equation and total:_________________ 6. Create a situation where the sum of the squares is less than the sum of the absolute

value. 7. Create a data set in which the least absolute value and least squares methods agree

on the line of best fit. 8. Place the following ordered pairs (4, 1), (4, 4), (-4, 0), and (-4, -3) in the table. Find

the line of best fit for each method. • Compare and contrast these two methods.



• How does squaring the residuals affect how individual data points contribute to the total error? Does squaring increase or decrease the effect of an individual residual on the total error? • What is the effect of an outlier point on each of the possible trend lines for each method? Further investigation Another method for finding regression lines is Chebyshev’s Best-Fit Line Method, also known as the MinMax Method, which finds the line with the minimum maximum residual. Chebyshev’s evaluates each line based on its largest residual and takes the

line with the smallest (largest residual ) as the regression line. • Use Chebyshev’s method in the previous graphs to determine a line of best fit. How does it compare to the least absolute value and least squares methods? How it is affected by outliers?



Activity 3 Absolute Value vs. Squaring – Selected Answers OBJECTIVE: It is important to understand the effect squaring has on the residuals and the placement of a regression line. In this activity, we will use an interactive java applet to investigate several data sets and contrast geometrically and numerically the effect of using the square of the residuals vs. the absolute value of the residuals. 1. Place three points forming a triangle on the graph. Select “plot line” and place a

trend line on the graph. 2. Select “Draw residuals.” Using the handle points, adjust your line to visually

minimize the length of the residuals.

Select “Show Trend Line Equation.” ____________________ 3. Select “Draw (residuals)2.” Using the handle points, adjust your line to visually

minimize the area of the squares.

Equation of the line: ____________________ 4. Now select “Sum of the |residuals|” and adjust your line to numerically minimize the

|residuals|. Record the equation and total: ___________________ 5. Now select “Sum of the (residuals)2” and adjust your line to numerically minimize the

(residuals)2. Record the equation and total:_________________ 6. Create a situation where the sum of the squares is less than the sum of the absolute

values. Participants should notice the effect squaring has on each residual. Place the points close to the line so that the residuals are less than 1.

7. Create a data set in which the least absolute value and least squares methods agree

on the line of best fit. Various possible answers 8. Place the following ordered pairs (4, 1), (4, 4), (-4, 0), and (-4, -3) in the table. Find

the line of best fit for each method. Note:The absolute value line is not unique. • Compare and contrast these two methods. Various answers: Note, both methods are valid. However, the absolute value method does not always give a unique regression line.



• How does squaring the residuals affect how individual data points contribute to the total error? Does squaring increase or decrease the effect of an individual residual on the total error? If the residual is less than one, squaring decreases it’s effect on the total squared residual. If the residual is greater than one, squaring increases it’s effect on the total squared residual. Thus, the squaring method rewards small errors and penalizes large residual errors. This penalizing and rewarding effect of the least squares method is often described as desirable by statisticians. The absolute value methods however treats all residuals the same (with equal contempt). • What is the effect of an outlier point on each of the possible trend lines for each method? Since squaring will give disproportion weight to the outlier when compared to the absolute value method it will have a greater effect on the sum errors of the least squares regression line. Further investigation Another method for finding regression lines is Chebyshev’s Best-Fit Line Method, also known as the MinMax Method, which finds the line with the minimum maximum residual. Chebyshev’s evaluates each line based on its largest residual and takes the line with

the smallest (largest residual ) as the regression line. • Use Chebyshev’s method in the previous graphs to determine a line of best fit. How does it compare to the least absolute value and least squares methods? How it is affected by outliers?



Supplemental Material

Two ways to minimize the sum of the squared residuals

The key to solving this or any problem is understanding exactly for what you are looking. Our model, or line of “best fit”, bmxxf +=)( , will be one that minimizes the sum of the squares of the vertical distances between the actual points and the predicted ones, i.e., the residuals = )( ii xfy − . It can be written 2))(( xfyL −= ∑ or

2))((∑ +−= bmxyL .

What kind of equation is 2))((∑ +−= bmxyL ? That’s right, quadratic. And we actually know enough about quadratics from Algebra II to solve this problem. But, one of the key words in the above paragraph is minimize, which should also make you think Calculus! This gives us an easy alternative approach.

Let us examine this quadratic more closely.

2))((∑ +−= bmxyL

)222( 2222 ybymyxbbmxxm +−−++=∑

It may look daunting, but remember, m and b are the only unknowns here. x and y are just numbers supplied by each of the actual points in our scatterplot.

Expanding L farther, ∑ ∑ ∑ ∑ ∑+−−++= 2222 222 yybxymnbxbmxmL

(You might want to double check all this! Why let someone else have all the fun?)

Remember that the summations are just constants! So now we have a choice to use calculus to find its minimum or use Algebra II to find its vertex.

Let’ try the Calculus!

In calculus, the minimum occurs here where the derivative is equal to zero. Since we have two variables, m and b, we will want to take the derivative of each variable separately. (These are called partial derivatives.)

∑ ∑ ∑ =−+=∂∂ 0222 2 xyxbxmmL



∑ ∑ =−+=∂∂ 0222 ynbxm

bL

All that’s left is to solve this system of equations by elimination or substitution. Take your pick. Using substitution, b in the second equation looks easiest to solve for. So, we get

nxmy

b ∑∑ −= . Substituting for b into the first equation and simplifying, we get

22 )(∑∑∑ ∑∑

−

+=

xxnyxxyn

m .

And that’s it. Your calculator or computer just sums the x’s, the y’s, the xy’s, etc. and out pops the slope and y-intercept of your regression equation. It is not hard, but certainly tedious when done by hand. (You may wonder how we know it is a minimum and not a maximum. The second derivative is 2; a positive second derivative means it must be a minimum.) Let us try it with Algebra!

Here we go. Remember that we want to find the minimum of

∑ ∑ ∑ ∑ ∑+−−++= 2222 222 yybxymnbxbmxmL

and that all of those summations are just constants. Thus, L is a quadratic with respect to m or b. This can be seen easily by rearranging.

=)(mL ∑ )( 2x m2 + )22( ∑∑ − xyxb m )2( 22 nbyyb ++− ∑ ∑

L(b)= n b2+ )22( ∑∑ − yxm b )2( 222 ∑∑∑ −−+ yxymxm



Do they open up or down? The leading coefficients, ∑ 2x and n, are both positive, so the answer is up.

From Algebra II, we know the vertex of Ax2 + Bx + C occurs at AB

2− .

So m = ∑

∑∑ −−22

)22(x

xyxb =

∑∑∑ −

2xxbxy

, and

b = n

yxm2

)22( ∑∑ −− =

nxmy ∑∑ −

.

Substituting one into the other, we get 22 )(∑∑∑ ∑∑

−

+=

xxnyxxyn

m and

b = 22

2

)(∑∑∑∑∑∑

+

−

xxnxyxxy

. This is exactly the same result as before.

Some Historical Notes

Who invented the method of least squares? It is not clear. Often credit is given to Karl Friedrich Gauss (1777–1855), who was first published on this subject in 1809. But the Frenchman Adrien Marie Legendre (1752–1833) published a clear example of the method four years earlier. Legendre was in charge of setting up the new metric system of measurement, and the meter was to be one ten-millionth of the distance from the North Pole through Paris to the Equator. Surveyors had measured portions of the arc but to get the best measurement for the whole arc, Legendre developed the method of least squares. He would probably use GPS today, but he was still amazingly accurate.

Where does the term "regression” come from? The term was first used by Sir Francis Galton (1822-1911) in his hereditary studies. He wanted to predict the heights of sons from their father’s heights. He learned that a tall father tended to have sons shorter than himself, and a short father tended to have sons taller than himself. The heights of sons thus regressed towards the mean height of the population over several generations. The term "regression” is now used for many types of prediction problems, and does not merely apply to regression towards the mean.


Understanding Correlation Properties with a Visual Model 3-34

Activity: Understanding Correlation Properties with a Visual Model Overview: This activity encourages participants to visually explore the meaning of

correlation and to recognize correlation patterns. . TEKS: This activity supports teacher content knowledge underlying the

following TEKS: §111.32. Algebra I (a) Basic understandings. (5) Tools for algebraic thinking. Techniques for working with functions

and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology, (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.


The student is expected to: (D) collect and organize data, make and interpret scatter plots

(including recognizing positive, negative, or no correlation for data approximating linear situations), and model, predict, and make decisions and critical judgments in problem situations.

Vocabulary: correlation, regression, Pearson Product moment correlation, causation Procedure: Participants use a computer to investigate correlation values and to

practice estimating correlation values for scatterplots.

After completing the activity, participants should have a visual feel for numerical correlation values, and should also be able to relate numerical values of correlation to contextual situations. Participants are also encouraged to investigate and understand the relationship between correlation and causation.

Materials: Computer with internet access and Java 1.4

PowerPoint slides: Correlation vs. Causation Handout 1 (pages 3-42 – 3-46) Handout 2 (pages 3-53 – 3-55) Handout 3 (pages 3-61 – 3-64) Handout 4 (page 3-65) Handout 5 (pages 3-66 – 3-67) Yard stick Photocopy of a forearm.



Grouping: Individually or pairs Time: 2 - 2½ hours Lesson:

Procedures Notes Activity 1 CSI Correlation Part A: Participants use a computer to investigate how the modeling process is used to generate new knowledge.

Distribute Handouts 1 and 2, Activities 1 and 2, (pages 3-42 – 3-46 and pages 3-53 – 3-55). Read the crime scene scenario. Participants will collect data from 8 people using a yard stick. Participants will use a computer to investigate correlation values. Using the applet Correlation. Hand out the photo copy of the assailants forearm. The participants will then extrapolate the assailants height.

The forearm should be measured from the elbow to the wrist. Participants should discuss measuring techniques and degree of accuracy. You will need a photocopy of the assailants forearm to distribute to each group. If possible use someone who is a bit out of the normal range. For example, the tallest or shortest participant. This will cause the participants to extrapolate instead of interpolate.

Part B: A Closer Look: Participants use a computer to investigate correlation values. Have participants open the applet Correlation. Changes in the data set are investigated. Outliers, changes in scale, and the geocenter of a set of data are discussed.

After completing Activity 1, participants should have a visual feel for numerical correlation values and should also be able to relate numerical values of correlation to contextual situations.



Procedures Notes Activity 2 Part A: The goal of this activity is to gain an intuitive understanding of r. Using the web applet Correlation, scatterplots are easily constructed. By clicking and dragging points, participants can change the data sets and investigate the effect on the correlation.

The dynamic nature of the applet allows you to see how the correlation changes when a data point is added or moved. Without technology, such intuition would take years to develop.

Part B: The r Game Have participants play a game with several classmates to develop deeper understanding of correlations, leverage points, and geocenters. Participants use the web applet Correlation, to create scatterplots with a specific correlation. (See Part B handout for further directions.)

When interpreting the correlation coefficient, you should always look at the scatterplot first to see if the relationship is linear. If it is, you may calculate the correlation coefficient. Always remember that a visual analysis of data is quite valuable in addition to a numerical analysis.

The participants should play several times until they have a good intuition of how each point’s relationship with the others affects the correlation.

Activity 3 Correlation vs. Causation This activity explores the relationship between correlation and causation.

Part A: Give out Handout 3, Activity 3 - Part A, (pages 3-61 – 3-64) or use the Power Point provided and lead a class discussion of correlation and causation.



Procedures Notes

Correlation vs. Correlation vs. CausationCausation

In a Gallup poll, surveyors asked, “Do you believe correlation implies causation?’”

64% of American’s answered “Yes” .38% replied “No”. The other 8% were undecided.

There is a humorous article discussing this poll in the appendix.

Ice-cream sales are strongly correlated with crime rates.

Therefore, ice-cream causes crime.

If correlation implies causation, this would be a fabulous finding! To reduce or eliminate crime, all we would have to do is stop selling ice cream. Even though the two variables are strongly correlated, assuming that one causes the other would be erroneous. What are some possible explanations for the strong correlation between the two? One possibility might be that high temperatures increase crime rates (presumably by making people irritable) as well as ice-cream sales.



Procedures Notes

The Simpsons(Season 7, "Much Apu About Nothing")

Homer:Not a bear in sight. The "Bear Patrol" is working like a charm!

Lisa: That's specious reasoning, Dad. Homer:[uncomprehendingly] Thanks,

honey. Lisa: By your logic, I could claim that

this rock keeps tigers away. Homer:Hmm. How does it work? Lisa: It doesn't work; it's just a

stupid rock! Homer:Uh-huh. Lisa: But I don't see any tigers

around, do you? Homer:(pause) Lisa, I want to buy your

rock.

An entertaining demonstration of this fallacy once appeared in an episode of The Simpsons (Season 7, "Much Apu About Nothing"). The city had just spent millions of dollars creating a highly sophisticated "Bear Patrol" in response to the sighting of a single bear the week before.

Without prope r inte rpre tation, Without prope r inte rpre tation, causation causation should notshould not be be assumed, or e ve n implie d.assumed, or e ve n implie d.

Cons ider the following res earch Cons ider the following res earch undertaken by the Univers ity of Texas undertaken by the Univers ity of Texas Health S c ience Center at S an Antonio Health S c ience Center at S an Antonio appearing to s how a link between appearing to s how a link between cons umption of cons umption of dietdiet s oda and weight s oda and weight gain.gain.

The s tudy of more than 600 normalThe s tudy of more than 600 normal--weight weight people found, eight years later, that they people found, eight years later, that they were 65 percent more likely to be were 65 percent more likely to be overweight if they drank one diet s oda a overweight if they drank one diet s oda a day than if they drank none. And if they day than if they drank none. And if they drank two or more diet s odas a day, they drank two or more diet s odas a day, they were even more likely to become were even more likely to become overweight or obes e.overweight or obes e.

Our students and the general public often take such relationships as causal. By no means does this state that diet soda causes obesity - but there is a strange pattern at play here.

A relationship other than causal might exist between the two variables. It is possible that there is some other variable or factor that is causing the outcome. This is sometimes referred to as the "third variable" or "missing variable" problem.

• What are some other possible plausible alternative explanations to our diet soda/obesity research example?



Procedures Notes

A re lationship othe r than causal A re lationship othe r than causal might e xist be twe en the two might e xist be twe en the two

variables. It's possible that the re variables. It's possible that the re is some othe r variable or factor is some othe r variable or factor

that is causing the outcome . This that is causing the outcome . This is some times re fe rred to as the is some times re fe rred to as the

"third variable " or "missing "third variable " or "missing variable " problem.variable " problem.

Ice cream sales and the number of shark Ice cream sales and the number of shark attacks on swimmers are correlated. attacks on swimmers are correlated.

Skirt lengths and stock prices are highly Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt correlated (as stock prices go up, skirt lengths get shorter).lengths get shorter).

The number of cavities in elementary The number of cavities in elementary school children and vocabulary size are school children and vocabulary size are strongly correlated.strongly correlated.

We must be very careful in interpreting correlation coefficients. Just because two variables are highly correlated does not mean that one causes the other. In statistical terms, we simply say that correlation does not imply causation. There are many good examples of correlation which are nonsensical when interpreted in terms of causation.

The re are two re lationships which can be mistake n for causation:

1 . Common response2. Confounding



Procedures Notes

1 . Common Response :

Both Xand Yrespond to changes in some unobse rve d variable , Z. All thre e of our pre vious e xample s are e xamples of common re sponse .

2. Confounding

The e ffe ct of Xon Yis indistinguishable from the e ffe cts of othe r e xplanatory variable s on Y. When studying medical tre atments, the “place bo e ffe ct” is an e xample of confounding.

The placebo effect is the phenomenon that a patient's symptoms can be alleviated by an otherwise ineffective treatment, since the individual expects or believes that it will work. For example, if we are studying the effects of Tylenol on reducing pain, and we give a group of pain-sufferers Tylenol and record how much their pain is reduced, the effect of Tylenol is confounded with the effect of giving them any pill. Many people will report a reduction in pain by simply being given a sugar pill with no medication.

Unless data have been gathered by experimental Unless data have been gathered by experimental means and confounding variables have been means and confounding variables have been eliminated, correlation eliminated, correlation never never implies causation.implies causation.

Controlled experiments Controlled experiments must be performed.must be performed.

→→ YXZ &→

When can we imply causation?

Experimental research attempts to understand and predict causal relationships. Since correlations can be created by an antecedent, Z, which causes both X and Y, or by confounding variables, controlled experiments are performed to remove these possibilities.

Still the great Scottish philosopher David Hume has argued that we can only perceive correlation, and causality can never truly be known or proven.



Procedures Notes Part B: Headlines Distribute Handout 4, Part B, (page 3-65). Participants brainstorm common causes of confounding variables for various headlines and related correlations.

Power point slides of the headlines are included to help in a summary discussion of this activity.

Within your group, brainstorm common causes or confounding variables. Write your ideas below and be prepared to share.

Handout 5, Supplemental Reading, (pages 3-66 – 3-67). This is a humorous article discussing the correlation, causation debate.


Handout 1-1 Understanding Correlation Properties with a Visual Model 3-42

ACTIVITY 1 This module opens with an explanation of the way that paired measurements can be plotted in two-dimensional space. Next, positive and negative relationships are discussed and participants are asked to predict values using a regression equation. It concludes with a discussion of outliers. PART A Consider the following. At approximately 6:45 a.m., Tuesday morning, Principal Espinoza saw something strange as he opened the backdoor to B. Wyatt High School. As he entered the hallway, he immediately discovered the broken glass from the classroom door. It was a 9th grade Math classroom. The computers were missing, the desks were overturned, and the prized school banner was torn from the wall. The perpetrators were long gone, but they had left something behind. Next to the desk, where Mrs. Joe’s computer once sat, was the imprint of a forearm on the board. When the police arrived, they immediately began to gather forensic evidence. Mr. Espinosa, knowing your love of CSI and Numb3rs, asks you to help gather data to help identify the bandits.

Bones of the arm can reveal interesting facts about an individual. But can they reveal a person's height? Forensic anthropologists team up with law enforcers to help solve crimes. Let us combine math with forensics to see how.

Collect data for 8 people.

Person Forearm Length (inches)

Height (inches)



1. From the table, describe any relationships you see between the variables forearm length and height.

Making a scatter plot can provide a useful summary of a set of bivariate data (two variables). It gives a good visual picture of the relationship between the two variables and aids in the interpretation of the correlation coefficient and regression model. The scatterplot should always be drawn before working out a linear correlation coefficient or fitting a regression line. A positive association is indicated on a scatterplot by an upward trend (positive slope), where larger x-values correspond to larger y-values and smaller x-values correspond to smaller y-values. A negative association would be indicated by the opposite effect (negative slope) where the higher x-values would correspond to lower y-values. Or, there might not be any notable linear association. 2. We will use the web applet Correlation for further investigation in the following

exercises. Enter the forearm length and height data into the table and examine the scatterplot.

In 1896, Karl Pearson gave the formula for calculating the correlation coefficient known as r. (To see it, select show equation for r.) He argued that it was the best indicator of linear relationships. It derives its name from linear, meaning “straight line,” and co-relation meaning to "go together." The drudgery of computing the correlation coefficient by hand is quite ominous. However, today’s calculators can easily compute r. It is often referred to as the Pearson Product Moment Correlation Coefficient. We can generally categorize the strength of correlation as follows:

• Strong |r| > 0.8

• Moderate: 0.5< |r |<0.8

• Weak: |r| < 0.5



If variables are strongly correlated, we often use one to predict the other. A gross example from forensic science is using the size and larva stage of maggots to predict time of death. Linear regression is the method used to create these mathematical prediction models. Given X, we can predict Y. If the correlation is high enough, record the function for the regression line.

3. Using the information you collected, try predicting the height of our assailant for Mr. Espinosa. A copy of the police imprint from our assailant is attached.

● What would increase your confidence in this prediction?

In real life, mathematics always begins with a question. What do you want to know? This is followed by data collection. If it is bivariate data, scatterplots are drawn to give the “big picture.” If the relationship looks linear, the correlation coefficient is calculated to quantify the relationship. If the r value is reasonable, a linear function can be found that is used to predict what has not been observed; in our case, the height of the assailant.



PART B – A Closer Look

Now let us look more closely at how we measure the strength of associations between data sets. The correlation coefficient can range from -1 to 1. ( ± 1 being a perfect linear correlation between the two variables.) If the variables are completely independent, the correlation is 0. However, the converse is not true since the correlation coefficient detects only linear dependencies between two variables. Let us investigate changes in our data set.

1. Click and drag one point of your scatterplot until the correlation is 0.3. Record the coordinates.

● Is the placement of this point unique? ● What does the new point represent in terms of the context? An outlier is an observation that lies an abnormal distance from other values in a sample. In a sense, this definition leaves it up to you, the analyst, to decide what will be considered abnormal. Before abnormal observations can be singled out, it is necessary to characterize normal observations. If the data point is in error, it should be corrected if possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. ● Would you consider your point an outlier? Why? 2. Suppose a “mistake” was made. All the forearm sizes were reported in centimeters

(1 in. = 2.54 cm.), and all the heights were recorded in inches. A student tells you that the correlation will be too low saying that increasing the forearm data by a factor greater than 1 will spread the points in a graph. Do you agree with the student? How would you explore this issue?



● What do you suppose would happen to our correlation value if we changed to different height scale?

3. Delete the outlying point from your table. Now, add two additional points to make a

correlation of 0.99. Discuss the placement of your points.

The geocenter, also called the center of mass or centroid is the “average” point of the data. If we have the points (x1,y1), (x2,y2) (x3,y3), and (x4,y4) then the coordinates of the geocenter would be ⎟

⎠⎞

⎜⎝⎛ ++++++

44321,

44321 yyyyxxxx . The further a point is from the

geocenter of the data the more “leverage” it has. (Note: The regression line always passes through this point.)

Students often have a naïve sense of correlation. We should look to extend their understandings. Dynamic applications such as Geometers Sketch Pad and web applets open up new avenues for exploration and deeper understandings. By allowing students to explore and test their own conjectures, they take ownership of their mathematical understandings.



ACTIVITY 1 - Selected Answers This module opens with an explanation of the way that paired measurements can be plotted in two-dimensional space. Next, positive and negative relationships are discussed and participants are asked to predict values using a regression equation. It concludes with a discussion of outliers. PART A Consider the following. At approximately 6:45 a.m., Tuesday morning, Principal Espinoza saw something strange as he opened the backdoor to B. Wyatt High School. As he entered the hallway, he immediately discovered the broken glass from the classroom door. It was a 9th grade Math classroom. The computers were missing, the desks were overturned, and the prized school banner was torn from the wall. The perpetrators were long gone, but they had left something behind. Next to the desk, where Mrs. Joe’s computer once sat, was the imprint of a forearm on the board. When the police arrived, they immediately began to gather forensic evidence. Mr. Espinosa, knowing your love of CSI and Numb3rs, asks you to gather data to help identify the bandits.

Bones of the arm can reveal interesting facts about an individual. But can they reveal a person's height? Forensic anthropologists team up with law enforcers to help solve crimes. Let us combine math with forensics to see how.

Collect data for 8 people. (The number can vary, use at least 7.)

Person Forearm Length (inches)

Height (inches)

10.5 63

10 66

11.5 72

10.25 62

10.5 66

11.5 71

12.5 76



1. From the table, describe any relationships you see between the variables forearm length and height.

As forearm length increases, the height increases.

Making a scatter plot can provide a useful summary of a set of bivariate data (two variables). It gives a good visual picture of the relationship between the two variables and aids in the interpretation of the correlation coefficient and regression model. The scatterplot should always be drawn before working out a linear correlation coefficient or fitting a regression line. A positive association is indicated on a scatterplot by an upward trend (positive slope) where larger x-values correspond to larger y-values and smaller x-values correspond to smaller y-values. A negative association would be indicated by the opposite effect (negative slope), where the higher x-values would have lower y-values. Or, there might not be any notable linear association. 2. We will use the web applet Correlation for further investigation in the following

exercises. Enter the forearm length and height data into the table and examine the scatterplot.



In 1896, Karl Pearson gave the formula for calculating correlation coefficients known as r. (To see it, select show equation for r.) He argued that it was the best indicator of linear relationships. It derives its name from linear, meaning “straight line,” and co-relation meaning to "go together." The drudgery of computing them by hand is quite ominous. However, today’s calculators can easily compute them. It is often referred to as the Pearson Product Moment Correlation. We can generally categorize the strength of correlation as follows:

• Strong: |r| > 0.8

• Moderate: 0.5< |r |<0.8

• Weak: |r| < 0.5 If variables are strongly correlated, we often use one to predict the other. A gross example from forensic science is using the size and larva stage of maggots to predict time of death. Linear regression is the method used to create these mathematical prediction models. Given X, we can predict Y. If your correlation is high enough, record the function for your regression line.

3. Using the information you collected, try predicting the height of our assailant for Mr. Espinosa. A copy of the police imprint from our assailant is attached.

Forearm length is approximately 12.5 inches.

Sample regression equation is y=5.29x+10.01 with a r-value of .92

y = 5.29(12.5) +10.01=76.135. The assailant is approx. 6’ 4”.

● What would increase your confidence in this prediction? Collecting more data More accurate measurements A higher r-value

Further Questions: Suppose you sample another 8 people's forearms and heights and find a regression line to that data. Would the new line be the same as before? Which of these lines should you use to predict height from forearm? How reliable is our equation for predicting any person's height given we know



their forearm? These questions of reliability are very important in applying the ideas of this activity to data from any real life experiment.

In real life, mathematics always begins with a question. What do you want to know? This is followed by data collection. If it is bivariate data, scatterplots are drawn to give the “big picture.” If the relationship looks linear, the correlation coefficient is calculated to quantify the relationship. If the r value is reasonable, a linear function can be found that is used to predict what has not been observed, in our case, the height of the assailant. PART B – A Closer Look -Selected Answers Now let us look more closely at how we measure the strength of association between data sets. The correlation coefficient can range from -1 to 1. ( ± 1 being a perfect linear correlation between the two variables.) If the variables are completely independent, the correlation is 0. However, the converse is not true since the correlation coefficient detects only linear dependencies between two variables. . Let us investigate changes in our data set. 1. Click and drag one point of your scatterplot until the correlation is 0.3. Record the

coordinates. 2.

Ex.(8.07, 72.6) Answers may vary



● Is the placement of this point unique? No, it is not. Encourage participants to find several.

Further Questions: How many places can the point be placed? 3, 4, infinite?

● What does the new point represent in terms of the context?

Answers may vary. The example point (8.07, 72.6) represents a person that is approximately 6 feet and ½ inches tall with 8 inch forearms.

An outlier is an observation that lies an abnormal distance from other values in a sample. In a sense, this definition leaves it up to you, the analyst, to decide what will be considered abnormal. Before abnormal observations can be singled out, it is necessary to characterize normal observations. If the data point is in error, it should be corrected if possible. If there is no reason to believe that the outlying point is in error, it should not be deleted without careful consideration. ● Would you consider your point an outlier? Why?

Answers may vary. Yes. The point represents an abnormal situation. He is 6 feet tall and has the arms of a child or possibly an amputee.

2. Suppose a “mistake” was made. All the forearm sizes were reported in centimeters (1 in. = 2.54 cm.) and all the heights were recorded in inches. A student tells you that the correlation will be too low saying that increasing the forearm data by a factor greater than 1 will spread the points in a graph. Do you agree with the student? How would you explore this issue?

Encourage participants to try it. This question should encourage a healthy discussion of important misconceptions. How does the visual spread of the data affect the correlation? What if both height and forearm are recorded in centimeters? How would this dilation affect the correlation? This confronts the students’ belief that if we spread the data, the correlation should diminish. However, changes in scale do not affect correlations.

● What do you suppose would happen to our correlation value if we changed to a different height scale?

It should not affect the correlation. Changes in scale do not affect correlations.

3. Delete the outlying point from your table. Now, add two additional points to make a correlation of 0.99. Discuss the placement of your points.



The geocenter, also called the center of mass or centroid is the “average” point of the data. If we have the points (x1,y1), (x2,y2) (x3,y3), and (x4,y4) then the coordinates of the geocenter would be

⎟⎠⎞

⎜⎝⎛ ++++++

44321,

44321 yyyyxxxx . The further a point is from the geocenter

of the data the more “leverage” it has. (Note: The regression line always passes through this point.)

Students often have a naïve sense of correlation. We should look to extend their understandings. Dynamic applications such as Geometers Sketch Pad and web applets open up new avenues for exploration and deeper understandings. By allowing students to explore and test their own conjectures, they take ownership of their mathematical understandings.



ACTIVITY 2

H.G Wells once said,”Statistical thinking will one day be as necessary as the ability to read and write.”

The goal of this activity is to gain an intuitive understanding of r. Using the web applet Correlation, scatterplots can easily be constructed. The dynamic nature of the applet allows you to see how the correlation changes when a data point is added or moved. PART A 1. Clear your table and place two points on the graph. Note the correlation. ● Would any two points have the same value? Explain. A student remarks that “when r is undefined, it means there is no linear model for the data.” Do you agree? How would you explore/explain this? 2. Make a lower left to upper right pattern of 10 points with a correlation of 0.7. 3. Make a vertical stack of 9 data points on the left side of the window. Add a 10th point

somewhere to the right and drag it until the correlation again reaches 0.7. Is its placement unique?

4. Make another scatter plot with 10 data points in a curved pattern that starts at the

lower left, rises to the right, then falls again at the far right. Adjust the points until you have a smooth curve with a correlation close to 0.7.

● Does any other curved pattern have this same correlation? ● What can you conclude about the numerical value of a correlation?



Juan and Kaylee have collected data for a class experiment. They correctly found an r-value of .15. Juan claims that no function will model the data. But, Kaylee says that the r-value is wrong because she has found a good one. Is this possible? How would you help these students? 5. Make a data plot with a correlation of 0 by placing 8 to 10 points on the graph. 6. Enter 4 points in the table to make a perfect rectangle. Note the correlation value. 7. Create several other data sets with a horizontal or vertical line of symmetry and note

the correlation value.

Let us take a closer look at the numerical value of r by investigating the equation that produces this quantity. n is the number of points.

8. Select “Show equation for r” and examine the formula to determine why and when the correlation is undefined. (Hint: use two points)

So far, we have developed some intuitions about r. Its formal definition is quite complex. However, r2 is much simpler. So we mention it here. r2 is the fraction of total variation in the y variable that can be explained by the regression equation. The rest of the variation is due to randomness or some other factors. For example, if the correlation coefficient is 0.7 then r2= 0.49 meaning that 49% of the variation in the y-variable can be explained by the regression equation. The other 51% is due to some other factors. How does this affect your understanding of how the strength of correlations are categorized in part A of activity 1? Consider some other “strong,” “moderate,” and “weak” r-values.



PART B – The r Game

One basic rule when interpreting the correlation coefficient is to “First look at the scatterplot to see if the relationship between variables is linear.” If it is, you may calculate the correlation coefficient. Always remember that a visual analysis of data is quite valuable in addition to a numerical analysis.

To understand r, it is important to understand how individual points affect the value of correlations. The relationship of outliers, leverage points and non-leverage points to the geocenter of a set of data are explored in this simple exercise. Use the web applet Correlation to practice creating scatterplots with a specific correlation. 1. Challenge your classmates to place seven points on the graph that have a correlation

of 0.7. 2. You are not allowed to delete or drag points once they are placed on the graph. 3. Price is Right Rules – The closest r-value without going over wins! Play several times varying the number of points and r-value.



ACTIVITY 2 -Selected Answers

H.G Wells once said, ”Statistical thinking will one day be as necessary as the ability to read and write.”

The goal of this activity is to gain an intuitive understanding of r. Using the web applet Correlation, scatterplots can easily be constructed. The dynamic nature of the applet allows you to see how the correlation changes when a data point is added or moved. PART A

1. Clear your table and place two points on the graph. Note the correlation.

● Would any two points have the same value? Explain.

No, depending on the placement of the points, it may be 1, -1, or undefined. A student remarks that “when r is undefined, it means there is no linear model for the data.” Do you agree? How would you explore/explain this?

No, consider the points (1, 4) and (3,4) modeled by y= 4. There is no correlation for these points since when the x-values increase, the y-values neither increase or decrease. The correlation is not zero since the variables are not independent of one another.



2. Make a lower left to upper right pattern of 10 points with a correlation of 0.7.

3. Make a vertical stack of 9 data points on the left side of the window. Add a 10th point

somewhere to the right and drag it until the correlation again reaches 0.7. Is its placement unique? No



4. Make another scatter plot with 10 data points in a curved pattern that starts at the lower left, rises to the right, then falls again at the far right. Adjust the points until you have a smooth curve with a correlation close to 0.7.

● Does any other curved pattern have this same correlation? ● What can you conclude about the numerical value of a correlation? Juan and Kaylee have collected data for a class experiment. They correctly found an r-value of .15. Juan claims that no function will model the data well. But, Kaylee says that the r-value is wrong because she has found a good model. Is this possible? How would you help these students?

It is possible. Since correlation measures only a linear relationship, to have r close to or equal to zero does not mean that there is no relationship between X and Y! For example, a relationship might be quadratic.



5. Make a data plot with a correlation of 0 by placing 8 to 10 points on the graph.

6. Enter 4 points in the table to make a perfect rectangle. Note the correlation value.



7. Create several other data sets with a horizontal or vertical line of symmetry and note the correlation value.

Let us take a closer look at the numerical value of r by investigating the equation that produces this quantity. n is the number of points.

8. Select “Show equation for r” and examine the formula to determine why and when the correlation is undefined and zero. (Hint: use two points)

So far, we have developed some intuitions about r. Its formal definition is quite complex. However, r2 is much simpler, so we mention it here. r2 is the fraction of total variation in the y variable that can be explained by the regression equation. The rest of the variation is due to randomness or other factors. For example, if the correlation coefficient is 0.7 then r2= 0.49 meaning that 49% of the variation in the y-variable can be explained by the regression equation. The other 51% is due to other factors. How does this affect your understanding of how the strength of correlations are categorized in part A of activity 1? Consider some other “strong,” “moderate,” and “weak” r-values.



ACTIVITY 3 - Correlation vs. Causation Objective To explore the relationship between correlation and causation. PART A In a Gallup poll, surveyors asked, “Do you believe correlation implies causation?” 64% of American’s answered “Yes” while only 38% replied “No”. The other 8% were undecided. Consider the following:

Ice-cream sales are strongly correlated with crime rates. Therefore, ice-cream causes crime.

If correlation implies causation, this would be a fabulous finding! To reduce or eliminate crime, all we would have to do is stop selling ice cream. Even though the two variables are strongly correlated, assuming that one causes the other would be erroneous. What are some possible explanations for the strong correlation between the two? One possibility might be that high temperatures increase crime rates (presumably by making people irritable) as well as ice-cream sales.

An entertaining demonstration of this fallacy once appeared in an episode of The Simpsons (Season 7, "Much Apu About Nothing"). The city had just spent millions of dollars creating a highly sophisticated "Bear Patrol" in response to the sighting of a single bear the week before.



Homer: Not a bear in sight. The "Bear Patrol" is working like a charm!

Lisa: That's specious reasoning, Dad. Homer: [uncomprehendingly] Thanks, honey. Lisa: By your logic, I could claim that this rock keeps tigers away. Homer: Hmm. How does it work? Lisa: It doesn't work; it's just a stupid rock! Homer: Uh-huh. Lisa: But I don't see any tigers around, do you? Homer: (pause) Lisa, I want to buy your rock.

Correlations are often reported inferring causation in newspaper articles, magazines, and television news. But, without proper interpretation, causation should not be implied or assumed.

Consider the following research undertaken by the University of Texas Health Science Center at San Antonio, appearing to show a link between consumption of diet soda and weight gain.

The study of more than 600 normal-weight people found, eight years later, that they were 65 percent more likely to be overweight if they drank one diet soda a day than if they drank none. And if they drank two or more diet sodas a day, they were even more likely to become overweight or obese.

Our students and the general public often take such relationships as causal. By no means does this state that diet soda causes obesity - but there is a strange pattern at play here.

A relationship other than causal might exist between the two variables. It is possible that there is some other variable or factor that is causing the outcome. This is sometimes referred to as the "third variable" or "missing variable" problem.

• What are some other possible plausible alternative explanations to our diet soda/obesity research example?

We must be very careful in interpreting correlation coefficients. Just because two variables are highly correlated does not mean that one causes the other. In statistical terms, we simply say that correlation does not imply causation. There are many



good examples of correlation which are nonsensical when interpreted in terms of causation.

For example, each of the following are strongly correlated:

• Ice cream sales and the number of shark attacks on swimmers. • Skirt lengths and stock prices (as stock prices go up, skirt lengths get shorter). • The number of cavities in elementary school children and vocabulary size. • Peanut butter sales and the economy. Two relationships which can be mistaken for causation are: 1. Common response: Both X and Y respond to changes in some unobserved variable,

Z. All three of our examples above are examples of common response.

2. Confounding variables: The effect of X on Y is hopelessly mixed up with the effects

of other variables on Y. When studying medical treatments, the placebo effect is an example of confounding. The placebo effect is the phenomenon that a patient's symptoms can be alleviated by an otherwise ineffective treatment, since the individual expects or believes that it will work. For example, if we are studying the effects of Tylenol on reducing pain, and we give a group of pain-sufferers Tylenol and record how much their pain is reduced, the effect of Tylenol is confounded with giving them any pill. Many people will report a reduction in pain by simply being given a sugar pill with no medication. False causes can be ruled out using the scientific method. This is done through a designed experiment.

In practice, three conditions must be met in order to conclude that X causes Y, directly or indirectly:

1) X must precede Y 2) Y must not occur when X does not occur 3) Y must occur whenever X occurs



Experimental research attempts to understand and predict causal relationships (X→Y). Since correlations can be created by an antecedent, Z, which causes both X and Y (Z →X & Y), or by confounding variables, controlled experiments are performed to remove these possibilities. Unless data has been gathered by experimental means and confounding variables have been eliminated, one can not infer causation.

Still the great Scottish philosopher David Hume has argued that we can only perceive correlation, and causality can never truly be known or proven.


Handout 4 Understanding Correlation Properties with a Visual Model 3-65

PART B - Headlines Consider the following headlines and their matching correlations from various sources. Accepted uncritically, each might be used to "prove" one’s point of view in an article. Within your group, brainstorm common causes or confounding variables. Write your ideas below and be prepared to share. Correlated variables Causation factors 1. Kids’ TV Habits Tied to Lower IQ Scores IQ scores and hours of TV time (r = -0.54) 2. Eating Pizza ‘Cuts Cancer Risk’ Pizza consumption and cancer rate (r = -0.59) 3. Gun bill introduced to ward off crime Gun ownership and crime (r = 0.71) 4. Reading Fights Cavities Number of cavities in elementary school children and their vocabulary size (r = 0.67) 5. Graffiti Linked to Obesity in City Dwellers BMI and amount of graffiti and litter (r =0.45) 6. Stop Global Warming: Become a Pirate Average global temperature and number of pirates ( r = -0.93)



Supplemental Reading NEW POLL SHOWS CORRELATION IS CAUSATION

WASHINGTON (AP) The results of a new survey conducted by pollsters suggest that, contrary to common scientific wisdom, correlation does in fact imply causation. The highly reputable source, Gallup Polls, Inc., surveyed 1009 Americans during the month of October and asked them, "Do you believe correlation implies causation?" An overwhelming 64% of American's answered "YES", while only 38% replied "NO". Another 8% were undecided. This result threatens to shake the foundations of both the scientific and mainstream community. "It is really a mandate from the people." commented one pundit who wished to remain anonymous. "It says that The American People are sick and tired of the scientific mumbo-jumbo that they keep trying to shove down our throats, and want some clear rules about what to believe. Now that correlation implies causation, not only is everything easier to understand, it also shows that even Science must answer to the will of John and Jane Q. Public." Others are excited because this new, important result actually gives insight into why the result occurred in the first place. "If you look at the numbers over the past two decades, you can see that Americans have been placing less and less faith in the old maxim 'Correlation is not Causation' as time progresses." explained pollster and pop media icon Sarah Purcell. "Now, with the results of the latest poll, we are able to determine that people's lack of belief in correlation not being causal has caused correlation to now become causal. It is a real advance in the field of meta-epistemology." This major philosophical advance is, surprisingly, looked on with skepticism amongst the theological community. Rabbi Marvin Pachino feels that the new finding will not affect the plight of theists around the world. "You see, those who hold a deep religious belief have a thing called faith, and with faith all things are possible. We still fervently believe, albeit contrary to strong evidence, that correlation does not imply causation. Our steadfast and determined faith has guided us through thousands of years of trials and tribulations, and so we will weather this storm and survive, as we have survived before." Joining the theologists in their skepticism are the philosophers. "It's really the chicken and the egg problem. Back when we had to worry about causation, we could debate which came first. Now that correlation IS causation, I'm pretty much out of work." philosopher-king Jesse "The Mind" Ventura told reporters. "I've spent the last fifteen years in a heated philosophical debate about epistemics, and then all of the sudden Gallup comes along and says, "Average household consumption of peanut butter is up, people prefer red to blue, and...by the way, CORRELATION IS CAUSATION. Do you know what this means? This means that good looks actually make you smarter! This means that Katie Couric makes the sun come up in the morning! This means that Bill Gates was right and the Y2K bug is Gregory's fault." Ventura was referring to Pope Gregory XIII, the 16th century pontiff who introduced the "Gregorian Calendar" we use today, and who we now know is to blame for the year 2000. The scientific community is deeply divided on this matter. "It sure makes my job a lot easier." confided neuroscientist Thad Polk. "Those who criticize my work always point out that, although highly correlated, cerebral blood flow is not 'thought'. Now that we know correlation IS causal, I can solve that pesky mind-body problem and conclude that thinking is merely the dynamic movement of blood within cerebral tissue. This is going to make getting tenure a piece of cake!"



Anti-correlationist Travis Seymour is more cynical. "What about all the previous correlational results? Do they get grandfathered in? Like, the old stock market/hemline Pearson's rho is about 0.85. Does this mean dress lengths actually dictated the stock market, even though they did it at a time when correlation did not imply causation? And what about negative and marginally significant correlations? These questions must be answered before the scientific community will accept the results of the poll wholeheartedly. More research is definitely needed." Whether one welcomes the news or sheds a tear at the loss of the ages-old maxim that hoped to eternally separate the highly correlated from the causal, one must admit that the new logic is here and it's here to stay. Here to stay, of course, until next October, when Gallup, Inc. plans on administering the poll again. But chances are, once Americans begin seeing the entrepeneurial and market opportunities associated with this major philosophical advance, there will be no returning to the darker age when causal relationships were much more difficult to detect.

http://www.obereed.net/hh/correlation.html


Tab 4: Algebra II: Table of Contents 4-i

Tab 4: Algebra II Table of Contents


Understanding Functions 4-1 Handout 1: Understanding Functions 4-13 Handout 2: Recording Sheet A 4-17 Handout 3: Recording Sheet B 4-18 Handout 4: Recording Sheet C 4-19 Handout 5 TEKS/TAKS Recording Sheet 4-20


Tab 4: Algebra II: Master Materials List 4-ii

Tab 4: Algebra II Master Materials List

Pencil Markers (red, green, and blue) Calculator with graphic and table generating capabilities Understanding Functions: Handouts The following materials are not in the notebook. They can be accessed on the CD through the links below. PowerPoint: Understanding Functions


Understanding Functions 4-1

Activity: Understanding Functions TEKS: This activity covers teacher content knowledge underlying:

§111.32. Algebra I (a) Basic understandings. (5) Tools for algebraic thinking. Techniques for working with functions and

equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numerical, symbolic, graphical, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems.

§111.33. Algebra II (a) Basic understandings. (3) Functions, equations, and their relationships. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations.

Overview: We will gain a deeper understanding of equations and functions and how

they are related. We will take a simple situation, create an equation from the situation, and solve it. We will then take a closer look at the roll that functions play in this process.

Materials: Pencil

Markers (red, green, and blue) Calculator with graphic and table generating capabilities PowerPoint: Understanding Functions Handout 1 (pages 4-13 – 4-16) Handout 2 (page 4-17) Handout 3 (page 4-18) Handout 4 (page 4-19) Handout 5 (pages 4-20 – 4-21)

Grouping: 3-5 Time: 1 ½ to 2 hours



Lesson: Procedures Notes

Have the participants discuss the following two questions.

What is an equation?What is an equation?

What is a root?What is a root?

Have the groups write down their answers to these questions. Then have each group record their answers on the board or large sticky notes.

Have them do a gallery walk of all the responses. They may discuss within their groups, but avoid a whole class discussion at this time.

Leave these answers on the board so at the end of the activity participants can look back and see if their understanding of these questions has changed. Also, don't lead the participants at all in the answering of these questions. Let their understanding grow as the activity progresses.

After they have spent sufficient time answering the above questions, distribute Handout 1, Understanding Functions (pages 4-13 – 4-16). Participants should read the introductory material on functions and discuss their understanding of functions as you go through the power point slides.

Be sure and emphasize the input/output relation in each of the different representations of the function.



Procedures Notes Process definition of function

The process definition treats the function as an input/output machine. This machine accepts inputs and produces well-defined, corresponding outputs. Hence, when working with a function, always be asking yourself: what are the acceptable inputs (the domain) and what are corresponding outputs (the range)? The more abstract Bourbaki and Dirichlet definitions printed in most texts are useful when studying higher mathematics such as Real Analysis, but do not add to the learner’s intuitions about mathematics at this level and should be avoided.

Common representations of functions

Throughout this exercise, you will be asked to study in detail four representations of the function: the tabular, the graphical, the algebraic, and the contextual. It is important to realize that none of these are really the function. They are just the most popular ways that we use to convey the idea of how the function is producing outputs from inputs.

Visualize the process in each representation.

To understand functions, students must focus on the input/output connection in each of the representations. It helps to visualize the input/output arrows of the function machine.



Procedures Notes After the discussion on functions, let everyone continue on with Activity 1.

Make sure that each group has the solution to the equation before you continue, because we do not want the groups wondering what the answer to this question is. We want them to discover what the answer to this equation really means in relationship to the functions that produced it.

After all groups have the solution, distribute Recording Sheets A, B, and C [Handouts 2 (page 4-17), 3 (page 4-18), and 4 (page 4-19)].

Make sure that each group understands how to create tables and graphs of functions on the calculators. The recording sheets are self-explanatory, and the solutions are provided in the folders. You may want to read over the solutions beforehand, but do not lead your participants too much. Let them discover the answers on their own.

After all groups have completed recording sheets A, B, and C, hand out the paper labeled TEKS TAKS Recording Sheet. [Handout 5 (pages 4-20 – 4-21)]

All groups must have precise answers to the bulleted questions. ● What is an equation? ● What is a root? ● What is a zero or x-intercept?

Also, have them look back on the board to their original answers and discuss any changes in their understanding of these terms. (The questions are answered on the last three power point slides.) After they discuss the questions, have them continue on with the recording sheet.



Procedures Notes

What is an equation?What is an equation?

An equation is a An equation is a questionquestion asked asked about a function or functions.about a function or functions.

This conceptualization of equation allows one to center algebra on the idea of functions. Thus the function concept becomes the foundation for understanding the structure of algebra.

What is a root?What is a root?

A root is an input that answers the A root is an input that answers the question asked by an equation.question asked by an equation.

Once one understands that an equation is a question about functions, then one has a better and more precise understanding of the terms used in algebra. Students often misuse the term roots when discussing functions. What are the roots of the function f(x) = x2+3x+2? Rephrase this, ‘What are the solutions of f(x) = x2+3x+2?’ and one can easily see that functions do not have roots/solutions. The phrase “roots of a function” thus must imply “the solution to the equation f(x) = 0” . In such cases, to say the “zero of the function” is better use of language.

What is a zero or xWhat is a zero or x--intercept?intercept?A zero or xA zero or x--intercept is an input that intercept is an input that produces an output of zero.produces an output of zero.

X-intercept refers to the graphical representation. Zeros refers to the numeric representation.



Activity 1: Solutions Equations: a question about functions Consider the following situation: Roger throws a ball of string upward from a downward moving construction elevator. The ball is modeled by the function -16t2+52t+140. The elevator is modeled by the function -10t+132. The units for this problem are in feet and seconds. 1) Write an equation to answer the question: When will the ball be at the same height as the elevator? -16t2+52t+140 = -10t+132 2) Solve this equation algebraically. -16t2+52t+140 = -10t+132 -16t2+52t+8 = -10t -16t2+62t+8 = 0 (t-4)(-16t-2) = 0 Property of zeros: t-4 = 0 so t = 4 and -16t-2 = 0 so t = -1/8 The problem is defined only for time greater than zero, so -1/8 is out of the contextual domain of this problem; hence 4 is the only solution to this problem.



Comparison of functions: f(x) = g(x) Let's unfold this problem by considering it from the point of view of a comparison of functions. First, let us go through the standard algebraic steps used in answering the question: When will the elevator and the ball be at the same height? To find the input time when the two functions output the same height, place the functions equal to each other. Stage 1: -16t2 + 52t + 140 = -10t + 132 f1(t) = g1(t) The next algebraic step is to combine any constants, so subtract 132 from both sides of the equation. Stage 2: -16t2 + 52t +8 = -10t f2(t) = g2(t) The next algebraic step is to combine the first order terms, so add 10t to both sides. Stage 3: -16t2 + 62t +8 = 0 f3(t) = g3(t) The next step would be to factor and use the property of zero to find that the answers to this question, or the roots to the equation, are -1/8 and 4 seconds after the ball was thrown. In this problem, we are assuming that time is positive, so -1/8 is out of the domain of this problem. Hence the only solution (root) to this problem is 4 seconds after the ball was thrown.



Recording Sheet A Solutions

Now let's study the functions that were produced at each stage of the problem and bring in the other representations of the functions to help us analyze what these functions are really telling us. 1) On recording sheet B, use the functions from each stage to complete the tables, and then use the tables to answer the questions. 2) On recording sheet C graph the functions. 3) Use a green marker to circle all the roots/solutions that you find on recording sheet B and recording sheet C. 4) Explain contextually what the root/solution is telling you at each stage.

Stage 1: The number of seconds it took, after the ball was thrown, for the ball and the elevator to be the same height above the ground.

Stage 2: The number of seconds it took, after the ball was thrown, for the

ball and the elevator to be the same height above the elevator's initial position.

Stage 3: The number of seconds it took, after the ball was thrown, for the

ball and the elevator to have no vertical separation. 5) Use a red marker to circle all the zeros of the functions on recording sheet B. Also circle all the x-intercepts of the functions on recording sheet C. 6) How do roots and zeros/x-intercepts differ? Explain. A root is a solution to an equation. A zero/x-intercept is an input that produces an output of zero. An equation (question) does not have a zero; it has a solution/root (answer). A function does not have a solution, but it may have an input value that makes it produce an output value of zero. 7) Take a blue marker and on recording sheet C, draw a line vertically through all the roots. What stays the same and why does it not change? The roots stay the same. This happens because the roots are initially the only input values that produced the same output values for both functions of the equation. Hence as you algebraically manipulate the functions in the equation (doing the same thing on both sides of the equation), the output values of the roots will continue to be the same.



Recording Sheet B Solutions



Recording Sheet C Solutions



TEKS/TAKS Recording Sheet Answers Connecting the language of functions with the TEKS refinement. ● What is an equation? A question asked about a function or functions. ● What is a root? A solution to an equation. (an answer to the question asked by an equation) ● What is a zero or x-intercept? An input that produces an output of zero. Zeros are used more in the language of tables, and x-intercepts are used more in the language of graphs. With your understanding of the language of roots, zeros, and x-intercepts, let's analyze some TEKS and TAKS. 1) Revise this TEKS item so that the language is correct. The TEKS, Algebra I (b) Knowledge and Skills (A.10) B, was changed. Before refinement the TEKS item reads: The student relates the solutions of quadratic equations to the roots of their functions. The student relates the solutions of quadratic equations to the zeros of their functions.



The TEKS item after refinement reads: The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. 2) How does this compare to the revision you wrote? They did a little extra. The only thing that was wrong was that a function does not have a root, but it may have a zero. 3) Revise this TEKS item so that the language is correct. The TEKS, Algebra II (b) Knowledge and Skills (2A.6) C, reads: The student is expected to determine a quadratic function from its roots or a graph. The student is expected to determine a quadratic function from its zeros or a graph. 4) Revise this TAKS question so that the language is correct. 42 Which ordered pair represents one of the roots of the function f(x) = 2x2 + 3x - 20? F (-5/2, 0) G (-4, 0) H (-5, 0) J (-20, 0) 42 Which ordered pair represents one of the x-intercepts of the function f(x) = 2x2 + 3x - 20?


Handout 1-1 Understanding Functions 4-13

Understanding Functions

Objective To gain a deeper understanding of equations and functions and how they are related. Materials Pencil, markers (red, green, and blue), and a calculator with graphic and table generating capabilities. Introduction The intent of this exercise is to increase your understanding of functions. The function is the single most important concept in all of mathematics. It enables us to mathematically model and describe the world around us. Hence, to understand and use mathematics correctly, it is paramount that you understand functions. In this exercise, we use the process definition of the function, which we believe is the best suited definition of function for teaching and understanding algebra. The process definition treats the function as an input/output machine. This machine accepts inputs and produces well defined, corresponding outputs.



Hence, when working with a function, always be asking yourself: what are the acceptable inputs (the domain) and what are the corresponding outputs (the range)? Throughout this exercise, you will be asked to study in detail four representations of the function: the tabular, the graphical, the algebraic, and the contextual.

It is important to realize that none of these is really the function. These representations are just the most popular ways that we use to convey the idea of how functions produce outputs from inputs.



Activity 1 Equation: a question about functions Consider the following situation: Roger throws a ball of string upward from a downward moving construction elevator. The ball is modeled by the function -16t2+52t+140. The elevator is modeled by the function -10t+132. The units for this problem are in feet and seconds. 1) Write an equation to answer the question: When will the ball be at the same height as the elevator? 2) Solve this equation algebraically.



Comparison of functions: f(x) = g(x) Let us unfold this problem by considering it from the point of view of a comparison of functions. First, let us go through the standard algebraic steps used in answering the question: When will the elevator and the ball be at the same height? To find the input time when the two functions output the same height, place the functions equal to each other. Stage 1: -16t2 + 52t + 140 = -10t + 132 f1(t) = g1(t) The next algebraic step is to combine any constants, so subtract 132 from both sides of the equation. Stage 2: -16t2 + 52t +8 = -10t f2(t) = g2(t) The next algebraic step is to combine the first order terms, so add 10t to both sides. Stage 3: -16t2 + 62t +8 = 0 f3(t) = g3(t) The next step would be to factor and use the property of zero to find that the answers to this question, or the roots to the equation, are -1/8 and 4 seconds after the ball was thrown. In this problem, we are assuming that time is positive, so -1/8 is out of the domain of this problem. Hence the only solution (root) to this problem is 4 seconds after the ball was thrown.


Handout 2 Understanding Functions 4-17

Recording Sheet A

Now let us study the functions that were produced at each stage of the problem and bring in the other representations of the functions to help us analyze what these functions are really telling us. 1) On recording sheet B, use the functions from each stage to complete the tables, and then use the tables to answer the questions. 2) On recording sheet C graph the functions. 3) Use a green marker to circle all the roots/solutions that you find on recording sheet B and recording sheet C. 4) Explain contextually what the root/solution is telling you at each stage. Stage 1: Stage 2: Stage 3: 5) Use a red marker to circle all the zeros of the functions on recording sheet B; also circle all the x-intercepts of the functions on recording sheet C. 6) How do roots and zeros/x-intercepts differ? Explain. 7) Take a blue marker and on recording sheet C, draw a line vertically through all the roots. What stays the same and why does it not change?



Recording Sheet B



Recording Sheet C Stage 1 Stage 2 Stage 3



TEKS/TAKS Recording Sheet

Connecting the language of functions with the TEKS refinement. ● What is an equation? ● What is a root? ● What is a zero or x-intercept? With your understanding of the language of roots, zeros, and x-intercepts, let's analyze some TEKS and TAKS. 1) Revise this TEKS item so that the language is correct. The TEKS, Algebra I (b) Knowledge and Skills (A.10) B, was changed. Before refinement, the TEKS item reads: The student relates the solutions of quadratic equations to the roots of their functions.



The TEKS item after refinement reads: The student is expected to make connections among the solutions (roots) of quadratic equations, the zeros of their related functions, and the horizontal intercepts (x-intercepts) of the graph of the function. 2) How does this compare to the revision you wrote? 3) Revise this TEKS item so that the language is correct. The TEKS, Algebra II (b) Knowledge and Skills (2A.6) C, reads: The student is expected to determine a quadratic function from its roots or a graph. 4) Revise this TAKS question so that the language is correct.

42 Which ordered pair represents one of the roots of the function f(x) = 2x2 + 3x - 20? F (-5/2, 0) G (-4, 0) H (-5, 0) J (-20, 0)


Tab 5: Geometry: Table of Contents 5-i

Tab 5: Geometry Table of Contents


Picture This! 5-1 Handout 1: Part I Diagrams 5-4 Handout 2: Drawings in Two-Dimensions 5-6 Handout 3: Isometric Drawings 5-9 Transparency 1 5-11 Transparency 2 5-12 Transparency 3 5-13 Transparency 4 5-14 Transparency 5 5-15 Transparency 6 5-16 Transparency 7 5-17 Transparency 8 5-18

Texas “T” Activity 5-19 Handout 1: Part I 5-22 Handout 2: Part I 5-23 Transparency 1: Part II Activity A 5-24 Transparency 2: Part II Activity A 5-25 Transparency 3: Part II Activity B 5-26 Transparency 4: Part II Activity B 5-27 Transparency 5: Part II Activity C 5-28 Transparency 6: Part II Activity C 5-29 Handout 3: Part III 5-30


Tab 5: Geometry: Master Materials List 5-ii

Tab 5: Geometry Master Materials List

Three-dimensional cube model Grid paper Snap cubes Rulers Calculator Markers Picture This!: Transparencies and handouts Texas “T” Activity: Transparencies and handouts The following materials are not in the notebook. They can be accessed on the CD through the links below. Isometric dot paper (1 per participant) and transparencies (1 per group)


Picture This! 5-1

Activity: Picture This! TEKS: This activity supports teacher content knowledge needed for: (G.6) Dimensionality and the geometry of location. The student

analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

The student is expected to: (C) use orthographic and isometric views of three-dimensional

geometric figures to represent and construct three-dimensional geometric figures and solve problems.

Overview: This activity encourages participants to explore and draw orthographic

and isometric views. It explores some non-technical aspects of orthographic drawings and the relationship between isometric and orthographic drawings.

Materials: Isometric dot paper

Grid paper Snap cubes Rulers Transparencies: 1-8 (pages 5-11 – 5-18) Isometric dot paper (1 per participant) and transparencies (1 per group) Three-dimensional cube model Handout 1 (pages 5-4 – 5-5) Handout 2 (pages 5-6 – 5-8) Handout 3 (pages 5-9 – 5-10)

Vocabulary: axonometric

isometric diametric trimetric orthographic perspective vanishing point

Grouping: Each participant should complete these activities. Participants may

work in groups of 3-4 so that they can help one another. Time: 1 to 1 ½ hours


Picture This! 5-2

Lesson: Procedures Notes

Part I: 1. Distribute Handout 1, Part 1 Diagrams,

(pages 5-4 – 5-5) to participants and at least one model of a cube to each group. Have participants determine which of the diagrams could represent a cube.

Participants should be in groups for discussion; allow time for discovery that all of these diagrams represent a cube. Ask the participants why the diagrams of the cube in Part I look so different. The diagrams show the cube from different angles and perspectives.

2. Provide participants with Handout 2, Drawings in Two-Dimensions, (pages 5-6 – 5-8) terms with definitions. Discuss and define perspective views, vanishing point, axonometric views and orthographic views.

Have participants decide in their groups which type of drawing each diagram from Handout 1 represents. Groups should be prepared to share their choices with the whole group.

(See discussion of vocabulary below) Transparencies 1-6 (pages 5-11 – 5-16) for each diagram are provided.

Part II: 3. Distribute two pre-made snap cube

examples, a transparency of isometric dot paper (see Materials List for a link to this), and transparency pens to each group; also provide a sheet of isometric dot paper for each participant.

Have each participant sketch the isometric view of the given example, and then have the group sketch the view onto their transparency. Each group will share its result.

Be sure to bring the participants back together for a whole class discussion. The trainer can make two of the given sample shapes, or make two simple shapes of his/her own. Optional: To save prep time, have the participants build the models shown on Transparency 7 (page 5-17).

4. Give participants a sheet of grid paper and have them sketch the three orthographic views (front, side and top).

Give each group a new set of pre-made (more challenging) examples and repeat the process.

Volunteers may share their sketches with the class. Trainer can choose any of the remaining sample shapes, or create their own.


Picture This! 5-3

Procedures Notes Part III: 6. Distribute Handout 3, Isometric

Drawings, (pages 5-9 – 5-10). Have participants practice creating isometric drawings from orthographic views without the aid of three-dimensional models.

Bring the participants back together for a whole class discussion and to analyze their drawings. Questions to ask: “What difference do you see between

the orthographic drawings of diagram 1 and diagram 2?”

“How can the orthographic drawings be changed to include additional information differentiating diagram 1 and diagram 2?”

On Handout 3, what is the difference between the orthographic drawings of diagram 1 and diagram 2? (see page 5-10) They are the same! There are easy solutions to this dilemma. Try adding segments in the drawing to outline the cube faces seen in each view. Indicate different distances from the viewer to different parts of the orthogonal drawings by using different line weights. Faces nearest the viewer can be outlined with dark segments. Faces one cube farther away can be a medium weight and faces two cubes farther away can be a light or dashed segment. Alternatively, add segments in the top view to outline the cube faces and number the squares to indicate the number of cubes in the “stack.” (See Transparency 8 (page 5-18) for solutions) These drawings are called “mats.” Show a Transparency 8 after discussion.

Extensions: After students have learned to draw orthographic and isometric figures,

they can investigate surface area and volume. Initially, they can use snap cube models, but as visualization skills improve, there should be less dependence on models and more interpretation from the diagrams.


Handout 1-1 Picture This! 5-4

PICTURE THIS! Part 1 Diagrams

Which of these two-dimensional pictures could be used to represent a cube?



PICTURE THIS! Part 1 Diagram Solutions

Which of these two-dimensional pictures could be used to represent a cube?

Orthographic

Orthographic

Dimetric Perspective

Dimetric

Isometric

156o



PICTURE THIS! Drawings in Two-Dimensions

Use the definitions below to determine which of these descriptions best fits each of the sketches given in Part I. Perspective View: The technique of portraying solid objects and spatial relationships on

a flat surface. There are numerous methods of depicting an object, depending upon the purpose of the drawing.

Vanishing Point: In an artistic perspective drawing, receding parallel lines (lines that run

away from the viewer) converge at a vanishing point on the horizon line. This maintains a realistic appearance of the object depicted, even as the vantage point changes.

Orthographic Drawing: Ortho means “straight” and the views in orthographic drawings show the faces of a solid as though you are looking at them “head-on” from the top, front or side. Orthographic drawings are used in engineering drawings to convey all the necessary information of how to make the part to the manufacturing department. The line of sight is perpendicular to the surfaces of the object. Two conventions are used in technical drawings. These are first angle and third angle, which differ only in position of the plan (top, front, and side views). They are derived from the method of projection used to transfer two-dimensional views onto an imaginary, transparent box surrounding the object being drawn. This is not a distinction the state of Texas has seen fit to recognize in the TEKS.

appears as

right sidefronttop



A common use of a “top” orthographic view is the floor plan of a house or other building.

Axonometric drawing: A two-dimensional drawing showing three dimensions of an

object. The vantage point is not perpendicular to a surface of the object being drawn. Axonometric means, “to measure along the axes.” There are three types: isometric, dimetric, and trimetric.

Isometric Drawing: A two-dimensional drawing that shows three sides of an object in one view. The vantage point is 45° to the side and above the object being viewed. The resulting angle between any two axes appears to be 120° . Isometric means, “one measure.” In this view, 90° angles appear to be 120° or 60° .

Courtesy Electronic Arts, Inc.

Video games, such as SimCity 2000, frequently use isometric drawings

120°

120° 120°



Dimetric drawing: A two-dimensional drawing similar to an isometric drawing, but with only two of the resulting angles between the axes having the same apparent measure. Dimetric means, “two measures.”

Trimetric drawing: Again, A two-dimensional drawing similar to an isometric drawing, but with none of the resulting angles between the axes having the same apparent measure. Trimetric means, “three measures.”

X Y

Z

100°

130° 130°



PICTURE THIS! Isometric Drawings

Make orthographic drawings of each diagram. You should include drawings from the top, front, and right side.

Diagram 2Diagram 1



PICTURE THIS! Isometric Drawings Solutions

top front right sideright sidefronttop

The orthogonal drawings are the same! There are easy solutions to this dilemma. Try adding segments in the drawing to outline the cube faces seen in each view. Indicate different distances from the viewer to different parts of the orthogonal drawings by using different line weights. Faces nearest the viewer can be outlined with dark segments. Faces one cube farther away can be a medium weight and faces two cubes farther away can be a light or dashed segment. Alternatively, add segments in the top view to outline the cube faces and number the squares to indicate the number of cubes in the “stack.”


Transparency 1 Picture This! 5-11



PICTURE THIS! Materials: snap cube shapes, grid paper, isometric paper, and rulers Prepare orthographic and isometric drawings of each shape. Sample shapes:



PICTURE THIS! Isometric Drawings

top front right sideright sidefronttop

Mats


Texas “T” Activity 5-19

Activity: Texas “T” Activity TEKS: This activity supports teacher content knowledge needed for: (G.2) Geometric Structure. The student analyzes geometric

relationships in order to make and verify conjectures. The student is expected to: (A) use constructions to explore attributes of geometric figures

and to make conjectures about geometric relationships. (G.3) Geometric Structure. The student applies logical reasoning to

justify and prove mathematical statements. The student is expected to: (D) use inductive reasoning to formulate a conjecture.

(G.5) Geometric Patterns. The student uses a variety of

representations to describe geometric relationships and solve problems.

The student is expected to: (A) use numeric and geometric patterns to develop algebraic

expressions representing geometric properties. (B) use numeric and geometric patterns to make generalizations

about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.

(G.6) Dimensionality and the geometry of location. The student

analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses these representations to solve problems.

The student is expected to: (B) use nets to represent and construct three-dimensional

geometric figures.


The student is expected to: (A) use and extend similarity properties and transformations to

explore and justify conjectures about geometric figures. (B) use ratios to solve problems involving similar figures. (D) describe the effect on perimeter, area, and volume when one

or more dimensions of a figure are changed and apply this idea in solving problems.



Overview: This activity encourages participants to explore patterns in height, surface area, and volume of similar figures when one or more dimensions are changed.

Materials: Snap cubes

Calculator Transparencies 1-6 (pages 5-24 – 5-29) Markers Handout 1 (page 5-22) Handout 2 (page 5-23) Handout 3 (pages 5-30 – 5-34)

Grouping: Groups of 3-4 Time: 1 ½ hours Lesson:

Procedures Notes Part I: 1. Distribute Handout 1, Part I, (page 5-22) of the

activity to each participant. Distribute approximately 250 snap cubes and a calculator to each group. Each participant builds his/her own “T” #1 and completes handout 1 of Part I of the activity. As a group, participants build additional “Ts” and complete Handout 2 of Part I (page 5-23).

This activity will help participants to discover the relationship between linear ratios, area ratios, and volume ratios when similar three-dimensional objects are built. Participants should work alone to build the first “T” and answer the questions, then discuss their answers in their groups. Participants should then discuss their answers to page 1 before beginning to work as a group to complete page 2.

2. Bring the class together for a whole group discussion of the findings for Part I. Questions to ask: “Did anyone build all 4 ‘Ts’? If not, why not?” “Were you able to complete the table without

building all the models?” “Did you notice any patterns as you completed

the table?” “How did you describe the patterns in terms of

n?”



Procedures Notes Part II: 3. Using a jigsaw cooperative learning procedure,

assign each group Activity A, Activity B, or Activity C [Transparencies 1 & 2 (pages 5-24 – 5-25), Transparencies 3 & 4 (pages 5-26 – 5-27), or Transparencies 5 & 6 (pages 5-28 – 5-29)] and provide each group with the appropriate transparencies for its assigned activity. Each group should select a reporter to complete the transparency for the assigned activity, and be prepared to present to the entire group.

Select one group per activity to actually present its results. Ask for discussion from other groups who completed the same activity.

Part III: 4. Distribute Handout 3, Part III, (pages 5-30 – 5-

34), “One and Two-Dimensional Change,” to each participant. Have participants work in their groups to complete Part III. Then bring the class back together to report their findings and discuss them as a whole group. Questions to ask:

“How did what you discovered in Activities A, B, and C relate to what you did in Part III?”

“When is the change in dimension(s) linear? Quadratic?”

Assign each group a piece of Part III to report out. Select groups that did not report out for the previous parts in step 3.

5. Lead participants to begin a discussion about the “Think About It” questions in Part III.

After the group determines that a change in the radius of a can would not be a linear change in volume, ask, “What would happen if we made a one-dimensional change in a sphere?”


Handout 1 Texas “T” Activity 5-22

TEXAS “T”: Part I Using snap cubes with the edge of one cube representing 1 unit of length, build the first “T” like the one shown in the illustration. The “T” is 4 units tall and 3 units wide. Use the “T” to complete the table below:

What do you think will happen to the height of the “T” when the dimensions of each part

are doubled? ___________________ tripled?________________________

What do you think will happen to the surface area of the “T” when the dimensions of

each part are doubled? _________________ tripled?________________________

What do you think will happen to the volume of the “T” when the dimensions of each

part are doubled? ___________________ tripled?________________________

N = “T” NUMBER HEIGHT OF “T” TOTAL SURFACE

AREA VOLUME

“T” #1


Handout 2 Texas “T” Activity 5-23

TEXAS “T”: Part I In your group, build at least the next two “Ts”, where n represents the height of the “T.” To build each “T”, take the dimensions of “T” #1 and multiply by n (n=2, n=3, n=4).

N = “T” NUMBER HEIGHT OF “T” TOTAL SURFACE

AREA VOLUME

“T” #2

“T” #3

“T” #4

“T” N


Transparency 1 Texas “T” Activity 5-24

TEXAS “T”: Part II Activity A: Comparing Height in Similar Figures

Using the completed tables from Part I, fill in the ratios for the following table. Record fractions in lowest terms. Use the table data to answer the questions below.

RATIO OF FIGURES RATIO OF HEIGHT “T” #1:”T” #2

“T” #2:”T” #3

“T” #1:”T” #3

“T” #3:”T” #4

“T” #1:”T” #4

“T” #2:”T” #5

1. Was your prediction in Part I about the height of the “Ts” accurate? Explain your

answer.

2. What pattern do you observe about the simplified ratios of the heights?

3. Using what you observed about the patterns, can you determine the ratio of the height of “T” #20 to the height of “T” #12?

4. If the heights of the “Ts” were in the ratio of 100/40, which “Ts” would you be comparing? Explain your answer.

5. Determine which “T” has a height of 60 units. Explain your answer.



6. What is the ratio of the height of “T” #48 to “T” #16?

7. If you know the height of “T” #1, how can you find the height of “T” #N? Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heights

would be _______________.

Write a rule based on your results: If the ratio of the heights in two similar figures is m/n, the ratio of their heights would be

______________.



TEXAS “T”: Part II Activity B: Comparing Height and Surface Area in Similar Figures

Using the completed table from Part I, fill in the ratios for the following table. Record fractions in lowest terms. Use the table data to answer the questions below.

RATIO OF FIGURES RATIO OF HEIGHT RATIO OF SURFACE AREA “T” #1:”T” #2

“T” #2:”T” #3

“T” #1:”T” #3

“T” #3:”T” #4

“T” #1:”T” #4

“T” #2:”T” #5

1. Was your prediction in Part I about the surface areas of the “Ts” accurate? Explain your answer.

2. What pattern do you observe about the simplified ratios of the heights and surface areas?

3. Using what you observed about the patterns, can you determine the ratio of the

surface area of “T” #20 to the surface area of “T” #12?

4. If the surface area of the “Ts” were in the ratio of 256/49, which “Ts” would you be comparing? Explain your answer.

5. Determine which “T” has a surface area of 3,744 square units. Explain your answer.



6. What is the ratio of the surface area of “T” #50 to “T” #15?

7. If you know the surface area of “T” #1, how can you find the surface area of “T” #N?

Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heights

would be _______________ and the ratio of the surface area would be

______________.

Write a rule based on your results: If the ratio of the heights in two similar figures is m/n, then the ratio of their surface area

would be _________________.



TEXAS “T”: Part II Activity C: Comparing Total Surface Area and Volume in Similar Figures

Using the completed table from Part I, fill in the ratios for the following table. Record fractions in lowest terms. Use the table data to answer the questions below.

RATIO OF FIGURES RATIO OF HEIGHT

RATIO OF SURFACE AREA

RATIO OF VOLUME

“T” #1:”T” #2

“T” #2:”T” #3

“T” #1:”T” #3

“T” #3:”T” #4

“T” #1:”T” #4

“T” #2:”T” #5

1. Was your prediction in Part I about the volumes of the “Ts” accurate? Explain your answer.

2. What pattern do you observe about the simplified ratios of the heights, surface

areas and volumes? 3. Using what you observed about the patterns, can you determine the ratio of the

volume of “T” #20 to the height of “T” #12?

4. If the volume of the “Ts” were in the ratio of 1000/512, which “Ts” would you be comparing? Explain your answer.



5. Determine which “T” has a volume of 34,992 cubic units? Explain your answer.

6. What is the ratio of the volume of “T” #50 to “T” #15?

7. If you know the volume of “T” #1, how can you find the volume of “T” #N?

8. What happens to the volume if all the dimensions of the “T” are reduced by one-half? Explain your answer.

Based upon your results, if the ratio of the heights of two similar figures is 4/5, the ratio

of the surface area would be ________________, and the ratio of the volume would be

___________________.

Write a rule based on your results:

If the ratio of the heights in two similar figures is m/n, the ratio of their surface area would be _______________ and the ratio of their volume would be ______________.


Handout 3-1 Texas “T” Activity 5-30

TEXAS “T”: Part III One and Two-Dimensional Change

Thus far we have examined the effect on surface area and volume of a figure if all dimensions are changed by the same factor. What do you suppose will happen to surface area and volume if only one dimension is changed? For this exercise, let’s use a simpler figure. l = length w = width h = height Volume of a rectangular prism: V = lwh Surface area of a rectangular prism: SA = 2(lw) + 2(wh) + 2(hl) Volume: A. Change in one dimension:

Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length and width, change the height by a factor of n and

complete the following table.

n Volume

1 4 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data.



3. Write in words the pattern you observe in the volumes.

4. Write an algebraic function for the pattern.

5. Without changing the length and height, change the width by a factor of n and complete the following table.

n Volume

1 4 2 3 4 5 6

n



8. Why is the pattern the same as #3?



B. Change in two dimensions Given a rectangular prism with a length of 2, width of 2 and a height of 1; 1. Without changing the length, change the height and width by a factor of n and


n Volume 1 4 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L3 and plot the data. 3. Write in words the pattern you observe in the volumes.


5. Write a function to predict the volume of our figure if all three dimension change

by factor of n.



Surface Area: A. Change in one dimension:


1. Without changing the length and width, change the height by a factor of n and


n Surface Area

1 2 3 4 5 6

n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data. 3. Write in words the pattern you observe in the surface area.

___________________________________________________________________

___________________________________________________________________

4. Write an algebraic function for the pattern._______________________________



B. Change in two dimensions: Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length, change the height and width by a factor of n and


n Surface Area

1 2 3 4 5 6

n



___________________________________________________________________

___________________________________________________________________

4. Write an algebraic function for the pattern._______________________________

Think about it: Does a one dimensional change in any 3-dimensional figure create a linear change in volume? What if the figure were a can?



TEXAS “T”: Part I SOLUTIONS

N = “T” NUMBER HEIGHT OF “T”

TOTAL SURFACE

AREA VOLUME

“T” #1

4

26

6

“T” #2

8

104 (26*22)

48 (6*23)

“T” #3

12

234 (26*32)

162 (6*33)

“T” #4

16

416 (26*42)

384 (6*43)

“T” N

4N

26N2

6N3



Part II Activity A: Comparing Height in Similar Figures

RATIO OF FIGURES RATIO OF HEIGHT “T” #1:”T” #2

4/8 = 1/2

“T” #2:”T” #3

8/12 = 2/3

“T” #1:”T” #3

4/12 = 1/3

“T” #3:”T” #4

12/16 = 3/4

“T” #1:”T” #4

4/16 = 1/4

“T” #2:”T” #5

8/20=2/5

1. Was your prediction about the height of the “Ts” accurate? Explain your

answer. Participants should predict that the height doubles when the dimensions are doubled and triples when the dimensions are tripled.

2. What pattern do you observe about the simplified ratios of the heights? The simplified ratio is the same as the ratio of the “Ts”. For example, “T” 2:”T”3 = 8/12 = 2/3

3. Using what you observed about the patterns, can you determine the ratio of

the height of “T” 20 to the height of “T” 12? The ratio is 20/12.

4. If the heights of the “Ts” were in the ratio of 100/40, which “Ts” would you be comparing? Explain your answer. You would be comparing “T” 25 to “T” 10. 100/4 = 25 and 40/4 = 10.

5. Determine which “T” has a height of 60 units. Explain your answer. “T” 15

has a height of 60 since 60 divided by 4 equals 15.

6. What is the ratio of the perimeters of “T” 48 to “T” 16? The ratio is 48/16 which simplifies to 3/1.



7. If you know the height of “T” 1, how can you find the height of “T” N? N times the height of “T” 1

Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heights would be ____4/5___________. Write a rule based on your results: If the ratio of two similar figures is m/n, the ratio of their heights would be ____m/n__________.



Part III Activity B: Comparing Height and Surface Area in Similar Figures

1. Was your prediction about the surface areas of the “Ts” accurate? Explain

your answer. The participants should predict that the surface area will increase by the original surface area times the square of the dimensional change (height).

2. What pattern do you observe about the simplified ratios of the heights and

surface areas? The ratio of the surface area is equal to the ratio of the heights squared.

3. Using what you observed about the patterns, can you determine the ratio of

the surface area of “T” 20 to the surface area of “T” 12? 400/144 = 25/9

4. If the surface area of the “Ts” were in the unsimplified ratio of 256/49, which “Ts” would you be comparing? Explain your answer. You would be comparing “T” 16 to “T” 7. You take the square root of 256 and 49 to determine the answer.

5. Determine which “T” has a surface area of 3,744 square units. Explain your

answer. 3,744 = 26 N2 Divide by 26 and take the square root. “T” 12 has a surface are of 3,744.

6. What is the ratio of the surface area of “T” 50 to “T” 15? 2500/225 = 100/9

RATIO OF FIGURES RATIO OF HEIGHT


“T” 1:”T” 2

4/8 = 1/2

26/104= 1/4

“T” 2:”T” 3

8/12 = 2/3

104/234 = 4/9

“T” 1:”T” 3

4/12 = 1/3

26/234 = 1/9

“T” 3:”T” 4

12/16 = 3/4

234/416 = 9/16

“T” 1:”T” 4

4/16 = 1/4

26/416 = 1/16

“T” 2:”T” 5

8/20 = 2/5

104/650 = 4/25



7. If you know the surface area of “T” 1, how can you find the surface area of

“T” N? 26N2 Based upon your results, if the ratio of two similar figures is 4/5, the ratio of the heights would be ____4/5___ and the ratio of the surface area would be ___16/25___________. Write a rule based on your results: If the ratio of two similar figures is m/n, the ratio of their heights would be ___m/n____ and the ratio of their surface area would be __m2/n2____.



Part II Activity C: Comparing Total Surface Area and Volume of Similar Figures

1. Was your prediction about the volumes of the “Ts” accurate? Explain your answer. The participants should predict that the volume will increase by the original volume times the cube of the dimensional change (height).

2. What pattern do you observe about the simplified ratios of the heights,

surface areas and volumes? The ratio of the surface area is equal to the ratio of the heights squared. The ratio of the volume is equal to the ratio of the heights cubed.

3. Using what you observed about the patterns, can you determine the ratio

of the volume of “T” 20 to the volume of “T” 12? 8000/1728 = 125/27

4. If the volume of the “Ts” were in the unsimplified ratio of 1000/512, which “Ts” would you be comparing? Explain your answer. You would be comparing “T” 10 to “T” 8. You take the cube root of 1000 and 512 to determine the answer.

5. Determine which “T” has a volume of 34,992 cubic units. Explain your

answer. 34,992 = 6N 3 Divide by 6 and take the cube root. “T” 18 has a volume of 34,992.

RATIO OF FIGURES

RATIO OF HEIGHT


RATIO OF VOLUME

“T” 1:”T” 2

4/8 = 1/2

26/104 = 1/4

6/48 = 1/8

“T” 2:”T” 3

8/12 = 2/3

104/234 = 4/9

48/162 = 8/27

“T” 1:”T” 3

4/12 = 1/3

26/234 = 1/9

6/162 = 1/27

“T” 3:”T” 4

12/16 = 3/4

234/416 = 9/16

162/384 = 27/64

“T” 1:”T” 4

4/16 = 1/4

26/416= 1/16

6/384 = 1/64

“T” 2:”T” 5

8/20 = 2/5

104/650 = 4/25

48/750 = 8/125



6. What is the ratio of the volume of “T” 50 to “T” 15? 125,000/3375 = 1000/27 7. If you know the volume of “T” 1, how can you find the volume of “T” N?

6N3

8. What happens to the volume if all the dimensions of the “T” are reduced by one-half? Explain your answer. The volume will reduce to 1/8 of the original volume. (1/2)3 = 1/8

Based upon your results, if the ratio of the heights of two similar figures is 4/5, the ratio of the surface area would be ___16/25___, and the ratio of the volume would be ___64/125______. Write a rule based on your results: If the ratio of the heights in two similar figures is m/n, the ratio of their surface area would be __m2/n2___ and the ratio of their volume would be ___m3/n3___.



TEXAS “T” Part III

One and Two-Dimensional Change Thus far we have examined the effect on surface area and volume of a figure if all dimensions are changed by the same factor. What do you suppose will happen to surface area and volume if only one dimension is changed? For this exercise, let us use a simpler figure. l = length w = width h = height Volume of a rectangular prism: V = lwh Surface area of a rectangular prism: SA = 2(lw) + 2(wh) + 2(hl) Volume: A. Change in one dimension:

Given a rectangular prism with a length of 2, width of 2 and a height of 1; 1. Without changing the length and width, change the height by a factor of n

and complete the following table.

n Volume

1 4 2 8 3 12 4 16 5 20 6 24

n

4n




3. Write in words the pattern you observe in the volumes. Each increase of n causes the volume to increase by a factor of 4.

4. Write an algebraic function for the pattern. V = 4n 5. Without changing the length and height, change the width by a factor of n and complete the following table.

n Volume

1 4 2 8 3 12 4 16 5 20 6 24

n

4n

6. Write in words the pattern you observe in the volumes. Each increase of n causes the volume to increase by a factor of 4.

7. Write an algebraic function for the pattern. V = 4n

8. Why is the pattern the same as #3? Because the change is only in one dimension, the effect on volume is linear. It does

not matter which dimension is changed.



B. Change in two dimensions Given a rectangular prism with a length of 2, width of 2 and a height of 1; 1. Without changing the length, change the height and width by a factor of n and complete the following table.

n Volume 1 4 2 16 3 36 4 64 5 100 6 144

n

4n2


3. Write in words the pattern you observe in the volumes. Volume increases by 4n2

4. Write an algebraic function for the pattern. V = 4n2 5. Write a function to predict the volume of our figure if all three dimensions change by a factor of n: V=4n3



Surface Area: A. Change in one dimension:


1. Without changing the length and width, change the height by a factor of n and complete the following table.

n Surface Area

1 16 2 24 3 32 4 40 5 48 6 54

n

8n+8

2. Enter the data into lists in your calculator as L1 and L2 and plot the data. 3. Write in words the pattern you observe in the surface area. The change is linear, increasing by eight each time the factor increases. Given that

the surface area equals 2(lw)+2(wh)+2(hl), the only parenthesis that does not

change when the height is increased by a given factor is 2(lw). This is your

constant. The sum of the remaining parenthesis [{(wh)+(hl)] is multiplied times the

factor. The surface area for a 2x2x1 rectangular prism where only the height is

changed by a given factor can be found by the equation: n[2((wh)+(hl))] + 2(lw),

where n is the given factor. For problems 1-4, the equation is SA =

n[2((2x1)+(1x2))]+2(2x2) = 8n + 8.

4. Write an algebraic function for the pattern. SA = 8n+8



B. Change in two dimensions: Given a rectangular prism with a length of 2, width of 2 and a height of 1;

1. Without changing the length, change the height and width by a factor of n and complete the following table.

n Surface Area

1 16 2 40 3 72 4 112 5 160 6 216

n

4n2 +12n

2. Enter the data into lists in your calculator as L1 and L2 and plot the data. 3. Write in words the pattern you observe in the surface areas. The change is quadratic. The parenthesis where both dimensions change is

multiplied times the square of the given factor. The sum of the remaining

parenthesis is multiplied times the given factor. The surface area for a 2x2x1

rectangular prism where the width and height are changed by the same given factor

can be found by the equation: n2 (2(wh))+n[2((lw+hl))], where n is the given factor.

For problems 1-4, the equation is SA = n2(2(2x1)) + n[2((2x2) + (1x2))] = 4n2+12n.

4. Write an algebraic function for the pattern. SA = 4n2 + 12n Think about it: Does a one dimensional change in any 3-dimensional figure create a linear change in volume? What if the figure was a can? No. For example, a change in the radius of a cylinder produces a quadratic change in volume.


Tab 6: Assessment: Table of Contents 6-i

Tab 6: Assessment Table of Contents


Opening Activity – Assessment Pyramid 6-1

What’s Your Problem? 6-10 Handout 1-Sample Assessment Items-Roger Throws a Ball 6-25 Handout 2-Three Problem Types Labels 6-27 Handout 3-What’s Your Problem? Items 6-30 Handout 4-Three Problem Types – How to Write 6-58

The Power of Creating 6-59 Handout 1-Sample Class Responses 6-63 Transparency 1-Slope of 4 6-64 Transparency 2-Y Intercept of 4 6-65

Closing Activity –Assessment Should Drive Instruction 6-66

Resources 6-72


Tab 6: Assessment: Master Materials List 6-ii

Tab 6: Assessment Master Materials List

Blank paper Chart paper or white board space Current textbook(s) or local curriculum (optional) Copies of all the PowerPoints with space for note taking What’s Your Problem? Handouts Generation Power Handout and Transparencies The following materials are not in the notebook. They can be accessed on the CD through the links below.

PowerPoint: Opener

PowerPoint: What’s Your Problem? PowerPoint: The Power of Creating PowerPoint: Closer



Activity: Opening Activity – Assessment Pyramid Overview: Assessment happens in many forms of classroom interaction:

questioning, homework, quizzes, tests, projects, classroom tasks. All of these forms of assessment have their own important place in the mathematics classroom. They all have in common the idea of ascertaining what students know and what they do not know. These assessments can be thought of in three dimensions: level of reasoning required, level of difficulty, and degree of skill or conceptual understanding required. The Assessment pyramid can be helpful in providing language and a perspective when looking at specific items, at a complete test, and at an entire course. Over time, all classroom assessments should generally fill the pyramid. The pyramid is not a rectangular prism, suggesting equal amounts of low and high level questions because it takes fewer high level reasoning questions to assess mathematical understanding. It takes more low level reasoning questions to assess mathematical understanding.

Participants will examine and discuss the Assessment Pyramid. They will

consider several assessment items and their approximate positions on the pyramid. Participants will reflect on their own assessment practices and consider several guiding questions, including how to change their existing questions. This activity should prime them for the next section – how to change questions.

Materials: PowerPoint: Opener

Copies of the PowerPoint with space for note taking

Grouping: Tables of 4 Time: 30 minutes Lesson: Distribute copies of the PowerPoint to help participants focus on important

ideas in the presentation as they take notes. Show the PowerPoint presentation Opener. Use the following notes pages for elaboration.

Procedures Notes Slide

1

Assessment

Mathematics TEKSRefinement Project




2 AssessmentIntroduce the Assessment Pyramid (adapted from de Lange’s Assessment Pyramid.) Note that the pyramid is a graphic that helps identify qualities or properties or dimensions of assessment items. Point out the two primary sections:

• concepts • mathematical skills

Level of difficulty runs from front (easy) to back (difficult). Levels of reasoning go from bottom (reproduction) to top (higher level). The line separating concepts and skills gets thinner as you move up the pyramid, suggesting that the lines blur between concepts and skills as you raise the level of reasoning required to solve problems. The next few slides give the opportunity to discuss the levels of reasoning in more depth. Be very clear that participants should not get bogged down with the pyramid. It can be useful and helpful but it should not cause arguments through the rest of the training about just how high or deep an item should be placed, etc. Its purpose is to get the conversation going and provide some vocabulary and perspective.

Slide 3 Assessment

Lower level - reproduction,procedures, concepts,definitions

Throughout the discussion, be sure to juxtapose the ideas of low level to high level reasoning versus easy to difficult. Many teachers talk past each other by using the same words to discuss different qualities of assessment items. Some teachers say, “I do not understand why my students are not successful. I ask them really hard questions. I do not know how to better prepare them for the hard/difficult questions they see on TAKS or other tests.”



Procedures Notes But is there a difference between hard or difficult questions and questions that require a higher level of reasoning? Now look at each level of reasoning separately. Lower level reasoning items “deal with knowing facts, representing, recognizing equivalents, recalling mathematical objects and properties, performing routine procedures, applying standard algorithms, and developing technical skills, as well as dealing and operating with statements and expressions that contain symbols and formulas in ‘standard’ form. Test items at this level are often similar to those on standardized tests and on chapter tests related to conventional curricula. These are familiar tasks for teachers and tend to be the types of tasks they are able to create.” (Romberg, 17)

Slide 4 Assessment

Middle level - connectionsand integration for problemsolving

“At this level, students start making connections within and between the different domains of mathematics, integrate information in order to solve simple problems, have a choice of strategies, and have a choice in the mathematical tools. At this level, students also are expected to handle representations according to situations and purpose and need to be able to distinguish and relate a variety of statements (eg., definitions, claims, examples, conditioned assertion, proofs). Items at this level often are placed within a context and engage students in mathematical decision making. These tasks tend to be open and similar to instructional activities.” (Romberg, 18) Note that at this level, the lines blur between concepts and skills. It is more about connections. Also, it takes fewer middle level reasoning items to assess mathematical achievement. A student’s answers to items higher up the pyramid



Procedures Notes give a more complete picture of that student’s understanding and skill. Therefore teachers need less higher level reasoning questions to ascertain a student’s mathematical achievement. Also, those higher level reasoning items generally take more time and involve more work.

Slide 5 Assessment

Higher level -mathematization,mathematical thinking,generalization, insight

“At this level, students are asked to mathematize situations: recognize and extract the mathematics embedded in the situation and use mathematics to solve the problem; analyze; interpret; develop models and strategies; and make mathematical arguments, proofs, and generalizations. Items at this level involve extended-response questions with multiple answers.” (Romberg, 18) Over time, a complete assessment program should “fill” the pyramid.

Slide 6 Consider the following:

A rectangular prism is 2cm x 4cm by 6cm. One dimension isenlarged by a scale factor of 3. What is the volume of the enlargedfigure?A rectangular prism is 2.7cm x 0.45cm by 609.01cm. Onedimension is enlarged by a scale factor of 3.5. What is the volumeof the enlarged figure?When a figure is dilated by a scale factor k to form a similar figure,the ratio of the areas of the two figures is ___ : ___ .A certain rectangular prism can be painted with n liters of paint.The factory enlarged it by a scale factor of 3 to make a similarprism. How much paint do they need to paint the larger box?

Ask participants to consider the four items on Slide 6. How would they rate the items? Easy, hard? What other kinds of words would they use to describe the items? Have them discuss in their groups.

Slide 7 Assessment Items - Where?

Where might the items fit in the pyramid?




8 Assessment Items - Where?A rectangular prism is 2cm x 4cm by 6cm.One dimension is enlarged by a scale factorof 3. What is the volume of the enlargedfigure?A rectangular prism is 2.7cm x 0.45cm by609.01cm. One dimension is enlarged by ascale factor of 3.5. What is the volume of theenlarged figure?When a figure is dilated by a scale factor k toform a similar figure, the ratio of the areas ofthe two figures is ___ : ___ .A certain rectangular prism can be paintedwith n liters of paint. The factory enlarged itby a scale factor of 3 to make a similar prism.How much paint do they need to paint thelarger box?

Ask participants to consider where each of the four items might sit in the pyramid. Have them discuss this with their group. Then ask groups to share their thinking with the whole group. #1 - Find the area is a fairly easy skill question with lower level thinking needed. #2 – This question is still a skill question with lower-level thinking but it is difficult because of the crazy numbers. Do we as teachers sometimes confuse “difficult” with higher-level thinking? Do we sometimes create difficult question because of the computation and fail to create higher level thinking questions? #3 – This could be considered a concept question but one that only requires memory and therefore easy and low level. This does depend highly on how the concept has been taught. If students have not had explicit experience with abstracting this concept – thus they would have to do that here in this item, that would change the item to being a much higher level problem. #4 – This requires a higher level of thinking and a conceptual knowledge of the fact that painting requires the knowledge of surface area. If a student has a good understanding of scale factors and the result to area, this is actually a fairly easy question.

Slide 9 Your Assessment Items - Where?

Teacher questioning?Homework?Quizzes?Tests?

Assessment is a broad term that for many has different implications. Is it all about grades? Is it a continual process that informs instructional decisions? At this point, ask participants to consider what they deem “assessment” and where their assessments might fit in the pyramid. While assessment includes teacher



Procedures Notes questioning, homework, quizzes, tests, and more, the discussion that follows will focus on individual assessment items - specific questions, tasks, problems - and how teachers can differentiate where the items are on the pyramid so that teachers can make better assessment decisions.

Slide 10 Guiding Questions

How can I ask questions for which students can not justmemorize their way through? How can I ask questionsthat demand that students actually understand what isgoing on?How can I ask questions that students can learn fromwhile answering?How can I make sure that I have higher level reasoningquestions and not just computationally more difficultquestions?

Ask participants to consider these guiding questions as we continue.

Slide 11 Passive Assessment Expertise

Understanding the role of the problem contextJudging whether the task format fits the goal of theassessmentJudging the appropriate level of formality (ie., informal,preformal, or formal)Judging the level of mathematical thinking involved inthe solution of an assessment problem

Feijs, de Lange, Standards-Based Mathematics Assessment in Middle School

The goal of this assessment discussion is to help teachers to select the assessment items that fit their needs and purposes, as they consider their entire assessment program. It is not to teach teachers to create such items, but to judge the appropriateness of those items from which they are selecting.

Slide 12 The Assessment Principle

Assessment should become a routine partof the ongoing classroom activity ratherthan an interruption.

NCTM’s Principles and Standards for School Mathematics (2000)

Before participants try to answer the guiding questions together, look at the Assessment Principle from the NCTM standards. Assessment should be a routine part, not an interruption. How can participants do that better? The next activities will give participants some ideas.



Procedures Notes Slide 13 TAKS Item 9th grade 2004

Tony and Edwin each built a rectangulargarden. Tony’s garden is twice as longand twice as wide as Edwin’sgarden. If the area of Edwin’s garden is600 square feet, what is the area ofTony’s garden?

Here is a TAKS released item from 2004 – Ninth grade. Ask participants: Which of the items that we looked at would better prepare students for this item? Specifically, would the computationally more difficult question better prepare students for this more abstract TAKS question? Or would the more open ended, higher level thinking question better prepare students for this question? It is not possible to predict all of the ways that a TEKS will be assessed. Would higher level thinking questions open the door for students to at least be thinking in that direction?

Slide 14 Our focus

Think about current classroomassessmentsHow can they improve?

Again, focusing on classroom assessment items we raise the question, “How can they improve?” Don’t answer this yet.

Slide 15 Take one typical assessment

What is the purpose of the assessment?Where are the items in the pyramid?Are you satisfied with the balance?

Here Trainers could have participants choose one of their classroom’s assessments (assignments, quizzes, tests) and answer these questions. This could be a group activity where the group looks at a common assessment, perhaps a textbook assessment or shared exam. Trainers could also have each teacher individually select a quiz he/she has given and then share with others an example of an item from 3 different places in the pyramid.



Procedures Notes Slide 16 Changing existing questions

to higher leveling reasoningto concept questionsmaintain balance between concept andskill questionsshift focus from what students do notknow to what they do know

This is a transition slide to get participants primed for the next activity, which is changing and improving items.

Slide 17 Targeted Content

(A.2) Foundations for functions. The student uses theproperties and attributes of functions. The student isexpected to: (D) collect and organize data, make andinterpret scatterplots (including recognizing positive,negative, or no correlation for data approximatinglinear situations) , and model predict, and makedecisions and critical judgments in problem situations .[vocabulary of zeros of functions, intercepts, roots]

Note that the items and discussion will be based on targeted content - the TEKS listed in the following slides.


(A.4)(C) connect equation notation withfunction notation, such as y = x + 1 andf(x) = x + 1.


(G.11) Similarity and the geometry ofshape. The student applies the conceptsof similarity and justifies properties offigures and solves problems. The studentis expected to: (D) describe the effect onperimeter, area, and volume when one ormore dimensions of a figure arechanged and apply this idea in solvingproblems.



Procedures Notes Slide 20 Targeted Content

(G.6) Dimensionality and the geometry of location. Thestudent analyzes the relationship between three-dimensional geometric figures and related two-dimensional representations and uses theserepresentations to solve problems. The student isexpected to: C) use orthographic and isometricviews of three-dimensional geometric figures torepresent and construct three-dimensional geometricfigures and solve problems.

Slide 21

So, let’s look at some ways toimprove …

Slide 22 Consider the following:

Factor: x2 - 5x - 6Factor: 36x2 + 45x - 25A soccer goalie kicks the ball from the ground.It lands after 2 seconds, reaching a maximumheight of 4.9 meters. Write the function thatmodels the relationship (time, height).Define “root” of an equation.

Here is the optional beginning for algebra teachers.

Resources: Romberg, Thomas A. ed., Standards-Based Assessment in Middle

School: Rethinking Classroom Practice. New York: Teachers College Press. 2004.

Verhage, H., & de Lange, J. (1997, April). Mathematics education and

assessment. Pythagoras, 42, 14-20.


What’s Your Problem? 6-10

Activity: What’s Your Problem? Overview: Examination, discussion, and writing of three types of assessment items:

snapshot problems, un-doing problems, and error analysis problems. Materials: Handout 1-Sample Assessment Items-Roger Throws a Ball

(pages 6-25 – 6-26) Handout 2-Three Problem Types Labels, 1 for large group

(pages 6-27 – 6-29) Handout 3-What’s Your Problem? Items (pages 6-30 – 6-57) Handout 4-Three Problem Types –How to Write (page 6-58)

PowerPoint: What’s Your Problem? Copies of the PowerPoint with space for note taking

Grouping: Groups of 3 Time: 1.5 hours Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation What’s Your Problem? Use the following notes pages to elaborate on the content of each slide.


1

What’s Your Problem?

The purpose of this PowerPoint is to give participants examples and experience with alternate problem types.

Slide 2 Ways to Modify Questions

Given limited timeFocus on three categoriesNot the only onesPrompt other methods

Discuss the disclaimers: We only have a limited amount of time. Therefore we are going to focus on three categories. They are certainly not the only ways to turn lower level thinking, closed questions into higher level thinking questions – there are other ways for sure, but these are helpful



generalities to look at. They are a place to start. This discussion will certainly prompt other methods and that is also one of the goals – to get teachers to consider alternatives.

Slide 3 Three Ways to Modify Questions

Un-DoingError AnalysisSnap Shot

There are three possibilities we will refer to loosely as “Un-doing,” “Error Analysis,” and “Snap Shot.” These are not technical names, just general descriptors of broad categories, again with the intent of giving teachers alternatives.

Slide 4

Examples of the Three Types

Slide 5 A Typical Textbook Item

Solve for x:

x2 + 9x − 36 = 0

A typical item is a quadratic equation, and students are asked to solve for x. Do students need to be able to do this? Of course. What other kinds of questions could we ask so that students learn more about quadratic equations and solving them?



Slide 6 What might it look like ….

as an Un-Doing problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 7 Un-Doing Example

Write and graph 2 quadratic functionsthat have zeros x = -12, x = 3

Have participants briefly discuss this problem with a partner or group. In mathematics, we often do something and then un-do it. We multiply binomials, we factor the product. We add, we subtract. Here, instead of solving a quadratic equation to find the zeros of the function, we give students the zeros and ask them to go backwards to find and graph the functions. If they can do it one way, often it is a good assessment to see if they can back up, un-do the process, start with the answer and work back to find the problem.


Write and graph a quadratic function thathas

a. one x-intercept.b. two x-intercepts.c. no x-intercepts

Here is another “Un-Doing” example. Have participants briefly discuss it. Ask them to consider how a student might think differently to solve these problems. What extra or additional parts about quadratics might be embedded in this problem beyond what students have to think about to do the original “solve for x” problem? Did this and the previous example seem more open? Might this possibly allow the teacher to see a greater variety of solutions and strategies that when discussed can build strength in connecting the different approaches?




as a snap shot problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 10 Snap Shot Example

Yesterday in class we solved some equations graphically. What was the equation we were solving below? What was the solution?

The TEKS call for multiple representations and for the use of graphing technology. How does one assess this? One example is to take a snap shot of your classroom instruction and ask about it. The day before this example was presented, the class had used graphing technology to solve quadratic equations like x2+9x-36=0 by graphing y1= x2+9x-36 and y2=0 and finding the intersection. Were students just pushing buttons, memorizing a series of steps, or were they looking at the graph and realizing that they were looking for the x values for which the y’s were 0? One way to assess would have been to give them more equations to solve. Another example is to give them the graphing windows and ask about them. This shifts the focus from button pushing to connecting.


Chelsea dropped her homework in a puddle.Help her reconstruct the blurry areas. Whatquestion was she answering?

Another way to open up the discussion is to take a procedure and hide carefully selected parts of it. This way, students have to think about someone else’s strategies. This is a way to see if students really know what is going on - the reasons for the steps - or if they are stuck in one way of “doing” it. This snap shot example is also a way to assess a student’s understanding of a



manipulative model without the teacher having to stand over the student’s shoulder and watch him/her use algebra tiles.


as an error analysis problem?

Solve for x:

x2 + 9x − 36 = 0

Slide 13 Error Analysis Example

Ralph:Factor

Eleanor:Solve:

Sanna: Solve

x2 + 9x − 36 = 0(x +12)(x − 3) = 0x = −12, 3

x2 + 9x − 36x2 + 9x − 36(x +12)(x − 3)

x2 + 9x − 36 = 0

x2 + 9x − 36 = 0x2 = −9x + 36

x2 = −9x + 36x = −3, 6

Each of these three examples has a different commonly made error. Students are asked to examine these erroneous processes and find the error(s). Ask teachers to consider the common errors of their own students. Suggest that instead of only re-teaching the correct method, they might also consider asking students to analyze the common errors made in their own classrooms. Here we see that Ralph solved instead of factored, Eleanor factored instead of solved, and Sanna’s first error was to think that the square root of a sum was the sum of the square roots.


Diane: Graph

x2 + 9x − 36 = 0(x +12)(x − 3) = 0x = 12,−3

y = x2 + 9x − 36

Another commonly made error: Diane did not set each factor equal to zero to find the value of x when the product equals zero.



Slide 15 TAKS Item (9th grade 2004)

The area of a rectangle is 3x 2 + 14x + 8,and the width is x + 4. Which expressionbest describes the rectangle’s length?

3x + 22x + 42x + 23x − 2

a.

b.

c.

d.

Here is a released item from the 2004 9th grade TAKS. Ask participants to consider the various problems at which they have just looked and how these problems might have prepared students for this item and the next two. Compare the original, typical question’s place on the pyramid with the locations of the other questions.


What are the roots of the functiongraphed below?

a. (-1, -9) and (0, -8)

b. (0, -4) and (2, 0)

c. (-4, 0) and (2, 0)

d. (0, 2) and (0, -4)

For extra credit, ask participants to find the error in the wording of this TAKS question.


Which ordered pair represents one of the rootsof the function ?

a. (-5/2, 0)

b. (-4, 0)

c. (-5, 0)

d. (-20, 0)

f x( )= 2x2 + 3x − 20

For extra credit, ask participants to find the error in the wording of this TAKS question.

Slide 18 Earlier with Roger

Earlier we looked at a series of graphs andtables that modeled Roger throwing abaseball upward from a downward movingelevator. Based on that work, answer thefollowing. The graphs and tables belowrepresent Roger starting from a differentheight, throwing at a different initial velocity.

A more effective discussion of assessment begins with a common experience to discuss. After participants have done the Roger Throws a Ball activity (page 4-15), they can now have a rich discussion about how to assess it. Ask participants to brainstorm how they might assess the Roger Throws a Ball activity.




1. Match the graph with the appropriate table.

2. What are the roots, solutions shown?

Here are several possibilities, ranging from low level to mid level reasoning questions. Ask participants to generally categorize the problem as Un-Do, Error Analysis, Snap Shot or other. (These examples are provided in Handout 1-Sample Assessment Items-Roger Throws a Ball, (pages 6-25 – 6-26) 1 and 2: These can be considered Snap Shots with an un-doing feel: take a snap shot of the activity, and ask students to un-do, or reverse, the procedures they did in Roger Throws a Ball.


3. When will Roger and the baseball be at the same height?

4. Equations have ____ or ____, whereas functions have

______ or _______.

3: Snap Shot. Were students paying attention in class, and did they make the necessary connections during the activity? 4: Low level vocabulary recall. Some things just have to be memorized. Ideally, teachers give students enough experience with concepts that the labels become automatic, but they are labels none the less and must be memorized.


5. What is Roger’s new starting height?

5: Snap Shot. Were students paying attention in class, and did they make the necessary connections during the activity?


Yesterday we looked at a series of graphs and tables thatmodeled Roger throwing a baseball upward from adownward moving elevator. Based on that work, Rauland Marco answered the following question. Who isright and why?

1. Given the table and graph representing Roger throwinga baseball upward from a downward moving elevator, whatare the roots, solutions shown?

RaulRoots and solutions are the same as zeros.Y1 is zero at x = 0 and in between x = 3and x = 4, so the roots are x = 0 and x isabout 3.5

MarcoA roo t is the so lution t o an equa t ion sow e are l ooking fo r w here Y1 = Y2 . I nt h e t ab le , both Y1 and Y2 have t he sameva lue s a t x = 0 and x = 4 . On the g r aph ,Y1 and Y2 in t er sec t at the se sa me xva lue s x = 0 and x = 4 .

Error Analysis




Bo was absent yesterday. When you startedtelling him about the Roger-throwing-the-ball-elevator activity, he said, ŅWhatÕs the big deal?Roots, zeros, solutions, x-intercepts Š theyare all the same thing.ÓHow do you respond to Bo?A complete answer includes graphs, tables,equations and discussion.

Error Analysis: This is a fine example of a problem where one can see more clearly what a student knows, instead of just what the student does not know. Wrap up: Teachers need all kinds of assessment all over the pyramid. These items have just provided a look at three broad categories that might help participants open up their options. These three categories should not be considered the only kinds of assessments. Trainers should point out that looking at these three categories should only just be considered a way to get the conversation started.

Slide 24 Part 2: As A Class

Everyone should have one problem fromthe setDiscuss the problems in your group.Decide where the items would best fit.Post your problemGallery walk - do you agree?Choose one to discuss as a group

Have participants look at some more items to continue to get a better sense of ways to alter and adjust classroom assessment for better student learning. Use Handout 2 (pages 6-27 – 6-29) to label sections of the room as Un-Do, Snap Shot, and Error Analysis. Distribute the problem items (Handout 3 pages 6-30 – 6-57), one per person if possible. Have participants work as a group and post each item under previously labeled sections of the room (Un-Do, Snap Shot, and Error Analysis). After items are posted, participants consider if they agree as they take a gallery walk. For items they think are posted incorrectly, participants could flag them with a red flag. The purpose here is not so much the three problem types, it is more to expose participants to alternative ways to assess. As a whole group, discuss the groupings. Bring out the following points: Un-Doing: Much of mathematics is doing something and then un-doing it. Many times a great question to assess if students



“got it” when doing something is to ask students to un-do it, to back up from the answer, to come at it from a different direction or representation. Some of the Un-Doing questions are a specific type – a creating type. This is a fine time to discuss this type that, for our purposes, is included in the Un-Doing group. Note: In the creating type of Un-Doing questions, students are asked to create or generate different answers. Posing questions where the answer becomes the question opens up the social “space” in the classroom to allow all students the opportunity to participate and makes them accountable for the content they are learning. “Generative design centers on taking tasks that typically converge to one outcome and turning them into tasks where students can create a space of responses.” Stroup, Ares, Hurford, 2005 Error analysis: Taking common misconceptions and mistakes and putting them up front for students to consider and explain. Snapshots: Taking a snapshot out of the middle of a process or solution or activity and asking students about it.

Slide 25 Discussion

Un-DoingError AnalysisSnap Shot

Have groups share out the one or two problems they found most interesting or compelling. Use the red flags, if any, to generate more conversation about the problems. It is less important if everyone agrees on the type of item. It is more important to discuss how the items assess student thinking.



Slide 26 Advantages and Disadvantages

GradingConceptual understandingMemorization

If they have not come up already, briefly discuss the advantages and disadvantages of these kinds of assessment items (Un-Do, Snap Shot, and Error Analysis). One disadvantage:

• Grading - many of the problems are more open ended. This may be a barrier for participants who have little experience or few resources to deal with more open-ended questions. In the Closer section there are resources to help.

Two advantages: • Conceptual understanding: These

assessments demand more conceptual understanding than many typical textbook bare problems.

• Memorization: Students cannot just memorize their way through these problems. They actually have to know what is going on.

Slide 27 Write your own

Choose a TEKS statementWrite a typical question to assess it.Write it as an Un-Doing questionWrite it as an Error Analysis questionWrite it as a Snap Shot question

Distribute Handout 4-Three Problem Types: How to Write (page 6-58) and blank paper. With participants in groups, have them create some assessment items on the blank paper using the How to Write handout. Participants can consult their textbooks for typical questions. They can also do a “search and rescue” as they search the textbook for examples of the three types of items. Have each group choose one or two and share out with the whole group. Collect, copy, and then hand them out so that everyone in the group will have more examples. Ask participants to label the items in some way as to suggest where they might use each item (on which assignment, test, project, what time of the year, etc.)



Slide 28 Snap Shot Problems

What are two ideas, processes, orrepresentations that students mix up?Juxtapose them and ask which is which.What part of a large activity can you grabto assess if students got the gist of thelarge activity?

Trainers might show this slide while participants are sharing Snap Shot problems they created.

Slide 29 Un-Doing Problems

Can you start with the answer?Can you start in the middle?Can you change one constraint?Can you start with a differentrepresentation?Ask students to create or invent thebeginning of a problem.

Trainers might show this slide while participants are sharing Un-Doing problems they created.

Slide 30 Error Analysis

What are the typical errors that studentsmake?Pose an incorrect solutionAsk students to explain what went wrong.Sometimes show the incorrect process,sometimes just show the incorrect answer

Trainers might show this slide while participants are sharing Error Analysis problems they created.




Trainers can use the following: 1 - if you have an audience of mostly

geometry teachers 2 - if you want more examples using a

different strand



Slide 32 Another Example

The following slides begin with a differentstem problem based on geometry andscale factors.

Slide 33 A Typical Textbook Item

The length of a rectangle is 8cm and the width is 6 cm. Findthe perimeter of the newrectangle created by when theoriginal width is dilated by ascale factor of 4.

A typical item: Use the given information to find the new perimeter. Do students need to be able to do this? Of course. What other kinds of questions could teachers ask so that students learn more about scale factors and the relationships with surface area and volume?


as an Un-Doing problem?



If the volume is increased by a factor of8, what is the change in the length of theside of a cube?

Have participants briefly discuss this problem with a partner or group. In mathematics, we often do something and then un-do it. We multiply binomials, we factor the product. We add, we subtract. Here, instead of giving students the original information and having them use the scale factor to find the new dimensions and then the volume, students are given information about the volume and asked to go backwards to find the length. Do they understand the relationship



between the volume and the side length or have they just memorized a set of steps to do (to find the volume)?


Below is Craig’s work. What might havebeen the question?

16

3

34

8+6=144x3=12

Here is another “Un-Doing” example. Have participants briefly discuss this example, and ask them to consider how a student might think differently to solve this problem. Did this and the previous example seem more open? Might this possibly allow the teacher to see a greater variety of solutions and strategies that when discussed can build strength in connecting the different approaches? If students can do it one way, often it is a good assessment to see if they can back up, un-do the process, start with the answer and work back to find the problem.


as a snap shot problem?



Yesterday in classwe exploredfigures formed bydilations. Abbydropped herpaper in a puddle.Help her fill in themissing titles andvalues.

22

332 4 96 64

216

2

Another way to open up the discussion is to take a procedure or process and hide carefully selected parts of it. In this way students have to think about someone else’s strategies. This example also has an Un-Doing part to it. The third row asks them to find the scale factor given the original and new side length.




as an error analysis problem?



The length of rectangleOLDR was enlarged by ascale factor of 3 to createrectangle NEWS. If OLDRhas a width of 3 cm andperimeter of 16 cm, what isthe area of NEWS?Sandi wrote the following.What do you say to her?

35

L

DR

O

9

15E

WS

N

So, A= 9x15 = 90+45 = 135 cm 2

Here is a commonly made error. Both the length and the width were mistakenly enlarged instead of just the width as instructed. In this type of problem, students are asked to examine erroneous processes and find the error(s). Ask participants to consider the common errors of their own students. Suggest that instead of only re-teaching the correct method, they might also consider asking students to analyze the common errors made in their own classrooms.


a. 1/3

b. 1/2

c. 2/7

d. 5

What scale factorwas used totransform ² MNP to² RST?

Here is a released item from the 2004 9th grade TAKS. Ask participants to consider the various items at which they have just looked, and how they might have prepared students for this item and the next two. Note that this TAKS item asks students to find the scale factor given the figure’s dimensions. This is an un-doing question. Trainers might compare the original, typical item’s place on the pyramid with the locations of the other items.




a. The length is 2 times the original length.

b. The length is 4 times the original length.

c. The length is 6 times the original length.

d. The length is 8 times the original length.

If the surface area of a cube is increased by afactor of 4, what is the change in the length of thesides of the cube?

Here is a released item from the 2004 11th grade TAKS. Ask participants to consider the various items at which they have just looked, and how they might have prepared students for this item and the next. Trainers might compare the original, typical item’s place on the pyramid with the locations of the other items.


A rectangle has a length of 4 feet anda perimeter of 14 feet. What is theperimeter of a similar rectangle with awidth of 9 feet?a. 36 ft.

b. 42 ft.

c. 108 ft.

d. 126 ft.


Handout 1-1 What’s Your Problem? 6-25

Sample Assessment Items – Roger Throwing a Ball Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

1. Match the graph with the appropriate table. 2. What are the roots, solutions shown? ____________ 3. When will Roger and the baseball be at the same height?

_______ 4. Equations have _______ or ________, whereas functions

have _______ or _______. 5. What is Roger’s new starting height? ______

___a.

i.

___b.

ii.

___c.

iii



Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, Raul and Marco answered the following question. Who is right and why? 1. Given the table and graph representing Roger throwing a

baseball upward from a downward moving elevator, what are the roots, solutions shown?

Raul

Roots and solutions are the same as zeros. Y1 is zero at x = 0 and in between x = 3 and x = 4, so the roots are x = 0 and x is about 3.5

Marco

A root is the solution to an equation so we are looking for where Y1 = Y2. In the table, both Y1 and Y2 have the same values at x = 0 and x = 4. On the graph, Y1 and Y2 intersect at these same x values x = 0 and x = 4.

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6-29



What are 2 possible sets of lists that would produce the following graph:

Fill in the lists and note your window. Possibility 1: Possibility 2:



Rectangle RECT has an area of 4 cm2. It is enlarged to make the rectangle BIGS. BIGS has an area of 16. What was the scale factor used to enlarge RECT to get BIGS?



Sketch the view of a cube below, but make it appear three-dimensional by making some of the lines dashed.



Square SQAR has side length s. Choose your favorite scale factor to dilate SQAR to a similar square and label new square HUGE. Label the side lengths of HUGE in terms of s and find the area of HUGE in terms of s.



Write two quadratic functions with zeros x = 2, x = -3



A cube, P, has volume 1000 cm3. It is dilated by a scale factor of 0.5 to form a similar cube, S. What is the side length of cube S?



The clear cube shown has the letters DOT printed on one face. When a light is shined on that face, the image of DOT appears on the opposite face. The image of DOT on the opposite face is then painted. Copy the net of the cube and sketch the painted image of the word, DOT, on the correct square and in the correct position.

Discovering Geometry



The cube has designs on three faces. When unfolded, which figure at right could it become?




Find the equations of two linear functions that intersect at the point (2, 3)



Find the equations of two linear functions that do not intersect.



Write your two favorite linear functions with a y-intercept of 4



Write your two favorite linear functions with a slope of 4.



The green prism below right was built from the two solids below left. Copy the figure on the right onto isometric dot paper and shade in one of the two pieces to show how the complete figure was created.




Sketch the solid shown, but with the two blue cubes removed and the red cube moved to cover the visible face of the green cube.




Given f x( )= x2 + 3, find f 2( ) Cameron got the following answer. What do you think he did? What should he have done?

f 2( )= 2x2 + 6



The width of rectangle DAWN was enlarged by a scale factor of 2:3 to form a new rectangle RISE. What is the perimeter of RISE?

Justin’s work is below. What do you say to Justin?

10 to 15, 20 to 30 so 2(15) + 2 (30) = 30 + 60 = 90

90 cm

10 cm

20 cm D A

W N



Sanna solved the equation below as follows. Is she correct? If not, explain to her why.

x2 + 9x − 36 = 0x2 = −9x + 36

x2 = −9x + 36x = −3, 6



José answered the following question below. How do you respond to his answer? Include the correct response.

Yesterday we looked at a series of graphs and tables that modeled Roger throwing a baseball upward from a downward moving elevator. Based on that work, answer the following. The graphs and tables below represent Roger starting from a different height, throwing at a different initial velocity.

What are the roots, solutions shown? _______

“The graph intersects the x-axis around 4.5, so the root (solution) is x = 4.5.”



For the figure shown, Kamisha drew the following orthographic drawing. How would you help her fix it?

Front Side Top



Lakrea created the net below. She is convinced that it will fold into a cube. Will it? If not, how would you convince her otherwise?



McKay says that the scatter plot below shows positive correlation because “the data goes up” as he pointed up from left to right. What do you say to McKay?



Bo was absent yesterday. When you started telling him about the Roger-throwing-the-ball-elevator activity, he said, “What’s the big deal? Roots, zeros, solutions, x-intercepts – they are all the same thing.” How do you respond to Bo? A complete answer includes graphs, tables, equations and discussion.



Yesterday we added marbles to cups hanging by rubber bands. We measured 3 distances, graphed the data, and found trend lines. Two groups results are below.

Group A

Group B

1. What can you conclude about Group 1’s marbles and rubber band compared to that of Group 2?

2. What can you conclude about Group 1’s table and cup compared to that of Group 2?



Yesterday we graphed different classes heights and arm spans. Two different classes are represented below.

Group A

Group B

1. Name one difference between the people measured by Group A and the people measured by Group B and explain how you know.

2. Name another difference between the people measured by Group A and the people measured by Group B and explain how you know.



Marisol spilled soda on her homework. Fill in the two missing steps covered by the stain.



For a week we have been bouncing balls under motion detectors and finding function rules to model one complete bounce. Here is some similar data. What function rule would you write to model the first complete bounce (the one the trace cursor is on)?



If the ratio of the heights in two similar figures is m/n, the ratio of their perimeters would be ______________, the ratio of their surface area would be _________________ and the ratio of their volume would be ____________________.



One of the following solids cannot be represented by the orthographic drawings. Change the orthographic drawings so that they represent the misfit and not the others.

top front side

a

b

c

d


Handout 4 What’s Your Problem? 6-58

Three Problem Types – How to Write

Snap Shot Problems: What are two ideas, processes, or representations that students mix up? Juxtapose them and ask which is which. What part of a large activity can you grab to assess if students got the gist of the large activity? Un-Doing Problems: Can you start with the answer? Can you start in the middle? Can you change one constraint? Can you start with a different representation? Ask students to create or invent the beginning of a problem. Error Analysis What are the typical errors that students make? Pose an incorrect solution. Ask students to explain what went wrong. Sometimes show the incorrect process; sometimes just show the incorrect answer.


The Power of Creating 6-59

Activity: The Power of Creating Overview: Teachers explore the power of creating problems. Materials: Two places to record answers (chart paper or white board space), each

with a coordinate axes and space for function rules Handout 1-Sample Class Responses, optional for use by Trainer

(page 6-63) Transparency 1-Slope of 4, optional one per group of 4 participants

(page 6-64) Transparency 2-Y Intercept of 4, optional one per group of 4 participants

(page 6-65) PowerPoint: The Power of Creating

Grouping: Partners Time: 30 minutes Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation The Power of Creating. Use the following note pages to elaborate on the content of each slide.


1

The Power of Creating

What do you mean?

In the set of problems that the participants classified as Un-Doing problems, there was a subset of those assessment items that are uniquely designed to teach while assessing. This subset is referred to here as “Creating Problems” because the students are asked to create or generate a response that contributes to a space of responses that as a whole increase students’ depth of understanding.

Slide 2 What’s Your Line?

Sketch a graph of your favorite line with ay-intercept of 4 and write the function rule.Sketch a graph of your favorite line with aslope of 4 and write the function rule.Post your graphs under the correspondingheadings and write the function rules.

Explain the 3 steps. Participants should work together in pairs to sketch the graph and write the function rule. Have one partner post the y-intercept graph and function rule, and the other partner post the slope graph and function rule in the appropriate places. As Trainers circulate, they should encourage some groups to be clever in their



choices. For instance, ask groups to consider negative numbers and all four quadrants. Ask some to use “complicated” numbers (fractions, decimals, really large, really small, etc.) On the board or on chart paper, designate two locations: one labeled “y-intercept of 4” and the other “slope of 4”. Or use the transparencies (pages 6-64 & 6-65) – one per group. Each group fills out one line with a function and graphs the same function. Then overlay them on the overhead projector.

Slide 3 Y-intercept and Slope

What do you see?

With both charts next to each other, discuss. Ask: What do you think?

What can you tell me about these two charts?”

Have participants focus on strategies. Ask: How did you find your line?

What were you thinking? Did anyone have a specific strategy

that you think will work every time?

Did anyone guess and check, using a graphing calculator?

Would your strategy work if I had asked for a y-intercept of 14 instead of a y-intercept of 4?”

Have participants share their strategies. Focus on the “y-intercept of 4” chart and ask the following questions if they did not already come up.

Ask: What can you tell me about these lines?” (They all intersect the y-axis at 4, the function rules all have a +4; they have different slopes; the function rules all have x1 (unless someone



chose y=4); they are all lines, …)

Focus on the “slope of 4” chart and ask the following questions if they did not already come up.

Ask: What do these sets of numbers have in common?” (They all have a middle number that is 5; they do not all have the same sum, …)

Slide 4 Summative

Find three equations of lines where theslope is -2 and three equations of lineswhere the y-intercept is -2. Graph thelines and label them with the functionrules.

Ask participants to find 3 equations of lines where the slope is –2 and 3 equations of different lines where the y-intercept is –2. Have them graph the lines and label them. This is a summative kind of task.

Slide 5 The Power of Creating

Powerful by themselvesMore powerful when put together to lookat commonalities.

Discussion: Creating Problems are powerful by themselves but even more powerful when the generated answers are compared and commonalities are found. This is a participant discussion. Get out of the content and broaden the discussion to Creating Problems in general.

Slide 6 The Power of Creating

Generative design centers ontaking tasks that typically convergeto one outcome and turning theminto tasks where students cancreate a space of responses.

Stroup, Ares &Stroup, Ares & Hurford Hurford, 2005, 2005

Ask participants to comment on this description of Creating Problems. What is a “space of responses”? How does this “space of responses” enhance learning?



Slide 7 Open Ended Questions

http://books. heinemann .com/math/

If your participants need support in using open ended questions, this sight has excellent resources that are free for teachers.

Slide 8 Rubrics

http://www. mathbenchmarks .org/rubric. htm

Grading help can be found at the Region 4 site: http://www.mathbenchmarks.org/rubric.htm This is the student rubric for grades 6-8.

Slide 9 Rubrics

http://www. mathbenchmarks .org/rubric. htm

This is the more detailed teacher rubric.

Slide 10 Other Examples:

Create the equation and graph of aquadratic function with y-intercept of 4and then again with a vertical stretchfactor of 4.Create a system of linear equationswhose solution is (3,2).

These are just two examples of Creating Problems. Ask participants to brainstorm with others and share. This same kind of activity could be done with quadratics. Find a quadratic with zeros x= 2, 4. What do they all have in common? What about all of the quadratics with zeros x= -2, -4? What about quadratics with y-intercepts of 2? What do their function rules have in common?


Handout 1 The Power of Creating 6-63

Sample Class Responses

y-intercept of 4 Function Rule Graph

1. y = 4 2. y = 4 − x 3. y = 4 + x 4. y = 3x − 4 5. y = −4 + 3x 6. y = 4 − 9x 7. y = 4 + 50x 8. y = 4 − 0.2x 9. y = 4 + 0.4x 10. 11. 12.

Slope of 4

Function Rule Graph 1. y = 4x 2. y = 2 + 4x 3. y = −2 + 4x 4. y = −15 + 4x 5. y = 12 + 4x 6. y = −8 + 4x 7. y = 7 + 4x 8. y = 0.5 + 4x 9. y = 18 + 4x 10. 11. 12.

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Closing Activity – Assessment Should Drive Instruction 6-66

Activity: Closing Activity – Assessment Should Drive Instruction Overview: Participants will discuss common definitions of Diagnostic, Formative, and

Summative assessment. Formative assessment, being defined as assessment that informs instruction, should dominate and be an integral part of mathematics classrooms. Several definitions are provided for discussion. Participants end the section by participating in an assessment scavenger hunt, looking for assessment items that appear all over the Assessment Pyramid.

Materials: PowerPoint: Closer

Current textbook(s) or local curriculum, optional Grouping: Tables of 4 Time: 30 minutes Lesson: Distribute the PowerPoint copies to participants to help them focus on the

important ideas from the PowerPoint presentation as they take notes. Show the PowerPoint presentation Closer. Use the following notes pages to elaborate on the content of each slide.


1

Assessment

Mathematics TEKSRefinement Project

Briefly recall the Assessment Pyramid and the guiding questions. Ask participants to reflect on their current assessment practices. How do they use assessment to inform instruction?

Slide 2 Assessment

Ask participants to reflect on the assessment items they have seen in the presentation and to reflect on these guiding questions. • Do you have a clearer picture of the

difference between levels of reasoning and levels of difficulty, and how they interact?

• What items exemplify the kind of question for which students cannot just memorize their way through?



• What items exemplify the kind of question from which students can learn while answering?

Slide 3 Guiding Questions

How can I ask questions for which students can not justmemorize their way through? How can I ask questionsthat demand that students actually understand what isgoing on?How can I ask questions that students can learn fromwhile answering?How can I make sure that I have higher level reasoningquestions and not just more computationally difficultquestions?

Discuss the descriptions of diagnostic, formative, and summative assessment. Whichever it is called, assessment should be a routine part of the classroom, not an interruption. Formative – assessment that helps teachers make decisions about the content or form of instruction. Summative – used to judge students’ attainment. (Principles and Standards, p.24) “Some identify classroom assessment with formative assessment. We agree with Biggs (1998) that formative assessment and summative assessment are not mutually exclusive, as suggested by Black and Wiliam (1998). Their argument is that feedback concerning the gap between what is and what should be is regarded as formative only when comparison of actual and reference levels yields information that is then used to alter the gap. But if the information cannot lead to appropriate action, then it is not formative. Summative assessment in the form of end-of-year tests gives teachers the proof of how well they handled the formative assessment, assuming that the underlying philosophy is coherent and consequent. The differences in formative and summative assessment within the classroom are more related to timing and the amount of cumulation than anything else. Needed for both, of course, is that the assessment is criterion-referenced, incorporating the curriculum and resulting in aligned assessment.” (DeLange “Framework”, p. 4)



Slide 4 Assessment

Before the lesson (diagnostic)assessmentDuring the lesson (formative)assessmentAfter the lesson (summative)assessment

Use the rest of the slides to clarify any terms as they come up in the discussion.

Slide 5 Formative

Assessment should be more than merely a test at the endof instruction to see how students perform under specialconditions; rather, it should be an integral part ofinstruction that informs and guides teachers as they makeinstructional decisions. Assessment should not merely bedone to students; rather, it should also be done forstudents, to guide and enhance their learning (TheAssessment Principle, ¶ 1).


Slide 6 Formative

When the results of those activities are used inthis way—to adapt the teaching and learningpractice—we speak of formative classroomassessment

deLange

Slide 7 Formative

"When the cook tastes the soup, that'sformative assessment; when the customertastes the soup, that's summativeassessment."

Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment inEducation, 5, 7-74.



Slide 8 Summative

Judging students’ progress/ attainment

Slide 9 Degrees of Openess

Closed Task - one correct answer, oneroute to arriving at that answerOpen-Middled Task - one correct answerbut many routes to arriving at that answer.Open-Ended Task - several correctanswers and many routes to arriving atthose answers.

Slide 10 Assessing Mathematical Skills

“An assessment task that focus primarily onmathematical skills gives students a chanceto apply a well-practiced and importantprocedure or algorithm.”

Mathematics Assessment: A Practical Handbook, NCTM, 2000

Slide 11 Assessing Mathematical Skills

“These tasks are usually-routine;short;based upon recalling a well-known procedure;cast in a simple context or no context at all;focused on a single correct answer.”



Slide 12 Assessing Conceptual Understanding

“Assessment tasks that focus primarily onmathematical concepts give students achance to apply a concept in a newsituation, to reformulate it, and to express itin their own terms. These tasks probe theunderstanding of an idea.”


Slide 13 Assessing Conceptual Understanding

“They are usually-non-routine;short;based upon reconstruction, rather thanmemorizationcast in a context;focused on representation and explanation ofthe solution.”

Slide 14 Assessing Problem Solving

“An assessment task that focuses primarilyon mathematical problem solving givesstudents a chance to select and useproblem-solving strategies.”


Slide 15 Assessing Problem Solving

“Problem solving tasks are usually-non-routine;long;Predicated on the high-level use of facts,concepts, and skillscast in a context;focused on the students’ abilities to developand use strategies to solve.”






How has this presentation affected your view of assessment?

Slide 17 Scavenger Hunt

Using your current text and assessmentsfind examples all over the pyramid,justify your examples,share with the group.


Resources 6-72

Resources deLange, Jan. FRAMEWORK FOR CLASSROOM ASSESSMENT IN MATHEMATICS,

Freudenthal Institute & National Center for Improving Student Learning and Achievement in Mathematics and Science, September 1999.

National Council of Teachers of Mathematics (NCTM). Mathematics Assessment: A

Practical handbook For Grades 6-8. Reston, Va.: NCTM, 2000. National Council of Teachers of Mathematics (NCTM). Mathematics Assessment: A

Practical handbook For Grades 9-12. Reston, Va.: NCTM, 2000. National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation

Standards for School Mathematics. Reston, Va.: NCTM, 1989. National Council of Teachers of Mathematics (NCTM).. Principles and Standards for

School Mathematics. Reston, Va.: NCTM, 2000. Romberg, Thomas, A. ed., Standards-Based Mathematics Assessment in Middle

School: Rethinking Classroom Practice; New York: Teachers College Press. 2004. Serra, Michael, Discovering Geometry: An Inductive Approach. Emeryville, CA, Key

Curriculum Press. 2003. Stroup, W., Ares, N., & Hurford, A., (2005). A Dialectic Analysis of Generativity: Issues

of Network Supported Design in Mathematics and Science. Journal of Mathematical Thinking and Learning,7(3), 181–206.

7 - 1

Math for English Language Learners (MELL): Complexities and Strategies

Intent of Activity: To encourage participants to consider the special learning needs of English Language Learners in math and to explore various instructional strategies, classroom practices, and resources that can enhance student learning. Participants will also receive information about the Math for English Language Learners Initiative, a collaborative endeavor of the Texas Education Agency, the Texas State University System, and the five TSUS universities. Approximate Length: 60-90 minutes Accompanying Handouts: A Place To Begin

MELL Classroom Practices Framework MELL Products and Resources

Accompanying Slides: Series of MELL PowerPoint Slides Recommended Grouping Arrangement for this Activity: Ideally participants would be seated in small groups of 4-6 participants.

Directions To Trainer Engage: Using the first two slides of “A Place to Begin: Underlying Assumptions” introduce the topic of Math for English Language Learners. Then ask participants to take a couple of minutes to use the bottom half of their first page of the MELL handout to identify some of the strategies and classroom practices that they have found helpful in teaching English Language Learners (ELLs).

Explore: Trainer should then give participants 5-10 minutes to share with their group the strategies they have identified. Trainer can then debrief discussion by having each group share one strategy discussed at their table, while a scribe captures these on chart paper. After each group has contributed one, discussion can be opened up for participants to add additional strategies. Explain: Trainer should then use the slides to share information about the work of the TSUS MELL Initiative and some of the insights and products derived from this work. After introducing the MELL Classroom Practices Framework and its various components, the trainer should have participants identify the Framework in their handout and give them an opportunity to review it briefly. Trainer should answer any clarifying questions and then tell participants, “let’s try it out with some real data.” Elaborate: The trainer should then ask participants to work in their group to again examine the strategies they have identified and to see which of the MELL CPF

7 - 2

components each of the strategies fit into. Trainer should explain that the CPF is an “evolving” document so participants may identify strategies or components that should be added to the MELL CPF. The trainer should conclude this lesson segment with a large group discussion in which participants share which components they identified most often and least often. Participants should also have the opportunity to give input on how the CPF might be refined and how it might be used by classroom teachers. Trainer should then briefly introduce the MELL Teachers Guides by using the Teachers Guide Excerpt pages in the handout. Participants should be given 10 minutes to review material included for Objective 9 and instructed to identify as many strategies as they can that support the English Language Learner. At the conclusion of 10 minutes, the trainer should debrief the activity by allowing each table to contribute 1-2 strategies they identified. Evaluate: Trainer should continue the slide presentation, pointing out the page in the handout that describes the various MELL products that were developed in 2004-05 and then MELL brochure and resource list. The trainer should pause the slides on the one that says “Q & A Time,” and allow participants to ask questions and make comments. Trainer should use these questions and comments to evaluate the group’s understanding and acceptance of the concepts, and to make culminating and synthesizing remarks. The presentation should then end on the “For More Information …” slide.

7 - 3

Math for English Language Learners (MELL):

Complexities and Strategies

A Place To Begin: Underlying Assumptions

• Teachers have very “full plates” and many demands

• TAKS standards are substantive and challenging

• ELL students face additional challenges that are currently reflected in TAKS results

• Texas educators are working hard to improve the performance of all students,

including those who are English Language Learners

• Educators are interested in acquiring additional knowledge and skills to better address the needs of ELL students

Think about your own experiences and those of other teachers who have been successful in teaching math to English Language Learners. What are some of the instructional strategies and/or classroom practices that enhanced the learning experience of these students?

7 - 4

MELL Classroom Practices Framework

Developed by

Texas State University – San Marcos Joyce Fischer, Ph.D.

Christopher Johnson, M.S. Leslie Huling, Ed.D.

For

Math for English Language Learners (MELL) Initiative

A Texas State University System (TSUS) and

Texas Education Agency (TEA) Collaborative

Participating TSUS Institutions Include: Angelo State University

Lamar University Sam Houston State University

Sul Ross State University Texas State University

June, 2005

7 - 5

MELL Classroom Practices Framework (CPF)

The MELL Classroom Practices Framework is a synthesis document compiled by the Texas State University System (TSUS) Math for English Language Learners (MELL) Initiative funded by a grant from the Texas Education Agency. In the summer of 2004 TEA, in response to the lingering achievement gap in mathematics between Limited English Proficient (LEP) students and other students, worked with TSUS and its five partner institutions to establish the MELL Initiative. The primary purpose of the MELL Initiative is to develop resources for professional development targeted at improving mathematics instruction for English Language Learners, especially those at the secondary level. In Phase I of the Year 1 scope of work, several avenues were simultaneously pursued in order to identify specific needs related to math instruction for English Language Learners and to identify existing resources. An extensive review of research and literature was coordinated by Sul Ross State University, while mathematicians at Texas State University explored 12 different professional development models designed to support the math instruction for struggling students. Lamar University gathered information directly from teachers through focus groups and survey instruments to identify their views about professional development needed to support math instruction for English Language Learners. Statisticians at Sam Houston State University worked with TAKS results to further analyze achievement trends by groups of students and by geographic and ESC regions. Angelo State University analyzed the preparation program for math teachers at each of the TSUS institutions to identify current practices in preparing math teachers to deal with the specific needs of English Language Learners. Periodic meetings involving TEA staff, faculty from the participating TSUS institutions, and experts in the field, were conducted throughout the year to guide and shape the work and to keep all parties updated on the progress of others. Each of these Phase I avenues of investigation has resulted in a specific MELL product and information about these products is available through the MELL website at www.tsusmell.org. Phase II of the Year 1 scope of work is devoted to developing various professional development resources designed to address the specific needs identified in Phase I. MELL and TEA staff both identified the need for a concise document that could not only capture the essence of the Phase I work, but could also provide a roadmap for use in Phase II products. The MELL Classroom Practices Framework (CPF) was developed in response to this need. The MELL CPF was generated collaboratively by MELL and TEA staff and was guided by the question of “What do the findings of our Phase I investigations suggest in regard to classroom practices that contribute to successful math instruction for English Language Learners. This framework represents the current collective thinking of MELL partners about what Phase I investigations revealed, and it is our intention that MELL professional development products support teachers in implementing these classroom practices. Over time, as additional insights are gleaned from ongoing work, it is likely that this evolving framework will be revised.

7 - 6

Reaching consensus on this framework was a lengthy and labor-intensive process and our group understandably has reservations about suggesting specific classroom practices in the absence of a definitive body of research. Admittedly, there is limited “hard data” to document that specific achievement gains are the result of specific classroom practices. Additional research in this area is critically needed and contributing to this body of research is a major goal of MELL. In the meantime, there is a pressing need to serve ELL students better, especially in mathematics. There is also a growing understanding among educational practitioners of the instructional needs of such students and how to address these needs. The MELL initiative is an attempt to connect this emerging understanding to the pressing need in a format that can be readily communicated. Much, perhaps most, of this framework is comprised of elements of effective instruction appropriate for all students, and clearly students would be well-served by these suggested practices, regardless of their language proficiency. It appears, however, from our investigations, that the success of ELL students is more highly dependent on receiving instruction geared to their specific needs. In other words, while many students who are not experiencing a language barrier might be able to experience success with less than optimal instructional practices, few ELL students can thrive in such an environment. For this reason, creating a rich classroom experience for ELL students is not simply desirable, but rather is necessary if they are to have a chance to succeed. The MELL Framework is targeted at achieving this goal.

1. Learning Atmosphere & Physical Environment

1.1. A caring classroom atmosphere of mutual respect and support is facilitated by the teacher who: 1.1.1 Knows each child as an individual, 1.1.2 Embraces languages, customs, and cultures of ELL students, 1.1.3 Provides culturally rich learning materials, 1.1.4 Encourages self-expression and provides positive recognition, 1.1.5 Builds student confidence and esteem, 1.1.6 Fosters an emotionally safe environment that allows students to feel

secure and to take risks.

1.2. The classroom is visually rich to support student learning. 1.2.1 Incorporates displays of student produced work, whenever possible, 1.2.2 Is colorful and thought stimulating, 1.2.3 Contains pertinent, real-world information and applications, 1.2.4 Reinforces math-specific vocabulary and concepts, 1.2.5 Provides color-coded learning supports when appropriate.

1.3. Room arrangement facilitates student interaction and group work.

7 - 7

2. Instructional Practices

2.1. Instructional practices foster cooperation and collaboration.

2.2. Concepts are presented accurately, logically, and in engaging ways.

2.3. Multiple representations incorporate mathematics learning levels: concrete, semi-concrete, and abstract.

2.4. The teacher employs student-centered instructional practices. 2.4.1 Approaches content from a concept-oriented constructivist method, 2.4.2 Surrounds students with different modalities (e.g., aural, visual,

kinesthetic), 2.4.3 Connects new concepts to prior learning, 2.4.4 Encourages students to refine and reflect about their own work and

verbalize concept understanding “in their own words”, 2.4.5 Chooses homework to optimize individual content development, 2.4.6 Provides extra help and resources on an individual basis.

2.5. Students are frequently partnered with peer learners to enhance learning opportunities. 2.5.1 To develop math content, 2.5.2 To aid English language development, 2.5.3 To insure sustained active participation in the class, 2.5.4 To welcome new students into an established learning community.

2.6. Instructional activities are varied and support diverse learning styles and multiple intelligences, including for instance: 2.6.1 Frequent use of models, 2.6.2 Music as a motivator and anchor, 2.6.3 Mind maps, poster-walks, and word walls 2.6.4 Key vocabulary and cognates presented in different forms, 2.6.5 Vivid adjectives.

3. Mathematics Content & Curriculum

3.1. Glossary of mathematical terms is always available for reference.

3.2. Content is aligned to appropriate grade-level, mathematics TEKS and professional standards.

3.3. Content is based on diagnosed student needs.

3.4. Content is systematically designed to incorporate sound learning principles. 3.4.1 To incorporate increased complexity, 3.4.2 To present a cohesive big-picture through chunking, 3.4.3 To connect concepts through bridging and scaffolding,

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3.4.4 To emphasize multidisciplinary understandings, 3.4.5 To reflect on inherent patterns by comparing and contrasting concepts.

3.5. Curriculum is challenging, relevant, age-appropriate, and well-paced 3.5.1 To include contextually-based problems, 3.5.2 To incorporate student realities, 3.5.3 To involve interactive problem solving.

4. Language Practices

4.1. Language support is offered without supplanting English instruction.

4.2. Support is aligned with student’s diagnosed language needs.

4.3. Language used is appropriate to age and grade level and presented in a socially meaningful context.

4.4. Mathematics-specific vocabulary is explicitly and implicitly taught and reinforced through repetition.

4.5. Teachers are knowledgeable about the second language acquisition theories and best practices embodied in Texas Administrative Code, Title 19, Part II, Chapter 128.

4.6. Ideally, dual language instructional support should be offered.

4.7. When dual language teachers are not available, sheltered instruction should be utilized to provide strong language support by addressing content through ESL.

5. Family & Community Involvement

5.1. Schools connect to student’s family-life by embedding contextual experiences and skills in teaching and curriculum.

5.2. Projects are relevant and promote family interaction.

5.3. Opportunities are available for English-speaking higher grade-level students to mentor ELL lower grade-level students either in an in-school or after-school program, as appropriate.

5.4. Teacher engages in frequent communication with families 5.4.1 About activities and events in which parents can participate, 5.4.2 About student progress.

5.5. Teacher utilizes services provided by a community liaison and is knowledgeable about community resources.

5.6. Parents are informed about the benefits of using their most cognitively advanced language at home.

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6. Assessment of Student Learning

6.1. Classroom assessment is designed to foster student success.

6.2. Assessment methods allow students frequent opportunities to demonstrate mastery in a variety of ways.

6.3. Various assessment techniques are used to measure student understandings.

6.4. Grades are oriented to promote and emphasize valid step-by-step logical reasoning processes.

6.5. Assessment data and results shape instructional planning.

6.6. Flexible time allotments are given to demonstrate concept mastery.

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Teachers Guide to Teaching Mathematics for English Language

Learners

Teachers Guide created by the SHSU MELL group, November, 2005, in collaboration with the Texas State University System and the Texas Education

Agency.

Dr. Bill Jasper, Project Leader ([email protected])

Daniel Reeves Center for Mathematics Education

Dr. John Huber, Director

([email protected])

Sam Houston State University

Mathematics for English Language Learner (MELL) Group

Dr. Bill Jasper, Dr. John Huber, Dr. Sylvia Taube, Dr. Felicia Taylor, Dr. Mark Klespis, Dr. Mary Swarthout, Carolyn Long, Eldaa Avalos, Lolita Gerardo, Susan Bohan

November 2005

A Texas State University System (TSUS) and

Texas Education Agency (TEA) Collaborative

Applicable Website: www.tsusmell.org

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TABLE OF CONTENTS SECTION 1 – PURPOSE AND STRUCTURE OF THE Page 2 TEACHERS GUIDE

The purpose and goals of this teachers guide are stated in this section, along with comments on the format of the teachers guide. SECTION 2 – BACKGROUND Page 4

Understanding the problems that English Language Learners sometimes encounter when learning mathematics in their second language. SECTION 3 – GENERAL TEACHING STRATEGIES Page 8

Strategies that include research-based “what works” teacher actions for English Language Learners as well as for all students who often struggle when learning mathematics concepts. Strategies focus on the learning of mathematics by Hispanic students. SECTION 4 – TEACHING MATHEMATICS CONCEPTS Page 16 ON THE EXIT-LEVEL TAKS TEST

Detailed strategies for teaching mathematics concepts that are tested on the Grade 11 Exit Level TAKS Mathematics Test for English Language Learners. Using each of the 10 mathematics concept objectives from the exit test as focus items, this section includes necessary mathematics vocabulary, specific teaching strategies, examples of performance tasks and projects, and appropriate assessment methods for English Language Learners. Links to other valuable statewide resources are included. APPENDIX A – MELL CLASSROOM PRACTICES FRAMEWORK Page 92

Covers the background for the MELL project and provides the conceptual framework for project activities.

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SECTION 4 -TEACHING MATHEMATICS CONCEPTS ON THE EXIT-LEVEL TAKS TEST

This section contains the specific applications of the previous strategies (see Sections 1 and 3) to the mathematics content of the Texas Grade 11 Exit Level TAKS Mathematics Test for English language learners (ELLs). Using each of the mathematics TAKS content objectives for the exit test as focus items, this section includes necessary mathematics vocabulary, specific teaching strategies, examples of performance tasks and projects, and appropriate assessment methods for ELLs. To access this information booklet on the Internet, go to www.tea.state.tx.us/student.assessment/taks/booklets/index.html and click on the exit level test.

The Grade 11 Exit Level TAKS Mathematics Test information booklet contains ten TAKS objectives. These objectives state that the student will be able to:

1. describe functional relationships in a variety of ways; 2. demonstrate an understanding of the properties and attributes of functions; 3. demonstrate an understanding of linear functions; 4. formulate and use linear equations and inequalities; 5. demonstrate an understanding of quadratic and other nonlinear functions; 6. demonstrate an understanding of geometric relationships and spatial reasoning; 7. demonstrate an understanding of two-and three-dimensional representations of

geometric relationships and shapes; 8. demonstrate an understanding of the concepts and uses of measurement and

similarity; 9. demonstrate an understanding of percents, proportional relationships; probability,

and statistics in application problems; and 10. demonstrate an understanding of the mathematical processes and tools used in

problem solving. The following information is provided for each of the above objectives:

Mathematics Content: A mathematical development of the topic and how it relates to previous learning and future learning in mathematics.

Mathematics Vocabulary: A list of appropriate English-Spanish mathematical terms and their meaning, and strategies for developing this vocabulary.

Teaching Strategies: A variety of appropriate teaching strategies for the mathematical content with an emphasis on good practices for English language learners and good strategies for teaching mathematics.

Assessment of Mathematics: Examples of performance tasks and/or projects to connect the mathematics content to contextual experiences and family-life of the English language learner.

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OBJECTIVE 9 (pp. 77-83)

DEMONSTRATE AN UNDERSTANDING OF PERCENTS, PROPORTIONAL RELATIONSHIPS, PROBABILITY, AND STATISTICS IN APPLICATION PROBLEMS

Mathematics content:

Prior mathematics knowledge requirements:

1 represent relationships between numbers as fractions, percents, and decimals 2 convert numbers from one form to another (fractions, ratios, percents, and

decimals) 3 identify, set up, and solve proportional relationships 4 list possible outcomes for probability problems 5 find measures of central tendency (mean, median, mode) 6 use appropriate terms to describe probabilities of events 7 represent and interpret data using circle graphs, bar graphs, and histograms Teachers need to prepare students to become informed consumers who can describe data and interpret statistical information. Students need to determine percents, predict results of probability experiments, account for all possible outcomes of a given situation, and to apply proportional reasoning in everyday problems. Objective 9 logically follows Objective 8, where proportional reasoning was necessary to solve similarity problems. Further applications of mathematical thinking and problem solving will be covered in Objective 10.

Minimum mathematics vocabulary needed for Objective 9: Strategies for learning this vocabulary:

English term Spanish term Description/meaning Drawing/example dependent event

evento dependiente eh-ben’-to deh-pehn-dee-ehn’-te

in probability, an event that depends on some previous outcome

pick a card from a deck of playing cards, do not replace it, and then pick another card. The pick of the second card is dependent on which card was picked on the first draw

event evento eh-ben’-to

a subset of a sample space

rolling a die and getting a 4 is an event

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English term Spanish term Description/meaning Drawing/example experimental probability

probabilidad experimental proh-bah-bee-lee-dahd ehks-peh-ree-mehn-tahl

the probability of an event determined by observation or measurement

toss a coin 100 times and you had 60 heads and 40 tails. The experimental probability for getting a head is 60/100 or 3/5

independent event

evento independiente eh-ben’to een-deh-pehn-dee-ehn’-te

in probability, an event that does not depend on some previous outcome

pick a card from a deck of playing cards, replace it , and then pick another card. The pick of the second card does not depend on which card was picked on the first draw

mean media meh-dee-ah

the calculated average of a set of numbers, found by adding all of the numbers in the set together and dividing by the number of numbers in the set

for the data set {80, 76, 89, 91} the mean is (80 + 76 + 89 + 91) 4 336/4 = 84

median mediana meh-dee-ah-nah

the middle value (or the average of the middle two values) of a set of data arranged in numerical order

for the data set {2, 3, 4, 6, 8, 10, 15}, the median is 6

mode moda moh-thah in a data set, the number or element that occurs most often.

for the data set {5, 8, 6, 7, 8, 1, 4, 8}, the mode is 8

possible outcome

resultado posible reh-sool-tah’-tho poh-see’-bleh

one of the elements or events in the sample space.

if you pull a card from a deck of 52 cards, an ace of spades is a possible outcome.

probability probabilidad proh-bah-bee-lee-dahd’

a measure of the chance or likelihood of an event to occur.

the probability of drawing a heart out of a standard deck of playing cards is 13/52 or 1/4

proportion proporci—n proh-pohr-seeohn’

an equation that states that two ratios are equal

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English term Spanish term Description/meaning Drawing/example ratio relaci—n —

raz—n reh-lah-seeohn’ oh rah-sohn’

a comparison of two or more quantities.

sales tax impuesto eem-poo-ehs’-toh

an amount added to the cost of items, calculated as a percentage of the cost

a sales tax of 8.5% on a $50 purchase would be $4.25 (50 x 0.085)

sample space espacio muestral ehs-pah’-seeoh moo-ehs-trahl’

the set of all possible outcomes for an experiment.

the sample space for tossing a 6-face die is S = {1, 2, 3, 4, 5, 6}

similar figures figuras similares fee-goo’-rahs see-mee-lah’-rehs

figures that have the same shape but may have different sizes. The lengths of corresponding sides of similar figures are proportional

theoretical probability

probabilidad te—rica proh-bah-bee-lee-dahd’ teh-oh’-ree-cah

the probability of an event determined by the ratio of the number of favorable outcomes to the number of possible outcomes in the sample space

when a coin is tossed, the theoretical probability of getting a head is 1/2

1 write definitions in everyday language while still following correct mathematics; 2 use previously defined or common words in definitions and explanations; 3 have students develop self-made glossaries of new vocabulary in journals, picture

cards, or charts; 4 as new vocabulary is introduced, add words and definitions with

illustrations/explanations to classroom word wall; 5 repeatedly connect the words to mathematical symbols and examples; 6 tape record mathematical words, definitions and verbal examples, for students to

play back when needed for extra support; and 7 examine words from Greek and Latin prefixes, roots, and suffixes. Teaching strategies and examples for Objective 9

1. identify proportional relationships and use them to solve problems.

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Example: An antique rectangular table has a width of 60 inches and a length of 72 inches as shown below. Raul wanted to make an exact scale model of the table and he wants the scale model to fit into his hand. What would be a good size (width and length) for the table to be for the scale model? Justify your answer.

2. find probabilities of compound events.

Example: Aar—n wants to go out with his friends on Friday night, and he estimates that he has a 50% chance that his parents will let him. If he does get to go out, there is a probability of 1/3 that his friends will go to the mall. What is the overall probability that Aar—n will end up at the mall?

3. have students use theoretical probabilities and experimental results to make predictions and decisions.

Example: Ana has a six-sided cube that has sides that are painted red, green or yellow. She tosses the cube 50 times and records 28 reds, 15 greens and 7 yellows. She gives these results to her teacher (who can’t see the cube) and asks her to predict how many sides on her colored cube are red, green, and yellow. What should her teacher say? Why?

4. select an appropriate measure of central tendency to describe a set of data.

Example: Create a data set with at least 10 homework grades so that the mean is 85 and the median is 80. Explain how you completed this problem. Do you think that your teacher will use the mean or the median to determine your homework grade? Why?

5. construct circle graphs, bar graphs, and histograms, with and without technology.

Example: Carlos makes $120 per week working at McDonalds. Even though his parents provide him with a room and food at home, he must pay for all of his other expenses out of his salary. Develop a reasonable monthly budget for Carlos, including gasoline for his pickup truck (his parents pay for repairs and other car costs), clothing, food (lunches, hamburgers, snacks, sodas, etc.), music CD’s, movie theater tickets, girl friend expenses for dates and gifts, other entertainment, and other miscellaneous expenses. Represent each category of expenses on both a circle graph and a bar graph. The highest grades on this assignment will be awarded to students who create their graphs on a computer or graphing calculator (both will be available in our classroom before and after school, if needed).

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6. recognize misuses of graphical or numerical information and evaluate predictions and conclusions based on data analysis.

Example: Diego and Geraldo are working on a project for their mathematics class, which requires them to find statistical errors in newspapers, magazines, or on Internet reports. For each of the following, decide whether there is a statistical error or not, and justify your response.

a. On a line graph, the horizontal axis has increments of 10, while the vertical axis has increments of 100.

b. A circle graph shows percentages of different types of restaurants in town, with four sections for Mexican (22%), Italian (18%), Seafood 11%, and Steakhouses 24%.

c. A line graph starts at 0 on the horizontal axis, and starts at 1000 on the vertical axis.

d. A histogram showing ages of students in the high school has intervals (such as 15 to 16.9 years old) on the horizontal axis.

e. A picture graph shows the average number of cans of soda bought by students aged 14 to 18 in 1995 and 2005. The average doubled, so the picture of the can of soda representing 2005 is twice as tall and twice as wide as the picture of the can representing 1995.

Assessment for Objective 9

General strategies for assessment:

1 allow students frequent opportunities to demonstrate mastery in a variety of ways;

2 provide sufficient time for ELL students to complete assessment tasks; 3 use assessment results to design instructional planning forremediation if needed; 4 assign projects for students to work together with their partners; 5 have students write their thoughts and problem-solving actions in a journal; 6 design performance measures with visuals to check conceptunderstanding; 7 design assessments to measure mathematical understanding, not reading

comprehension; 8 ensure assignments are as free of bias as possible; and 9 make assignments that require writing explanations in English.

Specific examples for assessment

1. Task students to design a probability experiment, using playing cards, dice, coins or chips of different colors. The experiment must involve 50 trials. Compute the theoretical probability for the experiment, and then have your partner or family member perform the experiment. Compare and discuss the experimental results with the theoretical probability. Were the experimental results surprising? How likely were the results from the experiment? How do you account for any differences between the theoretical and the experimental probabilities from the

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experiment? Write a report of your findings and conclusions, and discuss any changes you would make to the design of the experiment before you would perform it again.

2. Use traditional assessment methods, including multiple-choice questions, to

measure mathematics understanding. Students need to practice solving mathematics problems in the same format of the TAKS test questions. When discussing these problems in class, have students analyze why one answer is correct and the others are incorrect. A sample problem could be:

A data set gives the blood pressure readings for 12 people, where the mean is 140. One of the 12 people had a reading of 180. If the person with this reading is eliminated from the group, which of the following will decrease?

a. The mode b. The median c. The mean d. It cannot be determined from the information given

Additional problems can be found on the Texas Education Agency (TEA) website (www.tea.state.tx.us) from the TAKS information booklets (www.tea.state.tx.us/student.assessment/taks/booklets) and from TAKS released tests (www.tea.state.tx.us/student.assessment/resources/release/taks/index.html). Also on the TEA website, there is a link to the TAKS Study Guide for Grade 11 Exit Level Mathematics and Science: A Student and Family Guide, which explains the key concepts under each objective and gives examples (see www.tea.state.tx.us/student.assessment/resources/guides/study/index.html). There are additional multiple-choice problems for each objective in this guide. Although it is not designed especially for ELL students, it is a very helpful resource in preparing to take the TAKS test.

3. Design projects that involve the families of students. For example: have students

ask their family members to predict how many minutes of a half-hour television program will be advertisements. After recording predictions for all family members, the family will need to watch a half-hour program (on a channel such as NBC or Telemundo if available) and record the number of seconds for every commercial or advertisement. Next, the family will need to watch a half-hour local news program, and again record the number of seconds for each commercial or advertisement. Students and their families will then calculate the percentage of time for commercials and the actual program, and compare the results. A short paper will be submitted describing this experiment and the results, as well as comparing the results with the original predictions and contrasting the two types of programs.

4. Let students work with a partner in class. Provide extra time for students to talk

together about an assigned mathematics problem, decide how to approach it,

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and make a summary paper on the problem. Provide a format sheet for their mini-reports, such as a) what is the problem, b) what do we need to find out, c) how do we get started, c) how do we solve the problem, d) what is our solution, and e) how can I describe what we did? Allow time for both partners to discuss their report together, before presenting it to the class.

5. Make sure that all students have the resources available to accomplish every

assignment. For example, do not assign projects that involve working on the Internet as a homework assignment, since not every student has access to a computer at home. If you want students to measure something, provide them with rulers or the tools to do the measuring. If you assign a project that requires the use of graphing calculators, provide the calculators and make it an in-class assignment, since many students will not have access to graphing calculators outside the classroom.

Teachers Guide created by the SHSU MELL group, November, 2005, in collaboration with the Texas State University System and the Texas Education Agency.

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MELL Products and Resources

For Practitioners

The following practitioner products were developed in 2004-05 by the TSUS Math for English Language Learners (MELL) Initiative and can be accessed through the web addresses listed below. Review of the Literature: Mathematics and English Language Learners Research and literature on Mathematics and English Language Learners have been analyzed and synthesized in this document. Areas of consensus and discrepancies within the literature and research are discussed, as well as gaps and directions for future research. A Summary of Mathematics Professional Development Models Used in the State of Texas. This report identifies twelve models of professional development in the area of mathematics that are prevalent in the state of Texas. Each model is summarized in a consistent fashion to facilitate a comparison between the various models. Each summary includes two main sections: an Overview and an Analysis. This product is especially relevant to math facilitators and staff development personnel. MELL TAKS Analysis Tool This product is primarily for researchers and educational evaluation personnel who are familiar with SPSS (Statistical Package for the Social Sciences). TAKS data have been compiled into a format that can be easily and quickly manipulated with SPSS to generate detailed within -district, across- district , regional and state comparisons. MELL Classroom Practices Framework The development of this product was guided by the question of “What do the findings of the Phase I MELL investigations (those listed above) suggest in regard to classroom practices that contribute to successful math instruction for English Language Learners?” The framework is organized into six components, each of which is comprised of series of enabling instructional approaches that support student success.

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An Analysis of Existing Professional Development To Support Mathematics for English Language Learners (MELL) Using the MELL Classroom Practices Framework. This product consists of a secondary analysis of the twelve MELL-identified math professional development models using the six components of the MELL Classroom Practices Framework. Math facilitators and staff developers can use this product to select specific professional development when targeting various components of classroom practice. A Compilation of Survey Data and Focus Group Findings: Texas Secondary Mathematics Teachers’ Views of Needed Professional Development to Support Mathematics Instruction for English Language Learners This product summarizes the perceptions of secondary mathematics teachers regarding their professional development experiences and needed professional development to support the math instruction of English Language Learners. Data were gathered through surveys and focus groups conducted throughout the state during Spring 2005. Teachers Guide to Teaching Mathematics for English Language Learners The teachers’ guide is designed to help teachers make mathematics more meaningful and understandable for the English Language Learners (ELLs). The initial focus of this guide is for grades7-11 and includes general teaching strategies that help the ELL when learning mathematics content. There is a detailed plan for teaching the ten mathematics objectives covered on the 11th grade TAKS Test. MELL Quick Start Module The MELL Quick Start Module is a professional development tool designed to explore the issues surrounding the teaching of mathematics to English Language Learners and strategies to enhance learning. The module parallels the MELL Classroom Practices Framework and consists of various professional readings, activities, and resources. The module is offered for graduate credit through Sul Ross State University and a non-credit version is available through the MELL website (www.tsusmell.org). The MELL products will be permanently housed on the MELL website at www.tsusmell.org and are temporarily housed at http://www.education.txstate.edu/epic/index.htm

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