Mathematics Curriculum Guidelines · 2020. 9. 25. · Mathematics Curriculum Guidelines and Common...

Diocese of Metuchen

Mathematics Curriculum Guidelines

Kindergarten – Grade 8

DIOCESE OF METUCHEN

ELLEN F. AYOUB SUPERINTENDENT OF SCHOOLS

TEL: 732.562.2443 FAX: 732.562.1016

September 2013 Dear Colleague, Catholic schools over the years have been recognized as a vital tool in the teaching mission of the Church. Throughout our history, our schools have been called upon to provide an excellent academic program within a faith-filled setting. An important component of our success has been our commitment to consistently provide the tools for educators to use as they guide each student both spiritually and academically. Elementary Mathematics Curriculum Guidelines is just such a tool. The Common Core Standards are reflected in the curriculum guide, enabling students to gain a deeper, richer understanding of mathematical concepts as they progress in their learning. Additionally, this guide is designed to assist educators in creating lessons that will truly represent the world in which our students live by integrating 21st century technology throughout the curriculum. The Office of Schools of the Diocese of Metuchen is most grateful to the Archdiocese of Newark for allowing us to use the curriculum guidelines they created as the basis for the document you have received. The collaboration among the Catholic Schools Offices in the dioceses of New Jersey provides the opportunity for us to adapt materials that are created and tailor them to the needs of our students. What a great gift that is for all of the students in our state. We also wish to thank the Archdiocese of Philadelphia for the extended resources that they provided. It is our prayer that the Elementary Mathematics Curriculum Guidelines will truly be a resource for each person as we enhance the educational experience of each of the students entrusted to us. Sincerely,

Ellen F. Ayoub

__________________________________________________________________________________________________________________

The St. John Neumann Pastoral Center P.O. Box 191 Metuchen, New Jersey 08840-0191 (732) 562-1990

TABLE OF CONTENTS

General Content

Vision and Goals……………………………...

Use of Technology……………………………

Diocesan Mathematics Curriculum Guidelines

Overview of Cluster Objectives………………

Standards for Mathematical Practice………….

The Language of Mathematics………………..

Standards

Kindergarten ………………………………….

Grade One……………………………………..

Grade Two…………………………………….

Grade Three…………………………………...

Grade Four…………………………………….

Grade Five…………………………………….

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1

2-4

5

6-11

12

13

1 – 25

1 – 32

1 – 43

1 – 39

1 – 40

1 – 45

Standards

Grade Six………………………………..........

Grade Seven……………………………..........

Grade Eight……………………………...........

Accelerated Grade 7……………………..........

Accelerated Grade 8 - Algebra I……………………………..

Appendixes

Appendix A: Glossary

Appendix B: Internet Resources

Page

1 – 43

1 – 37

1 – 33

1 – 21

1 – 22

Mathematics Education in the Elementary Schools in the Diocese of Metuchen

Vision Statement

A rigorous and coherent mathematics program will ensure that students acquire the mathematical skills and understanding that will enable

them to function competently and ethically in the 21st century. Through the study of mathematics students will develop an appreciation of

the order and unity of God’s creation and the relevance of mathematics in their daily lives. Students will be able to flexibly use

mathematical concepts and procedures to complete daily tasks, reason mathematically, and analyze and respond to various situations in

logical and innovative ways.

Goals

All students will:

1) value mathematics and appreciate its relevance in daily life.

2) recognize the relationship between mathematical ideas and apply them to real-life situations and other areas of the curriculum.

3) exhibit confidence and competence in their application of mathematical skills and knowledge.

4) use mathematical skills and understanding flexibly to solve problems.

5) reason and communicate mathematically.

6) apply ethical principles when formulating solutions to problems and communicating mathematical relationships.

7) use technology mindfully and responsibly to deepen understanding of mathematical concepts and facilitate mathematical processes.

Transforming Learning Environments with Technology

Reprinted with permission from National Educational Technology Standards for Students, Second Edition, ©2007, ISTE®

(International Society for Technology in Education), www.iste.org. All rights reserved.

Today’s educators must provide a learning environment that takes students beyond the walls of their classrooms and into a world of endless

opportunities. Technology standards promote this classroom transformation by ensuring that digital-age students are empowered to learn, live,

and work successfully today and tomorrow.

National Educational Technology Standards for Students 2007 (NETS)

1. Creativity and Innovation

Students demonstrate creative thinking, construct knowledge, and develop innovative products and processes using technology.

Students:

a. apply existing knowledge to generate new ideas, products, or processes.

b. create original works as a means of personal or group expression.

c. use models and simulations to explore complex systems and issues.

d. identify trends and forecast possibilities.

2. Communication and Collaboration

Students use digital media and environments to communicate and work collaboratively, including at a distance, to support individual

learning and contribute to the learning of others.

Students:

a. interact, collaborate, and publish with peers, experts, or others employing a variety of digital environments and media.

b. communicate information and ideas effectively to multiple audiences using a variety of media and formats.

c. develop cultural understanding and global awareness by engaging with learners of other cultures.

d. contribute to project teams to produce original works or solve problems.

http://www.iste.org/

3. Research and Information Fluency

Students apply digital tools to gather, evaluate, and use information.

Students:

a. plan strategies to guide inquiry.

b. locate, organize, analyze, evaluate, synthesize, and ethically use information from a variety of sources and media.

c. evaluate and select information sources and digital tools based on the appropriateness to specific tasks.

d. process data and report results.

4. Critical Thinking, Problem Solving, and Decision Making

Students use critical thinking skills to plan and conduct research, manage projects, solve problems, and make informed decisions

using appropriate digital tools and resources.

Students:

a. identify and define authentic problems and significant questions for investigation.

b. plan and manage activities to develop a solution or complete a project.

c. collect and analyze data to identify solutions and/or make informed decisions.

d. use multiple processes and diverse perspectives to explore alternative solutions.

5. Digital Citizenship

Students understand human, cultural, and societal issues related to technology and practice legal and ethical behavior.

Students:

a. advocate and practice safe, legal, and responsible use of information and technology.

b. exhibit a positive attitude toward using technology that supports collaboration, learning, and productivity.

c. demonstrate personal responsibility for lifelong learning.

d. exhibit leadership for digital citizenship.

6. Technology Operations and Concepts

Students demonstrate a sound understanding of technology concepts, systems, and operations.

Students:

a. understand and use technology systems.

b. select and use applications effectively and productively.

c. troubleshoot systems and applications.

d. transfer current knowledge to learning of new technologies.

Calculator Use in Elementary Grades A Position of the National Council of Teachers of Mathematics

July 2011

NCTM Position

Calculators have an important role in supporting and advancing elementary mathematics learning. The benefits of their selective

and strategic use are twofold. Calculators can promote the higher-order thinking and reasoning needed for problem solving in our

information- and technology-based society, and they can also increase students’ understanding of and fluency with arithmetic

operations, algorithms, and numerical relationships.

Although calculators—from simple four-function versions to programmable graphing models—are used routinely outside school for a variety

of purposes, their specific use within the mathematics classroom must be selective and strategic, with attention to how such a tool will support

and advance learning. More important, the use of calculators does not supplant the need for students to develop proficiency with efficient,

accurate methods of mental and pencil-and-paper calculation and in making reasonable estimations. (NCTM, 2011)

Diocesan Mathematics Curriculum Guidelines

and

Common Core State Standards

The Diocesan Mathematics Curriculum Guidelines for Elementary Schools are based on the Common Core State Standards (CCSS) which

were released in June 2010 and have been adopted by more than 45 states, including the state of New Jersey. The Common Core State

Standards are the results of a national effort led by the National Governors Association Center for Best Practices and the Council of Chief

State School Officers to establish consistent standards for what students should know and be able to do by the time they graduate from high

school.

The standards outline the content of the mathematics curriculum by stating what students should know and be able to do by the time they

complete each grade level. The content standards are arranged by ―domains‖ with ―clusters‖ of related standards within each domain. The

charts on the following pages are designed to provide an overview. The specific standards for moving students towards the ―cluster objective‖

can be found in the guidelines for each grade level, along with suggestions for strategies and assessments. The Standards for Mathematical

Practice are provided at the beginning of the guidelines for each grade level. The content standards identify the skills and knowledge teachers

should ensure students are learning. The practice standards indicate the practices and habits of mathematical thinking that teachers need to

model for and expect of their students in order to facilitate deeper levels of understanding.

The domains into which the standards for Grades K-5 are arranged differ from the domains for Grades 6-8. Therefore, the first chart provides

an overview for Grades K-5 and the second chart provides an overview for Grades 6-8. All teachers should be aware of the progression of

knowledge and skills from Kindergarten through Grade 8 and work collaboratively to develop a rigorous and coherent mathematics

curriculum.

Overview of Cluster Objectives for Kindergarten through Grade 5

Domain: Counting and Cardinality (CC)

Kindergarten Grade 1 Grade 2 Cluster Objectives:

Know number names and

the count sequence

Count to tell the number

of objects

Compare numbers

Cluster Objectives:

Continue to develop and

extend concepts taught at

previous grade levels

Cluster Objectives:

Continue to develop and

extend concepts taught at

previous grade levels

Domain: Operations and Algebraic Thinking (OA)

Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Grade 5 Cluster Objectives:

Understand addition as

putting together and

adding to, and understand

subtraction as taking apart

and taking from

Cluster Objectives:

Represent and solve

problems involving

addition and subtraction

Understand and apply

properties of operations

and the relationship

between addition and

subtraction

Add and subtract within

20

Work with addition and

subtraction equations

Cluster Objectives:

Represent and solve

problems involving


Add and subtract within

20 (Know from memory

all sums of two 1-digit

numbers)

Work with equal groups

of objects to gain

foundations for

multiplication

Cluster Objectives:

Represent and solve

problems involving

multiplication and

division

Understand properties of

multiplication and the

relationship between

multiplication and

division

Multiply and divide

within 100

Solve problems involving

the four operations and

explain patterns in

arithmetic

Cluster Objectives:

Use the four operations

with whole numbers to

solve problems

Gain familiarity with

factors and multiples

Generate and analyze

patterns

Cluster Objectives:

Write and interpret

numerical expressions

Analyze patterns and

relationships

Domain: Number and Operations in Base Ten (NBT)

Kindergarten Grade1 Grade 2 Grade 3 Grade 4 Grade 5 Cluster Objectives:

Work with numbers 11-19

to gain foundations for

place value

Cluster Objectives:

Extend the counting

sequence

Understand place value

(2 digit numbers)

Use place value

understanding and


to add and subtract

Cluster Objectives:

Understand place value

(3 digit numbers)

Use place value

understanding and


to add and subtract

Cluster Objectives:

Use place value

understanding and


to perform multi-digit

arithmetic

Cluster Objectives:

Generalize place value

understanding for multi-

digit whole numbers

Use place value

understanding and


to perform multi-digit

arithmetic

Cluster Objectives:

Understand the place

value system

Perform operations with

multi-digit whole

numbers and with

decimals to hundredths

Domain: Number and Operations – Fractions (NF)


Domain not included at

this grade level

Cluster Objectives:


this grade level

Cluster Objectives:


this grade level

Cluster Objectives:

Develop understanding of

fractions as numbers

Cluster Objectives:

Extend understanding of

fraction equivalence and

ordering

Build fractions from unit

fractions by applying and

extending previous

understandings of

operations on whole

numbers

Understand decimal

notation for fractions, and

compare decimal

fractions

Cluster Objectives:

Use equivalent fractions

as a strategy to add and

subtract fractions

Apply and extend

previous understandings

of multiplication and

division to multiply and

divide fractions

Domain: Measurement and Data (MD)


Describe and compare

measurable attributes

Classify objects and count

the number of objects in

categories

Cluster Objectives:

Measure lengths

indirectly and by iterating

length units

Tell and write time

Collect, organize, display

and interpret data

Cluster Objectives:

Measure and estimate

lengths in standard units

Relate addition and

subtraction to length

Work with time and

money

Cluster Objectives:


measurement and

estimation of liquid

volumes, masses of

objects and intervals

of time

Represent and interpret

data

Geometric measurement:

Understand concepts of

area and relate area to

multiplication and

addition

Recognize perimeter as

an attribute of plane

figures and distinguish

between linear and area

measures

Cluster Objectives:


measurement and

conversion of

measurements from a

larger unit to a smaller

unit

Represent and interpret

data



angles and angle

measurement

Cluster Objectives:

Convert like

measurement units within

a given measurement

system

Represent, interpret and

analyze data



volume and relate

volume to multiplication

and addition

Domain: Geometry (G)


Identify and describe two-

dimensional and three-

dimensional shapes

Analyze, compare,

create, and compose

shapes

Cluster Objectives:

Reason with shapes and

their attributes

Cluster Objectives:


their attributes

Cluster Objectives:


their attributes

Cluster Objectives:

Draw and identify lines

and angles, and classify

shapes by properties of

their lines and angles

Cluster Objectives:

Graph points on the

coordinate plane to solve

real-world and

mathematical problems

Classify two-dimensional

figures into categories

based on their properties

Overview of Cluster Objectives for Grade 6 through Grade 8

Domain: Ratios and Proportional Relationships (RP)

Grade 6 Grade 7 Grade 8 Cluster Objectives:

Understand ratio concepts and use ratio reasoning to

solve problems

Cluster Objectives:

Analyze proportional relationships and use them to

solve real-world and mathematical problems

Cluster Objectives:

Domain: The Number System (NS)


Apply and extend previous understandings of

multiplication and division to divide fractions by

fractions

Compute fluently with multi-digit numbers and find

common factors and multiples

Apply and extend previous understandings of numbers

to the system of rational numbers

Cluster Objectives:


operations with fractions to add, subtract, multiply,

and divide rational numbers

Cluster Objectives:

Know that there are numbers that are not rational,

and approximate them by rational numbers

Domain: Expressions and Equations (EE)



arithmetic to algebraic expressions

Reason about and solve one-variable equations and

inequalities

Represent and analyze quantitative relationships

between dependent and independent variables

Cluster Objectives:

Use properties of operations to generate equivalent

expressions

Solve real-life and mathematical problems using

numerical and algebraic expressions and equations

Cluster Objectives:

Work with radicals and integer exponents

Understand the connections between proportional

relationships, lines, and linear equations

Analyze and solve linear equations and pairs of

simultaneous linear equation

Domain: Functions


Domain not included at this grade level

Cluster Objectives:

Domain not included at this grade level

Cluster Objectives:

Define, evaluate, and compare functions

Use functions to model relationships between

quantities

Domain: Geometry (G)


Solve real-world and mathematical problems

involving area, surface area, and volume

Cluster Objectives:

Draw, construct and describe geometrical

figures and describe the relationships between

them

Solve real-life and mathematical problems

involving angle measure, area, surface area,

and volume

Cluster Objectives:

Understand congruence and similarity using physical

models, transparencies, or geometry software

Understand and apply the Pythagorean Theorem

Solve real-world and mathematical problems

involving volume of cylinders, cones and spheres

Domain: Statistics and Probability (SP)


Develop understanding of statistical variability

Summarize and describe distributions

Cluster Objectives:

Use random sampling to draw inferences about a

population

Draw informal comparative inferences about two

populations

Investigate chance processes and develop, use,

and evaluate probability models

Cluster Objectives:

Investigate patterns of association in bi-variate data

Standards for Mathematical Practice (From the Common Core State Standards)

The Standards for Mathematical Practice describe ways in which developing students should engage with the subject matter as they grow in

mathematical maturity and expertise throughout the elementary, middle and high school years. Activities and opportunities that foster the

development of these skills and proficiencies should underlie the mathematics curriculum at all grade levels.

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

A full explanation of each of the Standards for Mathematical Practice is provided at the beginning of the curriculum guidelines for each

grade.

The Language of Mathematics The chart below identifies key terms used in mathematics instruction. The chart indicates the grade level at which students are introduced to the meaning

of the term in relation to mathematical skills and knowledge. The understanding of terms is developed as students progress from grade to grade. Terms

should be used appropriately and consistently to communicate mathematical ideas and processes. The list below is provided for general guidance and is

not intended to be a complete list of mathematical terms. Definitions for many of these terms can be found in the Common Core State Standards Glossary

provided in Appendix A.

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

= Equal to x Area x

< Less than x Array x

> Greater than x Associative Property x

¢ Cent x Attributes x

$ Dollar x Average x

2-Digit number x Axis x

2-Dimensional Figure x Balance scale x

3-Dimensional Figure x Bar Graph x

A.M. x Base (exponents) x

Absolute Value x Base (figures) x

Acute Angle x Bills (currency) x

Add x Bivariate x

Addend x Box and Whisker Plot x

Addition x Braces { } x

Addition Sentence x Brackets [ ] x

Additive Inverse x Calculate x

Adjacent x Calculator

Adjacent Angles x Calendar

Algebraic Expression x Capacity x

Algorithm x Category x

Alike x Celsius x

Analog Clock x Cent x

Angle x Center x

x

x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Centimeter x Coordinate Plane x

Chart x Coordinates x

Chord x Corner x

Circle x Corresponding x

Circle Graph x Counter Clockwise x

Circumference x Counting Numbers x

Classify x Counting On x

Clockwise x Cube x

Closed Figure x Cube Root x

Coefficient x Cubic Unit x

Coin x Curve x

Column x Cylinder x

Common Factor x Data x

Common Multiple x Decimal x

Commutative Property x Decimal Point x

Compare x Decompose x

Compass x Degree (angle measure) x

Compatible Numbers x Degrees (temperature) x

Complementary Angles x Denominator x

Complex Fractions x Dependent Variable x

Compose x Deviation x

Composite Number x Diagonal x

Compound Event x Diameter x

Compute/Computation x Difference (subtraction) x

Cone x Different x

Congruent x Digit x

Consecutive x Digital Clock x

Constant x Dilation x

Convert/Conversion x Dime x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Dimension x Factor x

Dimensional Analysis x Factoring x

Distributive Property x Fahrenheit x

Divide x Flat x

Dividend x Formula x

Divisibilty x Fourths x

Division x Fraction x

Divisor x Frequency Table x

Dollar x Function x

Domain x Function Notation f(x) x

Double x Function Table x

Edge x Graph x

Elapsed Time x Greater than/more than x

Endpoint x Greatest Common Factor x

Equal x Half/Halves x

Equal Shares x Half Circle x

Equation x Half Hour x

Equilateral Triangle x Height x

Equivalent/Equivalence x Hexagon x

Estimate x Histogram x

Evaluate x Horizontal x

Even Number x Hour x

Expanded Form x Hour Hand x

Experimental Probability x Hundreds (place value) x

Exponent (power) x Hundredths x

Expression x Hypotenuse x

Exterior Angle x Improper Fraction x

Face x Independent Variable x

Fact Family x Inequality x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Input/Output x Minus x

Integer x Minute x

Interquartile Range x Minute Hand x

Intersecting Lines x Mixed Number x

Intersection x Mode x

Interval x Money x

Inverse x Multi Digit x

Irrational Number x Multiple x

Isosceles Triangle x Multiplication x

Key x Multiply x

Laws of Exponents x Negative Number x

Least Common Multiple x Net x

Length x Nickel x

Less than x Notation x

Line x Number x

Line Graph x Number Line x

Line of Symmetry x Number Sentence x

Line Plot x Numeral x

Line Segment x Numerator x

Linear Equation x Numerical Expression x

Linear Units x Obtuse Angle x

Liquid Volume x Octagon x

Mass x Odd Number x

Maximum x Ones x

Mean x Operation x

Measure x Opposite x

Median x Order x

Metric Units x Order of Operations x

Minimum x Ordered pairs x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Ordinal Number x Prism x

Origin x Probability x

Outcome x Product x

Outlier x Proper Fraction x

P.M. x Property x

Parallel x Proportion x

Parallel Lines x Proportional Relationship x

Parallelogram x Protractor x

Parentheses x Pythagorean Theorem x

Partition x Quadrant x

Pattern x Quadratic Equation x

Penny x Quadrilateral x

Pentagon x Quantitative x

Percent x Quarter x

Perimeter x Quartile x

Perpendicular x Quotient x

Pi (π) x Radical x

Picture Graph x Radius/Radii x

Place Value x Random Sample x

Plane Figure x Range x

Plus x Rate x

Point x Ratio x

Polygon x Rational Number x

Polyhedron x Ray x

Polynomial x Reasonableness x

Positive Number x Reciprocal x

Prime Factor x Rectangle x

Prime Factorization x Rectangular Prism x

Prime Number x Reflection (Flip) x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Regroup x Sort x

Remainder x Sphere x

Rhombus x Square (shape) x

Right Angle x Square Number x

Right Triangle x Square Root x

Rotation (Turn) x Square Unit x

Round x Standard Form x

Rounding x Standard Unit x

Row x Stem and Leaf Plot x

Rule x Straight Angle x

Ruler x Straight Edge x

Sample Space x Subtract x

Scale (model, map) x Subtraction x

Scalene Triangle x Sum x

Scatter Plot x Supplementary Angles x

Scientific Notation x Surface x

Seconds x Surface Area x

Sequence x Survey x

Set x Symbol x

Shape x Symmetry x

Side x Systems of Equations x

Similar x Tally x

Simplest Form x Tangram x

Simplify x Temperature x

Slope x Tens x

Solid x Tenths x

Solid Figure x Theoretical Probability x

Solution x Thermometer x

Solve x Thirds x

K 1 2 3 4 5 6 7 8 Alg. K 1 2 3 4 5 6 7 8 Alg.

Time x Vertical x

Total x Vertical Angles x

Transformation x Volume x

Translation (Slide) x Weight x

Transversal x Whole x

Trapezoid x Whole Numbers x

Tree Diagram x Width x

Triangle Word Form x

Unit x x-axis (horizontal axis) x

Unit Fraction x x-coordinate x

Unit Rate x x-intercept x

Units of Measurement x y-axis (vertical axis) x

Unknown x y-coordinate x

Value x y-intercept x

Variable x Zero x

Venn Diagram x Zero Property of Addition x

Vertex x Zero Property of Multiplication x


Kindergarten

Kindergarten

Page 1

Standards for Mathematical Practice (from the Common Core State Standards)




1) Make sense of problems and persevere in solving them.Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a

solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler

forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if

necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on

their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations,

verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or

trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically

proficient students check their answers to problems using a different method, and they continually ask themselves, ―Does this make sense?‖

They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2) Reason abstractly and quantitatively.Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary

abilities to bear on problems involving quantitative relationships: the ability to de-contextualize in order to abstract a given situation and

represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their

referents; and the ability to contextualize or pause as needed during the manipulation process in order to probe into the referents for the

symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units

involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of

operations and objects.

Kindergarten

Page 2

3) Construct viable arguments and critique the reasoning of others.Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing

arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to

analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate

them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account

the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible

arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is.

Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can

make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains

to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask

useful questions to clarify or improve the arguments.

4) Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the

workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might

apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to

solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who

can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these

may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as

diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They

routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving

the model if it has not served its purpose.

5) Use appropriate tools strategically.Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and

paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic

geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about

when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically

proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by

strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable

them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient

Kindergarten

Page 3

students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and

use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6) Attend to precision.Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in

their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They

are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate

accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary

grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims

and make explicit use of definitions.

7) Look for and make use of structure.Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and

seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.

Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the

expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a

geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift

perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.

For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than

5 for any real numbers x and y.

8) Look for and express regularity in repeated reasoning.Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary

students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a

repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with

slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when

expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x

3 + x

2 + x + 1) might lead them to the general formula for the sum of a geometric

series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details.

They continually evaluate the reasonableness of their intermediate results.

Kindergarten

Page 4

Kindergarten

Introduction (from Common Core State Standards)

Instructional time in Kindergarten focuses on two critical areas: (1) representing and comparing whole numbers, initially with sets of objects;

(2) describing shapes and space. More learning time in Kindergarten should be devoted to number than to other topics.

(1) Students use numbers, including written numerals, to represent quantities and to solve quantitative problems, such as counting objects

in a set; counting out a given number of objects; comparing sets or numerals; and modeling simple joining and separating situations with

sets of objects, or eventually with equations such as 5 + 2 = 7 and 7 – 2 = 5. (Kindergarten students should see addition and subtraction

equations, and student writing of equations in kindergarten is encouraged, but it is not required.) Students choose, combine, and apply

effective strategies for answering quantitative questions, including quickly recognizing the cardinalities of small sets of objects, counting

and producing sets of given sizes, counting the number of objects in combined sets, or counting the number of objects that remain in a set

after some are taken away.

(2) Students describe their physical world using geometric ideas (e.g., shape, orientation, spatial relations) and vocabulary. They identify,

name, and describe basic two-dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons, presented in a variety of

ways (e.g., with different sizes and orientations), as well as three-dimensional shapes such as cubes, cones, cylinders, and spheres. They

use basic shapes and spatial reasoning to model objects in their environment and to construct more complex shapes.

Kindergarten

Page 5

Kindergarten Standards for Mathematical Content

Counting and Cardinality (K.CC)

Standards Strategies and Assessments

K.CC All students will know number names and the count sequence.

1. Count to 100 by ones and by tens.

2. Count forward beginning from a given number within the known sequence (instead

of having to begin at 1).

3. Recognize and write numerals from 0 to 20. Represent a number of objects with a

written numeral 0-20 (with 0 representing a count of no objects).

K.CC All students will count to tell the number of objects.

4. Demonstrate an understanding of the relationship between numbers and quantities;

connect counting to cardinality.

a. When counting objects, say the number names in the standard order, pairing

each object with one and only one number name and each number name

with one and only one object.

Mathematics instruction must include:

abundant and meaningful use of physical

and visual representations, modeling, small

group activities, integration of technology,

connections to daily experiences, and

writing about mathematics.

Count orally using the hundred chart and

number line; count from various starting

points.

Have students put a set of 1-20 number cards

in order.

Identify the number that comes before and

after a given number.

Use stories, songs, dances, rhymes and

games that incorporate number concepts.

Give each student a card with an ordinal

number and have students line up in the

correct positions.

Sort and count sets of objects (cubes,

counters, bears) to 10.

Kindergarten

Page 6


b. Understand that the last number name said tells the number of objects

counted. The number of objects is the same regardless of their arrangement

or the order in which they were counted.

c. Understand that each successive number name refers to a quantity that is one

larger.

5. a. Count to answer ―how many?‖ questions about:

o as many as 20 things arranged in a line, a rectangular array, or a circle,

o as many as 10 things in a scattered configuration

b. Count out a requested number of objects from 1-20.

Display a chart showing numbers, with their

corresponding number words and quantities.

Use 5-frames and 10-frames to help develop

initial ideas of small numbers. By filling up

a 5-cell grid with counters first, and then

putting 2 more while trying to show ―7 in

all,‖ the child learns not only ―7,‖ but also its

relationship to ―5.‖

Arrange a given number of objects in various

arrays or configurations.

Give student a card with a number from

1-20 on it and have them create a set of

objects with the given number or identify the

set with the correct number of objects.

K.CC All students will be able to compare numbers

6. Identify whether the number of objects in one group is greater than, less than,

or equal to the number of objects in another group, by using matching and counting

strategies.

7. Compare two numbers between 1 and 10 presented as written numerals.

Use dot pattern cards or dominoes to practice

more, less, and same. For example, given a

card with 6 dots on it, students use counters

to make a set that is more, another that is

less, and one that is the same.

Kindergarten

Page 7

Operations and Algebraic Thinking (K.OA)


K.OA All students will understand addition as putting together and adding to, and

understand subtraction as taking apart and taking from.

1. Represent addition and subtraction with objects, fingers, mental images, drawings,

sounds (e.g., claps) acting out situations, verbal explanations, expressions, or

equations.

2. Solve addition and subtraction word problems, and add and subtract within 10,

(e.g., by using objects or drawings to represent the problem).

3. Decompose numbers less than or equal to 10 into pairs in more than one way,

(e.g., by using objects or drawings, and record each decomposition by a drawing

or equation such as, 5 = 2 + 3 and 5 = 4 + 1).

4. For any number from 1 to 9, find the number that makes 10 when added to the given

number, (e.g., by using objects or drawings, and record the answer with a drawing or

equation.).

5. Fluently add and subtract within 5.

Use real-life situations from the classroom,

playground and home to discover the

meaning of addition and subtraction.

Listen to Adding Animals by Colin Hawkins

while counting animals and learning about

addition.

Use unifix cube towers of two colors to show

all the ways to make a number.

Example: 7=3+4, 7= 2+5, 7=0+7, and so on.

Use concrete objects to add and subtract

within 10.

Kindergarten

Page 8

Numbers and Operations in Base Ten (K.NBT)


K.NBT All students will gain a foundation for understanding place value.

1. Compose and decompose numbers from 11 to 19 into ten ones and some further

ones (e.g., by using objects or drawings, and record each composition or

decomposition by a drawing or equation such as, 18 = 10 + 8); understand that these

numbers are composed of a group of ten ones and one, two, three, four, five, six,

seven, eight or nine ones.

During daily calendar activity a popsicle

stick is added to a collection representing all

of the days of school to date. Whenever 10

single sticks are available, they are bundled

with a rubber band and are thereafter counted

as a ten. On the hundredth day of school, the

ten tens are wrapped together to make a

hundred.

Measurement and Data (K.MD)


K.MD All students will describe and compare measurable attributes.

1. Describe measurable attributes of objects, such as length or weight.

Describe several measurable attributes of a single object.

2. Directly compare two objects with a measurable attribute in common, to see

which object has ―more of‖/―less of‖ the attribute, and describe the difference.

Example: Directly compare the heights of two children and describe one child as

taller/shorter.

Create a list showing the length and weight

of various classroom objects (various

books, crayon box, piece of paper, etc.).

Have students hold an object in each hand

to determine which feels heavier. Use a

balance scale to compare weights of small

items.

Arrange several items in the classroom by

height. Classify objects into given categories; count the numbers of objects ineach category and sort the categories by count. (Limit category counts to lessthan or equal to 10.)

3.

Use colored tiles, cubes or round chips and sort by color or shape; identify which group has the most or least objects.

Kindergarten

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Geometry (K.G)


K.G All students will identify and describe shapes (squares, circles, triangles,

rectangles, hexagons, cubes, cones, cylinders, and spheres).

1. Describe objects in the environment using names of shapes, and describe the

relative positions of these objects using terms such as above, below, beside, in front

of, behind, next to, inside, and outside.

2. Correctly name shapes regardless of their orientations or overall size.

3. Identify shapes as two-dimensional (lying in a plane, ―flat‖) or three-dimensional

(―solid‖).

K.G All students will analyze, compare, create, and compose shapes.

4. Analyze and compare two- and three-dimensional shapes, in different sizes and

orientations, using informal language to describe their similarities, differences, parts

(e.g., number of sides and vertices/―corners‖) and other attributes (e.g., having sides

of equal length).

5. Model shapes in the world by building shapes from components (e.g., sticks and

clay, balls) and drawing shapes.

6. Compose simple shapes to form larger shapes. Example: “Can you join these two

triangles with full sides touching to make a rectangle?”

Place different number cards on various

items in the classroom and conduct a Shape

Hunt. Give students oral directions for

locating a shape using relative position

terms. Students identify the shape by its

number card.

Distribute models of different two and three

dimensional shapes. Have the student stand

when you name the shape they have.

Use shape models to explore and identify

parts of various shapes.

Compose a Shape Diary made up of pictures

or simple descriptions of shapes observed

during the course of a day.

Give each student three of the same size

sheets of construction paper. Have students

identify the shape. Fold one rectangle in

halves, one in thirds, and one in fourths.

Analyze and discuss results.

Kindergarten Page 10

Extended Resources – Kindergarten Mathematics

Domain: Counting and Cardinality (K.CC) - Know number names and the count sequence.

Essential Questions What should I be able to answer? What guides my thinking?

Why is counting important? How do I count forward? How do I count by ones? How do I count by tens? How do I write numbers? Do I know the difference between a number and a letter?

Assessment What will I be expected to know, understand, and be able to do in order to demonstrate my learning?

Skills What skills do I need to have in order to answer the essential questions?

Content What content do I need to know in order to answer the essential questions?

Formative:Students will count objects in a collection and know counting words. Students will count out objects for a given number. Students will write numbers to 20. Draw pictures to represent a given number. Summative:Individual assessment of counting skills presented1.Count to 100 by ones and by tens.2.Count forward beginning from a given number within the known sequence (instead of having tobegin at 1). 3.Write the numbers from 0 to 20. Represent a number of objects with a written numeral 0- 20(with 0 representing a count of no objects).

Understanding of numbers How to represent numbers Counting begins with 0


Integration of Learning How does this learning connect to my other areas (subjects) of learning?

Tools for Learning Which tools will I use that will assist me in my learning?

ELA- Children’s Literature using counting words and symbols in titles..The Three Bears….5 Little

Monkeys

Music- Songs with numbers in lyrics This Old Man….One Little,Two Little, Three Little…The Ants

Go Marching

Circle Time-Calendar Look Whose Here Today

Manipulatives (Counting Bears, Links, Cubes, Buttons)

Counting Board Games and/or Computer Games

10 Frames

Number Puzzles

Domain: Counting and Cardinality (K.CC) - Count to tell the number of objects.


Do I know the value of a number? Can I count how many objects are in a set? Do I know which numeral goes with a specific set? (1-20) Can I count by ones to 20?



Formative:Count the objects in a collection and say how many using number names. Know that the last counting word tells how many items are in the collection. That the arrangement of objects does not change the number counted. Summative:Individual assessment of skills presented: drawing/showing counting/ordering1.Understand the relationship between numbers and quantities; connect countingto cardinality. a. When counting objects, say the number names in standard order, pairing each object with oneand only one number name and each number name with one and only one object. b. Understand that the last number name said tells the number of objects counted. The number ofobjects is the same regardless of their arrangement or the order in which they were counted. c. Understand that each successive number name refers to a quantity that is one larger.


2. Count to answer “how many?” questions about as many as 20 things arranged in a line, arectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects. Rote counting numbers 1-20 Identifying numbers 1-20 Using manipulatives to represent a given number



Draw pictures or symbols to represent how many in a collection Read counting books..”5 Chinese Brothers” Songs “5 Little Monkeys” “Four Little Freckled Frogs” Count Classmates… chairs…tables Calendar Counting


Manipulatives (Counting Bears, Links, Cubes, Buttons) Counting Board Games and/or Computer Games Big Books Number Cards Number Puzzles Smart Boards


Domain: Counting and Cardinality (K.CC) - Compare numbers




What are some ways I can express relationships between 2 numbers? How can I identify groups with more or less objects than a given set? Do I know which numbers/objects are less, more, equal? Do I know how graphing helps me identify less/more/equal? Formative: Students will determine how many more, less, or equal by verbally countingnumber sequence. Students will compare, match, and count objects in a group. Summative:Teacher observation of students engaged in individual and group graphing activities. 1. Identify whether the numbers of objects in one group is greater than, less than or equal tothe number of objects in another group, e.g. by using matching and counting strategies. (Include groups with up to ten objects.)2. Compare two numbers between 1 and 10 presented as written numerals.


Counting from 0 to 10 Demonstrate meaning of more, less, equal One to one correspondence


Circle Time: Grouping of boys and girls, birthday months graphing Art: Draw pictures of objects showing more/less Sorting materials for a craft Counting out paper, crayons, markers needed Sorting by color Counting beanbags for games


Math symbol cards > < = Vocabulary cards Number cards Graphs for pictographs Trade books Smart Boards Technology


Domain:Operations and Algebraic Thinking (K.OA) - Understand addition as putting together and adding to, andunderstand subtraction as taking apart and taking from.


Do I know what is more? What is less? How do I draw an equation? What objects can I draw/show/clap to show I can make a set more/less than 10? How can I put different manipulatives together to show “how many I have? What happens when I remove objects from my set or cross out drawings from my set? Can I orally explain my equation/word problem?


Formative:Use manipulatives/drawings to demonstrate, read, and explain addition and subtraction sentences

involving numbers 0-10.

Orally solve simple addition and subtraction word problems 0-5.

Summative:Demonstrate problem solving strategies using manipulatives/drawings/number cards.


1. Represent addition and subtraction with objects, fingers, mental images, drawings{Drawings need not show details but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)}, sounds (e.g., claps). Acting out situations, verbal explanations, expressions, or equations. 2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by usingobjects or drawings to represent the problem. 3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by usingobjects or drawings, and record each decomposition by a drawing or equation (e.g., 5=2+3 and 5=4+1). 4. For any number from 1 to 9, find the number that makes 10 when added to the given number,e.g., by using objects or drawings, and record the answer with a drawing or equation.Fluently add and subtract within 5.



Understanding of number…more/less

Understanding that addition is putting together/adding to

Understanding that subtraction is taking apart/taking from

Understanding of “combine”/”decompose”

Understanding of the commutative property of addition


Classroom materials…Do we need more/less? Books/papers/crayons/pencils

Oral word problems We have ____plates and Jeff brought in ____plates. How many plates do we

have?

Hooks for clothing

Chairs at a table


Counting cubes/Math manipulatives (Bears, links, cubes)

Pencils/crayons/paper

Counting Books

Computer Games

Smart Boards


Standard : Numbers and Operations in Base Ten (K.NBT) - Work with numbers 11-19 to gain foundations for place value.





How can I make a group of 10? How does making groups of tens and ones help me create two-digit numbers? What strategies (drawings or equations) can I use to learn how to add or subtract two-digit numbers? Formative:Count out and/or draw a group of ten. Using groups of ten, add a specific number to the group by counting/drawing. Summative:Student work samples…Worksheets, drawings and manipulatives showing tens and ones (grouping) 1. Compose and decompose numbers from 11-19 into ten ones and some further ones, e.g., byusing objects or drawings, and record each composition or decomposition by drawing or equation (e.g., 18=10+8); Understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones. Understanding of the base ten system Know that composing is putting together or adding to Know that decomposing is taking apart or taking from


Learning to count numbers/objects/people quickly…skip counting Estimating number of materials needed Calendar counting of the days of the week


Manipulatives Calendars Tens/Ones Chart Popsicle sticks, rubber bands, labeled cups (tens, ones) Computer programs Smart Boards Bean sticks


Domain: Measurement and Data (K.MD) - Describe and compare measurable attributes.


Why is measurement important in my everyday life? What tools can I use to measure more/less, taller/shorter? Why is it necessary for me to have a standard and/or non-standard unit to measure length or weight, etc. of an object?




Formative:Use standard and non-standard units to measure both vertical and horizontal lengths. Recognize the attributes of length and weight using everyday language. Use comparative words to describe the relationship of objects to one another. Sort, categorize, or classify objects by more than one attribute. Summative:Student work samples showing attributes of objects. (Drawings comparing the heights of two children. Comparing different containers to show holding more/less) 1. Describe measurable attributes of objects, such as length or weight. Describe severalmeasurable attributes of a single object. 2. Drectly compare two objects with a measurable attribute in common, to see which object has

“more of”/”less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter. Meaning of height, weight, length, width The difference between standard and non-standard units of measurement Name standard and non-standard units of measurement


Organization and classification of science activities/materials Graphing Understanding vocabulary in Fiction and Non-fiction trade books Lining up in height order



Unifix cubes Blocks Links Balance Scale Weight Scale Fingers, feet Paper clips, measuring cups, etc. Rulers, Measuring tapes, String, Sand/Water table, Smart Board,Technology


Domain: Measurement and Data (K.MD) - Classify objects and count the number of objects in each category.


How do I sort/group objects? How do sorting/grouping objects help me count? Can I tell if objects are the same or different?



Formative:Objects can be organized into groups with similar characteristics. Grouping helps people find things and understand relationships. Graphing helps organize and display relevant data. Summative:Teacher observation of students participating in graphing activities of different categories. (Eye color/hair color ,height, favorite candy/color) 1. Classify objects into given categories; count the number of objects in each category and sortthe categories by count. (Limit category counts to be less than or equal to 10.)


Objects can be grouped according to various categories…number, shape, size, color. Tell why objects are grouped the same or different. Counting 10 of a specified objects Number Sense


Classroom Centers: Kitchen dishes, cups, utensils Art: Sort crayons/markers by color Science: Sink or Float Grouping Graphing of favorite_______(Colors, toys, video games)



Sorting mats/trays Picture Cards Manipulatives: Bears, buttons, shapes, links Graphs Trade Books “Caps For Sale” Smart Boards Technology


Domain: Geometry (K.G) - Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes,cones, cylinders, and spheres).


Can I identify a geometric shape? Where do I find geometric shapes in the real world? What words do I use to describe a geometric shape? What are the ways I can move a geometric shape? Do I know the difference between “flat” and “solid” in a shape? What words do I use to describe the position of an object (above, below, next to, etc.)?



Formative:Recognize and name geometric shapes using correct vocabulary.

Experiment with slides, flips, and turns of two-dimensional shapes.

Build/draw/compare shapes using various materials.

Use objects to show relative positions in space (above, behind, next to, etc.)

Summative:Student work samples and activities using various shapes found in the environment. (Buildings,

maps, toys, signs)

1. Describe objects in the environment using names of shapes, and describe the relative

positions of these objects using terms such as, above, below, beside, in front of, behind, and

next to.

2. Correctly name shapes regardless of their orientations or overall size.

3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).


Recognize and name basic shapes

Know the difference between flat and solid

Know that shapes can be two or three dimensional



Science: Go on a shape walk…indoors or outdoors

Gross Motor: Use bean bags to show positions e.g., above your head, below your chin

Art: Use various art materials to draw pictures using shapes

Literature: Trade Books “The Three Billy Goats Gruff”

Informational Books “Tall Buildings” “Under the Sea”


Attribute Blocks Puzzles Informational Trade Books..buildings, bridges

Geoboards

Building Blocks, Legos

Songs “Teddy Bear, Teddy Bear”

Smart Boards Technology


Domain: Geometry (K.G) - Analyze, compare, create, and compose shapes.





How do I match similar shapes?

Does the size of a shape change its name?

Can I combine simple shapes to make other shapes?

Can the shapes I know have similar characteristics?

Formative:Compare and contrast a sphere and a circle, a square and a cube.

Know vocabulary and show sides, corners, round, straight, curved.

Be able to describe objects in the environment using shape names.

Match shapes and parts of shapes.

Summative: Demonstrate knowledge of shapes by finding two and three dimensional shapes in

the classroom. Bring in shapes from home to share with classmates.

1. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations,

using informal language to describe their similarities, differences, parts (e.g., number of sides and

vertices/”corners”) and other attributes (e.g., having sides of equal length).

2. Model shapes in the world by building shapes from components (e.g., sticks and clay balls)

and drawing shapes.

3. Compose simple shapes to form larger shapes. For example, “Can you join two triangles with

full sides touching to make a rectangle?”

Know what shapes have four sides (square, rectangle, rhombus)

Know the difference between a square and a cube, a circle and a sphere

Know that shapes can be matched to make other shapes



Use shapes to build buildings and bridges…draw the finished project

Find examples of geometry in nature, art, and architecture

Cut folded paper to make objects…butterflies, snowflakes


Clay Play Dough

Pipe Cleaners

Pattern Blocks

Legos, Blocks Tinker Toys Building Logs

Popsicle Sticks

Smart Boards



Grade 1

Grade One

Page 1

Grade 1


Instructional time in Grade 1 focuses on four critical areas: (1) developing understanding of addition, subtraction, and strategies for addition

and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones;

(3) developing understanding of linear measurement and measuring lengths as iterating length units; and (4) reasoning about attributes of, and

composing and decomposing geometric shapes.

(1) Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a

variety of models, including discrete objects and length-based models (e.g., cubes connected to form lengths), to model add-to, take-from,

put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction, and to develop

strategies to solve arithmetic problems with these operations. Students understand connections between counting and addition and

subtraction (e.g., adding two is the same as counting on two). They use properties of addition to add whole numbers and to create and use

increasingly sophisticated strategies based on these properties (e.g., ―making tens‖) to solve addition and subtraction problems within 20.

By comparing a variety of solution strategies, children build their understanding of the relationship between addition and subtraction.

(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to add within 100 and subtract multiples of 10. They

compare whole numbers (at least to 100) to develop understanding of and solve problems involving their relative sizes. They think of

whole numbers between 10 and 100 in terms of tens and ones (especially recognizing the numbers 11 to 19 as composed of a ten and

some ones). Through activities that build number sense, they understand the order of the counting numbers and their relative magnitudes.

(3) Students develop an understanding of the meaning and processes of measurement, including underlying concepts such as iterating (the

mental activity of building up the length of an object with equal-sized units) and the transitivity principle for indirect measurement.

(4) Students compose and decompose plane or solid figures (e.g., put two triangles together to make a quadrilateral) and build

understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine shapes, they

recognize them from different perspectives and orientations, describe their geometric attributes, and determine how they are alike and

different, to develop the background for measurement and for initial understandings of properties such as congruence and symmetry.

Grade One

Page 2

Standards for Mathematical Practice(from the Common Core State Standards)





















Grade One

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4) Model with mathematics.Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the


















Grade One

Page 4




















3 + x




Grade One

Page 5

Grade One Standards for Mathematical Content

Counting and Cardinality (1.CC) Standards Strategies and Assessments

Continue to develop the skills and concepts identified for this domain in

Kindergarten guidelines.





connections to daily experiences, and writing

about mathematics.

Operations and Algebraic Thinking (1.OA)


1.OA All students will be able to represent and solve problems involving addition and

subtraction.

1. Use addition and subtraction within 20 to solve word problems involving

situations of adding to, taking from, putting together, taking apart, and

comparing, with unknowns in all positions (e.g., by using objects, drawings,

number lines, and equations with a symbol for the unknown number to

represent the problem).

2. Solve word problems that call for addition of three whole numbers whose

sum is less than or equal to 20 (e.g., by using objects, drawings, and equations

with a symbol for the unknown number to represent the problem).

Use two colors of connecting cubes or links

to represent addition and subtraction

problems.

Cover one group of objects in the

representation of a problem and ask the

student how many objects are in the hidden

group. Have the child explain his/her answer.

Use Scrabble tiles to make three-letter words.

Add letter values and compare sum.

Use a Double Ten Frame mat. Show the

addition of two numbers on the mat and ask

how many more are needed to get to 20.

Grade One

Page 6


1.OA All students will understand and apply properties of operations and the

relationship between addition and subtraction.

3. Apply properties of operations as strategies to add and subtract.

Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known (Commutative

property of addition). To add 2 + 6 + 4, the second two numbers can be added to

make a ten, so 2 + 6 + 4 = 2 + 10 = 12 (Associative property of addition).

4. Understand subtraction as an unknown-addend problem.

Example: Subtract 10 – 8 by finding the number that makes 10 when added to 8.

1.OA All students will add and subtract within 20.

5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

6. Add and subtract within 20, demonstrating fluency for addition and subtraction

within 10, using strategies such as:

o counting on;

o making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);

o decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9);

o using the relationship between addition and subtraction (e.g., knowing that

8 + 4 = 12, one knows 12 – 8 = 4);

o creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating

the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Show addition and subtraction in horizontal

and vertical form.

Use specific numbers to illustrate general

principles and properties of operations.

Make booklets containing original word

problems that illustrate different addition or

subtraction situations that occur through

the course of a day.

Take time to explore addition and subtraction

facts and to develop a math “tool kit‖ of

strategies for addition and subtraction. Ask

students to identify which ―tool‖ or strategy

will help them find the sum or difference.

Grade One

Page 7


1.OA All students will work with addition and subtraction equations.

7. Understand the meaning of the equal sign, and determine if equations

involving addition and subtraction are true or false. Example: Which

of the following equations are true and which are false? 6 = 6, 7 = 8 – 1,

5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

8. Determine the unknown whole number in an addition or subtraction equation

relating three whole numbers. Example: Determine the unknown number that

makes the equation true in each of the equations 8 +? = 11, 5 = � – 3, 6 + 6 = �.

Give each student a set of cards with the

numbers 0-20. Have them hold up the card

with the number that makes an equation with

an unknown true.

Number and Operations in Base Ten (1.NBT)


1.NBT All students will be able to extend the counting sequence.

1. Count to 120, starting at any number less than 120. In this range, read and write

numerals and represent a number of objects with a written numeral.

1.NBT All students will understand place value.

2. Understand that the two digits of a two-digit number represent amounts

of tens and ones. Understand the following as special cases:

a) 10 can be thought of as a bundle of ten ones — called a ―ten.‖

b) The numbers from 11 to 19 are composed of a ten and one, two,

three, four, five, six, seven, eight, or nine ones.

Count in sequence by ones, twos, fives, and

tens.

Use hundred chart to identify patterns in skip

counting.

Count out more than 10 objects. Show as

groups of tens and remaining ones.

Identify the number represented by a place

value model.

Grade One

Page 8


c) The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two,

three, four, five, six, seven, eight, or nine tens (and 0 ones).

3. Compare two two-digit numbers based on meanings of the tens and ones

digits, recording the results of comparisons with the symbols >, =, and <.

Have student choose four cards from a set of

cards numbered 0-9. Make two 2-dgit

numbers then compare the numbers using the

less than (<) or greater than (>) symbol.

Explain that the symbol points to the number

that is less.

1.NBT All students will use place value understanding and properties of operations to

add and subtract.

4. a. Add within 100, including: adding a two-digit number and a one-digit number;

and adding a two-digit number and a multiple of 10, using concrete models or

drawings and strategies based on place value, properties of operations, and/or the

relationship between addition and subtraction; relate the strategy to a written

method and explain the reasoning used.

b. Understand that in adding two-digit numbers, one adds tens and tens, ones

and ones; and sometimes it is necessary to compose a ten.

5. Given a two-digit number, mentally find 10 more or 10 less than the number,

without having to count; explain the reasoning used.

6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the

range 10-90 (positive or zero differences), using concrete models or

drawings and strategies based on place value, properties of operations,

and/or the relationship between addition and subtraction; relate the

strategy to a written method and explain the reasoning used.

Use popsicle sticks, tiles, linking cubes,

counters, and other visual representations

to model addition of 2-digit numbers and re-

grouping into tens. Connect these models to

the written method.

Apply various ―math tool kit‖ strategies to the

process of adding two digit numbers.

Use a hundred chart to show 10 more than a

given number and 10 less than a given

number.

Use models and visual representations

to show subtraction of multiples of 10.

Connect the models to the written method.

Place signs showing each of the multiples of

ten (0-100) in order across the front of the

room. Give each student a card with a two

digit number written on it. Have them

place themselves in front of the multiple of

ten closest to their number.

Grade One

Page 9

Measurement and Data (1.MD)


1.MD All students will measure lengths indirectly and by iterating length units.

1. Order three objects by length; compare the lengths of two objects indirectly by

using a third object.

2. Express the length of an object as a whole number of length units, by laying

multiple copies of a shorter object (the length unit) end to end. Understand that

the length measurement of an object is the number of same-size length units

that span it with no gaps or overlaps. (Limit to contexts where the object being

measured is spanned by a whole number of length units with no gaps or overlaps.)

1.MD All students will tell and write time.

3. Tell and write time in hours and half-hours using analog and digital clocks.

1.MD All students will collect, organize, display and interpret data.

4. Organize, represent, and interpret data with up to three categories using tallies,

charts, tables, bar graphs, pictographs, and Venn diagrams; ask and answer questions

about the total number of data points, how many in each category, and how many

more or less are in one category than in another.

Place three books or pencils in order by

length. Discuss how the lengths of the objects

compare to each other. Which is longest?

Which is shortest?

Have students use their feet to measure

distance across the classroom. Compare

number of student ―feet‖ needed to number of

teacher ―feet‖.

Make predictions about how many length

units (e.g., paper clips) are needed to measure

the length of various objects. Measure to

check predictions.

Create a schedule showing the time of several

daily activities (e.g., start of school day,

lunch, dismissal) on a picture of a digital and

an analog clock.

Keep track of weather conditions for each

day. At the end of the month organize and

graph the results. Discuss the data depicted on

the graph.

Have students compose a survey question

with 3 or 4 response choices. Pose the

question to 12 or more people, record the

responses, and present the data in a graph.

Grade OnePage 10

Geometry (1.G)


1.G All students will reason with shapes and their attributes,

1. Distinguish between defining attributes (e.g., triangles are closed and three-sided)versus non-defining attributes (e.g., color, orientation, overall size); build anddraw shapes to possess defining attributes.

2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangularprisms, right circular cones, and right circular cylinders) to create a compositeshape, and compose new shapes from a composite shape..

3. Partition circles and rectangles into two and four equal shares, describe the sharesusing the words halves, fourths, and quarters, and use the phrases half of, fourthof, and quarter of. Describe the whole as two of, or four of the shares. Understandfor these examples that decomposing into more equal shares creates smallershares.

Provide materials such as geo-boards, play-dough, clay, toothpicks, pipe cleaners, or other suitable materials and allow students to create models of various shapes.

Display shapes in various sizes, and colors. Describe a shape based on its attributes and have the students identify the shape that matches the description.

Use circles and rectangles cut out of construction paper or cardboard. Give students two copies of the same shape. Cut one in half, and one in fourths. Compare the size of the halves and fourths.

Extended Resources – Grade 1 Mathematics



What is addition?

How does addition affect me and other people?

What is subtraction?

How does subtraction affect me and other people?

Formative: students write word problems; think, pair, share; write an addition or subtraction word problem; act out an

addition or subtraction word problem; teacher observation of children playing addition and subtraction

games and solving word problems in order to see how students use mental math, paper and pencil,

manipulatives, or a combination of strategies; Think aloud; collaborative pairs; Show Me What You Know/

Hold Up the Answer; Agree or Disagree; Show Me/Tell Me; I Learned Statements; exit card; Rubric

Novice/Apprentice/Practitioner

Summative: Solve teacher developed addition and subtraction word problems; students develop addition

and subtraction word problems; performance assessment e.g. using a picture of the shelves in a toy store

or in a food market or using prepared plastic bags of manipulatives which will be used to solve addition

and subtraction problems based on the materials

Domain: Operations and Algebraic Thinking (1.OA) - Represent and solve problems involving addition and subtraction

Grade One Page 11


1. Use addition and subtraction within 20 to solve word problems involving situations of adding to,taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using

objects, drawings, and equations with a symbol for the unknown number to represent the problem.

2. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent

the problem.



Numbers 0-20: represent a number of objects with a written numeral 0-20; the relationship between

number and quantities; represent addition and subtraction with objects, fingers, mental images, drawings,

verbal explanations, expressions or equations; four main types of addition and subtraction situations

Vocabulary: add, addend, addition sentence, count on/up, count back, difference, doubles, doubles + 1,

doubles - equals, fact family, join, minus, missing addend, number sentence, order, part + part = whole,

plus, related addition facts, related subtraction facts, subtract, subtraction sentence, sum, take away, ten-

frame, & whole – part = part

Children’s literature; real life problems related to sports scores, collections, family members;

ELA writing to explain addition and subtraction; poetry about addition and subtraction; write word

problems

Science using addition and subtraction in science investigations and experiments

Grade One Page 12


Manipulatives e.g. inch and centimeter cubes, straws, buttons, pattern blocks,; Board Games;

Computer Games

Grade One Page 13

Domain: Operations and Algebraic Thinking (1.OA) - Understand and apply properties of operations and the relationship between addition and subtraction.Essential Questions What should I be able to answer? What guides my thinking?


Formative:Students write word problems; think, pair, share, write an addition or subtraction word problem;

act out an addition or subtraction word problem; teacher observation of children playing addition

and subtraction games and solving word problems in order to see how students use mental math,

paper and pencil, manipulatives, or a combination of strategies; Think Aloud; Collaborative Pairs;

Show Me What You Know/Hold Up the Answer; Agree or Disagree; Show Me/Tell Me; I Learned

Statements; Exit Card; Rubric-- Novice/Apprentice/Practitioner

Summative:Write a letter to a friend explaining how addition can be helpful when subtracting. Teacher made

test; create a poster with examples and labels; apply knowledge to real life task; model

operations with concrete objects; timed math fact practice

How can I use what I know about addition to help me subtract?What strategies can I use to add and subtract?Why do we need efficient methods to add and subtract?What is the quickest way I can add?What is the quickest way I can subtract?

Grade One Page 14


1. Apply properties of operations as strategies to add and subtract. [Students need not use

formal terms for these properties.] Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also

known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be

added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

2. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by

finding the number that makes 10 when added to 8. Add and subtract within 20.

3. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

4. Add and subtract within 20, demonstrating fluency for addition and subtraction

Use strategies such as counting (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number

leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and

subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 =4; and creating equivalent but easier

or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). Work

with addition and subtraction equations.

5. Understand the meaning of the equal sign, and determine if equations involving addition

and subtraction are true or false. For example, which of the following equations are true and

which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

6. Determine the unknown whole number in an addition or subtraction equations;

For example, determine the unknown number that makes the equation true

in each of the equations 8 + ? = 11, 5 = □ – 3, 6 + 6 = □.

Grade One Page 15




Numbers 0-20 - represent a number of objects with a written numeral that shows

the relationship between number and quantities; represent addition and subtraction with objects,

fingers, mental images, drawings, verbal explanations, expressions or equations; four main types of

addition and subtraction situations

Vocabulary: add, addend, addition sentence, count on/up, count back, difference, doubles, doubles

+ 1, doubles - equals, fact family, join, minus, missing addend, number sentence, order, part + part

= whole, plus, related addition facts, related subtraction facts, subtract, subtraction sentence, sum,

take away, ten-frame, & whole – part = part

Children’s literature; real life problems related to classroom life; ELA letter writing, writing

word problems ; Science activities; Religion count days of Advent or Lent answer the question How

many more days to Christmas or Easter?

Balance scales, number lines, random number generators, playing cards, board games, flash

cards, Math manipulatives e.g. inch and centimeter cubes, straws, buttons, pattern blocks,;

Board Games; Computer Games

Grade One Page 16

Domain: Number and Operations in Base Ten (1.NBT) - Extend the counting sequence.



Where do we see numerals in the real world?

How do we use counting in everyday life?

What do numerals represent?

Formative :

Observe students counting orally; use counting jars; observe students making sets and hopping

forward and back on a number-line; journal entries related to counting; develop and label

displays of materials illustrating various quantities; Think, Pair, Share; write a counting word

problem; teacher observation of children using counting games and manipulatives; Think Aloud;

collaborative pairs; Show Me What You Know/Hold Up the Answer; Agree or Disagree; Show

Me/Tell Me; I Learned Statements; Exit Card; I Can./Not Yet; Rubric--


Summative:Performance assessment - counting and labeling given collections; write and model numbers

using base ten blocks

Grade One Page 17


1. Count to 120, starting at any number less than 120. In this range, read and write numerals

and represent a number of objects with a written numeral.


Count orally by ones ( may use 2s, 5s, 10s); use counters to count out a specified number; count

forward and backward; identify sets with one-to-one correspondence; read the symbols for the

numbers; write numerals; use objects, words, &/or symbols to express understanding of

numbers; count from different starting points e.g. begin at 47, 82, etc.



Teach the correct formation of numerals in handwriting; Count out supplies, snacks, specified

amounts of items for a party; identify the number of objects used in a building project, Answer

questions such as How many tickets are needed to get on an amusement park ride, What is

the temperature recorded on a thermometer?

Children’s Literature

Board Games: e.g. Chutes and Ladders; Dino Math Tracks; Mancala; Collect 15; Race to 100

Card and Dice Games

Grade One Page 18


Why do we have numbers with more than one digit?

How can you tell the value of a digit from its placement?

How are place value patterns repeated in large numbers?


Formative:Observe students comparing the value of one, two and three digit numbers; journal entries related

to place value; develop and label displays of materials illustrating various quantities; Think, Pair,

Share; teacher observation of children using counting games and manipulatives; Think Aloud;

Collaborative Pairs; Show Me What You Know/Hold Up the Answer; Agree or Disagree; Show

Me/Tell Me; I Learned Statements; Exit Card; I Can./Not Yet;

Rubric-- Novice/Apprentice/Practitioner

Summative:Draw pictures representing given quantities; Create a number book; performance assessment

where children model numbers using manipulatives and write about their model; create a poster

of numbers using the symbols >, =, and < to illustrate the comparison of the numbers

Domain: Number and Operations in Base Ten (1.NBT) - Understand place value

Grade One Page 19


1. Understand that the two digits of a two-digit number represent amounts of tens and

ones. Understand the following as special cases:

a. 10 can be thought of as a bundle of ten ones — called a “ten.”

b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,

eight, or nine ones.

c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,



eight, or nine tens (and 0 ones).

2. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the


results of comparisons with the symbols >, =, and <.

Understanding the value of 10; counting and grouping/bundling 10 objects; comparing the value of

various two- and three-digit numbers by looking at the numbers that have the highest place

values; common terms e.g. about, after, before, equal, near, closer to, between, a little less than,

more than, equivalent, non-equivalent, tens, hundreds; use the number line to compare numbers;

use symbols for comparison +, - =,<,>

Use real life number lines e.g. rulers, thermometers; organize supply boxes to have a given

amount of items; Money number of pennies in a dime, dimes in a dollar

Children’s Literature

Manipulatives, base-ten blocks, ten-frames,

Board Games e.g. Dino Math Tracks, Treasure Chest

Grade One Page 20

Domain: Number and Operation (1.NBT) - Use place value understanding and properties of operations to add and subtract.



How does understanding place value help me solve double digit addition and subtraction

problems?

How can I use what I know about tens and ones to add and subtract two-digit numbers?

Formative:Students write word problems; think, pair, share, write an addition or subtraction word problem; act out an

addition or subtraction word problem; teacher observation of children playing addition and subtraction games

and solving word problems in order to see how students use mental math, paper and pencil, manipulatives, or

a combination of strategies; Think aloud; collaborative pairs; Show Me What You Know/Hold Up the

Answer; Agree or Disagree; Show Me/Tell Me; I Learned Statements; exit card; I Can. Not Yet; Rubric

Novice/Apprentice/Practitioner; use Base Ten Blocks (rods, flats and cubes)

Summative: Model and record addition and subtraction in a variety of ways.

Record strategies for solving, combining, and separating problems using pictures, numbers,

equations & words

Performance Assessment solving addition and subtraction word problems based on one and two

digit numbers within 100

Grade One Page 21



1. Add within 100, including adding a two-digit number and a one-digit number, and adding a two

-digit number and a multiple of 10, using concrete models or drawings and strategies based on

place value, properties of operations, and/or the relationship between addition and subtraction;

relate the strategy to a written method and explain the reasoning used. Understand that in adding

two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to

compose a ten.

2. Given a two-digit number, mentally find 10 more or 10 less than the number, without having to

count; explain the reasoning used.

3. Subtract multiples of 10 in the range of 10-90: from multiples of 10 in the range of 10-90:

(positive or zero differences), using concrete models or drawings and strategies based on place

value, properties of operations, and/or the relationship between addition and subtraction; relate

the strategy to a written method and explain the reasoning used.

Numbers 0-120 represent a number of objects with a written numeral 0-120 the relationship

between number and quantities; represent addition and subtraction with objects, fingers, mental

images, drawings, verbal explanations, expressions or equations; Fluency with addition and

subtraction facts; knowledge of place value; apply place value skills to decompose numbers into

10s and 1s; add and subtract multiples of ten; counting up and back by 10s

Vocabulary: add, addend, sum, count on/up, count back (by ones, by tens), difference, equals,

digit, 2-digit number, 3-digit number, 10 ones = 1 ten, 10 tens = 1 hundred, 1 hundred = 10 tens

Grade One Page 22


Children’s literature; real life problems e.g. comparing temperatures, comparing dates on a calendar, comparing ages of adults; ELA writing word problems, journal entries explaining the addition and subtraction of two digit numbers


Manipulatives e.g. inch and centimeter cubes, straws, buttons, pattern blocks, base ten blocks; Board Games e.g. Dino Math Tracks; Computer Games

Grade One Page 23




How do we measure the world around us?

What are different ways to collect, display, and analyze data?

What types of problems are solved with measurement?

What are tools of measurement and how are they used?

Formative:

Students write word problems; think, pair, share, write a measurement word problem; teacher

observation of children measuring and solving word problems in order to see how students use

measurement tools; Think aloud; collaborative pairs; Show Me What You Know/Hold Up the

Answer; Agree or Disagree; Show Me/Tell Me; I Learned Statements; exit card; Rubric Novice/

Apprentice/Practitioner

Summative:

Performance assessment which involves ordering and measuring specified items:

1. Order three objects by length; compare the lengths of two objects indirectly by using a third object.

2. Express the length of an object as a whole number of length units, by laying multiple copies of a

shorter object (the length unit) end to end; understand that the length measurement of an object is the

number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the

object being measured is spanned by a whole number of length units with no gaps or overlaps.

Domain: Measurement and Data (1.MD) Measure lengths indirectly and by iterating length units.

Grade One Page 24




Vocabulary: long, short, longer, shorter, wide, length, tall, short, long, far, close height, depth

The measuring tools we use have agreed upon standards. Measurements are estimates and

are close enough rather than exact.

Widespread use in daily life e.g. how deep is the swimming pool; how long is a shark; how tall is

John or Marie; which is longer the table or the door

Science investigations and activities

Non-standard units of measurement e.g. paper clips, unifix cubes, markers; Standard

measurement tools e.g. ruler, yard stick, meter stick, tape measures; Literature; Websites

e.g.http://cfbstaff.cfbisd.edu/library/Great%20Weblinks/math%20page.htm#Counting

http://nlvm.usu.edu/en/nav/topic_t_1.html

www.primarygames.com/curriculum/math.htm coolmath.com/

Grade One Page 25

Domain: Measurement and Data (1.MD) -Tell and write time.


What do we do with time?

How does time influence our lives?

What would happen without time?

How would we live without clocks?



Formative: students think, pair, share, teacher observation of children using clocks in order to see how

students represent time on analog and digital clocks; paper and pencil representation of time on

clocks and in sentences; Think aloud; collaborative pairs; Show Me What You Know/Hold Up the

Answer; Agree or Disagree; Show Me/Tell Me; I Learned Statements; exit card; Rubric


Summative: Performance Assessment in which children display and name times displayed on digital and analog

clocks

1.Tell and write time in hours and half-hours using analog and digital clocks.


Design of analog clock and digital clock

Vocabulary: before, after, early, late, morning, afternoon, evening, night, second, minute,

hour, half-hour, o’clock, hour hand, minute hand, second hand, 24 hours in a day, A.M., P.M.

Grade One Page 26



Anticipate and record daily activities in hours and minutes; record daily schedule in hours

and minutes

Clocks are important in business, science, travel, cooking, sports

Moveable analog and digital clocks; timers, stop watches, sand timers, sun dials;

children’s literature; Websites

e.g.http://cfbstaff.cfbisd.edu/library/Great%20Weblinks/math%20page.htm#Counting

http://nlvm.usu.edu/en/nav/topic_t_1.html

www.primarygames.com/curriculum/math.htm coolmath.com/

Grade One Page 27

Domain: Measurement and Data ( 1.MD) - Represent and interpret data.




Who needs to gather, record and to organize information?

Why do I and other people need to gather, record and to organize information?

How can I gather, record, and organize information?

Formative:

Observe, listen to, and interact with students as they work to gather and use data, generate

representations and descriptions, create posters, graphs, Think aloud; collaborative pairs; Show

Me What You Know/Hold Up the Answer; Agree or Disagree; Show Me/Tell Me; I Learned

Statements; exit card; Rubric Novice/Apprentice/Practitioner; create object graphs, tally charts

Summative:

Construct a graph, label the information, and make up questions about the graph. Ask another

student to answer the questions. Given a collection of items (e.g. buttons, bread bag tabs,

stickers, coins), student will group, display and interpret information by sorting and arranging data

to construct a real graph

1. Organize, represent, and interpret data with up to three categories; ask and answer questions

about the total number of data points, how many in each category, and how many more or less

are in one category than in another.

Grade One

Page 28



Identify and describe attributes of various materials (e.g. size, color shape, similar, alike, different,

most, least, order of days and months); tally; tally chart; pictograph; make a plan for gathering

and recording data; create survey questions; sort and categorize data; explain results of surveys;

present data to others in a way that communicates information

Keeping track of student attendance, weather, student birthdays, lunch orders

Science e.g. weather tracking, leaf identification, seed sorting, floating and sinking

Social Studies e.g. voting in response to a specific question


Random generators; stick-on notes; drawing paper; items to sort and classify e.g. attribute

blocks, buttons, stickers, bread bag tabs, lids, coins

Grade One Page 29

Domain: Geometry (1.G) - Reason with shapes and their attributes.



How do we use geometry in our lives?

How do we use geometry to solve problems?

What jobs use geometry?

Formative:Journal entries; develop and label displays of materials illustrating various two and three

dimensional shapes; Observe students e.g. using pattern blocks and geo-blocks to manipulate

basic geometric figures; to build new shapes from other shapes; using fraction manipulatives such

as fraction bars and playing fraction games e.g. Pizza Party; partitioning groups to show fractions

of a set or fair shares

Summative:Using paper models of two dimensional shapes students will build new shapes

Students will create a shape design according to specific directions e.g. use three squares, two

triangles, four circles and three rectangles. Have a square overlap a triangle, have two triangles

under a circleT ( have plenty of extra paper for those who will use trial and error)

Students will create a poster or flip books illustrating fractions of a shape or fractions of a set;

students will use pre-cut 2 dimensional shapes to create the scene of a story e.g. rectangles and

circles will form a wagon on a road

Grade One Page 30


1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus

non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess

defining attributes.

2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles,

and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular

cones, and right circular cylinders) to create a composite shape, and compose new shapes from

the composite shape. [Students do not need to learn formal names such as “right rectangular

prism.”]

3. Partition circles and rectangles into two and four equal shares, describe the shares using

the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.

Describe the whole as two of, or four of the shares. Understand for these examples that

decomposing into more equal shares creates smaller shares.


Name, identify and compare two dimensional and three dimensional shapes; Recognize two

dimensional and three dimensional shapes in the environment; compare size, shape, attributes;

identify open and closed shapes; know and apply position words, faces, flat surface, fraction,

halves, fourths, quarter

Grade One Page 31



Art: make geometric designs; use three dimensional shapes to build a building or a community;

use yarn to make open and closed shapes

ELA: Dictate or write a class letter to another class to tell how it is possible to make new

shapes from other shapes e.g. using Writing clear instructions and describing how one folds

and cuts to make given shapes

Social Studies: compare the use of geometric shapes in architecture around the world; relate to

Japan and the art of Origami and Kiragami; relate to quilting blocks

Building blocks, geo-blocks, pattern blocks, Lego©, tangrams, geoboards, fraction bars, string,

toothpicks, boxes, photos and magazine pictures which illustrate two and three dimensional

shapes

Grade One Page 32


Grade 2

Grade Two

Page 1

Grade 2


Instructional time in Grade 2 focuses on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with

addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes.

(1) Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds,

tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to

1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g.,

853 is 8 hundreds + 5 tens + 3 ones).

(2) Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within

1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and

generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value

and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved

to mentally calculate sums and differences for numbers with only tens or only hundreds.

(3) Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with

the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need

to cover a given length.

(4) Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about

decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes,

students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

Grade Two

Page 2






















Grade Two

Page 3











4) Model with mathematics.Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the


















Grade Two

Page 4




















3 + x




Grade Two

Page 5

Grade Two Standards for Mathematical Content



2.OA All students will represent and solve problems involving addition and subtraction.

1. Use addition and subtraction within 100 to solve one- and two-step word problems

involving situations of adding to, taking from, putting together, taking apart, and

comparing, with unknowns in all positions (e.g., by using drawings and equations

with a symbol for the unknown number to represent the problem).

2.OA All students will add and subtract within 20.

2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know

from memory all sums of two one-digit numbers.

2.OA All students will work with equal groups of objects to gain foundations for

multiplication.

3. Determine whether a group of objects (up to 20) has an odd or even number of members,

(e.g., by pairing objects or counting them by 2s; write an equation to express an even

number as a sum of two equal addends).

4. Use addition to find the total number of objects arranged in rectangular arrays with up to

5 rows and up to 5 columns; write an equation to express the total as a sum of equal

addends.



and visual representations, modeling,

small group activities, integration of

technology, connections to daily

experiences, and writing about

mathematics.

Work with problems involving: addition as

combination (plus, combined with, more);

subtraction as comparison (how much less,

how much more), equalizing (how many

more are needed to make these equal), and

separation (how much remaining).

Students make booklets containing original

word problems that are based on situations

from their daily lives that involve addition

or subtraction.

Students pair up and each child is given

half of a deck of playing cards with the face

cards removed. They each turn over a card

and the person who wins the turn is the first

to say the sum (or difference) of the two

numbers showing.

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Number and Operations in Base Ten (2.NBT) Standards Strategies and Assessments

2.NBT All students will understand place value.

1. Understand that the three digits of a three-digit number represent amounts of hundreds,

tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the

following as special cases:

a) 100 can be thought of as a bundle of ten tens — called a ―hundred.‖

b) The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to

one, two, three, four, five, six, seven, eight, or nine hundreds (and 0

tens and 0 ones).

2. Count within 1000; skip-count by 5s, 10s, and 100s.

3. Read and write numbers to 1000 using base-ten numerals, number names, and

expanded form.

4. Compare two three-digit numbers based on meanings of the hundreds, tens,

and ones digits, using >, =, and < symbols to record the results of comparisons.

2.NBT All students will use place value understanding and properties of operations to

add and subtract.

5. Fluently add and subtract within 100 using strategies based on place value, properties

of operations, and/or the relationship between addition and subtraction.

6. Add up to four two-digit numbers using strategies based on place value and properties

of operations.

Use a variety of visual representations to

model place value concepts.

Have students model 100 as ten 10’s in a

variety of ways. Display ten paper chains of

ten loops each, ten necklaces of ten beads,

ten chains with ten paper clips, ten trees

with ten leaves on each, etc.

Display several 3-digit numbers. Ask

children to write the number that matches

your oral description. For example, ―I am a

number that has two hundreds, five tens,

and three ones‖; I am a number that has

six ones, nine hundreds and no tens.‖

Display three 1-digit numbers. Have

students place each number in a column on

a three-column place value chart. This

becomes the student’s number for the

activity. Select a child to display his or her

3-digit number. This number becomes the

target number. Ask students to stand if their

number is greater than (or less than or equal

to) the target number. Repeat with different

numbers and different comparisons.

Have students identify the addition and

subtraction strategies they apply when

adding or subtracting 2 -digit numbers.

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7. Add and subtract within 1000, using concrete models or drawings and strategies

based on place value, properties of operations, and/or the relationship between addition

and subtraction; relate the strategy to a written method. Understand that in adding or

subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens

and tens, ones and ones; and sometimes it is necessary to compose or decompose

tens or hundreds.

8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100

from a given number 100–900.

9. Explain why addition and subtraction strategies work, using place value and the

properties of operations. (Explanations may be supported by visual representations.)

Demonstrate with place value models (one

units, ten rods and hundred flats) and place-

value charts; stress adding or subtracting

ones first, then tens, then hundreds.

Use a 2-row chart to help students see the

change in numbers when 10 or 100 is added

or subtracted.

265 383 502 650

+ 10 275 393 512 660



2.MD All students will measure and estimate lengths in standard units.

1. Measure the length of an object by selecting and using appropriate tools such as rulers,

yardsticks, meter sticks, and measuring tapes.

2. Measure the length of an object twice, using length units of different lengths for

the two measurements; describe how the two measurements relate to the size of

the unit chosen.

3. Estimate lengths using units of inches, feet, centimeters, and meters.

Help students learn the length of an inch,

centimeter, foot, yard and meter by

identifying common objects as referents for

particular lengths.

Use books such as: Jim and The Beanstalk

by Raymond Briggs; How Big Is A Foot? by

Rolf Myller; Inch by Inch by Leo Lionni to

explore the concept of measurement.

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4. Measure to determine how much longer one object is than another, expressing the

length difference in terms of a standard length unit.

Send students on a scavenger hunt. Give

students a list of lengths. Have them find

objects in the classroom that match each

length. Students should record the name of

the object and its length. Objects might

include tables, books, paper clips, doorways.

2.MD All students will relate addition and subtraction to length.

5. Use addition and subtraction within 100 to solve word problems involving lengths

that are given in the same units (e.g., by using drawings, such as drawings of rulers)

and equations with a symbol for the unknown number to represent the problem.

6. Represent whole numbers as lengths from 0 on a number line diagram with equally

spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number

sums and differences within 100 on a number line diagram.

2.MD All students will work with time and money.

7. Tell and write time from analog and digital clocks to the nearest five minutes,

using a.m. and p.m.

8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies,

using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies,

how many cents do you have?

Use the number line as a visual

representation when adding or subtracting

the length of two or more items.

Write various times in digital form on index

cards and put them in a bag. Have students

pick an index card from the bag and show

the time on an analog clock.

Attach price tags to items or pictures of

items. Give each student a collection of play

money (different amounts). Ask students

various questions regarding the items: Which

of these items can be bought for exactly the

amount of money that you have (requiring

no change)? Which items can you buy and

have some money left over? Which of these

items cannot be bought because you do not

have enough money?

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2.MD All students will represent and interpret data.

9. Generate measurement data by measuring lengths of several objects to the nearest

whole unit, or by making repeated measurements of the same object. Show the

measurements by making a line plot, where the horizontal scale is marked off in

whole-number units.

10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set

with up to four categories. Solve simple put together, take-apart, and compare problems

using information presented in a bar graph.

When working with graphs make sure that

all graphs have a title. The horizontal and

vertical axes should be clearly labeled and

the data displayed accurately.

Have pairs of students measure each other’s

arm height or shoe length. Record the

measurements in inches on a chart. Have

students work in pairs to display results on a

line plot.

Survey students about their favorite sport

(fruit, season of the year, etc.) and display

results in a chart. Have students create a bar

graph and a picture graph to represent data.

Geometry (2.G) Standards Strategies and Assessments

2.G All students will reason with shapes and their attributes.

1. Recognize and draw shapes having specified attributes, such as a given number of

angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons,

hexagons, and cubes.

2. Partition a rectangle into rows and columns of same-size squares and count to find the

total number of them.

Display various two and three-dimensional

shapes. Have students identify the shape

based on the description of it attributes.

Sort shapes into two different groups and

ask student to tell what attributes were used

to sort the shapes.

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3. Partition circles and rectangles into two, three, or four equal shares, describe the shares

using the words halves, thirds, half of, a third of, etc., and describe the whole as two

halves, three thirds, four fourths. Recognize that equal shares of identical wholes need

not have the same shape.

Have students fold a rectangular sheet of

paper or a cutout of a circle to show various

fractional parts.

Emphasize that fractions refer to equal parts

of a whole. Have students tell whether a

shape is divided into equal or unequal parts.



• Where and how do I use math and numbers in the real world?

• What are different ways to count?

• How can I use manipulatives and drawings to help me solve a problem?

• How can making a list or table help me solve a problem?


Formative:

• “Problem of the Day” – During Morning Meeting/Circle Time or as a warm-up to the daily

Math lesson, the class can solve one of the four types of addition/subtraction problems.

• “Learning Centers” – Teacher observation of children playing addition and subtraction

games involving word problems in Learning Centers.

• “Role Play” – Students can act out an addition or subtraction word problem.

Summative:

• Math Journals – Students can solve a weekly word problem in their Math Journals. In these

journals, students should be able to show their work as well as explain how they solved the

problem.

Domain: Operations and Algebraic Thinking (2.OA) - Represent and solve problemsinvolving addition and subtraction.

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1. Use addition and subtraction within 100 to solve one and two step word problems involving

situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all

positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the

problem.

Math Vocabulary: add, addend, addition sentence, count back, count on/up, difference,

doubles fact, doubles + 1, doubles , equals, fact family, join, minus, missing addend, number

sentence, order, part + part = whole, plus, related addition facts, related subtraction facts,

subtract, subtraction sentence, sum, take away, ten frame, & whole – part = part

Math Concepts: Understand and apply properties of operations and the relationship between


Students should be exposed to the 4 main types of addition and subtraction situations:

• Take-from example: David had 63 stickers. He gave 37 to Susan. How many stickers does

David have now? 63 – 37 =

• Add to example: David had $37. His grandpa gave him some money for his birthday. Now

he has $63. How much money did David’s grandpa give him? $37 + = $63

• Compare example: David has 63 stickers. Susan has 37 stickers. How many more stickers

does David have than Susan? 63 – 37 =

o Even though the modeling of the two problems above is different, the equation,

63 - 37 = ?, can represent both situations (How many more do I need to make 63?)

• Take-from (Start Unknown) David had some stickers. He gave 37 to Susan. Now he has 26

stickers. How many stickers did David have before? - 37 = 26

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Students can create their own word problems about topics in ELA, Religion, Science, & Social Studies.


Children’s Literature:

First in Math Website (http://www.firstinmath.com)

Manipulatives: connecting cubes, counters, dominoes, flash cards, hundreds chart,

interactive white board, number cards, number line, spinners, stickers, ten frame

Technology: Ed U Smart (http://ed u smart.com/2ndAl.aspx)

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Domain: Operations and Algebraic Thinking (2.OA) - Add and subtract within 20.



• How can I use the words before, after, and between to describe number order?

• How can I use what I know about addition to help me subtract?


Formative:

• “Around the World” – Using flash cards, pairs of students must correctly solve an addition orsubtraction fact. The student who correctly solves the problem first, moves on to faceanother student while the student who solved the problem wrong, must sit down.

• “Calendar Time” – During Circle Time or Morning Meeting, incorporate number activities.For example, if the date is April 8th students should describe 8 as 4+4, 15-7, 3+3+2, etc.Students can also describe 8 as the number that comes after 7, before 9, or between 7 & 9.

• “Case of the Missing Pennies” – Each pair of student must be given a cup and 20 pennies orcounters. Students must lay the pennies or counters out so all can be seen. One student,Detective A, covers some pennies or counters with a cup while the other student, DetectiveB, looks away. Once the pennies or counters are covered with the cup, Detective B mustwrite a number sentence to solve the problem. For example, if there are only 7 pennies orcounters left, the number sentence would be 20 – 7 = 13. 13 pennies or counters werecovered by the cup.

• “Clip the Sum” – Make game boards with the numbers 0 – 9 in a random order.

3 7 0 2 5

1 4 8 6 9

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Have students work in groups of 3 or 4. Each student must choose a goal sum between 2 – 18. When it is the player’s turn, 2 paper clips or counters are tossed onto the board.Whatever the numbers the paper clips or counters land on, the student must create a number sentence. For example, 7 + 9 = 16. Whoever’s goal sum in the group is closest to 16 gets a point. The game continues until a player reaches 3. Then, a new goal sum is chosen.

Summative:

• Math Journals – Students should be given various problems involving sum and difference to20 to solve in their Math Journals.

• Timed Fact Practice – Since this standard emphasize fluency, students should constantly beattempting to better their time with flash cards for addition/subtraction or using computersoftware/games. In the beginning, students can use manipulatives to show problems but bythe end of 2nd Grade, students should know their facts to 20 without the use ofmanipulatives.


1. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know frommemory all sums of two one digit numbers.


Math Vocabulary: add, addend, addition sentence, count back, count on/up, difference,

doubles fact, doubles + 1, doubles, equals, fact family, join, minus, missing addend, number

sentence, order, part + part = whole, plus, related addition facts, related subtraction facts,

subtract, subtraction sentence, sum, take away, ten frame, & whole – part = part

Math Concepts: 1. Represent and solve problems involving addition and subtraction .

Understand and apply properties of operations and the relationship between addition and

subtraction.

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ELA: Students should use magazines or newspapers to find examples of basic addition and

subtraction facts and how they are used in real life.

Religion: Students can use the words before and after to describe the order of the parts of the

Mass.

Science: Use the words before and after to describe the position of the planets in the solar

system.

Children’s Literature: Annie’s One to Ten by Annie Owen; Cats Add Up! by Diane Ochiltree; The

Philharmonic Gets Dressed by Karla Kuskin; Pondlarker by Fred Gwynne; & 12 Ways to Get to 11

by Eve Mirriam

First in Math Website (http://www.firstinmath.com): Skill Sets, More or Less, First in Tens, Ten

Wheels, Grand Slam Addition & Subtraction, First to 20, K2 3 to 9, K2 More or Less, K2 Skip

Game Manipulatives: connecting cubes, counters, dominoes, flash cards, hundreds chart,

interactive white board, number cards, number line, spinners, ten frame

Technology: (http://ed u smart.com/2ndMathFacts.aspx)

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Domain: Operations and Algebraic Thinking Standard (2.OA) - Work with equal groups of objects to gain foundations for multiplication.


• How can I tell if a number is even or odd?

• How is multiplication related to addition?

• Where can I use multiplication in the real world?


Formative:

• “Even or Odd” – Give each student 15 connecting cubes. Have the students make a train with 8cubes. Then, disconnect the 8 cube train into groups of two. Students should recognize that fourequal groups of two are made. Repeat the activity for 9, 12, & 15. Emphasize that some trains willhave left over cubes. On a hundreds chart, students should color even numbers one color and oddnumbers another color. This visual representation will show students that even numbers end in 0, 2,4, 6, & 8 while odd numbers end in 1, 3, 5, 7, & 9. In their Math Journals, students should writeabout things that come in pairs – gloves, mittens, shoes, socks, twins, etc. Teach the students asong to remember even and odd numbers.

• “Amazing Arrays” – Allow students to create arrays using Dot Art/BINGO stamps and stickers.Students can create these arrays on large chart paper. Emphasize the necessity of keeping the DotArt/BINGO stamps and stickers in neat rows and columns. Once the arrays are created studentscan write multiplication sentences. Students can also switch arrays to solve problems. Studentscan also use Geoboards to form rectangular arrays.

Summative:

• “Scrumptious Snack” – For snack, give students a baggie of cereal, crackers fruit snacks, popcorn,pretzels, raisins, etc. First, have students count how many pieces of food they have. They shoulddetermine if they have an even or odd amount. Then, have the children make an array to match aflashcard. Challenge students to see how many arrays they can build in 10 minutes. Whenfinished, they can share their arrays with another class so they can explain their mathematicalreasoning. Of course, they can eat their arrays when finished!

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1. Determine whether a group of objects (up to 20) has an odd or even number of members,

e. g. by pairing objects or counting them by 2s; write an equation to express an even number as a

sum of two equal addends.

2. Use addition to find the total number of objects arranged in rectangular arrays with up to 5

rows and up to 5 columns; write an equation to express the total as a sum of equal addends.


Math Vocabulary: array, column, equal groups, even, factor, hundreds chart; model,

multiplication sign, multiply, odd, pair, product, repeated addition, & row

Math Concepts: Knowledge of Addition; Knowledge of Shapes/Attributes


Art/ELA: Allow students the opportunity to create a classroom floor plan for desks/furniture.

Once the designs are completed, students should present their ideas to the class and attempt to

persuade their classmates why their design works best in the classroom. Students could vote for

their favorite arrangement which would be used in the classroom for a set period of time.

Science: Use pictures of mammals, reptiles, insects, or spiders when making arrays.

Social Studies: Students could use their knowledge of arrays to create a community.

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Children’s Literature: Amanda Bean’s Amazing Dream by Cindy Neuschwander; Bats on

Parade by Kathi Appelt; Bunches and Bunches of Bunnies by Louise Matthews; Corkscrew

Counts: A Story about Multiplication by Donna Jo Napoli & Richard Tchen; The Doorbell Rang by

Pat Hutchins; Emma’s Christmas by Irene Trivas; Even Steven, Odd Todd by Kathryn Cristaldi;

Hershey’s Milk Chocolate Multiplication Book By Jerry Pallotta & Rob Bolster; Two of Everything

by Lily Toy Hong; 2 X 2 = Boo: A Set of Spooky Multiplication Stories by Loreen Leedy

First in Math Website (http://www.firstinmath.com): Just the Facts: Multiply; Whole Numbers:

Multiply Gym

Manipulatives: connecting cubes, counters, Dot Art, geoboards, interactive white board,

stickers Technology: (http://ed u smart.com/2ndNumbers.aspx)

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Domain: Number and Operations in Base Ten (2.NBT) Understand place value.


• Why is zero so important to our number system?• How can I tell if a number is even or odd?• How does place value help me understand numbers?• How can estimating help me work with large numbers?• How can I recognize patterns in numbers?


Formative:

• “Base 10 Blocks” – Throughout this unit, students should have daily practice with Base 10Blocks. Children should form numbers with the blocks and recognize opportunities to makeexchanges.

• “Calendar Time” – During Circle Time or Morning Meeting, incorporate place value activitieseach day. For example, students can count the day using Base 10 Blocks. If the date is the23rd, students should put 2 rods (groups of 10) and 3 units (1 each) into the Unit, Rod, & Flatcups. Each day this should be done which will lead to exchanges being made on certaindays. Also, if the date is the 23rd, students can say that “Today’s date is March 23rd. Thenumber 23 is greater than 20 but less than 30. 23 comes between 22 and 24. There are 2tens and 3 ones in the number 23. 23 is an odd number. The expanded form for 23 is 20 +3.”

• “Number Mania” – post a 2-digit or 3-digit number in your classroom each day. Vary theform in which you present the number (i.e. expanded form, visual picture of its place on anumber line, number word, ordinal form, etc.) Throughout the day, allow children to giveother versions of that number. Students can also discuss what number comes before andafter as well as if the number is even or odd.

• “Place Value Houses” – Give students the opportunity to create their own place valuehouses (i.e. each house is one period containing 3 spots – ones, tens, & hundreds). Afterstudents had ample exposure to Base 10 Blocks, the children should make numbers on theirhouses and practice saying numbers correctly without using the word “and”.

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Summative:

• “My Number” – students should select a meaningful 2-digit or 3-digit number in their life suchas a house number, favorite number, sports team number, etc. Allow children theopportunity to create a poster or use Paint in Computer class to give differentrepresentations of the chosen number. Students should use vocabulary words from the unitsuch as place value, tens and ones, number words, expanded form, ordinals, etc.


1. Understand that the three digits of a three digit number represent amounts of hundreds,

tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as

special cases:

a. 100 can be thought of as a bundle of ten tens — called a “hundred.”

b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five,

six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2. Count within 1000; skip count by 5s, 10s, and 100s.

3. Read and write numbers to 1000 using base ten numerals, number names, and

expanded form.

Compare two three digit numbers based on meanings of the hundreds, tens, and ones

digits, using >, =, and < symbols to record the results of comparisons.

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Math Vocabulary: after, before, between, equal to, estimate, even numbers, expanded form, flat,

greater than >, greatest, halfway, hundreds, least, less than <, number words, odd numbers, ones,

ordinal numbers, rod, round, place value chart, standard form, tens, unit, 2 digit number, & 3 digit

number

Math Concepts: Extend the counting sequence and understand place value. Use place value

understanding and properties of operations to add and subtract.

ELA: Since students will be comparing numbers in Math, incorporate compare/contrast writing at

this time in ELA. For example, students could read the Frog and Toad books. Students could use

a Venn Diagram to compare/contrast the characters Frog & Toad.

Religion: In the unit on 7 Sacraments, discuss how 7 is an odd number and how Jesus had 12

Apostles which is an even number.

Science: Continue the theme of compare/contrast when teaching students the differences

between reflection and refraction of light.

Social Studies: Using toothpicks and gumdrops/marshmallows, students can create 3-D

geometric shapes in order to form a neighborhood or community. Students should label the house

numbers in their community. Students should be able to explain how even number houses are on

side and odd number houses are on another side. Reason with Shapes and their Attributes. In

addition, students could compare/contrast rural, suburban, and urban communities.

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Children’s Literature: How Much is a Million by David M. Schwartz

First in Math Website (http://www.firstinmath.com):

Manipulatives: Base 10 Blocks (units, rods, flats, & cubes), connecting cubes, counters, hundred

chart, interactive white board, number line, place value mat

Technology: Ed U Smart (http://ed u smart.com/2ndPlaceValue.aspx & http://ed u

smart.com/2ndGradeExplore.aspx)

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Domain: Number and Operations in Base Ten (2.NBT) - Use place value understanding and properties of operations to add and subtract.


• How does learning how to add/subtract large numbers help me in the real world?

• How does regrouping and understanding place value help me to add and subtract large

numbers?


Formative:

• “Base 10 Blocks” – Throughout this unit, students should have daily practice with Base 10Blocks. Children should form numbers with the blocks and recognize opportunities to makeexchanges.

• “Calendar Time” – During Circle Time or Morning Meeting, incorporate place value activitieseach day. For example, students can count the day using Base 10 Blocks. If the date is the23rd, students should put 2 rods (groups of 10) and 3 units (1 each) into the Unit, Rod, & Flatcups. Each day this should be done which will lead to exchanges being made on certaindays. Also, if the date is the 23rd, students can say that “Today’s date is March 23rd. Thenumber 23 is greater than 20 but less than 30. 23 comes between 22 and 24. There are 2tens and 3 ones in the number 23. 23 is an odd number. The expanded form for 23 is 20 +3.”

• “Number Mania” – post a 2-digit or 3-digit number in your classroom each day. Vary theform in which you present the number (i.e. expanded form, visual picture of its place on anumber line, number word, ordinal form, etc.) Throughout the day, allow children to giveother versions of that number. Students can also discuss what number comes before andafter as well as if the number is even or odd.

• “Place Value Houses” – Give students the opportunity to create their own place valuehouses (i.e. each house is one period containing 3 spots – ones, tens, & hundreds). Afterstudents had ample exposure to Base 10 Blocks, the children should make numbers on theirhouses and practice saying numbers correctly without using the word “and”.

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Summative:

• “My Number” – students should select a meaningful 2-digit or 3-digit number in their life suchas a house number, favorite number, sports team number, etc. Allow children theopportunity to create a poster or use Paint in Computer class to give differentrepresentations of the chosen number. Students should use vocabulary words from the unitsuch as place value, tens and ones, number words, expanded form, ordinals, etc.


1. Fluently add and subtract within 100 using strategies based on place value, properties of

operations, and/or the relationship between addition and subtraction.

2. Add up to four two digit numbers using strategies based on place value and properties of

operations.

3. Add and subtract within 1000, using concrete models or drawings and strategies based on

place value, properties of operations, and/or the relationship between addition and subtraction;

relate the strategy to a written method. Understand that in adding or subtracting three-digit

numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and

sometimes it is necessary to compose or decompose tens or hundreds.

4. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from

a given number 100–900.

5. Explain why addition and subtraction strategies work, using place value and the properties

of operations. [Explanations may be supported by drawings or objects.]


Math Vocabulary: addend, algorithm, difference, digit, estimate, regroup, round, sum, tens, 3-digit

number, 10 ones = 1 ten, 10 tens = 1 hundred, 1 hundred = 10 tens

Math Concepts: Fluency with addition and subtraction facts to 20 & Knowledge of Place Value

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ELA: Students should use magazines or newspapers to find examples of larger numbers and

how they are used in real life.

Children’s Literature:


Manipulatives: addition frame, Base 10 Blocks, coins/bills, hundred chart, interactive white board,

number line, money addition/subtraction frames, place value mat, subtraction frame, 3 digit

number cards

Technology: Ed U Smart (http://ed u smart.com/2ndNumbers.aspx & http://ed u

smart.com/2ndMathFacts.aspx)

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Domain: Measurement and Data ((2.MD) - Measure and estimate lengths in standard units.


• Why do we need to learn how to measure in everyday life?

• Why is it important to measure using a standard tool?


Formative:

• Allow students the opportunity to measure various classroom and home objects using anonstandard tool (i.e. counting cube, each student’s footprint, paper clip, pencil, etc.) Havestudents compare measurements. At the end of the lesson, have students write on an ExitCard why it is important to have a standard unit of measurement.

• “Which tool should I use?” – Give students different examples of items to measure. Havestudents decide what is the best tool to use – measuring tape, meter stick, ruler, or yardstick.

• “Long Jump” – Tape 2 meter sticks end to end on the floor. Mark a starting place withmasking tape. Students can estimate how far they think they can jump. After the estimatesare recorded, students should jump and compare their estimates with the exactmeasurements. Students can calculate the difference.

• “Let’s Race” – Using pull back cars, students can record the distance traveled by the cars.First, students can estimate how far the cars will travel in inches and centimeters. Then, thecars will race. Students can record the distances in both the Customary & Metric Systems.The children can calculate the differences between their estimates and actualmeasurements.

Summative:

• “Metric Olympics” – see lesson at http://www2.scholastic.com/browse/article.jsp?id=7516Focus only on the events where students will measure in centimeters and meters.

• “Measure This…” – In their Math Journals, students can create a Venn Diagram that showsthe similarities and differences between the Customary System & Metric System.Afterwards, students can give examples of when it would be appropriate to use a centimeter,meter, inch, foot, or yard.

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1. Measure the length of an object by selecting and using appropriate tools such as

rulers, yardsticks, meter sticks, and measuring tapes.

2. Measure the length of an object twice, using length units of different lengths for the two

measurements; describe how the two measurements relate to the size of the unit chosen.

3. Estimate lengths using units of inches, feet, centimeters, and meters.


4. Measure to determine how much longer one object is than another, expressing the

length difference in terms of a standard length unit.

Math Vocabulary: centimeter, Customary System, difference, estimate, half inch, feet,

foot, length, meter, Metric System, unit of measure, & yard

Math Concepts: Measure lengths indirectly and by iterating length units


ELA: Students should look through various magazines to cut out pictures of items that have measurements in meters, feet, or yards.

Religion: Allow students to create a diorama of the inside of a Church to display the altar and sacramentals. Students will need to use their knowledge of measurement to position objects within their diorama.

Science: During a unit on life science, students can plant their own seed in a container with potting soil. The class can chart the growth of the plants by keeping a journal of the plant’s measurements as they grow.

Social Studies: Using rulers, graph paper, and furniture catalogs, invite children to design a bookcase for a dollhouse after first examining a bookcase in a furniture catalog. Children can draw their bookcase and shelves on graph paper. When finished, students can label the height and width of the bookcase.

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Children’s Literature: The Biggest Fish by Shelia Keenan; How Big is a Foot?; Inch by Inch by

Leo Lionni; Moira’s Birthday by Robert Munsch; Ten Beads Tall by Pam Adams; & Twelve Snails

to One Lizard by Susan Hightower


Manipulatives: centimeter grid paper, connecting cubes, graph paper, inch grid paper,

interactive white board, measuring tape, meter stick, objects to measure, ruler, yard stick

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Domain: Measurement and Data (2.MD) - Relate addition and subtraction to length.Essential Questions What should I be able to answer? What guides my thinking?





Formative:

• Allow students the opportunity to measure various classroom and home objects using a nonstandardtool (i.e. counting cube, each student’s footprint, paper clip, pencil, etc.) Have students comparemeasurements. At the end of the lesson, have students write on an Exit Card why it is important tohave a standard unit of measurement.

• “Which tool should I use?” – Give students different examples of items to measure. Have studentsdecide what is the best tool to use – measuring tape, meter stick, ruler, or yard stick.

• “Long Jump” – Tape 2 meter sticks end to end on the floor. Mark a starting place with masking tape.Students can estimate how far they think they can jump. After the estimates are recorded, studentsshould jump and compare their estimates with the exact measurements. Students can calculate thedifference.

• “Let’s Race” – Using pull back cars, students can record the distance traveled by the cars. First,students can estimate how far the cars will travel in inches and centimeters. Then, the cars will race.Students can record the distances in both the Customary & Metric Systems. The children cancalculate the differences between their estimates and actual measurements.

Summative:

• “Metric Olympics” – see lesson at http://www2.scholastic.com/browse/article.jsp?id=7516Focus only on the events where students will measure in centimeters and meters.

• “Measure This…” – In their Math Journals, students can create a Venn Diagram that shows thesimilarities and differences between the Customary System & Metric System. Afterwards, studentscan give examples of when it would be appropriate to use a centimeter, meter, inch, foot, or yard.

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1. Use addition and subtraction within 100 to solve word problems involving lengths that are

given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a

symbol for the unknown number to represent the problem.

2. Represent whole numbers as lengths from 0 on a number line diagram with equally

spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and

differences within 100 on a number line diagram.


Math Vocabulary: centimeter, Customary System, difference, estimate, feet, foot, half inch,

interactive white board, length, meter, Metric System, unit of measure, & yard

Math Concepts: Measure lengths indirectly and by iterating length units; Knowledge of

Addition & Subtraction; Fluency of facts to 20


ELA: Use content area material in creating original word problems using these concepts.



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Domain: Measurement and Data (2.MD) - Work with time and money.


• Why do I need to tell time in daily life?

• What is the relationship between analog and digital clocks?

• How does recognizing coins and bills help me in real life?

• How does skip counting help me to count money and tell time?


Formative:

• “Calendar Time” – During Circle Time or Morning Meeting, incorporate money and time

activities each day. For example, students can count the day with money. If the date is the

24th, students should place 2 dimes and 4 pennies in the Penny, Nickel, Dime, Quarter, Half

Dollar, & Dollar cups. Exchanges will arise on different days. For time, ask students

throughout the day what time it is when beginning or ending certain activities.

• “Classroom Piggy Banks” – Create a large piggy bank on pink paper and laminate for

each student. Throughout the unit on money, students can use coins to form various

amounts and make change on the piggy bank mat. Students should be given multiple

opportunities to form equivalent amounts using coins and bills. (Example: 76 cents – 3

quarters & 1 penny; 7 dimes, 1 nickel, & 1 penny; 76 pennies; 15 nickels & 1 penny, etc.)

• “Human Clock” – create a “Human Clock” in your classroom using students. Twelve

students should be given a number from 1 – 12 to wear on their chest. Evenly space these

students in a circle to create a clock. Place 2 additional students in the center of the clock to

act as the hour and minute hand. The clock hand students should be given a time to form

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on the clock. Other students in the classroom can form the time on individual Judy clocks.

Students can be rotated so they can be both a clock number and clock hand.

• “Quick Write” - Provide students with a large sticky note, an index card, or half sheet of

paper. Pose a question to the students from the time and money unit. Students can write

their reflections.

Summative:

• “Let’s Go Shopping” – Given a circular for a store, give students a budget to buy items.

Students will have to select their items to purchase and choose how they will pay for items

with bills and coins. Students can also calculate if they were under or over budget.

• “Geometry Money” – Allow students the opportunity to create a picture with pattern blocks

or draw a picture using circles, squares, triangles, and rectangles. Assign a monetary value

(1 cent, 5 cents, 10 cents, 25 cents, etc.) and ask the students to calculate how much their

pictures are worth.

• “It’s About Time” – Create a Flipbook where students can chart an entire day with their

activities and the time they completed the activity. Students should begin with the time they

wake until the time when they go to bed. All activities should be charted. Each page in the

Flipbook should represent a different time and activity. Students should form the time on an

analog clock on each page and draw a picture of the activity. In addition, students should

include 1 – 2 sentences that describe the activity. Emphasize the two cycles of 12 hours in

a day – A.M./P.M.

• “Time Flies..” - In their Math Journals, have students write about three activities in our world

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that would be very different if we had no way of telling time.


1. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m.

and p.m.

2. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $

and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents

do you have?


Math Vocabulary: A.M., analog clock, cent symbol, change, decimal point, digital clock, dime,

dollar, dollar sign, equal amounts, half hour, half past, hour, hour hand, minute, minute hand,

nickel, P.M., penny, quarter, quarter hour, second, & total

Math Concepts: Knowledge of Addition & Subtraction; Problem Solving Strategies; Skip Counting

Students need practice with common time phrases such as quarter till, quarter after, ten till, ten

after, and half past.

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ELA: Have students brainstorm places to travel for a class trip. Once a destination is chosen,

have students create an itinerary with specific times for activities on the field trip. After the trip,

have students write about if they were able to follow the schedule and if enough time was given for

activities on the trip.

Religion: Read the book, Alexander, Who Used to be Rich Last Sunday. Discuss the concepts of

saving and spending money. Students can create a chart with three columns (Spend, Save, &

Donate). Students can create this chart in Microsoft Word or on large chart paper. In the Spend &

Save columns, students can write or draw pictures of things they would like to buy or things they

would like to save their money. In the Donate column, students should brainstorm activities of how

they can help the less fortunate. The class could choose student ideas for a class Service Project.

Science: Read the book, The Grouchy Ladybug. Each student can choose an animal from the

story to research.

Social Studies: Create a class store where students can serve as customers/cashiers in order to

practice using money and making change; discuss the terms producers/consumers; introduce

students to the concept of credit/debit cards as well as checking/savings accounts.

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Children’s Literature: Alexander, Who Used to Be Rich Last Sunday by Judith Viorst; Benny’s

Pennies by Pat Brisson; A Chair for My Mother by Vera Williams; The Grouchy Ladybug by Eric

Carle; How the Second Grade Got $8,205.50 to Visit the Statue of Liberty by Nathan Zilmelman; If

You Made a Million by David M. Schwartz

First in Math Website (http://www.firstinmath.com): Measurement World (“Equal Pay”, “Time

More or Less”, “Equal Time”, & “Passing Time”); K 2 World K2 Skip Game

Manipulatives: coins/bills, interactive white board, Judy Clocks

Technology: United States Mint http://www.usmint.gov/kids;

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Domain: Measurement and Data (2.MD) - Represent and interpret data.




• How do I ask appropriate questions to find out specific data?

• How can we collect data to learn about each other?

• How do tally marks help us to organize data?

• How can I organize large amounts of data using a pictograph or bar graph?


Formative:

• Create a “Measurement Station” in your classroom. In this area, have a “Question of the

Day” where students can answer via tally mark or by adding his/her name square to create a

bar graph or pictograph. In addition, in this measurement station have a weekly item that

students can measure and a monthly item where students can estimate the number of items

in a container.

• Allow students the opportunity to measure various classroom and home objects using a

nonstandard tool (i.e. counting cube, each student’s footprint, paper clip, pencil, etc.) Have

students compare measurements. At the end of the lesson, have students write on an Exit

Card why it is important to have a standard unit of measurement.

Summative:

• “Graphing Guru” – Have students create a question which they will use to survey classmates

and members of their families. Once the question is formulated, students should survey a

given number of people and collect data using tally marks. Afterwards, the students should

create a pictograph and bar graph to represent the data. Students should be exposed to

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both horizontal and vertical bar graphs.

• “Picture This…” – In their Math Journals, students can create a Venn Diagram that shows

the similarities and differences between a pictograph and bar graph. Afterwards, students

can give examples of when it would be appropriate to create a pictograph and when it would

appropriate to create a bar graph.


1. Generate measurement data by measuring lengths of several objects to the nearest whole

unit, or by making repeated measurements of the same object. Show the measurements by

making a line plot, where the horizontal scale is marked off in whole number units.

2. Draw a picture graph and a bar graph (with single unit scale) to represent a data set with

up to four categories. Solve simple put-together, take-apart, and compare problems using

information presented in a bar graph.


Math Vocabulary: bar graph, data, horizontal, key, length, line plot, pictograph, symbol, survey,

tally chart, & vertical

Math Concepts: Knowledge of Addition & Subtraction; Problem solving strategies


ELA: Create a bar graph/pictograph to demonstrate the number of books read or favorite genre.

Religion: Create a bar graph/pictograph to display the students’ favorite story of forgiveness.

Science: Create a bar graph/pictograph to show local weather patterns for a specified time; Have

students create a chart showing the basic food groups. Each student should tally how many

servings of each food group he/she eats in one day. Then, the results could be graphed. As ELA

extension, students could write about if he/she is eating in a healthy way.

Social Studies: Create a personal timeline of important events in a student’s life.

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Children’s Literature: The Biggest Fish by Shelia Keenan; How Big is a Foot?; Inch by Inch by

Leo Lionni; Lemonade for Sale by Stuart J. Murphy; Moira’s Birthday by Robert Munsch; Ten

Beads Tall by Pam Adams; & Twelve Snails to One Lizard by Susan Hightower

First in Math Website (http://www.firstinmath.com): Know & Show 3 (“Picture This”)

Manipulatives: classroom objects to measure, graphing materials (crayons, colored pencils, graph

paper, interactive white board, markers, paper, stickers/symbols for pictograph, tally chart, etc.),

meter stick, yard stick, ruler

Technology: Ed U Smart (http://ed u smart.com/2ndGradeData.aspx); Microsoft Excel (Bar

Graphs)

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Domain: Geometry (2.G) - Reason with shapes and their attributes.Essential Questions What should I be able to answer? What guides my thinking?

• How are shapes represented in our world? What are the differences between two and three

dimensional shapes?

• How can recognizing patterns help me solve problems?

• How can I find smaller shapes in a larger picture?

• How can I create new shapes from other shapes?

• How can I divide a shape into equal parts?


Formative:

• “Shape Sorter” – Cut basic shapes in various sizes from cardboard or foam. Place the

shapes in a bag. Have students place their hands in the bag and choose a shape. Before

pulling the shape out of the bag, have the student describe as many

characteristics/properties as possible. If possible, the student should attempt to name the

shape.

• “Geometric War” – Provide each small group of students with a deck of geometric shape

cards. Player #1 draws a card and describes the shape to the group. The first child to

correctly identify the shape takes the card and adds it to his/her pile. This procedure

continues until all cards have been used. The winner of the game will be the student with

the most cards.

• “Scavenger Hunt” – Students should locate various shapes in school and at home. Students

can compare the objects that they found. Afterwards, students can form these shapes using

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a geoboard.

• “Fraction Frenzy” – Drop a handful of pennies or other coins on a table. Have students

observe how many coins landed heads-up and how many landed tails-up. Students could

determine the fraction of coins that is head-up and the fraction of coins that is tails-up. This

activity can also be used with M&Ms or Skittles. Students can estimate what fractions of the

candies are red, yellow, green, orange, brown, etc.

Summative:

• “Baking with Fractions” – In the kitchen at school or at home, allow students the opportunity

to make a recipe and do the measuring. For example, if the class makes brownies allow the

children to determine how the brownies would be cut so each student gets an equal share.

• “The Riddler” – Have students create riddles about real life solid shapes. Compile these

riddles into a class booklet.

• “Geometry in Our World” – Have students in your class create a booklet or PowerPoint with

pictures of real world items with different shapes. Students can find pictures using the

Internet or taking their own pictures with a camera.


1. Recognize and draw shapes having specified attributes, such as a given number of

angles or a given number of equal faces. [Sizes are compared directly or visually, not compared

by measuring.] Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

2. Partition a rectangle into rows and columns of same size squares and count to find the

total number of them.

3. Partition circles and rectangles into two, three, or four equal shares, describe the shares

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using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three

thirds, four fourths. Recognize that equal shares of identical wholes need not have the same

shape.


Math Vocabulary: angle, attribute, circle, corner, cube, edge, equal share, face, flat surface,

fraction, fourths, half, halves, hexagon, partition, pentagon, quadrilateral, rectangle, thirds, triangle,

& vertex/vertices

Math Concepts: Ability to differentiate between defining attributes and non-defining attributes;

Knowledge of two & three dimensional shapes; Knowledge of halves and quarters


ELA:

Religion: Use the book 3 in 1: A Picture Book of God by Joanne Markhausen to teach about the

Holy Trinity; Have students create a liturgical calendar that demonstrates the seasons of the

Church year.

Science: Using Oreo cookies, allow children to create the phases of the moon. By creating the

phases of the moon on the cookies, students practice working with fractions.

Social Studies: Using toothpicks and gumdrops/marshmallows, students can create 3-D

geometric shapes in order to form a neighborhood or community. Students should label the house

numbers in their community. Students should be able to explain how even number houses are on

side and odd number houses are on another side. This activity ties in with Standard M.2.D.

Understand Place Value.

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Children’s Literature: Circles, Triangles, and Squares by Tana Hoban;Color Zoo by Lois Ehlert;

Eating Fractions by Bruce McMillan; Fraction Action by Loreen Leedy; Fraction Fun by David A.

Adler; Gator Pie by Louise Matthews; The Greedy Triangle by Marilyn Burns; Listen to a Shape by

Grace Maccarone; The Village of Round and Square Houses by Ann Grifalconi

First in Math Website (http://www.firstinmath.com): K2 Pundi’s Puzzle & Triplets, Shape

Shuffle, Speed Shuffle

Manipulatives: fraction bars/circles, geoboards, interactive white board, pattern blocks

Technology: Ed U Smart (http://ed u smart.com/2ndGradeGeometry.aspx; http://ed u

smart.com/2ndFract.aspx)

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Grade 3

Grade Three

Page 1

Grade 3


Instructional time in Grade 3 focuses on four critical areas: (1) developing understanding of multiplication and division and strategies for

multiplication and division within 100; (2) developing understanding of fractions, especially unit fractions (fractions with numerator 1);

(3) developing understanding of the structure of rectangular arrays and of area; and (4) describing and analyzing two-dimensional shapes.

(1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems

involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown

factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown

group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based

on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution

strategies, students learn the relationship between multiplication and division.

(2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of

unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a

fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint

in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts

are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than,

and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on

noticing equal numerators or denominators.

(3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of

same size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for

measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By

decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to

determine the area of a rectangle.

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(4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and

angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of

a shape as a unit fraction of the whole.

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Grade Three

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Grade Three

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expression x2

+ 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a










3 + x




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Grade Three Standards for Mathematical Content

Operations and Algebraic Thinking (3.OA) Standards Strategies and Assessments

3. OA All students will represent and solve problems involving multiplication and

division.

1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of

objects in 5 groups of 7 objects each. Example: Describe a context in which

a total number of objects can be expressed as 5 × 7.

2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the

number of objects in each share when 56 objects are partitioned equally into 8 shares,

or as a number of shares when 56 objects are partitioned into equal shares of 8 objects

each. Example: Describe a context in which a number of shares or a number of

groups can be expressed as 56 ÷ 8.

3. Use multiplication and division within 100 to solve word problems in situations

involving equal groups, arrays, and measurement quantities, e.g., by using drawings

and equations with a symbol for the unknown number to represent the problem.

4. Determine the unknown whole number in a multiplication or division equation

relating three whole numbers. Example: Determine the unknown number that

makes the equation true in each of the equations 8×? = 48, 5 = ÷ 3, 6 × 6 = ?.


abundant and meaningful use of

physical and visual representations,

modeling, small group activities,

integration of technology, connections

to daily experiences, and writing about

mathematics.

Use a variety of manipulatives and visual

representations to explore multiplication.

For example: connecting cubes, tiles,

counters in the sections of an egg carton,

arrays, pictures, etc. Write addition and

multiplication sentences to match

situation.

Give students a large number of counters,

tiles, connecting cubes, etc. Describes

situations that require a group of objects

to be divided into equal shares. Use

different numbers and include some

situations that involve remainders.

Students explore division and

multiplication by reading books such as

The Doorbell Rang by Pat Hutchins; One

Hundred Hungry Ants by Elinor Pinczes,

Lucy and Tom’s 1, 2, 3 by Shirley

Hughes or Number Families by Jane

Srivastava.

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3.OA All students will understand properties of multiplication and the relationship

between multiplication and division.

5. Apply properties of operations as strategies to multiply and divide.

Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known (Commutative

Property of Multiplication). 3 × 5 × 2 can be found by 3× 5 = 15, then 15 × 2 = 30,

or by 5 × 2 = 10, then 3 × 10 = 30. (Associative Property of Multiplication). Knowing

that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) =

40 + 16 = 56. (Distributive Property).

6. Understand division as an unknown-factor problem. Example: Find 32 ÷ 8 by

finding the number that makes 32 when multiplied by 8.

3. OA All students will multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the relationship

between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows

40 ÷ 5 = 8) or properties of operations. (By the end of Grade 3, students should know

from memory all products of two one-digit numbers.)

3.OA All students will solve problems involving the four operations, and identify and

explain patterns in arithmetic.

8. Solve two-step word problems using the four operations. Represent these problems

using equations with a letter standing for the unknown quantity. Assess the

reasonableness of answers using mental computation and estimation strategies

including rounding.

9. Identify arithmetic patterns (including patterns in the addition table or multiplication

table), and explain them using properties of operations. Example: Observe that 4 times

a number is always even, and explain why 4 times a number can be decomposed into

two equal addends.

Prepare connecting cube towers of ones,

twos, threes, and fours. These will be

used to model the associative property of

multiplication. (2 x 4) x 3 =2 x (4 x 3).

Have students model two sets of four,

using towers. Next, following the

algorithm, students need to show two sets

of four three times. Students need to

count and recount (2 x 4) x 3 = 24.

Now, ask students to model four sets of 3

towers two times and record

2 x (4 x 3) = 24

Use base 10 blocks to model

multiplication sentences and properties.

Have students explore different ways to

group the blocks. For example, 3 x 12

may be modeled as (3 x 10) + (3 x 2) or

(3 x 5) + (3 x 7).

3 x 10 + 3 x 2

Have students circle or cross out the

multiples of a given factor on a hundred

chart and describe any pattern they see in

the numbers.

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Number and Operations in Base Ten (3.NBT)


3. NBT All students will use place value understanding and properties of operation topreform multi-digit arithmetic.

Use place value understanding to round whole numbers to the nearest 10 or 100..

1.

Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90(e.g., 9 × 80, 5 × 60) using strategies based on place value and properties ofoperations.

Use number lines to develop students’ understanding of rounding to the nearest multiple of 10 or 100.

Select and use a variety of strategies (e.g., rounding, regrouping, properties ) to estimate quantities, measures, and the results of whole-number computations up to three-digit whole numbers and amounts of money to $1000, and to judge the reasonableness of the answer. Have students explain the strategies they use. For example: 28 x 7 = (30 x 7) – (2 x 7) or 28 x 7 = (25 x 7) + (3 x 7).

2.

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Number and Operations—Fractions (3.NF) (Grade 3 expectations in this domain are limited to fractions with denominators of 2, 3, 4, 6, and 8.)


3.NF All students will develop an understanding of fractions as numbers.

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is

partitioned into b equal parts; understand a fraction a/b as the quantity formed

by a parts of size 1/b.

2. Understand a fraction as a number on the number line; represent fractions on a number

line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval

from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that

each part has size 1/b and that the endpoint of the part based at 0 locates the

number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths

1/b from 0. Recognize that the resulting interval has size a/b and that its

endpoint locates the number a/b on the number line.

3. Explain equivalence of fractions in special cases, and compare fractions by

reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the

same point on a number line.

b. Recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4, 4/6 = 2/3).

Explain why the fractions are equivalent (e.g., by using a visual fraction

model).

Use fraction bars, fraction circles, and

pattern blocks to model fractional parts of

a whole. Use counters, tiles, etc. to model

fractional parts of a set of objects.

Have students make fraction models of

rectangles (construction paper), circles

(paper plates or flattened cupcake papers).

Provide each student with multiple sheets

of the same shape and have students fold,

and cut into various fractional parts. One

rectangular sheet is cut in halves, another

sheet into thirds, another into fourths, etc.

Label part (i.e., 1/2, 1/4…).

Give students a 12 inch long paper strip

with a line drawn from left to right and the

endpoints label 0 and 1. Have students fold

the strip in half, keep folded and fold in

half two more times. Unfold and label in

intervals of 1/8 at each crease. Where

would 1/2 be? 1/4? 3/4?

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c. Express whole numbers as fractions, and recognize fractions that are

equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1;

recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line

diagram.

d. Compare two fractions with the same numerator or the same denominator

by reasoning about their size. Recognize that comparisons are valid only

when the two fractions refer to the same whole. Record the results of

comparisons with the symbols >, =, or <, and justify the conclusions

(e.g., by using a visual fraction model).

Use the rectangular fraction models made

by the children with the fractional pieces

cut apart to play a cover-up game. Make a

fraction die or set of index cards with

appropriate fractions on them for each

group of three or four children. The object

of the game is to be the first to completely

cover the ―whole‖ rectangle. If a student

rolls or picks 1/2 and only needs 1/4 to

complete a strip, allow the child to trade

the 1/2 piece for two that are 1/4 each.

Use a fraction bar chart as a visual model

for comparing fractional parts.

Measurement and Data (3.MD) Standards Strategies and Assessments

3.MD All students will solve problems involving measurement and estimation of

intervals of time, liquid volumes, and masses of objects.

1. Tell and write time to the nearest minute and measure time intervals in minutes.

Solve word problems involving addition and subtraction of time intervals in

minutes( e.g., by representing the problem on a number line diagram).

Use duration of classroom activities and

students’ daily schedule to create problems

involving time.

Have a collection of various size containers

available in the classroom so students can

explore different units for measuring liquid

volume.

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2. Measure and estimate liquid volumes and masses of objects using standard units

of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to

solve one-step word problems involving masses or volumes that are given in the

same units to represent the problem (e.g., by using drawings, such as a beaker with a

measurement scale).


3. Draw a scaled picture graph and a scaled bar graph to represent a data set with

several categories. Solve one- and two-step ―how many more‖ and ―how many less‖

problems using information presented in scaled bar graphs. Example: Draw a bar

graph in which each square in the bar graph might represent 5 pets.

4. Generate measurement data by measuring lengths using rulers marked with inches,

halves and fourths of an inch, or centimeters. Show the data by making a line plot,

where the horizontal scale is marked off in appropriate units—whole numbers,

halves, or quarters.

Help student establish referents for

weight/mass by creating and displaying a

chart showing common items and their

weight.

Have students practice reading the

measurement scale on a measuring cup or

beaker and comparing measurements.

Data investigations are logical ways to

integrate other areas of the curriculum by

having students gather data to answer a

number of questions:

• Science - weather records, plant growth

• Social Studies - states visited by students,

population of surrounding towns

• Physical Education - amount of time to

walk a given distance, number of

jumping jacks completed in a given time.

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3.MD Geometric Measurement: All students will understand concepts of area and

relate area to multiplication and to addition.

5. Recognize area as an attribute of plane figures and understand concepts of

area measurement.

a. A square with side length 1 unit, called ―a unit square,‖ is said to

have ―one square unit‖ of area, and can be used to measure area.

b. A plane figure which can be covered without gaps or overlaps by

n unit squares is said to have an area of n square units.

6. Measure areas by counting unit squares (square cm, square m, square inches,

square ft, and improvised units).

7. Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it,

and show that the area is the same as would be found by multiplying the

side lengths.

Display square units of various lengths:

a square inch, a square centimeter, a square

foot, a square meter. Discuss which unit

would be the best to use to determine the

area of various surfaces or regions.

Have students use grid paper to draw as

many different figures as they can with an

area of 12 units; 7 units.

Draw a number of rectangles of varying

sizes on grid paper. Identify the length and

width of each rectangle. Determine the

number of square units within the

rectangle. Display the information in a

table and have students determine the

relationship between length and width and

area.

Length Width Area 8 units 6 units 48 sq. units

3 units 7 units 21 sq. units

10 units 5 units 50 sq. units

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b. Multiply side lengths to find areas of rectangles with whole number

side lengths in the context of solving real world and mathematical problems,

and represent whole-number products as rectangular areas in mathematical

reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with

whole number side lengths a and b + c is the sum of a × b and a × c.

Use area models to represent the distributive property in mathematical

reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing

them into non-overlapping rectangles and adding the areas of the non-

overlapping parts, applying this technique to solve real world problems.

3. MD Geometric Measurement: All students will recognize perimeter as an attribute

of plane figures and distinguish between linear and area measures.

8. Solve real world and mathematical problems involving perimeters of polygons,

including finding the perimeter given the side lengths, finding an unknown side

length, and exhibiting rectangles with the same perimeter and different areas or

with the same area and different perimeters.

Have students cut out rectangles of various

sizes. Label each rectangle with its area.

Place the rectangles together on a mat to

create a floor plan. Determine the total

area of the space covered by the floor plan.

Determine the perimeter of polygons

created with tangram pieces placed on grid

paper.

Have students draw pictures on grid paper

to match descriptions of polygons based on

perimeter. Example: ―Draw a square with

a perimeter of 40 units.‖ ―Draw a

rectangle with a perimeter of 20 inches.

One of its sides is 6 inches.‖

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3.G All students will reason with shapes and their attributes.

1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and

others) may share attributes (e.g., having four sides), and that the shared attributes can

define a larger Domain (e.g., quadrilaterals). Recognize rhombuses, rectangles,

and squares as examples of quadrilaterals, and draw examples of quadrilaterals that

do not belong to any of these subcategories.

2. Partition shapes into parts with equal areas. Express the area of each part as a unit

fraction of the whole. Example: Partition a shape into 4 parts with equal area, and

describe the area of each part as 1/4 of the area of the shape.

Use a chart to compare attributes of various

quadrilaterals. Shape Four sides All sides = Angles =

Square Yes Yes Yes

Rhombus Yes Yes No

Rectangle Yes No Yes

Trapezoid Yes No No

Draw a shape on grid paper. Determine

number of square units covered by the

shape. Divide total square units by the

number of equal parts the shape is to be

divided into. Example: Shape covers 20

square units. Partition into 5 equal parts.

Each part is 4 square units. Each part is

1/5 of the area of the shape. Color each

equal part a different color.



Why is it important to have multiplication/division as a tool in my life?

Are there times when knowledge of multiplication/division will make my life easier?

How will fluency in multiplication and division frame my interaction with real life situations?

How can what I know about addition help me with multiplication?

How can what I know about subtraction help me with division?


Formative: I Can/Not Yet, Exit cards with examples of solved problems, Label groups in 3

categories; #1--I can do this and want word problems to practice; #2--need more practice and

#3-- have no idea, need teacher help. Math journaling

Find someone who….Review Students explain what they heard during the lesson to another

student, agree/disagree/discuss

Homework check/board work

Summative: Performance Assessments, Chapter test

Domain: Operations and Algebraic Thinking (3.OA) - Represent and solve problemsinvolving multiplication and division

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1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in

5 groups of 7 objects each. For example, describe a context in which a total number of objects

can be expressed as 5 × 7.

2. Interpret whole number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number

of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of

shares when 56 objects are partitioned into equal shares of 8 objects each. For example,

describe a context in which a number of shares or a number of groups can be expressed as 56 ÷

8.

3. Use multiplication and division within 100 to solve word problems in situations involving

equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a

symbol for the unknown number to represent the problem.

4. Determine the unknown whole number in a multiplication or division equation relating

three whole numbers. For example, determine the unknown number that makes the equation true

in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.


Adding equal groups, repeated addition, being able to break apart numbers into equal groups,

repeated subtraction, vocabulary-array, factor, product, divisor, dividend, quotient, multiples

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Real life, buying multiples dividing numbers of people into tables at parties, making equal groups

for field trips, games, etc.


Manipulatives, graph paper, multiplication table, calculator, math websites

Iknowthat.com; Literature: Hershey's Milk Chocolate Multiplication Book

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Domain: Operations and Algebraic Thinking (3.OA) - Understand properties of multiplication and the relationship between multiplication and division.


1. How does my knowledge about multiplication facts help me to solve problems?

2. How can I use the array model to explain multiplication?

3.When I manipulate numbers in multiplication does it make a difference in the

answer?


Formative: flashcards, Around the World, pair quizzing, show me with either manipulatives,

whiteboards, paper, pencil, teacher question, observation.

Summative: Create the Room – Children can create classroom arrangements using

different multiplication examples, Demonstration using manipulatives, i.e., gram cubes, etc.

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1. Apply properties of operations as strategies to multiply and divide. [Students need not use

formal terms for these properties.] Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also

known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 ×

2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8

× 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56.

(Distributive property.)

2. Understand division as an unknown factor problem. For example, find 32 ÷ 8 by finding

the number that makes 32 when multiplied by 8.


Essential multiplication facts, if you know, 3x5= 15, you also know 5x3=15 Commutative property

Mastering 5x12 is the same as 5x10+ 5x2 or 5(10+2)= Distributive Property



Planting in a garden, rows of plants, cupcakes in a box, desk arrangements

Planning arrangements for seating

Social Studies - reading graphs with a key meaning more than one Science

Calculating measurements

http://www.vtaide.com/png/twinfacts.htm

http://www.vtaide.com/png/multiplication.htm

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Domain: Operations and Algebraic Thinking (3.OA) - Multiply and divide within 100.


How does fluency in multiplication and division help me?

Why is it important for me to know multiplication and corresponding division facts?


Formative:Think, Pair, Share. Think out Loud, I Can, Not Yet. Self label groups-- 3 I can do

this and want word problems to practice, 2 need more practice ,1 have no idea /need teacher

help. Summative: Students will say a times table fluently.


Understand numbers, ways of representing numbers, relationships among numbers, and number

systems.

Understand that multiplication is repeated addition and is a shorter and faster way of computation.

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Fluently multiply and divide within 100, using strategies such as the relationship between

multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of

operations. By the end of Grade 3, know from memory all products of two one-digit numbers.



Anywhere-multiples-of-numbers-are-needed,-groups-of-equal-numbers.-

Science: multiples

Social-Studies: graphs

Multiplication-chart,-manipulatives-of-any-kind,-gram-cubes-are-great,-100-chart,-calculators-

http://www.iknowthat.com, http://www.math-drills.com/multiplication.shtml

Children’s literature website-http://mathforum.org/t2t/faq/gail/lit.html

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Domain: Operations and Algebraic Thinking Standard (3.OA) - Solve problems involving the four operations, and identify and explain patterns in arithmetic.


Are there times when I need to use more than one operation to solve a problem?

How can I figure out what operation to do and when to do it?

How will number patterns help me with word problems?



Formative: Exit cards with examples of solved problems on whiteboards, hold up your solutions.

Agree/disagree, discuss

Summative: Solve word problems using more than one operation. Explain how it was done

either written or orally.

1. Solve two step word problems using the four operations. Represent these problems using

equations with a letter standing for the unknown quantity. Assess the reasonableness of answers

using mental computation and estimation strategies including rounding. [This standard is limited

to problems posed with whole numbers and having whole-number answers; students should

know how to perform operations in the conventional order when there are no parentheses to

specify a particular order (Order of Operations).]

2. Identify arithmetic patterns (including patterns in the addition table or multiplication table),

and explain them using properties of operations. For example, observe that 4 times a number is

always even, and explain why 4 times a number can be decomposed into two equal addends.

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Addition, subtraction, multiplication, and division skills. Recognition of wording used for each

operation, ie, all together, how many are left, equal number of groups, how many in each

group, identifying 4 types of subtraction problems.

Calculating perimeter

Preparing materials in science for an experiment or science fair project

Prepare ingredients for baking at home for a class snack or dinner for multiple guests Tools for Learning Which tools will I use that will assist me in my learning? 24 cards, www.firstinmath.com , http://www.internet4classrooms.com/word_problems_quest.htm

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Domain: Numbers and Operations in Base Ten (3.NBT) - Use place value understanding and properties ofoperations to perform multi-digit arithmetic.


Why do I need to know the places for numbers?

How will knowing number places help me set up problems for different operations?

When will I round rather than calculate the exact answer in math?


Formative: Physical body use Using multiple sets of numbers with a start and end points the

children organize their numbers from least to greatest or greatest to least. .Decisions can be made

to decide which numbers are closer to the endpoints in the lineup? Hold up the answer, Yes or no

answer cards.

Summative: Given particular numbers, explain how they have to be ordered to that they show

greatest to least, or least to greatest.


1. Use place value understanding to round whole numbers to the nearest 10 or 100.

2. Fluently add and subtract within 1000 using strategies and algorithms based on place

value, properties of operations, and/or the relationship between addition and subtraction.

3. Multiply one digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 ×

60) using strategies based on place value and properties of operations.

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Understand comparing and ordering numbers, greater than and less than, wording “about how

many”.

Addition and subtraction skills

Multiplication tables

Understand that the four digits of a four digit number represent amounts of thousands,

hundreds, tens, and ones; e.g. 3,706 = 3000 + 700 + 6 = 3 thousands, 7 hundreds, 0 tens, and

6 ones.

Estimation of crowds at a gathering World Events, Olympics, Sports game

Estimation and calculation of money needed for a purchase.


Place value charts, http://www.superteacherworksheets.com/place-value.html

Ruler, meter stick, number lines

http://www.oswego.org/ocsd-web/games/DogBone/gamebone.html

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Domain: Numbers and Operations- Fractions (3.NF) - Develop understanding of fractions as numbers. Limited to denominators 2,3,4,6 and 8


How do I show a number when it’s smaller than one whole?



What are fractions? What can I do with them? Why do I need to know about them?

Formative: demonstration using equal size shapes, that ½ of a shape is the same size as 2/4 of the same

size shape, Show Me on whiteboards/chalkboards/Smartboard different fractions.

Summative: http://www.rda.aps.edu/mathtaskbank/pdfs/tasks/3 =/ t3LotsChoc.pdf

Create a chart showing how different names of fractions mean the same thing i.e., 5 equal circles the first

whole, the next showing ½, the next 2/4 the next 3/6 and the last 4/8 illustrating that they are all the same

size.

Make a fraction booklet with the same idea as above

Fold a piece of paper with 4 folds, showing 1/2 =2/4=4/8

Make a mobile using equal sized pieces of paper folded into the different equivalent fractions and a

hanger/rod

1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into bequal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

2. Understand a fraction as a number on the number line; represent fractions on a number line

diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and

partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part

based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that

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the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about

their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number

line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the

fractions are equivalent, e.g., by using a visual fraction model.

c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.

Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a

number line diagram.

d. Compare two fractions with the same numerator or the same denominator by reasoning about their

size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record

the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual

fraction model.

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vocabulary, parts of a whole, equal pieces, multiples

multiplication times tables, >, <


Science - dividing an amount of liquid or solid into equal pieces.

Sharing a whole pizza/cookie/project etc. equally among people.


The Hershey's Milk Chocolate Bar Fractions Book by Jerry Pallotta

http://www.songsforteaching.com/math/fractions/equivalentfractions.htm

Fraction Fun by David Adler

Rulers, meter stick, number line

Fraction bar pieces

http://nlvm.usu.edu/en/nav/frames_asid_105_g_2_t_1.html

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Domain: Measurement and Data (3.MD) - Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.


Is it important to tell time? How can knowing how to tell time help me?

How can I measure liquids? What do I use to measure liquids or solids?

Can I add or subtract amounts of liquids or solids?


Formative: Show me, write in math journals, explain/demonstrate measuring liquids/solids using

grams, kilograms, milliliters, and liters. Graduates and beakers and be used to show

measurements. Clock manipulatives: say a time, show a time, draw the time.

Summative: Tell the correct time at a designated time, estimate and calculate liquid and mass

measurements in a given experiment.


1. Tell and write time to the nearest minute and measure time intervals inminutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.

2. Measure and estimate liquid volumes and masses of objects using

standard units of grams (g), kilograms (kg), and liters (l). [Excludes compound units such as cm3and finding the geometric volume of a container.] Add, subtract, multiply, or divide to solve one step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem. [Excludes multiplicative comparison problems (problems involving notions of “times as much”]

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Need to know skip counting by 5, and 10, minutes in an hour, ½ hour and ¼ hour, grams,

kilograms, liters


Science: measuring liquids and solids. Time: reading schedules, calculating available time to

complete work and calculating how much time is left


Number lines, rulers, graduates, balances, beakers, scales, clock faces

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Domain: Measurement and Data (3.MD) - Represent and Interpret Data




What are graphs used for?

Can graphs tell us any information?

How can I make a picture graph or a bar graph?Formative: Show what you know! Thumbs up/thumbs down.

Summative: Create a survey, collect data and make a bar or pictograph using units meaning one

and/or parts representing more than one. Write coordinating word problems to compare/contrast

the data in the graph.

1. Draw a scaled picture graph and a scaled bar graph to represent a data set with several

categories. Solve one- and two-step “how many more” and “how many less” problems using

information presented in scaled bar graphs. For example, draw a bar graph in which each square

in the bar graph might represent 5 pets.

2. Generate measurement data by measuring lengths using rulers marked with halves and

fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off

in appropriate units—whole numbers, halves, or quarters.

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Greater than, less than signs, fractions. Addition and subtraction skills


Interpreting graphs in Social Studies, newspapers etc.


Rulers, graph or centimeter paper, Means of collecting data i.e., clipboard

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Domain: Measurement and Data (3.MD) - Geometric measurement: understand concepts of area and relate area to multiplication and to addition.




What is area used for?

What are some other ways I can use times tables to help calculate the area of a shape?

Formative: timed and untimed multiplication tests Show me what 3x5 looks like. Use counters

or gram cubes to create squares and rectangles with specific dimensions.

Summative: Math Journals Students use journal writing to demonstrate knowledge obtained.

Build arrays given specific directions ie. Create the floor of the new doghouse using these

dimensions Multiplication: times tables Distributive property

Vocabulary: array, plane figures, rectangle, square, column, row, dimension

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1. Recognize area as an attribute of plane figures and understand concepts of area

measurement.

a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of

area, and can be used to measure area.

b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have

an area of n square units.

2. Measure areas by counting unit squares (square cm, square m, square in, square ft, and

improvised units).

3. Relate area to the operations of multiplication and addition.

a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area

is the same as would be found by multiplying the side lengths.

b. Multiply side lengths to find areas of rectangles with whole-number side lengths in the context

of solving real world and mathematical problems, and represent whole-number products as

rectangular areas in mathematical reasoning.

c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side

lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive

property in mathematical reasoning.

d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-

overlapping rectangles and adding the areas of the non-overlapping parts, applying this

technique to solve real world problems.

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Measuring carpets, flooring, finding space available to use in a specific place

Engineering, architecture, design, sewing picture, framing


Graph paper, Times table charts, websites for timed practice

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Are shapes important in our world?

Can I create different shapes with lines? What are other shapes I can create?

How are shapes used in our world? How do I calculate the distance around and inside shapes?



Formative: Draw shapes with specific sides, construct shapes using gram cubes, make cards with

a specific perimeter.

Summative: Design a floor plan using given dimensions, calculate the area of the floor, walls etc.

“How much space do I have?” Mini Martha lives in a mini house, her floor is 9 X 12 feet, she needs

a 7x 5 space for her couch and a 4x4 space for each of her 2 chairs. She also has a coffee table

that needs a 4x6 space. Does she have enough space to fit everything?

Solve real world and mathematical problems involving perimeters of polygons, including

finding the perimeter given the side lengths, finding an unknown side length, and exhibiting

rectangles with the same perimeter and different areas or with the same area and different

perimeters.

Domain: Measurement and Data (3.MD) - Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

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Students will learn polygon shapes and definitions - triangle, square, rectangle, parallelogram,

pentagon, octagon

Students will learn that perimeter is the distance around a polygon and area is the distance inside

the shape.


Construction, buildings, design architecture, sewing


Graph paper, rulers, interactive boards, white boards

http://www.shodor.org/interactivate/lessons/LengthPerimeterArea/

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Domain: Geometry (3.G) - Reason with shapes and their attributes.




Are shapes of figures important? How are shapes represented in our world? How can I group

shapes to classify them? Does the shape of a figure matter in drawing or building? Can shapes be

combined to create new shapes?

Formative: Observe and interact with students to determine their level of understanding. Partner to

partner paraphrase, journal records, question/response, sketch understanding, pair share

summary, quizzes. Sort and classify shapes by their attributes.

Summative: “Which shape is it?” Using a list of criteria, the student will identify and/or draw the

shape based on the description. How many equal fourths, thirds, can you divide this shape into?

1. Understand that shapes in different categories (e.g., rhombuses,

rectangles, and others) may share attributes (e.g., having four sides), and that the shared

attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles,

and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not

belong to any of these subcategories.

2. Partition shapes into parts with equal areas. Express the area of each part

as a unit fraction of the whole. For example, partition a shape into 4 parts with equal

areand describe the area of each part as 1/4 of the area of the shape.

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Students should know polygon shapes and definitions of square, rectangle, parallelogram, and

others.

Recognize that 4 sided shapes have similarities and differences.

Vocabulary angle, acute, obtuse, quadrilateral halves, thirds, fourths

Knowledge of fractions


Architecture, engineering, landscaping, carpentry


Text books, trade books, graphic organizer, virtual manipulatives

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Grade 4

Grade Four

Page 1

Grade 4


Instructional time in Grade 4 focuses on three critical areas: (1) developing understanding and fluency with multi-digit multiplication,

and developing understanding of dividing to find quotients involving multi-digit dividends; (2) developing an understanding of fraction

equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers;

(3) understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular

sides, particular angle measures, and symmetry.

(1) Students generalize their understanding of place value to 1,000,000, understanding the relative sizes of numbers in each place. They

apply their understanding of models for multiplication (equal-sized groups, arrays, area models), place value, and properties of operations,

in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products

of multi-digit whole numbers. Depending on the numbers and the context, they select and accurately apply appropriate methods to

estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and

explain why the procedures work based on place value and properties of operations; and use them to solve problems. Students apply their

understanding of models for division, place value, properties of operations, and the relationship of division to multiplication as they

develop, discuss, and use efficient, accurate, and generalizable procedures to find quotients involving multi-digit dividends. They select

and accurately apply appropriate methods to estimate and mentally calculate quotients, and interpret remainders based upon the context.

(2) Students develop understanding of fraction equivalence and operations with fractions. They recognize that two different fractions can

be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions. Students extend previous

understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit

fractions, and using the meaning of fractions and the meaning of multiplication to multiply a fraction by a whole number.

(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional

shapes, students deepen their understanding of properties of two-dimensional objects and the use of them to solve problems involving

symmetry.

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expression x2

+ 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a










3 + x




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Grade Four Standards for Mathematical Content



4.OA All students will use the four operations with whole numbers to solve problems.

1. Interpret a multiplication equation as a comparison (e.g., interpret 35 = 5 × 7 as a

statement that 35 is 5 times as many as 7 and 7 times as many as 5).

Represent verbal statements of multiplicative comparisons as multiplication

equations.

2. Multiply or divide to solve word problems involving multiplicative

Comparison (e.g., by using drawings and equations with a symbol for the unknown

number to represent the problem) distinguishing multiplicative comparison from

additive comparison.

3. Solve multi-step word problems posed with whole numbers and having

whole-number answers using the four operations, including problems in which

remainders must be interpreted. Represent these problems using equations

with a letter standing for the unknown quantity. Assess the reasonableness of

answers using mental computation and estimation strategies including rounding.


abundant and meaningful use of physical and

visual representations, modeling, small group

activities, integration of technology,


about mathematics.

Broaden students’ understanding of

multiplication as repeated addition by dealing

with situations involving arrays, expansions,

and combinations. Multiplication can be used

to answer such questions as: How many

stickers are on a 7 by 8 sheet? How old is

Jordan if he is five times as old as his 2- year

old brother? How many outfits can you make

with 2 pairs of pants and 3 shirts? There are

12 girls in our class. This is twice the number

of girls in Grade 5. How many girls are there

in Grade 5?

Have students work with a set of problems that

involve the same division problem but have

different answers. Example: How many cars

will we need to transport 19 people if each car

holds 5? How many more packages of 5 ping-

pong balls can be made if there are 19 balls

left in the bin? How much does each of 5

children have to contribute to the cost of a $19

gift?

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4.OA All students will gain familiarity with factors and multiples.

4. Find all factor pairs for a whole number in the range 1–100. Recognize that a

whole number is a multiple of each of its factors. Determine whether a given

whole number in the range 1–100 is a multiple of a given one-digit number.

Determine whether a given whole number in the range 1–100 is prime or composite.

4.OA All students will be able to generate and analyze patterns.

5. Generate a number or shape pattern that follows a given rule. Identify apparent

features of the pattern that were not explicit in the rule itself.

Have students work together as ―Divisibility

Detectives.‖ Divide the class into groups.

Assign each group one of the following

numbers- 3, 4, 5, 6, or 9. Have them list ten

3-digit multiples of the number they’ve been

assigned. Tell them to examine their list of

multiples to see if they can detect a divisibility

rule for the multiples of their assigned number.

Given the rule ―Add 3‖ and the starting

number 1, generate terms in the resulting

sequence and observe that the terms appear to

alternate between odd and even numbers. Ask

students to explain why the numbers will

continue to alternate in this way.

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Number and Operations in Base Ten (4.NBT) (Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.)


4.NBT All students will generalize place value understanding for multi-digit whole

numbers.

1. Recognize that in a multi-digit whole number, a digit in one place represents ten

times what it represents in the place to its right. Example: Recognize that

700 ÷ 70 = 10 by applying concepts of place value and division.

2. Read and write multi-digit whole numbers using base-ten numerals, number names,

and expanded form. Compare two multi-digit numbers based on meanings of the

digits in each place, using >, =, and < symbols to record the results of comparisons.

3. Use place value understanding to round multi-digit whole numbers to any place.

4.NBT All students will use place value understanding and properties of operations

to perform multi-digit arithmetic.

4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

5. Multiply a whole number of up to four digits by a one-digit whole number, and

multiply two two-digit numbers, using strategies based on place value and the

properties of operations. Illustrate and explain the calculation by using equations,

rectangular arrays, and/or area models.

Point out place value patterns:

8 x 10 = 80

80 x 10 = 800

800 x 10 = 8,000

Students locate numbers as points on a

number line extending across the room,

continuing to attach labels as they learn

more about numbers. Paper clips or tape are

used to fasten equivalent forms of a number

to the same point.

Provide students with real life situations that

may require rounding. In a grocery store,

for example, when a person wants to be sure

there is enough money to pay for items that

cost $1.89, $2.95, and $4.45, the best

strategy may be to round each price up to

the next dollar and compare the estimated

total to the amount of money available to

spend.

Use open arrays to help students to model

multiplication problems. Example: 36 x 27

30 6

20 x 30 20 x 6

7 x 30 7 x 6

20

7

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6. Find whole-number quotients and remainders with up to four-digit dividends and

one-digit divisors, using strategies based on place value, the properties of

operations, and/or the relationship between multiplication and division. Illustrate

and explain the calculation by using equations, rectangular arrays, and/or area

models.

Use base ten blocks to help students

understand the process of division.

Use the problem 127 Build 127 with

base ten blocks using one flat (100), two

longs (10) and seven cubes (ones). In order

to divide the flat into six groups, it must first

be traded for ten longs. These ten longs

along with the two longs create 12. These 12

longs are distributed into six piles with two

in each. Now the seven cubes are distributed

so that one goes in each pile and there is one

left.

Number and Operations—Fractions (4.NF) (Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.)


4.NF All students will be able to extend understanding of fraction equivalence and

ordering.

1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual

fraction models, with attention to how the number and size of the parts differ even

though the two fractions themselves are the same size. Use this principle to

recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators

(e.g., by creating common denominators and renaming the numerators, or by

comparing to a benchmark fraction such as ½). Recognize that comparisons are

valid only when the two fractions refer to the same whole. Record the results of

comparisons with symbols >, =, or <, and justify the conclusions (e.g., by using a

visual fraction model).

The number ―one‖ comes in many forms.

Use it to make an equivalent fraction.

43 x 2

2 = 86 (one is in disguise as 2

2).

Have students fold a piece of paper in half,

and shade one half with crayon. Ask,

―What fractional part is shaded?‖ Fold the

paper in half again and ask,‖ How many

equal parts are there?‖ ―What fractional part

is shaded?‖ continue folding and discussing

the number of equal parts and the size of the

shaded part.

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Identify the numerator and denominator of a fraction and understand what each

refers to.

4.NF All students will build fractions from unit fractions by applying and extending

previous understandings of operations on whole numbers.

3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating

parts referring to the same whole.

b. Decompose a fraction into a sum of fractions with the same denominator

in more than one way, recording each decomposition by an equation.

Justify decompositions (e.g., by using a visual fraction model).

c. Add and subtract mixed numbers with like denominators (e.g., by

replacing each mixed number with an equivalent fraction) and/or by

using properties of operations and the relationship between addition and

subtraction.

d. Solve word problems involving addition and subtraction of fractions

referring to the same whole and having like denominators (e.g., by using

visual fraction models and equations to represent the problem).

4. Apply and extend previous understandings of multiplication to multiply a

fraction by a whole number.

a. Understand a fraction a/b as a multiple of 1/b.

Use number lines and fractions bars to

compare and order fractions.

Show three congruent rectangles: one with

1/4 shaded; one with 2/3 shaded; and one

with 7/8 shaded. Use this visual

representation to discuss how to determine

if a fraction is closer to 0, 1/2, or 1.

N ―u”merator (upstairs)

“D”enominator (downstairs)

Use a visual fraction model to represent 3/4

as the sum of 1/4 + 1/4 + 1/4 , recording the

conclusion by the equation 1/4 + 1/4 + 1/4

=3/4.

Have students identify and model different

combinations of fractions with like

denominators that equal the same sum

Examples:

3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8

2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

Use visual representations to show the

fraction equivalent of mixed numbers.

2 21 = 5/2

2/2 + 2/2 + 1/2 = 5/2

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b. Understand a multiple of a/b as a multiple of 1/b, and use this

understanding to multiply a fraction by a whole number.

(In general, n × (a/b) = (n × a)/b.)

c. Solve word problems involving multiplication of a fraction by a

whole number (e.g., by using visual fraction models and equations

to represent the problem). Example: If each person at a party will eat 3/8 of

a pound of roast beef, and there will be 5 people at the party, how many

pounds of roast beef will be needed? Between what two whole numbers

does your answer lie?

4.NF All students will understand decimal notation for fractions, and compare

decimal fractions.

5. Express a fraction with denominator 10 as an equivalent fraction with denominator

100, and use this technique to add two fractions with respective denominators 10

and 100.

6. Use decimal notation for fractions with denominators 10 or 100.

7. Compare two decimals to hundredths by reasoning about their size. Recognize that

comparisons are valid only when the two decimals refer to the same whole. Record

the results of comparisons with the symbols >, =, or <, and justify the conclusions

(e.g., by using a visual model).

Use a visual fraction model to represent 5/4

as the product 5 × (1/4 ), recording the

conclusion by the equation 5/4 = 5 × (1/4).

Refer to previous work with multiplication

of whole number wherein the sum of

6 + 6 + 6 is the same as 3 × 6.

1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 5 × (1/4) = 5/4.

Use a visual fraction model to express 3 ×

(2/5) as 6 × (1/5), recognizing this product

as 6/5.

Use a 10 x 10 centimeter grid to show that

3/10 is equivalent to 30/100. Add 3/10 +

4/100 = 30/100 + 4/100 = 34/100.

Rewrite 0.62 as 62/100; describe a length as

0.62 meters; locate 0.62 on a number line

diagram.

Journal writing using money: ―Would you

rather 1 dime (.10) or 1 penny (.01)? Why?‖

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4.MD All students will solve problems involving measurement and conversion of

measurements from a larger unit to a smaller unit.

1. Know relative sizes of measurement units within one system of units including km,

m, cm; kg, g; l, ml; yd., ft, in; lb, oz; gal, qt, pt, c; hr, min, sec. Within a single

system of measurement, express measurements in a larger unit in terms of a smaller

unit. Record measurement equivalents in a two column table.

2. Use the four operations to solve word problems involving distances, intervals of

time, liquid volumes, masses of objects, and money, including problems involving

simple fractions or decimals, and problems that require expressing measurements

given in a larger unit in terms of a smaller unit. Represent measurement quantities

using diagrams such as number line diagrams that feature a measurement scale.

3. Apply the area and perimeter formulas for rectangles in real world and

mathematical problems.


4. Make a line plot to display a data set of measurements in fractions of a unit (1/2,

1/4, 1/8). Solve problems involving addition and subtraction of fractions by using

information presented in line plots.

Know that 1 ft is 12 times as long as 1 in.

Express the length of a 4 ft. snake as 48 in.

Generate a conversion table for feet and

inches listing the number pairs (1, 12),

(2, 24), (3, 36), ...

Use the book Counting on Frank by Rod

Clement to generate discussion and activities

related to measurement.

Students find out how many inches long their

hand is. The class then generates a graph

showing the results and write word problems

based on the results presented in the graph.

Students investigate truth-in-packaging by

reading labels, estimating weights, and then

using balance scales to weigh foods.

Find the width of a rectangular room given

the area of the flooring and the length of the

room. Present the area formula as a

multiplication equation with an unknown

factor.

From a line plot find and interpret the

difference in length between the longest and

shortest specimens in an insect collection.

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Strategies and Assessments

4.MD Geometric Measurment: All students will understand concepts of anglesand angle measurement

5. Recognize angles as geometric shapes that are formed wherever two rays share acommon endpoint, and understand concepts of angle measurement.

a. An angle is measured with reference to a circle with its center at the commonendpoint of the rays, by considering the fraction of the circular arc between thepoints wher two rays intersect the circle. An angle that turns through 1/360 of acircle is called a "one degree angle", and can be used to measure angles

b. An angle that turns through n one-degree angles is said to have anangle measure of n degrees.

Explore angles using AngLegs or the virtual circular geo-board at: http://nlvm.usu.edu/en/nav category_g_2_t_3.html

Have students use a virtual protractor to practice measuring angles. www.mathplayground.com/measuringangles.html

4.MD Geometric Meassurment continued on nextpage.

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6. Measure angles in whole-number degrees using a protractor. Sketch angles of

specified measure.

7. Recognize angle measure as additive. When an angle is decomposed into non-

overlapping parts, the angle measure of the whole is the sum of the angle measures

of the parts. Solve addition and subtraction problems to find unknown angles on a

diagram in real world and mathematical problems: e.g., by using an equation with a

symbol for the unknown angle measure.

Students are given pictures of objects that

have right angles. A ray is drawn in the angle

from the vertex that will split the angle into

two angles. The students measure each angle

and see that the two angles add to 90°.

Vocabulary such as complementary, adjacent,

vertex and ray should be discussed.


4.G All students will draw and identify lines and angles, and classify shapes by

properties of their lines and angles.

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and

perpendicular and parallel lines. Identify these in two-dimensional figures.

2. Classify two-dimensional figures based on the presence or absence of parallel or

perpendicular lines, or the presence or absence of angles of a specified size.

Recognize right triangles as a category, and identify right triangles.

3. Recognize a line of symmetry for a two-dimensional figure as a line across the

figure such that the figure can be folded along the line into matching parts.

Identify line-symmetric figures and draw lines of symmetry.

―Geometry Memory‖: make sets of index

cards with picture on top and definition on

bottom. Cut in half and use the parts to play

the game.

Use straws, toothpicks or coffee stirrers to

form various two-dimensional figures;

identify different types of lines and angles in

the figures.

Create a model of basketball court, football

field, baseball field, soccer field or other

sport venues to illustrate and identify

attributes of geometric figures.

Use dot paper to create simple designs for

partners to match and complete second half

of design across a line of symmetry.


Domain: Operations and Algebraic Thinking (4.OA) - Use the four operations with wholenumbers to solve problems


How will knowledge of problems solving help me function in everyday life?

How do I choose which operation to use to solve a word problem?

What strategies can I use to help me solve word problems?


Formative: Homework, board work, multiplication drill or bee,

3 2 1 summarizer/ 3 list strategies, 2 give important things to look for when solving problem,

1 solution to the problem


Summative: Chapter Test, quiz, performance assessment

1.Interpret a multiplication equation as a comparison, e.g. interpret 35 = 5 x 7 as a statement that

35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of

multiplicative comparisons as multiplication equations.

2.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using

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drawings and equations with a letter standing for the unknown quantity. Assess the

reasonableness of answers using mental computation and estimation strategies including

rounding. 3. Solve multistep word problems posed with whole numbers and having whole number

answers using the four operations, including problems in which remainders must be interpreted.

Represent these problems using equations with a letter standing for the unknown quantity. Assess

the reasonableness of answers using mental computation and estimation strategies including

rounding.



Fluency in the multiplication facts 1 – 10, including visual and verbal representation e.g. If 3 boys

go to the store and buy 7 baseball cards and 7 boys go the store and buy 3 baseball cards. Did

they buy the same amount? Justify your answer.

Missing factors e.g. 45 chairs were arranged in 5 rows. How many in each row? 5 x __ = 45

Strategies and Steps in Problem solving – draw a picture or model, make a table, diagram or list,

Write an equation

Inverse operations

Estimation strategies – front end/clustering/rounding/rounding and adjusting/compatible numbers

Economics – money

Science – weights and measurements

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79


Flash cards

Dry Erase Board

Games –Acting Out

www.ixl.com/math/grade 4

The King’s Chessboard – David Birch

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Domain: Operations and Algebraic Thinking (4.OA) - Gain familiarity with factors and multiples.



What are the different ways I count?

Why is it important for me to skip count?

What are prime and composite numbers?

How are factors and multiples related?

Formative: Board work

Making a factor rainbow for 12 – connecting 1 and 12, 2 and 6, 3 and

4 Quick write : list different factor pairs


Mad Minute drill

Summative: Chapter Test, quiz, performance assessment

1.Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number isa multiple of each of its factors. Determine whether a given whole number in the range 1-100 is

a multiple of a given one digit number. Determine whether a given whole number in the range

1-100 is prime or composite.

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Multiplication facts

Vocabulary: prime, composite, factor x factor = product, product ÷factor = factor, multiple Multiple

vs. Product

Sieve of Eratosthenes

Odds, evens and squares

Area – build rectangles (arrays) with a given area can be 2 ways ex. 6 = 1x6 or 3x2: prime

numbers have only 1 rectangle

Ex. An ice cream truck visits a neighborhood every 4 days. She missed the truck today. When

can she expect it again?

Calendar – every 3rd day you cut the lawn

Science -- measurement


Multiplication Table

100 Chart

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Can I describe change mathematically?

Can I describe numbers or objects that repeat mathematically?



Formative: Explore pattern problems

Chapter test, quiz,Performance Assessment:

Summative: Every Beat of Your Heart - The heart of a child below the age of twelve beats about

100 times a minute. Have the children find their pulse at rest for 1 minute. Then have them do

some form of exercise, find their pulse for 1 minute. Then, have them run in place for 1 minute. Ask

follow up questions that predict what could happen for longer time or different activities.

1.Generate a number or shape pattern that follows a given rule. Identify apparent features of the

pattern that were not explicit in the rule itself.

For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting

sequence and observe that the terms appear to alternate between odd and even numbers.

Explain informally why the numbers will continue to alternate in this way.

Domain: Operations and Algebraic Thinking (4.OA) - Generate and analyze patterns.

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Vocabulary: skip counting, multiple counting, sequences ex. Rule start with 2 and multiple by 3

Model situations that the 4 operations of whole numbers as patterns. An operation can be “undone”

by its inverse.

Represent a variable as an unknown quantity using a letter or symbol.

Develop an understanding of the Commutative and Associative Property ex. 3 x 2 = 2 x 3

IN/OUT Tables

Growth in tables and graphs using real world events

How change is related to time (time tables)

Observing geometric patterns

Patterns in science

Pattern Blocks, or Tiles

Hundreds chart

Calendar

Number line

The Patchwork Quilt – Valorie Flournoy

Performance assessment example:

www.rda.aps.edu/mathtaskbank/starthtm. Skip the Movie or Betty’s Miniature Birdhouses

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Domain:Number and Operations in Base Ten (4.NBT) - Generalize place value understanding for multi-digit whole numbers.


Can I express order and compare numbers?

Do I know the position of a digit in a number affects its value?


Formative: Board work, exit cards with solved problems, homework


Summative: Chapter test,quiz, performance assessment

1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what it

represents in the place to its right. For example, recognize that 700÷70 = 10 by applying

concepts of place value and division.

2. Read and write multi-digit whole numbers using base ten numerals, number names, and

expanded form. Compare two multi digit numbers based on meanings of the digits in each

place, using >, =, < symbols to record the results of comparisons.

3. Use place value understanding to round multi-digit whole numbers to any place.

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Vocabulary: Place value names, period names, expanded form, rounding Discover patterns in

place value

Compare numbers using place value

Using rounding to the nearest dollar to estimate costs

Inverse operations

Social Studies: Population and Area

Almanac Data e.g. Tallest Buildings

Other number systems – Roman numerals, Chinese abacus Newspaper or magazine scavenger

hunt for large numbers

Consumer topics - estimating money

Place Value Chart

Number Line

Base Ten Blocks

Calculator

Napier Bones

Literature – How Much is a Million – David Schwartz

Anno’s Counting Book – Mitsumasa Anno

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Do I know efficient methods for finding sums, differences, products and quotients?

Do I know what makes a strategy both effective and efficient?

Are my answers reasonable?

Formative: board work, exit cards with problems, math journal writing, homework


Summative: chapter test, quiz, performance assessments

1. Fluently add and subtract multi-digit whole numbers using the standard algorithm.

2. Multiply a whole number of up to four digits by a one digit whole number, and multiply two two

digit numbers, using strategies based on place value and the properties of operations. Illustrate

and explain the calculation by using equations, rectangular arrays, and/or area models.

3. Find whole number quotients and remainders with up to four digit dividends and one digit

divisors, using strategies based on place value, the properties of operations, and/or the

relationship between multiplication and division. Illustrate and explain the calculation by using

equations, rectangular arrays and/or area models.

Domain: Number and Operations in Base Ten (4.NBT) - Use place value understanding and properties of operations to perform multi-digit arithmetic.

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Properties –commutative, associative and distributive

Model multiplication using patterns (multiples of 10, 100 and 1,000), graph paper

Model division with base ten blocks and/or graph paper

Vocabulary: factors, product, partial products, divisor, dividend, quotient, remainder, compatible

numbers

Using estimation to check reasonable answers

Break apart strategy ex. 3,456 3,000 + 400 + 50 + 6

+ 1,357 1,000 + 300 + 50 + 7

260÷4 = 200 ÷4 + 60 ÷4


Consumer topics – money, shopping using advertisements, catalogue order forms, seating

arrangements, compare shopping prices

Averages (mean)

Lattice multiplication as enrichment


Calculators – show how multiplication is repeated addition and division is repeated subtraction

Base ten blocks used as area models

Literature – Anno’s Mysterious Multiplying Jar – Anno

A Remainder of One – Pinczes

http://mason.gmu.edu/~mmankus/whole/base10/asmdb10.htm

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Domain: Number and Operations - Fractions (4.NF) - Extend understanding of fraction equivalence and ordering.




Do I know when fractions and whole numbers are used together in real life? ]Can I model, compare and order fractions?

Do I know how fractions are alike and different?

Formative: Quick Write/Quick Draw – on the right draw a picture of the fraction and on the right explain

what it means.

Model 1/2 is the same as 2/4,3/6 and 4/8 in arrays and on the number line

Students fold paper rectangles into thirds, shading 2/3 of the rectangles. Students refold the papers into

thirds and then in halves opening the rectangles to see that 4/6 are now shaded. Students look at the

displayed fractionsas 2/3, (2x2)(2x3), and 4/6 and reason that when the denominator and numerator

are multiplied by the same number the resulting fraction is equivalent to the original as shown by the

shaded rectangles. Summative: Chapter test, quiz, performance assessment

1. Explain why fraction a/b is equivalent to a fraction (n x a)/(n x b) by using visual fraction models, with

attention to how the number and size of the parts differ even though the two fractions themselves are the

same size. Use this principle to recognize and generate equivalent fractions.

2. Compare two fractions with different numerators and different denominators, e.g., by creating common

denominators or numerators, or by comparing to a benchmark fraction such as ½. Recognize that

comparisons are valid only when the two fractions refer to the same whole. Record the results of

comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction

model.

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Fractions are limited with denominators of 2,3,4,5,6,8,10,12, and 100.

Compare fractions with like denominators by comparing the numerators.

Compare fractions with different denominators by multiplying the numerator and denominator by 1

renaming as 2/2,3/3 etc.

Example: Is 5/6 > 1/2 ? 3/6 = ½ then 5/6 > 3/6 therefore 5/6 > 1/2

Visual pictures of 2 fractions that make a whole. e.g. ½ + ½ = 1 and ¼ + ¾ = 1

Example: If each person at a party will eat 3/8 of a pound of turkey and there are 5 people at the

party, how many pounds of turkey will be needed? Between what 2 whole numbers does your

answer lie?

Recipes

Carpentry


Fraction bars to compare Number lines, ruler

Pattern blocks Legos

Quilts visual fractions. com

Literature – Fraction Action by Leedy

Fraction Fun by Adler

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Domain: Number and Operations - Fractions (4.NF) - Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.



Do I know how fractions are represented in everyday life?

Do I know if the computation with fractions and whole numbers the same or different?

Formative: Modeling on white boards: have the children draw a variety of fractions as the sum

of unit fractions

Exit cards


Summative: Chapter tests, quiz performance assessment

1. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.

a. Understand addition and subtraction of fractions as joining and separating parts referring to the

same whole.

b. Decompose a fraction into a sum of fractions with the same denominator in more than one way,

recording each decomposition by an equation. Justify decompositions, e.g., by using a visual

fraction model. Examples: 3/8 = 1/8 +1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 +

1/8.

c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number

with an equivalent fraction, and/or by using properties of operations and the relationship between

addition and subtraction.

d. Solve word problems involving addition and subtraction of fractions referring to the same whole

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and having like denominators, e.g., by using visual fraction models and equations to represent the

problem.

2. Apply and extend previous understanding of multiplication to multiply a fraction by a whole

number.

a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to

represent 5/4 as the product 5 x ¼, recording the conclusion by the equation 5/4 = 5 x ¼ .

b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a

fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6 x

(1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a)/b.)

c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using


visual fraction models and equations to represent the problem. For example, if each person at a

party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many

pounds of roast beef will be needed? Between what two whole numbers does your answer lie?

Unit fractions

Add and subtract fractional numbers with like denominators, using equivalent fractions. Example:

1 ¼ ¾

If 4/4 + ¼ = 5/4 than 5/4 – ¾ = 2/4 = 1/2

Find fractional part of a given set or area. Model this example: 3 x 2/5 is the same as 6 x 1/5 = 6/5

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Recipes

Carpentry

Money

Probability


Graph paper, place value charts with decimals, number line

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Domain: Number and Operations - Fractions (4.NF) - Understand decimal notation for fractions, and compare decimal fractions.



How can numbers be expressed in multiple ways?

How are fractions and decimals related?

Formative: Whiteboard work showing fractions above a number line and decimal equivalent

below the line.


“I have, Who has”

Summative: Chapter test, quiz, performance assessment

1. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and

use this technique to add two fractions with respective denominators 10 and 100. [Students who

can generate equivalent fractions can develop strategies for adding fractions with unlike

denominators in general. But addition and subtraction with unlike denominators in general is not a

requirement at this grade.] For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

2. Use decimal notation for fractions with denominators 10 or 100. For examples, rewrite

0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

3. Compare two decimals to hundredths by reasoning about their size. Recognize that

comparisons are valid only when the two decimals refer to the same whole. Record the results of

comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

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Compare fractions and decimals using models of hundred squares and base ten blocks

Identify and understand the relationship between fractions and decimals

Use number line to connect fractions and decimals.

Use base ten blocks to connect 3/10 or 3 longs is the same as 30/100


Money

Batting averages

Use the computer to draw circle graphs – and the web site create a graph


Physical models, pictures, number lines, use both customary and centimeter rulers, coins

Base ten blocks

Graph paper

Nlvm.usu.edu

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Domain: Measurement and Data (4.MD) - Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.


How do I use measurements on a daily basis?

Are there different systems for measuring things?

Can I relate smaller units of measurement to larger ones?



Formative: Use a dry erase board to have the students display answers to equivalent

measurements. Example: How many inches = 1 foot? Can 3 feet be equivalent to another unit of

measurement? Have students model units of measurement.

Summative: Have students design a real life situation problem using finding the perimeter or area

of a rectangle. For example: fencing a yard, buying the correct amount of carpet for a room. Design

a blueprint for the first floor of a house. For instance, door size, window size, and room size.

1. Know relative sizes of measurement units within one system of units including km, m, cm;

kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements

in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.

For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48

in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3,

36), ...

2. Use the four operations to solve word problems involving distances, intervals of time,

liquid volumes, masses of objects, and money, including problems involving simple fractions or

decimals, and problems that require expressing measurements given in a larger unit in terms of a

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smaller unit. Represent measurement quantities using diagrams such as number line diagrams

that feature a measurement scale.

3. Apply the area and perimeter formulas for rectangles in real world and mathematical

problems. For example, find the width of a rectangular room given the area of the flooring and the

length, by viewing the area formula as a multiplication equation with an unknown factor.



Knowledge of customary and metric systems for length, time, volume, and mass. Relationship between smaller and larger units of measure. Example: 1 ft is 12 times as long as 1 i n.

How to apply the formula for area and perimeter of rectangles in real life situations. Example: Determining the amount of flooring for a room, or fencing for a yard. Make a conversion table for comparing related smaller and larger units. Vocabulary: metric, convert, conversion table, perimeter, area, mass, volume.

Science: measuring and finding the mass of objects. Comparing measurements such as 16 ounces

= 1 lb. 48 ounces = 3 lb. (3x16). Analyzing and interpreting data from experiments.


Conversion tables, metric stick, yardstick, websites, assorted books, manipulatives, Scholastic printable resources

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Domain: Measurement and Data (4.MD) - Represent and interpret data.




What Is a line plot?

How can I use a line plot diagram to show data?

Do I know how to add and subtract fractions?

What does a fraction represent?

Formative: Use a dry erase board to display a set of measurements in fractions of a unit. For

example: ½, ¼, 1/8.

Solve problems involving adding and subtraction fraction by using information presented in line

plots.

Summative: Project: Design a line plot to solve a word problem. For example: Find and interpret

the difference in length between the longest and shortest specimens in an insect collection.

Understanding of line plots

Fractions, operations on fractions

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Vocabulary: line plot, fraction, data set


Science: interpreting data and representing it by a line plot. Social Studies: representing with a line

plot how many years lapsed between two historical events located on the same time line


Websites

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Domain: Measurement and Data ( 4.MD) - Geometric measurement: Understanding concepts of angle and measure angles.


What geometric shapes do I know?

Where do I see geometric shapes everyday?

What is an angle? How do I form an angle? Can I see angles in my daily life?


Formative: Be able to identify different types of angles; acute, obtuse, and right angles.

Students should be able to draw and measure various kinds of angles and geometric figures.

Summative: Project: Identify and label geometric shapes found in a real world and every day

situation. For example: the classroom, street scene, the beach. The lifeguard stand has angles in

the rungs, a one way street sign could be a ray, a stop sign could be an octagon, the hands on a

clock could show an acute, obtuse, or right angle.

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1. Recognize angles as geometric shapes that are formed wherever two rays share a

common endpoint, and understand concepts of angle measurement:

a. An angle is measured with reference to a circle with its center at the common endpoint of the

rays, by considering the fraction of the circular arc between the points where the two rays

intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,”

and can be used to measure angles.

b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.

2. Measure angles in whole number degrees using a protractor. Sketch angles of specified


measure.

3. Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts,

the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and

subtraction problems to find unknown angles on a diagram in real world and mathematical

problems, e.g., by using an equation with a symbol for the unknown angle measure

Vocabulary: angle, acute angle, obtuse angle, right angle, rays, protractor

Knowledge of relationship between an angle being 1/360 of a circle.

Knowledge of variables and how to substituting them in solving addition and subtraction problems to

find an unknown angle.

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Science: interpreting data using circle graphs Social Studies: circle graphs


Protractors, manipulatives, websites, dry erase boards

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Domain: Geometry (4.G) Draw and identify lines and angles, and classify shapes by properties of their lines and angels.




What are lines made of? How can I draw a line?

Do I know of different types of lines?

Can I identify lines in my world?

Do I know what a line of symmetry is?

Formative: Use a dry erase board to practice drawing lines points, line segments, angles (right,

acute, obtuse), perpendicular and parallel lines. Students should be able to identify and represent

two dimensional figures.

Use models, concrete objects and manipulatives to represent symmetry and lines of symmetry.

Use protractors to draw and measure right angles and triangles.

Summative: Project -Identify lines in the world. Example: train tracks, intersections, and parallel

streets.

1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular

and parallel lines. Identify these in two dimensional figures.

2. Classify two dimensional figures based on the presence or absence of parallel or

perpendicular lines, or the presence or absence of angles of a specified size. Recognize right

triangles as a category, and identify right triangles.

3. Recognize a line of symmetry for a two dimensional figure as a line across the figure

such that the figure can be folded along the line into matching parts. Identify line-symmetric

figures and draw lines of symmetry.

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Vocabulary: point, line, parallel lines, perpendicular lines, symmetry, lines of symmetry, triangle,

right triangle, angels.


Science: constructing tables, charts to represent data

Social Studies: identifying borders, rivers, how the states fit with each other

Art: What’s in a name? Perspective drawing


Ruler, protractor, commercial websites

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Grade 5

Grade Five

Page 1

Grade 5


Instructional time in Grade 5 focuses on three critical areas: (1) developing fluency with addition and subtraction of fractions, and

developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole

numbers and whole numbers divided by unit fractions); (2) extending division to 2-digit divisors, integrating decimal fractions into the place

value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and

decimal operations; and (3) developing understanding of volume.

(1) Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike

denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions,

and make reasonable estimates of them. Students also use the meaning of fractions, of multiplication and division, and the relationship

between multiplication and division to understand and explain why the procedures for multiplying and dividing fractions make sense.

(Note: this is limited to the case of dividing unit fractions by whole numbers and whole numbers by unit fractions.)

(2) Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of

operations. They finalize fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of

models for decimals, decimal notation, and properties of operations to add and subtract decimals to hundredths. They develop fluency in

these computations, and make reasonable estimates of their results. Students use the relationship between decimals and fractions, as well

as the relationship between finite decimals and whole numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole

number), to understand and explain why the procedures for multiplying and dividing finite decimals make sense. They compute products

and quotients of decimals to hundredths efficiently and accurately.

(3) Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the

total number of same-size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-

unit cube is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve

estimating and measuring volume. They decompose three-dimensional shapes and find volumes of right rectangular prisms by viewing

them as decomposed into layers of arrays of cubes. They measure necessary attributes of shapes in order to determine volumes to solve

real world and mathematical problems.

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Grade Five

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3 + x




Grade Five

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Grade Five Standards for Mathematical Content



5.OA All students will be able to write and interpret numerical expressions.

1. Use parentheses, brackets, or braces in numerical expressions, and evaluate

expressions with these symbols.

2. Write simple expressions that record calculations with numbers, and interpret

numerical expressions without evaluating them. Example: Express the calculation

“add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 +

921) is three times as large as 18932 + 921,without having to calculate the

indicated sum or product.

5.OA All students will analyze patterns and relationships.

3. Generate two numerical patterns using two given rules. Identify apparent

relationships between corresponding terms. Form ordered pairs consisting of

corresponding terms from the two patterns, and graph the ordered pairs on a

coordinate plane. Example: Given the rule “Add 3” and the starting number0, and given the rule “Add 6” and the starting number 0, generate terms inthe resulting sequences, and observe that the terms in one sequence aretwice the corresponding terms in the other sequence.






about mathematics.

Teach students the mnemonic device for

remembering the Order of Operations ---

PEMDAS

Please Excuse My Dear Aunt Sally

Parentheses

Exponents

Multiplication and Division (in the order

in which they appear left to right)

Addition and Subtraction (in the order in

which they appear left to right)

Have students identify the next elements in a

pattern and explain the rule.

ABBCCC…

1, 5, 9, 13 …

… ...

Apply Order of Operation for expressions.

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Numbers and Operations in Base Ten (5.NBT)


5.NBT All students will understand the place value system.

1. Recognize that in a multi-digit number, a digit in one place represents 10 times as

much as it represents in the place to its right and 1/10 of what it represents in the

place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a

number by powers of 10, and explain patterns in the placement of the decimal

point when a decimal is multiplied or divided by a power of 10. Use whole-

number exponents to denote powers of 10.

3. Read, write, and compare decimals to thousandths.

a) Read and write decimals to thousandths using base-ten numerals, number

names, and expanded form, e.g., 347.392 = 3 × 100 + 4 ×

10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b) Compare two decimals to thousandths based on meanings of the digits

in each place, using >, =, and < symbols to record the results of

comparisons.

4. Use place value understanding to round decimals to any place.

Display a place value chart as a visual aid. Ten

thousands

10,000

104

Thousands 1,000

103

Hundreds 100

102

Tens 10

101

Ones 1

100

Tenths .1

10-1

2 20,000 2,000 200 20 2 .2

3 30,000 3,000 300 30 3 .3

7 70,000 7,000 700 70 7 .7

Provide examples to show the results of

multiplying and dividing by powers of 10.

35 x 102 = 35 x (10 x 10) = 35 x 100 = 3,500

35 x 103 = 35 x (10 x 10 x10) = 35 x 1,000 = 35,000

35 102 = 35 100 = 35/100 = .35

35 103 = 35 1000 = 35/1000 = .035

Use 10 x 10 centimeter grids to develop

students’ decimal sense. The 10 x 10 grid

represents one whole. Each column of 10

represents one tenth and each individual box

represents one hundredth. Demonstrate

equivalent decimals by shading.

Example: 2 shaded columns shows .2 and .20

Locate decimal numbers on a number line.

0 .1 .3 1 1.5 2

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5.NBT All students will be able to perform operations with multi-digit whole

numbers and with decimals to hundredths.

5. Fluently multiply multi-digit whole numbers using the standard algorithm.

6. Find whole-number quotients of whole numbers with up to four-digit dividends

and two-digit divisors, using strategies based on place value, the properties of

operations, and/or the relationship between multiplication and division. Illustrate

and explain the calculation by using equations, rectangular arrays, and/or area

models.

7. Add, subtract, multiply, and divide decimals to hundredths using concrete models

or drawings and strategies based on place value, properties of operations, and/or

the relationship between addition and subtraction; relate the strategy to a written

method and explain the reasoning used.

S1. Select and use appropriate operations (addition, subtraction, multiplication,

division) to solve problems, including those involving money.

Explore various strategies for simplifying

computation. Example: 225 x 12

225 x (10 + 2) or (200 + 25) x 12

Students should apply properties of operations

and engage in mathematical reasoning to

simplify computation.

Example: 12 x 3.4

(12 x 3) + (12 x .4)

36 + 4.8 = 40.8

Use money to teach decimal sense.

dollar, (1.00), dime( .10 = .1), nickel (.05),

penny (.01), quarter (.25)

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Number and Operations—Fractions (5.NF)


5.NF All students will use equivalent fractions as a strategy to add and subtract

fractions.

1. Add and subtract fractions with unlike denominators (including mixed numbers)

by replacing given fractions with equivalent fractions in such a way as to produce

an equivalent sum or difference of fractions with like denominators.

Example: 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)

2. Solve word problems involving addition and subtraction of fractions referring to

the same whole, including cases of unlike denominators, e.g., by using visual

fraction models or equations to represent the problem. Use benchmark fractions

and number sense of fractions to estimate mentally and assess the reasonableness

of answers. Example: Recognize an incorrect result 2/5 + 1/2 = 3/7, by observing

that 3/7 < 1/2.

5.NF All students will apply and extend previous understandings of multiplication

and division to multiply and divide fractions.

3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).

Solve word problems involving division of whole numbers leading to answers

in the form of fractions or mixed numbers by using visual fraction models

or equations to represent the problem. Example: Interpret 3/4 as the result of

dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes

are shared equally among 4 people each person has a share of size 3/4.

If 9 people want to share a 50-pound sack of rice equally by weight, how many

pounds of rice should each person get? Between what two whole numbers

does your answer lie?

Use fraction bars and number lines to find

and rename equivalent fractions. 1/3 1/3 1/3

1/4 1/4 1/4 1/4

1/4 1/4 1/4 1/4

2/3 + 5/4 = 23/12

Interpret 3/4 = 3 4, as 3 wholes divided

evenly into 4 groups: A, B, C, D.

A A B B C D

A B C C D D

Each group consists of 3/4.

2/3 = 8/12

5/4 =15/12

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4. Apply and extend previous understandings of multiplication to multiply a fraction

or whole number by a fraction.

a) Interpret the product (a/b) × q as a parts of a partition of q into b equal

parts; equivalently, as the result of a sequence of operations a × q ÷ b.

Example: Use a visual fraction model to show (2/3) × 4 = 8/3, and create a

story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In

general, (a/b) × (c/d) = ac/bd.)

b) Find the area of a rectangle with fractional side lengths by tiling it with

unit squares of the appropriate unit fraction side lengths, and show that

the area is the same as would be found by multiplying the side lengths.

Multiply fractional side lengths to find areas of rectangles, and represent

fraction products as rectangular areas.

5. Interpret multiplication as scaling (resizing) by:

a) Comparing the size of a product to the size of one factor on the basis of

the size of the other factor, without performing the indicated multiplication.

b) Explaining why multiplying a given number by a fraction greater than 1

results in a product greater than the given number (recognizing multiplication

by whole numbers greater than 1 as a familiar case); explaining why

multiplying a given number by a fraction less than 1 results in a product

smaller than the given number; and relating the principle of fraction

equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.

6. Solve real world problems involving multiplication of fractions and mixed

numbers,( e.g., by using visual fraction models or equations to represent

the problem).

A visual fraction model for 2/3 x 4/5

* * * *

* * * *

4/5

Explore fractions with a variety of virtual

manipulatives at:

http://nlvm.usu.edu/en/nav/vlibrary.html

Have students compare the products that

results from multiplying a given number by

various different factors. What do they

observe about the size of each of the different

factors the resulting product?

Example:

12 x 2 = 24

12 x 4 = 48

12 x 1 = 12

12 x ¾ = 9

2/3 of 4/5 = 8/15

When the size of one of the factors

doubles; the size of the product

doubles.

Multiplying a given number by a

number less than 1 results in a

product that is less than the given

number.

http://nlvm.usu.edu/en/nav/vlibrary.html

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7. Apply and extend previous understandings of division to divide unit fractions by

whole numbers and whole numbers by unit fractions.

a) Interpret division of a unit fraction by a non-zero whole number, and

compute such quotients. Example: Create a story context for (1/3) ÷ 4,

and use a visual fraction model to show the quotient. Use the relationship

between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because

(1/12) × 4 = 1/3.

b) Interpret division of a whole number by a unit fraction, and compute such

quotients. Example: Create a story context for 4 ÷ (1/5), and use a visual

fraction model to show the quotient. Use the relationship between

multiplication and division to explain that 4 ÷ (1/5) = 20 because

20 × (1/5) = 4.

c) Solve real world problems involving division of unit fractions by non-zero

whole numbers and division of whole numbers by unit fractions by using

visual fraction models and equations to represent the problem.

Example: How much chocolate will each person get if 3 people share 1/2 lb

of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Use visual fraction models. To show 1/3

divided by 4 = 1/12. Have students divide a

paper into thirds and then divide each third

into four equal pieces.

1/3

1/3 ÷ 4 = 1/12

Give examples of related multiplication and

division facts using whole numbers.

Example: 12 x 6 = 72

72 ÷ 6 = 12 and 72 ÷ 12 = 6

Guide students to apply this reasoning to

division with fractions.

20 x 1/5 = 4

4 ÷ 1/5 = 20 and 4 ÷ 20 = 1/5

How much chocolate will each person get if

3 people share 1/2 lb of chocolate equally?

1 lb.

1/2 of 1 lb

1/2 ÷ 3 =1/6 Each person gets 1/6 lb.

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5.MD

Provide students with basic equivalences within the customary measurement system and within the metric system and use these to convert to larger or smaller units within the same system.


5.MD All students will be able to represent, interpret, and analyze data.

2. Make a line plot to display a data set of measurements in fractions of a unit (1/2,1/4,1/8). Use operations on fractions for this grade to solve problems involvinginformation presented in line plots. Example: Given different measurments ofliquid in identical beakers, find if the amount in all the beakers wereredistributed equally.

Have students work in small groups to collect data through direct measurement or internet resources (e.g., student heights, daily high temperatures over a two week period, etc.) Use data collected to create tables, graphs, line plots. Have students order the values in a data set to determine Measures of Center: Mode, Median, Mean Measure of Variation: Range

Measurment and Data (5.MD)

All students will be able to convertlike measurement units within a gigen measurement system.

Convert among different-sized standard measurement units within a given measurement system (e.g.,convert 5 cm to 0.05m) and use these conversions in solving multi-step, real world problems.

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5. MD Geometric measurement: All students will understand concepts of volume and

relate volume to multiplication and to addition.

3. Recognize volume as an attribute of solid figures and understand concepts of

volume measurement.

a) A cube with side length 1 unit, called a ―unit cube,‖ is said to have ―one

cubic unit‖ of volume, and can be used to measure volume.

b) A solid figure which can be packed without gaps or overlaps using n unit

cubes is said to have a volume of n cubic units.

4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and

improvised units.

5. Relate volume to the operations of multiplication and addition, and solve real world

and mathematical problems involving volume.

a) Find the volume of a right rectangular prism with whole-number side lengths

by packing it with unit cubes, and show that the volume is the same as would

be found by multiplying the edge lengths, equivalently by multiplying the

height by the area of the base. Represent threefold whole-number products

as volumes (e.g., to represent the associative property of multiplication).

b) Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find

volumes of right rectangular prisms with whole number edge lengths in the

context of solving real world and mathematical problems.

Use connecting cubes and other small cubes

to develop students’ understanding of

volume measurement. Use boxes of various

sizes to explore how many cubes it takes to

fill each box.

Cubic inch

1 in3

Cubic centimeter

1 cm3

1 cm 1 inch

1 cm

1 inch

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c) Recognize volume as additive. Find volumes of solid figures composed of

two non-overlapping right rectangular prisms by adding the volumes of the

non-overlapping parts, applying this technique to solve real world problems.

2

V = l x w x h V =Area of Base x h

V= 3 x 2 x 5 V = (3 x 2) x 5

Have students explore the concept of volume

at: http://www.learner.org/interactives/

geometry/area_volume.html

3

5

http://www.learner.org/interactives/%20geometry/area_volume.html

http://www.learner.org/interactives/%20geometry/area_volume.html

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5.G All students will be able to graph points on the coordinate plane to solve

real-world and mathematical problems.

1. Use a pair of perpendicular number lines, called axes, to define a coordinate system,

with the intersection of the lines (the origin) arranged to coincide with the 0 on each

line and a given point in the plane located by using an ordered pair (x, y) of

numbers, called its coordinates. Understand that the first number indicates how far

to travel from the origin in the direction on the horizontal axis, and the second

number indicates how far to travel in the direction on the vertical axis, with the

convention that the names of the two axes and the coordinates correspond

(e.g., x-axis and x-coordinate, y-axis and y-coordinate).

2. Represent real world and mathematical problems by graphing points in the first

quadrant of the coordinate plane, and interpret coordinate values of points in the

context of the situation.

5.G All students will classify two-dimensional figures into categories based on their

properties.

3. Understand that attributes belonging to a category of two-dimensional figures also

belong to all subcategories of that category. Example: All rectangles have four right

angles and squares are rectangles, so all squares have four right angles.

Plot ordered pairs and connect dots to reveal

a shape. Example: A(2, 6); B(4, 6);

C(6, 3); D(2, 3). Be sure to connect point D

to point A.

Use a coordinate plane to show the location

of various places in relation to each other.

Have students tell which ordered pair

represents the location of a specific building

or place? Describe the path from one location

to another location.

0

y-axis

x-axis

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4. Classify two-dimensional figures in a hierarchy based on properties. Polygons

Quadrilaterals

Parallelograms Trapezoids Kites

Rectangle Rhombus

Square

http://www.nctm.org/standards/content.aspx?id=26772


Domain: Operations and Algebraic Thinking (5.OA) - Write and Interpret Numerical Expressions.Essential Questions What should I be able to answer? What guides my thinking?

How do I deal with questions that have multiple math operations?

Why is the strategy of order of operations both effective and efficient for me?

When am I going to see and use this skill?


Formative: Students can discuss in small groups what they have learned about numerical

expressions and order of operations.

Students can write in math journal, “I used to think I could solve a series of equations by working

from left to right. Now I know I have to follow order of operations because…”

Math Journal: Students will answer one or more of the Essential Questions.

Summative: Performance Assessment: In a long-range activity, student will devise an activity or

game (such as Hopscotch) that can use steps that will simulate order of operations. Explanations,

illustrations, and game rules will be included.

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1. Use parentheses, brackets, or braces in numerical expressions, and evaluate




expressions with these symbols.

2. Write simple expressions that record calculations with numbers, and interpret numerical

expressions without evaluating them. For example, express the calculation “add 8 and 7, then

multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 +

921, without having to calculate the indicated sum or product.

Simplify Order of Operations through the use of parentheses, braces, or brackets.

Do those operations that are in parentheses, braces, or brackets first.

PEMDASFLTR – Please Excuse My Dear Aunt Sally From Leaving The Room

Parentheses, Exponents, Multiplication OR Division, Addition OR Subtraction, From Left To Right

Use mathematical representation to solve problems.

Science, Computer Programming, and Accounting: Investigate how and why mathematical

calculations in these areas need order of operations.

Technology: Use several different types of calculators, e.g. four function, fraction, etc. Give

students a series of equations to be put into each type of calculator. Decide which type of

calculator automatically solves the equations using order of operations. Student will write the

results and decide why certain types of calculators do/do not automatically use order of

operations. Trade books, text books, calculators, computers, interactive boards

NCTM website, commercial websites

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Domain: Operations and Algebraic Thinking (5.OA) - Analyze patterns and relationships

Essential Questions What should I be able to answer? What guides my thinking? How can I relate mathematical patterns to patterns in life?

How do patterns shape my understanding of mathematics?

How can I change patterns in mathematics to rules of mathematics?


Formative: Students can use physical patterns such as in card games, checkers, chess, skip

counting, patterns in nature, etc., and relate them to mathematical number patterns.

In small groups, each student will create number patterns, graph these, then discuss these

patterns. Students can use pattern blocks to create tessellations.


Summative: Performance Assessment: In a long-range activity, students can create games that

use patterns. Explanations will include how the game is played, what patterns are used, how

patterns help to solve the game, etc. Illustrations should include the actual game and game pieces.

These can be made of cardboard or other simple materials. Graphs can be used to illustrate the

game.

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1. Generate two numerical patterns using two given rules. Identify apparent relationships between

corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and

graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting

number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting

sequences, and observe that the terms in one sequence are twice the corresponding terms in the other

sequence. Explain informally why this is so.

Vocabulary: patterns, ordered pairs, corresponding terms, coordinate plane, sequence. Compare patterns

of one sequence to patterns of another sequence. E.G., 0,3,5,7,9 (add 2 to each number). 0,4,8,12,16

(add 4 to each number).

Describe the relationship between the “add” numbers in the patterns. E.G., “add” 4 is twice as much as

“add” 2. Form ordered pairs, e.g., (2,4), then graph these on a coordinate plane. Social Science: Make

numerical and logical predictions for various real world activities.

History: Discuss patterns in history, such as wars, diseases, migration, inventions, medical advances,

etc. Science: Discuss patterns in science and nature, such as lunar cycles, scientific inventions, the

growth cycle of plants, etc.

text books, trade books, calculators, computers, pattern blocks, graphs, interactive boards NCTM, and

commercial websites

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Domain: Number and Operation in Base Ten (5.NBT) - Understand the Place Value System.


What does zero mean?

Where do I see and use zeros?

How do I compare powers of ten to the use of zeros in the place value system?

Where will I use decimals in real life situations?

How does expanding numbers help me to understand place value?


Formative: Students will model whole numbers and decimals using base ten blocks.

Students will use 100 grid paper to demonstrate decimals. Use several colors to show

comparison of decimals. In groups of two, students will write the expansion of whole numbers,

decimal numbers, and numbers with exponents. Each student will write a portion of the expansion

and will critique her/his partner’s written work.


Summative: Performance Assessment:: In a long range activity, students will use whole numbers,

money, and decimals in comparison shopping. In small groups, students will choose a grocery or

dry goods product. Compare the prices of the same item in three different stores. Investigate or

deduce why there is a difference in prices. Use store ads or actual trips to the store. Use rounding

of decimals and whole numbers.

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1. Recognize that in a multi digit number, a digit in one place represents 10 times as much as

it represents in the place to its right and 1/10 of what it represents in the place to its left.

2. Explain patterns in the number of zeros of the product when multiplying a number by

powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole number exponents to denote powers of

3. Read, write, and compare decimals to thousandths.

a. Read and write decimals to thousandths using base-ten numerals, number names, and

expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).

b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =,


and < symbols to record the results of comparisons.

4. Use place value understanding to round decimals to any place.

Vocabulary: decimal, decimal point, exponents, powers of 10, tenths, hundredths, thousandths,

expanded form. Stress: The word “and” is used when reading decimals only, not with large whole

numbers. Decimals are read as “and,” not as “point.”

1.E.G. Use the number 5327. Show how the 5 is 5x1000, the 3 is 3x100, etc. Show how each

decimal digit is 1/10 of the place to the left.

2.Show how multiplying by 102 will result in a product ending in two zeros. E.G. 525 x102 = 52500.

Explain how the decimal point is moved to the right when multiplying by powers of 10 and to the left

when dividing by powers of 10.

3.Expand whole numbers and decimal numbers. a) in tenths, hundredths, thousandths. b) compare

decimals to the thousandths place. E.G. 0.123 <0256. c) round decimals to any place.

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Investigate how science formulas, engineering, and computer science use exponents,

and expanded notation.

Health/Science: Use labels from cans, boxes that show volume or weight as decimal numbers.

Compare these numbers.

Social Studies: Compare the population of several United States cities. Make inferences as to

why the population in some cities is larger/smaller than others (round these numbers).

Investigate how science formulas, engineering, and computer science use exponents and

expanded notation.

Health/Science: Use labels from cans, boxes, etc. that show volume or weight as decimal

numbers (compare these numbers).

Social Studies: Investigate the populations of several cities. Make inferences as to why

populations in some cities are larger/smaller than other cities.

Text books, trade books, calculators, computers, graphs, base ten blocks, grid paper,

interactive boards, NCTM , and commercial websites

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Domain: Number and Operation in Base Ten (5.NBT) - Perform operations with multi-digit whole numbers and with decimals to hundredths.



Where will I use multiplication of whole numbers and decimals in real world situations?

How do I multiply multi-digit whole numbers?

How can I find the relationship between multiplication and division?

How can I use concrete models to demonstrate addition, subtraction, multiplication, and division of

decimals?

How can models help me to understand mathematical concepts?

Formative: Students will use concrete objects, such as base ten blocks to illustrate whole

numbers and decimals. Use grid paper and colored pencils/ crayons to illustrate whole numbers

and decimals. Write explanations of these in math journals.

Math Journal: Students will answer one or more of the Essentials Questions.

Summative: Performance Assessment: In a long range activity, students will create a school

bank, with deposits, loans, and interest, using addition, subtractions, multiplication, and division of

whole numbers, money, and decimals. Write an on-going explanation of what has been done and

keep records of each process.

In a long range activity, students will make flash cards. The cards will illustrate addition,

subtraction, multiplication, and division of multi-digit whole numbers and decimals. Each card will

list the concept and the process for solving the equation, and create pictures of the operations. A

word problem should be included on each card. One card can be used for each concept. The

flash cards can vary according to teacher specifications.

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1. Fluently multiply multi-digit whole numbers using the standard algorithm.

2. Find whole number quotients of whole numbers with up to four digit dividends and two digit

divisors, using strategies based on place value, the properties of operations, and/or the relationship

between multiplication and division. Illustrate and explain the calculation by using equations, rectangular

arrays, and/or area models.


3. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and

strategies based on place value, properties of operations, and/or the relationship between addition and

subtraction; relate the strategy to a written method and explain the reasoning used.

Vocabulary: properties, quotient, dividend, divisor, digit, equation, arrays, concrete model

algorithm (teacher use). Stress the word “and” not “point” when reading decimal numbers. When reading

numbers, the word “and” should be used with decimals only, not with large whole numbers.

1) Mastery of multi-digit whole numbers using standard form of multiplication.

2) Divide whole numbers with 4-digit dividends and 2-digit divisors. Show how division is the inverse

(opposite) of multiplication.

3) Addition, subtraction, multiplication, and division of decimals to the hundredths place using models,


drawings, and written explanations to relate this to the use of numbers.

Technology: In groups of three students, a series of math problems will be solved to determine the most

efficient method. Student 1 uses pencil and paper. Student 2 uses mental math. Student 3 uses a

calculator.

Students can interview people in professions that use decimals, such as engineering, carpentry,

plumbing, manufacturing to discover how math is used in those jobs.

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Social Science: Students can interview people in professions such as engineering, carpentry, plumbing,

and manufacturing, that use decimals to discover how math is used in these jobs.


Text books, trade books, calculators, computers, pattern blocks,

graphs, interactive board, NCTM, and commercial websites

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Where will I use portions of numbers?

Where do I see fractions or portions of objects/numbers used in real world situations?

Where can I use addition and subtraction of fractions in real world situations?

How do I use equivalent fractions used to add and subtract fractions with unlike denominators?

How can I use benchmark fractions to understand if my answer is logical?


Formative: Students will use fraction dominoes to create games that add and subtract fractions.

Use fraction strips to compare fractions, E.G., How many eighth strips equal a ¼ strip? Is 2/5

greater or less than 3/8? Use fractions strips to compare mixed numbers to improper fractions.

Measure objects to the nearest fraction.


Summative: Performance assessment: In a long-range activity, students will explore real-world

use of fractions. Obtain newspapers, magazines, TV advertisements to demonstrate sale items

with “fraction off.” E.G. The sales is ½ off the regular price. Student will write a report describing

this, and will use illustrations, computation, word problems, and written explanations.

Performance assessment: Students will investigate and create a report on occupations that use

fractions. Demonstrate how the occupation uses addition and subtraction of fractions.

Domain: Number and Operations-Fractions (5.NF) Use equivalent fractions as a strategy to add andsubtract fractions.

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1. Add and subtract fractions with unlike denominators (including mixed numbers) by

replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum

or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12.

(In general, a/b + c/d = (ad + bc)/bd.)


2. Solve word problems involving addition and subtraction of fractions referring to the same whole,

including cases of unlike denominators, e.g., by using visual fraction models or equations to

represent the problem. Use benchmark fractions and number sense of fractions to estimate

mentally and assess the reasonableness of answers. For example, recognize an incorrect result

2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.

Vocabulary: numerator, denominator, equivalent fractions, mixed numbers, sum, difference,

benchmark fractions, visual fraction model

1) Add and subtract unlike fractions and mixed numbers.

2) Use the least common denominator to make equivalent fractions.

3) Use representations, diagrams, illustrations, manipulatives and equations to solve word

problems. Use bench mark fractions and number line to decide if an answer is reasonable.


Science: Discuss how science formulas use fractions and decimals.

Language: Use a Venn diagram to determine if fractions that are greater than, less than, or

equal to a designated fractions.

Music: Compare whole, half, quarter, eighth, and sixteenth notes. Decide on various

arrangements of notes that will make a measure of four beats.

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Social Science/ Life Skills: Investigate how people in real life occupations, such as

carpentry, landscaping, automotive, flooring, painting, plumbing, etc. use fractions.


Text books, trade books, calculators, computers, pattern blocks, grid paper, base 10 blocks, interactive boards NCTM, and commercial websites

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Domain: Number and Operations-Fractions (5.NF) - Apply and extend previous understandings of multiplication and division to multiplication and division of fractions.



How can fraction models and drawings help me to understand multiplication and divisions of

fractions?

Where will I see portions of objects multiplied and divided in real life?

Where in real life will I see fractions multiplied and divided?

Formative: Students will use fraction dominoes to create games that use multiplication and

division of fractions. Use fraction strips to demonstrate multiplication and division of fractions.

With small groups of students, use dry erase boards to solve word problems: The first student

creates a word problem, and passes the board to the next person who decide which operation is

needed. The third student solves the equations. The fourth student describes why this solution is

correct or incorrect. Use grid paper to show multiplication of fractions. Use several colors to show

this. Math Journal: Students will answer one or more of the Essential Questions.

Summative: Performance Assessment: In a long range activity, each student will create a flip

book. Page 1 shows the word problem. Page 2 shows the steps needed to solve the problem.

Page 3 shows an illustrations of the solution. Page 4 shows the equation/s used to solve the

problem and the answer. Page 5 is a written explanation of the solution and why this solution was

chosen.

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1. Apply and extend previous understandings of multiplication and division to multiply and

divide fractions.

2. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve

word problems involving division of whole numbers leading to answers in the form of fractions or

mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For

example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and

that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9

people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should

each person get? Between what two whole numbers does your answer lie?

3. Apply and extend previous understandings of multiplication to multiply a fraction or whole

number by a fraction.

a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as

the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to

show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5)

= 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the

appropriate unit fraction side lengths, and show that the area is the same as would be found by

multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and

represent fraction products as rectangular areas.

4. Interpret multiplication as scaling (resizing), by:

a. Comparing the size of a product to the size of one factor on the basis of the size of the other

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factor, without performing the indicated multiplication.

b. Explaining why multiplying a given number by a fraction greater than 1 results in a product

greater than the given number (recognizing multiplication by whole numbers greater than 1 as a

familiar case); explaining why multiplying a given number by a fraction less than 1 results in a

product smaller than the given number; and relating the principle of fraction equivalence a/b =

(n×a)/(n×b) to the effect of multiplying a/b by 1.

5. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by

using visual fraction models or equations to represent the problem.

6. Apply and extend previous understandings of division to divide unit fractions by whole

numbers and whole numbers by unit fractions. [Students able to multiply fractions in general can

develop strategies to divide fractions in general, by reasoning about the relationship between

multiplication and division. But division of a fraction by a fraction is not a requirement at this

grade.]

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients.

For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the

quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12

because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For

example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the

quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20

because 20 × (1/5) = 4.

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c. Solve real world problems involving division of unit fractions by non-zero whole numbers and


division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to

represent the problem. For example, how much chocolate will each person get if 3 people share

1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Vocabulary: equivalent fractions, numerators, like and unlike denominators, sum, difference,

benchmark, benchmark fractions, visual fraction model

1) Use models and equations to understand the meaning of a fraction.

2) Multiply a fraction, or a whole number, by a fraction.

Use a visual model to show the equation.

Find the area of a rectangle whose sides are fractions. Use unit squares

3) Visualize the product by the size of the fraction e.g.

4/3 x 2 =2 2/3, which is greater than 2, 2/3 x 4 =

8/3 = 2 2/3, which is less than 4.

4) Use models in solving word problems.

5) Divide fractions by whole numbers. Divide whole numbers by fractions.

Use visuals and stories to explain division of a fraction by a whole number.

Use visuals and stories to explain division of a whole number by a fraction.

Use real world word problems in division of fractions.


Science: Investigate how and why fractions are used in science formulas.

Technology: Investigate the role fractions play in calculators and computers. Social Science/ Life

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Skills: Double or half recipes that make use of fractions in their ingredients.


Social Science/ Life Skills: Obtain newspaper ads showing the cost of products. Figure the

sales price of items on sale for ½ off, 1/3 off, ¼ off.

Text books, trade books, fractions strips, grid paper, dominoes, dry erase boards, calculators,

computers, interactive boards

NCTM , and commercial websites

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Where, in daily life and/or occupations, will I find it necessary to convert measurements?

Can I learn different systems, other than customary/standard measurement, for measuring things?

Will I find that metric measurement is needed more in today’s world than standard measurement

is?

Where will I see standard measurement and metric measurement used in real life?

Does it make sense to me to measure objects in a variety of ways to find the best method?

Formative: Students will work with partners: “My book is 30 cm long. How many mm is that? “

The partner answers, then makes up her/his own problem.

Students will create word problems using measurement in real life situations.

Math journal: Students will answer one or more of the essential Questions.

Summative: Performance Assessment: In a long range activity, students will make a booklet

using a theme, e.g., sports, toys, objects in the home, etc. The student will chose a variety of

objects, measure each in several different ways, using both standard and metric measurement.

The booklet will have illustrations of each object and its measurement conversion. The student

will create word problems and write explanations of the activity

Convert like measurement units within a given measurement system. 1. Convert among different sized standard measurement units within a given measurement

system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real

world problems.

Domain: Measurement and Data (5.MD) - Convert like measurement units within a given measurement system.

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Vocabulary: convert, all terms used in standard measurement, all terms used in metric

measurement

Convert measurements in multi step, real world problems.

Science: Investigate how measurement is used in science experiments.

Technology: Investigate how measurement is used in computers, art, and graphic arts.

Make measurements using virtual manipulatives.

Social Science/Life Skills: Discover ways in which the conversion of measurement is used in

occupations such as carpentry, flooring, plumbing, etc.

Social Science: Discuss what might happen if carpenters building a deck did not use accurate

measurements.

Text books, trade books, metric and standard measuring tools, calculators, computers,

interactive boards, virtual manipulatives

NCTM, and commercial websites

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Domain: Measurement and Data (5.MD) Represent and interpret data.Essential Questions What should I be able to answer? What guides my thinking?

How can line plots help me understand various types of information?

Where will I find line plots used in the real world?

Will I be able to use various methods to interpret data?


Formative: Math journal: Student will draw line plots showing various data, then write an

explanation of what the information shows.

Use grid paper to plot fractions. Place the number “one” on the far left, and the number “two” on

the far right. Given a specific denominator, the student will fill in as many fractions as possible,

placing them in an equidistant manner.


Summative: Performance Assessment: In a long range activity, the student will use line plots to

represent various fractions in real life situations, such as comparing measurements of objects.

Decide and explain how and why these measurements were chosen, and why precision is

necessary in real life measurements. Formulate word problems using data from line plots.


1. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use

operations on fractions for this grade to solve problems involving information presented in line

plots. For example, given different measurements of liquid in identical beakers, find the amount

of liquid each beaker would contain if the total amount in all the beakers were redistributed

equally.

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Vocabulary: line plot, data, represent, interpret

Make line plots using fractions, then interpret that data through models, written explanations.



Science / Social Science: Observe and count the birds at a bird feeder for five minutes. Distinguish

and count the types of birds. Plot these on a line plot. Also, note what kind and how many bird fly

away, then return to the feeder. This also can be plotted.

Social studies: Make line plots of the area and population of several cities and compare the results.

Text books, trade books, grid paper, metric and standard measuring tools, calculators, computers,

interactive boards, virtual manipulatives

NCTM, and commercial websites

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Domain: Measurement and Data. Standard (5.MD) - Geometric Measurement: Understand the concepts of volume and relate volume to multiplication and addition.




Where and how is volume used in real life situations?

When will I use volume?

How can models help me to understand volume?

Formative: Student, with a partner, obtains several square or rectangular boxes. Each person will

guess the number of unit cubes needed to fill the containers. Keep score as to which person has

the nearest estimate of the correct number of unit cubes.


Summative: Performance Assessment: In a long range activity, student will choose four

containers of various sizes and shapes and fill each with cubic units. Write a detailed report on the

amount of cubic units that are needed to fill each container. Decide which types of containers can

or cannot be fully packed with cubic units. Explain why this is so and how this conclusion was

reached. Use the mathematical formula for volume where it is applicable. Word problems should

be created to show the results of the experiment.

1. Recognize volume as an attribute of solid figures and understand concepts of volume measurement.

a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume,

and can be used to measure volume.

b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have

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a volume of n cubic units.

2. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised

units.

3. Relate volume to the operations of multiplication and addition and solve real world and

mathematical problems involving volume.

a. Find the volume of a right rectangular prism with whole-number side lengths by packing it with

unit cubes, and show that the volume is the same as would be found by multiplying the edge

lengths, equivalently by multiplying the height by the area of the base. Represent threefold

whole-number products as volumes, e.g., to represent the associative property of multiplication.

b. Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right

rectangular prisms with whole-number edge lengths in the context of solving real world and


c. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping

right rectangular prisms by adding the volumes of the non-overlapping parts, applying this

technique to solve real world problems.


Vocabulary: volume, solid figures, unit cubes, cubic cm, cubic in, cubic ft, base, rectangular prism,

right rectangular prism

1) Volume and volume measurement.

Use unit cubes to measure volume. If a solid figure is packed with 15 unit cubes, the

volume is 15 unit cubes.

2) Use standard measures as well as non-standard measures.

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3) Use real world word problems involving volume.

Compare the measuring of volume using unit cubes to the measuring of volume by

multiplying the three side lengths.

Use formulas for volume using real world word problems.

Add the volumes of two right rectangular prisms.

Real world: Find a way to ship two boxes which will fit into one box. Stack equal-sized

boxes.


Science / Life Skills: Given specific measurements (length, width, and height), calculate how

much concrete will be needed to make a patio.

Science / Life Skills: Given specific measurements (length, width, height), calculate how much soil

will be needed to make a raised garden bed.

Technology: Use virtual manipulatives to calculate the unit cubes needed to fill various types and

sizes of containers.


Text book, trade books, metric and standard measurement tools, calculators,

computers, virtual manipulatives, interactive boards, NCTM, and commercial websites

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Domain: Geometry (5.G) - Graph points on the coordinate plane to solve real world and mathematical problems.



Where will I see graphs used in the real world, and how are they used?

How will graphing in mathematics help me in understand graphs in other situations and other

subject areas?

Why is graphing important to me in solving mathematical problems?

Formative: In small groups, list as many real life situations in which graphs are useful.

Use grid paper to plot a given table of ordered pairs. This can be done with partners – one

person plots the X axis, the other plots the Y axis. Or one partner plots the ordered pair and the

other person checks for correctness.


Summative: Performance Assessment: In a long range activity, the student will choose a real life

political, economic, historical, or geographic situation and graph this. Make inferences as to why

the graph is representative of the situation, and explain these inferences. Make predictions

based on the graph as to the future of the situation. E.G., will a drop in the stock market over a

period of time cause a slow down in other areas, such as jobs?

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1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with

the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given

point in the plane located by using an ordered pair of numbers, called its coordinates. Understand

that the first number indicates how far to travel from the origin in the direction of one axis, and the

second number indicates how far to travel in the direction of the second axis, with the convention

that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-

axis and y-coordinate).


2. Represent real world and mathematical problems by graphing points in the first quadrant of the

coordinate plane, and interpret coordinate values of points in the context of the situation.

Vocabulary: perpendicular, axes, ordered pair, coordinates, coordinate plane, quadrant

1) Graph X and Y axes on a coordinate plane.

2) Use real world problems involving graphing on a coordinate plane.


Science: Set up an experiment testing the life of certain batteries. Graph the results.

Social Studies: Obtain and graph historical data on rain or snowfall in a series of years.

Social Studies: Trace family histories of the students in the class and graph the number of students

from various countries.

Technology: Use spreadsheets to predict the direction and placement of lines on a coordinate

plane. Write an explanation that hypothesizes about the direction and placement of these lines.

science, history, geography, economics, art, technology

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Text books, trade books, calculators, computers, graphs, interactive boards, virtual

manipulatives, NCTM, and commercial websites

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Where will I find two dimensional figures used in the real world?

Where, in the real world, do I see shapes that use two dimensional figures?

How can using models help me understand two dimensional figures?


Formative: The student will make a tree diagram of two dimensional figures based on their

properties. Make a chart, similar to a computer flow chart, of two dimensional figures based on

their properties. Students can discuss why engineers must study two dimensional figures as well

as the computation of numbers used in these figures.

Summative: Performance Assessment: In a long range activity, the student will design and build a

replica of a building which makes use of several types of two dimensional figures. The building

can be made of cardboard or some similar easily obtainable material. A written explanation of the

process of designing and the types of figures used will be included. In order to observe examples

needed to study the use of various types of figures in buildings, students can observe the variety of

buildings in a large city, such as Philadelphia.

Domain: Geometry (5.G) - Classify two-dimensional figures into categories based on their properties.

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122


1. Understand that attributes belonging to a category of two dimensional figures also belong

to all subcategories of that category. For example, all rectangles have four right angles and

squares are rectangles, so all squares have four right angles.

2. Classify two dimensional figures in a hierarchy based on properties.


Vocabulary: angle: acute, right, obtuse; two-dimensional figures, polygon, triangle: scalene,

isosceles, equilateral; quadrilateral, parallelogram, trapezoid, rectangle, rhombus, square,

pentagon, hexagon, heptagon, octagon, nonagon, hexagon

1) List attributes of two-dimensional figures.

2) Classify figures according to angles, sides, and shapes.


Art/Architecture: Use pattern blocks, or attribute blocks to create designs or tessellations of two-

dimensional figures.

Technology: Use virtual manipulatives to create designs of two-dimensional figures.

Art: Draw a chart showing hierarchy of figures, using colors to show comparison/contrast of those

figures. This can also be done using a computer.


Text books, trade books, calculators, computers, virtual manipulatives, pattern blocks, attribute blocks, interactive boards MCTM, and commercial websites

dmkanowitz

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Grade Five Page 45


Grade 6

Grade Six

Page 1

Grade 6


Instructional time in Grade 6 focuses on four critical areas: (1) connecting ratio and rate to whole number multiplication and division

and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of

number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations;

and (4) developing understanding of statistical thinking.

(1) Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent

ratios and rates as deriving from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings

that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus

students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and

fractions. Students solve a wide variety of problems involving ratios and rates.

(2) Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and

division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems.

Students extend their previous understandings of number and the ordering of numbers to the full system of rational numbers, which

includes negative rational numbers, and in particular negative integers. They reason about the order and absolute value of rational

numbers and about the location of points in all four quadrants of the coordinate plane.

(3) Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given

situations, evaluate expressions, and use expressions and formulas to solve problems. Students understand that expressions in different

forms can be equivalent, and they use the properties of operations to rewrite expressions in equivalent forms. Students know that the

solutions of an equation are the values of the variables that make the equation true. Students use properties of operations and the idea of

maintaining the equality of both sides of an equation to solve simple one-step equations. Students construct and analyze tables, such as

tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe relationships between quantities.

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(4) Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students

recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The

median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that

each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point.

Students recognize that a measure of variability (inter-quartile range or mean absolute deviation) can also be useful for summarizing data

because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to

describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data

were collected.

Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine

area, surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes,

rearranging or removing pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas

for areas of triangles and parallelograms. Students find areas of polygons and surface areas of prisms and pyramids by decomposing them

into pieces whose area they can determine. They reason about right rectangular prisms with fractional side lengths to extend formulas for

the volume of a right rectangular prism to fractional side lengths. They prepare for work on scale drawings and constructions in Grade 7

by drawing polygons in the coordinate plane.

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3 + x




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Grade Six Standards for Mathematical Content

Ratios and Proportional Relationships (6.RP)


6.RP All students will understand ratio concepts and use ratio reasoning to solve

problems.

1. Understand the concept of a ratio and use ratio language to describe a ratio

relationship between two quantities. Example: “The ratio of wings to beaks

in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.”

“For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0,

and use rate language in the context of a ratio relationship.

Example: “This recipe has a ratio of 3 cups of flour to 4 cups of sugar,

so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15

hamburgers, which is a rate of $5 per hamburger.”1

3. Use ratio and rate reasoning to solve real-world and mathematical problems,

e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number

line diagrams, or equations.

a) Make tables of equivalent ratios relating quantities with whole number

measurements, find missing values in the tables, and plot the pairs of values

on the coordinate plane. Use tables to compare ratios.

b) Solve unit rate problems including those involving unit pricing and

constant speed. Example: If it took 7 hours to mow 4 lawns, then

at that rate, how many lawns could be mowed in 35 hours? At what

rate were lawns being mowed?

1Expectations for unit rates in this grade are limited to non-complex fractions.







Students make a list of ratios they encounter

in daily life, such as: 5 lbs. for $3.00,

55 miles per hour, etc.

Students look through newspaper ads, store

flyers and catalogs to find ratios. Have

students write the ratio and solve to find unit

price.

Use double number lines to solve ratio

problems. Example: 6 pencils cost $1.00.

How many pencils can Sam buy with $2.50

$0.50 $1.00 $1.50 $2.00 $2.50 $3.00

6 pencils

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c) Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means

30/100 times the quantity); solve problems involving finding the whole when

given a part and the percent.

d) Use ratio reasoning to convert measurement units; manipulate and transform

units appropriately when multiplying or dividing quantities; solve problems

involving proportional relationships (e.g. scale models, maps, speed)

Use a tape diagram or bar model to solve

percent problems. Example: Fourteen

students in the class ride bikes to school.

This is 20% of the class. How many

students are in the class?

14 students

Use cross products rules and equivalent

fractions to solve problems involving

proportional relationships.

1 foot = 5 feet

12 inches x inches

The Number System (6.NS) Standards Strategies and Assessments

6.NS All students will apply and extend previous understandings of multiplication

and division to divide fractions by fractions.

1. Interpret and compute quotients of fractions, and solve word problems involving

division of fractions by fractions (e.g., by using visual fraction models and equations

to represent the problem). Example: Create a story context for (2/3) ÷ (3/4) and use a

visual fraction model to show the quotient; use the relationship between

multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is

2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person

How much chocolate will each person

get if 3 people share 1/2 lb of chocolate

equally?

1 lb of chocolate

1/2 of 1 lb

1/2 ÷ 3 =1/6 Each person gets 1/6 lb.

0% 20% 40% 60% 80% 100%

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Standards Strategies and Assessments get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings

are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with

length 3/4 mi and area 1/2 square mi?

6.NS All students will compute fluently with multi-digit numbers and find common

factors and multiples.

2. Fluently divide multi-digit numbers using the standard algorithm.

3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard

algorithm for each operation.

4. Find the greatest common factor of two whole numbers less than or equal to 100

and the least common multiple of two whole numbers less than or equal to 12.

Use the distributive property to express a sum of two whole numbers 1–100 with a

common factor as a multiple of a sum of two whole numbers with no common factor.

Example: Express 36 + 8 as 4 (9 + 2).

6.NS All students will apply and extend previous understandings of numbers to the

system of rational numbers.

Continue 6.NS Standard on next page

Use visual representations to show division

of fractions. Example: How many 3/4-cup

servings are in 2/3 of a cup of yogurt?

3/4 cup serving

3/4 cup serving

Use factor rainbows to find the factors of a

number.

The factors of 24

2/3 cup

of yogurt

yogurt

(2/3) ÷ (3/4) = 8/9

1 2 3 4 6 8 12 24

24

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6.NS All students will apply and extend previous understandings of numbers to thesystem of rational numbers5. Understand that positive and negative numbers are used together to describe

quantities having opposite directions or values; use positive and negative numbers to

represent quantities in real-world contexts, explaining the meaning of 0 in each

situation.

6. Understand a rational number as a point on the number line. Extend number line

diagrams and coordinate axes familiar from previous grades to represent points on

the line and in the plane with negative number coordinates.

a) Recognize opposite signs of numbers as indicating locations on opposite

sides of 0 on the number line; recognize that the opposite of the opposite

of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b) Understand signs of numbers in ordered pairs as indicating locations in

quadrants of the coordinate plane; recognize that when two ordered pairs

differ only by signs, the locations of the points are related by reflections

across one or both axes.

c) Find and position integers and other rational numbers on a horizontal or

vertical number line diagram; find and position pairs of integers and other

rational numbers on a coordinate plane.

7. Understand ordering and absolute value of rational numbers.

a) Interpret statements of inequality as statements about the relative

position of two numbers on a number line diagram.

Use vertical and horizontal number lines to

show opposites.

Present situations that involve positive and

negative numbers e.g., temperature

above/below zero, elevation above/below

sea level, credits/debits, positive/negative

electric charge.

Give each student an index card with a

rational number on it. Have students create a

human number line by placing themselves

in order from least to greatest.

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b) Write, interpret, and explain statements of order for rational numbers

in real-world contexts. Example: Write –3 C > –7 C to express

the fact that –3 C is warmer than –7 C.

c) Understand the absolute value of a rational number as its distance

from 0 on the number line; interpret absolute value as magnitude for a

positive or negative quantity in a real-world situation.

d) Distinguish comparisons of absolute value from statements about order.

Example: Recognize that an account balance less than –30

dollars represents a debt greater than 30 dollars.

8. Solve real-world and mathematical problems by graphing points in all four

quadrants of the coordinate plane. Include use of coordinates and absolute value

to find distances between points with the same first coordinate or the same second

coordinate.

Show –3 > –7 as a statement that –3 is

located to the right of –7 on a number line

oriented from left to right.

For an account balance of –30 dollars, write

|–30| = 30 to describe the size of the debt in

dollars.

Make sure students understand numbers in

the context of the situation they are used to

describe. Example: Although -30 < -40

an account balance of -$40 represents a

debt greater than -$30.

Have students plot various points to

determine how the signs of the coordinates

can be used to determine the quadrant a

point is located in.

x- coordinate y-coordinate Quadrant

+ + I

- + II

- - III

+ - IV

-10 -7 -3 0 10

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Expressions and Equations (6.EE) Standards Strategies and Assessments

6.EE All students will apply and extend previous understandings of arithmetic to

algebraic expressions.

1. Write and evaluate numerical expressions involving whole-number exponents.

2. Write, read, and evaluate expressions in which letters stand for numbers.

a) Write expressions that record operations with numbers and with letters

standing for numbers. Example: Express the calculation

“Subtract y from 5” as 5 – y.

b) Identify parts of an expression using mathematical terms (sum, term, product,

factor, quotient, coefficient); view one or more parts of an expression as a

single entity. Example: Describe the expression 2 (8 + 7) as a product of two

factors; view (8 + 7) as both a single entity and a sum of two terms.

c) Evaluate expressions at specific values of their variables. Include expressions

that arise from formulas used in real-world problems. Perform arithmetic

operations, including those involving whole number exponents, in the

conventional order when there are no parentheses to specify a particular

order (Order of Operations). Example: Use the formulas V = s3 and

A = 6s2 to find the volume and surface area of a cube with sides

of length s = 1/2.

3. Apply the properties of operations to generate equivalent expressions.

Example: Apply the distributive property to the expression 3 (2 + x) to produce the

equivalent expression 6 + 3x; apply the distributive property to the expression 24x +

18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations

to y + y + y to produce the equivalent expression 3y.

Students should recognize that quantities

written in exponential form contain two

parts: a base and an exponent. The

exponent indicates the number of time the

base is used as factor.

30

= 1

31 = 3

32 = 3 x 3 = 9

33 = 3 x 3 x 3 = 27

34 = 3 x 3 x 3 x 3 = 81

Provide students with several algebraic

expressions and have them select the one

that matches a given statement.

Have students identify the correct way of

―translating‖ an expression. Example: How

would you translate 2(8 + 7)

2 times 8 plus 7

the product of 2 and 8 plus 7

the product of 2 and the sum of 8 and 7

Understand that 3(2 + x) means:

(2 + x) + (2 + x) + (2 + x)

Apply previously learned strategies for

finding common factors when applying the

distributive property.

24x + 18y

(6 x 4)x + (6 x 3)y

6(4x + 3y)

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Standards Strategies and Assessments 4. Identify when two expressions are equivalent (i.e., when the two expressions name

the same number regardless of which value is substituted into them). Example: The

expressions y + y + y and 3y are equivalent because they name the same number

regardless of which number y stands for.

6.EE All students will reason about and solve one-variable equations and inequalities.

5. Understand solving an equation or inequality as a process of answering a question:

which values from a specified set, if any, make the equation or inequality true?

Use substitution to determine whether a given number in a specified set makes an

equation or inequality true.

6. Use variables to represent numbers and write expressions when solving a real-world

or mathematical problem; understand that a variable can represent an unknown

number, or, depending on the purpose at hand, any number in a specified set.

7. Solve real-world and mathematical problems by writing and solving equations

of the form x + p = q and px = q for cases in which p, q and x are all nonnegative

rational numbers.

8. Write an inequality of the form x > c or x < c to represent a constraint or condition

in a real-world or mathematical problem. Recognize that inequalities of the form

x > c or x < c have infinitely many solutions; represent solutions of such inequalities

on number line diagrams.

6.EE All students will represent and analyze quantitative relationships between

dependent and independent variables.

9. Use variables to represent two quantities in a real-world problem that change in

relationship to one another; write an equation to express one quantity, thought of as

the dependent variable, in terms of the quantity, thought of as the independent

Use input-output tables to determine if

expressions are equivalent.

y y + y + y 3y

5 15 15

2.5 7.5 7.5

3 ½ 10 ½ 10 ½

Give students one side of an equation and

have them provide the possibilities for the

other half using an expression with one

variable.

Example: ? = 25

Possible equations: 2x+ 5 =25

18 + x = 25

5(2.3 + x) =25

What value for x would make each equation

true?

Understand that statements of inequality

have many solutions.

Example: x + 5 < 8

x < 3

Have students list ten rational numbers that

are solutions to the inequality.

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Standards Strategies and Assessments variable. Analyze the relationship between the dependent and independent variables

using graphs and tables, and relate these to the equation. Example: In a

problem involving motion at constant speed, list and graph ordered pairs of

distances and times, and write the equation d = 65t to represent the relationship

between distance and time.

Generate different examples of situations

where one quantity is dependent on another,

e.g., taxable amount and tax charged,

quantity and total cost; temperature and

wind speed; number correct and total score;

hours worked and earnings, etc.

Geometry (6.G)


6.G All students will solve real-world and mathematical problems involving area,

surface area, and volume.

1. Find the area of right triangles, other triangles, special quadrilaterals, and

polygons by composing into rectangles or decomposing into triangles and

other shapes; apply these techniques in the context of solving real-world and


2. Find the volume of a right rectangular prism with fractional edge lengths by

packing it with unit cubes of the appropriate unit fraction edge lengths, and

show that the volume is the same as would be found by multiplying the edge

lengths of the prism. Apply the formulas V = l w h and V = B h to find volumes

of right rectangular prisms with fractional edge lengths in the context of solving


Use geo-boards, tangrams, or graph paper to

show that a triangle is ½ a quadrilateral.

Find the area of the isosceles trapezoid by

decomposing and rearranging it into a

rectangle with the same area.

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3. Draw polygons in the coordinate plane given coordinates for the vertices; use

coordinates to find the length of a side joining points with the same first

coordinate or the same second coordinate. Apply these techniques in the context

of solving real-world and mathematical problems.

4. Represent three-dimensional figures using nets made up of rectangles and

triangles, and use the nets to find the surface area of these figures. Apply these

techniques in the context of solving real-world and mathematical problems.

Fill cubes and prisms with small cubes or

sand to understand that volume means

capacity.

Draw polygons on a coordinate plane

by connecting points that have been plotted.

Explore three-dimensional figures at: http://www.learner.org/interactives/geometry/3d.html

http://www.mathsnet.net/geometry/solid/nets.html

Draw nets of a variety of three-dimensional

figures. Identify prisms and cubes from their

net.

http://www.learner.org/interactives/geometry/3d.html

http://www.mathsnet.net/geometry/solid/nets.html

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Statistics and Probability (6.SP) Standards Strategies and Assessments

6.SP All students will develop an understanding of statistical variability.

1. Recognize a statistical question as one that anticipates variability in the data

related to the question and accounts for it in the answers. Example: “How old am

I?” is not a statistical question, but “How old are the students in my school?”

is a statistical question because one anticipates variability in students’ ages.

2. Understand that a set of data collected to answer a statistical question has a

distribution which can be described by its center, spread, and overall shape.

3. Recognize that a measure of center for a numerical data set summarizes all of its

values with a single number, while a measure of variation describes how its values

vary with a single number.

6.SP All students will summarize and describe distributions.

4. Display numerical data in plots on a number line, including dot plots,

histograms, and box-and- whisker plots.

5. Summarize numerical data sets in relation to their context:

a. Reporting the number of observations.

b. Describing the nature of the attribute under investigation, including how it

was measured and its units of measurement.

c. Giving quantitative measures of center (median and/or mean) and variability

(interquartile range and/or mean absolute deviation), as well as describing

any overall pattern and any striking deviations from the overall pattern with

reference to the context in which the data were gathered.

A statistical question is one that collects

information that addresses differences in a

population. Have students formulate several

statistical questions they might ask of their

classmates. Have students use one of their

statistical questions to collect data from their

classmates.

The distribution is the arrangement of the

values of a data set. Have students order the

values in a data set to determine the median

and range of the data set.

Measures of Center: Mode, Median, Mean

Measure of Variation: Range

Students should be familiar with the various

ways data can be displayed and be able to

determine the best way to display a particular

data set.

Divide students into groups. Give each

group a different graph displaying a data set.

The graph should be untitled. Have the group

write a description of the data presented in

the graph: number of data points; median

and/or mean; range; deviations from the

general pattern.

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d. Relating the choice of measures of center and variability to the shape of the

data distribution and the context in which the data were gathered.

6.

Students should be able to use Excel

software to create tables and appropriate

graphs to display the data.

http://mathwire.com/games/datagames.html

Extended Resources– Grade 6 Mathematics

Domain: Ratio and Proportional Relationships (6.RP) - Understand ratio concepts anduse ratio reasoning to solve problems. Essential Questions What should I be able to answer? What guides my thinking?

What is a mathematical way to compare quantities?

How do mathematical models and/or tables shape my understanding of mathematics?

How can unit rates help me to make comparisons?

How does fluency with operations help me to solve ratio and percent problems?

How are ratios, rates, and percents connected/related?


Formative: Have students use online or “hands on” manipulatives (assorted shapes, colors) to

create representation of, describe and illustrate ratio concepts in a journal entry; Have students

demonstrate fluency with relating commonly used percents as fractions and decimals using

interactive web based activities or flash cards. Have students create a story problem for ratios

written on board (example: 9/7, 2:10, and 6 to 4).

Summative: Performance Assessment to include student creating, writing, and solving a “real

life” ratio problem: “Describe a situation in your life that represents a ratio…” Example: number of

boys to girls in your class or family; Sports number of hits to at bats in a baseball/softball game or

shots on goals to actual goals (use your own stats or research a professional player or team stats);

Music – number of rock songs to rap songs on your Ipod; Research a recipe and compare

ingredients (dry to liquid, teaspoons to tablespoons), etc. Students can extend and compute

percentages, and/or create graphs.

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Grade Six Page 17


1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two

quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings

there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the

context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is

3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

[Expectations for unit rates in this grade are limited to non-complex fractions.]

3. Use ratio and rate reasoning to solve real world and mathematical problems, e.g., by reasoning about tables

of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the

tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours

to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being

mowed?

c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve

problems involving finding the whole, given a part and the percent.

d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying

or dividing quantities.

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Grade Six Page 18




Use mathematical representation (equivalent fractions, tables) to solve unit, rate and/or ratio problems.

Use mathematical operations and properties to solve problems (Multiplication/Division, Cross Product Rule)

Writing math sentences in equation or proportion format and solve for missing value (16 is 50% of what number? 16

= .5n).

Vocabulary: ratio, ratio representation, rate, unit rate, proportion, cross product rule, percent, formula

Social Studies: use of mileage scale to interpret distances on maps, globes, &/or Google earth.

Consumer Science: computing unit costs and percentages; manipulating recipes to increase/decrease measurement

of ingredients

Science: Using formulas to compute rates, distance travelled, (example: mph, mpg, etc).; Physical Education: heart

rate, Art: perspective

Various commercial products and websites for practice activities (example: AIMS activities, Interactive board

activities), virtual manipulatives, math songs, and calculators

dmkanowitz

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Grade Six Page 19

Domain: The Number System (6.NS) - Apply and extend previous understandings of multiplication and division to divide fractions by fractions.



Why do I need to work with parts of numbers?

How can I use fractions efficiently to solve real world problems?

Why would I use division as a strategy to solve for a variable?

Formative: use models , concrete objects and/or on-line manipulatives to represent parts and

division of parts (fractions); Have students use fraction strips to model multiplication of fractions,

example: Two 1/4s are as long as ½; 1/6 is half of 1/3; 2/3 of 2/3 = 4/9. Have students fold a piece

of paper in half vertically twice and shade ¼ with a colored pencil. Then, have students fold paper

in half horizontally and shade ½ with a different colored pencil. The part shaded with both colors is

½ of ¼ or ½ times ½. What part of the whole is shaded with both colors? (1/2 x ¼ = 1/8).

In a journal entry, have students explain steps to divide fractions by fractions and simplify the

quotient to lowest terms using mathematical terms.

Summative: Performance based assessment to demonstrate dividing fractions using models; for

example, (1) in a lab environment, have students divide a recipe, illustrate or write steps used,

write equation and show work used to solve; (2) have each student write a fraction between 0 and

1 on a small piece of paper or index card and on a piece of paper that they will keep. Have each

student pass their fraction to another student and find the product of their fractions. Repeat with

division. Continue passing fractions and computing products and quotients as time allows. Have

student select one of each operation and create a word problem using the fractions.

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Grade Six Page 20


1. Interpret and compute quotients of fractions, and solve word problems involving division

of fractions by fractions, e.g., by using visual fraction models and equations to represent the

problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to

show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷

(3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will

each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in

2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2

square mi?



Previous mastery of multiplication of whole numbers, fractions, and mixed numbers and expressing

products in lowest terms; Measurement equivalents; Vocabulary: fraction, numerator,

denominator, equivalent fraction, dividend, divisor, quotient, mixed numbers, improper fractions,

bar notation, reciprocal, simplest form/lowest terms

Apply formulas to solve problems

Problem solving in multiple subject areas, including Science: measurement, Social Studies: size,

distances, Consumer Sciences: applying formulas to cooking, measurement,

Technology: using virtual manipulatives

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Grade Six Page 21


Various commercial products and websites for practice activities (example: AIMS

activities, Interactive board activities. NCTM), virtual manipulatives, math songs,

and calculators

dmkanowitz

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Grade Six Page 22


Why is it important for me to compute multi-digit numbers with fluency? How can I use common

factors and multiples to solve problems? Why is it important for me to be able to estimate

quantities involving multi-digit numbers and/or decimals? When would estimation of quantities

be helpful? How can I use models to understand operations with decimals? How can I use the

Distributive Property to simplify problem solving?


Formative: use on line and/or hands on base ten manipulatives or centimeter grid paper to model

decimals and decimal operations; Have students write a decimal (example: 0.43) and a mixed

decimal (2.067) on two strips of paper. Put all strips in a bag. Have each student pick 2 strips,

rewrite the numbers on a sheet of paper, being careful to line up the decimal points, and add; Have

students solve problems using the standard algorithm for all operations with multi digit numbers

and numbers containing decimals.

Summative: (1) Present completed problems containing errors to students either on paper or

interactive board. Have students identify the error and explain. Then, have student complete

problem correctly. (2) Describe several situations in everyday life where decimals are used;

Performance Assessment example: Compute the cost of a special event; Using the internet and/or

print media (advertisements, newspapers), have students prepare a “shopping list” for an event

and research costs associated with products to be purchased (for example, Back to School

supplies, Christmas shopping, Planning a party, a class trip, a family night out to a Yankees game,

etc.), and then present a budget to classmates for the event (Could extend and have students offer

two different budget plan choices.)

Domain: The Number System (6.NS) – Compute fluently with multi-digit numbers and find common factors and multiples.

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Grade Six Page 23


1. Fluently divide multi-digit numbers using the standard algorithm.

2. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard

algorithm for each operation.

3. Find the greatest common factor of two whole numbers less than or equal to 100 and the

least common multiple of two whole numbers less than or equal to 12. Use the distributive

property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a

sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).



Fluency with: multiplication facts of multi digit numbers; Estimation; Divisibility Rules for

2,3,4,5,6,9,10; Properties of Multiplication and Addition: Commutative, Associative, Distributive;

Prime Factorization of numbers; Vocabulary: factors, greatest common factor, products, multiples,

least common multiple, quotients, divisor, dividend, place value, decimal

Problem solving in multiple subject areas, including Science, Social Studies, Consumer Sciences,

Art, Technology, PE

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Various commercial products and websites for practice activities (example: AIMS activities,

Interactive board activities, NCTM), virtual manipulatives, math songs, and calculators

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96

Domain: The Number System (6.NS) - Apply and extend previous understandings of numbers to the system of rational numbers.



Why is important for me to know when a value is “above” or “below” zero? (Bank account,

Temperature, Sea level)

Why is – 12 less than 26? How can I use a number line to compare integers?

What is an absolute value and how can I use a number line to prove the meaning of the term,

absolute value?

Formative: Describe how to use on-line or hands on manipulatives (chips, beads, tiles) to

model working with integers; Have students create a number line from 29 to +20 (use a long

strip of paper, similar to a tape measure width) and model adding integers (examples:

3 + 5, 8 + 4, 6 + 6)

6 + 6); Have students read a temperature from a thermometer (can be drawn on board or

accessed through interactive technology), then read the opposite temperature ( the opposite of

positive 5⁰ is negative 5⁰. How far away from 0 is 5 and 5? (absolute value); Have students graph

ordered pairs (x,y) on a coordinate plane; Identify given points on a coordinate plane and indicate

quadrant; Have students plot and connect points which result in a picture.

Summative: Have students create their own picture utilizing all quadrants of the coordinate plane,

identify and document points to be plotted. Students trade with another student and complete each

other’s picture problem; Use manipulatives to demonstrate and solve problems for values

above/below zero; Cross Curricula Performance Assessment: Research a topic involving

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numbers above/below zero and prepare a informational presentation (for example, prepare a

presentation on a recently studied land mass in Social Studies; or have students plan an event and

create a budget to include estimating costs, paying vendors, and balancing the income and

expenditures.

1. Understand that positive and negative numbers are used together to describe quantities

having opposite directions or values (e.g., temperature above/below zero, elevation above/below

sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to

represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

2. Understand a rational number as a point on the number line. Extend number line

diagrams and coordinate axes familiar from previous grades to represent points on the line and in

the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the

number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –

(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the

coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of

the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line

diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

3. Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a

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number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the

right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts.

For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC.

c. Understand the absolute value of a rational number as its distance from 0 on the number line;

interpret absolute value as magnitude for a positive or negative quantity in a real-world

situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the

size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example,

recognize that an account balance less than –30 dollars represents a debt greater than 30

dollars.

4. Solve real world and mathematical problems by graphing points in all four quadrants of

the coordinate plane. Include use of coordinates and absolute value to find distances between

points with the same first coordinate or the same second coordinate.


Number sets (natural, whole, integers, rational, irrational); Symbols used to represent negative ( )

and positive (+), and absolute value / / ; Positive and negative numbers; number line containing

positive and negative numbers; coordinate plane; quadrants, ordered pairs (x,y) and graphing of

such on the coordinate plane

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Science (temperature, positive/negative electrical charge); Social Studies/Geography: above/

below sea level

Consumer Sciences: bank accounts, reconciling bank accounts Art: Contrast

Economics: Stock Market


Interactive board activities, virtual manipulatives, math songs, and calculators

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Domain: Expressions and Equations (6.EE) - Apply and extend previous understandings of arithmetic to algebraic expressions.


How would translating word sentences into mathematical sentences help me to solve problems?

How do algebraic and numerical expressions help me to solve problems?

Why can I use variables to represent a value?

How can I use properties to simplify expressions, solve for missing values, and show

equivalency?


Formative: Use manipulatives (algebra scales, tiles, fraction bars) to demonstrate equivalent

expressions; Journey entry/Exit card: Provide a word sentence and have students translate the

word sentence into an expression and/or equation using variables (Example: A number divided by

three OR Twice a number subtracted from ten is two.) ; Have students identify key words used in

word phrases (example key words for addition: increased by, sum, more than, etc); Use interactive

board to create two lists. One side containing word phrases and the other side containing

mathematical expressions. Have students “match” the phrase to the appropriate expression (OR

use “flash cards” for students to match).

Summative: Using real-life experiences, create a list of mathematical phrases and another list of

the correlating algebraic expressions. Have students match the phrase to the expression or vice

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versa. Then, provide phrases and have students write an expression to represent each phrase.


1. Write and evaluate numerical expressions involving whole number exponents.

2. Write, read, and evaluate expressions in which letters stand for numbers.

a. Write expressions that record operations with numbers and with letters standing for numbers.

For example, express the calculation “Subtract y from 5” as 5 – y.

b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient,

coefficient); view one or more parts of an expression as a single entity. For example, describe

the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a

sum of two terms.

c. Evaluate expressions at specific values of their variables. Include expressions that arise from

formulas used in real-world problems. Perform arithmetic operations, including those involving

whole-number exponents, in the conventional order when there are no parentheses to specify a

particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 find

the volume and surface area of a cube with sides of length s = 1/2.

3. Apply the properties of operations to generate equivalent expressions. For example,

apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6

+ 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent

expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent

expression 3y.

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4. Identify when two expressions are equivalent (i.e., when the two expressions name the

same number regardless of which value is substituted into them). For example, the expressions y

+ y + y and 3y are equivalent because they name the same number regardless of which number

y stands for.


Order of Operations; positive, whole number exponents, working with properties, Evaluating

formulas with given values; Apply formulas with and without exponents to solve for missing values

and/or calculate measurements;

Vocabulary: sum, term, like terms, product, factor, quotient, coefficient, formulas, equivalents,

base, exponent, power



Art, Technology



activities, Interactive board activities), virtual manipulatives, math songs, and calculators

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Domain: Expressions and Equations (6.EE) - Reason about and solve one-variableequations and inequalities.


How will fluency in solving for unknown values help me succeed in a global environment? Marketplace?

Community?

How can I represent an unknown value?

Why are variables used in solving equations and inequalities?

What strategies can I use to solve for unknowns in algebraic equations or inequalities?

How can I use inverse properties to get the variable alone?

How can a formula aid me in problem solving?


Formative: Practice writing equations/inequalities from word phrases and vice versa; Use manipulatives to

model equations (example: algebra tiles and scales); Provide equations and inequalities to students to solve

for unknown values using appropriate strategies and properties in a step by step procedure.

Summative: Provide equations and have students solve for missing values by “balancing” both sides of

the equation using the Properties of Equality; Have students count the number of pieces in a stained glass

window in your school chapel or church. Then, have students create an equation to determine how many

complete windows they could create with 650 pieces of stained glass (if 24 pieces in one window then 24x =

650).Performance based assessment: Have students create an original problem requiring they write a one

variable equation (or inequality) to represent the problem, solve for the variable using appropriate

strategies/properties, graph the solution on a number line and write an answer statement reflecting the

solution as described through the original problem. For example, Tim is expected to swim 20 laps during

practice on Wednesday and Thursday. He swam 9 laps on Wednesday. Write and solve an inequality to

show how many laps he had to swim on Thursday. x + 9 > 20; x > 11 , 11 laps or more. Make sure

number line graph has a solid point on 11 and is shaded in the correct direction, > 11.

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1. Understand solving an equation or inequality as a process of answering a question: which values from a

specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a

specified set makes an equation or inequality true.

2. Use variables to represent numbers and write expressions when solving a real world or mathematical

problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any

number in a specified set.

3. Solve real world and mathematical problems by writing and solving equations of the form x + p = q and px = q

for cases in which p, q and x are all nonnegative rational numbers.

4. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real world or

mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent

solutions of such inequalities on number line diagrams.



VOCABULARY: expression, equation, inequalities, evaluate, variable, coefficient, terms, like terms, constant terms,

inverse operation, solution, graph, function, linear function

APPLY PROPERTIES: Properties of Equality, Inverse Properties, Distributive Property; NUMBER SENSE: Numbers

can be represented in multiple ways; Being able to compute fluently means making smart choices about which

strategies and/or tools to use and when to use them; Knowing the reasonableness of an answer comes from using

good number sense and estimation strategies. GRAPHING SOLUTIONS: Interpreting results and graphing solutions

on a number line.

Formulas and equations are used extensively throughout the science curriculum, for example: measurement

conversion (metric/customary), distance, speed, and velocity problems.


Various commercial products and websites for practice activities (example: AIMS activities, Interactive board activities), virtual manipulatives (algebra tiles or scales), and graphing calculators

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Domain: Expressions and Equations (6.EE) - Represent and analyze quantitative relationships between dependent and independent variables

Essential Questions What should I be able to answer? What guides my thinking? How can I use equations, graphs, and tables to understand quantitative relationships?


Formative: Practice solving equations, graphing ordered pairs on a coordinate plane, and explain

the relationship between variables.

Summative: Performance assessment: Using a real world problem, have students create and

solve an equation; Graph results and explain relationships (i.e., Outside Temperature and air

conditioning and/or electric usage; Snowfall and Snow Removal costs, etc).


1. Use variables to represent two quantities in a real world problem that change in

relationship to one another; write an equation to express one quantity, thought of as the

dependent variable, in terms of the other quantity, thought of as the independent variable.

Analyze the relationship between the dependent and independent variables using graphs and

tables, and relate these to the equation. For example, in a problem involving motion at constant

speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to

represent the relationship between distance and time.

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Graphing on the coordinate plane

Solving equations containing variables

Appropriately displaying and/or using data in problem solving settings

Creating a table to display data

VOCABULARY: expression, equation, inequalities, evaluate, variable, coefficient, terms, like

terms, constant terms, inverse operation, solution, graph, function, linear function

APPLY PROPERTIES: Properties of Equality, Inverse Properties, Distributive Property

NUMBER SENSE: Numbers can be represented in multiple ways; Being able to compute

fluently means making smart choices about which strategies and/or tools to use and when to use

them; Knowing the reasonableness of an answer comes from using good number sense and

estimation strategies.

GRAPHING SOLUTIONS: Interpreting results and graphing solutions on a number line and

coordinate plane.

Formulas and equations are used extensively throughout the science curriculum, for example:

motion, distance, speed, and velocity problems; Social Studies; Technology (Using Microsoft

and/or web based products for graphing, tables, etc).


Interactive board activities), virtual manipulatives, math songs, and graphing calculators.

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Domain: Geometry (6.G) - Solve real-world and mathematical problems involving area, surface area, and volume.



Why do I need to identify the basic shape of a figure and use the appropriate formula to find the

area, surface area, and volume? What is Pi and how can I use it to solve problems? Why do I

need to find a missing dimension when given the area, surface area, and/or volume of a figure?

Formative: Have students use manipulatives to identify familiar polygons and three dimensional

figures; have students evaluate formulas when given measurements; Have students use

centimeter paper to construct parellelograms and estimate area, then compute area of

parellelograms when given measurements. Repeat with triangles, then use two congruent

triangles to construct a parallelogram and investigate relationship between the area of the triangles

and the parallelogram created. Write a journal entry to explain the relationship; Present irregular

figures to students and have them divide it into familiar shapes, compute the area of each shape,

and add the area of each part.

Summative: Performance Assessment: Have students use nets to create three dimensional

figures and compute surface area and volume; Have students use deconstruct/unfold boxes and

compute surface area of each box (example, boxes from cereal, snacks, etc.); Using different

shape containers of a similar size, have students predict which container has greater volume. Then

have students compute volume using appropriate formula(s); Contrast/compare results; Measure

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walls/ceiling of a room for painting (surface area) and/or new flooring (area) and compute

associated costs of supplies.


1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by

composing into rectangles or decomposing into triangles and other shapes; apply these

techniques in the context of solving real world and mathematical problems.

2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with

unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as

would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V =

b h to find volumes of right rectangular prisms with fractional edge lengths in the context of

solving real-world and mathematical problems.

3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates

to find the length of a side joining points with the same first coordinate or the same second

coordinate. Apply these techniques in the context of solving real-world and mathematical

problems.

4. Represent three dimensional figures using nets made up of rectangles and triangles, and

use the nets to find the surface area of these figures. Apply these techniques in the context of

solving real-world and mathematical problems.

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Classify polygons by sides and angles, Classifying three dimensional figures by faces, edges,

vertices; Applying formulas for Perimeter, Area, Surface Area, and Volume; Using nets to identify

and construct figures; Graphing ordered pairs on a coordinate plane and use coordinates to draw

polygons; Apply to real life problem solving.

Formulas are used extensively to solve problems involving Science, Architecture, Geography,

Social Studies; Technology (Using Microsoft and/or web based products for graphing, tables, etc).




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Domain: Statistics and Probability Standard (6.SP) - Develop understanding of statistical variability.Essential Questions What should I be able to answer? What guides my thinking?


Why is it important for me to be able to represent data in different formats?

How will understanding different representations of data help me to solve problems? Why are

some data collection methods more appropriate than others? How do I decide which method to

use? How can I compare measures of central tendency and distribution?

Formative: In a journal entry, have students interpret data based on plots/graphs provided (from a

newspaper, book, or online) – compare/contrast results and give reason for differences/

similarities. Summative: Performance Based Assessment: Have students collect and organize

data (for example, student heights in class/grade, sports team data – batting averages, points

scored/game, amount of sugar (grams) in different cereals, etc.); represent the data in an

appropriate format; and interpret results/trends/distributions (For example, measure heights of

students in a grade/class, and have students create a histogram and/or box and whisker plot using

height data collected). Skills What skills do I need to have in order to answer the essential questions?

1. Display numerical data in plots on a number line, including dot plots, histograms, and box

plots.

2. Summarize numerical data sets in relation to their context, such as by:

a. Reporting the number of observations.

b. Describing the nature of the attribute under investigation, including how it was measured and

its units of measurement.

c. Giving quantitative measures of center (median and/or mean) and variability (interquartile

range and/or mean absolute deviation), as well as describing any overall pattern and any

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striking deviations from the overall pattern with reference to the context in which the data were

gathered.

d. Relating the choice of measures of center and variability to the shape of the data distribution

and the context in which the data were gathered.


Data collection methods (i.e., survey, samples – random, unbiased); Using appropriate tools to

organize and represent data (tables, graphs, plots, etc); Measures of Central Tendency - used to

summarize data (Mean, Median, Mode, and Range); Developing “intervals” based on data

collected for histograms; Using a Box & Whisker Plot to organize data into four groups (quartiles).



Art, Technology




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Domain: Statistics and Probability (6.SP) - Summarize and describe distributions.




In what ways can sets of data be represented by statistical measures?

How do you collect, organize, and display data? What kinds of questions can be answered using

different data displays? How can you tell if the data collection is sufficient?

Formative: Have students work in small groups/partners to draft a school survey (use survey in

class setting to “test” effectiveness, ease of understanding, etc). Example survey topics: shoe

size, height, bed time, number of letters in first name. Students can “test” survey in their own

class and plot results to see the results (example, use a line plot).

Summative: Using small group survey activity results from above, have students finalize a school

survey and administer the survey; Using data gathered in the student administered school survey,

compute measures of Central Tendency, analyze data distribution, and display results in an

appropriate graph/plot. Compare/contrast results by grade level. 1. Recognize a statistical question as one that anticipates variability in the data related to the

question and accounts for it in the answers. For example, “How old am I?” is not a statistical

question, but “How old are the students in my school?” is a statistical question because one

anticipates variability in students’ ages.

2. Understand that a set of data collected to answer a statistical question has a distribution

which can be described by its center, spread, and overall shape.

3. Recognize that a measure of center for a numerical data set summarizes all of its values with

a single number, while a measure of variation describes how its values vary with a single number

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Data collection methods (i.e., survey, samples – random, unbiased); Using appropriate tools to

organize and represent data (tables, graphs, plots, etc); Measures of Central Tendency - used to

summarize data (Mean, Median, Mode, and Range); Understanding of ratios ; Fluency with all

operations



Art, Technology



Interactive board activities), virtual manipulatives, math songs, and calculators

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Grade 7

Grade Seven

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Grade 7


Instructional time in Grade 7 focuses on four critical areas: (1) developing understanding of and applying proportional relationships;

(2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems

involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems

involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

(1) Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems.

Students use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving

discounts, interest, taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding

lengths between the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students

graph proportional relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope.

They distinguish proportional relationships from other relationships.

(2) Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal

representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and

division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and

multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts

owed or temperatures below zero), students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative

numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations

to solve problems.

(3) Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface

area of three-dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among

two-dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships

between angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by

examining cross-sections. They solve real world and mathematical problems involving area, surface area, and volume of two- and three-

dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.

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(4) Students build on their previous work with single data distributions to compare two data distributions and address questions about

differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of

representative samples for drawing inferences.

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Grade Seven

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3 + x




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Grade Seven Standards for Mathematical Content

Ratios and Proportional Relationships (7.RP)


7.RP Analyze proportional relationships and use them to solve real world and


1. Compute unit rates associated with ratios of fractions, including ratios of lengths,

areas and other quantities measured in like or different units. Example: If a person

walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction1/2

/1/4 miles per hour, equivalently 2 miles per hour.

2. Recognize and represent proportional relationships between quantities.

a) Decide whether two quantities are in a proportional relationship e.g., by testing

for equivalent ratios in a table or graphing on a coordinate plane and observing

whether the graph is a straight line through the origin.

b) Identify the constant of proportionality (unit rate) in tables, graphs, equations,

diagrams, and verbal descriptions of proportional relationships.

c) Represent proportional relationships by equations. Example: If total

cost t is proportional to the number n of items purchased at a constant

price p, the relationship between the total cost and the number of items

can be expressed as t = pn.

d) Explain what a point (x, y) on the graph of a proportional relationship

means in terms of the situation, with special attention to the points (0, 0)

and (1, r) where r is the unit rate.







Students’ work with ratios is extended to

include ratios of fractions. Example: If a

person takes 3/4 of an hour to paint 2/3 of

the wall surface in a room, how much time

will it take to paint the entire wall surface in

the room? 3/4 = x 2/3(x) = 3/4 x=9/8

2/3 1

# of

items

Cost Unit rate

(Cost per item)

1 $.25 $.25

2 $.50 $.25

3 $.75 $.25

4 $1.00 $.25

Graph the ordered pairs of two quantities on

a coordinate plane. If the graph is a straight

line through the origin (0,0), then the two

quantities are proportional

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3. Use proportional relationships to solve multi-step ratio and percent problems.

Examples: Simple interest, tax, markups and markdowns, gratuities and

commissions, fees, percent of change (increase and decrease), percent error.

Explore proportional relationships at: http://education.ti.com/calculators/timathnspired/US/

Activities/Detail?sa=1008&t=9447&id=16893

Have students estimate the length of an item

in the classroom. Use the following formula

to calculate percent error:

Percent error = │approximate value – true value│

x 100 true value


7.NS All students will apply and extend previous understandings of operations with

fractions to add, subtract, multiply, and divide rational numbers.

1. Apply and extend previous understandings of addition and subtraction to add and

subtract rational numbers; represent addition and subtraction on a horizontal or

vertical number line diagram.

a) Describe situations in which opposite quantities combine to make zero.

Example: A hydrogen atom has a zero charge because its two constituents are

oppositely charged.

Have students provide real-world situations

in which they would use positive or negative

rational numbers. In each situation, they

should explain what the positive and

negative numbers represent. Examples:

weight gain and loss; elevator position;

loss or gain of yards in football; income and

expenses; temperature.

This visual can be used to illustrate that

opposite quantities combine to make 0.

3 + -2 = 1

+ + +

0 + 0 + (+1) = 1

http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9447&id=16893

http://education.ti.com/calculators/timathnspired/US/Activities/Detail?sa=1008&t=9447&id=16893

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b) Understand p + q as the number located a distance |q| from p, in the positive

or negative direction depending on whether q is positive or negative.

Show that a number and its opposite have a sum of zero (additive inverses).

Interpret sums of rational numbers by describing real world contexts.

c) Understand subtraction of rational numbers as adding the additive inverse,

p – q = p + (–q). Show that the distance between two rational numbers on

the number line is the absolute value of their difference, and apply this

principle in real world contexts.

d) Apply properties of operations as strategies to add and subtract rational

numbers.

2. Apply and extend previous understandings of multiplication, division, and fractions

to multiply and divide rational numbers.

a) Understand that multiplication is extended from fractions to rational numbers

by requiring that operations continue to satisfy the properties of operations,

particularly the distributive property, leading to products such as (–1)(–1) = 1

and the rules for multiplying signed numbers. Interpret products of rational

numbers by describing real world contexts.

b) Understand that integers can be divided, provided that the divisor is not zero,

and every quotient of integers (with non-zero divisor) is a rational number.

If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of

rational numbers by describing real world contexts.

4 units

4 units

Show that the distance between two rational

numbers on the number line is the absolute

value of their difference.

p q p - q Distance between

p and q

5 -7 12 12 units

-12 8 -20 20 units

3 5 ½ -2 ½ 2 ½ units

Multiplicative Property of -1

Demonstrate that the result of multiplying a

number by (-1) is the additive inverse of the

number being multiplied.

34 x (-1) = -34 34 + -34 = 0

-45.5 x (-1) = 45.5 -45.5 + 45.5 = 0

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c) Apply properties of operations as strategies to multiply and divide rational

numbers.

d) Convert a rational number to a decimal using long division; know that the

decimal form of a rational number terminates in zero or eventually repeats.

3. Solve real world mathematical problems involving rational numbers using the four

operations.

Demonstrate how properties of operations

can simplify computation with rational

numbers.

Examples:

Distributive Commutative

Property Property

3 ¾ x 12 -2 + 3.5 - 8

(3 + ¾) x 12 -2 + 3.5 + (-8)

(3 x 12) + (¾ x 12) -2 + (-8) + 3.5

6 + 9 = 45 -10 + 3.5 = -6.5

Have students explain how to write fractions

as decimals and how to write terminating

and repeating decimals as fractions.

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7.EE All students will use properties of operations to generate equivalent expressions.

1. Apply properties of operations as strategies to add, subtract, factor, and expand

linear expressions with rational coefficients.

2. Understand that rewriting an expression in different forms may enhance the

interpretation of the problem and how the quantities in it are related.

Example: a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply

by 1.05.”

S1. Apply the rule for Order of Operations to solve numerical and algebraic equations

and to simplify and evaluate expressions.

7.EE All students will solve real world and mathematical problems using numerical

and algebraic expressions and equations.

3. Solve multi-step real-life and mathematical problems posed with positive and

negative rational numbers in any form (whole numbers, fractions, and decimals),

using tools strategically. Apply properties of operations to calculate with numbers in

any form; convert between forms as appropriate; and assess the reasonableness of

answers using mental computation and estimation strategies. Example: If a woman

making $25 an hour gets a 10% raise, she will make an additional 1/10 of

her salary an hour, or $2.50, for a new salary of $27.50. If you want to place

a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide,

you will need to place the bar about 9 inches from each edge; this estimate can be

used as a check on the exact computation.

Students apply the properties of operations

to rational numbers (integers and positive /

negative fractions and decimals) as they

work with linear expressions.

Example: -5(3.5n -10)

-5[3.5n + (-10)]

(-5 x 3.5n) + [-5 x (-10)]

-17.5n + 50

Recognize the equivalency of expressions

written in different forms.

Example: The price of an item on sale for

20% off can also be calculated by finding

80% of the item’s price.

Price(p) – 20% x p = 80% x p

Practice translating verbal phrases into

algebraic expressions and algebraic

expressions into verbal phrases.

Use a variety of representations for

situations presented in word problems.

Summarize the method for solving a word

problem using algebraic expressions and

equations. Use previously learned properties

and strategies to simplify algebraic

expressions and solve algebraic equations.

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4. Use variables to represent quantities in a real world or mathematical problem,

and construct simple equations and inequalities to solve problems by reasoning

about the quantities.

a) Solve word problems leading to equations of the form px + q = r

and p(x + q) = r, where p, q, and r are specific rational numbers.

Solve equations of these forms fluently. Compare an algebraic solution

to an arithmetic solution, identifying the sequence of the operations used in

each approach. Example: The perimeter of a rectangle is 54 cm.

Its length is 6 cm. What is its width?

b) Solve word problems leading to inequalities of the form px + q > r

or px + q < r, where p, q, and r are specific rational numbers. Graph the

solution set of the inequality and interpret it in the context of the problem.

Example: As a salesperson, you are paid $50 per week plus $3 per sale. This

week you want your pay to be at least $100. Write an inequality for the

number of sales you need to make, and describe the solutions.

Use variables to represent unspecified

quantities in a word problem.

Example: The perimeter of a rectangle is 54

cm. Its length is 6 cm. What is its width?

Use the formula for perimeter P= 2 ( l + w)

54 = (2 x 6) + (2 x w)

Use inequalities to solve word problems that

have more than one possible solution.

Example: Paid $50 per week plus $3 per

sale; want week’s pay to be at least $100

50 + (3 x s) 100

The amount earned for sales (3 x s) has to

be equal to or more than $50. Have students

discuss the values for s (number of sales)

that would make (3 x s) equal to or greater

than 50. Which numbers make sense in the

context of the problem?


7.G All students will draw, construct, and describe geometrical figures and describe

the relationships between them.

1. Solve problems involving scale drawings of geometric figures, including

computing actual lengths and areas from a scale drawing and reproducing a

drawing at a different scale.

Have students make a scale drawing of

something in the classroom. Have them

include a key showing the scale as well as the

steps and calculations they used to determine

the various dimensions.

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2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes

with given conditions. Focus on constructing triangles from three measures of

angles or sides, noticing when the conditions determine a unique triangle, more than

one triangle, or no triangle.

3. Describe the two-dimensional figures that result from slicing three-dimensional

figures, as in plane sections of right rectangular prisms and right rectangular

pyramids.

7.G All students will solve real world and mathematical problems involving angle

measure, area, surface area, and volume.

4. Know the formulas for the area and circumference of a circle and use them to

solve problems. Give an informal derivation of the relationship between the

circumference and area of a circle.

5. Use facts about supplementary, complementary, vertical, and adjacent angles

in a multi-step problem to write and solve simple equations for an unknown angle

in a figure.

6. Solve real world and mathematical problems involving area, volume, and surface

area of two- and three-dimensional objects composed of triangles, quadrilaterals,

polygons, cubes, and right prisms.

Prepare index cards with written descriptions

of geometric shapes with specific conditions

(one shape per card.) Distribute a card to

each pair of students. Students draw the

described shape. Display the shapes,

redistribute the cards and have students

match each shape to its description.

View an interactive representation of slicing

of three-dimensional figures at:

http://www.learner.org/courses/learningmath/

geometry/session9/part_c/index.html

Have students explain what the symbol π

represents and how it is related to circles.

Have students trace circular objects on

graph paper. Estimate the area of the circle.

Cut the circle out and fold it in half. Identify

the diameter and determine its length. Fold

the circle in fourths. Identify the radius and

determine its length. Discuss the relationship

between the various measurements.

Have students work in pairs. Have one

student draw two pairs of parallel lines cut by

a transversal and number the angles. The

other student lists all the pairs of adjacent,

vertical, complementary, supplementary,

corresponding, and adjacent angles.

http://www.learner.org/courses/learningmath/%20geometry/session9/part_c/index.html


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Statistics and Probability (7.SP) Standards Strategies and Assessments

7.SP All students will use random sampling to draw inferences about a population.

1. Understand that statistics can be used to gain information about a population by

examining a sample of the population; generalizations about a population from a

sample are valid only if the sample is representative of that population. Understand

that random sampling tends to produce representative samples and support valid

inferences.

2. Use data from a random sample to draw inferences about a population with an

unknown characteristic of interest. Generate multiple samples (or simulated

samples) of the same size to gauge the variation in estimates or predictions.

Example: Estimate the mean word length in a book by randomly sampling

words from the book; predict the winner of a school election based on randomly

sampled survey data. Gauge how far off the estimate or prediction might be.

7.SP All students will be able to draw informal comparative inferences about two

populations.

3. Informally assess the degree of visual overlap of two numerical data distributions

with similar variabilities, measuring the difference between the centers by

expressing it as a multiple of a measure of variability. Example: The mean

height of players on the basketball team is 10 cm greater than the mean height of

players on the soccer team, about twice the variability (mean absolute deviation) on

either team; on a dot plot, the separation between the two distributions of heights is

noticeable.

The following websites provide resources for

teaching concepts related to statistics. http://msteacher.org/return_list_math.aspx?id=418

http://www.nctm.org/profdev/content.aspx?id=11688

Have students explain how they could best

find a random sample of their peers to survey

about their favorite sport. Would they

survey only those students on the soccer

team?

Discuss how to use proportions to create a

sample group that is representative of a larger

group. Use census information found at:

http://www.census.gov/schools/ to design a

sample group of 100 people that is

representative of the population of the

United States.

Have students predict how many drops of

water will fit on a penny; write predictions

on a post-it note along with an explanation of

his/her reasoning. Predictions are collected

and displayed on bar graphs or stem-and-leaf

plots. Students perform the experiment and

records results on another post-it note and

compare the data distribution for the

hypotheses with the data distribution for the

conclusions. (Combine with science lesson

on surface tension.)

http://msteacher.org/return_list_math.aspx?id=418


http://www.census.gov/schools

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4. Use measures of center and measures of variability for numerical data from random

samples to draw informal comparative inferences about two populations.

Example: Decide whether the words in a chapter of a seventh-grade science book

are generally longer than the words in a chapter of a fourth-grade science book.

7.SP All students will investigate chance processes and develop, use, and evaluate

probability models.

5. Understand that the probability of a chance event is a number between 0 and 1 that

expresses the likelihood of the event occurring. Larger numbers indicate greater

likelihood. A probability near 0 indicates an unlikely event, a probability around

1/2 indicates an event that is neither unlikely nor likely, and a probability near 1

indicates a likely event.

6. Approximate the probability of a chance event by collecting data on the chance

process that produces it and observing its long-run relative frequency, and predict

the approximate relative frequency given the probability. Example: When

rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly

200 times, but probably not exactly 200 times.

7. Develop a probability model and use it to find probabilities of events. Compare

probabilities from a model to observed frequencies; if the agreement is not good,

explain possible sources of the discrepancy.

a) Develop a uniform probability model by assigning equal probability to all

outcomes, and use the model to determine probabilities of events.

Example: If a student is selected at random from a class, find the probability

that Jane will be selected and the probability that a girl will be selected.

Have students find, describe, and interpret

appropriate measures of center (mean,

median, and mode) and measures of

variation/variability (range) that represent a

set of data. Example: Collect, display and

compare data for heights of 3rd grade

students vs. heights of 7th

grade students.

Experimental Probability

P(event )= number of times event occurs

number of trials

Example: Have students work in pairs

rolling a dice 20 times and recording the

outcome in a chart. Use a ratio to show how

many times each number was rolled out of

the 20 trials. Compare results recorded by

each pair of students. Then compare class

results to theoretical probability (number of

times event should occur.)

Theoretical Probability

P(event )= number of favorable outcomes

number of possible outcomes

Example:

Experiment: Rolling a dice

Possible Outcomes: 1, 2, 3, 4, 5, 6

Event: Rolling 3 or 6

Favorable Outcomes: 2 (Rolling 3 or 6)

Probability: P(3 or 6) = 2/6 = 33%

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b) Develop a probability model (which may not be uniform) by observing

frequencies in data generated from a chance process. Example: Find the

approximate probability that a spinning penny will land heads up or that a

tossed paper cup will land open-end down. Do the outcomes for the spinning

penny appear to be equally likely based on the observed frequencies?

8. Find probabilities of compound events using organized lists, tables, tree diagrams,

and simulation.

a) Understand that, just as with simple events, the probability of a compound

event is the fraction of outcomes in the sample space for which the compound

event occurs.

b) Represent sample spaces for compound events using methods such as

organized lists, tables, and tree diagrams. For an event described in everyday

language (e.g., ―rolling double sixes‖), identify the outcomes in the sample

space which compose the event.

c) Design and use a simulation to generate frequencies for compound events.

Example: Use random digits as a simulation tool to approximate the answer

to the question: If 40% of donors have type A blood, what is the probability

that it will take at least 4 donors to find one with type A blood?

For some probability activities go to: http://mathforum.org/library/topics/probability/

Students should be able to distinguish

between dependent events and independent

events when discussing the probability of

compound events.

What is the probability of two odd numbers

when spinning the two spinners?

Spinner A Spinner B

1 1

4 2 4 2

3 3

List of Possible Outcomes

1, 1 1, 2 1,3 1, 4

2, 1 2, 2 2, 3 2, 4

3, 1 3, 2 3, 3 3, 4

4, 1 4, 2 4, 3 4, 4

The probability of the spinning two odd

numbers is 4/16 or 1/4.

P(A and B) = P(A) x P(B)

P(odd and odd) = 2 x 2 = 4 or 1

4 4 16 4

http://mathforum.org/library/topics/probability/



How do I use ratio and proportion in every day life ?

How are scale drawings and models used in the real world ?

How do I use ratio and proportion to solve multistep problems involving percents ?

Where do I find proportional relationships in science and nature ?

Formative: Create word problems related to proportion and percent and give to other students to

solve.

Summative: Consumer math project involving a budget for vacation plans, or a budget for

groceries or monthly family budget.

Domain: Ratios and Proportional Relationships (7.RP) - Analyze proportional relationships and use them tosolve real-world and mathematical problems.

Extended Resources– Grade 7 Mathematics

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1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other

quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour,

compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.


a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a

table or graphing on a coordinate plane and observing whether the graph is a straight line through the

origin.

b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal

descriptions of proportional relationships.

c. Represent proportional relationships by equations. For example, if total cost t is proportional to the

number n of items purchased at a constant price p, the relationship between the total cost and the

number of items can be expressed as t = pn.

d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation,

with special attention to the points (0, 0) and (1, r) where r is the unit rate.

3. Use proportional relationships to solve multi step ratio and percent problems. Examples: simple

interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease,

percent error.


Use ratios, proportions and percents to represent relationships between quantities and

measures. Develop proportional thinking using concrete and graphic methods.

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Science using proportions to make solutions,and interpreting graphs with data from lab

experiments

Examples: Pulse Rates, time trials, measurement, skeletal proportion


Standard Specific tools and websitesRulers,calculators,graph paper, maps and scale drawings

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Domain: The Number System (7.NS) - Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.


How will I apply the understanding of the study, properties, and operations of rational numbers to the real world?



Formative: Use a unit collage – provide a list of key points to remember when computing rational

numbers

Summative: performance assessment: Gold Seal Lesson Love Letters and Numbers


subtract rational numbers; represent addition and subtraction on a horizontal or vertical number

line diagram.

a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen

atom has 0 charge because its two constituents are oppositely charged.

b. Understand p + q as the number located a distance |q| from p, in the positive or negative

direction depending on whether q is positive or negative. Show that a number and its opposite

have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-

world contexts.

c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).

Show that the distance between two rational numbers on the number line is the absolute value

of their difference, and apply this principle in real-world contexts.

d. Apply properties of operations as strategies to add and subtract rational numbers.

2. Apply and extend previous understandings of multiplication and division and of fractions

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a. Understand that multiplication is extended from fractions to rational numbers by requiring that

operations continue to satisfy the properties of operations, particularly the distributive property,

leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret

products of rational numbers by describing real-world contexts.

b. Understand that integers can be divided, provided that the divisor is not zero, and every

quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –

(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world contexts.

c. Apply properties of operations as strategies to multiply and divide rational numbers.

d. Convert a rational number to a decimal using long division; know that the decimal form of a

rational number terminates in 0s or eventually repeats.

3. Solve real world and mathematical problems involving the four operations with rational

numbers. [Computations with rational numbers extend the rules for manipulating fractions to

complex fractions.]


Operations with fractions, decimals, integers

Solve equations horizontally and vertically

Order of Operations; appropriateness of Distributive Property.

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Physical education: batting averages, gains, losses

Science: unit conversions

Finance: balance accounts.

Integer chips, number lines, fraction, decimal equivalencies, computer software, calculators

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Essential Questions What should I be able to answer? What guides my thinking? How do I use algebraic expressions to analyze or solve problems?

How do the properties contribute to algebraic understanding?



Formative: Play Silent Pass with variables. Fill in one answer and then pass to the next person

writing an expression or equation with a variable. Be sure to designate each person in the group to

use a different writing utensil or color.

Summative: make a foldable describing properties with an example using variables

1. Apply properties of operations as strategies to add, subtract, factor, and expand linear

expressions with rational coefficients.

2. Understand that rewriting an expression in different forms in a problem context can shed

light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a

means that “increase by 5%” is the same as “multiply by 1.05.”


Identify and use properties

Expand expressions

Evaluate expressions with addition and subtraction

Domain: Expressions and Equations (7.EE) - Use properties of operations to generate equivalent expressions.

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Science calculations


Calculator, pencil, paper, interactive boards

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Domain: Expressions and Equations ((7.EE) - Solve real-life mathematical problems using numerical and algebraic expressions and equations


How does the use of variables help me to represent real life and mathematical problems?

How do estimates help to determine if my answer is reasonable?

When do I use an inequality instead of an equation?

Will solving algebraic equations help me in higher levels of mathematics?


Formative: Students will create word problems that can be solved algebraically and exchange

with classmates.


Summative: To use sports statistics, roller coaster velocity and census numbers to

show understanding of expressions, equalities and inequalities.

1. Solve multi step real life and mathematical problems posed with positive and negative

rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically.

Apply properties of operations to calculate with numbers in any form; convert between forms as

appropriate; and assess the reasonableness of answers using mental computation and

estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will

make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to

place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will

need to place the bar about 9 inches from each edge; this estimate can be used as a check on

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the exact computation.

2. Use variables to represent quantities in a real world or mathematical problem, and

construct simple equations and inequalities to solve problems by reasoning about the quantities.

a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q,

and r are specific rational numbers. Solve equations of these forms fluently. Compare an

algebraic solution to an arithmetic solution, identifying the sequence of the operations used in

each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is

its width?

b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and

r are specific rational numbers. Graph the solution set of the inequality and interpret it in the

context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per

sale. This week you want your pay to be at least $100. Write an inequality for the number of

sales you need to make, and describe the solutions.


Solve multi-step equations using distributive property and order of operations.

Understand the importance of estimation to check answers and make predictions.

Represent real life situations with equations and inequalities and graphs.

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Science: use of equations and variables with formulas and balancing equations, interpret and

analyze results from experiments with equations and graphs

Standard Specific tools and websites

NETS tools and websites

Calculator

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Domain: Geometry (7.G) - Draw, construct, and describe geometrical figures and describe the relationships between them.


How will the mastery of geometric constructions contribute to my understanding of

geometry and its relevancy to the real world?


Formative: Make a scale drawing. Draw geometric shapes with given conditions

Summative: performance assessment. – Design a floor plan – use actual and scale

measures


1. Solve problems involving scale drawings of geometric figures, including computing actual

lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.

2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with

given conditions. Focus on constructing triangles from three measures of angles or sides, noticing

when the conditions determine a unique triangle, more than one triangle, or no triangle. 3.

Describe the two dimensional figures that result from slicing three dimensional figures, as in

plane sections of right rectangular prisms and right rectangular pyramids.


Demonstrate competence with scale, ratio, and proportion.

Identify and construct geometric figures.

Recognize two and three dimensional objects.

Manipulation of geometric tools and/or geometric software.

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Cartography: interpreting map distances

Art: enlarging pictures


Manipulate objects for geometry.: ruler, protractor, geoboards, three dimensional figures,

interactive boards

Geometric Computer Software

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Domain: Geometry (7.G) - Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.


How can finding area, surface area, and volume of a composite figure be useful in my life?



Formative : Students decompose the shapes displayed on the board/overhead into triangles and rectangles

and circles. Discuss how to determine the area of the shape in order to calculate the volume of a prism with

that shape for a base.

Summative: performance task packaging problems

1. Know the formulas for the area and circumference of a circle and use them to solve

problems; give an informal derivation of the relationship between the circumference and area of a

circle.

2. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-

step problem to write and solve simple equations for an unknown angle in a figure.

3. Solve real world and mathematical problems involving area, volume and surface area of

two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and

right prisms.


Identify three-dimensional figures

Use formulas for calculating area, surface area, and volume

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Art: design packaging,

Science: volume, density measures


Calculator, three dimensional figures, computer graphics

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Domain: Statistics and Probability (7.SP) Use random samplings to draw inferences about a population.


Why is it important to study a population?

How can likely outcomes or conclusions be made from a random sample?

How can statistical samplings help me make decisions?


Formative: To use surveys to infer information from the data and display in graphs


Summative: Performance assessment using survey data.


examining a sample of the population; generalizations about a population from a sample are valid

only if the sample is representative of that population. Understand that random sampling tends to

produce representative samples and support valid inferences.

2. Use data from a random sample to draw inferences about a population with an unknown

characteristic of interest. Generate multiple samples (or simulated samples) of the same size to

gauge the variation in estimates or predictions. For example, estimate the mean word length in a

book by randomly sampling words from the book; predict the winner of a school election based

on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

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Knowledge of central tendency and use of other graphs and tables.

Organize data from a random sample.

Arrive at valid conclusions based on the information.

Consult with Social Studies teacher: Population studies


Random number generator, random digit page, calculator, Excel spreadsheets and graphs


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Domain: Statistics and Probability (7.SP) - Draw informal comparative inferences about two populations.


What inferences and conclusions can I make using statistics and probability to further my

understanding of data in the real world?

What kind of relationship can be found when comparing two sets of data?



Formative: Make a dot plot to compare shoe sizes of boys and girls.

Summative: performance assessment – Use measures of central tendency to compare

the 5 largest cities of New Jersey.

1. Informally assess the degree of visual overlap of two numerical data distributions with

similar variabilities, measuring the difference between the centers by expressing it as a multiple

of a measure of variability. For example, the mean height of players on the basketball team is 10

cm greater than the mean height of players on the soccer team, about twice the variability (mean

absolute deviation) on either team; on a dot plot, the separation between the two distributions of

heights is noticeable.


samples to draw informal comparative inferences about two populations. For example, decide

whether the words in a chapter of a seventh-grade science book are generally longer than the

words in a chapter of a fourth-grade science book.

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Gather, collect, calculate data.

Graph data. Venn Diagram, dot plot

Apply operations to a set of values and determine a relationship between the values.

Make inferences and draw conclusions based on data.

Science: draw comparative inferences about wildlife populations

Social Studies: compare populations of cities, states, countries


Calculators, school, local, state data, graph paper, graphing calculator

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Domain: Statistics and Probability (7.SP) - Investigate chance processes and develop, use, and evaluate probability models.

:


What is the probability that I can make a correct decision given

partial information?

Is anything in nature truly random?


Formative: experimenting using dice, spinners, coins, colored chips


Summative: performance task--design and run a simulation, draw conclusions from use of

probability equation


expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A

probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is

neither unlikely nor likely, and a probability near 1 indicates a likely event.

2. Approximate the probability of a chance event by collecting data on the chance process

that produces it and observing its long-run relative frequency, and predict the approximate

relative frequency given the probability. For example, when rolling a number cube 600 times,

predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

3. Develop a probability model and use it to find probabilities of events. Compare probabilities

from a model to observed frequencies; if the agreement is not good, explain possible sources of

the discrepancy.

a. Develop a uniform probability model by assigning equal probability to all outcomes, and use

the model to determine probabilities of events. For example, if a student is selected at random

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from a class, find the probability that Jane will be selected and the probability that a girl will be

selected.

b. Develop a probability model (which may not be uniform) by observing frequencies in data

generated from a chance process. For example, find the approximate probability that a spinning

penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes

for the spinning penny appear to be equally likely based on the observed frequencies?

4. Find probabilities of compound events using organized lists, tables, tree diagrams, and

simulation.

a. Understand that, just as with simple events, the probability of a compound event is the fraction

of outcomes in the sample space for which the compound event occurs.

b. Represent sample spaces for compound events using methods such as organized lists, tables

and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”),

identify the outcomes in the sample space which compose the event.

c. Design and use a simulation to generate frequencies for compound events. For example, use

random digits as a simulation tool to approximate the answer to the question: If 40% of donors

have type A blood, what is the probability that it will take at least 4 donors to find one with type

A blood?

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Identify dependent and independent events, Use tree diagrams, tables, and

simulations, Calculate probability


Social Studies: election predictions through sampling


Calculator, dice, chips, coins, spinners, interactive boards

Data obtained either experimentally or through research

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Grade 8

Grade Eight

Page 1

Grade 8


Instructional time in Grade 8 focuses on three critical areas: (1) formulating and reasoning about expressions and equations, including

modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the

concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures

using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

(1) Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students

recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of

proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant

rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A.

Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for

students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in

the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the

relationship (such as slope and y-intercept) in terms of the situation.

Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they

use the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve

systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the

same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze

situations and solve problems.

(2) Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe

situations where one quantity determines another. They can translate among representations and partial representations of functions

(noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are

reflected in the different representations.

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(3) Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about

congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the

angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of

the angles created when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse,

and can explain why the Pythagorean Theorem holds, (e.g., by decomposing a square in two different ways). They apply the Pythagorean

Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work

on volume by solving problems involving cones cylinders, and spheres.

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3 + x




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Grade Eight Standards for Mathematical Content


8.NS All students will know that there are numbers that are not rational, and

approximate them by rational numbers.

1. Know that numbers that are not rational are called irrational. Understand informally

that every number has a decimal expansion; for rational numbers show that the

decimal expansion repeats eventually, and convert a decimal expansion which

repeats eventually into a rational number. (See example below.)

2. Use rational approximations of irrational numbers to compare the size of irrational

numbers, locate them approximately on a number line diagram, and estimate the

value of expressions (e.g., π2). Example: By truncating the decimal expansion

of 2 , show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how

to continue on to get better approximations.

Example for 8.NS.1: Use the following method to change the repeating

decimal 0. 2 to a fraction.

Let x = 0.2222…

Multiply both sides so that the repeating digit/s will be to the left of the decimal. In

this example, one digit repeats so both sides are multiplied by 10 resulting in 10x =

2.2222…

Subtract the original equation from the new equation.

10x = 2.2222…

- x = 0.2222…

9x = 2

Solve the equation to determine the equivalent fraction.

9x = 2; x =9

2







Provide students with a list of numbers and

have them identify each number as rational

or irrational and explain their answers.

Example:

Rational Irrational

25 8

0.245245245… 0.01020304…

94

1612

Explore patterns in particular families of

decimal expansions, such as those for the

fractions 1/7, 2/7, 3/7, ... or 1/9, 2/9, 3/9, ... .

Each student is given a card with a rational

or irrational number written on it. (For

example: 3

12, 1.1, 1.01, 2 , π, -2

2, -(2)

2,

3

1, 8 , 5.3 , etc.) Students then order

themselves from least to greatest along the

front of the classroom.

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8. EE All students will work with radicals and integer exponents.

1. Know and apply the properties of integer exponents to generate equivalent

numerical expressions: Example: 32×3

-5= 3

–3 = 1/3

3 = 1/27

2. Use square root and cube root symbols to represent solutions to equations of the

form x2 = p and x

3 = p, where p is a positive rational number. Evaluate square

roots of small perfect squares (at least up to 225) and cube roots of small perfect

cubes. Know that 2 is irrational.

3. Use numbers expressed in the form of a single digit times an integer power of 10

to estimate very large or very small quantities, and to express how many times as

much one is than the other. Example: Estimate the population of the United States as

3 × 108 and the population of the world as 7 × 10

9, and determine that the world

population is more than 20 times larger.

4. Perform operations with numbers expressed in scientific notation, including

problems where both decimal and scientific notation are used. Use scientific notation

and choose units of appropriate size for measurements of very large or very small

quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific

notation that has been generated by technology.

Use numbers to illustrate the properties of

integer exponents.

Laws of Integer Exponents

For any nonzero real numbers a and b

and integers n and m:

Law Example

a0= 1 3

0= 1

a-n

= 1/an 2

-3 = 1/2

3 = 1/8

ana

m = a

n+m 32 x 3

4= 3

(2+4) =3

6

(3 x 3) x (3 x 3 x 3 x 3) = 36

(an)m

= anm (52

)3 = 5

2 x 3 = 5

6 = 15,625

52 x 5

2 x 5

2

(5 x 5) x (5 x 5) x (5 x 5) = 56

anb

n = (ab)

n 23 x 5

3 = (2 x 5)

3 = 10

3 = 1,000

(2 x 2 x 2) x (5 x 5 x 5)

8 x 125 = 1,000

am

/ an= a

m-n 34

/32 = 3

(4-2) = 3

2 = 9

3 x 3 x 3 x 3 = 81 = 9

3 x 3 = 9 1

(a/b)n= a

n/b

n (2/5)3 = 2

3/5

3 = 8/125

2/5 x 2/5 x 2/5 = 8/125

View the film Powers of Ten at: http://micro.magnet.fsu.edu/primer/

java/scienceopticsu/powersof10/

http://micro.magnet.fsu.edu/primer/%20java/scienceopticsu/powersof10/


Grade Eight

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8.EE All students will understand the connections between proportional relationships,

lines, and linear equations.

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph.

Compare two different proportional relationships represented in different ways.

Example: Compare a distance-time graph to a distance-time equation to determine

which of two moving objects has greater speed.

6. Use similar triangles to explain why the slope m is the same between any two distinct

points on a non-vertical line in the coordinate plane; derive the equation y = mx for a

line through the origin and the equation y = mx + b for a line intercepting the vertical

axis at b.

S1. Recognize and understand the forms of linear equation:

slope-intercept form as y = mx + b, and standard form as Ax + By =C.

8.EE All students will analyze and solve linear equations and pairs of simultaneous

linear equations.

7. Solve linear equations in one variable.

a) Give examples of linear equations in one variable with one solution (x = a),

infinitely many solutions (a = a), or no solutions ( a = b where a and b are

different numbers). Show which of these possibilities is the case by

successively transforming the given equation into simpler forms.

b) Solve linear equations with rational number coefficients, including equations

whose solutions require expanding expressions using the distributive property

and collecting like terms.

Have students compare the rate of speed

represented graphically to the rate of speed

represented algebraically. Example: Which

shows a faster rate of speed?

Rate A Rate B

100

75

50

25

Using a graph, students construct triangles

between two points on a line and compare the

sides to understand that the slope (ratio of rise to

run) is the same between any two points on a

line.

Discuss the solution possibilities for linear

equations. Examples: One Solution - ¼(8x +12) = 5

x = -4

No Solution 6d – (-4 + 2) = 6d

2 0

Many Solutions 2a(4 x .25) = 2a

2a = 2a D

ista

nce

(m

i)

0 1 2 3 4 5

Time (h)

80 miles in 3 hours

D = rate x time

80 = r x 3

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8. Analyze and solve pairs of simultaneous linear equations.

a) Understand that solutions to a system of two linear equations in two variables

correspond to points of intersection of their graphs, because points of

intersection satisfy both equations simultaneously.

b) Solve systems of two linear equations in two variables algebraically, and

estimate solutions by graphing the equations. Solve simple cases by

inspection. Example: 3x + 2y = 5 and 3x + 2y = 6 have no solution

because 3x + 2y cannot simultaneously be 5 and 6.

c) Solve real-world and mathematical problems leading to two linear equations

in two variables. Example: Given coordinates for two pairs of points,

determine whether the line through the first pair of points intersects the line

through the second pair.

Graph systems of equations to show

solution possibilities.

Examples:

One solution

(2, -1)

3x – 3y = 9

and y = -x + 1

No solution -x +7- 4x

and 12 – 5x

Many

solutions

2x – y = -3

and 8x – 4y = -12

Identify real world problems that can be

solved using a system of two linear

equations. Example: Use a system of

equations to find out how many movies a

person would need to download in a month

to make the cost of Plan A a better deal than

Plan B.

Plan A: $6.00 monthly membership fee

and $1.25 per movie

Plan B: $2.00 per movie

No monthly membership fee.

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Functions (8.F) Standards Strategies and Assessments

8.F All students will define, evaluate, and compare functions.

1. Understand that a function is a rule that assigns to each input exactly one output.

The graph of a function is the set of ordered pairs consisting of an input and the

corresponding output.

2. Compare properties of two functions each represented in a different way

(algebraically, graphically, numerically in tables, or by verbal descriptions).

3. Interpret the equation y = mx + b as defining a linear function, whose graph is

a straight line; give examples of functions that are not linear.

Example: The function A = s2

giving the area of a square as a function of its

side length is not linear because its graph contains the points (1,1), (2,4) and (3,9),

which are not on a straight line.

Note: Function notation f (x) is not required in Grade 8.

8.F All students will use functions to model relationships between quantities.

4. Construct a function to model a linear relationship between two quantities.

Determine the rate of change and initial value of the function from a description of a

relationship or from two (x, y) values, including reading these from a table or from a

graph. Interpret the rate of change and initial value of a linear function in terms of the

situation it models, and in terms of its graph or a table of values.

5. Describe qualitatively the functional relationship between two quantities by

analyzing a graph (e.g., where the function is increasing or decreasing, linear or

nonlinear). Sketch a graph that exhibits the qualitative features of a function that has

been described verbally.

Given a linear function represented by a

table of values and a linear function

represented by an algebraic expression,

determine which function has the greater

rate of change. Example: Compare the

following functions to determine which has

the greater rate of change.

Function 1: y = 2x + 4

Function 2:

x y

-1 -6

0 -3

2 3

Have students graph non-linear functions.

Example: Have students consider what

happens if they start with two bacteria on a

kitchen counter and the number of bacteria

doubles every hour. Students make a table

and graph their results, noting that the

graph is not linear.

Students should work with a variety of

representations of functions (tables, graphs,

equations) and be able to classify and

describe the function as linear or non-linear,

increasing or decreasing.

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8. All students will be able to apply appropriate techniques, tools, and

formulas to determine measurements.

Given the formulas, convert from one system of measurement to another. Use

technology as appropriate.

Convert 12 ft to centimeters by using

multiple unit multipliers.

(Hint: 2.54 cm = 1 in and 12 in = 1 ft).


8.G All students will understand congruence and similarity using physical models,

transparencies, or geometry software.

1. Verify experimentally the properties of rotations, reflections, and translations:

a) Lines are taken to lines, and line segments to line segments of the same length.

b) Angles are taken to angles of the same measure.

c) Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent ( ) to another if the second

can be obtained from the first by a sequence of rotations, reflections, and

translations; given two congruent figures, describe a sequence that exhibits the

congruence between them.

Students use compasses, protractors and

ruler, or technology to explore figures

created from translations, reflections and

rotations. Characteristics of figures, such as

lengths of line segments, angle measures

and parallel lines are explored before the

transformation (pre-image) and after the

transformation (image). Students understand

that these transformations produce images

of exactly the same size and shape as the

pre-image and are known as rigid

transformations.

Explore geometric transformations at: http://www.mathsisfun.com/geometry/

transformations.html

http://nlvm.usu.edu/en/nav/category_g_3_t_3.html

http://www.mathsisfun.com/geometry/%20transformations.html



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3. Describe the effect of dilations, translations, rotations, and reflections on

two-dimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can

be obtained from the first by a sequence of rotations, reflections, translations, and

dilations; given two similar two-dimensional figures, describe a sequence that

exhibits the similarity between them.

5. Use informal arguments to establish facts about the angle sum and exterior angle

of triangles, about the angles created when parallel lines are cut by a transversal, and

the angle-angle criterion for similarity of triangles

8.G All students will understand and apply the Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem (a2

+ b2

= c2) and its converse.

7. Apply the Pythagorean Theorem to determine the unknown side lengths in right

triangles in real-world and mathematical problems in two and three dimensions.

8. Apply the Pythagorean Theorem to find the distance between two points in a

coordinate system.

8.G All students will solve real-world and mathematical problems involving volume

of cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use

them to solve real-world and mathematical problems.

Students apply transformations to figures

drawn on coordinate grids, record the

coordinates of the original figure and its

image, and look for patterns. They express

these patterns verbally and symbolically.

Examples: A reflection point across the

x-axis changes the sign of the y-coordinate

so that the point (x, y) moves to (x,-y).

A translation of 5 left and 2 up would

subtract 5 from the x-coordinate and

add 2 to the y-coordinate

D (-4, -3) D’ (-9, -1).

Arrange three copies of the same triangle so

that the sum of the three angles appears to

form a line, and give an argument in terms

of transversals why this is so.

Students use graph paper or geo-boards to

construct squares on each side of a right

triangle. They find the area of each square.

They record their results in a table and look

for a pattern, leading them to ―discover‖ the

Pythagorean Theorem.

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Statistics and Probability (8.SP)


8.SP All students will investigate patterns of association in bivariate data.

1. Construct and interpret scatter plots for bivariate measurement data to investigate

patterns of association between two quantities. Describe patterns such as clustering,

outliers, positive or negative association, linear association, and nonlinear

association.

2. Know that straight lines are widely used to model relationships between two

quantitative variables. For scatter plots that suggest a linear association, informally

fit a straight line, and informally assess the model fit by judging the closeness

of the data points to the line.

3. Use the equation of a linear model to solve problems in the context of bivariate

measurement data, interpreting the slope and intercept.

4. Understand that patterns of association can also be seen in bivariate categorical data

by displaying frequencies and relative frequencies in a two-way table. Construct and

interpret a two-way table summarizing data on two categorical variables collected

from the same subjects. Use relative frequencies calculated for rows or columns to

describe possible association between the two variables.

Construct a scatter plot to show the

relationship between number of cigarettes

smoked per year per adult and the alarming

rate of coronary heart disease in several

countries. Use World Almanac for

information.

In a linear model for a biology experiment,

interpret a slope of 1.5 cm/hr as meaning that

an additional hour of sunlight each day is

associated with an additional 1.5 cm in

mature plant height.

Collect data from students in your class on

whether or not they have a curfew on school

nights and whether or not they have assigned

chores at home. Is there evidence that those

who have a curfew also tend to have chores?


Why do I need rational and irrational numbers?


Formative: Place rational and irrational numbers on a number line

Summative: Make a visual display with explanation and examples of rational and irrational

numbers.


1. Know that numbers that are not rational are called irrational. Understand informally that

every number has a decimal expansion; for rational numbers show that the decimal expansion

repeats eventually, and convert a decimal expansion which repeats eventually into a rational

number.


numbers, locate them approximately on a number line diagram, and estimate the value of

expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is

between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better

approximations.

Domain: The Number System. Standard (8.NS) Know that there are numbers that are not rational, and approximate them by rational numbers.


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Placement of values (decimals, fractions, integers, squares, square roots, radicals) on the number

line.

Conversion of fraction, decimal, squares, square roots, radicals, repeating, terminating decimals

Science: measurement


Number line, calculator.

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Domain: Expressions and Equations (8.EE) - Work with radicals and integer exponents.Essential Questions What should I be able to answer? What guides my thinking?

How do I work with very large and very small numbers?

When am I going to use this?

How is this idea going to help me with my thinking?

When is scientific notation used and by whom, what careers?


Formative: Find Someone Who…Review Students will explain what they heard during the

lesson to another student, agree/disagree/discuss

Exit Cards – periodically through the unit

Summative: My Math Textbook Page create a page with explanation and examples and

problems on each skill listed above


1. Know and apply the properties of integer exponents to generate equivalent numerical

expressions.

2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p

and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares

and cube roots of small perfect cubes. Know that √2 is irrational.

3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate

very large or very small quantities, and to express how many times as much one is than the

other.

4. Perform operations with numbers expressed in scientific notation, including problems where both

decimal and scientific notation are used. Use scientific notation and choose units of appropriate

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Grade Eight Page 16

size for measurements of very large or very small quantities (e.g., use millimeters per year for

seafloor spreading). Interpret scientific notation that has been generated by technology.



Mathematical representation to solve problems

Representation of mathematical situations using algebraic symbols

Understanding of the interconnection of mathematical ideas

*Science: examples weights, distances, measurement

Economics – example – debt

*Confer with science teacher


Standard Specific tools and websites, NETS tools and websites, calculator

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Domain: Expressions and Equations (8.EE) - Understand the connections between proportional relationships, lines, and linear equations.



What would a graph look like for a linear relationship?

Formative: Place centimeter cubes on the graph to build the following slopes: m = 1, ½, 2/3

Use a ruler to sketch a line following the under side of the stairs. Trace the right triangle formed

under each step. Remove the centimeter cubes and compare any four triangles using the

ratios of corresponding sides to determine if the triangles are similar.


Summative: Students generate data by translating information from a verbal problem, record

results in tabular form, and write a rule for the number patterns discovered in the table. Students

plot a set of data points and connect the points to form a graph of the related rule.

1. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example,

compare a distance-time graph to a distance-time equation to determine which of two

moving objects has greater speed.


points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through

dmkanowitz

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the origin and the equation y = mx + b for a line intercepting the vertical axis at b.


Proportions, time, distance , rate and similar formulas; coordinate plane, slope


Art: perspective drawing

Science: graphing collected data


Graph paper, graphing calculators

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Domain: Expressions and Equations (8.EE) - Analyze and solve linear equations and pairs of simultaneous linear equations


How and in what ways do I use symbolic algebra to represent and explain mathematical

relationships?


How is linear programming used in the real world to solve problem?

What are the benefits of having different types of strategies to solve systems of equations

related to real-world situation?

Formative: Given a magic square, solve by writing and solving algebraic equations

Summative: performance assessment - Purchase a cell phone with systems of equations


1. Solve linear equations with one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions,

or no solutions. Show which of these possibilities is the case by successively transforming the

given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b

results (where a and b are different numbers).

b. Solve linear equations with rational number coefficients, including equations whose solutions

require expanding expressions using the distributive property and collecting like terms.


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a. Understand that solutions to a system of two linear equations in two variables correspond to

points of intersection of their graphs, because points of intersection satisfy both equations

simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by

graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x +

2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

c. Solve real-world and mathematical problems leading to two linear equations in two variables.

For example, given coordinates for two pairs of points, determine whether the line through the

first pair of points intersects the line through the second pair.


Solve one variable equations to include using the distributive property and combining like terms.

Solve equations with two variables


Science: graphing quantitative data


Graphing calculator, graph paper, ruler

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Domain: Functions (8.F) - Define, evaluate, and compare functions


How will the study of input and output help our understanding of mathematics ?

Why is it important to understand the relationships between inputs and outputs?

Where will I see these types of relationships in real life?


Formative: Provide students with completed function tables and have them write the rule

associated with the input and output data in the table (for example, x = :, 0, 1, 2 and y = 0, 2, 4, 6.

Rule: y = 2x+2); Have students interpret function tables and rules to identify which functions are

linear and which are not (for example: linear: y = 2x +3 ; y = 2x;non linear: y = x2

+ 3). Have

students explain their reasoning. Using interactive boards and graph paper, have students graph

linear functions and explain the slope, rate of change, etc.

Summative: Performance Assessment: Using a real life example, have students use data

provided to compare two linear functions and determine which one has a positive slope. Function

One: John has a $50 iTunes gift card. He spends $4.50 per week to download songs. Let y be the

amount remaining as a function of the number of weeks, x. Function Two: The local library rents

e Readers for $5 per month. The library collects a non refundable annual charge of $20. Write the

rule for the cost (c) of renting an e Reader as a function of the number of months (m). Function

One results in a negative slope (4.5) as John spends money each week, decreasing the value of

the gift card. Function Two results in a positive slope (5) which is the monthly rental fee. (Research

additional problem examples online and in text books).

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1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a

function is the set of ordered pairs consisting of an input and the corresponding output. [Function notation

is not required in Grade 8.]

2. Compare properties of two functions each represented in a different way (algebraically,

graphically, numerically in tables, or by verbal descriptions). For example, given a linear function

represented by a table of values and a linear function represented by an algebraic expression, determine

which function has the greater rate of change.

3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give

examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a

function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which

are not on a straight line.


Generate tables with x and y values from a linear equation.

Graph functions in slope intercept form where slope can be a rate of change.



Science connection:

Determine input and output with heat and heat transfer. Also determine the relationship

between voltage and current, light and sound waves.

Graphic calculator and graph paper


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Domain: Functions (8.F) - Use functions to model relationships between quantities.


How does analysis of models and graphs help us to understand number relationships?

What are the different ways I can represent a function?


Formative: Have students interpret linear function tables and write the rule represented by the

data in the table in the form y = mx + b. Write a journal entry to explain the relationship between

the cost of attending an event and purchasing “extras.” For example, research costs associated

with lower level seating vs. upper level seating at a sports arena or concert venue. Does the cost

increase or decrease as the seat location changes? Have students write a journal entry about real

life example in their lives.

Summative: Performance Assessment: Using a real life example, have students use data

provided to write a linear rule, create a function table, compute inputs and outputs using the rule,

and graph results. For example, have students compute the costs of renting a vehicle ($35 per

day cost plus one time charge of $20 for GPS). Write the expression for total cost in dollars, y, as

a function of the number of rental days, x. Another example, have students compute the cost of a

fitness/gym membership at $25 per month, (y), plus a one time joiner fee, $75, for x number of

months.

Have students enter data into graphing calculator, view graph, and explain relationship(s).

dmkanowitz

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Grade Seven Page 24


1. Construct a function to model a linear relationship between two quantities. Determine the rate of

change and initial value of the function from a description of a relationship or from two (x, y) values,

including reading these from a table or from a graph. Interpret the rate of change and initial value of a

linear function in terms of the situation it models, and in terms of its graph or a table of values.


analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).

Sketch a graph that exhibits the qualitative features of a function that has been described

verbally.


Understand linear equations in two variables, and be able to create a graph from the relationship

of x and y.



Science connection:

Some examples of rate of change might be heat and heat transfer, locomotion, flight and

relationship between current and voltage.

Graphic Calculators and graph paper


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Domain: Geometry Standard (8.G) - Understand congruence and similarity using physical models, transparencies, or geometric software.


How are similarity and congruence established? Why is this important?



Formative: experiment with geo boards, attribute tiles, virtual manipulative websites

Summative: performance assessment – Students apply transformations to figures drawn on

coordinate grids, record the coordinates of the original figure and its image, and look for patterns.

(They express these patterns verbally and symbolically.)


a. Lines are taken to lines, and line segments to line segments of the same length.

b. Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

2. Understand that a two dimensional figure is congruent to another if the second can be

obtained from the first by a sequence of rotations, reflections, and translations; given two

congruent figures, describe a sequence that exhibits the congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on two dimensional

figures using coordinates.

4. Understand that a two dimensional figure is similar to another if the second can be

obtained from the first by a sequence of rotations, reflections, translations, and dilations; given

two similar two-dimensional figures, describe a sequence that exhibits the similarity between

them.

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Grade Eight Page 26

5. Use informal arguments to establish facts about the angle sum and exterior angle of

triangles, about the angles created when parallel lines are cut by a transversal, and the angle-

angle criterion for similarity of triangles. For example, arrange three copies of the same triangle

so that the sum of the three angles appears to form a line, and give an argument in terms of

transversals why this is so.


Transformations congruency and similarity, sum of angles, interior and exterior angles, transversal


Art: design


Manipulatives, geo-boards, attribute tiles, graph paper

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Grade Eight Page 27

Domain: Geometry (8.G) - Understand and apply the Pythagorean Theorem.


How can my understanding of the Pythagorean Theorem affect my understanding of the

world around me?


Formative: Students use tangram pieces to build squares on each side of the middle sized

triangular tangram piece. They then describe the relationship among the areas of the three squares.

Summative: performance assessment - Have students work in groups to draw several right

triangles. They should measure the sides and verify that they satisfy the Pythagorean Theorem.


1. Explain a proof of the Pythagorean Theorem and its converse.

2. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in


real world and mathematical problems in two and three dimensions.

3. Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system

Pythagorean Theorem, coordinate plane

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Grade Eight Page 28


Art: design


Calculator, coordinate plane, graph paper, ruler

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Grade Eight Page 29

Domain: Geometry (8.G) - Solve real-world and mathematical problems volume of cylinders, cones, and spheres.


How does understanding volume affect our day to day lives?


Formative: Students will solve various volume problems.

Summative: A new soda can is designed to be twice as tall as the original can. How are the

volumes of the cans related?


Complete questions on volume and capacity of cylinder problems.

1. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve



Formula for volume of cones, cylinders, and spheres


Science: volume of flow rate of pipes

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Grade Eight Page 30


Discovery education video, calculator, cones, spheres, and cylinders to measure

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Grade Eight Page 31

Domain: Statistics and Probability (8.SP) - Investigate patterns of association in bivariate data.


Which relationships in the real world can be represented when graphing statistics and probability

values?

What conclusions and inferences will be deduced from the data and applied to real life?



Formative: collect data, construct and interpret a scatter plot

Summative: performance assessment. Model linear data in a variety of settings that range

from car repair costs to sports to medicine. Students can work alone or in small groups to

construct scatter plots, interpret data points and trends, and investigate the notion of line of

best fit, interpreting slope and y intercept.


patterns of association between two quantities. Describe patterns such as clustering, outliers,

positive or negative association, linear association, and nonlinear association.

2. Know that straight lines are widely used to model relationships between two quantitative

variables. For scatter plots that suggest a linear association, informally fit a straight line, and

informally assess the model fit by judging the closeness of the data points to the line.


measurement data, interpreting the slope and intercept. For example, in a linear model for a

biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight

each day is associated with an additional 1.5 cm in mature plant height.

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Grade Eight Page 32

4. Understand that patterns of association can also be seen in bivariate categorical data by

displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-

way table summarizing data on two categorical variables collected from the same subjects. Use

relative frequencies calculated for rows or columns to describe possible association between the

two variables. For example, collect data from students in your class on whether or not they have

a curfew on school nights and whether or not they have assigned chores at home. Is there

evidence that those who have a curfew also tend to have chores?


Coordinate Plane, Measures of Central Tendency, Linear Equations, Slope and y intercept, Scatter

Plot, Box and Whisker Plot.


Science: Lab Experiments; Social Studies - Politics – interpreting scatter plots and patterns of data


Graph paper, graphing calculators, computer programs (Excel).

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Grade Eight Page 33

Accelerated Grade 7

Page 1

High School Mathematics in Middle School (Adapted from the Common Core State Standards)

There are some students who are able to move through mathematics quickly. Students who are capable of moving more quickly deserve

thoughtful attention, both to ensure that they are challenged and that they are mastering the full range of mathematical content and skills—

without omitting critical concepts and topics. Care must be taken to ensure that students master and fully understand all important topics

in the mathematics curriculum, and that the continuity of the mathematics learning progression is not disrupted. Students should have

appropriate progressions of foundational content to maximize their likelihood of success in high school mathematics.

Students begin preparing for algebra in Kindergarten, as they start learning about the properties of operations. Furthermore, much of the

content central to typical Algebra I courses—namely linear equations, inequalities, and functions—is found in the 8th grade Common Core

State Standards. Students who are ready for Algebra I in eighth grade should be provided this opportunity. To prepare students for Algebra I

in eighth grade schools are encouraged to have a well-crafted sequence of compacted courses. The term ―compacted‖ means to compress

content, which requires a faster pace to complete, as opposed to skipping content. The following compacted sequence based on the Common

Core State Standards compacts grades 7, 8, and High School Algebra I into two years: ―Accelerated 7th Grade‖ and ―8th Grade Algebra I.‖

Accelerated Mathematics – Grade 7

Accelerated Grade 7

Page 2

Accelerated 7th Grade


This course differs from the non-accelerated 7th Grade course in that it contains content from 8th grade. While coherence is retained, in that it

logically builds from the 6th Grade, the additional content when compared to the non-accelerated course demands a faster pace for instruction

and learning. Content is organized into four critical areas, or units. The Mathematical Practice Standards apply throughout each course and,

together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of

their ability to make sense of problem situations. The critical areas are as follows:

Unit 1: Rational Numbers and Exponents Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal

representation), and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and

division to all rational numbers, maintaining the properties of operations and the relationships between addition and subtraction, and

multiplication and division. By applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts

owed or temperatures below zero) students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative

numbers. They use the arithmetic of rational numbers as they formulate expressions and equations in one variable and use these equations

to solve problems. They extend their mastery of the properties of operations to develop an understanding of integer exponents, and to

work with numbers written in scientific notation.

Unit 2: Proportionality and Linear Relationships

Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize

equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality

(m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so

that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m × A. Students strategically

choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of

equality and the concept of logical equivalence, they maintain the solutions of the original equation.

Unit 3: Introduction to Sampling Inference

Students build on their previous work with single data distributions to compare two data distributions and address questions about

differences between populations. They begin informal work with random sampling to generate data sets and learn about the importance of

representative samples for drawing inferences.

Accelerated Grade 7

Page 3

Unit 4: Creating, Comparing, and Analyzing Geometric Figures

Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area

of three-dimensional objects. In preparation for work on congruence and similarity, they reason about relationships among two-

dimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between

angles formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining

cross sections. They solve real-world and mathematical problems involving area, surface area, and volume of two-and three-dimensional

objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. Students use ideas about distance and angles, how they

behave under translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-

dimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line,

and that various configurations of lines give rise to similar triangles because of the angles created when a transversal cuts parallel lines.

Students complete their work on volume by solving problems involving cones, cylinders, and spheres.

Accelerated Grade 7

Page 4

Standards for Mathematical Practice(From the Common Core State Standards)





















Accelerated Grade 7

Page 5






























Accelerated Grade 7

Page 6




















3 + x




Accelerated Grade 7

Page 7

Accelerated Grade 7 Standards for Mathematical Content

Note: The numbering of standards in the compacted courses follows the numbering found in the non-compacted courses. Therefore,

numbering of standards in the Accelerated Grade 7 Mathematics course and the Grade 8 Algebra I course are not in sequential order.

Some standards appear in more than one unit and in some cases only certain standards within a cluster are included in a unit.

Notation for Common Core Standards in Grades 7 and 8

Domains

Ratios and Proportional Relationships (RP)

Number System (NS)

Expressions and Equations (EE)

Geometry (G)

Functions (F )

Statistics and Probability (SP)

Accelerated Grade 7

Page 8

Unit 1: Rational Numbers and Exponents Standards Strategies and Assessments

7.NS Students will apply and extend understandings of operations with fractions to

add, subtract, multiply, and divide rational numbers.


subtract rational numbers; represent addition and subtraction on a horizontal or

vertical number line diagram.

a) Describe situations in which opposite quantities combine to make zero.

Example: A hydrogen atom has a zero charge because its two constituents are

oppositely charged.

b) Understand p + q as the number located a distance |q| from p, in the positive

or negative direction depending on whether q is positive or negative.

Show that a number and its opposite have a sum of zero (additive inverses).

Interpret sums of rational numbers by describing real world contexts.

c) Understand subtraction of rational numbers as adding the additive inverse,

p – q = p + (–q). Show that the distance between two rational numbers on

the number line is the absolute value of their difference, and apply this

principle in real world contexts.

d) Apply properties of operations as strategies to add and subtract rational

numbers.

2. Apply and extend previous understandings of multiplication, division, and fractions


a) Understand that multiplication is extended from fractions to rational numbers

by requiring that operations continue to satisfy the properties of operations,







Have students provide real-world situations

in which they would use positive or negative

rational numbers. Examples: weight gain

and loss; elevator position; loss or gain of

yards in football; income and expenses;

temperature.

This visual can be used to illustrate that

opposite quantities combine to make 0.

3 + -2 = 1

+ + +

0 + 0 + (+1) = 1

Show that the distance between two rational

numbers on the number line is the absolute

value of their difference.

p q p - q Distance between

p and q

5 -7 12 12 units

-12 8 -20 20 units

3 5 ½ -2 ½ 2 ½ units

Accelerated Grade 7

Page 9

Standards Strategies and Assessments particularly the distributive property, leading to products such as (–1)(–1) = 1

and the rules for multiplying signed numbers. Interpret products of rational

numbers by describing real world contexts.

b) Understand that integers can be divided, provided that the divisor is not zero,

and every quotient of integers (with non-zero divisor) is a rational number.

If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of

rational numbers by describing real world contexts.

c) Apply properties of operations as strategies to multiply and divide rational

numbers.

d) Convert a rational number to a decimal using long division; know that the

decimal form of a rational number terminates in zero or eventually repeats.

3. Solve real world mathematical problems involving rational numbers using the four

operations with rational numbers.

7. All students will understand numbers, ways of representing numbers,

relationships among numbers, and number systems.

1. Represent numbers in scientific notation and use them in calculations and problem

situations.

2. Apply the rules of powers and roots to the solution of problems. Extend the Order of

Operations to include positive integer exponents and square roots.

3. Compare and order fractions, decimals, and percents efficiently and find their

approximate locations on a number line.

Multiplicative Property of -1

Demonstrate that the result of multiplying a

number by (-1) is the additive inverse of the

number being multiplied.

34 x (-1) = -34 34 + -34 = 0

-45.5 x (-1) = 45.5 -45.5 + 45.5 = 0

Demonstrate how properties of operations

can simplify computation with rational

numbers.

Examples:

Distributive Commutative

Property Property

3 ¾ x 12 -2 + 3.5 - 8

(3 + ¾) x 12 -2 + 3.5 + (-8)

(3 x 12) + (¾ x 12) -2 + (-8) + 3.5

6 + 9 = 45 -10 + 3.5 = -6.5

Have students explain how to write fractions

as decimals and how to write terminating

and repeating decimals as fractions.

Provide students with examples that

demonstrate a specific rule of powers.

Guide students to identify the pattern and

state the rule.

Example: 32 x 3

4 = 3

6

55 x 5

3=5

8

(-2)2 x (-2)

3= (-2)

5

Accelerated Grade 7

Page 10


8.NS Students will know that there are numbers that are not rational, and

approximate them by rational numbers.

1. Know that numbers that are not rational are called irrational. Understand informally

that every number has a decimal expansion; for rational numbers show that the

decimal expansion repeats eventually, and convert a decimal expansion which

repeats eventually into a rational number.


numbers, locate them approximately on a number line diagram, and estimate the

value of expressions (e.g., π2). Example: By truncating the decimal expansion

of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to

continue on to get better approximations.

Example for 8.NS.1: Use the following method to change the repeating

decimal 0. 2 to a fraction.

Let x = 0.2222…

Multiply both sides so that the repeating digit/s will be to the left of the decimal.

In this example, one digit repeats so both sides are multiplied by 10 resulting in

10x = 2.2222…

Subtract the original equation from the new equation.

10x = 2.2222…

- x = 0.2222…

9x = 2

Solve the equation to determine the equivalent fraction.

9x = 2

x =9

2

Provide students with a list of numbers and

have them identify each number as rational

or irrational and explain their answers.

Example:

Rational Irrational

25 8

0.245245245… 0.01020304…

94

1612

Explore patterns in particular families of

decimal expansions, such as those for the

fractions 1/7, 2/7, 3/7, ... or 1/9, 2/9, 3/9, ... .

Each student is given a card with a rational

or irrational number written on it. (For

example: 3

12, 1.1, 1.01, 2 , π, -2

2, -

(2)2,

3

1, 8 , 5.3 , etc.) Students then

order themselves from least to greatest

along the front of the classroom.

Accelerated Grade 7

Page 11


8.EE Students will work with radicals and integer exponents.

1. Know and apply the properties of integer exponents to generate equivalent

numerical expressions. Example: 32×3

-5= 3

–3 = 1/3

3 = 1/27.

2. Use square root and cube root symbols to represent solutions to equations of the

form x2 = p and x

3 = p, where p is a positive rational number. Evaluate square

roots of small perfect squares (at least up to 225) and cube roots of small perfect

cubes. Know that √2 is irrational.

3. Use numbers expressed in the form of a single digit times an integer power of 10

to estimate very large or very small quantities, and to express how many times as

much one is than the other. Example: Estimate the population of the United States as

3 × 108 and the population of the world as 7 × 10

9, and determine that the world

population is more than 20 times larger.

4. Perform operations with numbers expressed in scientific notation, including

problems where both decimal and scientific notation are used. Use scientific notation

and choose units of appropriate size for measurements of very large or very small

quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific

notation that has been generated by technology.

Use numbers to illustrate the properties of

integer exponents.

Laws of Integer Exponents

For any nonzero real numbers a and b

and integers n and m:

Law Example

a0= 1 3

0= 1

a-n

= 1/an 2

-3 = 1/2

3 = 1/8

ana

m = a

n+m 32 x 3

4= 3

(2+4) =3

6

(3 x 3) x (3 x 3 x 3 x 3) = 36

(an)m

= anm (52

)3 = 5

2 x 3 = 5

6 = 15,625

52 x 5

2 x 5

2

(5 x 5) x (5 x 5) x (5 x 5) = 56

anb

n = (ab)

n 23 x 5

3 = (2 x 5)

3 = 10

3 = 1,000

(2 x 2 x 2) x (5 x 5 x 5)

8 x 125 = 1,000

am

/ an= a

m-n 34

/32 = 3

(4-2) = 3

2 = 9

3 x 3 x 3 x 3 = 81 = 9

3 x 3 = 9 1

(a/b)n= a

n/b

n (2/5)3 = 2

3/5

3 = 8/125

2/5 x 2/5 x 2/5 = 8/125

View the film Powers of Ten at: http://micro.magnet.fsu.edu/primer/

java/scienceopticsu/powersof10/



Accelerated Grade 7

Page 12

Unit 2: Proportionality and Linear Relationships


7.RP Students will analyze proportional relationships and use them to solve real world

and mathematical problems.

1. Compute unit rates associated with ratios of fractions, including ratios of lengths,

areas and other quantities measured in like or different units. Example: If a person

walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction1/2

/1/4 miles per hour, equivalently 2 miles per hour.


a) Decide whether two quantities are in a proportional relationship, e.g., by testing

for equivalent ratios in a table or graphing on a coordinate plane and observing

whether the graph is a straight line through the origin.

b) Identify the constant of proportionality (unit rate) in tables, graphs, equations,

diagrams, and verbal descriptions of proportional relationships.

c) Represent proportional relationships by equations. Example: If total

cost t is proportional to the number n of items purchased at a constant

price p, the relationship between the total cost and the number of items

can be expressed as t = pn.

d) Explain what a point (x, y) on the graph of a proportional relationship

means in terms of the situation, with special attention to the points (0, 0)

and (1, r) where r is the unit rate.

3. Use proportional relationships to solve multi-step ratio and percent problems.

Examples: Simple interest, tax, markups and markdowns, gratuities and

commissions, fees, percent of change (increase and decrease), percent error.

Students’ work with ratios is extended to

include ratios of fractions. Example: If a person

takes 3/4 of an hour to paint 2/3 of the wall

surface in a room, how much time will it take to

paint the entire wall surface in the room?

3/4 = x 2/3(x) = 3/4 x=9/8

2/3 1

# of

items

Cost Unit rate

(Cost per item)

1 $.25 $.25

2 $.50 $.25

3 $.75 $.25

4 $1.00 $.25

Graph the ordered pairs of two quantities on a

coordinate plane. If the graph is a straight line

through the origin (0,0), then the two quantities

are proportional

Explore proportional relationships at: http://education.ti.com/calculators/timathnspired/

US/Activities/Detail?sa=1008&t=9447&id=16893

Have students estimate the length of an item in

the classroom. Use the following formula to

calculate percent error:

Percent error = │approximate value – true value│

x 100 true value

http://education.ti.com/calculators/timathnspired/%20US/Activities/Detail?sa=1008&t=9447&id=16893

http://education.ti.com/calculators/timathnspired/%20US/Activities/Detail?sa=1008&t=9447&id=16893

Accelerated Grade 7

Page 13


7.EE Students will use properties of operations to generate equivalent expressions.

1. Apply properties of operations as strategies to add, subtract, factor, and expand

linear expressions with rational coefficients.

2. Understand that rewriting an expression in different forms may enhance the

interpretation of the problem and how the quantities in it are related.

Example: a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply

by 1.05.”

3.. Apply the rule for Order of Operations to solve numerical and algebraic equations

and to simplify and evaluate expressions.

7.EE Students will solve real world and mathematical problems using numerical and

algebraic expressions and equations.

3. Solve multi-step real-life and mathematical problems posed with positive and

negative rational numbers in any form (whole numbers, fractions, and decimals),

using tools strategically. Apply properties of operations to calculate with numbers in

any form; convert between forms as appropriate; and assess the reasonableness of

answers using mental computation and estimation strategies. Example: If a woman

making $25 an hour gets a 10% raise, she will make an additional 1/10 of

her salary an hour, or $2.50, for a new salary of $27.50. If you want to place

a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide,

you will need to place the bar about 9 inches from each edge; this estimate can be

used as a check on the exact computation.

4. Use variables to represent quantities in a real world or mathematical problem,

and construct simple equations and inequalities to solve problems by reasoning

about the quantities.

Students apply the properties of operations

to rational numbers (integers and positive /

negative fractions and decimals) as they

work with linear expressions.

Example: -5(3.5n -10)

-5[3.5n + (-10)]

(-5 x 3.5n) + [-5 x (-10)]

-17.5n + 50

Recognize the equivalency of expressions

written in different forms.

Example: The price of an item on sale for

20% off can also be calculated by finding

80% of the item’s price.

Price(p) – 20% x p = 80% x p

Practice translating verbal phrases into

algebraic expressions and algebraic

expressions into verbal phrases.

Use a variety of representations for

situations presented in word problems.

Summarize the method for solving a word

problem using algebraic expressions and

equations. Use previously learned properties

and strategies to simplify algebraic

expressions and solve algebraic equations.

Accelerated Grade 7

Page 14


a) Solve word problems leading to equations of the form px + q = r

and p(x + q) = r, where p, q, and r are specific rational numbers.

Solve equations of these forms fluently. Compare an algebraic solution

to an arithmetic solution, identifying the sequence of the operations used in

each approach. Example: The perimeter of a rectangle is 54 cm.

Its length is 6 cm. What is its width?

b) Solve word problems leading to inequalities of the form px + q > r

or px + q < r, where p, q, and r are specific rational numbers. Graph the

solution set of the inequality and interpret it in the context of the problem.

Example: As a salesperson, you are paid $50 per week plus $3 per sale.

This week you want your pay to be at least $100. Write an inequality for

the number of sales you need to make, and describe the solutions.

8.EE Students will understand the connections between proportional relationships,

lines, and linear equations.

5. Graph proportional relationships, interpreting the unit rate as the slope of the graph.

Compare two different proportional relationships represented in different ways.

Example: Compare a distance-time graph to a distance-time equation to determine

which of two moving objects has greater speed.


points on a non-vertical line in the coordinate plane; derive the equation y = mx for a

line through the origin and the equation y = mx + b for a line intercepting the vertical

axis at b.

7. Recognize and understand the forms of linear equation:

slope-intercept form as y = mx + b, and standard form as Ax + By =C.

Use variables to represent unspecified

quantities in a word problem.

Example: The perimeter of a rectangle is 54

cm. Its length is 6 cm. What is its width?

Use the formula for perimeter P= 2 ( l + w)

54 = (2 x 6) + (2 x w)

Use inequalities to solve word problems that

have more than one possible solution.

Example: Paid $50 per week plus $3 per sale;

want week’s pay to be at least $100

50 + (3 x s) 100

The amount earned for sales (3 x s) has to be

equal to or more than $50. Have students

discuss the values for s (number of sales) that

would make (3 x s) equal to or greater

than 50. Which numbers make sense in the

context of the problem?

Have students compare the rate of speed

represented graphically to the rate of speed

represented algebraically. Example: Which

shows a faster rate of speed?

Rate A Rate B

100

75

50

25

0 1 2 3 4 5

Time (h)

80 miles in 3 hours

D = rate x time

80 = r x 3

Dis

tance

(m

i)

Accelerated Grade 7

Page 15


8.EE Students will analyze and solve linear equations and pairs of simultaneous linear

equations.

7. Solve linear equations in one variable.

a) Give examples of linear equations in one variable with one solution (x = a),

infinitely many solutions (a = a), or no solutions (a = b where a and b are

different numbers). Show which of these possibilities is the case by

successively transforming the given equation into simpler forms.

b) Solve linear equations with rational number coefficients, including equations

whose solutions require expanding expressions using the distributive property

and collecting like terms.

Using a graph, students construct triangles

between two points on a line and compare

the sides to understand that the slope (ratio

of rise to run) is the same between any two

points on a

line.

Discuss the solution possibilities for linear

equations. Examples:

One Solution - ¼(8x +12) = 5

x = -4

No Solution 6d – (-4 + 2) = 6d

2 0

Many Solutions 2a(4 x .25) = 2a

2a = 2a

Accelerated Grade 7

Page 16

Unit 3: Introduction to Sampling Inference Standards Strategies and Assessments

7.SP Students will use random sampling to draw inferences about a population.


examining a sample of the population; generalizations about a population from a

sample are valid only if the sample is representative of that population. Understand

that random sampling tends to produce representative samples and support valid

inferences.

2. Use data from a random sample to draw inferences about a population with an

unknown characteristic of interest. Generate multiple samples (or simulated samples)

of the same size to gauge the variation in estimates or predictions.

Example: Estimate the mean word length in a book by randomly sampling

words from the book; predict the winner of a school election based on randomly

sampled survey data. Gauge how far off the estimate or prediction might be.

7.SP Students will be able to draw informal comparative inferences about two

populations.

3. Informally assess the degree of visual overlap of two numerical data distributions

with similar variabilities, measuring the difference between the centers by expressing

it as a multiple of a measure of variability. Example: The mean height of players on

the basketball team is 10 cm greater than the mean height of players on the soccer

team, about twice the variability (mean absolute deviation) on either team; on a dot

plot, the separation between the two distributions of heights is noticeable.


samples to draw informal comparative inferences about two populations.

Example: Decide whether the words in a chapter of a seventh-grade science book are

generally longer than the words in a chapter of a fourth-grade science book.

The following websites provide resources for

teaching concepts related to statistics. http://msteacher.org/return_list_math.aspx?id=418


Have students explain how they could best find

a random sample of their peers to survey about

their favorite sport. Would they survey only

those students on the soccer team?

Use census information found at:

http://www.census.gov/schools/ to design a

sample group of 100 people that is

representative of the population of the United

States.

Have students predict how many drops of water

will fit on a penny; write predictions on a post-it

note along with an explanation of his/her

reasoning. Predictions are collected and

displayed on bar graphs or stem-and-leaf plots.

Students perform the experiment and records

results on another post-it note and compare the

data distribution for the hypotheses with the data

distribution for the conclusions. (Combine with

science lesson on surface tension.)

Have students find, describe, and interpret

appropriate measures of center (mean, median,

and mode) and measures of variation/variability

(range)that represent a set of data. Example:

Collect, display and compare data for heights of

3rd grade students vs. heights of 7th grade

students.

http://msteacher.org/return_list_math.aspx?id=418


http://www.census.gov/schools

Accelerated Grade 7

Page 17


7.SP Students will investigate chance processes and develop, use, and evaluate

probability models.


expresses the likelihood of the event occurring. Larger numbers indicate greater

likelihood. A probability near 0 indicates an unlikely event, a probability around

1/2 indicates an event that is neither unlikely nor likely, and a probability near 1

indicates a likely event.

6. Approximate the probability of a chance event by collecting data on the chance

process that produces it and observing its long-run relative frequency, and predict the

approximate relative frequency given the probability. Example: When

rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly

200 times, but probably not exactly 200 times.

7. Develop a probability model and use it to find probabilities of events. Compare

probabilities from a model to observed frequencies; if the agreement is not good,

explain possible sources of the discrepancy.

a) Develop a uniform probability model by assigning equal probability to all

outcomes, and use the model to determine probabilities of events.

Example: If a student is selected at random from a class, find the probability

that Jane will be selected and the probability that a girl will be selected.

b) Develop a probability model (which may not be uniform) by observing

frequencies in data generated from a chance process. Example: Find the

approximate probability that a spinning penny will land heads up or that a

tossed paper cup will land open-end down. Do the outcomes for the spinning

penny appear to be equally likely based on the observed frequencies?

Experimental Probability

P(event) = number of times event occurs

number of trials

Example: Have students work in pairs rolling a

dice 20 times and recording the outcome in a

chart. Use a ratio to show how many times each

number was rolled out of the 20 trials.

Compare results recorded by each pair. Then

compare class results to theoretical probability

(number of times event should occur.)

Theoretical Probability

P(event) = number of favorable outcomes

number of possible outcomes

Example:

Experiment: Rolling a dice

Possible Outcomes: 1, 2, 3, 4, 5, 6

Event: Rolling 3 or 6

Favorable Outcomes: 2 (Rolling 3 or 6)

Probability: P(3 or 6) = 2/6 = 33%

Students should be able to distinguish between

dependent events and independent events when

discussing the probability of compound events.

What is the probability of two odd numbers

when spinning the two spinners?

Spinner A Spinner B

1 1

4 2 4 2

3 3

Accelerated Grade 7

Page 18


8. Find probabilities of compound events using organized lists, tables, tree diagrams,

and simulation.

a) Understand that, just as with simple events, the probability of a compound

event is the fraction of outcomes in the sample space for which the compound

event occurs.

b) Represent sample spaces for compound events using methods such as

organized lists, tables, and tree diagrams. For an event described in everyday

language (e.g., ―rolling double sixes‖), identify the outcomes in the sample

space which compose the event.

c) Design and use a simulation to generate frequencies for compound events.

Example: Use random digits as a simulation tool to approximate the answer

to the question: If 40% of donors have type A blood, what is the probability

that it will take at least 4 donors to find one with type A blood?

List of Possible Outcomes

1, 1 1, 2 1,3 1, 4

2, 1 2, 2 2, 3 2, 4

3, 1 3, 2 3, 3 3, 4

4, 1 4, 2 4, 3 4, 4

The probability of the spinning two odd

numbers is 4/16 or 1/4.

P(A and B) = P(A) x P(B)

P(odd and odd)= 2 x 2 = 4 or 1

4 4 16 4

Accelerated Grade 7

Page 19

Unit 4: Creating, Comparing, and Analyzing Geometric Shapes Standards Strategies and Assessments

7.G Students will draw, construct, and describe geometrical figures and describe the

relationships between them.

1. Solve problems involving scale drawings of geometric figures, including

computing actual lengths and areas from a scale drawing and reproducing a

drawing at a different scale.

2. Draw (freehand, with ruler and protractor, and with technology) geometric shapes

with given conditions. Focus on constructing triangles from three measures of

angles or sides, noticing when the conditions determine a unique triangle, more than

one triangle, or no triangle.

3. Describe the two-dimensional figures that result from slicing three-dimensional

figures, as in plane sections of right rectangular prisms and right rectangular

pyramids.

7.G Students will solve real world and mathematical problems involving angle

measure, area, surface area, and volume.

4. Know the formulas for the area and circumference of a circle and use them to

solve problems. Give an informal derivation of the relationship between the

circumference and area of a circle.

5. Use facts about supplementary, complementary, vertical, and adjacent angles

in a multi-step problem to write and solve simple equations for an unknown angle

in a figure.

6. Solve real world and mathematical problems involving area, volume, and surface

area of two- and three-dimensional objects composed of triangles, quadrilaterals,

polygons, cubes, and right prisms.

Have students make a scale drawing of

something in the classroom. Have them

include a key showing the scale as well as

the steps and calculations they used to

determine the various dimensions.

Prepare index cards with written descriptions

of geometric shapes with specific conditions

(one shape per card.) Distribute a card to

each pair of students. Students draw the

described shape. Display the shapes,

redistribute the cards and have students

match each shape to its description.

View an interactive representation of slicing

of three-dimensional figures at:

http://www.learner.org/courses/learningmath/

geometry/session9/part_c/index.html

Have students explain what the symbol π

represents and how it is related to circles.

Have students trace circular objects on

graph paper. Estimate the area of the circle.

Cut the circle out and fold it in half. Identify

the diameter and determine its length. Fold

the circle in fourths. Identify the radius and

determine its length. Discuss the relationship

between the various measurements. Have

students work in pairs. Have one student



Accelerated Grade 7

Page 20


8.G Students will understand congruence and similarity using physical models,

transparencies, or geometry software.


a) Lines are taken to lines, and line segments to line segments of the same

length.

b) Angles are taken to angles of the same measure.

c) Parallel lines are taken to parallel lines.

2. Understand that a two-dimensional figure is congruent to another if the second

can be obtained from the first by a sequence of rotations, reflections, and

translations; given two congruent figures, describe a sequence that exhibits the

congruence between them.

3. Describe the effect of dilations, translations, rotations, and reflections on

two-dimensional figures using coordinates.

4. Understand that a two-dimensional figure is similar to another if the second can

be obtained from the first by a sequence of rotations, reflections, translations, and

dilations; given two similar two-dimensional figures, describe a sequence that

exhibits the similarity between them.

5. Use informal arguments to establish facts about the angle sum and exterior angle

of triangles, about the angles created when parallel lines are cut by a transversal, and

the angle-angle criterion for similarity of triangles.

draw two pairs of parallel lines cut by a

transversal and number the angles. The other

student lists all the pairs of adjacent, vertical,

complementary, supplementary,

corresponding, and adjacent angles.

Explore geometric transformations at: http://www.mathsisfun.com/geometry/

transformations.html


Students apply transformations to figures

drawn on coordinate grids, record the

coordinates of the original figure and its

image, and look for patterns. They express

these patterns verbally and symbolically.

Examples: A reflection point across the x-

axis changes the sign of the y-coordinate so

that the point (x, y) moves to (x,-y).

A translation of 5 left and 2 up would

subtract 5 from the x-coordinate and add 2

to the y-coordinate D (-4, -3) D’ (-9, -1).

Arrange three copies of the same triangle so

that the sum of the three angles appears to

form a line, and give an argument in terms of

transversals why this is so.




Accelerated Grade 7

Page 21


8.G Students will solve real-world and mathematical problems involving volume of

cylinders, cones, and spheres.

9. Know the formulas for the volumes of cones, cylinders, and spheres and use

them to solve real-world and mathematical problems.

Prepare a scrapbook illustrating use of all

volume formulae, finding examples in real

life.

Explore volume using the interactive

manipulative at: http://nlvm.usu.edu/en/nav/

frames_asid_275_g_3_t_3.html

http://nlvm.usu.edu/en/nav/%20frames_asid_275_g_3_t_3.html

http://nlvm.usu.edu/en/nav/%20frames_asid_275_g_3_t_3.html

Accelerated Mathematics - Grade 8

Grade 8 Algebra I

Page 1

Grade 8 Algebra I

Introduction (from Common Core State Standards

The fundamental purpose of 8th Grade Algebra I is to formalize and extend the mathematics that students learned through the end of seventh

grade. The critical areas, called units, deepen and extend understanding of linear and exponential relationships by contrasting them with each

other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using

quadratic functions. In addition, the units will introduce methods for analyzing and using quadratic functions, including manipulating

expressions for them, and solving quadratic equations. Students understand and apply the Pythagorean theorem, and use quadratic functions

to model and solve problems. The Mathematical Practice Standards apply throughout each course and, together with the content standards,

prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of

problem situations.

This course differs from High School Algebra I in that it contains content from 8th grade. While coherence is retained, in that it logically

builds from the Accelerated 7th Grade, the additional content when compared to the high school course demands a faster pace for instruction

and learning.

Unit 1: Relationships Between Quantities and Reasoning with Equations

Work with quantities and rates, including simple linear expressions and equations forms the foundation for this unit. Students use units to

represent problems algebraically and graphically, and to guide the solution of problems. Student experience with quantity provides a

foundation for the study of expressions, equations, and functions. Instruction builds on earlier experiences with equations by asking

students to analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between

various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and

apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations.

Unit 2: Linear and Exponential Relationships

Building on earlier work with linear relationships, students learn function notation and language for describing characteristics of

functions, including the concepts of domain and range. They explore many examples of functions, including sequences; they interpret

functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of

Grade 8 Algebra I

Page 2

various representations. They work with functions given by graphs and tables, keeping in mind that depending upon the context, these

representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by

formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason

with the units in which those quantities are measured. Students explore systems of equations and inequalities, and they find and interpret

their solutions. Students build on and informally extend their understanding of integral exponents to consider exponential functions. They

compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret

arithmetic sequences as linear functions and geometric sequences as exponential functions.

Unit 3: Descriptive Statistics

Students use regression techniques to describe relationships between quantities. They use graphical representations and knowledge of the

context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness

of fit.

Unit 4: Expressions and Equations

Students build on their previous experience where they extended the laws of exponents to rational exponents. Students apply this new

understanding of numbers and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and

solve equations, inequalities, and systems of equations involving quadratic expressions.

Unit 5: Quadratic Functions and Modeling

In preparation for work with quadratic relationships students explore distinctions between rational and irrational numbers. They consider

quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from

among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of

quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function.

Students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist,

analogous to the way in which extending the whole numbers to the negative numbers allows x+1= 0 to have a solution. Formal work with

complex numbers comes in Algebra II. Students expand their experience with functions to include more specialized functions—absolute

value, step, and those that are piecewise-defined.

Grade 8 Algebra I

Page 3


The Standards for Mathematical Practice describe ways in which developing students increasingly ought to engage with the subject matter as

they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Activities and opportunities that

foster the development of these skills and proficiencies should underlie the mathematics curriculum at all grade levels.


















Grade 8 Algebra I

Page 4




























Grade 8 Algebra I

Page 5






















3 + x




Grade 8 Algebra I

Page 6

Grade 8 Algebra 1 Standards for Mathematical Content Note: The numbering of standards in the compacted courses follows the numbering found in the non-compacted courses. Therefore,

numbering of standards in the Accelerated Grade 7 Mathematics course and the Grade 8 Algebra I course are not in sequential order.

Some standards appear in more than one unit and in some cases only certain standards within a cluster are included in a unit.

Instructional notes from the Common Core State Standards document are included to provide guidance as to which elements of the

standard should be emphasized in the various units.

Notations for Algebra I Standards

Domain: Number and Quantity (N)

Clusters: N.RN The Real Number System

N.Q Quantities

Domain: Algebra (A)

Clusters: A.SSE Seeing Structure in Expressions

A.APR Arithmetic with Polynomials and Rational Expressions

A.CED Creating Equations

A.REI Reasoning with Equations and Inequalities

Domain: Functions (F)

Clusters: F.IF Interpreting Functions

F.BF Building Functions

F.LE Linear, Quadratic, and Exponential Models

Domain: Statistics and Probability (S)

Cluster: S.ID Interpreting Categorical and Quantitative Data

★The content standards must be connected to the Standards for Mathematical Practice to ensure that the skills needed for later success are

developed. In particular, Modeling (defined by a ★ in the CCSS) is defined as both a conceptual category for high school mathematics

and a mathematical practice and is an important avenue for motivating students to study mathematics, for building their understanding of

mathematics, and for preparing them for future success.

Grade 8 Algebra I

Page 7

Unit 1: Relationships Between Quantities and Reasoning with Equations Standards Instructional Notes and Examples

N.Q Students will reason quantitatively and use units to solve problems.

1. Use units as a way to understand problems and to guide the solution of multi-step

problems; choose and interpret units consistently in formulas; choose and interpret

the scale and the origin in graphs and data displays.

2. Define appropriate quantities for the purpose of descriptive modeling.

3. Choose a level of accuracy appropriate to limitations on measurement when

reporting quantities.

A.SSE Students will interpret the structure of expressions.

1. Interpret expressions that represent a quantity in terms of its context. ★

a) Interpret parts of an expression, such as terms, factors, and coefficients.

b) Interpret complicated expressions by viewing one or more of their parts as a

single entity.







Instructional Note for N.Q.1, 2, and 3: Working with quantities and the

relationships between them provides

grounding for work with expressions,

equations, and functions.

Instructional Note for A.SSE.1: Limit to

linear expressions and to exponential

expressions with integer exponents.

Example: Students should recognize that in

the expression 4x + 8, ―4‖ is the coefficient,

―4‖ and ―x‖ are factors, and ―8‖ is a

constant, as well as ―4x‖ and ―8‖ being

terms of the binomial expression.

Example: Interpret P(1+r)n as the product of

P and a factor not depending on P.

Grade 8 Algebra I

Page 8

Standards Instructional Notes and Examples

A.CED Students will create equations that describe numbers or relationships.

1. Create equations and inequalities in one variable and use them to solve problems.

Include equations arising from linear and quadratic functions, and simple rational

and exponential functions.

2. Create equations in two or more variables to represent relationships between

quantities; graph equations on coordinate axes with labels and scales.

3. Represent constraints by equations or inequalities, and by systems of equations

and/or inequalities, and interpret solutions as viable or non-viable options in a

modeling context. Example: Represent inequalities describing nutritional and cost

constraints on combinations of different foods.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as

in solving equations.

A.REI Students will understand solving equations as a process of reasoning and be

able to explain the reasoning.

1. Explain each step in solving a simple equation as following from the equality of

numbers asserted at the previous step, starting from the assumption that the original

equation has a solution. Construct a viable argument to justify a solution method.

Instructional Note for A.CED: Limit A.CED.1

and A.CED.2 to linear and exponential

equations, and, in the case of exponential

equations, limit to situations requiring

evaluation of exponential functions at integer

inputs. Limit A.CED.3 to linear equations and

inequalities. Limit A.CED.4 to formulas which

are linear in the variables of interest.

Example: Represent inequalities describing

nutritional and cost constraints on combinations

of different foods.

Example: An equation to model the maximum

area of a rectangle when the perimeter of a

rectangle is 16 inches should recognize that y =

x(8 – x) only makes sense when 0 < x < 8. This

restriction on the domain is necessary because

the side of a rectangle under these conditions

cannot be less than or equal to 0, but must be

less than 8.

Example: Rearrange Ohm’s law

V = IR (Voltage = Current x Resistance)

to highlight resistance

R = V / I (Resistance = Voltage Current)

Instructional Note for A.REI.1: Students

should focus on and masterA.REI.1 for linear

equations and be able to extend and apply their

reasoning to other types of equations in future

units and courses.

Grade 8 Algebra I

Page 9


A.REI Students will be able to solve equations and inequalities in one variable.

3. Solve linear equations and inequalities in one variable, including equations with

coefficients represented by letters.

Instructional Note for A.REI.3: Extend

earlier work with solving linear equations to

solving linear inequalities in one variable

and to solving literal equations that are

linear in the variable being solved for.

Include simple exponential equations that

rely only on application of the laws of

exponents, such as 5x = 125 or 2

x = 16

1 .

Unit 2: Linear and Exponential Functions Standards Instructional Notes and Examples

N.RN Students will be able to extend the properties of exponents to rational

exponents.

1. Explain how the definition of the meaning of rational exponents follows from

extending the properties of integer exponents to those values, allowing for a

notation for radicals in terms of rational exponents.

2. Rewrite expressions involving radicals and rational exponents using the properties

of exponents.

Instructional Note for N.RN.1 and 2: In

implementing the standards in curriculum,

these standards should occur before

discussing exponential models with

continuous domains.

Example: We define 5 1/3

to be the cube root

of 5 (i.e., 51/3

= 3 5 ) because we want

(51/3

)3

= 5(1/3

)3 to hold, so (5

1/3)3 must

equal 5.

nm

a = ( n a )m = n ma

Grade 8 Algebra I

Page 10


8.EE Students will analyze and solve linear equations and pairs of simultaneous

linear equations.


a) Understand that solutions to a system of two linear equations in two variables

correspond to points of intersection of their graphs, because points of

intersection satisfy both equations simultaneously.

b) Solve systems of two linear equations in two variables algebraically, and

estimate solutions by graphing the equations. Solve simple cases by

inspection.

c) Solve real-world and mathematical problems leading to two linear equations

in two variables. Example: Given coordinates for two pairs of points,

determine whether the line through the first pair of points intersects the line


A.REI Students will solve systems of equations.

5. Prove that, given a system of two equations in two variables, replacing one

equation by the sum of that equation and a multiple of the other produces a system

with the same solutions (substitution and elimination).

6. Solve systems of linear equations exactly and approximately (e.g., with graphs),

focusing on pairs of linear equations in two variables.

Instructional Note for 8.EE.8: While this

content is likely subsumed by A.REI.3, 5,

and 6, it could be used for scaffolding

instruction to the more sophisticated content

found there.

Example: Graph systems of equations to

show solution possibilities.

One solution

(2, -1)

3x – 3y = 9

and y = -x + 1

No solution -x +7- 4x

and 12 – 5x

Many

solutions

2x – y = -3

and 8x – 4y = -12

Example: 3x + 2y = 5 and 3x + 2y = 6 have

no solution because 3x + 2y cannot

simultaneously be 5 and 6.

Example: Given coordinates for two pairs of

points, determine whether the line through

the first pair of points intersects the line


Instructional Note for A.REI.5 and 6: Include cases where two equations describe

the same line (yielding infinitely many

solutions) and cases where two equations

describe parallel lines (yielding no solution).

Grade 8 Algebra I

Page 11


A. REI Students will represent and solve equations and inequalities graphically.

10. Understand that the graph of an equation in two variables is the set of all its

solutions plotted in the coordinate plane, often forming a curve (which could be a

line).

11. Explain why the x-coordinates of the points where the graphs of the equations

y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the

solutions approximately, (e.g., using technology to graph the functions, make tables

of values, or find successive approximations.) Include cases where f(x) and/or g(x)

are linear, polynomial, rational, absolute value, exponential, and logarithmic

functions. ★

12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding

the boundary in the case of a strict inequality), and graph the solution set to a

system of linear inequalities in two variables as the intersection of the

corresponding half-planes.

8.F Students will define, evaluate, and compare functions.

1. Understand that a function is a rule that assigns to each input exactly one output.

The graph of a function is the set of ordered pairs consisting of an input and the

corresponding output.



3. Interpret the equation f(x) = mx + b as defining a linear function whose graph is a

straight line; give examples of functions that are not linear.

Instructional Note for A.REI.10: Focus on

linear and exponential equations and be able to

adapt and apply that learning to other types of

equations in future courses.

For A.REI.11: Focus on cases where f(x) and

g(x) are linear or exponential.

Example: Graph 5x − y ≥ 3

Instructional Note for 8.F.1, 2 and 3: While

this content is likely subsumed by F.IF.1-3 and

F.IF.7a, it could be used for scaffolding


found there.

Example: Given a linear function represented by

a table of values and a linear function

represented by an algebraic expression,

determine which function has the greater rate of

change.

Example: The function f(A) = s2 giving the area

of a square as a function of its side length is not

linear because its graph contains the points

(1,1), (2,4) and (3,9), which are not on a straight

line.

Grade 8 Algebra I

Page 12

Standards Instructional Notes and Examples F.IF Students will understand the concept of a function and use function notation.

1. Understand that a function from one set (called the domain) to another set (called

the range) assigns to each element of the domain exactly one element of the range.

If f is a function and x is an element of its domain, then f(x) denotes the output of f

corresponding to the input x. The graph of f is the graph of the equation y = f(x).

2. Use function notation, evaluate functions for inputs in their domains, and interpret

statements that use function notation in terms of a context.

3. Recognize that sequences are functions, sometimes defined recursively, whose

domain is a subset of the integers.

8.F Students will use functions to model relationships between quantities.

4. Construct a function to model a linear relationship between two quantities.

Determine the rate of change and initial value of the function from a description of

a relationship or from two (x, y) values, including reading these from a table or

from a graph. Interpret the rate of change and initial value of a linear function in

terms of the situation it models, and in terms of its graph or a table of values.


analyzing a graph (e.g., where the function is increasing or decreasing, linear or

nonlinear). Sketch a graph that exhibits the qualitative features of a function that

has been described verbally.

Instructional Note for F.IF.1, 2, and 3: Students should experience a variety of types of

situations modeled by functions. Detailed

analysis of any particular class of function at

this stage is not advised. Students should apply

these concepts throughout their future

mathematics courses.

Constrain examples to linear functions and

exponential functions having integral domains.

In F.IF.3, draw connection to F.BF.2, which

requires students to write arithmetic and

geometric sequences

Example: The Fibonacci sequence is defined

recursively by f(0) = f(1) = 1,

f(n+1) = f(n)+ f(n-1) for n ≥ 1.

Instructional Note for 8.F.4 and 5: While this

content is likely subsumed by F.IF.4 and

F.BF.1a, it could be used for scaffolding


found there.

Example: Compare the following functions to

determine which has the greater rate of change.

Function 1: y = 2x + 4 Function 2:

x y

-1 -6

0 -3

2 3

Grade 8 Algebra I

Page 13


F.IF Students will be able to interpret functions that arise in applications in terms

of a context.

4. For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing

key features given a verbal description of the relationship. Key features include:

intercepts; intervals where the function is increasing, decreasing, positive, or

negative; relative maximums and minimums; symmetries; end behavior; and

periodicity.★

5. Relate the domain of a function to its graph and, where applicable, to the

quantitative relationship it describes.★

6. Calculate and interpret the average rate of change of a function (presented

symbolically or as a table) over a specified interval. Estimate the rate of change

from a graph. ★

F.IF Students will analyze functions using different representations.

7. Graph functions expressed symbolically and show key features of the graph, by

hand in simple cases and using technology for more complicated cases. ★

a) Graph linear and quadratic functions and show intercepts, maxima, and

minima.

e) Graph exponential and logarithmic functions, showing intercepts and end

behavior, and trigonometric functions, showing period, midline, and

amplitude.



Instructional Note for F.IF.4 and 5: Focus

on linear and exponential functions.

For F.IF.6: Focus on linear functions and

exponential functions whose domain is a

subset of the integers. Unit 5 in this course

addresses other types of functions.

Example: If the function h(n) gives the

number of person-hours it takes to assemble

n engines in a factory, then the positive

integers would be an appropriate domain for

the function.

Instructional Note for F.IF.7a, 7e, and 9: Focus on linear and exponential functions.

Include comparisons of two functions

presented algebraically. For example,

compare the growth of two linear functions,

or two exponential functions such as y=3n

and y=100·2n.

Example: Given a graph of one quadratic

function and an algebraic expression for

another, say which has the larger maximum.

Grade 8 Algebra I

Page 14


F.BF Students will be able to build a function that models a relationship between two

quantities.

1. Write a function that describes a relationship between two quantities. ★

a) Determine an explicit expression, a recursive process, or steps for calculation

from a context.

b) Combine standard function types using arithmetic operations.

2. Write arithmetic and geometric sequences both recursively and with an explicit

formula, use them to model situations, and translate between the two forms.★

F.BF Students will build new functions from existing functions.

3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and

f(x + k) for specific values of k (both positive and negative); find the value of k

given the graphs. Experiment with cases and illustrate an explanation of the effects

on the graph using technology. Include recognizing even and odd functions from

their graphs and algebraic expressions for them.

F.LE Students will construct and compare linear, quadratic, and exponential models

and solve problems.

1. Distinguish between situations that can be modeled with linear functions and with

exponential functions.

a) Prove that linear functions grow by equal differences over equal intervals

and that exponential functions grow by equal factors over equal intervals.

Instructional Note for F.BF.1 and 2: Limit

F.BF.1a, 1b, and 2 to linear and exponential

functions. In F.BF.2. connect arithmetic

sequences to linear functions and geometric

sequences to exponential functions.

Example: Build a function that models the

temperature of a cooling body by adding a

constant function to a decaying exponential,

and relate these functions to the model.

Instructional Note for F.BF.3: Focus on

vertical translations of graphs of linear and

exponential functions. Relate the vertical

translation of a linear function to its

y-intercept. While applying other

transformations to a linear graph is

appropriate at this level, it may be difficult

for students to identify or distinguish

between the effects of the other

transformations included in this standard.

Grade 8 Algebra I

Page 15


b) Recognize situations in which one quantity changes at a constant rate per unit

interval relative to another.

c) Recognize situations in which a quantity grows or decays by a constant

percent rate per unit interval relative to another.

2. Construct linear and exponential functions, including arithmetic and geometric

sequences, given a graph, a description of a relationship, or two input-output pairs

(include reading these from a table).

3. Observe using graphs and tables that a quantity increasing exponentially eventually

exceeds a quantity increasing linearly, quadratically, or (more generally) as a

polynomial function.

F.LE Students will be able to interpret expressions for functions in terms of the

situation they model.

5. Interpret the parameters in a linear or exponential function in terms of a context.

Instructional Note for F.LE.3: Limit to

comparisons between linear and exponential

models.

Example: Have students consider what

happens if they start with two bacteria on a

kitchen counter and the number of bacteria

doubles every hour. Students make a table

and graph their results, noting that the graph

is not linear.

Example: Students should work with a

variety of representations of functions

(tables, graphs, equations) and be able to

classify and describe the function as linear

or non-linear, increasing or decreasing.

Instructional Note for F.LE.5: Limit

exponential functions to those of the form

f(x) = bx + k

Grade 8 Algebra I

Page 16

Unit 3: Descriptive Statistics Standards Instructional Notes and Examples

S.ID Students will summarize, represent, and interpret data on a single count or

measurement variable.

1. Represent data with plots on the real number line (dot plots, histograms, and

box-and-whisker plots).

2. Use statistics appropriate to the shape of the data distribution to compare center

(median, mean) and spread (inter-quartile range, standard deviation) of two or more

different data sets.

3. Interpret differences in shape, center, and spread in the context of the data sets,

accounting for possible effects of extreme data points (outliers).

8.SP Students will investigate patterns of association in bivariate data.


patterns of association between two quantities. Describe patterns such as clustering,

outliers, positive or negative association, linear association, and nonlinear

association.

2. Know that straight lines are widely used to model relationships between two

quantitative variables. For scatter plots that suggest a linear association, informally

fit a straight line, and informally assess the model fit by judging the closeness

of the data points to the line.


measurement data, interpreting the slope and intercept.

Instructional Note for S.ID.1, 2, and 3: In grades 6 and 7, students describe center

and spread in a data distribution. Here they

choose a summary statistic appropriate to

the characteristics of the data distribution,

such as the shape of the distribution or the

existence of extreme data points.

Example: For the data set {1, 3, 6, 7, 10, 12,

14, 15, 22, 120}, the interquartile range is

15 – 6 = 9.

Instructional Note for 8.SP.1, 2, 3, and 4:: While this content is likely subsumed by

S.ID.6-9, it could be used for scaffolding


found there.

Example: Construct a scatter plot to show

the relationship between number of

cigarettes smoked per year per adult and the

alarming rate of coronary heart disease in

several countries. Use World Almanac for

information.

Example: In a linear model for a biology

experiment, interpret a slope of 1.5 cm/hr as

meaning that an additional hour of sunlight

each day is associated with an additional

1.5 cm in mature plant height.

Grade 8 Algebra I

Page 17

Standards Instructional Notes and Examples 4. Understand that patterns of association can also be seen in bivariate categorical data

by displaying frequencies and relative frequencies in a two-way table. Construct

and interpret a two-way table summarizing data on two categorical variables

collected from the same subjects. Use relative frequencies calculated for rows or

columns to describe possible association between the two variables.

S.ID Students will summarize, represent, and interpret data on two categorical and

quantitative variables.

5. Summarize categorical data for two categories in two-way frequency tables.

Interpret relative frequencies in the context of the data (including joint, marginal,

and conditional relative frequencies). Recognize possible associations and trends in

the data.

6. Represent data on two quantitative variables on a scatter plot, and describe how the

variables are related.

a) Fit a function to the data; use functions fitted to data to solve problems in the

context of the data. Use given functions or choose a function suggested by the

context. Emphasize linear and exponential models.

b) Informally assess the fit of a function by plotting and analyzing residuals.

c) Fit a linear function for a scatter plot that suggests a linear association.

S.ID Students will interpret linear models.

7. Interpret the slope (rate of change) and the intercept (constant term) of a linear

model in the context of the data.

8. Compute (using technology) and interpret the correlation coefficient of a linear fit.

9. Distinguish between correlation and causation.

Example: Collect data from students in your

class on whether or not they have a curfew

on school nights and whether or not they

have assigned chores at home. Is there

evidence that those who have a curfew also

tend to have chores?

Instructional Note S.ID.5 and 6: Students

take a more sophisticated look at using a

linear function to model the relationship

between two numerical variables. In

addition to fitting a line to data, students

assess how well the model fits by analyzing

residuals. S.ID.6b should be focused on

linear models, but may be used to preface

quadratic functions in the Unit 6 of this

course.

Instructional Note for S.ID.7, 8 and 9: Build on students’ work with linear

relationship and introduce the correlation

coefficient. The focus here is on the

computation and interpretation of the

correlation coefficient as a measure of how

well the data fit the relationship. The

important distinction between a statistical

relationship and a cause-and-effect

relationship arises in S.ID.9.

Grade 8 Algebra I

Page 18

Unit 4: Expressions and Equations Standards Instructional Notes and Examples

A.SSE Students will interpret the structure of expressions.

1. Interpret expressions that represent a quantity in terms of its context.★

a) Interpret parts of an expression, such as terms, factors, and coefficients.

b) Interpret complicated expressions by viewing one or more of their parts as a

single entity.

2. Use the structure of an expression to identify ways to rewrite it.

A.SSE Students will write expressions in equivalent forms to solve problems.

3. Choose and produce an equivalent form of an expression to reveal and explain

properties of the quantity represented by the expression. ★

a) Factor a quadratic expression to reveal the zeros of the function it defines.

b) Complete the square in a quadratic expression to reveal the maximum or

minimum value of the function it defines.

c) Use the properties of exponents to transform expressions for exponential

functions.

A.APR Students will perform arithmetic operations on polynomials.

1. Understand that polynomials form a system analogous to the integers, namely, they

are closed under the operations of addition, subtraction, and multiplication; add,

subtract, multiply, and divide polynomials.

Instructional Note A.SSE.1 and 2: Focus

on quadratic and exponential expressions.

For A.SSE.1b, exponents are extended from

integers found in Unit 1 to rational

exponents focusing on those that represent

square roots and cube roots.

Example: Interpret P(1+r)n as the product of

P and a factor not depending on P.

Example: See x4 – y

4 as (x

2)2 – (y

2)2, thus

recognizing it as a difference of squares that

can be factored as

(x2 – y

2)(x

2 + y

2) =(x + y)( x - y)(x

2 + y

2.)

Instructional Note for A.SSE.3: Consider extending this unit to include the

relationship between properties of

logarithms and properties of exponents.

Example: The expression 1.15t can be

rewritten as (1.15 1/12

)12t

≈ 1.01212t

to reveal

the approximate equivalent monthly interest

rate if the annual rate is 15%.

Instructional Note for A.APR.1: Focus on

polynomial expressions that simplify to

forms that are linear or quadratic in a

positive integer power of x.

Grade 8 Algebra I

Page 19


A.CED Students will create equations that describe numbers or relationships.

1. Create equations and inequalities in one variable and use them to solve problems.

Include equations arising from linear and quadratic functions, and simple rational and

exponential functions.

2. Create equations in two or more variables to represent relationships between

quantities; graph equations on coordinate axes with labels and scales.

4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in

solving equations.

A.REI Students will solve equations and inequalities in one variable.

4. Solve quadratic equations in one variable.

a) Use the method of completing the square to transform any quadratic equation in

x into an equation of the form (x – p)2 = q that has the same solutions. Derive

the quadratic formula from this form

Quadratic Formula: x = a

acbb

2

42

b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,

completing the square, the quadratic formula and factoring, as appropriate to

the initial form of the equation. Recognize when the quadratic formula gives

complex solutions and write them as a ± bi for

real numbers a and b.

A.REI Students will solve systems of equations.

7. Solve a simple system consisting of a linear equation and a quadratic equation in two

variables algebraically and graphically.

Instructional Note for A.CED.1, 2, and 4::

Extend work on linear and exponential

equations in Unit 1 to include quadratic

equations. Extend A.CED.4 to formulas

involving squared variables.

Example: Rearrange Ohm’s law

V = IR (Voltage = Current x Resistance)

to highlight resistance

R = V / I (Resistance = Voltage Current)

Instructional Note for A.REI.4:: Students

should learn of the existence of the complex

number system, but will not solve quadratics

with complex solutions until Algebra II.

Instructional Note for A.REI.7: Include

systems consisting of one linear and one

quadratic equation. Include systems that lead

to work with fractions. For example, finding

the intersections between x2+y

2=1 and y =

(x+1)/2 leads to the point (3/5, 4/5) on the

unit circle, corresponding to the Pythagorean

triple 32 + 4

2 = 5

2.

Example: Find the points of intersection

between the line y = –3x and the circle x2 +

y2 = 3.

Grade 8 Algebra I

Page 20

Unit 5: Quadratic Functions and Modeling Standards Instructional Notes and Examples

N.RN Students will use properties of rational and irrational numbers.

3. Explain why the sum or product of two rational numbers is rational; that the sum of a

rational number and an irrational number is irrational; and that the product of a

nonzero rational number and an irrational number is irrational.

8.G Students will understand and apply the Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem (a2

+ b2

= c2) and its converse.

7. Apply the Pythagorean Theorem to determine the unknown side lengths in right

triangles in real-world and mathematical problems in two and three dimensions.

8. Apply the Pythagorean Theorem to find the distance between two points in a

coordinate system.

F.IF Students will interpret functions that arise in applications in terms of a context.

4. For a function that models a relationship between two quantities, interpret key

features of graphs and tables in terms of the quantities, and sketch graphs showing key

features given a verbal description of the relationship. Key features include:

intercepts; intervals where the function is increasing, decreasing, positive, or

negative; relative maximums and minimums; symmetries; end behavior; and

periodicity. ★

5. Relate the domain of a function to its graph and, where applicable, to the quantitative

relationship it describes. ★

6. Calculate and interpret the average rate of change of a function (presented

symbolically or as a table) over a specified interval. Estimate the rate of change from

a graph. ★

Instructional Note for N.RN.3: Connect

N.RN.3 to physical situations, (e.g., finding

the perimeter of a square of area 2).

Instructional Note for 8.G.6, 7, and 8: Discuss applications of the Pythagorean

Theorem and its connections to radicals,

rational exponents, and irrational numbers.

Example: Students use graph paper or geo-

boards to construct squares on each side of

a right triangle. They find the area of each

square. They record their results in a table

and look for a pattern, leading them to

―discover‖ the Pythagorean Theorem.

Instructional Note for F.IF.4, 5, and 6:

Focus on quadratic functions; compare

with linear and exponential functions

studied in Unit 2.

Example: If the function h(n) gives the

number of person-hours it takes to

assemble n engines in a factory, then the

positive integers would be an appropriate

domain for the function.

Grade 8 Algebra I

Page 21


F.IF Students will analyze functions using different representations.

7. Graph functions expressed symbolically and show key features of the graph, by hand

for simple cases and using technology for more complicated cases. ★

a) Graph linear and quadratic functions and show intercepts, maxima, and minima.

b) Graph square root, cube root, and piecewise-defined functions, including

step functions and absolute value functions.

8. Write a function defined by an expression in different but equivalent forms to reveal

and explain different properties of the function.

a) Use the process of factoring and completing the square in a quadratic function

to show zeros, extreme values, and symmetry of the graph, and interpret these in

terms of a context.

b) Use the properties of exponents to interpret expressions for exponential

functions.



Instructional Note for F.IF.7 and 8:

Compare and contrast absolute value, step

and piecewise- defined functions with

linear, quadratic, and exponential

functions. Highlight issues of domain,

range, and usefulness when examining

piecewise-defined functions. Note that this

unit, and in particular in F.IF.8b, extends

the work begun in Unit 2 on exponential

functions with integral exponents.

For F.IF.9: Focus on expanding the types

of functions considered to include linear,

exponential, and quadratic. Extend work

with quadratics to include the relationship

between coefficients and roots, and that

once roots are known, a quadratic equation

can be factored.

Example: Identify percent rate of change in

functions such as y = (1.02)t, y = (0.97)

t, y

= (1.01)12t

, y = (1.2)t/10

, and classify them

as representing exponential growth or

decay.

Example: Given a graph of one quadratic

function and an algebraic expression for

another, say which has the larger

maximum.

Grade 8 Algebra I

Page 22


F.BF Students will build a function that models a relationship between two quantities.

1. Write a function that describes a relationship between two quantities. ★

a) Determine an explicit expression, a recursive process, or steps for calculation

from a context.

b) Combine standard function types using arithmetic operations.

F.BF Students will build new functions from existing functions.

3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and

f(x + k) for specific values of k (both positive and negative); find the value of k given

the graphs. Experiment with cases and illustrate an explanation of the effects on the

graph using technology. Include recognizing even and odd functions from their graphs

and algebraic expressions for them.

4. Find inverse functions.

a) Solve an equation of the form f(x) = c for a simple function f that has an inverse

and write an expression for the inverse.

Example: f(x) =2x3

for x > 0 or f(x) = (x+1)/(x-1) for x ≠ 1.

F.LE Students will construct and compare linear, quadratic, and exponential models

and solve problems.

3. Observe using graphs and tables that a quantity increasing exponentially eventually

exceeds a quantity increasing linearly, quadratically, or (more generally) as a

polynomial function.


situations that exhibit a quadratic

relationship.

Example: Build a function that models the

temperature of a cooling body by adding a

constant function to a decaying exponential

and relate these functions to the model.


quadratic functions, and consider including

absolute value functions.

For F.BF.4a: Focus on linear functions but

consider simple situations where the

domain of the function must be restricted in

order for the inverse to exist, such as f(x) =

x2, x >0.

Instructional Note for F.LE.3: Compare

linear and exponential growth to growth of

quadratic growth.


Page 1

Glossary Reprinted from the Common Core State Standards

www.corestandards.org

Addition and subtraction within 5, 10, 20, 100, or 1000 Addition or subtraction of two whole numbers with whole number answers, and with sum or minuend in the range 0-5, 0-10, 0-20, or 0-100, respectively. Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a subtraction within 20, and 55 – 18 = 37 is a subtraction within 100.

Additive inverses Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and – 3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.

Associative property of addition See Table 3 in this Glossary.

Associative property of multiplication See Table 3 in this Glossary.

Bi-variate data Pairs of linked numerical observations. Example: a list of heights and weights for each player on a football team.

Box plot A method of visually displaying a distribution of data values by using the median, quartiles, and extremes of the data set. A box shows the middle 50% of the data.

1

Commutative property See Table 3 in this Glossary.

Complex fraction A fraction A/B where A and/or B are fractions (B nonzero).

Computation algorithm A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy. __________ 1Adapted from Wisconsin Department of Public Instruction, http://dpi.wi.gov/

standards/mathglos.html, accessed March 2, 2010.

Computation strategy Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. See also: computation algorithm.

Congruent Two plane or solid figures are congruent if one can be obtained from the other by rigid motion (a sequence of rotations, reflections, and translations).

Counting on A strategy for finding the number of objects in a group without having to count every member of the group. For example, if a stack of books is known to have 8 books and 3 more books are added to the top, it is not necessary to count the stack all over again. One can find the total by counting on—pointing to the top book and saying “eight,” following this with “nine, ten, eleven. There are eleven books now.”

Dot plot See: line plot.

Dilation A transformation that moves each point along the ray through the point emanating from a fixed center, and multiplies distances from the center by a common scale factor.

Expanded form A multi-digit number is expressed in expanded form when it is written as a sum of single-digit multiples of powers of ten. For example, 643 = 600 + 40 + 3.

Expected value For a random variable, the weighted average of its possible values, with weights given by their respective probabilities.


Page 2

First quartile For a data set with median M, the first quartile is the median of the data values less than M. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.

2 See also: median, third quartile, interquartile

range.

Fraction A number expressible in the form a/b where a is a whole number and b is a positive whole number. (The word fraction in these standards always refers to a non-negative number.) See also: rational number.

Identity property of 0 See Table 3 in this Glossary.

Independently combined probability models Two probability models are said to be combined independently if the probability of each ordered pair in the combined model equals the product of the original probabilities of the two individual outcomes in the ordered pair.

Integer A number expressible in the form a or –a for some whole number a.

Interquartile Range A measure of variation in a set of numerical data, the interquartile range is the distance between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the interquartile range is 15 – 6 = 9. See also: first quartile, third quartile.

Line plot A method of visually displaying a distribution of data values where each data value is shown as a dot or mark above a number line. Also known as a dot plot.

3

Mean A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the list.

4 Example:

For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean is 21.

2Many different methods for computing quartiles are in use. The method

defined here is sometimes called the Moore and McCabe method. See Langford, E., “Quartiles in Elementary Statistics,” Journal of Statistics Education Volume 14, Number 3 (2006).

Mean absolute deviation A measure of variation in a set of numerical data, computed by adding the distances between each data value and the mean, then dividing by the number of data values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20.

Median A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of the list—or the mean of the two central values, if the list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.

Midline In the graph of a trigonometric function, the horizontal line halfway between its maximum and minimum values.

Multiplication and division within 100 Multiplication or division of two whole numbers with whole number answers, and with product or dividend in the range 0-100. Example: 72 ÷ 8 = 9.

Multiplicative inverses Two numbers whose product is 1 are multiplicative inverses of one another. Example: 3/4 and 4/3 are multiplicative inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1.

Number line diagram A diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.

Percent rate of change A rate of change expressed as a percent. Example: if a population grows from 50 to 55 in a year, it grows by 5/50 = 10% per year.

Probability distribution The set of possible values of a random variable with a probability assigned to each.

Properties of operations See Table 3 in this Glossary.

3Adapted from Wisconsin Department of Public Instruction, op. cit.

4To be more precise, this defines the arithmetic mean.


Page 3

Properties of equality See Table 4 in this Glossary.

Properties of inequality See Table 5 in this Glossary.

Properties of operations See Table 3 in this Glossary.

Probability A number between 0 and 1 used to quantify likelihood for processes that have uncertain outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a target, or testing for a medical condition).

Probability model A probability model is used to assign probabilities to outcomes of a chance process by examining the nature of the process. The set of all outcomes is called the sample space, and their probabilities sum to 1. See also: uniform probability model.

Random variable An assignment of a numerical value to each outcome in a sample space.

Rational expression A quotient of two polynomials with a non-zero denominator.

Rational number A number expressible in the form a/b or – a/b for some fraction a/b. The rational numbers include the integers.

Rectilinear figure A polygon all angles of which are right angles.

Rigid motion A transformation of points in space consisting of a sequence of one or more translations, reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.

Repeating decimal The decimal form of a rational number. See also: terminating decimal.

Sample space In a probability model for a random process, a list of the individual outcomes that are to be considered.

Scatter plot A graph in the coordinate plane representing a set of bivariate data. For example, the heights and weights of a group of people could be displayed on a scatter plot.

5

Similarity transformation A rigid motion followed by a dilation.

Tape diagram A drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model.

Terminating decimal A decimal is called terminating if its repeating digit is 0.

Third quartile For a data set with median M, the third quartile is the median of the data values greater than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15. See also: median, first quartile, interquartile range.

Transitivity principle for indirect measurement If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of object A is greater than the length of object C. This principle applies to measurement of other quantities as well.

Uniform probability model A probability model which assigns equal probability to all outcomes. See also: probability model.

Vector A quantity with magnitude and direction in the plane or in space, defined by an ordered pair or triple of real numbers.

Visual fraction model A tape diagram, number line diagram, or area model.

Whole numbers The numbers 0, 1, 2, 3, ….

___________ 5Adapted from Wisconsin Department of Public Instruction, op. cit.


Page 4

Table 1. Common Addition and Subtraction Situations (from the Common Core State Standards)

Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3Total

Total Unknown Addend Unknown Both Addends Unknown1

Put together/ Take apart2

Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 =

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1, 5 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have?

2 + 3 = ?, 3 + 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have?

5 – 3 = ?, ? + 3 = 5 1These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean “is the same number as”. 2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. 3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller unknown). The other versions are more difficult.


Page 5

Table 2. Common Multiplication and Division Situations (from the Common Core State Standards)

Unknown Product Group Size Unknown (“How many in each group?” Division)

Number of Groups Unknown (“How many groups”? Division)

3 x 6 =? 3 x ? = 18, and 18 3 = ? ? x 6 = 18, and 18 6 = ?

Equal Groups

There are 3 bags with 6 plums in each bag. How many plums are there in all?

Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether?

If 18 plums are shared equally into 3 bags, then how many plums will be in each bag?

Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be?

If 18 plums are to be packed 6 to a bag, then how many bags are needed?

Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have?

Arrays4, Areas5

There are 3 rows of apples with 6 apples in each row. How many apples are there?

Area example. What is the area of a 3 cm by 6 cm rectangle?

If 18 apples are arranged into 3 equal rows, how many apples will be in each row?

Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it?

If 18 apples are arranged into equal rows of 6 apples, how many rows will there be?

Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it?

Compare

A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost?

Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long?

A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost?

Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first?

A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat?

Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first?

General a x b = ? a x ? = p, and p a = ? ? x b = p, and p b = ? 4The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable. 5Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially important measurement situations.


Page 6

Table 3. The Properties of Operations (from the Common Core State Standards) Here a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.

Associative property of addition

Commutative property of addition

Additive identity property of 0

Existence of additive inverses

Associative property of multiplication

Commutative property of multiplication

Multiplicative identity property of 1

Existence of multiplicative inverses

Distributive property of multiplication over addition

(a + b) + c = a + (b + c)

a + b = b + a

a + 0 = 0 + a = a

For every a there exists –a so that a + (–a) = (–a) + a = 0.

(a × b) × c = a × (b × c)

a × b = b × a

a × 1 = 1 × a = a

For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.

a × (b + c) = a × b + a × c

Table 4. The Properties of Equality (from the Common Core State Standards) Here a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality

Symmetric property of equality

Transitive property of equality

Addition property of equality

Subtraction property of equality

Multiplication property of equality

Division property of equality

Substitution property of equality

a = a

If a = b, then b = a.

If a = b and b = c, then a = c.

If a = b, then a + c = b + c.

If a = b, then a – c = b – c.

If a = b, then a × c = b × c.

If a = b and c ≠ 0, then a ÷ c = b ÷ c.

If a = b, then b may be substituted for a in any expression containing a.


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Table 5. The Properties of Inequality (from the Common Core State Standards) Here a, b and c stand for arbitrary numbers in the rational or real number systems.

Exactly one of the following is true: a < b, a = b, a > b.

If a > b and b > c then a > c.

If a > b, then b < a.

If a > b, then –a < –b.

If a > b, then a ± c > b ± c.

If a > b and c > 0, then a × c > b × c.

If a > b and c < 0, then a × c < b × c.

If a > b and c > 0, then a ÷ c > b ÷ c.

If a > b and c < 0, then a ÷ c < b ÷ c.

Appendix B: Internet Resources

Internet Resources for Mathematics

Common Core State Standards

www.corestandards.org

Interactive Lessons and Activities

http://illuminations.nctm.org/

www.thinkfinity.org

www.mathwarehouse.com/interactive

Virtual Manipulatives and Math Tools

www.nlvm.usu.edu/en/nav/vlibrary.html

www.mathplayground.com/geoboard.html

Real World Applications of Mathematics

www.thefutureschannel.com/index.php

Interactive Whiteboard Resources

www.misterteacher.com/index.html

Animations Demonstrating Use of Visual Aids

www.nzmaths.co.nz/equipment-animations

Geometry and Measurement

http://images.rbs.org/content_strands/visualization.shtml

www.learner.org/interactives/geometry/index.html

Integrating Math and Literature

www.love2learn2day.blogspot.com/p/math-book-lists-tba.html

www.teacher.scholastic.com/reading/bestpractices/math.htm

Teaching Financial Literacy

www.edutopia.org/financial-literacy-resources

Math Vocabulary

www.spellingcity.com/math-vocabulary.html

Teacher Resources

www.K-5MathTeachingResources.com

www.mathfactcafe.com

www.mathforum.org/teachers

www.mathwire.com/index.html

www.pbs.org/teachers

Practice Activities for Students

www.aaamath.com

www.abc.net.au//countusin/default.htm

www.softschools.com/math

www.aplusmath.com

www.figurethis.org/index.html

Mathematics Curriculum Guidelines · 2020. 9. 25. · Mathematics Curriculum Guidelines and Common...

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