Teacher Training Revised ELA and Math Standards Math 9–12...• K–12 learning progressions •...

Teacher TrainingRevised ELA and Math Standards

Math 9–12Tennessee Department of Education | 2017 Summer Teacher Training

Welcome, Teachers!

1

We are excited to welcome you to this summer’s teacher training on the revised math standards. We appreciate your dedication to the students in your classroom and your growth as an educator. As you interact with the math standards over the next two days, we hope you are able to find ways to connect this new content to your own classroom. Teachers perform outstanding work every school year, and our hope is that the knowledge you gain this week will enhance the high-quality instruction you provide Tennessee's children every day.

We are honored that the content of this training was developed by and with Tennessee educators for Tennessee educators. We believe it is important for professional development to be informed by current educators, who work every day to cultivate every student’s potential.

We’d like to thank the following educators for their contribution to the creation and review of this content:

Dr. Holly Anthony, Tennessee Technological University Michael Bradburn, Alcoa City Schools Dr. Jo Ann Cady, University of Tennessee Sherry Cockerham, Johnson City Schools Dr. Allison Clark, Arlington Community Schools Kimberly Herring, Cumberland County Schools Dr. Joseph Jones, Cheatham County Schools Dr. Emily Medlock, Lipscomb University

Part 1: The Standards

3

Module 1: Standards Review Process

Module 2: Tennessee Academic Standards

Module 3: Summary of Revisions

Part 2: Developing a Deeper Understanding

Module 4: Diving into the Standards (KUD)

Part 3: Instructional Shifts

Module 5: Revisiting the Shifts and SMP’s

Module 6: Literacy Skills for Mathematical Proficiency

Part 4: Assessment and Materials

Module 7: Connecting Standards and Assessment

Module 8: Evaluating Instructional Materials

Part 5: Putting it All Together

Module 9: Instructional Planning

Notes

Time Content

8–11:15(includes break)

Part 1: The Standards• M1: Standards Review Process• M2: TN Academic Standards• M3: Summary of Revisions

11:15–12:30 Lunch (on your own)

12:30–4(includes break)

Part 2: Developing a Deeper Understanding• M4: Diving into the Standards (KUD)

Part 3: Instructional Shifts• M5: Revisiting the Shifts and SMP’s• M6: Literacy Skills for Mathematical Proficiency

Agenda: Day 1

Goals: Day 1• Review the standards revision process.

• Highlight changes/revisions to standards.

• Use a KUD exercise to deepen our understanding of the expectations of the standards.

• Explore the Literacy Skills for Mathematical Proficiency.

5

• Discuss the instructional shifts and the Standards for Mathematical Practice (SMPs).

Agenda: Day 2

Time Content

8–11:15(includes break)

Part 4: Aligned Materials and Assessments• M7: Assessing Student Understanding

11:15–12:30 Lunch (on your own)

12:30–4(includes break)

• M8: Evaluating Instructional Materials

Part 5: Putting it All Together• M9: Instructional Planning

Goals: Day 2• Examine best practices for assessing student learning.

• Develop a process for evaluating instructional materials.

• Connect standards and assessment through instructional planning.

7

Appointment TimeMake four appointments to meet with fellow participants throughout the training to discuss the content. Record participants’ names in the form below and bookmark this page for your reference.

1 2

3 4

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Key Ideas for Teacher Training

Strong Standards High Expectations

Instructional Shifts

Aligned Materials and Assessments

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High Expectations

We have a continued goal to prepare students to be college and career ready.

We know that Tennessee educators are working hard and striving to get better. This summer’s teacher training is an exciting opportunity to learn about our state’s newly adopted math and ELA standards and ways to develop a deeper understanding of the standards to improve classroom instructional practices. The content of this training is aligned to the standards and is designed to address the needs of educators across our state.

Throughout this training, you will find a series of key ideas that are designed to focus our work on what is truly important. These key ideas align to the training objectives and represent the most important concepts of this course.

Strong Standards

Standards are the bricks that should be masterfully laid through quality instruction to ensure that all students reach the expectation of the standards.


The instructional shifts are an essential component of the standards and provide guidance for how the standards should be taught and implemented.


Educators play a key role in ensuring that our standards, classroom instructional materials, and assessments are aligned.

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Strong Standards

High Expectations

Instructional ShiftsAligned

Materials and Assessments

Part 1: The StandardsModule 1: The Standards Review Process

TAB PAGE

Standards Review Process The graphic below illustrates Tennessee’s standards review process. Here you can see the various stakeholders involved throughout the process.

• The process begins with a website for public feedback. • Tennessee educators who are experts in their content area and grade band serve

on the advisory panels. These educators review all the public feedback and the current standards, then use their content expertise and knowledge of Tennessee students to draft a revised set of standards.

• The revised standards are posted for a second feedback collection from Tennessee’s stakeholders.

• The Standards Recommendation Committee (SRC) consists of 10 members appointed by legislators. This group looks at all the feedback from the website, the current standards, and revised drafts. Recommendations are then made for additional revisions if needed.

• The SRC recommends the final draft to the State Board of Education for approval.

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Educator Advisory Team MembersEvery part of the state was represented with multiple voices.

47Timeline of Standards Adoptions and Aligned Assessments Implementation

Subject

Math

ELA

Science

Social Studies

Standards Proposed to State Board

for Final Read

April 2016

April 2016

October 2016

July 2017

LEAs: New Standards in Classrooms

2017–18

2017–18

2018–19

2019–20

Aligned Assessment

2017–18

2017–18

2018–19

2019–20

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Standards Revision Key Points

• within a single grade level, and• between multiple grade levels.

“Districts and schools in Tennessee will exemplify excellence and equity such that

all students are equipped with the knowledge and skills to successfully

embark upon their chosen path in life.”

What is your role in ensuring that all students are college and career ready?

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• The instructional shifts remain the same and are still the focus of the standards.• The revised standards represent a stronger foundation that will support the progression of rigorous standards throughout the grade levels.

• The revised standards improve connections:

Strong Standards

High Expectations



Part 1: The StandardsModule 2: The Tennessee Mathematics Academic Standards

TAB PAGE

Goals• Reinforce the continued expectations of the Tennessee Math Academic

Standards.

• Revisit the three instructional shifts and their continued and connected role in the revised standards.

• Review the overarching changes of the revised Tennessee Math Academic Standards.

High Expectations


Strong Standards

Standardsarethebricksthatshouldbemasterfullylaidthroughqualityinstructiontoensurethatallstudentsreachtheexpectationofthestandards.


Theinstructionalshiftsareanessentialcomponentofthestandardsandprovideguidanceforhowthestandardsshouldbetaughtandimplemented.


Educatorsplayakeyroleinensuringthatourstandards,classroominstructionalmaterials,andassessmentsarealigned.

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Setting the StageDirections:

2. After reading and annotating the two parts, write the sentence or phrase you felt was the most important in the box below and your rationale for choosing it.

Most Important Idea:

Rationale:

Key Ideas from Discussion:

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1. Read and annotate the General Introduction to the TN Math Standards (pages 1–2) focusing on the “Mathematically Prepared” and “Conceptual Understanding, Procedural Fluency, and Application” sections.

What Has NOT Changed

• Students prepared for college and career

• K–12 learning progressions

• Traditional and integrated pathways (for high school)

• Standards for Mathematical Practice

• Instructional shifts

What HAS Changed• Category Change

• Revised Structured

• Coding & Nomenclature

• Literacy Skills for Mathematical Proficiency

Notes:

25

What HAS Changed

Category Change

Supporting Work of the Grade

Major Work of the Grade

Major Work of the Grade

Additional Work of the Grade

Supporting Work of the

Grade

Former Current

Notes:

26

What HAS Changed

Revised Structure

Notes:

27

What HAS Changed

Revised Structure

Notes:

28

What HAS Changed

Coding and Nomenclature

Notes:

A2.N.RN.A.1

M1

A

SSE

A

1

M1.A.SSE.A.1

A2

N

RN

A

1

29

Communication in mathematics requires literacy skills in reading, vocabulary, speaking, listening, and writing. Students must be able to:

What HAS Changed

Literacy Skills for Mathematical Proficiency

Notes:

30

Module 2 Review• Reinforce the continued expectations of the Tennessee Math Academic

Standards.

• Revisit the three instructional shifts and their continued and connected role in the revised standards.

• Review the overarching changes of the revised Tennessee Math Academic Standards.

Strong Standards


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Strong Standards

High Expectations



Part 1: The StandardsModule 3: Summary of Revisions

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TAB PAGE

Goals

• Review a summary of revisions to the math standards by grade band.

• Compare 2016–17 standards to 2017–18 standards.

High Expectations


Strong Standards






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“To assess student achievement accurately, teachers and administrators must know and understand the content standards that their students are to master. Again, we cannot teach or assess achievement that we have not defined.”

—S. Chappuis, Stiggins, Arter, and J. Chappuis, 2006

Why Standards?

What about this quotation sticks out to you?

Notes:

36

Revisions to the Math Standards

Specific to K–5

• Refined for clarity

• Increased fluency expectations

• Revised examples

Overarching Revisions

• Supporting and additional work of the grade is combined as supporting work of the grade


Increased Fluency Expectations

Former Standard Current Standard

Kindergarten K.OA.5 Fluently add and subtract within 5.

K.OA.A.5 Fluently add and subtract within 10 using mental strategies.

FirstGrade

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

and subtraction within 10.

1.OA.C.6 Fluently add and subtract within 20 using mental strategies. By

the end of Grade 1, know from memory all sums up to 10.

SecondGrade

2.OA.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.B.2 Fluently add and subtract within 30 using mental strategies. By

the end of Grade 2, know from memory all sums of two one-digit numbers and related subtraction

facts.

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Specific to K–5





• Added/shifted a small number of standards to strengthen coherence across grade levels

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Specific to K–5





• Added/shifted a small number of standards to strengthen coherence across grade levels

39


Specific to K–5




Overarching Revisions• Revised language to provide clarity and continuity

• Highlighted chart for–grade level mastery expectation for addition, subtraction, multiplication and division

Former Standard 2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

Current Standard 2.NBT.A.3 Read and write numbers to 1000 using standard form, word form, and expanded form.

Former Standard 4.NBT.3 Use place value understanding to round multi-digit whole numbers to any place.

Current Standard 4.NBT.A.3 Round multi-digit whole numbers to any place (up to and including the hundred-thousand place) using understanding of place value.

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Specific to 6–8

• Refined major work of the grade

• Revised supporting work of the grade, especially in statistics and probability


• Slight revisions made to geometry in grade 8


• Revised language to provide clarity and continuity

Former Standard 6.SP.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.

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Current Standard 6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center (mean, median, mode), spread (range), and overall shape.


Specific to 6–8




• Revised a small number of standards to strengthen coherence by condensing, expanding, and removing standards

• Revised a small number of statistics and probability standards

Former Standard 6.EE.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. For example, Susan is putting money in her savings account by depositing a set amount each week (50). Represent her savings account balance with respect to the number of weekly deposits (s = 50w, illustrating the relationship between balance amount s and number of weeks w). 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.

Current Standard 6.EE.C.9 Use variables to represent two quantities in a real-world problem that change in relationship to one another. For example, Susan is putting money in her savings account by depositing a set amount each week (50). Represent her savings account balance with respect to the number of weekly deposits (s = 50w, illustrating the relationship between balance amount s and number of weeks w).

a. 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.

b. Analyze the relationship between the dependent and independentvariables using graphs and tables, and relate these to the equation.

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Removed Standard 7.G.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.


Specific to 6–8




• Revised a small number of standards to strengthen coherence by condensing, expanding, and removing standards

• Revised a small number of statistics and probability standards

Former Standard6.SP.5c Summarize numerical data sets in relation to their context, such as by: 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.

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Current Standard 6.SP.B.5c Summarize numerical data sets in relation to their context, such as by: c. Giving quantitative measures of center (median and/or mean) and variability (range), as well as describing any overall pattern with reference to the context in which the data were gathered.


Specific to 9–12

• Refined and revised scope and clarifications

• Revisions for Algebra II and Integrated Math III

• Restructured additional math courses to reflect college and career readiness



• Removed or shifted a small number of standards to the major work of the grade to streamline vertical progression

• Revised language and examples to provide clarity and continuity

Former StandardG.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Moved StandardA2/M3.F.TF.5 to P.F.TF.A.4 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

This standard moved from Algebra II/Integrated III to Pre-Calculus.

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• Shifted a small number of supporting work of the grade standards to additional mathematics courses

Current Standard G.SRT.C.8a Know and use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.


Specific to 9–12

• Refined and revised scope and clarifications

• Revisions for Algebra II and Integrated Math III


• Restructured additional mathematics courses to reflect college and career readiness by removing three courses and adding “Applied Mathematical Concepts”

Rationale:

• High expectations

• Retention of rigorous standards

• Clearly defined and coherent pathways

• Equity and opportunity

• Aligned with student interest in postsecondary fields

• Shift to a discipline and career based pathway

Former:• Advanced Algebra and Trigonometry• Discrete Math• Finite Math• Bridge Math • Pre-Calculus• Statistics• Calculus

Current:• Applied Mathematical Concepts• Bridge Math• Pre-Calculus• Statistics• Calculus

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• Restructured additional courses to reflect college and career readiness


New Applied Mathematical Concepts Course• For students interested in careers that use applied mathematics such as banking,

industry, or human resources

• Rich problem solving experience

• Combines standards from Senior Finite Math and Discrete Mathematics

• Designed with industry needs in mind

• Alignment with first three math courses and ACT college and career readiness

• Possible dual credit exam

Problems in Applied Mathematical ConceptsAM.G.L.A.3: Solve a variety of logic puzzles

What's the easiest way to heat a pan of water for 9 minutes when you have only a 6-minute hour-glass timer and a 21-minute hour-glass timer?

AM.D.ID.A.2: Use a variety of counting methods to organize information, determine probabilities, and solve problems.

Given a group of students: G = {Allen, Brenda, Chad, Dorothy, Eric} list and count the different ways of choosing the following officers or representatives for student congress. Assume that no one can hold more than one office.

A president, a secretary, and a treasurer, if the president must be a woman and the other two must be men.

AM.N.NQ.B.6: Solve contextual problems involving financial decision-making.

The cash price of a fitness system is $659.99. The customer paid $115 as a down payment. The remainder will be paid in 36 monthly installments of $19.16 each. Find the amount of the finance charge.

46


Standards Comparison Activity

Compare the former standards to the current standards

Directions:

1. Highlight any changes you notice between the former standards and the current standards in the column on the right.

Notes:

47

2. Use the included chart to compare the former standards with the current standards.

Notation

TN Standards through M

ay 2017 Standard Clarification

Revised TN Standards

Standard Clarification

A1.N.RN.A.3 3. Explain w

hy the sum or

product of two rational

numbers is rational; and that

they product of a nonzero rational num

ber and an irrational num

ber is irrational.

There is no additional scope or clarification inform

ation for this standard.

A1.N.Q.A.1

1. Use units as a w

ay to understand problem

s and to guide the solution of m

ulti-step problem

s; choose and interpret units consistently in form

ulas; choose and interpret the scale and the origin in graphs and data displays.



A1.N.Q.A.1 U

se units as a way to

understand problems and to guide

the solution of multi-step problem

s; choose and interpret units consistently in form

ulas; choose and interpret the scale and the origin in graphs and data displays.

There are no assessment lim

its for this standard. The entire standard is assessed in this course.

A1.N.Q.A.2

2. Define appropriate

quantities for the purpose of descriptive m

odeling.

This standard will be assessed in

Algebra I by ensuring that some

modeling tasks (involving A

lgebra I content or securely held content

from grades 6-8) require the student

to create a quantity of interest in the situation being described (i.e., a

quantity of interest is not selected for the student by the task). For exam

ple, in a situation involving data, the student m

ight autonom

ously decide that a m

easure of center is a key variable in a situation, and then choose to

work w

ith the mean

A1.N.Q.A

.2 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m

odeling.

Descriptive m

odeling refers to understanding and interpreting graphs; identifying extraneous inform

ation; choosing appropriate units; etc. There are no assessm

ent limits for this

standard. The entire standard is assessed in this course.

A1.N.Q.A.3

3. Choose a level of accuracy

appropriate to limitations on

measurem

ent when reporting

quantities.



A1.N.Q.A.3 C

hoose a level of accuracy appropriate to lim

itations on m

easurement w

hen reporting quantities.



48

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imits

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. The

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is

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in th

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e.

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.A.2

2.

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an

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–as

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). Se

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the

stru

ctur

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an

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to id

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to

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xam

ple,

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.

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f the

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.B.3

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and

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func

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itde

fines

.

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ompl

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the

squa

re in

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adra

tic e

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ssio

n in

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form

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+Bx

+ C

whe

reA

= 1

to re

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.

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1.A.

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B.3c

:

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mpl

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owth

of b

acte

ria

can

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odel

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y ei

ther

f(t)

= 3(t+

2) o

r g

(t) =

9(3

t ) b

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se th

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pres

sion

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(t+2)

can

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have

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al-w

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text

. As

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ribed

in th

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here

is a

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and

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bout

the

situ

atio

n.

49

http://api.gmath.guru/cgi-bin/gmath?%5Cdpi%7B480%7Dx%5E4

http://api.gmath.guru/cgi-bin/gmath?%5Cdpi%7B480%7Dx%5E4

http://api.gmath.guru/cgi-bin/gmath?%5Cdpi%7B480%7Dy4

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c.Use the properties of

exponents to transformexpressions for exponentialfunctions. For exam

ple theexpression 1.15

t can berew

ritten as (1.151/12) 12t ≈

1.01212t to reveal the

approximate equivalent

monthly interest rate if the

annual rate is 15%.

c.Use the properties of exponents

to rewrite exponential expressions.

ii)Tasks are limited to exponential

expressions with integer exponents.

A1.A.APR.A.1 1.U

nderstand thatpolynom

ials form a system

analogous to the integers,nam

ely, they are closed underthe operations of addition,subtraction, and m

ultiplication;add, subtract, and m

ultiplypolynom

ials.



A1.A.APR.A.1 Understand that

polynomials form

a system

analogous to the integers, namely,

they are closed under the operations of addition, subtraction, and m

ultiplication; add, subtract, and m

ultiply polynomials.



A1.A.APR.B.3 3.Identify zeros ofpolynom

ials when suitable

factorizations are available,and use the zeros to constructa rough graph of the functio nd efined by the polynom

ial.

i)Tasks are limited to quadratic and

cubic polynomials in w

hich linearand quadratic factors are available.For exam

ple, find the zeros of (x -2)(x 2 - 9).

A1.A.APR.B.2 Identify zeros of polynom

ials when suitable

factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom

ial.

Graphing is lim

ited to linear and quadratic polynom

ials

A1.A.CED.A.1 1.C

reate equations andinequalities in one variableand use them

to solveproblem

s. Include equationsarising from

linear andquadratic functions, andsim

ple rational and

i)Tasks are limited to linear,

quadratic, or exponential equationsw

ith integer exponents.

A1.A.CED.A.1 Create equations and

inequalities in one variable and use them

to solve problems.

Tasks are limited to linear, quadratic,

or exponential equations with integer

exponents.

50

expo

nent

ial f

unct

ions

.

A1.A

.CED

.A.2

2.

Cre

ate

equa

tions

in tw

o or

m

ore

varia

bles

to re

pres

ent

rela

tions

hips

bet

wee

n qu

antit

ies;

gra

ph e

quat

ions

on

coor

dina

te a

xes

with

labe

ls

and

scal

es.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd

A1.A

.CED

.A.2

Cre

ate

equa

tions

in

two

or m

ore

varia

bles

to re

pres

ent

rela

tions

hips

bet

wee

n qu

antit

ies;

gr

aph

equa

tions

with

two

varia

bles

on

coo

rdin

ate

axes

with

labe

ls a

nd

scal

es.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

A1.A

.CED

.A.3

3.

Rep

rese

nt c

onst

rain

ts b

y eq

uatio

ns o

r ine

qual

ities

, and

by

sys

tem

s of

equ

atio

ns

and/

or in

equa

litie

s, a

nd

inte

rpre

t sol

utio

ns a

s vi

able

or

nonv

iabl

e op

tions

in a

m

odel

ing

cont

ext.

For

exam

ple,

repr

esen

t in

equa

litie

s de

scrib

ing

nutri

tiona

l and

cos

t con

stra

ints

on

com

bina

tions

of d

iffer

ent

food

s.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd

A1.A

.CED

.A.3

Rep

rese

nt

cons

train

ts b

y eq

uatio

ns o

r in

equa

litie

s an

d by

sys

tem

s of

eq

uatio

ns a

nd/o

r ine

qual

ities

, and

in

terp

ret s

olut

ions

as

viab

le o

r no

nvia

ble

optio

ns in

a m

odel

ing

cont

ext.

For e

xam

ple,

repr

esen

t ine

qual

ities

de

scrib

ing

nutri

tiona

l and

cos

t co

nstra

ints

on

com

bina

tions

of

diffe

rent

food

s.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

A1.A

.CED

.A.4

4.

Rea

rrang

e fo

rmul

as to

hi

ghlig

ht a

qua

ntity

of i

nter

est,

usin

g th

e sa

me

reas

onin

g a

sin

solv

ing

equa

tions

. For

ex

ampl

e, re

arra

nge

Ohm

’s la

w

V =

IR to

hig

hlig

ht re

sist

ance

R

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd

A1.A

.CED

.A.4

Rea

rran

ge fo

rmul

as

to h

ighl

ight

a q

uant

ity o

f int

eres

t, us

ing

the

sam

e re

ason

ing

as in

so

lvin

g eq

uatio

ns.

i) Ta

sks

are

limite

d to

linea

r, qu

adra

tic,

piec

ewis

e, a

bsol

ute

valu

e, a

nd

expo

nent

ial e

quat

ions

with

inte

ger

expo

nent

s . i

i) Ta

sks

have

a re

al-w

orld

con

text

.

A1.A

.REI

.A.1

1.

Exp

lain

eac

h st

ep in

sol

ving

a

sim

ple

equa

tion

as fo

llow

ing

from

the

equa

lity

of n

umbe

rs

asse

rted

at th

e pr

evio

us s

tep,

st

artin

g fro

m th

e as

sum

ptio

n th

at th

e or

igin

al e

quat

ion

has

a so

lutio

n. C

onst

ruct

a v

iabl

e ar

gum

ent t

o ju

stify

a s

olut

ion

met

hod.

i) Ta

sks

are

limite

d to

qua

drat

ic

equa

tions

A1

.A.R

EI.A

.1 E

xpla

in e

ach

step

in

solv

ing

an e

quat

ion

as fo

llow

ing

from

the

equa

lity

of n

umbe

rs

asse

rted

at th

e pr

evio

us s

tep,

st

artin

g fro

m th

e as

sum

ptio

n th

at th

e or

igin

al e

quat

ion

has

a so

lutio

n.

Con

stru

ct a

via

ble

argu

men

t to

just

ify a

sol

utio

n m

etho

d.

Task

s ar

e lim

ited

to li

near

, qua

drat

ic,

piec

ewis

e, a

bsol

ute

valu

e, a

nd

expo

nent

ial e

quat

ions

with

inte

ger

expo

nent

s.

51

A1.A.REI.B.3 3. Solve linear equations and inequalities in one variable, including equations w

ith coefficients represented by letters.



A1.A.REI.B.2 Solve linear equations and inequalities in one variable, including equations w




A1.A.REI.B.4 4. Solve quadratic equations in one variable. a. U

se 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 sam

e solutions. Derive

the quadratic formula from

this form

. b. Solve quadratic equations by inspection (e.g., for x

2 = 49), taking square roots, com

pleting the square, the quadratic form

ula and factoring, as appropriate to the initial form

of the equation. R

ecognize when the quadratic

formula gives com

plex solutions and w

rite them as a

± bi for real numbers a and b.

For A-REI.4b:

i) Tasks do not require students to w

rite solutions for quadratic equations that have roots w

ith nonzero im

aginary parts. How

ever, tasks can require the student to recognize cases in w

hich a quadratic equation has no real solutions. N

ote, solving a quadratic equation by factoring relies on the connection betw

een zeros and factors of polynom

ials (cluster A-APR.B).

Cluster AAPR

.B is formally

assessed in A2.

A1.A.REI.B.3 Solve quadratic equations and inequalities in one variable. a. U

se the method of com

pleting the square to rew

rite 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.

b. Solve quadratic equations by inspection (e.g., for x

2 = 49), taking square roots, com

pleting the square, know

ing and applying the quadratic form

ula, and factoring, as appropriate to the initial form

of the equation. R

ecognize when the

quadratic formula gives com

plex solutions.

For A1.A.REI.B.3b:

Tasks do not require students to write

solutions for quadratic equations that have roots w

ith nonzero imaginary

parts. How

ever, tasks can require the student to recognize cases in w

hich a quadratic equation has no real solutions. N

ote: solving a quadratic equation by factoring relies on the connection betw

een zeros and factors of polynom

ials. This is formally assessed

in Algebra II.

A1.A.REI.C.4 W

rite and solve a system

of linear equations in context.

Solve systems both algebraically and

graphically. System

s are limited to at m

ost two

equations in two variables.

A1.A.REI.C.5 5. Prove that, given a system

of tw

o equations in two

variables, replacing one equation by the sum

of that equation and a m

ultiple of the other produces a system

with

the same solutions.



52

A1.A

.REI

.C.6

6.

Sol

ve s

yste

ms

of li

near

eq

uatio

ns e

xact

ly a

nd

appr

oxim

atel

y (e

.g.,

with

gr

aphs

), fo

cusi

ng o

n pa

irs o

f lin

ear e

quat

ions

in tw

o va

riabl

es.

i) Ta

sks

have

a re

al-w

orld

con

text

. ii)

Tas

ks h

ave

hallm

arks

of

mod

elin

g as

a m

athe

mat

ical

pr

actic

e (le

ss d

efin

ed ta

sks,

mor

e of

th

e m

odel

ing

cycl

e, e

tc.).

A1.A

.REI

.D.1

0 10

. Und

erst

and

that

the

grap

h of

an

equa

tion

in tw

o va

riabl

es

is th

e se

t of a

ll its

sol

utio

ns

plot

ted

in th

e co

ordi

nate

pla

ne,

ofte

n fo

rmin

g a

curv

e (w

hich

co

uld

be a

line

).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.A

.REI

.D.5

Und

erst

and

that

the

grap

h of

an

equa

tion

in tw

o va

riabl

es

is th

e se

t of a

ll its

sol

utio

ns p

lotte

d in

th

e co

ordi

nate

pla

ne, o

ften

form

ing

a cu

rve

(whi

ch c

ould

be

a lin

e).

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e

A1.A

.REI

.D.1

1 11

. Exp

lain

why

the

x-co

ordi

nate

s of

the

poin

ts

whe

re th

e gr

aphs

of t

he

equa

tions

y =

f(x)

and

y =

g(x

) in

ters

ect a

re th

e so

lutio

ns o

f th

e eq

uatio

n f(x

) = g

(x);

find

the

solu

tions

app

roxi

mat

ely,

e.

g., u

sing

tech

nolo

gy to

gra

ph

the

func

tions

, mak

e ta

bles

of

valu

es, o

r fin

d su

cces

sive

ap

prox

imat

ions

. Inc

lude

cas

es

whe

re f(

x) a

nd/o

r g(x

) are

lin

ear,

poly

nom

ial,

ratio

nal,

abso

lute

val

ue, e

xpon

entia

l, an

d lo

garit

hmic

func

tions

. ★

i) Ta

sks

that

ass

ess

conc

eptu

al

unde

rsta

ndin

g of

the

indi

cate

d co

ncep

t may

invo

lve

any

of th

e fu

nctio

n ty

pes

men

tione

d in

the

stan

dard

exc

ept e

xpon

entia

l and

lo

garit

hmic

func

tions

. ii)

Fin

ding

the

solu

tions

ap

prox

imat

ely

is li

mite

d to

cas

es

whe

re f(

x) a

nd g

(x) a

re p

olyn

omia

l fu

nctio

ns.

A1.A

.REI

.D.6

Exp

lain

why

the

x-co

ordi

nate

s of

the

poin

ts w

here

the

grap

hs o

f the

equ

atio

ns y

= f(

x) a

nd

y =

g(x)

inte

rsec

t are

the

solu

tions

of

the

equa

tion

f(x) =

g(x

); fin

d th

e ap

prox

imat

e so

lutio

ns u

sing

te

chno

logy

. ★

Incl

ude

case

s w

here

f(x)

and

/or g

(x)

are

linea

r, qu

adra

tic, a

bsol

ute

valu

e,

and

expo

nent

ial f

unct

ions

. For

ex

ampl

e, f(

x) =

3x

+ 5

and

g(x)

= x

2 +

1.

Expo

nent

ial f

unct

ions

are

lim

ited

to

dom

ains

in th

e in

tege

rs.

A1.A

.REI

.D.1

2 12

. Gra

ph th

e so

lutio

ns to

a

linea

r ine

qual

ity in

two

varia

bles

as

a ha

lf-pl

ane

(exc

ludi

ng th

e bo

unda

ry in

the

case

of a

stri

ct in

equa

lity)

, and

gr

aph

the

solu

tion

set t

o a

syst

em o

f lin

ear i

nequ

aliti

es in

tw

o va

riabl

es a

s th

e in

ters

ectio

n of

the

corr

espo

ndin

g ha

lf-pl

anes

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.A

.REI

.D.7

Gra

ph th

e so

lutio

ns to

a

linea

r ine

qual

ity in

two

varia

bles

as

a ha

lf-pl

ane

(exc

ludi

ng th

e bo

unda

ry

in th

e ca

se o

f a s

trict

ineq

ualit

y), a

nd

grap

h th

e so

lutio

n se

t to

a sy

stem

of

linea

r ine

qual

ities

in tw

o va

riabl

es a

s th

e in

ters

ectio

n of

the

corre

spon

ding

ha

lf-pl

anes

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

53

A1.F.IF.A.1 1.U

nderstand that a functionfrom

one set (called thedom

ain) to another set (calledthe range) assigns to eachelem

ent of the domain exactly

one element of the range. If f

is a function and x is a nel em

ent of its domain, then

f(x) denotes the output of fcorresponding to the input x.The graph of f is the graph ofthe equation y = f(x).



A1.F.IF.A.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 elem

ent of the range. If f is a function and x is an elem

ent 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).



A1.F.IF.A.2 2.U

se function notation,evaluate functions for inputs intheir dom

ains, and interpretstatem

ents that use functio nnotation in term

s of a context.



A1.F.IF.A.2 Use function notation,

evaluate functions for inputs in their dom

ains, and interpret statements

that use function notation in terms of

a context.



A1.F.IF.A.3 3.R

ecognize that sequencesare functions, som

etimes

defined recursively, whose

domain is a subset of the

integers. For example, th e

F ibonacci sequence is definedrecursively by f(0) = f(1) = 1,f(n+1) = f(n) + f(n-1) for n ≥ 1.

i)This standard is part of the Major

work in Algebra I and w

ill beassessed accordingly.

A1.F.IF.B.4 4.For a function that m

odels arelationship betw

een two

quantities, interpret keyfeatures of graphs and tablesin term

s of the quantities, andsketch graphs show

ing keyfeatures given a verbaldescription of the relationship.Key features include:intercepts; intervals w

here th ef unction is increasing,decreasing, positive, ornegative; relative m

aximum

s

i)Tasks have a real-world context.

ii)Tasks are limited to linear

functions, quadratic functions,square root functions, cube rootfunctions, piecew

ise-definedf unctions (including step functionsand absolute value functions), andexponential functions w

ith domains

in the integers.

Com

pare note (ii) with standard F-

IF.7. The function types listed here are the sam

e as those listed in the

A1.F.IF.B.3 For a function that m

odels a relationship between tw

o quantities, interpret key features of graphs and tables in term

s of the quantities, and sketch graphs show

ing key features given a verbal description of the relationship. ★

Key features include: intercepts; intervals w

here the function is increasing, decreasing, positive, or negative; relative m

aximum

s and m

inimum

s; symm

etries; and end behavior.


ii)Tasks are limited to linear functions,

quadratic functions, absolute val uef unctions, and exponential functionsw

ith domains in the integers.

54

and

min

imum

s; s

ymm

etrie

s;

end

beha

vior

; and

per

iodi

city

. ★

Alge

bra

I col

umn

for s

tand

ards

F-

IF.6

and

F-IF

.9.

A1.F

.IF.B

.5

5. R

elat

e th

e do

mai

n of

a

func

tion

to it

s gr

aph

and,

w

here

app

licab

le, t

o th

e qu

antit

ativ

e re

latio

nshi

p it

desc

ribes

. For

exa

mpl

e, if

the

func

tion

h(n)

giv

es th

e nu

mbe

r of

per

son-

hour

s it

take

s to

as

sem

ble

n en

gine

s in

a

fact

ory,

then

the

posi

tive

inte

gers

wou

ld b

e an

ap

prop

riate

dom

ain

for t

he

func

tion.★

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.F

.IF.B

.4 R

elat

e th

e do

mai

n of

a

func

tion

to it

s gr

aph

and,

whe

re

appl

icab

le, t

o th

e qu

antit

ativ

e re

latio

nshi

p it

desc

ribes

. ★

For e

xam

ple,

if th

e fu

nctio

n h(

n) g

ives

th

e nu

mbe

r of p

erso

n-ho

urs

it ta

kes

to

asse

mbl

e n

engi

nes

in a

fact

ory,

then

th

e po

sitiv

e in

tege

rs w

ould

be

an

appr

opria

te d

omai

n fo

r the

func

tion.

T

here

are

no

asse

ssm

ent l

imits

for

this

sta

ndar

d. T

he e

ntire

sta

ndar

d is

as

sess

ed in

this

cou

rse.

A1.F

.IF.B

.6

6. C

alcu

late

and

inte

rpre

t the

av

erag

e ra

te o

f cha

nge

of a

fu

nctio

n (p

rese

nted

sy

mbo

lical

ly o

r as

a ta

ble)

ov

er a

spe

cifie

d in

terv

al.

Estim

ate

the

rate

of c

hang

e fro

m a

gra

ph.★

i) Ta

sks

have

a re

al-w

orld

con

text

. ii)

Tas

ks a

re lim

ited

to li

near

fu

nctio

ns, q

uadr

atic

func

tions

, sq

uare

root

func

tions

, cub

e ro

ot

func

tions

, pie

cew

ise-

defin

ed

func

tions

(inc

ludi

ng s

tep

func

tions

an

d ab

solu

te v

alue

func

tions

), an

d ex

pone

ntia

l fun

ctio

ns w

ith d

omai

ns

in th

e in

tege

rs.

The

func

tion

type

s lis

ted

here

are

th

e sa

me

as th

ose

liste

d in

the

Alge

bra

I col

umn

for s

tand

ards

F-

IF.4

and

F-IF

.9.

A1.F

.IF.B

.5 C

alcu

late

and

inte

rpre

t th

e av

erag

e ra

te o

f cha

nge

of a

fu

nctio

n (p

rese

nted

sym

bolic

ally

or

as a

tabl

e) o

ver a

spe

cifie

d in

terv

al.

Estim

ate

the

rate

of c

hang

e fro

m a

gr

aph.★

i) Ta

sks

have

a re

al-w

orld

con

text

. ii

) Tas

ks a

re li

mite

d to

line

ar fu

nctio

ns,

quad

ratic

func

tions

, pie

cew

ise-

defin

ed

func

tions

(inc

ludi

ng s

tep

func

tions

and

ab

solu

te v

alue

func

tions

), an

d ex

pone

ntia

l fun

ctio

ns w

ith d

omai

ns in

th

e in

tege

rs.

A1.F

.IF.C

.7

7. G

raph

func

tions

exp

ress

ed

sym

bolic

ally

and

sho

w k

ey

feat

ures

of t

he g

raph

, by

hand

in

sim

ple

case

s an

d us

ing

tech

nolo

gy fo

r mor

e co

mpl

icat

ed c

ases

. ★

a. G

raph lin

ear

and

quad

ratic

func

tions

and

sho

w

inte

rcep

ts, m

axim

a, a

nd

min

ima.

b

. Gra

ph s

quar

e ro

ot, c

ube

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.F

.IF.C

.6 G

raph

func

tions

ex

pres

sed

sym

bolic

ally

and

sho

w

key

feat

ures

of t

he g

raph

, by

hand

an

d us

ing

tech

nolo

gy.

a. G

raph

line

ar a

nd q

uadr

atic

fu

nctio

ns a

nd s

how

inte

rcep

ts,

max

ima,

and

min

ima.

b

. Gra

ph s

quar

e ro

ot, c

ube

root

, an

d pi

ecew

ise-

defin

ed fu

nctio

ns,

incl

udin

g st

ep fu

nctio

ns a

nd

abso

lute

val

ue fu

nctio

ns.

Task

s in

A1.

F.IF

.C.6

b ar

e lim

ited

to

piec

ewis

e, s

tep

and

abso

lute

val

ue

func

tions

.

55

root, and piecewise-defined

functions, including step functions and absolute value functions

A1.F.IF.C.8 8. W

rite a function defined by an expression in different but equivalent form

s to reveal and explain different properties of the function. a. U

se the process of factoring and com

pleting the square in a quadratic function to show

zeros, extrem

e values, and sym

metry of the graph, and

interpret these in terms of a

context.



A1.F.IF.C.7 Write a function defined

by an expression in different but equivalent form

s to reveal and explain different properties of the function. a. U

se the process of factoring and com

pleting the square in a quadratic function to show

zeros, extrem

e values, and sym

metry of the graph,

and interpret these in term

s of a context.



A1.F.IF.C.9 9. C

ompare properties of tw

o functions each represented in a different w

ay (algebraically, graphically, num

erically in tables, or by verbal descriptions). For exam

ple, given a graph of one quadratic function and an algebraic expression for another, say w

hich has the larger m

aximum

.

i) Tasks are limited to linear

functions, quadratic functions, square root functions, cube root functions, piecew

ise-defined functions (including step functions and absolute value functions), and exponential functions w

ith domains

in the integers. The function types listed here are the sam

e as those listed in the Algebra I colum

n for standards F-IF.4 and F-IF.6.

A1.F.IF.C.8 Com

pare properties of tw

o functions each represented in a different w

ay (algebraically, graphically, num

erically in tables, or by verbal descriptions).

i) Tasks have a real-world context.

ii) Tasks are limited to linear functions,

quadratic functions, piecewise-defined

functions (including step functions and absolute value functions), and exponential functions w

ith domains in

the integers.

A1.F.BF.A.1 1. W

rite a function that describes a relationship betw

een two quantities.

★ a

. Dete

rmin

e a

n e

xplicit

expression, a recursive process, or steps for calculation from

a context.


ii) Tasks are limited to linear

functions, quadratic functions, and exponential functions w

ith domains

in the integers.

A1.F.BF.A.1 Write a function that

describes a relationship between

two quantities.★

a. D

etermine an explicit


a context.


ii) Tasks are limited to linear functions,

quadratic functions, and exponential functions w


56

A1.F

.BF.

B.3

3. Id

entif

y th

e ef

fect

on

the

grap

h of

repl

acin

g f(x

) by

f(x) +

k,

k f(

x), f

(kx)

, and

f(x

+ k)

for

spec

ific

valu

es o

f k (b

oth

posi

tive

and

nega

tive)

; fin

d th

e va

lue

of k

giv

en th

e gr

aphs

. Ex

perim

ent w

ith c

ases

and

ill

ustra

te a

n ex

plan

atio

n of

the

effe

cts

on th

e gr

aph

usin

g te

chno

logy

. Inc

lude

re

cogn

izin

g ev

en a

nd o

dd

func

tion

from

thei

r gra

phs

and

alge

brai

c ex

pres

sion

s fo

r th

em.

i) Id

entif

ying

the

effe

ct o

n th

e gr

aph

of re

plac

ing

f(x) b

y f(x

) + k

, k f(

x),

f(kx)

, and

f(x+

k) fo

r spe

cific

val

ues

of k

(bot

h po

sitiv

e an

d ne

gativ

e) is

lim

ited

to li

near

and

qua

drat

ic

func

tions

. ii

) Exp

erim

entin

g w

ith c

ases

and

illu

stra

ting

an e

xpla

natio

n of

the

effe

cts

on th

e gr

aph

usin

g te

chno

logy

is li

mite

d to

linea

r fu

nctio

ns, q

uadr

atic

func

tions

, sq

uare

root

func

tions

, cub

e ro

ot

func

tions

, pie

cew

ise-

defin

ed

func

tions

(inc

ludi

ng s

tep

func

tions

an

d ab

solu

te v

alue

func

tions

), an

d ex

pone

ntia

l fun

ctio

ns w

ith d

omai

ns

in th

e in

tege

rs.

iii) T

asks

do

not i

nvol

ve re

cogn

izin

g ev

en a

nd o

dd fu

nctio

ns.

The

func

tion

type

s lis

ted

in n

ote

(ii)

are

the

sam

e as

thos

e lis

ted

in th

e Al

gebr

a I c

olum

n fo

r sta

ndar

ds F

-IF

.4, F

-IF.6

, and

F-IF

.9.

A1.F

.BF.

B.2

Iden

tify

the

effe

ct o

n th

e gr

aph

of re

plac

ing

f(x) b

y f(x

) + k

, k

f(x),

f(kx)

, and

f(x

+ k)

for s

peci

fic

valu

es o

f k (b

oth

posi

tive

and

nega

tive)

; fin

d th

e va

lue

of k

giv

en

the

grap

hs. E

xper

imen

t with

cas

es

and

illus

trate

an

expl

anat

ion

of th

e ef

fect

s on

the

grap

h us

ing

tech

nolo

gy.

i) Id

entif

ying

the

effe

ct o

n th

e gr

aph

of

repl

acin

g f(x

) by

f(x) +

k, k

f(x)

, and

f(x

+k) f

or s

peci

fic v

alue

s of

k (b

oth

posi

tive

and

nega

tive)

is li

mite

d to

lin

ear,

quad

ratic

, and

abs

olut

e va

lue

func

tions

. ii

) f(k

x) w

ill no

t be

incl

uded

in A

lgeb

ra

1. It

is a

ddre

ssed

in A

lgeb

ra 2

. ii

i) Ex

perim

entin

g w

ith c

ases

and

illu

stra

ting

an e

xpla

natio

n of

the

effe

cts

on th

e gr

aph

usin

g te

chno

logy

is

limite

d to

line

ar fu

nctio

ns, q

uadr

atic

fu

nctio

ns, a

bsol

ute

valu

e, a

nd

expo

nent

ial f

unct

ions

with

dom

ains

in

the

inte

gers

. iv

) Tas

ks d

o no

t inv

olve

reco

gniz

ing

even

and

odd

func

tions

.

A1.F

.LE.

A.1

1. D

istin

guis

h be

twee

n si

tuat

ions

that

can

be

mod

eled

w

ith li

near

func

tions

and

with

ex

pone

ntia

l fun

ctio

ns.

a. P

rove

that

line

ar fu

nctio

ns

grow

by

equa

l diff

eren

ces

over

eq

ual i

nter

vals

, and

that

ex

pone

ntia

l fun

ctio

ns g

row

by

equa

l fac

tors

ove

r equ

al

inte

rval

s.

b. R

ecog

nize

situ

atio

ns in

w

hich

one

qua

ntity

cha

nges

at

a co

nsta

nt ra

te p

er u

nit

inte

rval

rela

tive

to a

noth

er.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.F

.LE.

A.1

Dis

tingu

ish

betw

een

situ

atio

ns th

at c

an b

e m

odel

ed w

ith

linea

r fun

ctio

ns a

nd w

ith e

xpon

entia

l fu

nctio

ns.

a. R

ecog

nize

that

line

ar fu

nctio

ns

grow

by

equa

l diff

eren

ces

over

equ

al

inte

rval

s an

d th

at e

xpon

entia

l fu

nctio

ns g

row

by

equa

l fac

tors

ove

r eq

ual i

nter

vals

. b.

Rec

ogni

ze s

ituat

ions

in w

hich

one

qu

antit

y ch

ange

s at

a c

onst

ant r

ate

per u

nit i

nter

val r

elat

ive

to a

noth

er.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

57

c. Recognize situations in

which a quantity grow

s or decays by a constant percent rate per unit interval relative to another.

c. Recognize situations in w

hich a quantity grow

s or decays by a constant factor per unit interval relative to another.

A1.F.LE.A.2 2. C

onstruct linear and exponential functions, including arithm

etic and geom

etric sequences, given a graph, a description of a relationship, or tw

o input-output pairs (include reading these from

a table).

i) Tasks are limited to constructing

linear and exponential functions in sim

ple context (not multi-step).

A1.F.LE.A.2 Construct linear and

exponential functions, including arithm

etic and geometric sequences,

given a graph, a table, a description of a relationship, or input-output pairs.

Tasks are limited to constructing linear

and exponential functions in simple

context (not multi-step).

A1.F.LE.A.3 3. O

bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m

ore generally) as a polynom

ial function.



A1.F.LE.A.3 Observe using graphs

and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (m


ial function.



A1.F.LE.B.5 5. Interpret the param

eters in a linear or exponential function in term

s of a context.


ii) Exponential functions are limited

to those with dom

ains in the integers.

F.LE.B.4 Interpret the parameters in

a linear or exponential function in term

s of a context.

For example, the total cost of an

electrician who charges 35 dollars for a

house call and 50 dollars per hour w

ould be expressed as the function y = 50x + 35. If the rate w

ere raised to 65 dollars per hour, describe how

the function w

ould change. i) Tasks have a real-w

orld context. ii) Exponential functions are lim

ited to those w


58

A1.S

.ID.A

.1

1. R

epre

sent

dat

a w

ith p

lots

on

the

real

num

ber l

ine

(dot

pl

ots,

his

togr

ams,

and

box

pl

ots)

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.S

.ID.A

.1 R

epre

sent

sin

gle

or

mul

tiple

dat

a se

ts w

ith d

ot p

lots

, hi

stog

ram

s, s

tem

plo

ts (s

tem

and

le

af),

and

box

plot

s.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

A1.S

.ID.A

.2

2. U

se s

tatis

tics

appr

opria

te to

th

e sh

ape

of th

e da

ta

dist

ribut

ion

to c

ompa

re c

ente

r (m

edia

n, m

ean)

and

spr

ead

(inte

rqua

rtile

rang

e, s

tand

ard

devi

atio

n) o

f tw

o or

mor

e di

ffere

nt d

ata

sets

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.S

.ID.A

.2 U

se s

tatis

tics

appr

opria

te to

the

shap

e of

the

data

di

strib

utio

n to

com

pare

cen

ter

(med

ian,

mea

n) a

nd s

prea

d (in

terq

uarti

le ra

nge,

sta

ndar

d de

viat

ion)

of t

wo

or m

ore

diffe

rent

da

ta s

ets.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

A1.S

.ID.A

.3

3. In

terp

ret d

iffer

ence

s in

sh

ape,

cen

ter,

and

spre

ad in

th

e co

ntex

t of t

he d

ata

sets

, ac

coun

ting

for p

ossi

ble

effe

cts

of e

xtre

me

data

poi

nts

(out

liers

).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.S

.ID.A

.3 In

terp

ret d

iffer

ence

s in

sh

ape,

cen

ter,

and

spre

ad in

the

cont

ext o

f the

dat

a se

ts, a

ccou

ntin

g fo

r pos

sibl

e ef

fect

s of

ext

rem

e da

ta

poin

ts (o

utlie

rs).

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is

asse

ssed

in th

is c

ours

e.

A1.S

.ID.B

.5

5. S

umm

ariz

e ca

tego

rical

dat

a fo

r tw

o ca

tego

ries

in tw

o-w

ay

frequ

ency

tabl

es. I

nter

pret

re

lativ

e fre

quen

cies

in th

e co

ntex

t of t

he d

ata

(incl

udin

g jo

int,

mar

gina

l, an

d co

nditi

onal

re

lativ

e fre

quen

cies

). R

ecog

nize

pos

sibl

e as

soci

atio

ns a

nd tr

ends

in th

e da

ta

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arific

atio

n in

form

atio

n fo

r thi

s st

anda

rd.

A1.S

.ID.B

.6

6. R

epre

sent

dat

a on

two

quan

titat

ive

varia

bles

on

a sc

atte

r plo

t, an

d de

scrib

e ho

w

the

varia

bles

are

rela

ted.

a.

Fit

a fu

nctio

n to

the

data

; us

e fu

nctio

ns fi

tted

to d

ata

to

solv

e pr

oble

ms

in th

e co

ntex

t of

the

data

. Use

giv

en

func

tions

or c

hoos

e a

func

tion

sugg

este

d by

the

cont

ext.

For S

-ID.6

a:

i) T

asks

hav

e a

real

-wor

ld c

onte

xt.

ii) E

xpon

entia

l fun

ctio

ns a

re li

mite

d to

thos

e w

ith d

omai

ns in

the

inte

gers

.

A1.S

.ID.B

.4 R

epre

sent

dat

a on

two

quan

titat

ive

varia

bles

on

a sc

atte

r pl

ot, a

nd d

escr

ibe

how

the

varia

bles

ar

e re

late

d.

a. F

it a

func

tion

to th

e da

ta; u

se

func

tions

fitte

d to

dat

a to

sol

ve

prob

lem

s in

the

cont

ext o

f the

dat

a.

Use

giv

en fu

nctio

ns o

r cho

ose

a fu

nctio

n su

gges

ted

by th

e co

ntex

t.

Emph

asiz

e lin

ear m

odel

s, q

uadr

atic

m

odel

s, a

nd e

xpon

entia

l mod

els

with

do

mai

ns in

the

inte

gers

. F

or A

1.S.

ID.B

.4a:

i)

Tas

ks h

ave

a re

al-w

orld

con

text

. ii

) Exp

onen

tial f

unct

ions

are

lim

ited

to

thos

e w

ith d

omai

ns in

the

inte

gers

.

59

Emphasize linear, quadratic,

and exponential models.

b. Informally assess the fit of

a function by plotting andanalyzing residuals.

c.Fit a linear function for ascatter plot that suggests alinear association.

b.Fit a linear function for a scatterplot that suggests a linearassociation.

A1.S.ID.C.7 7.Interpret the slope (rate ofchange) and the intercept(constant term

) of a linearm

odel in the context of th edata.



A1.S.ID.C.5 Interpret the slope (rate of change) and the intercept (constant term

) of a linear model in

the context of the data.



A1.S.ID.C.8 8.C

ompute (using technology)

and interpret the correlationcoefficient of a linear fit.



A1.S.ID.C.6 Use technology to

compute and interpret the correlation

coefficient of a linear fit.



A1.S.ID.C.9 9.D

istinguish between

correlation and causation.There is no additional scope or clarification inform


A1.S.ID.C.7 Distinguish betw

een correlation and causation.



Major W

ork of the Grade

Supporting Work

60

Not

atio

n TN

Sta

ndar

ds th

roug

h M

ay 2

017

Stan

dard

Cla

rific

atio

n Re

vise

d TN

Sta

ndar

ds

Stan

dard

Cla

rifi

cati

on

A2.N

.RN.

A.1

1.

Exp

lain

how

the

defin

ition

of

the

mea

ning

of r

atio

nal

expo

nent

s fo

llow

s fro

m

exte

ndin

g th

e pr

oper

ties

of

inte

ger e

xpon

ents

to th

ose

valu

es, a

llow

ing

for a

not

atio

n fo

r rad

ical

s in

term

s of

ratio

nal

expo

nent

s. F

or e

xam

ple,

we

defin

e 51/

3 to

be

the

cube

root

of

5 b

ecau

se w

e w

ant (

51/3 )3 =

5 (

1/3 )3 t

o ho

ld, s

o (5

1/3 )3 m

ust

equa

l 5.

Th

ere

is n

o ad

ditio

nal s

cope

or

clar

ifica

tion

info

rmat

ion

for t

his

stan

dard

.

A2.N

.RN.

A.1

Expl

ain

how

the

defin

ition

of t

he m

eani

ng o

f rat

iona

l ex

pone

nts

follo

ws

from

ext

endi

ng

the

prop

ertie

s of

inte

ger e

xpon

ents

to

thos

e va

lues

, allo

win

g fo

r a

nota

tion

for r

adic

als

in te

rms

of

ratio

nal e

xpon

ents

.

For e

xam

ple,

we

defin

e 51/

3 to

be th

e cu

be ro

ot

of 5

bec

ause

we

wan

t (51/

3 )3 =

5(⅓

)3 to

hol

d, s

o (5

1/3 )

3 mus

t equ

al 5

. Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd. T

he e

ntire

sta

ndar

d is

ass

esse

d in

th

is c

ours

e.

A2.N

.RN.

A.2

2.

Rew

rite

expr

essi

ons

invo

lvin

g ra

dica

ls a

nd ra

tiona

l ex

pone

nts

usin

g th

e pr

oper

ties

of e

xpon

ents

.

Th

ere

is n

o ad

ditio

nal s

cope

or

clar

ifica

tion

info

rmat

ion

for t

his

stan

dard

.

A2.N

.RN.

A.2

Rew

rite

expr

essi

ons

invo

lvin

g ra

dica

ls a

nd ra

tiona

l ex

pone

nts

usin

g th

e pr

oper

ties

of

expo

nent

s.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is a

sses

sed

in

this

cou

rse.

A2.N

.NQ

.A.2

2.

Def

ine

appr

opria

te

quan

titie

s fo

r the

pur

pose

of

desc

riptiv

e m

odel

ing.

This

sta

ndar

d w

ill be

ass

esse

d in

A

lgeb

ra II

by

ensu

ring

that

som

e m

odel

ing

task

s (in

volv

ing

Alg

ebra

II

cont

ent o

r sec

urel

y he

ld c

onte

nt

from

pre

viou

s gr

ades

and

co

urse

s) re

quire

the

stud

ent t

o cr

eate

a q

uant

ity o

f int

eres

t in

the

situ

atio

n be

ing

desc

ribed

(i.e

., th

is

is n

ot p

rovi

ded

in th

e ta

sk).

For

exam

ple,

in a

situ

atio

n in

volv

ing

perio

dic

phen

omen

a, th

e st

uden

t m

ight

aut

onom

ousl

y de

cide

that

am

plitu

de is

a k

ey v

aria

ble

in a

si

tuat

ion,

and

then

cho

ose

to w

ork

with

pea

k am

plitu

de.

A2.N

Q.A

.1 Id

entif

y, in

terp

ret,

and

just

ify a

ppro

pria

te q

uant

ities

for t

he

purp

ose

of d

escr

iptiv

e m

odel

ing.

Des

crip

tive

mod

elin

g re

fers

to u

nder

stan

ding

an

d in

terp

retin

g gr

aphs

; ide

ntify

ing

extra

neou

s in

form

atio

n; c

hoos

ing

appr

opria

te u

nits

; etc

. Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd. T

he e

ntire

sta

ndar

d is

ass

esse

d in

th

is c

ours

e.

A2.N

.CN.

A.1

1. K

now

ther

e is

a c

ompl

ex

num

ber i

suc

h th

at i2 =

–1,

and

ev

ery

com

plex

num

ber h

as th

e fo

rm a

+ b

i with

a a

nd b

real

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.N

.CN.

A.1

Know

ther

e is

a

com

plex

num

ber i

suc

h th

at i2 =

–1,

an

d ev

ery

com

plex

num

ber h

as th

e fo

rm a

+ b

i with

a a

nd b

real

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire s

tand

ard

is a

sses

sed

in

this

cou

rse.

61

A2.N.CN.A.2 2. U

se the relation i 2 = –1 and the com

mutative, associative,

and distributive properties to add, subtract, and m

ultiply com

plex numbers.



A2.N.CN.A.2 Know and use the

relation i 2= –1 and the com

mutative, associative, and

distributive properties to add, subtract, and m

ultiply complex

numbers.



A2.N.CN.B.7 7. Solve quadratic equations w

ith real coefficients that have com

plex solutions.



A2.N.CN.B.3 Solve quadratic equations w

ith real coefficients that have com

plex solutions.



A2.A.SSE.A.2 2. U

se the structure of an expression to identify w

ays to rew

rite it. For example, see

x 4 – y 4 as (x 2) 2 – (y 2) 2, thus recognizing it as a difference of squares that can be factored as (x 2 – y 2)(x 2 + y 2).

i) Tasks are limited to polynom

ial, rational, or exponential expressions. ii) E

xamples: see x

4 – y4 as (x

2 ) 2 – (y 2) 2 , thus recognizing it as a difference of squares that can be factored as (x

2 – y2 )(x

2 + y2 ). In

the equation x2 + 2x + 1 + y

2 = 9, see an opportunity to rew

rite the first three term

s as (x+1) 2 , thus recognizing the equation of a circle w

ith radius 3 and center (-1, 0). S

ee (x2 + 4)/(x

2 + 3) as ( (x 2+3) + 1 )/(x

2 +3), thus recognizing an opportunity to w

rite it as 1 + 1/(x

2 +3).

A2.A.SSE.A.1 Use the structure of

an expression to identify ways to

rewrite it.

For example, see 2x

4 + 3x2 – 5 as its factors (x

2 – 1) and (2x2 + 5); see x

4 – y4 as (x 2 ) 2 – (y 2 ) 2 , thus

recognizing it as a difference of squares that can be factored as (x 2 – y 2 ) (x

2 + y 2 ); see (x 2 + 4)/(x 2 + 3) as ((x 2+ 3) + 1 )/(x

2 + 3), thus recognizing an opportunity to w

rite it as 1 + 1/(x 2 + 3). Tasks are lim

ited to polynom

ial, rational, or exponential expressions.

A2.A.SSE.B.3 3. C

hoose and produce an equivalent form

of an expression to reveal and explain properties of the quantity represented by the expression.★

c. U

se the properties of exponents to transform

expressions for exponential


As described in the standard,

there is an interplay between the

mathem

atical structure of the expression and the structure of the situation such that choosing and producing an equivalent form

of the expression reveals som

ething about the situation. ii) Tasks are lim

ited to exponential

A2.A.SSE.B.2 Choose and

produce an equivalent form of an

expression to reveal and explain properties of the quantity represented by the expression.★

a. U

se the properties of exponents to rew

rite expressions for exponential functions.

For example the expression 1.15

t can be rewritten as

((1.15) 1/12) 12t ≈ 1.01212t to reveal that the

approximate equivalent m

onthly interest rate is 1.2%

if the annual rate is 15%.

i) Tasks have a real-world context. A

s described in the standard, there is an interplay betw

een the m

athematical structure of the expression and the

structure of the situation such that choosing and producing an equivalent form

of the expression

62

func

tions

. For

exa

mpl

e th

e ex

pres

sion

1.15

t can

be

rew

ritte

n as

(1.1

51/12

)12t ≈

1.01

212t t

o re

veal

the

appr

oxim

ate

equi

vale

nt m

onth

ly in

tere

st ra

te

if th

e an

nual

rate

is 1

5%.

expr

essi

ons

with

ratio

nal o

r rea

l ex

pone

nts.

re

veal

s so

met

hing

abo

ut th

e si

tuat

ion.

i i)Ta

sks

are

limite

d to

exp

onen

tial e

xpre

ssio

ns w

ithra

tiona

l or r

eal e

xpon

ents

.

A2.A

.SSE

.B.4

4.

Der

ive

the

form

ula

for t

hesu

m o

f a fi

nite

geo

met

ricse

ries

(whe

n th

e co

mm

on ra

ti ois

not

1),

and

use

the

form

ula

to s

olve

pro

blem

s. F

or e

xam

ple,

calcu

late

mor

tgag

e pa

ymen

ts.★

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.A

.SSE

.B.3

Rec

ogni

ze a

fini

te

geom

etric

ser

ies

(whe

n th

e co

mm

on ra

tio is

not

1),

and

know

an

d us

e th

e su

m fo

rmul

a to

sol

ve

prob

lem

s in

con

text

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.A

.APR

.A.2

2.

Know

and

app

ly th

eR

emai

nder

The

orem

: For

apo

lyno

mia

l p(x

) and

a n

umbe

ra,

the

rem

aind

er o

n di

visi

on b

yx

– a

is p

(a),

so p

(a) =

0 if

and

only

if (x

– a

) is

a fa

ctor

of p

(x).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.A

.APR

.A.1

Kno

w a

nd a

pply

the

Rem

aind

er T

heor

em: F

or a

po

lyno

mia

l p(x

) and

a n

umbe

r a,

the

rem

aind

er o

n di

visi

on b

y x

– a

is p

(a),

so p

(a) =

0 if

and

onl

y if

(x –

a)

is a

fact

or o

f p(x

).

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.A

.APR

.A.3

3.

Iden

tify

zero

s of

poly

nom

ials

whe

n su

itabl

ef a

ctor

izat

ions

are

ava

ilabl

e,an

d us

e th

e ze

ros

to c

onst

ruct

a ro

ugh

grap

h of

the

func

tion

defin

ed b

y th

e po

lyno

mia

l.

i)Ta

sks

incl

ude

quad

ratic

, cub

ic,

and

quar

tic p

olyn

omia

ls a

ndpo

lyno

mia

ls fo

r whi

ch fa

ctor

s ar

eno

t pro

vide

d. F

or e

xam

ple,

find

the

zero

s of

(x2 -

1)(

x2 + 1

)

A2.A

.APR

.A.2

Iden

tify

zero

s of

po

lyno

mia

ls w

hen

suita

ble

fact

oriz

atio

ns a

re a

vaila

ble,

and

us

e th

e ze

ros

to c

onst

ruct

a ro

ugh

grap

h of

the

func

tion

defin

ed b

y th

e po

lyno

mia

l.

Task

s in

clud

e qu

adra

tic, c

ubic

, and

qua

rtic

poly

nom

ials

and

pol

ynom

ials

for w

hich

fact

ors

are

not p

rovi

ded.

For

exa

mpl

e, fi

nd th

e ze

ros

of

(x 2

- 1)(

x 2 +

1).

A2.A

.APR

.B.4

4.

Prov

e po

lyno

mia

l ide

ntiti

esan

d us

e th

em to

des

crib

enu

mer

ical

rela

tions

hips

. For

ex

ampl

e, th

e po

lyno

mia

l ide

ntity

(x

2 +

y2 ) 2 =

(x2 –

y2 )

2 +(2

xy) 2

can

be u

sed

to g

ener

ate

Pyth

agor

ean

trip

les.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.A

.APR

.B.3

Kno

w a

nd u

se

poly

nom

ial i

dent

ities

to d

escr

ibe

num

eric

al re

latio

nshi

ps.

For e

xam

ple,

com

pare

(31)

(29)

= (3

0 +

1) (3

0 –

1) =

30

2 –

1 2

with

(x +

y) (

x –

y) =

x2 –

y 2 .

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.A

.APR

.C.6

6.

Rew

rite

sim

ple

ratio

nal

expr

essi

ons

in d

iffer

ent f

orm

s;w

rite

a(x)

/b(x

) in

the

form

q(x

) +r(

x)/b

(x),

whe

re a

(x),

b(x)

, q(x

),an

d r(x

) are

pol

ynom

ials

with

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.A

.APR

.C.4

Rew

rite

ratio

nal

expr

essi

ons

in d

iffer

ent f

orm

s.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

63

the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the m

ore complicated

examples, a com

puter algebra system

.

A2.A.CED.A.1 1. C

reate equations and inequalities in one variable and use them

to solve problem

s. Include equations arising from

linear and quadratic functions, and sim

ple rational and exponential functions.

i) Tasks are limited to exponential

equations with rational or real

exponents and rational functions. ii) Tasks have a real-w

orld context.

A2.A.CED.A.1 Create equations

and inequalities in one variable and use them

to solve problems.

Include equations arising from linear and quadratic

functions, and rational and exponential functions. Tasks have a real-w

orld context.

A2.A.CED.A.2 R

earrange formulas

to highlight a quantity of interest, using the sam

e reasoning as in solving equations.

i) Tasks are limited to square root, cube root,

polynomial, rational, and logarithm

ic functions. ii) Tasks have a real-w

orld context

A2.A.REI.A.1 1. Explain each step in solving a sim

ple equation as following

from the equality of num

bers asserted at the previous step, starting from

the assumption

that the original equation has a solution. C

onstruct a viable argum

ent to justify a solution m

ethod.

i) Tasks are limited to sim

ple rational or radical equations

A2.A.REI.A.1 Explain each step in solving an equation as follow

ing from

the equality of numbers

asserted at the previous step, starting from

the assumption that

the original equation has a solution. C

onstruct a viable argument to

justify a solution method.

Tasks are limited to square root, cube root,

polynomial, rational, and logarithm

ic functions.

A2.A.REI.A.2 2. Solve sim

ple rational and radical equations in one variable, and give exam

ples show

ing how extraneous

solutions may arise.



A2.A.REI.A.2 Solve rational and radical equations in one variable, and identify extraneous solutions w

hen they exist.



A2.A.REI.B.4 4. Solve quadratic equations in

i) In the case of equations that have roots w

ith nonzero imaginary

A2.A.REI.B.3 Solve quadratic In the case of equations that have roots w

ith nonzero im

aginary parts, students write the solutions as a ±

64

one

varia

ble.

b.

Sol

ve q

uadr

atic

equ

atio

ns

by in

spec

tion

(e.g

., fo

r x2 =

49

), ta

king

squ

are

root

s,

com

plet

ing

the

squa

re, t

he

quad

ratic

form

ula

and

fact

orin

g, a

s ap

prop

riate

to th

e in

itial

form

of t

he e

quat

ion.

R

ecog

nize

whe

n th

e qu

adra

tic

form

ula

give

s co

mpl

ex

solu

tions

and

writ

e th

em a

s a

± bi

for r

eal n

umbe

rs a

and

b.

parts

, stu

dent

s w

rite

the

solu

tions

as

a ±

bi f

or re

al n

umbe

rs

a an

d b.

equa

tions

and

ineq

ualit

ies

in o

ne

varia

ble.

a.

Sol

ve q

uadr

atic

equ

atio

ns b

y in

spec

tion

(e.g

., fo

r x 2 =

49)

, tak

ing

squa

re ro

ots,

com

plet

ing

the

squa

re, k

now

ing

and

appl

ying

the

quad

ratic

form

ula,

and

fact

orin

g, a

s ap

prop

riate

to th

e in

itial

form

of t

he

equa

tion.

Rec

ogni

ze w

hen

the

quad

ratic

form

ula

give

s co

mpl

ex

solu

tions

and

writ

e th

em a

s a

± bi

fo

r rea

l num

bers

a a

nd b

.

bi fo

r rea

l num

bers

a a

nd b

A2.A

.REI

.C.6

6.

Sol

ve s

yste

ms

of li

near

eq

uatio

ns e

xact

ly a

nd

appr

oxim

atel

y (e

.g.,

with

gr

aphs

), fo

cusi

ng o

n pa

irs o

f lin

ear e

quat

ions

in tw

o va

riabl

es.

i) Ta

sks

are

limite

d to

3x3

sy

stem

s.

A2.A

.REI

.C.4

Writ

e an

d so

lve

a sy

stem

of l

inea

r equ

atio

ns in

co

ntex

t.

Whe

n so

lvin

g al

gebr

aica

lly, t

asks

are

lim

ited

to

syst

ems

of a

t mos

t thr

ee e

quat

ions

and

thre

e va

riabl

es. W

ith g

raph

ic s

olut

ions

, sys

tem

s ar

e lim

ited

to o

nly

two

varia

bles

.

A2.A

.REI

.C.7

7.

Sol

ve a

sim

ple

syst

em

cons

istin

g of

a li

near

equ

atio

n an

d a

quad

ratic

equ

atio

n in

tw

o va

riabl

es a

lgeb

raic

ally

and

gr

aphi

cally

. For

exa

mpl

e, fi

nd

the

poin

ts o

f int

erse

ctio

n be

twee

n th

e lin

e y

= –3

x and

the

circ

le x

2 + y

2 = 3

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s st

anda

rd.

A2.A

.REI

.C.5

Sol

ve a

sys

tem

co

nsis

ting

of a

line

ar e

quat

ion

and

a qu

adra

tic e

quat

ion

in tw

o va

riabl

es a

lgeb

raic

ally

and

gr

aphi

cally

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.A

.REI

.D.1

1 11

. Exp

lain

why

the

x-co

ordi

nate

s of

the

poin

ts

whe

re th

e gr

aphs

of t

he

equa

tions

y

= f(x

) and

y =

g(x

) int

erse

ct

are

the

solu

tions

of t

he

equa

tion

f(x) =

g(x

); fin

d th

e so

lutio

ns a

ppro

xim

atel

y, e

.g.,

usin

g te

chno

logy

to g

raph

the

func

tions

, mak

e ta

bles

of

valu

es, o

r fin

d su

cces

sive

ap

prox

imat

ions

. Inc

lude

cas

es

i) Ta

sks

may

invo

lve

any

of th

e fu

nctio

n ty

pes

men

tione

d in

the

stan

dard

.

A2.A

.REI

.D.6

Exp

lain

why

the

x-co

ordi

nate

s of

the

poin

ts w

here

the

grap

hs o

f the

equ

atio

ns y

= f(

x) a

nd

y =

g(x)

inte

rsec

t are

the

solu

tions

of

the

equa

tion

f(x) =

g(x

); fin

d th

e ap

prox

imat

e so

lutio

ns u

sing

te

chno

logy

. ★

Incl

ude

case

s w

here

f(x)

and

/or g

(x) a

re li

near

, po

lyno

mia

l, ra

tiona

l, ab

solu

te v

alue

, exp

onen

tial,

and

loga

rithm

ic fu

nctio

ns.

Tas

ks m

ay in

volv

e an

y of

the

func

tion

type

s m

entio

ned

in th

e st

anda

rd.

65

where f(x) and/or g(x) are

linear, polynomial, rational,

absolute value, exponential, and logarithm

ic functions.★

A2.F.IF.A.3 3. R

ecognize that sequences are functions, som

etimes

defined recursively, whose

domain is a subset of the

integers. For example, the

Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

i) This standard is Supporting

work in A

lgebra II. This standard should support the M

ajor work in

F-BF.2 for coherence.

A2.F.IF.B.4 4. For a function that m

odels a relationship betw

een two

quantities, interpret key features of graphs and tables in term


ing key features given a verbal description of the relationship. Key features include: intercepts; intervals w


aximum

s and minim

ums;

symm

etries; end behavior; and periodicity.★

i) Tasks have a real-world context

ii) Tasks may involve polynom

ial, exponential, logarithm

ic, and trigonom

etric functions. Compare

note (ii) w

ith standard F-IF.7. The function types listed here are the sam

e as those listed in the Algebra II colum


A2.F.IF.A.1 For a function that m

odels a relationship between tw

o quantities, interpret key features of graphs and tables in term



Key features include: intercepts; intervals w


aximum

s and minim

ums;

symm

etries; and end behavior. i) Tasks have a real-w

orld context. ii) Tasks m

ay involve square root, cube root, polynom

ial, exponential, and logarithmic functions.

A2.F.IF.B.6 6. C

alculate and interpret the average rate of change of a function (presented sym

bolically or as a table) over a specified interval. Estim

ate the rate of change from

a graph. ★



ial, exponential, logarithm

ic, and trigonom

etric functions. The function types listed here are the sam



A2.F.IF.A.2 Calculate and interpret

the average rate of change of a function (presented sym


ate the rate of change from a

graph.★



ial, exponential, and logarithm

ic function s.

A2.F.IF.C.7 7. G

raph functions expressed sym

bolically and show key

features of the graph, by hand



A2.F.IF.B.3 Graph functions

expressed symbolically and show

key features of the graph, by hand

A2.F.IF.B

.3a: Tasks are limited to square root and

cube root functions. The other functions are assessed in A

lgebra 1 .

66

in s

impl

e ca

ses

and

usin

g te

chno

logy

for m

ore

com

plic

ated

cas

es.★

c.G

raph

pol

ynom

ial f

unct

ions

,id

entif

ying

zer

os w

hen

suita

ble

fact

oriz

atio

ns a

re a

vaila

ble,

and

show

ing

end

beha

vior

.e.

Gra

ph e

xpon

entia

l and

loga

rithm

ic fu

nctio

ns, s

how

ing

i nte

rcep

ts a

nd e

nd b

ehav

ior,

and

trigo

nom

etric

func

tions

,sh

owin

g pe

riod,

mid

line,

and

ampl

itude

.

and

usin

g te

chno

logy

.★

a.G

raph

squ

are

root

, cub

e ro

ot,

and

piec

ewis

e de

fined

func

tions

,in

clud

ing

step

func

tions

and

abso

lute

val

ue fu

nctio

ns.

b.G

raph

pol

ynom

ial f

unct

ions

,id

entif

ying

zer

os w

hen

suita

ble

fact

oriz

atio

ns a

re a

vaila

ble

and

show

ing

end

beha

vior

.

c.G

raph

exp

onen

tial a

ndlo

garit

hmic

func

tions

, sho

win

gin

terc

epts

and

end

beh

avio

r.

A2.F

.IF.C

.8

8.W

rite

a fu

nctio

n de

fined

by

an e

xpre

ssio

n in

diff

eren

t but

equi

vale

nt fo

rms

to re

veal

and

expl

ain

diffe

rent

pro

perti

es o

fth

e fu

nctio

n.b.

Use

the

prop

ertie

s of

expo

nent

s to

inte

rpre

tex

pres

sion

s fo

r exp

onen

tial

func

tions

. For

exa

mpl

e,id

entif

y pe

rcen

t rat

e of

cha

nge

in fu

nctio

ns s

uch

as y

=(1

.02)

t , y

= (0

.97)

t ,y

= (1

.01)

12t ,

y =

(1.2

)t/10 ,

and

clas

sify

them

as

repr

esen

ting

expo

nent

ial g

row

th o

r dec

ay.

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

A2.F

.IF.B

.4 W

rite

a fu

nctio

n de

fined

by

an e

xpre

ssio

n in

di

ffere

nt b

ut e

quiv

alen

t for

ms

to

reve

al a

nd e

xpla

in d

iffer

ent

prop

ertie

s of

the

func

tion.

a.Kn

ow a

nd u

se th

e pr

oper

ties

ofex

pone

nts

to in

terp

ret e

xpre

ssio

nsf o

r exp

onen

tial f

unct

ions

.

For e

xam

ple,

iden

tify

perc

ent r

ate

of c

hang

e in

fu

nctio

ns s

uch

as y

= 2

x , y

= (1

/2)x ,

y =

2-x ,

y =

(1/2

)-x .

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.F

.IF.C

.9

9.C

ompa

re p

rope

rties

of t

wo

func

tions

eac

h re

pres

ente

d in

a di

ffere

nt w

ay (a

lgeb

raic

ally

,gr

aphi

cally

, num

eric

ally

i nt a

bles

, or b

y ve

rbal

desc

riptio

ns).

For e

xam

ple,

give

n a

grap

h of

one

qua

drat

icfu

nctio

n an

d an

alg

ebra

icex

pres

sion

for a

noth

er, s

ayw

hich

has

the

larg

erm

axim

um.

i)Ta

sks

may

invo

lve

poly

nom

ial,

expo

nent

ial,

loga

rithm

ic, a

nd tr

igon

omet

ricfu

nctio

ns.

The

func

tion

type

s lis

ted

here

are

the

sam

e as

thos

e lis

ted

int h

e Al

gebr

a II

colu

mn

for

stan

dard

s F-

IF.4

and

F-IF

.6.

A2.F

.IF.B

.5 C

ompa

re p

rope

rties

of

two

func

tions

eac

h re

pres

ente

d in

a

diffe

rent

way

(alg

ebra

ical

ly,

grap

hica

lly, n

umer

ical

ly in

tabl

es, o

r by

ver

bal d

escr

iptio

ns).

Task

s m

ay in

volv

e po

lyno

mia

l, ex

pone

ntia

l, an

d lo

garit

hmic

func

tions

.

67

A2.F.BF.A.1 1. W


een two quantities.★

a. D



a context. b. C

ombine standard function

types using arithmetic

operations. For 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 m

odel.

For F-BF.1a: i) Tasks have a real-w

orld context ii) Tasks m

ay involve linear functions, quadratic functions, and exponential functions.

A2.F.BF.A.1 Write a function that

describes a relationship between

two quantities.★

a. D



a context. b. C

ombine standard function types

using arithmetic operations.

For example, given cost and revenue functions,

create a profit function. For A

2.F.BF.A

.1a: i) Tasks have a real-w

orld context. I i) Tasks m

ay involve linear functions, quadratic functions, and exponential functions.

A2.F.BF.A.2 2. W

rite arithmetic and

geometric sequences both

recursively and with an explicit

formula, use them

to model

situations, and translate betw

een the two form

s.★



A2.F.BF.A.2 Know and w

rite arithm

etic and geometric

sequences with an explicit form

ula and use them

to model situations.★



A2.F.BF.B.3 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. Experim

ent 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

.

) Tasks may involve

polynomial, exponential,

logarithmic, and trigonom

etric functions ii) Tasks m

ay involve recognizing even and odd functions. The function types listed in note (i) are the sam


n for standards F-IF.4, F-IF.6, and F-IF.9.

A2.F.BF.B.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. Experim

ent with

cases and illustrate an explanation of the effects on the graph using technology.

i) Tasks may involve polynom


ic functions. ii) Tasks m

ay involve recognizing even and odd functions.

A2.F.BF.B.4 4. Find inverse functions. a. Solve an equation of the form

f(x) = c for a simple

function f that has an inverse



A2.F.BF.B.4 Find inverse functions. a. Find the inverse of a function



68

and

writ

e an

exp

ress

ion

for t

he

inve

rse.

For

exa

mpl

e, f(

x) =

2x3

or f(

x) =

(x+1

)/(x–

1) fo

r x =

1.

whe

n th

e gi

ven

func

tion

is o

ne-to

-on

e.

A2.F

.LE.

A.2

2. C

onst

ruct

line

ar a

nd

expo

nent

ial f

unct

ions

, in

clud

ing

arith

met

ic a

nd

geom

etric

seq

uenc

es, g

iven

a

grap

h, a

des

crip

tion

of a

re

latio

nshi

p, o

r tw

o in

put-

outp

ut p

airs

(inc

lude

read

ing

thes

e fro

m a

tabl

e).

i) Ta

sks

will

incl

ude

solv

ing

mul

ti-st

ep p

robl

ems

by

cons

truct

ing

linea

r and

ex

pone

ntia

l fun

ctio

ns.

A2.F

.LE.

A.1

Con

stru

ct li

near

and

ex

pone

ntia

l fun

ctio

ns, i

nclu

ding

ar

ithm

etic

and

geo

met

ric

sequ

ence

s, g

iven

a g

raph

, a ta

ble,

a

desc

riptio

n of

a re

latio

nshi

p, o

r in

put-o

utpu

t pai

rs.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.F

.LE.

A.4

4. F

or e

xpon

entia

l mod

els,

ex

pres

s as

a lo

garit

hm th

e so

lutio

n to

abct

= d

whe

re a

, c,

and

d ar

e nu

mbe

rs a

nd th

e ba

se b

is 2

, 10,

or e

; eva

luat

e th

e lo

garit

hm u

sing

te

chno

logy

.

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d

A2.F

.LE.

A.2

For e

xpon

entia

l m

odel

s, e

xpre

ss a

s a

loga

rithm

the

solu

tion

to a

bc t =

d w

here

a, c

, and

d

are

num

bers

and

the

base

b is

2,

10, o

r e; e

valu

ate

the

loga

rithm

us

ing

tech

nolo

gy.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is c

ours

e.

A2.F

.LE.

B.5

5. In

terp

ret t

he p

aram

eter

s in

a

linea

r or e

xpon

entia

l fun

ctio

n in

term

s of

a c

onte

xt.

i) Ta

sks

have

a re

al-w

orld

co

ntex

t. ii)

Tas

ks a

re lim

ited

to

expo

nent

ial f

unct

ions

with

do

mai

ns n

ot in

the

inte

gers

.

A2.F

.LE.

B.3

Inte

rpre

t the

pa

ram

eter

s in

a li

near

or

expo

nent

ial f

unct

ion

in te

rms

of a

co

ntex

t.

For e

xam

ple,

the

equa

tion

y =

5000

(1.0

6)x m

odel

s th

e ris

ing

popu

latio

n of

a c

ity w

ith 5

000

resi

dent

s w

hen

the

annu

al g

row

th ra

te is

6 p

erce

nt. W

hat w

ill be

the

effe

ct o

n th

e eq

uatio

n if

the

city

's g

row

th ra

te

was

7 p

erce

nt in

stea

d of

6 p

erce

nt?

Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

cou

rse.

A2.F

.TF.

A.1

1. U

nder

stan

d ra

dian

mea

sure

of

an

angl

e as

the

leng

th o

f th

e ar

c on

the

unit

circ

le

subt

ende

d by

the

angl

e x.

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

A2.F

.TF.

A.1

Und

erst

and

and

use

radi

an m

easu

re o

f an

angl

e.

a. U

nder

stan

d ra

dian

mea

sure

of

an a

ngle

as

the

leng

th o

f the

arc

on

the

unit

circ

le s

ubte

nded

by

the

angl

e.

b. U

se th

e un

it ci

rcle

to fi

nd s

in θ

, co

s θ,

and

tan

θ w

hen

θ is

a

com

mon

ly re

cogn

ized

ang

le

betw

een

0 an

d 2π

.

Com

mon

ly re

cogn

ized

ang

les

incl

ude

all m

ultip

les

nπ /6

and

nπ

/4, w

here

n is

an

inte

ger.

Ther

e ar

e no

as

sess

men

t lim

its fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

cou

rse.

A2.F

.TF.

A.2

2. E

xpla

in h

ow th

e un

it ci

rcle

Th

ere

is n

o ad

ditio

nal s

cope

A2

.F.T

F.A.

2 Ex

plai

n ho

w th

e un

it Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd.

69

in the coordinate plane enables the extension of trigonom

etric functions to all real num

bers, interpreted as radian m

easures of angles traversed counterclockw

ise around the unit circle.

or clarification information for

this standard. circle in the coordinate plane enables the extension of trigonom





The entire standard is assessed in this course.

A2.F.TF.B.5 5. C

hoose trigonometric

functions to model periodic

phenomena w

ith specified am

plitude, frequency, and m

idline.★



A2.F.TF.C.8 8. Prove the Pythagorean identity sin

2(θ) + cos2(θ) = 1

and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.



A2.F.TF.B.3 Know

and use trigonom

etric identities to find values of trig functions. a. G

iven a point on a circle centered at the origin, recognize and use the right triangle ratio definitions of sin θ, cos θ, and tan θ to evaluate the trigonom

etric functions. b. G

iven the quadrant of the angle, use the identity sin

2 θ + cos2 θ = 1

to find sin θ given cos θ, or vice versa.

Com

monly recognized angles include all m

ultiples nπ

/6 and nπ /4, w

here n is an integer. There are no assessm

ent limits for this standard.


A2.GPE.A.2

2. Derive the equation of a

parabola given a focus and directrix.



A2.S.ID.A.4 4. U

se the mean and standard

There is no additional scope A2.S.ID.A.1 U

se the mean and



70

devi

atio

n of

a d

ata

set t

o fit

it

to a

nor

mal

dis

tribu

tion

and

to

estim

ate

popu

latio

n pe

rcen

tage

s. R

ecog

nize

that

th

ere

are

data

set

s fo

r whi

ch

such

a p

roce

dure

is n

ot

appr

opria

te. U

se c

alcu

lato

rs,

spre

adsh

eets

, and

tabl

es to

es

timat

e ar

eas

unde

r the

no

rmal

cur

ve.

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

stan

dard

dev

iatio

n of

a d

ata

set t

o fit

it to

a n

orm

al d

istri

butio

n an

d to

es

timat

e po

pula

tion

perc

enta

ges

usin

g th

e Em

piric

al R

ule.

A2.S

.ID.B

.6

6.R

epre

sent

dat

a on

two

quan

titat

ive

varia

bles

on

asc

atte

r plo

t, an

d de

scrib

e ho

wth

e va

riabl

es a

re re

late

d.a.

Fit a

func

tion

to th

e da

ta;

use

func

tions

fitte

d to

dat

a to

s olv

e pr

oble

ms

in th

e co

ntex

tof

the

data

. Use

giv

enfu

nctio

ns o

r cho

ose

a fu

nctio

ns u

gges

ted

by th

e co

ntex

t.Em

phas

ize

linea

r, qu

adra

tic,

and

expo

nent

ial m

odel

s.

i)Ta

sks

have

a re

al-w

orld

cont

ext.

ii)Ta

sks

are

limite

d to

expo

nent

ial f

unct

ions

wit h

dom

ains

not

in th

e in

tege

rsan

d tri

gono

met

ricfu

nctio

ns.

A2.S

.ID.B

.2 R

epre

sent

dat

a on

two

quan

titat

ive

varia

bles

on

a sc

atte

r pl

ot, a

nd d

escr

ibe

how

the

varia

bles

are

rela

ted.

a.Fi

t a fu

nctio

n to

the

data

; use

func

tions

fitte

d to

dat

a to

sol

vepr

oble

ms

in th

e co

ntex

t of t

he d

ata.

Use

giv

en fu

nctio

ns o

r cho

ose

a fu

nctio

n su

gges

ted

by th

e co

ntex

t.

Em

phas

ize

linea

r, qu

adra

tic, a

nd e

xpon

entia

l m

odel

s.

i)Ta

sks

have

a re

al-w

orld

con

text

.ii)

Task

s ar

e lim

ited

to e

xpon

entia

l fun

ctio

ns w

ithdo

mai

ns n

ot in

the

inte

gers

.

A2.S

.IC.A

.1

1.U

nder

stan

d st

atis

tics

as a

proc

ess

for m

akin

g in

fere

nces

abou

t pop

ulat

ion

para

met

ers

base

d on

a ra

ndom

sam

ple

from

that

pop

ulat

ion.

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

A2.S

.IC.A

.2

2.D

ecid

e if

a sp

ecifi

ed m

odel

is c

onsi

sten

t with

resu

lts fr

oma

give

n da

ta-g

ener

atin

gpr

oces

s, e

.g.,

usi n

gs i

mul

atio

n. F

or e

xam

ple,

am

odel

say

s a

spin

ning

coi

nfa

lls h

eads

up

with

pro

babi

lity

0.5.

Wou

ld a

resu

lt of

5 ta

ils in

a ro

w c

ause

you

to q

uest

ion

the

mod

el?

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

A2.S

.IC.B

.3

3.R

ecog

nize

the

purp

oses

of

and

diffe

renc

es a

mon

g sa

mpl

eTh

ere

is n

o ad

ditio

nal s

cope

or

cla

rific

atio

n in

form

atio

n fo

r A2

.S.IC

.A.1

Rec

ogni

ze th

e pu

rpos

es o

f and

diff

eren

ces

amon

g Fo

r exa

mpl

e, in

a g

iven

situ

atio

n, is

it m

ore

appr

opria

te to

use

a s

ampl

e su

rvey

, an

expe

rimen

t,

71

surveys, experiments, and

observational studies; explain how

randomization relates to

each.

this standard. sam

ple surveys, experiments, and

observational studies; explain how

randomization relates to each.

or an observational study? Explain how

random

ization affects the bias in a study. There are no assessm



A2.S.IC.B.4 4. U

se data from a sam

ple survey to estim

ate a population m

ean or proportion; develop a m

argin of error through the use of sim

ulation m

odels for random sam

pling.



A2.S.IC.A.2 Use data from

a sam

ple survey to estimate a

population mean or proportion; use

a given margin of error to solve a

problem in context



A2.S.IC.B.5 5. U

se data from a

randomized experim

ent to com

pare two treatm

ents; use sim

ulations to decide if differences betw

een param

eters are significant.



A2.S.IC.B.6 6. Evaluate reports based on data.



A2.S.CP.A.1 1. D

escribe events as subsets of a sam

ple space (the set of outcom

es) using characteristics (or categories) of the outcom

es, or as unions, intersections, or com

plements

of other events (“or,” “and,” “not”).



A2.S.CP.A.1 Describe events as

subsets of a sample space (the set

of outcomes) using characteristics

(or categories) of the outcomes, or

as unions, intersections, or com

plements of other events (“or,”

“and,” “not”).



A2.S.CP.A.2 2. U

nderstand that two events

A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determ

ine if they are independent.



A2.S.CP.A.2 Understand that tw

o events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determ

ine if they are independent.


its for this standard. The entire standard is assessed in this course

A2.S.CP.A.3 3. U

nderstand the conditional There is no additional scope

A2.S.CP.A.3 Know and understand


its for this standard.

72

prob

abilit

y of

A g

iven

B a

s P(

A an

d B)

/P(B

), an

d in

terp

ret

inde

pend

ence

of A

and

B a

s sa

ying

that

the

cond

ition

al

prob

abilit

y of

A g

iven

B is

the

sam

e as

the

prob

abilit

y of

A,

and

the

cond

ition

al p

roba

bilit

y of

B g

iven

A is

the

sam

e as

th

e pr

obab

ility

of B

.

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

the

cond

ition

al p

roba

bilit

y of

A

give

n B

as P

(A a

nd B

)/P(B

), an

d in

terp

ret i

ndep

ende

nce

of A

and

B

as s

ayin

g th

at th

e co

nditi

onal

pr

obab

ility

of A

giv

en B

is th

e sa

me

as th

e pr

obab

ility

of A

, and

the

cond

ition

al p

roba

bilit

y of

B g

iven

A

is th

e sa

me

as th

e pr

obab

ility

of B

.

The

entir

e st

anda

rd is

ass

esse

d in

this

cou

rse

A2.S

.CP.

A.4

4.C

onst

ruct

and

inte

rpre

t tw

o-w

ay fr

eque

ncy

tabl

es o

f dat

aw

hen

two

cate

gorie

s ar

eas

soci

ated

with

eac

h ob

ject

bein

g cl

assi

fied.

Use

the

two-

way

tabl

e as

a s

ampl

e sp

ace

to d

ecid

e if

even

ts a

rein

depe

nden

t and

toap

prox

imat

e co

nditi

onal

prob

abilit

ies.

For

exa

mpl

e,co

llect

dat

a fro

m a

rand

omsa

mpl

e of

stu

dent

s in

you

rsc

hool

on

thei

r fav

orite

sub

ject

amon

g m

ath,

sci

ence

, and

Engl

ish.

Est

imat

e th

epr

obab

ility

that

a ra

ndom

lyse

lect

ed s

tude

nt fr

om y

our

scho

ol w

ill fa

vor s

cien

ce g

iven

that

the

stud

ent i

s in

tent

hgr

ade.

Do

the

sam

e fo

r oth

ersu

bjec

ts a

nd c

ompa

re th

ere

sults

.

Ther

e is

no

addi

tiona

l sco

pe

or c

larif

icat

ion

info

rmat

ion

for

this

sta

ndar

d.

A2.S

.CP.

A.5

5.R

ecog

nize

and

exp

lain

the

conc

epts

of c

ondi

tiona

lpr

obab

ility

and

inde

pend

ence

in e

very

day

lang

uage

and

e ver

yday

situ

atio

ns. F

orex

ampl

e, c

ompa

re th

e ch

ance

of h

avin

g lu

ng c

ance

r if y

ouar

e a

smok

er w

ith th

e ch

anc e

of b

eing

a s

mok

er if

you

hav

elu

ng c

ance

r.

Ther

e is

no

addi

tiona

l sc

ope

or c

larif

icat

ion

info

rmat

ion

for t

his

stan

dard

.

A2.S

.CP.

A.4

Rec

ogni

ze a

nd

expl

ain

the

conc

epts

of c

ondi

tiona

l pr

obab

ility

and

inde

pend

ence

in

ever

yday

lang

uage

and

eve

ryda

y si

tuat

ions

.

For e

xam

ple,

com

pare

the

chan

ce o

f hav

ing

lung

ca

ncer

if y

ou a

re a

sm

oker

with

the

chan

ce o

f bei

ng

a sm

oker

if y

ou h

ave

lung

can

cer.

The

re a

re n

o as

sess

men

t lim

its fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

cou

rse.

73

A2.S.CP.B.6 6. Find the conditional probability of A given B as the fraction of B’s outcom

es that also belong to A, and interpret the answ

er in terms of the

model.



A2.S.CP.B.5 Find the conditional probability of A given B as the fraction of B’s outcom

es that also belong to A and interpret the answ

er in terms of the m

odel.

For example, a teacher gave tw

o exams. 75 percent

passed the first quiz and 25 percent passed both. W

hat percent who passed the first quiz also passed

the second quiz? There are no assessm



A2.S.CP.B.7 7. Apply the Addition R

ule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answ


odel.



A2.S.CP.B.6 Know and apply the

Addition Rule, P(A or B) = P(A) +

P(B) – P(A and B), and interpret the answ


odel.

For example, in a m

ath class of 32 students, 14 are boys and 18 are girls. O

n a unit test 6 boys and 5 girls m

ade an A. If a student is chosen at random

from

a class, what is the probability of choosing a girl

or an A student?



M

ajor Work of the G

rade

Supporting Work

74

Not

atio

n TN

Sta

ndar

ds th

roug

h M

ay 2

017

Revi

sed

TN S

tand

ards

St

anda

rd

C

lari

ficat

ion

G.CO

.A.1

Kn

ow p

reci

se d

efin

ition

s of a

ngle

, circ

le,

perp

endi

cula

r lin

e, p

aral

lel l

ine,

and

line

se

gmen

t, ba

sed

on th

e un

defin

ed n

otio

ns

of p

oint

, lin

e, d

istan

ce a

long

a li

ne, a

nd

dist

ance

aro

und

a ci

rcul

ar a

rc.

G.CO

.A.1

Kno

w p

reci

se d

efin

ition

s of a

ngle

, ci

rcle

, per

pend

icul

ar li

ne, p

aral

lel l

ine,

and

lin

e se

gmen

t, ba

sed

on th

e un

defin

ed

notio

ns o

f poi

nt, l

ine,

pla

ne, d

istan

ce a

long

a

line,

and

dist

ance

aro

und

a ci

rcul

ar a

rc.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.CO

.A.2

R

epre

sent

tran

sfor

mat

ions

in th

e pl

ane

usin

g, e

.g.,

tran

spar

enci

es a

nd g

eom

etry

so

ftwar

e; d

escr

ibe

tran

sfor

mat

ions

as

func

tions

that

take

poi

nts i

n th

e pl

ane

as

inpu

ts a

nd g

ive

othe

r poi

nts a

s out

puts

. Co

mpa

re tr

ansf

orm

atio

ns th

at p

rese

rve

dist

ance

and

ang

le to

thos

e th

at d

o no

t (e

.g.,

tran

slatio

n ve

rsus

hor

izont

al

stre

tch)

.

G.CO

.A.2

Rep

rese

nt tr

ansf

orm

atio

ns in

the

plan

e in

mul

tiple

way

s, in

clud

ing

tech

nolo

gy.

Desc

ribe

tran

sfor

mat

ions

as f

unct

ions

that

ta

ke p

oint

s in

the

plan

e (p

re-im

age)

as i

nput

s an

d gi

ve o

ther

poi

nts (

imag

e) a

s out

puts

. Co

mpa

re tr

ansf

orm

atio

ns th

at p

rese

rve

dist

ance

and

ang

le m

easu

re to

thos

e th

at d

o no

t (e.

g., t

rans

latio

n ve

rsus

hor

izont

al

stre

tch)

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.CO

.A.3

G

iven

a re

ctan

gle,

par

alle

logr

am,

trap

ezoi

d, o

r reg

ular

pol

ygon

, des

crib

e th

e ro

tatio

ns a

nd re

flect

ions

that

car

ry it

ont

o its

elf.

G.CO

.A.3

Giv

en a

rect

angl

e, p

aral

lelo

gram

, tr

apez

oid,

or r

egul

ar p

olyg

on, d

escr

ibe

the

rota

tions

and

refle

ctio

ns th

at c

arry

the

shap

e on

to it

self.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.CO

.A.4

D

evel

op d

efin

ition

s of r

otat

ions

, re

flect

ions

, and

tran

slatio

ns in

term

s of

angl

es, c

ircle

s, pe

rpen

dicu

lar l

ines

, par

alle

l lin

es, a

nd li

ne se

gmen

ts.

G.CO

.A.4

Dev

elop

def

initi

ons o

f rot

atio

ns,

refle

ctio

ns, a

nd tr

ansla

tions

in te

rms o

f an

gles

, circ

les,

perp

endi

cula

r lin

es, p

aral

lel

lines

, and

line

segm

ents

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.CO

.A.5

G

iven

a g

eom

etric

figu

re a

nd a

rota

tion,

re

flect

ion,

or t

rans

latio

n, d

raw

the

tran

sfor

med

figu

re u

sing,

e.g

., gr

aph

pape

r, tr

acin

g pa

per,

or g

eom

etry

so

ftwar

e. S

peci

fy a

sequ

ence

of

tran

sfor

mat

ions

that

will

carr

y a

give

n fig

ure

onto

ano

ther

.

G.C

O.A

.5 G

iven

a g

eom

etric

figu

re a

nd a

rig

id m

otio

n, d

raw

the

imag

e of

the

figur

e in

m

ultip

le w

ays,

incl

udin

g te

chno

logy

. Spe

cify

a

sequ

ence

of r

igid

mot

ions

that

will

carr

y a

give

n fig

ure

onto

ano

ther

.

Rigi

d m

otio

ns in

clud

e ro

tatio

ns, r

efle

ctio

ns,

and

tran

slatio

ns.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

75

G.CO.B.6

Use geom

etric descriptions of rigid m

otions to transform figures and to

predict the effect of a given rigid motion

on a given figure; given two figures, use

the definition of congruence in terms of

rigid motions to decide if they are

congruent.

G.CO.B.6 U

se geometric descriptions of rigid

motions to transform

figures and to predict the effect of a given rigid m

otion on a given figure; given tw

o figures, use the definition of congruence in term

s of rigid motions to

determine inform

ally if they are congruent.



G.CO.B.7

Use the definition of congruence in term

s of rigid m

otions to show that tw

o triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.CO.B.7 U

se the definition of congruence in term

s of rigid motions to show

that two

triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.



G.CO.B.8

Explain how the criteria for triangle

congruence (ASA, SAS, and SSS) follow

from the definition of congruence in term

s of rigid m

otions.

G.CO.B.8 Explain how

the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow

from


rigid motions.



G.CO.C.9

Prove theorems about lines and angles.

Theorems include: vertical angles are

congruent; when a transversal crosses

parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segm

ent are exactly those equidistant from

the segment’s

endpoints.

G.CO.C.9 Prove theorem

s about lines and angles.

Proving includes, but is not limited to,

completing partial proofs; constructing tw

o-colum

n or paragraph proofs; using transform

ations to prove theorems; analyzing

proofs; and critiquing completed proofs.

Theorems include but are not lim

ited to: vertical angles are congruent; w

hen a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segm


the segm

ent’s endpoints.

76

G.CO

.C.1

0 P

rove

theo

rem

s abo

ut tr

iang

les.

Th

eore

ms i

nclu

de: m

easu

res o

f int

erio

r an

gles

of a

tria

ngle

sum

to 1

80°;

base

an

gles

of i

sosc

eles

tria

ngle

s are

co

ngru

ent;

the

segm

ent j

oini

ng m

idpo

ints

of

two

sides

of a

tria

ngle

is p

aral

lel t

o th

e th

ird si

de a

nd h

alf t

he le

ngth

; the

med

ians

of

a tr

iang

le m

eet a

t a p

oint

.

G.CO

.C.1

0 Pr

ove

theo

rem

s abo

ut tr

iang

les.

Prov

ing

incl

udes

, but

is n

ot li

mite

d to

, co

mpl

etin

g pa

rtia

l pro

ofs;

cons

truc

ting

two-

colu

mn

or p

arag

raph

pro

ofs;

usin

g tr

ansf

orm

atio

ns to

pro

ve th

eore

ms;

ana

lyzin

g pr

oofs

; and

criti

quin

g co

mpl

eted

pro

ofs.

Th

eore

ms i

nclu

de b

ut a

re n

ot li

mite

d to

: m

easu

res o

f int

erio

r ang

les o

f a tr

iang

le su

m

to 1

80°;

base

ang

les o

f iso

scel

es tr

iang

les a

re

cong

ruen

t; th

e se

gmen

t joi

ning

mid

poin

ts o

f tw

o sid

es o

f a tr

iang

le is

par

alle

l to

the

third

sid

e an

d ha

lf th

e le

ngth

; the

med

ians

of a

tr

iang

le m

eet a

t a p

oint

G.CO

.C.1

1

Pro

ve th

eore

ms a

bout

par

alle

logr

ams.

Theo

rem

s inc

lude

: opp

osite

side

s are

co

ngru

ent,

oppo

site

angl

es a

re co

ngru

ent,

the

diag

onal

s of a

par

alle

logr

am b

isect

ea

ch o

ther

, and

conv

erse

ly, r

ecta

ngle

s are

pa

ralle

logr

ams w

ith co

ngru

ent d

iago

nals.

G.CO

.C.1

1 P

rove

theo

rem

s abo

ut

para

llelo

gram

s.

Prov

ing

incl

udes

, but

is n

ot li

mite

d to

, co

mpl

etin

g pa

rtia

l pro

ofs;

cons

truc

ting

two-

colu

mn

or p

arag

raph

pro

ofs;

usin

g tr

ansf

orm

atio

ns to

pro

ve th

eore

ms;

ana

lyzin

g pr

oofs

; and

criti

quin

g co

mpl

eted

pro

ofs.

Th

eore

ms i

nclu

de b

ut a

re n

ot li

mite

d to

: op

posit

e sid

es a

re co

ngru

ent,

oppo

site

angl

es

are

cong

ruen

t, th

e di

agon

als o

f a

para

llelo

gram

bise

ct e

ach

othe

r, an

d co

nver

sely

, rec

tang

les a

re p

aral

lelo

gram

s w

ith co

ngru

ent d

iago

nals.

77

G.CO.D.12

Make form

al geometric constructions

with a variety of tools and m

ethods (com

pass and straightedge, string, reflective devices, paper folding, dynam

ic geom

etric software, etc.). Copying a

segment; copying an angle; bisecting a

segment; bisecting an angle; constructing

perpendicular lines, including the perpendicular bisector of a line segm

ent; and constructing a line parallel to a given line through a point not on the line.

G.CO.D.12 M

ake formal geom

etric constructions w

ith a variety of tools and m

ethods (compass and straightedge, string,

reflective devices, paper folding, dynamic

geometric softw

are, etc.).

Constructions include but are not limited to:

copying a segment; copying an angle;

bisecting a segment; bisecting an angle;

constructing perpendicular lines, including the perpendicular bisector of a line segm

ent; constructing a line parallel to a given line through a point not on the line, and constructing the follow

ing objects inscribed in a circle: an equilateral triangle, square, and a regular hexagon.

G.CO.D.13

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G.SRT.A.1 Verify experim

entally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segm

ent is longer or shorter in the ratio given by the scale factor.

G.SRT.A.1 Verify informally the properties of

dilations given by a center and a scale factor.

Properties include but are not limited to: a

dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center of the dilation unchanged; the dilation of a line segm


G.SRT.A.2 Given tw

o figures, use the definition of sim

ilarity in terms of sim

ilarity transform

ations to decide if they are sim

ilar; explain using similarity

transformations the m

eaning of similarity

for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G.SRT.A.2 Given two figures, use the

definition of similarity in term

s of similarity

transformations to decide if they are sim

ilar; explain using sim

ilarity transformations the

meaning of sim

ilarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.



78

G.SR

T.A.

3U

se th

e pr

oper

ties o

f sim

ilarit

y tr

ansf

orm

atio

ns to

est

ablis

h th

e AA

cr

iterio

n fo

r tw

o tr

iang

les t

o be

sim

ilar.

G.SR

T.A.

3 U

se th

e pr

oper

ties o

f sim

ilarit

ytr

ansf

orm

atio

ns to

est

ablis

h th

e AA

crite

rion

for t

wo

tria

ngle

s to

be si

mila

r.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.SR

T.B.

4 P

rove

theo

rem

s abo

ut tr

iang

les.

Th

eore

ms i

nclu

de: a

line

par

alle

l to

one

side

of a

tria

ngle

div

ides

the

othe

r tw

o pr

opor

tiona

lly, a

nd co

nver

sely

; the

Py

thag

orea

n Th

eore

m p

rove

d us

ing

tria

ngle

sim

ilarit

y.

G.SR

T.B.

4 Pr

ove

theo

rem

s abo

ut si

mila

rtr

iang

les.

Prov

ing

incl

udes

, but

is n

ot li

mite

d to

, co

mpl

etin

g pa

rtia

l pro

ofs;

cons

truc

ting

two-

colu

mn

or p

arag

raph

pro

ofs;

usin

g tr

ansf

orm

atio

ns to

pro

ve th

eore

ms;

ana

lyzin

g pr

oofs

; and

criti

quin

g co

mpl

eted

pro

ofs.

Theo

rem

s inc

lude

but

are

not

lim

ited

to: a

lin

e pa

ralle

l to

one

side

of a

tria

ngle

div

ides

th

e ot

her t

wo

prop

ortio

nally

, and

conv

erse

ly;

the

Pyth

agor

ean

Theo

rem

pro

ved

usin

g tr

iang

le si

mila

rity.

G.SR

T.B.

5 U

se co

ngru

ence

and

sim

ilarit

y cr

iteria

for

tria

ngle

s to

solv

e pr

oble

ms a

nd to

pro

ve

rela

tions

hips

in g

eom

etric

figu

res.

G.SR

T.B.

5 U

se co

ngru

ence

and

sim

ilarit

ycr

iteria

for t

riang

les t

o so

lve

prob

lem

s and

toju

stify

rela

tions

hips

in g

eom

etric

figu

res.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.SR

T.C.

6 U

nder

stan

d th

at b

y sim

ilarit

y, si

de ra

tios

in ri

ght t

riang

les a

re p

rope

rtie

s of t

he

angl

es in

the

tria

ngle

, lea

ding

to

defin

ition

s of t

rigon

omet

ric ra

tios f

or

acut

e an

gles

.

G.SR

T.C.

6 U

nder

stan

d th

at b

y sim

ilarit

y, si

dera

tios i

n rig

ht tr

iang

les a

re p

rope

rtie

s of t

hean

gles

in th

e tr

iang

le, l

eadi

ng to

def

initi

ons

of tr

igon

omet

ric ra

tios f

or a

cute

ang

les.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.SR

T.C.

7 E

xpla

in a

nd u

se th

e re

latio

nshi

p be

twee

n th

e sin

e an

d co

sine

of c

ompl

emen

tary

an

gles

.

G.SR

T.C.

7 Ex

plai

n an

d us

e th

e re

latio

nshi

pbe

twee

n th

e sin

e an

d co

sine

ofco

mpl

emen

tary

ang

les.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

79

G.SRT.C.8 U

se trigonometric ratios and the

Pythagorean Theorem to solve right

triangles in applied problems. ★

G.SRT.C.8 Solve triangles. ★

a. Know

and use trigonometric ratios

and the Pythagorean Theorem to

solve right triangles in applied problem

s.

b. Know

and use the Law of Sines and

Law of Cosines to solve problem

s in real life situations. Recognize w

hen it is appropriate to use each.

Ambiguous cases w

ill not be included in assessm

ent.

G.C.A.1 Prove that all circles are sim

ilar. G.C.A.1 Recognize that all circles are sim

ilar.



G.C.A.2 Identify and describe relationships am

ong inscribed angles, radii, and chords. Include the relationship betw

een central, inscribed, and circum

scribed angles; inscribed angles on a diam

eter are right angles; the radius of a circle is perpendicular to the tangent w

here the radius intersects the circle.

G.C.A.2 Identify and describe relationships am

ong inscribed angles, radii, and chords.

Include the relationship between central,

inscribed, and circumscribed angles; inscribed

angles on a diameter are right angles; the

radius of a circle is perpendicular to the tangent w

here the radius intersects the circle, and properties of angles for a quadrilateral inscribed in a circle.

G.C.A.3 Construct the inscribed and circum

scribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

G.C.A.5 Derive using sim

ilarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian m

easure of the angle as the constant of proportionality; derive the form

ula for the area of a sector.

80

G.C.

A.3

Con

stru

ct th

e in

cent

er a

ndci

rcum

cent

er o

f a tr

iang

le a

nd u

se th

eir

prop

ertie

s to

solv

e pr

oble

ms i

n co

ntex

t.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.C.

B.4

Know

the

form

ula

and

find

the

area

of a

sect

or o

f a ci

rcle

in a

real

-wor

ld co

ntex

t.

For e

xam

ple,

use

pro

port

iona

l rel

atio

nshi

ps

and

angl

es m

easu

red

in d

egre

es o

r rad

ians

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.GP

E.A.

1 D

eriv

e th

e eq

uatio

n of

a c

ircle

of g

iven

ce

nter

and

radi

us u

sing

the

Pyth

agor

ean

Theo

rem

; com

plet

e th

e sq

uare

to fi

nd th

e ce

nter

and

radi

us o

f a ci

rcle

giv

en b

y an

eq

uatio

n.

G.GP

E.A.

1 Kn

ow a

nd w

rite

the

equa

tion

of a

circ

le o

f giv

en ce

nter

and

radi

us u

sing

the

Pyth

agor

ean

Theo

rem

.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

G.GP

E.B.

4 U

se co

ordi

nate

s to

prov

e sim

ple

geom

etric

theo

rem

s alg

ebra

ical

ly. F

or

exam

ple,

pro

ve o

r disp

rove

that

a fi

gure

de

fined

by

four

giv

en p

oint

s in

the

coor

dina

te p

lane

is a

rect

angl

e; p

rove

or

disp

rove

that

the

poin

t (1,

√3,

) lie

s on

the

circ

le ce

nter

ed a

t the

orig

in a

nd

cont

aini

ng th

e po

int (

0, 2

).

G.GP

E.B.

2 U

se c

oord

inat

es to

pro

ve si

mpl

ege

omet

ric th

eore

ms a

lgeb

raic

ally

.

For e

xam

ple,

pro

ve o

r disp

rove

that

a fi

gure

de

fined

by

four

giv

en p

oint

s in

the

coor

dina

te

plan

e is

a re

ctan

gle;

pro

ve o

r disp

rove

that

th

e po

int (

1, √

3) li

es o

n th

e ci

rcle

cent

ered

at

the

orig

in a

nd co

ntai

ning

the

poin

t (0,

2).

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

. G.

GPE.

B.5

Pro

ve th

e slo

pe cr

iteria

for p

aral

lel a

nd

perp

endi

cula

r lin

es a

nd u

se th

em to

solv

e ge

omet

ric p

robl

ems (

e.g.

, fin

d th

e eq

uatio

n of

a li

ne p

aral

lel o

r pe

rpen

dicu

lar t

o a

give

n lin

e th

at p

asse

s th

roug

h a

give

n po

int).

G.GP

E.B.

3 Pr

ove

the

slope

crit

eria

for p

aral

lel

and

perp

endi

cula

r lin

es a

nd u

se th

em to

solv

e ge

omet

ric p

robl

ems.

For e

xam

ple,

find

the

equa

tion

of a

line

pa

ralle

l or p

erpe

ndic

ular

to a

giv

en li

ne th

at

pass

es th

roug

h a

give

n po

int.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

81

G.GPE.B.6 6. Find the point on a directed line segm

ent between tw

o given points that partitions the segm

ent in a given ratio.

G.GPE.B.4 Find the point on a directed line segm

ent between tw

o given points that partitions the segm

ent in a given ratio.



G.GPE.B.7 7. U

se coordinates to compute perim

eters of polygons and areas of triangles and rectangles, e.g., using the distance form

ula. ★

G.GPE.B.5 Know and use coordinates to

compute perim

eters of polygons and areas of triangles and rectangles.★

For example, use the distance form

ula. There are no assessm



G.GMD.A.1

1. Give an informal argum

ent for the form

ulas for the circumference of a circle,

area of a circle, volume of a cylinder,

pyramid, and cone. U

se dissection argum

ents, Cavalieri’s principle, and inform

al limit argum

ents.

G.GMD.A.1 Give an inform

al argument for

the formulas for the circum

ference of a circle and the volum

e and surface area of a cylinder, cone, prism

, and pyramid.

Informal argum

ents may include but are not

limited to using the dissection argum

ent, applying Cavalieri’s principle, and constructing inform

al limit argum

ents. There are no assessm

ent limits for this

standard. The entire standard is assessed in this course

G.GMD.A.3

3. Use volum

e formulas for cylinders,

pyramids, cones, and spheres to solve

problems. ★

G.GMD.A.2 Know

and use volume and

surface area formulas for cylinders, cones,

prisms, pyram

ids, and spheres to solve problem

s.★



G.GMD.A.4

4. Identify the shapes of two-dim

ensional cross-sections of three dim

ensional objects, and identify three-dim

ensional objects generated by rotations of tw

o-dim

ensional objects.

82

G.

MG.

A.1

1. U

se g

eom

etric

shap

es, t

heir

mea

sure

s, an

d th

eir p

rope

rtie

s to

desc

ribe

obje

cts

(e.g

., m

odel

ing

a tr

ee tr

unk

or a

hum

an

tors

o as

a c

ylin

der)

. ★

G.M

G.A.

1 U

se g

eom

etric

shap

es, t

heir

mea

sure

s, an

d th

eir p

rope

rtie

s to

desc

ribe

obje

cts.★

For e

xam

ple,

mod

el a

tree

trun

k or

a h

uman

to

rso

as a

cylin

der.

Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd. T

he e

ntire

stan

dard

is a

sses

sed

in

this

cour

se.

G.M

G.A.

2 2.

App

ly co

ncep

ts o

f den

sity

base

d on

ar

ea a

nd v

olum

e in

mod

elin

g sit

uatio

ns

(e.g

., pe

rson

s per

squa

re m

ile, B

TUs p

er

cubi

c foo

t). ★

G.M

G.A.

3 3.

App

ly g

eom

etric

met

hods

to so

lve

desig

n pr

oble

ms (

e.g.

, des

igni

ng a

n ob

ject

or

stru

ctur

e to

satis

fy p

hysic

al co

nstr

aint

s or

min

imize

cost

; wor

king

with

ty

pogr

aphi

c grid

syst

ems b

ased

on

ratio

s).

★

G.M

G.A.

2 Ap

ply

geom

etric

met

hods

to so

lve

real

-wor

ld p

robl

ems.★

Geom

etric

met

hods

may

inclu

de b

ut a

re n

ot

limite

d to

usin

g ge

omet

ric sh

apes

, the

pr

obab

ility

of a

shad

ed re

gion

, den

sity,

and

de

sign

prob

lem

s.

Ther

e ar

e no

ass

essm

ent l

imits

for t

his

stan

dard

. The

ent

ire st

anda

rd is

ass

esse

d in

th

is co

urse

.

Maj

or C

onte

nt

Su

ppor

ting

Con

tent

83

Notation 2016-17 Standard

2016-17 Scope and Clarification 2017-18 Standard

2017-18 Scope & Clarifications

N-Q.A.1

1. Use units as a w



ulti-step problems; choose and

interpret units consistently in formulas;

choose and interpret the scale and the origin in graphs and data displays.

There is no additional scope or clarification for this standard.

M1.N.Q

.A.1 Use units as a w



ulti-step problems;

choose and interpret units consistently in form

ulas; choose and interpret the scale and the origin in graphs and data displays

There are no assessment

limits for this standard. The

entire standard is assessed in this course.

N-Q.A.2

2. Define appropriate quantities for the

purpose of descriptive modeling.


Math I by ensuring that som

e m

odeling tasks (involving Math I

content or securely held content from

grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For exam


ight autonomously

decide that a measure of center is

a key variable in a situation, and then choose to w

ork with the

mean.

M1.N.Q

.A.2 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m

odeling.

Clarification: D

escriptive m


ation; choosing appropriate units; etc. Tasks are lim

ited to linear or exponential equations w


N-Q.A.3

3. Choose a level of accuracy,

appropriate to limitations on

measurem

ent when reporting quantities.


M1.N.Q

.A.3 Choose a level of

accuracy appropriate to limitations on

measurem

ent when reporting

quantities.




A-SSE.A.1 1.Interpret expressions that represent a quantity in term

s of its context. ★

a. Interpret parts of an expression, such as term

s, factors, and coefficients. b. Interpret com

plicated expressions by view

ing one or more of their parts as a

single entity. For example, interpret

P(1+r) n as the product of P and a factor

not depending on P.

i) Tasks are limited to exponential

expressions, including related num

erical expressions.

M1.A.SSE.A.1 Interpret expressions

that represent a quantity in terms of

its context. ★ a. Interpret parts of an expression, such as term

s, factors, and coefficients. b. Interpret com

plicated expressions by view

ing one or more of their parts

as a single entity.

For example, interpret P

(1 + r) n as the product of P

and a factor not depending on P

. Tasks are lim

ited to linear and exponential expressions, including related num

erical expressions

84

A-SS

E.B.

33.

Cho

ose

and

prod

uce

an e

quiv

alen

tfo

rm o

f an

expr

essi

on to

reve

al a

ndex

plai

n pr

oper

ties

of th

e qu

antit

yre

pres

ente

d by

the

expr

essi

on.★

c.

Us e

the

prop

ertie

s of

exp

onen

ts to

trans

form

exp

ress

ions

for e

xpon

entia

lfu

nctio

ns. F

or e

xam

ple

the

expr

ess i

on1.

15t c

an b

e re

writ

ten

as (1

.151/

12)12

t ≈1.

01212

t to re

veal

the

appr

oxim

ate

equi

v ale

nt m

onth

ly in

tere

st ra

te if

the

annu

al ra

te is

15%

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

A.SS

E.B.

2 C

hoos

e an

d pr

oduc

e an

equ

ival

ent f

orm

of a

n ex

pres

sion

to

reve

al a

nd e

xpla

in p

rope

rties

of t

he

quan

tity

repr

esen

ted

by th

e ex

pres

sion

. ★

a.U

se th

e pr

oper

ties

of e

xpon

ents

tore

writ

e ex

pone

ntia

l exp

ress

ions

.

For

M1.

A.S

SE.

B.2

a: F

or

exam

ple,

the

grow

th o

f ba

cter

ia c

an b

e m

odel

ed b

y ei

ther

f(t

) = 3

(t+2)

or

g(t)

= 9

(3t )

bec

ause

the

expr

essi

on 3

(t+2)

can

be

rew

ritte

n as

(3t )

(32 )

= 9

(3t ).

Task

s ha

ve a

real

-wor

ld

cont

ext.

As

desc

ribed

in th

e st

anda

rd, t

here

is a

n in

terp

lay

betw

een

the

mat

hem

atic

al

stru

ctur

e of

the

expr

essi

on

and

the

stru

ctur

e of

the

situ

atio

n su

ch th

at c

hoos

ing

and

prod

ucin

g an

equ

ival

ent

form

of t

he e

xpre

ssio

n re

veal

s so

met

hing

abo

ut th

e si

tuat

ion.

A-CE

D.A.

11.

Cre

ate

equa

tions

and

ineq

ualit

ies

inon

e va

riabl

e an

d us

e th

em to

sol

vepr

oble

ms.

Incl

ude

equa

tions

aris

ing

from

linea

r and

qua

drat

ic fu

nctio

ns, a

ndsi

mpl

e ra

tiona

l and

exp

onen

t ial

f unc

tions

.

i)Ta

sks

are

limite

d to

line

ar o

rex

pone

ntia

l equ

atio

ns w

ith in

tege

rex

pone

nts.

ii) T

asks

hav

e a

real

-w

orld

con

text

. iii)

In th

e lin

ear

case

, tas

ks h

ave

mor

e of

the

hal lm

arks

of m

odel

ing

as a

mat

hem

atic

al p

ract

ice

(less

defin

ed ta

sks,

mor

e of

the

mod

elin

g cy

cle,

etc

.).

M1.

A.CE

D.A.

1 C

reat

e eq

uatio

ns a

nd

ineq

ualit

ies

in o

ne v

aria

ble

and

use

them

to s

olve

pro

blem

s.

i)Ta

sks

are

limite

d to

line

aror

exp

onen

tial e

quat

ions

with

int e

ger e

xpon

ents

.ii)

Task

s ha

ve a

real

-wor

ldc o

ntex

t.iii)

In th

e lin

ear c

ase,

task

sha

ve m

ore

of th

e ha

llmar

ks o

fm

odel

ing

as a

mat

hem

atic

alpr

actic

e (le

ss d

efin

ed ta

sks,

mor

e of

the

mod

elin

g cy

cle,

etc.

).

A-CE

D.A.

22.

Cre

ate

equa

tions

in tw

o or

mor

eva

riabl

es to

repr

esen

t rel

atio

nshi

psbe

twee

n qu

antit

ies;

gra

ph e

quat

ions

on

coor

dina

te a

xes

with

labe

ls a

nd s

cale

s.

i)Ta

sks

are

limite

d to

line

areq

uatio

ns ii

) Tas

ks h

ave

a re

al-

wor

ld c

onte

xt. i

ii) T

asks

hav

e th

eha

llmar

ks o

f mod

elin

g as

am

athe

mat

ical

pra

ctic

e (le

ssde

f ined

task

s, m

ore

of th

em

odel

ing

cycl

e, e

tc.).

M1.

A.CE

D.A.

2 C

reat

e eq

uatio

ns in

tw

o or

mor

e va

riabl

es to

repr

esen

t re

latio

nshi

ps b

etw

een

quan

titie

s;

grap

h eq

uatio

ns w

ith tw

o va

riabl

es o

n co

ordi

nate

axe

s w

ith la

bels

and

sc

ales

.

i)Ta

sks

are

limite

d to

line

areq

uatio

nsii)

Task

s ha

ve a

real

-wor

ldc o

ntex

t.iii)

Task

s ha

ve th

e ha

llmar

ksof

mod

elin

g as

am

athe

mat

ical

pra

ctic

e (le

ssde

fined

task

s, m

ore

of th

em

odel

ing

cycl

e, e

tc.).

85

A-CED.A.3 3. R

epresent constraints by equations or inequalities, and by system

s of equations and/or inequalities, and interpret solutions as viable or nonviable options in a m

odeling context. For exam

ple, represent inequalities describing nutritional and cost constraints on com

binations of different foods.


M1.A.CED.A.3 R

epresent constraints by equations or inequalities and by system

s of equations and/or inequalities, and interpret solutions as viable or nonviable options in a m

odeling context.

For example, represent

inequalities describing nutritional and cost constraints on com

binations of different foods. There are no assessm

ent lim


A-CED.A.4 4. R

earrange formulas to highlight a

quantity of interest, using the same

reasoning as in solving equations. For exam

ple, rearrange Ohm

’s law V = IR

to highlight resistance R

.


equations ii) Tasks have a real-w

orld context.

M1.A.CED.A.4 R

earrange formulas




equations. ii) Tasks have a real-w

orld context.

A-REI.A.3 3. Solve linear equations and inequalities in one variable, including equations w



M1.A.REI.A.1 Solve linear equations

and inequalities in one variable, including equations w





A-REI.B.5 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 w

ith the same solutions.


A-REI.B.6 6. Solve system

s of linear equations exactly and approxim

ately (e.g., with

graphs), focusing on pairs of linear equations in tw

o variables.


M1.A.REI.B.2 W

rite and solve a system

of linear equations in context. S

olve systems both

algebraically and graphically. S

ystems are lim

ited to at m

ost two equations in tw

o variables.

A-REI.C.10 10. U

nderstand that the graph of an equation in tw

o variables is the set of all its solutions plotted in the coordinate plane, often form

ing a curve (which

could be a line).


M1.A.REI.C.3 U

nderstand that the graph of an equation in tw

o variables is the set of all its solutions plotted in the coordinate plane, often form

ing a curve (w

hich could be a line).




A-REI.C.11 11. Explain w

hy the x-coordinates of the points w

here the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find

i) Tasks that assess conceptual understanding of the indicated concept m

ay involve any of the function types m

entioned in the

M1.A.REI.C.4 Explain w


here the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of

Include cases where f(x)

and/or g(x) are linear, absolute value, and exponential functions. For

86

the

solu

tions

app

roxi

mat

ely,

e.g

., us

ing

tech

nolo

gy to

gra

ph th

e fu

nctio

ns, m

ake

tabl

es o

f val

ues,

or f

ind

succ

essi

ve

appr

oxim

atio

ns. I

nclu

de c

ases

whe

re f(

x)

and/

or g

(x) a

re li

near

, pol

ynom

ial,

ratio

nal,

abso

lute

val

ue, e

xpon

entia

l, an

d lo

garit

hmic

func

tions

.★

stan

dard

exc

ept e

xpon

entia

l and

lo

garit

hmic

func

tions

. ii)

Find

ing

the

solu

tions

app

roxi

mat

ely

is

limite

d to

cas

es w

here

f(x)

and

g(

x) a

re p

olyn

omia

l.

the

equa

tion

f(x) =

g(x

); fin

d th

e ap

prox

imat

e so

lutio

ns u

sing

te

chno

logy

. ★

exam

ple:

f(

x) =

3x

+ 5.

i)

Task

s th

at a

sses

s co

ncep

tual

und

erst

andi

ng o

f th

e in

dica

ted

conc

ept m

ay

invo

lve

any

of th

e fu

nctio

n ty

pes

men

tione

d in

the

stan

dard

exc

ept e

xpon

entia

l an

d lo

garit

hmic

func

tions

. ii)

Fin

ding

the

solu

tions

ap

prox

imat

ely

is li

mite

d to

ca

ses

whe

re f(

x) a

nd g

(x) a

re

poly

nom

ial.

iii) T

asks

are

lim

ited

to li

near

an

d ab

solu

te v

alue

func

tions

.

A-RE

I.C.1

2 12

. Gra

ph th

e so

lutio

ns to

a li

near

in

equa

lity

in tw

o va

riabl

es a

s a

halfp

lane

(e

xclu

ding

the

boun

dary

in th

e ca

se o

f a

stric

t ine

qual

ity),

and

grap

h th

e so

lutio

n se

t to

a sy

stem

of l

inea

r ine

qual

ities

in

two

varia

bles

as

the

inte

rsec

tion

of th

e co

rresp

ondi

ng h

alf-

plan

es.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

A.RE

I.C.5

Gra

ph th

e so

lutio

ns to

a

linea

r ine

qual

ity in

two

varia

bles

as

a ha

lf-pl

ane

(exc

ludi

ng th

e bo

unda

ry

in th

e ca

se o

f a s

trict

ineq

ualit

y), a

nd

grap

h th

e so

lutio

n se

t to

a sy

stem

of

linea

r ine

qual

ities

in tw

o va

riabl

es a

s th

e in

ters

ectio

n of

the

corre

spon

ding

ha

lf-pl

anes

.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

F-IF

.A.1

1.

Und

erst

and

that

a fu

nctio

n fro

m o

ne

set (

calle

d th

e do

mai

n) to

ano

ther

set

(c

alle

d th

e ra

nge)

ass

igns

to e

ach

elem

ent o

f the

dom

ain

exac

tly o

ne

elem

ent o

f the

rang

e. If

f is

a fu

nctio

n an

d x

is a

n el

emen

t of

its

dom

ain,

then

f(x)

den

otes

the

outp

ut o

f f c

orre

spon

ding

to th

e in

put x

. Th

e gr

aph

of f

is th

e gr

aph

of th

e eq

uatio

n

y =

f(x).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

F.IF

.A.1

Und

erst

and

that

a

func

tion

from

one

set

(cal

led

the

dom

ain)

to a

noth

er s

et (c

alle

d th

e ra

nge)

ass

igns

to e

ach

elem

ent o

f the

do

mai

n ex

actly

one

ele

men

t of t

he

rang

e. If

f is

a fu

nctio

n an

d x

is a

n el

emen

t of i

ts d

omai

n, th

en f(

x)

deno

tes

the

outp

ut o

f f c

orre

spon

ding

to

the

inpu

t x. T

he g

raph

of f

is th

e gr

aph

of th

e eq

uatio

n y

= f(x

).

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

F-IF

.A.2

2.

Use

func

tion

nota

tion,

eva

luat

e fu

nctio

ns fo

r inp

uts

in th

eir d

omai

ns, a

nd

inte

rpre

t sta

tem

ents

that

use

func

tion

no

tatio

n in

term

s of

a c

onte

xt.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

F.IF

.A.2

Use

func

tion

nota

tion,

ev

alua

te fu

nctio

ns fo

r inp

uts

in th

eir

dom

ains

, and

inte

rpre

t sta

tem

ents

th

at u

se fu

nctio

n no

tatio

n in

term

s of

a

cont

ext.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

87

F-IF.A.33.R

ecognize that sequences arefunctions, som

etimes defined

recursively, whose dom

ain is a subset ofthe integers. For exam

ple, the Fibonaccisequence is defined recursively by f(0) =f(1) = 1,f(n+1) = f(n) +f(n-1) for n ≥ 1.


F-IF.B.44.For a function that m

odels arelationship betw

een two quantities,

interpret key features of graphs andt ables in term

s of the quantities, andsketch graphs show

ing key featuresgiven a verbal description of t her elationship. K

ey features include:intercepts; intervals w

here the function isincreasing, decreasing, positive, ornegative; relative m

aximum

s andm

inimum

s; symm

etries; end behavior;and periodicity. ★



functions, square root functions,cube root functions, piecew

ise-defined functions (including st epfunctions and absolute val uefunctions), and exponentialfunctions w

ith domains in t he

integers. The function types list edhere are the sam

e as those listedin the M

ath I column for standards

F-IF.6 and F-IF.9.

M1.F.IF.B.3 For a function that

models a relationship betw

een two




Key features include:

intercepts; intervals where

the function is increasing, decreasing, positive, or negative; relative m

aximum

s and m

inimum

s; symm

etries; and end behavior.

i)Tasks have a real-wor ld

c ontext.ii)Tasks are lim

ited to linearfunctions, absolute value,and exponential functionsw


F-IF.B.55.R

elate the domain of a function to its

graph and, where applicable, to the

quantitative relationship it describes. Forexam

ple, if the function h(n) gives t henum

ber of person-hours it takes toassem

ble n engines in a factory, t hent he positive integers w

ould be anappropriate dom

ain for the function. ★



functions, square root functions,cube root functions, piecew

ise-defined functions (including stepfunctions and absolute val uef unctions), and exponentialfunctions w

ith domains in t he

i ntegers.

M1.F.IF.B.4 R

elate the domain of a

function to its graph and, where

applicable, to the quantitative relationship it describes. ★

For example, if the function

h(n) gives the number of

person-hours it takes to assem

ble n engines in a factory, then the positive integers w

ould be an appropriate dom

ain for the function.


c ontext.ii)Tasks are lim

ited to linearfunctions, piecew

ise functions(including step functions andabsolute value functions),and exponential functionsw


88

F-IF

.B.6

6.

Cal

cula

te a

nd in

terp

ret t

he a

vera

ge

rate

of c

hang

e of

a fu

nctio

n (p

rese

nted

sy

mbo

lical

ly o

r as

a ta

ble)

ove

r a

spec

ified

inte

rval

. Est

imat

e th

e ra

te o

f ch

ange

from

a g

raph

.★

i) Ta

sks

have

a re

al-w

orld

con

text

. ii)

Tas

ks a

re li

mite

d to

line

ar

func

tions

, squ

are

root

func

tions

, cu

be ro

ot fu

nctio

ns, p

iece

wis

e-de

fined

func

tions

(inc

ludi

ng s

tep

func

tions

and

abs

olut

e va

lue

func

tions

), an

d ex

pone

ntia

l fu

nctio

ns w

ith d

omai

ns in

the

inte

gers

. The

func

tion

type

s lis

ted

here

are

the

sam

e as

thos

e lis

ted

in th

e M

ath

I col

umn

for s

tand

ards

F-

IF.4

and

F-IF

.9.

M1.

F.IF

.B.5

Cal

cula

te a

nd in

terp

ret

the

aver

age

rate

of c

hang

e of

a

func

tion

(pre

sent

ed s

ymbo

lical

ly o

r as

a ta

ble)

ove

r a s

peci

fied

inte

rval

. Es

timat

e th

e ra

te o

f cha

nge

from

a

grap

h. ★

i) Ta

sks

have

a re

al-w

orld

co

ntex

t. ii)

Tas

ks a

re li

mite

d to

line

ar

func

tions

, pie

cew

ise

func

tions

(in

clud

ing

step

func

tions

and

ab

solu

te v

alue

func

tions

), an

d ex

pone

ntia

l fun

ctio

ns

with

dom

ains

in th

e in

tege

rs.

F-IF

.C.7

7.

Gra

ph fu

nctio

ns e

xpre

ssed

sy

mbo

lical

ly a

nd s

how

key

feat

ures

of

the

grap

h, b

y ha

nd in

sim

ple

case

s an

d us

ing

tech

nolo

gy fo

r mor

e co

mpl

icat

ed

case

s.★

a. G

raph

line

ar a

nd q

uadr

atic

func

tions

an

d sh

ow in

terc

epts

, max

ima,

and

m

inim

a.

i) Ta

sks

are

limite

d to

line

ar

func

tions

. M

1.F.

IF.C

.6 G

raph

func

tions

ex

pres

sed

sym

bolic

ally

and

sho

w k

ey

feat

ures

of t

he g

raph

, by

hand

and

us

ing

tech

nolo

gy.

a. G

raph

line

ar a

nd q

uadr

atic

fu

nctio

ns a

nd s

how

inte

rcep

ts,

max

ima,

and

min

ima.

Task

s ar

e lim

ited

to li

near

fu

nctio

ns.

F-IF

.C.9

9.

Com

pare

pro

perti

es o

f tw

o fu

nctio

ns

each

repr

esen

ted

in a

diff

eren

t way

(a

lgeb

raic

ally

, gra

phic

ally

, num

eric

ally

in

tabl

es, o

r by

verb

al d

escr

iptio

ns).

For

exam

ple,

giv

en a

gra

ph o

f one

qua

drat

ic

func

tion

and

an a

lgeb

raic

exp

ress

ion

for

anot

her,

say

whi

ch h

as th

e la

rger

m

axim

um.

i) Ta

sks

have

a re

al-w

orld

con

text

. ii)

Tas

ks a

re li

mite

d to

line

ar

func

tions

, squ

are

root

func

tions

, cu

be ro

ot fu

nctio

ns, p

iece

wis

e-de

fined

func

tions

(inc

ludi

ng s

tep

func

tions

and

abs

olut

e va

lue

func

tions

), an

d ex

pone

ntia

l fu

nctio

ns w

ith d

omai

ns in

the

inte

gers

. The

func

tion

type

s lis

ted

here

are

the

sam

e as

thos

e lis

ted

in th

e M

ath

I col

umn

for s

tand

ards

F-

IF.4

and

F-IF

.6

M1.

F.IF

.C.7

Com

pare

pro

perti

es o

f tw

o fu

nctio

ns e

ach

repr

esen

ted

in a

di

ffere

nt w

ay (a

lgeb

raic

ally

, gr

aphi

cally

, num

eric

ally

in ta

bles

, or

by v

erba

l des

crip

tions

). .

i) Ta

sks

have

a re

al-w

orld

co

ntex

t. ii)

Tas

ks a

re li

mite

d to

line

ar

func

tions

, pie

cew

ise

func

tions

(in

clud

ing

step

func

tions

and

ab

solu

te v

alue

func

tions

), an

d ex

pone

ntia

l fun

ctio

ns

with

dom

ains

in th

e in

tege

rs

F-BF

.A.1

1.

W

rite

a fu

nctio

n th

at d

escr

ibes

a

rela

tions

hip

betw

een

two

quan

titie

s.★

a.

D

eter

min

e an

exp

licit

expr

essi

on, a

re

curs

ive

proc

ess,

or s

teps

for

calc

ulat

ion

from

a c

onte

xt.

i) Ta

sks

have

a re

al-w

orld

co

ntex

t. ii)

Ta

sks

are

limite

d to

line

ar

func

tions

and

exp

onen

tial

func

tions

with

dom

ains

in th

e in

tege

rs.

M1.

F.BF

.A.1

Writ

e a

func

tion

that

de

scrib

es a

rela

tions

hip

betw

een

two

quan

titie

s.★

a. D

eter

min

e an

exp

licit

expr

essi

on, a

recu

rsiv

e pr

oces

s, o

r st

eps

for c

alcu

latio

n fro

m a

con

text

.

i) Ta

sks

have

a re

al-w

orld

co

ntex

t. ii)

Tas

ks a

re li

mite

d to

line

ar

func

tions

and

exp

onen

tial

func

tions

with

dom

ains

in th

e in

tege

rs.

89

F-BF.A.2 2. W

rite arithmetic and geom

etric sequences both recursively and w

ith an explicit form

ula, use them to m

odel situations, and translate betw

een the two

forms. ★


M1.F.BF.A.2 W

rite arithmetic and

geometric sequences w

ith an explicit form

ula and use them to m

odel situations. ★




F-LE.A.1 1. D

istinguish between situations that

can be modeled w

ith linear functions and w

ith 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. b. R

ecognize situations in which one

quantity changes at a constant rate per unit interval relative to another. c. R

ecognize situations in which a

quantity grows or decays by a constant

percent rate per unit interval relative to another.


M1.F.LE.A.1 D

istinguish between

situations that can be modeled w

ith linear functions and w

ith exponential functions. a. R

ecognize that linear functions grow

by equal differences over equal intervals and that exponential functions grow

by equal factors over equal intervals. b. R

ecognize situations in which one

quantity changes at a constant rate per unit interval relative to another. c. R

ecognize situations in which a

quantity grows or decays by a

constant factor per unit interval relative to another.




F-LE.A.2 2. C


etic and geom

etric sequences, given a graph, a description of a relationship, or tw

o input-output pairs (include reading these from

a table).


M1.F.LE.A.2 C


etic and geometric sequences,

given a graph, a table, a description of a relationship, or input-output pairs.




F-LE.A.3 3. O



ial function.


M1.F.LE.A.3 O

bserve using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.

Tasks are limited linear and

exponential functions.

F-LE.B.5 5. Interpret the param


s of a context.


M1.F.LE.B.4 Interpret the param


s of a context.

For example, the total cost of

an electrician who charges 35

dollars for a house call and 50 dollars per hour w

ould be expressed as the function y = 50x + 35. If the rate w

ere raised to 65 dollars per hour,

90

desc

ribe

how

the

func

tion

wou

ld c

hang

e.

Task

s ha

ve a

real

-wor

ld

cont

ext.

G-C

O.A

.1

1. K

now

pre

cise

def

initi

ons

of a

ngle

, ci

rcle

, pe

rpen

dicu

lar l

ine,

par

alle

l lin

e,

and

line

segm

ent,

base

d on

the

unde

fined

not

ions

of p

oint

, lin

e, d

ista

nce

alon

g a

line,

and

dis

tanc

e ar

ound

a

circ

ular

arc

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.A

.1 K

now

pre

cise

de

finiti

ons

of a

ngle

, circ

le,

perp

endi

cula

r lin

e, p

aral

lel l

ine,

and

lin

e se

gmen

t, ba

sed

on th

e un

defin

ed

notio

ns o

f poi

nt, l

ine,

pla

ne, d

ista

nce

alon

g a

line,

and

dis

tanc

e ar

ound

a

circ

ular

arc

.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

G-C

O.A

.2

2. R

epre

sent

tran

sfor

mat

ions

in th

e pl

ane

usin

g, e

.g.,

trans

pare

ncie

s an

d ge

omet

ry s

oftw

are;

des

crib

e tra

nsfo

rmat

ions

as

func

tions

that

take

po

ints

in th

e pl

ane

as in

puts

and

giv

e ot

her p

oint

s as

out

puts

. C

ompa

re tr

ansf

orm

atio

ns th

at p

rese

rve

dist

ance

and

ang

le to

thos

e th

at d

o no

t (e

.g.,

trans

latio

n ve

rsus

hor

izon

tal

stre

tch)

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.A

.2 R

epre

sent

tra

nsfo

rmat

ions

in th

e pl

ane

in

mul

tiple

way

s, in

clud

ing

tech

nolo

gy.

Des

crib

e tra

nsfo

rmat

ions

as

func

tions

th

at ta

ke p

oint

s in

the

plan

e (p

re-

imag

e) a

s in

puts

and

giv

e ot

her

poin

ts (i

mag

e) a

s ou

tput

s. C

ompa

re

trans

form

atio

ns th

at p

rese

rve

dist

ance

and

ang

le m

easu

re to

thos

e th

at d

o no

t (e.

g., t

rans

latio

n ve

rsus

ho

rizon

tal s

tretc

h).

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

G-C

O.A

.3

3. G

iven

a re

ctan

gle,

par

alle

logr

am,

trape

zoid

, or r

egul

ar p

olyg

on, d

escr

ibe

the

rota

tions

and

refle

ctio

ns th

at c

arry

it

onto

itse

lf.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.A

.3 G

iven

a re

ctan

gle,

pa

ralle

logr

am, t

rape

zoid

, or r

egul

ar

poly

gon,

des

crib

e th

e ro

tatio

ns a

nd

refle

ctio

ns th

at c

arry

the

shap

e on

to

itsel

f.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

G-C

O.A

.4

4. D

evel

op d

efin

ition

s of

rota

tions

, re

flect

ions

, and

tran

slat

ions

in te

rms

of

angl

es, c

ircle

s, p

erpe

ndic

ular

line

s,

para

llel l

ines

, and

line

seg

men

ts.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.A

.4 D

evel

op d

efin

ition

s of

ro

tatio

ns, r

efle

ctio

ns, a

nd tr

ansl

atio

ns

in te

rms

of a

ngle

s, c

ircle

s,

perp

endi

cula

r lin

es, p

aral

lel l

ines

, and

lin

e se

gmen

ts.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

G-C

O.A

.5

5. G

iven

a g

eom

etric

figu

re a

nd a

ro

tatio

n, re

flect

ion,

or t

rans

latio

n, d

raw

th

e tra

nsfo

rmed

figu

re u

sing

, e.g

., gr

aph

pape

r, tra

cing

pap

er, o

r geo

met

ry

softw

are.

Spe

cify

a s

eque

nce

of

trans

form

atio

ns th

at w

ill ca

rry a

giv

en

figur

e on

to a

noth

er.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.A

.5 G

iven

a g

eom

etric

fig

ure

and

a rig

id m

otio

n, d

raw

the

imag

e of

the

figur

e in

mul

tiple

way

s,

incl

udin

g te

chno

logy

. Spe

cify

a

sequ

ence

of r

igid

mot

ions

that

will

ca

rry a

giv

en fi

gure

ont

o an

othe

r.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

91

G-C0.B.6

6. Use geom

etric descriptions of rigid m

otions to transform figures and to

predict the effect of a given rigid motion

on a given figure; given two figures, use


rigid motions to decide if they are

congruent.


M1.G

.CO.B.6 U

se geometric

descriptions of rigid motions to

transform figures and to predict the

effect of a given rigid motion on a

given figure; given two figures, use

the definition of congruence in terms

of rigid motions to determ

ine inform

ally if they are congruent.




G-C0.B.7

7. Use the definition of congruence in

terms of rigid m

otions to show that tw



M1.G

.CO.B.7 U

se the definition of congruence in term

s of rigid motions

to show that tw





G-CO

.B.8 8. Explain how

the criteria for triangle congruence (ASA, SAS, and SSS) follow

from

the definition of congruence in term

s of rigid motions.


M1.G

.CO.B.8 Explain how

the criteria for triangle congruence (ASA, SAS, AAS, and SSS) follow

from the

definition of congruence in terms of

rigid motions.




G-CO

.C.9 9. Prove theorem

s about lines and angles. Theorem

s include: vertical angles are congruent; w



the segment’s

endpoints.


M1.G

.CO.C.9 Prove theorem

s about lines and angles.

Proving includes, but is not

limited to, com

pleting partial proofs; constructing tw

o-colum

n or paragraph proofs; using transform

ations to prove theorem

s; analyzing proofs; and critiquing com

pleted proofs. Theorem

s include but are not lim

ited to: vertical angles are congruent; w



the segm

ent’s endpoints.

92

G-C

O.C

.10

10.P

rove

theo

rem

s ab

out t

riang

les.

Theo

rem

s in

clud

e: m

easu

res

of in

terio

ran

gles

of a

tria

ngle

sum

to 1

80°;

bas

ean

gles

of i

sosc

eles

tria

ngle

s ar

ec o

ngru

ent;

the

segm

ent j

oini

ngm

idpo

ints

of t

wo

side

s of

a tr

iang

le is

para

llel t

o th

e th

ird s

ide

and

half

t he

leng

t h; t

he m

edia

ns o

f a tr

iang

le m

eet a

ta

poin

t.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.C

.10

Prov

e th

eore

ms

abou

t tri

angl

es.

.

Pro

ving

incl

udes

, but

is n

ot

limite

d to

, com

plet

ing

parti

al

proo

fs; c

onst

ruct

ing

two-

colu

mn

or p

arag

raph

pro

ofs;

us

ing

trans

form

atio

ns to

pr

ove

theo

rem

s; a

naly

zing

pr

oofs

; and

crit

iqui

ng

com

plet

ed p

roof

s.

Theo

rem

s in

clud

e bu

t are

not

lim

ited

to: m

easu

res

of

inte

rior a

ngle

s of

a tr

iang

le

sum

to 1

80°;

bas

e an

gles

of

isos

cele

s tri

angl

es a

re

cong

ruen

t; th

e se

gmen

t jo

inin

g m

idpo

ints

of t

wo

side

s of

a tr

iang

le is

par

alle

l to

the

third

sid

e an

d ha

lf th

e le

ngth

; th

e m

edia

ns o

f a tr

iang

le

mee

t at a

poi

nt.

G-C

O.C

.11

11.P

rove

theo

rem

s ab

out

para

llelo

gram

s. T

heor

ems

incl

ude:

oppo

site

sid

es a

re c

ongr

uent

, opp

osite

angl

es a

re c

ongr

uent

, the

dia

gona

ls o

f apa

ralle

logr

am b

isec

t eac

h ot

her,

and

conv

erse

ly, r

ecta

ngle

s ar

epa

ralle

logr

ams

with

con

grue

nt d

iago

nals

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

G.C

O.C

.11

Prov

e th

eore

ms

abou

t pa

ralle

logr

ams.

P

rovi

ng in

clud

es, b

ut is

not

lim

ited

to, c

ompl

etin

g pa

rtial

pr

oofs

; con

stru

ctin

g tw

o-co

lum

n or

par

agra

ph p

roof

s;

usin

g tra

nsfo

rmat

ions

to

prov

e th

eore

ms;

ana

lyzi

ng

proo

fs; a

nd c

ritiq

uing

co

mpl

eted

pro

ofs.

Theo

rem

s in

clud

e bu

t are

not

lim

ited

to: o

ppos

ite s

ides

are

co

ngru

ent,

oppo

site

ang

les

are

cong

ruen

t, th

e di

agon

als

of a

par

alle

logr

am b

isec

t ea

ch o

ther

, and

con

vers

ely,

re

ctan

gles

are

par

alle

logr

ams

with

con

grue

nt d

iago

nals

.

S-ID

.A.1

1.R

epre

sent

dat

a w

ith p

lots

on

the

real

num

ber l

ine

(dot

plo

ts, h

isto

gram

s, a

ndbo

x pl

ots)

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M1.

S.ID

.A.1

Rep

rese

nt s

ingl

e or

m

ultip

le d

ata

sets

with

dot

plo

ts,

hist

ogra

ms,

ste

m p

lots

(ste

m a

nd

leaf

), an

d bo

x pl

ots.

Ther

e ar

e no

ass

essm

ent

limits

for t

his

stan

dard

. The

en

tire

stan

dard

is a

sses

sed

in

this

cou

rse.

93

S-ID.A.2 2. U

se statistics appropriate to the shape of the data distribution to com

pare center (m

edian, mean) and spread (interquartile

range, standard deviation) of two or

more different data sets.


M1.S.ID.A.2 U

se statistics appropriate to the shape of the data distribution to com

pare center (m

edian, mean) and spread

(interquartile range, standard deviation) of tw

o or more different

data sets.




S-ID.A.3 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extrem

e data points (outliers).


M1.S.ID.A.3 Interpret differences in

shape, center, and spread in the context of the data sets, accounting for possible effects of extrem

e data points (outliers).




S-ID.B.5 5. Sum

marize categorical data for tw

o categories in tw

o-way frequency tables.

Interpret relative frequencies in the context of the data (including joint, m

arginal, and conditional relative frequencies). R

ecognize possible associations and trends in the data.


S-ID.B.6 6. R

epresent 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 problem

s in the context of the data. U

se given functions or choose a function suggested by the context. E

mphasize linear,

quadratic, and exponential models.

c. Fit a linear function for a scatter plot that suggests a linear association.

i) Tasks have real-world context.

ii) Tasks are limited to linear

functions and exponential functions w

ith domains in the

integers.

M1.S.ID.B.4 R

epresent data on two

quantitative variables on a scatter plot, and describe how



se given functions or choose a function suggested by the context. b. Fit a linear function for a scatter plot that suggests a linear association.

i) Tasks have real-world

context. ii) Tasks are lim

ited to linear functions and exponential functions w

ith domains in the

integers.

S-ID.C.7 7. Interpret the slope (rate of change) and the intercept (constant term

) of a linear m

odel in the context of the data.



M1.S.ID.C.5 Interpret the slope (rate

of change) and the intercept (constant term

) of a linear model in

the context of the data.




S-ID.C.8 8. C

ompute (using technology) and

interpret the correlation coefficient of a linear fit.



M1.S.ID.C.6 C

ompute (using

technology) and interpret the correlation coefficient of a linear fit.




94

S-ID

.C.9

9.D

istin

guis

h be

twee

n co

rrela

tion

and

c aus

atio

n.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

M1.

S.ID

.C.7

Dis

tingu

ish

betw

een

corre

latio

n an

d ca

usat

ion.

Th

ere

are

no a

sses

smen

t lim

its fo

r thi

s st

anda

rd. T

he

entir

e st

anda

rd is

ass

esse

d in

th

is c

ours

e.

★*

= M

odel

ing

Stan

dard

95

Notation 2016-17 Standard

2016-17 Scope and Clarification 2017-18 Standard

2017-18 Scope & Clarifications

N-RN.A.1 1. Explain how

the definition of the m

eaning 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. For example, w

e define 5

1/3 to be the cube root of 5 because we

want (5

1/3) 3 = 5( 1/3) 3 to hold, so (51/3) 3

must equal 5.


M2.N.RN.A.1 Explain how

the definition of the m

eaning of rational exponents follow

s from extending

the properties of integer exponents to those values, allow

ing for a notation for radicals in term

s of rational exponents.

For example, w

e define 5 1/3 to be the cube root of 5 because w

e want (5

1/3) 3 = 5( 1/3) 3 to hold, so (5

1/3) 3 must equal 5.



N-RN.A.2 2. R

ewrite expressions involving radicals

and rational exponents using the properties of exponents.


M2.N.RN.A.2 R

ewrite expressions

involving radicals and rational exponents using the properties of exponents.



N-RN.A.3 3. Explain w

hy the sum or product of tw

o rational num

bers 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.


N-Q.A.2

2. Define appropriate quantities for the

purpose of descriptive modeling.


Math II by ensuring that som

e m

odeling tasks (involving Math I

content or securely held content from

grades 6-8) require the student to create a quantity of interest in the situation being described (i.e., a quantity of interest is not selected for the student by the task). For exam


ight autonom

ously decide that a measure

of center is a key variable in a situation, and then choose to w

ork w

ith the mean.

M2.N.Q

.A.1 Identify, interpret, and justify appropriate quantities for the purpose of descriptive m

odeling.

Descriptive m


ation; choosing appropriate units; etc. Tasks are lim

ited to linear or exponential equations w


N-CN.A.1 1. Know

there is a complex num

ber i such that i 2 = –1, and every com

plex num

ber has the form a + bi w

ith a and b real.


M2.N.CN.A.1 Know

there is a com

plex number i such that i 2 = –1,

and every complex num

ber has the form

a + bi with a and b real.



97

N-C

N.A

.22.

Use

the

rela

tion

i 2 =

–1

and

the

com

mut

ativ

e, a

ssoc

iativ

e, a

nd

dist

ribut

ive

prop

ertie

s to

add

, sub

tract

, an

d m

ultip

ly c

ompl

ex n

umbe

rs.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

N.C

N.A

.2 K

now

and

use

the

rela

tion

i 2 =

–1

and

the

com

mut

ativ

e,

asso

ciat

ive,

and

dis

tribu

tive

prop

ertie

s to

add

, sub

tract

, and

m

ultip

ly c

ompl

ex n

umbe

rs.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

N-C

N.B

.77.

Sol

ve q

uadr

atic

equ

atio

ns w

ith re

al

coef

ficie

nts

that

hav

e co

mpl

ex s

olut

ions

.Th

ere

is n

o ad

ditio

nal s

cope

or

clar

ifica

tion

for t

his

stan

dard

.M

2.N

.CN

.B.3

Sol

ve q

uadr

atic

eq

uatio

ns w

ith re

al c

oeffi

cien

ts th

at

have

com

plex

sol

utio

ns.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

A-SS

E.A.

11.

Inte

rpre

t exp

ress

ions

that

repr

esen

t a

quan

tity

in te

rms

of it

s co

ntex

t. ★

b. In

terp

ret c

ompl

icat

ed e

xpre

ssio

ns b

y vi

ewin

g on

e or

mor

e of

thei

r par

ts a

s a

sing

le e

ntity

. For

exa

mpl

e, in

terp

ret

P(1

+r) n

as

the

prod

uct o

f P a

nd a

fact

or

not d

epen

ding

on

P.

i) Ta

sks

are

limite

d to

qua

drat

ic

expr

essi

ons.

M2.

A.SS

E.A.

1 In

terp

ret e

xpre

ssio

ns

that

repr

esen

t a q

uant

ity in

term

s of

its

con

text

. ★

a. In

terp

ret c

ompl

icat

ed e

xpre

ssio

ns

by v

iew

ing

one

or m

ore

of th

eir p

arts

as

a s

ingl

e en

tity.

For e

xam

ple,

inte

rpre

t P(1

+ r)

n as

the

prod

uct o

f P a

nd a

fact

or

not d

epen

ding

on

P.

Task

s ar

e lim

ited

to q

uadr

atic

ex

pres

sion

s.

A-SS

E.A.

2 U

se th

e st

ruct

ure

of a

n ex

pres

sion

to

iden

tify

way

s to

rew

rite

it. F

or e

xam

ple,

s

ee

x 4

– y

4 as

(x 2 )

2 –

(y 2 )

2 , th

us re

cogn

izin

g it

as

a di

ffere

nce

of s

quar

es th

at c

an b

e fa

ctor

ed a

s (x

2 –

y 2 )(

x 2

+ y

2 ).

i) Ta

sks

are

limite

d to

qua

drat

ic

and

expo

nent

ial e

xpre

ssio

ns,

incl

udin

g re

late

d nu

mer

ical

ex

pres

sion

s.ii)

Exa

mpl

es: S

ee a

n op

portu

nity

to

rew

rite

a 2

+ 9a

+ 1

4 as

(a+7

)(a+

2).

Rec

ogni

ze 5

3 2

- 47

2 as

a d

iffer

ence

of

squ

ares

and

see

an

oppo

rtuni

ty

to re

writ

e it

in th

e ea

sier

-to-

eval

uate

form

(53+

47)(

53-4

7).

M2.

A.SS

E.A.

2 U

se th

e st

ruct

ure

of

an e

xpre

ssio

n to

iden

tify

way

s to

re

writ

e it.

For e

xam

ple,

reco

gniz

e 5

3 2

- 47

2 as

a d

iffer

ence

of

squa

res

and

see

an

oppo

rtuni

ty to

rew

rite

it in

the

easi

er-to

-eva

luat

e fo

rm

(53

+ 47

) (53

– 4

7).

See

an

oppo

rtuni

ty to

rew

rite

a 2

+ 9a

+ 1

4 as

(a +

7) (

a +

2).

Task

s ar

e lim

ited

to n

umer

ical

ex

pres

sion

s an

d po

lyno

mia

l ex

pres

sion

s in

one

var

iabl

e

A-SS

E.B

.33.

Cho

ose

and

prod

uce

an e

quiv

alen

t fo

rm o

f an

expr

essi

on to

reve

al a

nd

expl

ain

prop

ertie

s of

the

quan

tity

repr

esen

ted

by th

e ex

pres

sion

. ★a.

Fac

tor a

qua

drat

ic e

xpre

ssio

n to

re

veal

the

zero

s of

the

func

tion

it de

fines

.b.

Com

plet

e th

e sq

uare

in a

qua

drat

ic

expr

essi

on to

reve

al th

e m

axim

um o

r m

inim

um v

alue

of t

he fu

nctio

n it

defin

es.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

A.SS

E.B

.3 C

hoos

e an

d pr

oduc

e an

equ

ival

ent f

orm

of a

n ex

pres

sion

to

reve

al a

nd e

xpla

in p

rope

rties

of

the

quan

tity

repr

esen

ted

by th

e ex

pres

sion

. ★a.

Fac

tor a

qua

drat

ic e

xpre

ssio

n to

re

veal

the

zero

s of

the

func

tion

it de

fines

.b.

Com

plet

e th

e sq

uare

in a

qu

adra

tic e

xpre

ssio

n in

the

form

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

Ax

2 + Bx + C

where A = 1 to reveal

the maxim

um or m

inimum

value of the function it defines.

A-APR.A.11.U

nderstand that polynomials form

asystem

analogous to the integers,nam

ely, they are closed under theoperations of addition, subtraction,and m

ultiplication; add, subtract, andm

ultiply polynomials


M2.A.APR.A.1 U

nderstand that polynom

ials form a system

analogous to the integers, nam

ely, they are closed under the operations of addition, subtraction, and m

ultiplication; add, subtract, and m

ultiply polynomials.



A-CED.A.11.C

reate equations and inequalities inone variable and use them

to solveproblem

s. Include equations arising fromlinear and quadratic functions, andsim

ple rational and exponentialfunctions.

i)Tasks are limited to quadratic

and exponential equations.ii)Tasks have a real-w

orld context.iii) In sim

pler cases (such asexponential equations w

ith integerexponents), tasks have m

ore of t hehal lm

arks of modeling as a

mathem

atical practice (less definedtasks, m

ore of the modeling cycle,

etc.).

M2.A.CED.A.1 C

reate equations and inequalities in one variable and use them

to solve problems.

Include equations arising from

linear and quadratic functions and rational and exponential functions. Tasks have a real-w

orld context.

A-CED.A.22.C

reate equations in two or m

orevariables to represent relationshipsbetw

een quantities; graph equations onc oordinate axes w

ith labels and scales.


equations.ii)Tasks have a real-w

orld context.iii) Tasks have the hallm

arks ofm

odeling as a mathem

aticalpractice (less defined tasks, m

or eof the m

odeling cycle, etc.).

M2.A.CED.A.2 C

reate equations in tw

o or more variables to represent

relationships between quantities;

graph equations with tw

o variables on coordinate axes w

ith labels and scales.


equationsii)Tasks have a real-w

orldcontext.iii)Tasks have the hallm

arks ofm

odeling as a mathem

atic alpr actice (less defined tasks,m


etc.).

A-CED.A.44.R

earrange formulas to highlight a

quantity of interest, using the same

reasoning as in solving equations. Forexam

ple, rearrange Ohm

’s law V = IR

tohighlight resistance R

.


equations.ii)Tasks have a real-w

orld context.

M2.A.CED.A.3 R

earrange formulas



i)Tasks are limited to quadratic,

square root, cube root, andpiec ew

ise functions.ii) Tasks have a real-w

orld context.

i)Tasks are limited to

quadratic, square root, cuberoot, and piecew

ise functions.ii)Tasks have a real-w

orldcontext.iii)Tasks have the hallm

arks ofm

odeling as a mathem

atic alpr actice (less defined tasks,m


etc.).

A-RE

I.A.1

1.

Exp

lain

eac

h st

ep in

sol

ving

a s

impl

e eq

uatio

n as

follo

win

g fro

m th

e eq

ualit

y of

nu

mbe

rs a

sser

ted

at th

e pr

evio

us s

tep,

st

artin

g fro

m th

e as

sum

ptio

n th

at th

e or

igin

al e

quat

ion

has

a so

lutio

n.

Con

stru

ct a

via

ble

argu

men

t to

just

ify a

so

lutio

n m

etho

d.

i) Ta

sks

are

limite

d to

qua

drat

ic

equa

tions

. M

2.A.

REI.A

.1 E

xpla

in e

ach

step

in

solv

ing

an e

quat

ion

as fo

llow

ing

from

the

equa

lity

of n

umbe

rs

asse

rted

at th

e pr

evio

us s

tep,

st

artin

g fro

m th

e as

sum

ptio

n th

at th

e or

igin

al e

quat

ion

has

a so

lutio

n.

Con

stru

ct a

via

ble

argu

men

t to

just

ify a

sol

utio

n m

etho

d.

Task

s ar

e lim

ited

to li

near

, qu

adra

tic, e

xpon

entia

l eq

uatio

ns w

ith in

tege

r ex

pone

nts,

squ

are

root

, cub

e ro

ot, p

iece

wis

e, a

nd

expo

nent

ial f

unct

ions

.

A-RE

I.B.4

4.

Sol

ve q

uadr

atic

equ

atio

ns in

one

va

riabl

e.

a. U

se th

e m

etho

d of

com

plet

ing

the

squa

re to

tran

sfor

m a

ny q

uadr

atic

eq

uatio

n in

x in

to a

n eq

uatio

n of

the

form

(x

– p

)2 = q

that

has

the

sam

e so

lutio

ns.

Der

ive

the

quad

ratic

form

ula

from

this

fo

rm.

b. S

olve

qua

drat

ic e

quat

ions

by

insp

ectio

n (e

.g.,

for x

2 = 4

9), t

akin

g sq

uare

root

s, c

ompl

etin

g th

e sq

uare

, the

qu

adra

tic fo

rmul

a an

d fa

ctor

ing,

as

appr

opria

te to

the

initi

al fo

rm o

f the

eq

uatio

n. R

ecog

nize

whe

n th

e qu

adra

tic

form

ula

give

s co

mpl

ex s

olut

ions

and

w

rite

them

as

a ±

bi fo

r rea

l num

bers

a

and

b.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

A.RE

I.B.2

Sol

ve q

uadr

atic

eq

uatio

ns a

nd in

equa

litie

s in

one

va

riabl

e.

a. U

se th

e m

etho

d of

com

plet

ing

the

squa

re to

rew

rite

any

quad

ratic

eq

uatio

n in

x in

to a

n eq

uatio

n of

the

form

(x –

p)2 =

q th

at h

as th

e sa

me

solu

tions

. Der

ive

the

quad

ratic

fo

rmul

a fro

m th

is fo

rm.

b. S

olve

qua

drat

ic e

quat

ions

by

insp

ectio

n (e

.g.,

for x

2 = 4

9), t

akin

g sq

uare

root

s, c

ompl

etin

g th

e sq

uare

, kn

owin

g an

d ap

plyi

ng th

e qu

adra

tic

form

ula,

and

fact

orin

g, a

s ap

prop

riate

to th

e in

itial

form

of t

he

equa

tion.

Rec

ogni

ze w

hen

the

quad

ratic

form

ula

give

s co

mpl

ex

solu

tions

and

writ

e th

em a

s a

± bi

for

real

num

bers

a a

nd b

.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

A-RE

I.B.7

7.

Sol

ve a

sim

ple

syst

em c

onsi

stin

g of

a

linea

r equ

atio

n an

d a

quad

ratic

equ

atio

n in

two

varia

bles

alg

ebra

ical

ly a

nd

grap

hica

lly. F

or e

xam

ple,

find

the

poin

ts

of in

ters

ectio

n be

twee

n th

e lin

e y

= –3

x an

d th

e ci

rcle

x2 +

y2 =

3.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

A.RE

I.C.3

Writ

e an

d so

lve

a sy

stem

of l

inea

r equ

atio

ns in

co

ntex

t.

Whe

n so

lvin

g al

gebr

aica

lly,

task

s ar

e lim

ited

to s

yste

ms

of

at m

ost t

hree

equ

atio

ns a

nd

thre

e va

riabl

es. W

ith g

raph

ic

solu

tions

sys

tem

s ar

e lim

ited

to

only

two

varia

bles

.

M2.

A.RE

I.C.4

Sol

ve a

sys

tem

co

nsis

ting

of a

line

ar e

quat

ion

and

a qu

adra

tic e

quat

ion

in tw

o va

riabl

es

alge

brai

cally

and

gra

phic

ally

.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

99

F-IF.B.4 4. For a function that m


een two quantities,

interpret key features of graphs and tables in term


ing key features given a verbal description of the relationship. K

ey features include: intercepts; intervals w


aximum

s and m

inimum

s; symm

etries; end behavior; and periodicity. ★

I)Tasks have a real-world context.

ii) Tasks are limited to quadratic

and exponential functions. The function types listed here are the sam

e as those listed in Math II

column for standards F-IF.6 and F-

IF.9.

M2.F.IF.A.1 For a function that


een two


s of the quantities and sketch graphs show



intercepts; intervals where the

function is increasing, decreasing, positive, or negative; relative m

aximum

s and m

inimum

s; symm


orld context. ii) Tasks are lim

ited to quadratic, exponential functions w

ith integer exponents, square root, and cube root functions.

F-IF.B.5 5.R

elate the domain of a function to its

graph and, where applicable, to the

quantitative relationship it describes. For exam

ple, if the function h(n) gives the num

ber of person-hours it takes to assem



ain for the function. ★



functions.

M2.F.IF.A.2 R

elate the domain of a

function to its graph and, where

applicable, to the quantitative relationship it describes. ★

For example, if the function

h(n) gives the number of

person-hours it takes to assem



ain for the function. Tasks are lim

ited to quadratic, square root, cube root, piecew

ise, and exponential functions.

F-IF.B.6 6. C




a graph. ★



and exponential functions. The function types listed here are the sam

e as those listed in the M

ath II column for standards F-IF.4

and F-IF.9.

M2.F.IF.A.3 C




graph. ★

i) Tasks have a real-world

context. ii) Tasks m

ay involve quadratic, square root, cube root, piecew

ise, and exponential functions.

F-IF.C.7 7. G


bolically and show key features of

the graph, by hand in simple cases and

using technology for more com

plicated

For F-IF.7a: i) Tasks are lim

ited to quadratic functions.

M2.F.IF.B.4 G


bolically and show

key features of the graph, by hand and using technology. ★

M2.F.IF.B.4a – Tasks are

limited to quadratic functions.

M2.F.IF.B.4c – Tasks are

case

s.★

a.

Gra

ph li

near

and

qua

drat

ic fu

nctio

nsan

d sh

ow in

terc

epts

, max

ima,

and

min

ima.

b.G

raph

squ

are

root

, cub

e ro

ot, a

ndpi

ecew

ise-

defin

ed fu

nctio

ns, i

nclu

ding

step

func

tions

and

abs

olut

e va

lue

func

tions

.e.

Gra

ph e

xpon

entia

l and

loga

rithm

i cfu

nctio

ns, s

how

ing

inte

rcep

ts a

nd e

ndbe

hav i

or, a

nd tr

igon

omet

ric fu

nctio

ns,

show

ing

perio

d, m

idlin

e, a

nd a

mpl

itude

.

For F

-IF.7

e:

i)Ta

sks

are

limite

d to

exp

onen

tial

func

tions

.

a.G

raph

line

ar a

nd q

uadr

atic

func

tions

and

sho

win

terc

epts

, max

ima,

and

min

ima.

b.G

raph

squ

are

root

, cub

er o

ot, a

nd p

iece

wis

e-de

fined

func

tions

, inc

ludi

ng s

t ep

f unc

tions

and

abs

olut

e va

lue

f unc

tions

.c.

c. G

raph

exp

onen

ti al a

ndl o

garit

hmic

func

tions

,sh

owin

g in

terc

epts

and

end

beha

vior

.

limite

d to

exp

onen

tial f

unct

ions

.

8.W

rite

a fu

nctio

n de

fined

by

anex

pres

sion

in d

iffer

ent b

ut e

quiv

alen

tfo

rms

to re

veal

and

exp

lain

diff

eren

tpr

oper

ties

of th

e fu

nctio

n.a.

Use

the

proc

ess

of fa

ctor

ing

and

c om

plet

ing

the

squa

re in

a q

uadr

atic

func

tion

to s

how

zer

os, e

xtre

me

valu

es,

and

sym

met

ry o

f the

gra

ph, a

nd in

terp

ret

thes

e in

term

s of

a c

onte

xt.

b.U

se th

e pr

oper

ties

of e

xpon

ents

toin

terp

ret e

xpre

ssio

ns fo

r exp

onen

tial

f unc

tions

. For

exa

mpl

e, id

entif

y pe

rcen

tra

te o

f cha

nge

in fu

nctio

ns s

uch

as y

=(1

.02)

t , y

= (0

.97)

t , y

= (1

.01)

12t ,

y =

(1.2

)t/10 ,

and

clas

sify

them

as

repr

esen

ting

expo

nent

ial g

row

th o

rde

cay.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

F.IF

.B.5

Writ

e a

func

tion

defin

ed

by a

n ex

pres

sion

in d

iffer

ent b

ut

equi

vale

nt fo

rms

to re

veal

and

ex

plai

n di

ffere

nt p

rope

rties

of t

he

func

tion.

a.

Use

the

proc

ess

of fa

ctor

ing

and

c om

plet

ing

the

squa

rein

a q

uadr

atic

func

tion

t osh

ow z

eros

, ext

rem

e va

lues

,an

d sy

mm

etry

of t

he g

raph

,an

d in

terp

ret t

hese

in te

rms

of a

con

text

.b.

b. K

now

and

use

the

prop

ertie

s of

exp

onen

ts to

i nte

rpre

t exp

ress

ions

for

expo

nent

ial f

unct

ions

.

For e

xam

ple,

iden

tify

perc

ent

rate

of c

hang

e in

func

tions

su

ch a

s y

= 2x

, y

= (1

/2)x ,

y =

2-x,

y =

(1/2

)-x.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

F-IF

.C.9

9.C

ompa

re p

rope

rties

of t

wo

func

tions

each

repr

esen

ted

in a

diff

eren

t way

(alg

ebra

ical

ly, g

raph

ical

ly, n

umer

ical

ly in

tabl

es, o

r by

verb

al d

escr

iptio

ns).

For

exam

ple,

giv

en a

gra

ph o

f one

qua

drat

icfu

nctio

n an

d an

alg

ebra

ic e

xpre

ssio

n fo

ran

othe

r, sa

y w

hich

has

the

larg

erm

axim

um.

i)Ta

sks

are

limite

d to

on

quad

ratic

and

expo

nent

ial f

unct

ions

. ii)

Task

sdo

not

hav

e a

real

-wor

ld c

onte

xt.

The

func

tion

type

s lis

ted

here

ar e

the

sam

e as

thos

e lis

ted

in t h

eM

ath

II co

lum

n fo

r sta

ndar

ds F

-IF.4

and

F-IF

.6.

M2.

F.IF

.B.6

Com

pare

pro

perti

es o

f tw

o fu

nctio

ns e

ach

repr

esen

ted

in a

di

ffere

nt w

ay (a

lgeb

raic

ally

, gr

aphi

cally

, num

eric

ally

in ta

bles

, or

by v

erba

l des

crip

tions

).

i)Ta

sks

do n

ot h

ave

a re

al-

wor

ld c

onte

xt.

ii)Ta

sks

may

invo

lve

quad

ratic

,sq

uare

root

, cub

e ro

ot,

piec

ewis

e, a

nd e

xpon

entia

lfu

nctio

ns.

101

F-BF.A.1 1. W


een two quantities. ★

b. Com

bine standard function types using arithm

etic operations. For exam

ple, 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 m

odel.

For F-BF.1a:

i) Tasks have real-world content.

ii) Tasks may involve linear

functions, quadratic functions, and exponential functions.

M2.F.BF.A.1 W


een tw

o quantities. ★ a. D

etermine an explicit expression,

a recursive process, or steps for calculation from

a context. b. C

ombine standard function types

using arithmetic operations.

For M2.F.B

F.A.1a:



ay involve linear and quadratic functions.

F-BF.B.3 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. Experim

ent with cases

and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from

their graphs an d algebraic expressions for them

.

i) Identifying 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) is lim

ited to linear and quadratic functions. ii) Experim

enting with

cases and illustrating an explanation of the effects on the graph using technology is lim

ited to linear functions, quadratic functions, square root functions, cube root functions, piecew

ise-defined functions (including step functions and absolute value functions), and exponential functions. iii) Tasks do not involve recognizing even and odd functions.

M2.F.BF.B.2 Identify the effect on

the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experim

ent with cases

and illustrate an explanation of the effects on the graph using technology.

i) Identifying the effect on the graph of replacing f(x) by f(x) + k, k f(x), and f(x+k) for specific values of k (both positive and negative) is lim

ited to linear, quadratic, and absolute value functions. ii) E

xperimenting w

ith cases and illustrating an explanation of the effects on the graph using technology is lim

ited to linear, quadratic, square root, cube root, and exponential functions. iii) Tasks do not involve recognizing even and odd functions.

G-SRT.A.1

1. Verify experimentally the properties of

dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segm



M2.G

.SRT.A.1 Verify informally the

properties of dilations given by a center and a scale factor.



G-S

RT.A

.2

2. G

iven

two

figur

es, u

se th

e de

finiti

on o

f si

mila

rity

in te

rms

of s

imila

rity

trans

form

atio

ns to

dec

ide

if th

ey a

re

sim

ilar;

expl

ain

usin

g si

mila

rity

trans

form

atio

ns th

e m

eani

ng o

f sim

ilarit

y fo

r tria

ngle

s as

the

equa

lity

of a

ll co

rresp

ondi

ng p

airs

of a

ngle

s an

d th

e pr

opor

tiona

lity

of a

ll co

rresp

ondi

ng p

airs

of

sid

es.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

G.S

RT.A

.2 G

iven

two

figur

es,

use

the

defin

ition

of s

imila

rity

in

term

s of

sim

ilarit

y tra

nsfo

rmat

ions

to

deci

de if

they

are

sim

ilar;

expl

ain

usin

g si

mila

rity

trans

form

atio

ns th

e m

eani

ng o

f sim

ilarit

y fo

r tria

ngle

s as

th

e eq

ualit

y of

all

corre

spon

ding

pa

irs o

f ang

les

and

the

prop

ortio

nalit

y of

all

corre

spon

ding

pa

irs o

f sid

es.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

G-S

RT.A

.3

3. U

se th

e pr

oper

ties

of s

imila

rity

trans

form

atio

ns to

est

ablis

h th

e AA

cr

iterio

n fo

r tw

o tri

angl

es to

be

sim

ilar.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

G.S

RT.A

.3 U

se th

e pr

oper

ties

of

sim

ilarit

y tra

nsfo

rmat

ions

to

esta

blis

h th

e AA

crit

erio

n fo

r tw

o tri

angl

es to

be

sim

ilar.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

G-S

RT.B

.4

4. P

rove

theo

rem

s ab

out t

riang

les.

Th

eore

ms

incl

ude:

a li

ne p

aral

lel t

o on

e si

de o

f a tr

iang

le d

ivid

es th

e ot

her t

wo

prop

ortio

nally

, and

con

vers

ely;

the

Pyt

hago

rean

The

orem

pro

ved

usin

g tri

angl

e si

mila

rity.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

G.S

RT.B

.4 P

rove

theo

rem

s ab

out s

imila

r tria

ngle

s.

Pro

ving

incl

udes

, but

is n

ot

limite

d to

, com

plet

ing

parti

al

proo

fs; c

onst

ruct

ing

two-

colu

mn

or p

arag

raph

pro

ofs;

us

ing

trans

form

atio

ns to

pro

ve

theo

rem

s; a

naly

zing

pro

ofs;

an

d cr

itiqu

ing

com

plet

ed

proo

fs.

Th

eore

ms

incl

ude

but a

re n

ot

limite

d to

: a li

ne p

aral

lel t

o on

e si

de o

f a tr

iang

le d

ivid

es th

e ot

her t

wo

prop

ortio

nally

, and

co

nver

sely

; the

Pyt

hago

rean

Th

eore

m p

rove

d us

ing

trian

gle

sim

ilarit

y.

G-S

RT.B

.5

5. U

se c

ongr

uenc

e an

d si

mila

rity

crite

ria

for t

riang

les

to s

olve

pro

blem

s an

d to

pr

ove

rela

tions

hips

in g

eom

etric

figu

res.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

G.S

RT.B

.5 U

se c

ongr

uenc

e an

d si

mila

rity

crite

ria fo

r tria

ngle

s to

so

lve

prob

lem

s an

d to

just

ify

rela

tions

hips

in g

eom

etric

figu

res.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

G-S

RT.C

.6

6. U

nder

stan

d th

at b

y si

mila

rity,

sid

e ra

tios

in ri

ght t

riang

les

are

prop

ertie

s of

th

e an

gles

in th

e tri

angl

e, le

adin

g to

de

finiti

ons

of tr

igon

omet

ric ra

tios

for

acut

e an

gles

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

G.S

RT.C

.6 U

nder

stan

d th

at b

y si

mila

rity,

sid

e ra

tios

in ri

ght t

riang

les

are

prop

ertie

s of

the

angl

es in

the

trian

gle,

lead

ing

to d

efin

ition

s of

tri

gono

met

ric ra

tios

for a

cute

ang

les.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

103

G.SRT.C.7

7. Explain and use the relationship betw

een the sine and cosine of com

plementary angles.


M2.G

.SRT.C.7 Explain and use the relationship betw

een the sine and cosine of com

plementary angles.



G-SRT.C.8

8. Use trigonom

etric ratios and the Pythagorean Theorem

to solve right triangles in applied problem

s. ★


M2.G

.SRT.C.8 Solve triangles. ★ a. Know

and use trigonometric ratios

and the Pythagorean Theorem to

solve right triangles in applied problem

s. b. Know

and use the Law of Sines

and the Law of C

osines to solve triangles in applied problem

s. R

ecognize when it is appropriate to

use each.

Am

biguous cases will not be

included in assessment.

G-G

MD.A.1

1. Give an inform

al argument for the

formulas for the circum

ference of a circle, area of a circle, volum

e of a cylinder, pyram

id, and cone. Use

dissection arguments, C

avalieri’s principle, and inform

al limit argum

ents.


M2.G

.GM

D.A.1 Give an inform

al argum

ent for the formulas for the

circumference of a circle and the

volume and surface area of a

cylinder, cone, prism, and pyram

id.

Informal argum

ents may

include but are not limited to

using the dissection argument,

applying Cavalieri’s principle,

and constructing informal lim

it argum

ents.

G-G

MD.A.3

3. Use volum

e formulas for cylinders,

pyramids, cones, and spheres to solve

problems. ★


M2.G

.GM

D.A.2 Know and use

volume and surface area form

ulas for cylinders, cones, prism

s, pyram

ids, and spheres to solve problem

s. ★



S-ID.B.6 6. R


variables on a scatter plot, and describe how



se given functions or choose a function suggested by the context. E

mphasize linear,


b. Informally assess the fit of a function

by plotting and analyzing residuals.

For S-ID.A.6a:

i) Tasks have real-world context.


functions. For S-ID

.A.6b: i) Tasks have a real-w

orld context. ii) Tasks are lim

ited to linear functions, quadratic functions, and exponential functions w

ith domains

in the integers.

M2.S.ID.A.1 R




s in the context of the data.

Use given functions or choose

a function suggested by the context. E

mphasize linear,

quadratic, and exponential m

odels. Exponential functions

are limited to those w

ith dom

ains in the integers. Tasks have a real-w

orld context.

S-CP

.A.1

1.

Des

crib

e ev

ents

as

subs

ets

of a

sa

mpl

e sp

ace

(the

set o

f out

com

es)

usin

g ch

arac

teris

tics

(or c

ateg

orie

s) o

f th

e ou

tcom

es, o

r as

unio

ns,

inte

rsec

tions

, or c

ompl

emen

ts o

f oth

er

even

ts (“

or,”

“and

,” “n

ot”).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

S.CP

.A.1

Des

crib

e ev

ents

as

subs

ets

of a

sam

ple

spac

e (th

e se

t of

out

com

es) u

sing

cha

ract

eris

tics

(or c

ateg

orie

s) o

f the

out

com

es, o

r as

uni

ons,

inte

rsec

tions

, or

com

plem

ents

of o

ther

eve

nts

(“or,”

“a

nd,”

“not

”).

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

S-CP

.A.2

2.

Und

erst

and

that

two

even

ts A

and

B

are

inde

pend

ent i

f the

pro

babi

lity

of A

an

d B

occ

urrin

g to

geth

er is

the

prod

uct

of th

eir p

roba

bilit

ies,

and

use

this

ch

arac

teriz

atio

n to

det

erm

ine

if th

ey a

re

inde

pend

ent.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

S.CP

.A.2

Und

erst

and

that

two

even

ts A

and

B a

re in

depe

nden

t if

the

prob

abili

ty o

f A a

nd B

occ

urrin

g to

geth

er is

the

prod

uct o

f the

ir pr

obab

ilitie

s, a

nd u

se th

is

char

acte

rizat

ion

to d

eter

min

e if

they

ar

e in

depe

nden

t.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

S-CP

.A.3

3.

Und

erst

and

the

cond

ition

al p

roba

bilit

y of

A g

iven

B a

s P

(A a

nd B

)/P(B

), an

d in

terp

ret i

ndep

ende

nce

of A

and

B a

s sa

ying

that

the

cond

ition

al p

roba

bilit

y of

A

giv

en B

is th

e sa

me

as th

e pr

obab

ility

of

A, a

nd th

e co

nditi

onal

pro

babi

lity

of B

gi

ven

A is

the

sam

e as

the

prob

abilit

y of

B

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M2.

S.CP

.A.3

Kno

w a

nd u

nder

stan

d th

e co

nditi

onal

pro

babi

lity

of A

giv

en

B a

s P

(A a

nd B

)/P(B

), an

d in

terp

ret

inde

pend

ence

of A

and

B a

s sa

ying

th

at th

e co

nditi

onal

pro

babi

lity

of A

gi

ven

B is

the

sam

e as

the

prob

abili

ty o

f A, a

nd th

e co

nditi

onal

pr

obab

ility

of B

giv

en A

is th

e sa

me

as th

e pr

obab

ility

of B

.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

st

anda

rd is

ass

esse

d in

this

co

urse

.

S-CP

.A.4

4.

Con

stru

ct a

nd in

terp

ret t

wo-

way

fre

quen

cy ta

bles

of d

ata

whe

n tw

o ca

tego

ries

are

asso

ciat

ed w

ith e

ach

obje

ct b

eing

cla

ssifi

ed. U

se th

e tw

o-w

ay

tabl

e as

a s

ampl

e sp

ace

to d

ecid

e if

even

ts a

re in

depe

nden

t and

to

appr

oxim

ate

cond

ition

al p

roba

bilit

ies.

Fo

r exa

mpl

e, c

olle

ct d

ata

from

a ra

ndom

sa

mpl

e of

stu

dent

s in

you

r sch

ool o

n th

eir f

avor

ite s

ubje

ct a

mon

g m

ath,

sc

ienc

e, a

nd E

nglis

h. E

stim

ate

the

prob

abilit

y th

at a

rand

omly

sel

ecte

d st

uden

t fro

m y

our s

choo

l will

favo

r sc

ienc

e gi

ven

that

the

stud

ent i

s in

tent

h gr

ade.

Do

the

sam

e fo

r oth

er s

ubje

cts

and

com

pare

the

resu

lts.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

105

★* = M

odeling Standard

S-CP.A.55.R

ecognize and explain the conceptsof conditional probability andindependence in everyday language ande veryday situations. For exam

ple,com

pare the chance of having lungcancer if you are a sm

oker with t he

c hance of being a smoker if you have

lung cancer.


M2.S.CP.A.4 R

ecognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.



S-CP.A.66.Find the conditional probability of Agiven B

as the fraction of B’s outcomes

that also belong to A, and interpret t heansw


odel.


M2.S.CP.B.5 Find the conditional

probability of A given B as the fraction of B’s outcom

es that also belong to A and interpret the answ

er in term

s of the model.

For example, a teacher gave

two exam

s. 75 percent passed the first exam

and 25 percent passed both. W

hat percent w

ho passed the first exam also

passed the second exam?



S-CP.A.77.Apply the Addition R

ule, P(A or B) =P(A) + P(B) – P(A and B), and interpretthe answ


odel.


M2.S.CP.B.6 Know

and apply the Addition R

ule, P

(A or B

) = P(A) + P

(B) –P

(A and

B), and interpret the answ

er in terms

of the model.

For example, in a m

ath class of 32 students, 14 are boys and 18 are girls. O

n a unit test 6 boys and 5 girls m

ade an A. If a student is chosen at random

from

a class, what is the

probability of choosing a girl or an A

student?



Nota

tion

2016

-17

Stan

dard

20

16-1

7 Sc

ope

and

Clar

ifica

tion

2017

-18

Stan

dard

20

17-1

8 Sc

ope

& Cl

arifi

catio

ns

N-Q

.A.2

2.

Def

ine

appr

opria

te q

uant

ities

for t

he

purp

ose

of d

escr

iptiv

e m

odel

ing.

Th

is s

tand

ard

will

be a

sses

sed

in

Mat

h III

by

ensu

ring

that

som

e m

odel

ing

task

s (in

volv

ing

Mat

h III

co

nten

t or s

ecur

ely

held

con

tent

fro

m p

revi

ous

grad

es a

nd c

ours

es)

requ

ire th

e st

uden

t to

crea

te a

qu

antit

y of

inte

rest

in th

e si

tuat

ion

bein

g de

scrib

ed (i

.e.,

a qu

antit

y of

in

tere

st is

not

sel

ecte

d fo

r the

st

uden

t by

the

task

). Fo

r exa

mpl

e,

in a

situ

atio

n in

volv

ing

perio

dic

phen

omen

a, th

e st

uden

t mig

ht

auto

nom

ousl

y de

cide

that

am

plitu

de is

a k

ey v

aria

ble

in a

si

tuat

ion,

and

then

cho

ose

to w

ork

with

pea

k am

plitu

de.

M3.

N.Q

.A.1

Iden

tify,

inte

rpre

t, an

d ju

stify

app

ropr

iate

qua

ntiti

es fo

r the

pu

rpos

e of

des

crip

tive

mod

elin

g.

Des

crip

tive

mod

elin

g re

fers

to

unde

rsta

ndin

g an

d in

terp

retin

g gr

aphs

; ide

ntify

ing

extra

neou

s in

form

atio

n; c

hoos

ing

appr

opria

te u

nits

; etc

. Th

ere

are

no a

sses

smen

t lim

its

for t

his

stan

dard

. The

ent

ire

stan

dard

is a

sses

sed

in th

is

cour

se.

A-SS

E.B.

2

2. U

se th

e st

ruct

ure

of a

n ex

pres

sion

to

iden

tify

way

s to

rew

rite

it. F

or e

xam

ple,

se

e x4 –

y4 a

s (x

2 )2 –

(y2 )

2 , th

us re

cogn

izin

g it

as a

diff

eren

ce o

f squ

ares

that

can

be

fact

ored

as

(x2 –

y2 )

(x2 +

y2 )

.

i) Ta

sks

are

limite

d to

pol

ynom

ial

and

ratio

nal e

xpre

ssio

ns.

ii) E

xam

ples

: see

x4 –

y4 a

s (x

2 )2 –

(y

2 )2 ,

thus

reco

gniz

ing

it as

a

diffe

renc

e of

squ

ares

that

can

be

fact

ored

as

(x2 –

y2 )

(x2 +

y2 )

. In

the

equa

tion

x2 +

2x

+ 1

+ y2 =

9, s

ee a

n op

portu

nity

to re

writ

e th

e fir

st th

ree

term

s as

(x+1

)2 , th

us r

ecog

nizi

ng

the

equa

tion

of a

circ

le w

ith ra

dius

3

and

cent

er

(-1, 0

). Se

e (x

2 + 4

)/(x2 +

3) a

s

( (x2 +

3) +

1 )/

(x2 +

3), t

hus

reco

gniz

ing

an o

ppor

tuni

ty to

writ

e it

as

1 +

1/(x

2 + 3

).

M3.

A.SS

E.A.

1 U

se th

e st

ruct

ure

of

an e

xpre

ssio

n to

iden

tify

way

s to

re

writ

e it.

For e

xam

ple,

see

2x4 +

3x2 –

5

as it

s fa

ctor

s (x

2 – 1

) and

(2x2 +

5)

; see

x4 –

y4 as

(x2 )

2 –

(y2 )

2, t

hus

reco

gniz

ing

it as

a d

iffer

ence

of

squa

res

that

can

be

fact

ored

as

(x2 –

y2 )

(x2 +

y2 )

; see

(x2 +

4)/(

x2 + 3

)

as ((

x2 + 3

) + 1

)/(x2 +

3),

thus

re

cogn

izin

g an

opp

ortu

nity

to

writ

e it

as 1

+ 1

/(x2 +

3).

Task

s ar

e lim

ited

to

poly

nom

ial,

ratio

nal,

or

expo

nent

ial e

xpre

ssio

ns.

M

3.A.

SSE.

B.2

Cho

ose

and

prod

uce

an e

quiv

alen

t for

m o

f an

expr

essi

on

to re

veal

and

exp

lain

pro

perti

es o

f th

e qu

antit

y re

pres

ente

d by

the

expr

essi

on.★

For e

xam

ple

the

expr

essi

on

1.15

t can

be

rew

ritte

n as

((1

.15)

1/12

)12t ≈

1.0

1212

t to

reve

al th

at th

e ap

prox

imat

e eq

uiva

lent

mon

thly

inte

rest

rate

107

a. Use the properties of exponents

to rewrite expressions for

exponential functions.

is 1.2% if the annual rate is

15%.


context. As described in the standard, there is an interplay betw

een the mathem

atical structure of the expression and the structure of the situation such that choosing and producing an equivalent form

of the expression reveals som

ething about the situation.

ii) Tasks are limited to

exponential expressions with

rational or real exponents

A-SSE.B.4

4. Derive the form

ula for the sum of a

finite geometric series (w

hen the com

mon ratio is not 1), and use the

formula to solve problem

s. For example,

calculate mortgage paym

ents. ★


M3.A.SSE.B.3 R

ecognize a finite geom

etric series (when the com

mon

ratio is not 1), and know and use the

same form

ula to solve problems in

context.



A-APR.A.2 2. Know

and apply the Rem

ainder Theorem

: For a polynomial p(x) and a

number a, the rem

ainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).


M3.A.APR.A.1 Know

and apply the R

emainder Theorem

: For a polynom

ial p(x) and a num

ber a, the remainder on division

by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).



A-APR.A.3 3. Identify zeros of polynom

ials when

suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom

ial.


M3.A.APR.A.2 Identify zeros of

polynomials w

hen suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynom

ial.

For example, find the zeros of

(x2 - 1)(x

2 + 1). There are no assessm

ent limits

for this standard. The entire standard is assessed in this course.

A-APR.B.4 4. Prove polynom

ial identities and use them

to describe numerical relationships.

For example, the polynom

ial identity (x

2 + y2) 2 = (x

2 – y2) 2 + (2xy) 2 can be

used to generate Pythagorean triples.


M3.A.APR.B.3 Know

and use polynom

ial identities to describe num

erical relationships.

For example, com

pare (31)(29) = (30 + 1)(30 – 1) = 30

2 – 12

with (x + y)(x – y) = x

2 – y2.

1

A-AP

R.C.

66.

Rew

rite

sim

ple

ratio

nal e

xpre

ssio

ns in

diffe

rent

form

s; w

rite

a(x)

/b(x

) in

the

form

q(x)

+ r(

x)/b

(x),

whe

re a

(x),

b(x)

, q(x

),an

d r(x

) are

pol

ynom

ials

with

the

degr

eeof

r(x)

less

than

the

degr

ee o

f b(x

), us

ing

insp

ectio

n, lo

ng d

ivis

ion,

or,

for t

he m

ore

c om

plic

ated

exa

mpl

es, a

com

pute

ral

gebr

a sy

stem

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

A.AP

R.C.

4 R

ewrit

e ra

tiona

l ex

pres

sion

s in

diff

eren

t for

ms.

Th

ere

are

no a

sses

smen

t lim

its

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is

cour

se.

A-CE

D.A.

11.

Cre

ate

equa

tions

and

ineq

ualit

ies

inon

e va

riabl

e an

d us

e th

em to

sol

vepr

oble

ms.

Incl

ude

equa

tions

aris

ing

from

linea

r and

qua

drat

ic fu

nctio

ns, a

ndsi

mpl

e ra

tiona

l and

exp

onen

t ial

f unc

tions

.

i)Ta

sks

are

limite

d to

sim

ple

ratio

nal o

r exp

onen

tial e

quat

ions

.ii)

Task

s ha

ve a

real

-wor

ld c

onte

xt.

M3.

A.CE

D.A.

1 C

reat

e eq

uatio

ns

and

ineq

ualit

ies

in o

ne v

aria

ble

and

use

them

to s

olve

pro

blem

s.

i)Ta

sks

are

limite

d to

poly

nom

ial,

ratio

nal,

abso

lute

v alu

e, e

xpon

entia

l, or

loga

rithm

ic fu

nctio

ns.

ii)Ta

sks

have

a re

al-w

orld

cont

ext.

A-CE

D.A.

22.

Cre

ate

equa

tions

in tw

o or

mor

eva

riabl

es to

repr

esen

t rel

atio

nshi

psbe

twee

n qu

antit

ies;

gra

ph e

quat

ions

on

coor

dina

te a

xes

with

labe

ls a

nd s

cale

s.

i)Ta

sks

are

limite

d to

sim

ple

poly

nom

ial,

ratio

nal,

or e

xpon

entia

leq

uat io

ns ii

) Tas

ks h

ave

a re

al-

wor

ld c

onte

xt.

M3.

A.CE

D.A.

2 C

reat

e eq

uatio

ns in

tw

o or

mor

e va

riabl

es to

repr

esen

t re

latio

nshi

ps b

etw

een

quan

titie

s;

grap

h eq

uatio

ns w

ith tw

o va

riabl

es

on c

oord

inat

e ax

es w

ith la

bels

and

sc

ales

.

i)Ta

sks

have

a re

al-w

orld

cont

ext.

ii)Ta

sks

are

limite

d to

poly

nom

ial,

ratio

nal,

abso

lute

valu

e, e

xpon

entia

l, or

loga

rithm

ic fu

nctio

ns.

M3.

A.CE

D.A.

3 R

earra

nge

form

ulas

to

hig

hlig

ht a

qua

ntity

of i

nter

est,

usin

g th

e sa

me

reas

onin

g as

in

solv

ing

equa

tions

.

i)Ta

sks

have

a re

al-w

orld

cont

ext.

ii)Ta

sks

are

limite

d to

poly

nom

ial,

ratio

nal,

abso

lute

valu

e, e

xpon

entia

l, or

loga

rithm

ic fu

nctio

ns.

A-RE

I.A.1

1.Ex

plai

n ea

ch s

tep

in s

olvi

ng a

sim

ple

equa

tion

as fo

llow

ing

from

the

equa

lity

ofnu

mbe

rs a

sser

ted

at th

e pr

evio

us s

tep,

star

ting

from

the

assu

mpt

ion

that

the

orig

inal

equ

atio

n ha

s a

solu

tion.

Con

stru

ct a

via

ble

argu

men

t to

just

ify a

s olu

tion

met

hod.

i)Ta

sks

are

limite

d to

sim

ple

ratio

nal o

r rad

ical

equ

atio

ns.

M3.

A.RE

I.A.1

Exp

lain

eac

h st

ep in

so

lvin

g an

equ

atio

n as

follo

win

g fro

m th

e eq

ualit

y of

num

bers

as

serte

d at

the

prev

ious

ste

p,

star

ting

from

the

assu

mpt

ion

that

the

orig

inal

equ

atio

n ha

s a

solu

tion.

C

onst

ruct

a v

iabl

e ar

gum

ent t

o ju

stify

a s

olut

ion

met

hod.

Task

s ar

e lim

ited

to s

impl

e ra

tiona

l or r

adic

al e

quat

ions

A-RE

I.A.2

2.So

lve

sim

ple

ratio

nal a

nd ra

dica

leq

uatio

ns in

one

var

iabl

e, a

nd g

ive

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

A.RE

I.A.2

Sol

ve ra

tiona

l and

ra

dica

l equ

atio

ns in

one

var

iabl

e,

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd. T

he e

ntire

109

examples show

ing how extraneous

solutions may arise.

and identify extraneous solutions w

hen they exist. standard is assessed in this course.

A-REI.D.11 11. Explain w


here 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 approxim

ately, e.g., using technology to graph the functions, m

ake tables of values, or find successive approxim

ations. Include cases where f(x )

and/or g(x) are linear, polynomial,

rational, absolute value, exponential, and logarithm

ic functions. ★

i) Tasks may involve any of the

function types mentioned in the

standard.

M3.A.REI.B.3 Explain w


here 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 approxim

ate solutions using technology. *

Tasks may include cases

where f(x) and/or g(x) are

linear, polynomial, rational,

absolute value, exponential, or logarithm

ic functions.

F-IF.B.4 4. For a function that m


een two quantities,

interpret key features of graphs and tables in term


ing key features given a verbal description of the relationship. K

ey features include: intercepts; intervals w


aximum

s and m

inimum

s; symm

etries; end behavior; and periodicity. ★



ial, logarithm

ic, and trigonometric

functions. The function types listed here are the sam

e as those listed in the M

ath III column for standards

F-IF.6 and F-IF.9.

M3.F.IF.A.1 For a function that


een two


s of the quantities and sketch graphs show



intercepts; intervals where the

function is increasing, decreasing, positive, or negative; relative m

aximum

s and m

inimum

s; symm


orld context. ii) Tasks m

ay involve polynom


ic functions.

F-IF.B.6 6. C




a graph. ★



ial, logarithmic,

and trigonometric functions. The function types

listed here are the same as those listed in the

Math III colum


M3.F.IF.A.2 C




graph. *



ay involve polynom


ic functions.

F-IF.C.7 7. G


bolically and show key features of

the graph, by hand in simple cases and

using technology for more com

plicated cases. ★

For F-IF.7e: i) Tasks are lim

ited to logarithmic and

trigonometric functions.

M3.F.IF.B.3 G


bolically and show

key features of the graph, by hand and using technology. ★

* a.

Graph linear and quadratic



c. G

raph

pol

ynom

ial f

unct

ions

, id

entif

ying

zer

os w

hen

suita

ble

fact

oriz

atio

ns a

re a

vaila

ble,

and

sho

win

g en

d be

havi

or.

e. G

raph

exp

onen

tial a

nd lo

garit

hmic

fu

nctio

ns, s

how

ing

inte

rcep

ts a

nd e

nd

beha

vior

, and

trig

onom

etric

func

tions

, sh

owin

g pe

riod,

mid

line,

and

am

plitu

de.

func

tions

and

sho

w

inte

rcep

ts, m

axim

a, a

nd

min

ima.

b.

G

raph

squ

are

root

, cub

e ro

ot, a

nd p

iece

wis

e-de

fined

fu

nctio

ns, i

nclu

ding

ste

p fu

nctio

ns a

nd a

bsol

ute

valu

e fu

nctio

ns.

c.

Gra

ph p

olyn

omia

l fun

ctio

ns,

iden

tifyi

ng z

eros

whe

n su

itabl

e fa

ctor

izat

ions

are

av

aila

ble

and

show

ing

end

beha

vior

. d.

G

raph

exp

onen

tial a

nd

loga

rithm

ic fu

nctio

ns,

show

ing

inte

rcep

ts a

nd e

nd

beha

vior

.

F-IF

.C.9

9.

Com

pare

pro

perti

es o

f tw

o fu

nctio

ns

each

repr

esen

ted

in a

diff

eren

t way

(a

lgeb

raic

ally

, gra

phic

ally

, num

eric

ally

in

tabl

es, o

r by

verb

al d

escr

iptio

ns).

For

exam

ple,

giv

en a

gra

ph o

f one

qua

drat

ic

func

tion

and

an a

lgeb

raic

exp

ress

ion

for

anot

her,

say

whi

ch h

as th

e la

rger

m

axim

um.

i) Ta

sks

have

a re

al-w

orld

con

text

. ii)

Tas

ks m

ay in

volv

e po

lyno

mia

l, lo

garit

hmic

, and

trig

onom

etric

fu

nctio

ns.

The

func

tion

type

s lis

ted

here

are

th

e sa

me

as th

ose

liste

d in

the

Mat

h III

col

umn

for s

tand

ards

F-

IF.4

and

F-IF

.6.

M3.

F.IF

.B.4

Com

pare

pro

perti

es o

f tw

o fu

nctio

ns e

ach

repr

esen

ted

in a

di

ffere

nt w

ay (a

lgeb

raic

ally

, gr

aphi

cally

, num

eric

ally

in ta

bles

, or

by v

erba

l des

crip

tions

).

Task

s m

ay in

volv

e po

lyno

mia

l, ex

pone

ntia

l, an

d lo

garit

hmic

fu

nctio

ns.

F-BF

.B.3

3.

Iden

tify

the

effe

ct o

n th

e gr

aph

of

repl

acin

g f(x

) by

f(x) +

k, k

f(x)

, f(k

x), a

nd

f(x +

k) f

or s

peci

fic v

alue

s of

k (b

oth

posi

tive

and

nega

tive)

; fin

d th

e va

lue

of k

gi

ven

the

grap

hs. E

xper

imen

t with

cas

es

and

illus

trate

an

expl

anat

ion

of th

e ef

fect

s on

the

grap

h us

ing

tech

nolo

gy.

Incl

ude

reco

gniz

ing

even

and

odd

fu

nctio

ns fr

om th

eir g

raph

s an

d al

gebr

aic

expr

essi

ons

for t

hem

.

i) Ta

sks

are

limite

d to

exp

onen

tial,

poly

nom

ial,

loga

rithm

ic, a

nd

trigo

nom

etric

func

tions

. ii)

Tas

ks m

ay in

volv

e re

cogn

izin

g ev

en a

nd o

dd fu

nctio

ns. T

he

func

tion

type

s lis

ted

in n

ote

(i) a

re

the

sam

e as

thos

e lis

ted

in th

e M

ath

III c

olum

n fo

r sta

ndar

ds F

-IF

.4, F

-IF.6

and

F-IF

.9.

M3.

F.BF

.A.1

Iden

tify

the

effe

ct o

n th

e gr

aph

of re

plac

ing

f(x) b

y f(x

) + k

, kf

(x),

f(kx)

, and

f(x

+ k)

for s

peci

fic

valu

es o

f k (b

oth

posi

tive

and

nega

tive)

; fin

d th

e va

lue

of k

giv

en

the

grap

hs. E

xper

imen

t with

cas

es

and

illus

trate

an

expl

anat

ion

of th

e ef

fect

s on

the

grap

h us

ing

tech

nolo

gy.

i) Ta

sks

may

invo

lve

poly

nom

ial,

expo

nent

ial,

and

loga

rithm

ic fu

nctio

ns.

ii) T

asks

may

invo

lve

reco

gniz

ing

even

and

odd

fu

nctio

ns.

F-BF

.B.4

4.

Fin

d in

vers

e fu

nctio

ns.

a. S

olve

an

equa

tion

of th

e fo

rm f(

x) =

c

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n in

form

atio

n fo

r thi

s M

3.F.

BF.A

.2 F

ind

inve

rse

func

tions

. a.

Fin

d th

e in

vers

e of

a fu

nctio

n Th

ere

are

no a

sses

smen

t lim

its

for t

his

stan

dard

. Th

e en

tire

111

for a simple function f that has an

inverse and write an expression for the

inverse. For example,

f(x) =2x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

standard. w

hen the given function is one-to-one.

standard is assessed in this course.

M3.F.LE.A.1 O



ial function.



F-LE.B.4 4. For exponential m

odels, express as a logarithm

the solution to abct = d w

here a, c, and d are num

bers and the base b is 2, 10, or e; evaluate the logarithm

using technology.


M3.F.LE.A.2 For exponential

models, express as a logarithm

the solution to ab

ct = d where a, c, and d

are numbers and the base b is 2, 10,

or e; evaluate the logarithm using

technology.



F-TF.A.1 1. U

nderstand radian measure of an

angle as the length of the arc on the unit circle subtended by the angle.


M3.F.TF.A.1 U

nderstand and use radian m

easure of an angle. a. U

nderstand radian measure of an

angle as the length of the arc on the unit circle subtended by the angle. b. U

se the unit circle to find sin θ, cos θ, and tan θ w

hen θ is a com

monly recognized angle

between 0 and 2π

.

Com

monly recognized angles

include all multiples of nπ

/6 and nπ /4, w

here n is an integer. There are no assessm

ent limits


F-TF.A.2 2. Explain how

the unit circle in the coordinate plane enables the extension of trigonom






M3.F.TF.A.2 Explain how

the unit circle in the coordinate plane enables the extension of trigonom







F-TF.B.5 5. C

hoose trigonometric functions to

model periodic phenom

ena with

specified amplitude, frequency, and

midline. ★


F-TF

.C.8

8.

Pro

ve th

e Py

thag

orea

n id

entit

y si

n2 (θ)

+

cos2 (

θ) =

1 a

nd u

se it

to fi

nd s

in(θ

), co

s(θ)

, or t

an(θ

) giv

en s

in(θ

), co

s(θ)

, or

tan(

θ) a

nd th

e qu

adra

nt o

f the

ang

le.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

F.TF

.B.3

Use

trig

onom

etric

id

entit

ies

to fi

nd v

alue

s of

trig

fu

nctio

ns.

a. G

iven

a p

oint

on

a ci

rcle

cen

tere

d at

the

orig

in, r

ecog

nize

and

use

the

right

tria

ngle

ratio

def

initi

ons

of s

in

θ, c

os θ

, and

tan

θ to

eva

luat

e th

e tri

gono

met

ric fu

nctio

ns.

b. G

iven

the

quad

rant

of t

he a

ngle

, us

e th

e id

entit

y si

n2 θ +

cos

2 θ =

1 to

fin

d si

n θ

give

n co

s θ,

or v

ice

vers

a.

Com

mon

ly re

cogn

ized

ang

les

incl

ude

all m

ultip

les

of n

π /6

an

d

nπ /4

, whe

re n

is a

n in

tege

r.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

F-CO

.D.1

2 12

. Mak

e fo

rmal

geo

met

ric c

onst

ruct

ions

w

ith a

var

iety

of t

ools

and

met

hods

(c

ompa

ss a

nd s

traig

hted

ge, s

tring

, re

flect

ive

devi

ces,

pap

er fo

ldin

g,

dyna

mic

geo

met

ric s

oftw

are,

etc

.).

Cop

ying

a s

egm

ent;

copy

ing

an a

ngle

; bi

sect

ing

a se

gmen

t; bi

sect

ing

an a

ngle

; co

nstru

ctin

g pe

rpen

dicu

lar l

ines

, inc

ludi

ng th

e pe

rpen

dicu

lar b

isec

tor o

f a li

ne s

egm

ent;

and

cons

truct

ing

a lin

e pa

ralle

l to

a gi

ven

line

thro

ugh

a po

int n

ot o

n th

e lin

e.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.C

O.A

.1 M

ake

form

al

geom

etric

con

stru

ctio

ns w

ith a

va

riety

of t

ools

and

met

hods

(c

ompa

ss a

nd s

traig

hted

ge, s

tring

, re

flect

ive

devi

ces,

pap

er fo

ldin

g,

dyna

mic

geo

met

ric s

oftw

are,

etc

.).

Con

stru

ctio

ns in

clud

e bu

t are

no

t lim

ited

to: c

opyi

ng a

se

gmen

t; co

pyin

g an

ang

le;

bise

ctin

g a

segm

ent;

bise

ctin

g an

ang

le; c

onst

ruct

ing

perp

endi

cula

r lin

es, i

nclu

ding

th

e pe

rpen

dicu

lar b

isec

tor o

f a

line

segm

ent;

cons

truct

ing

a lin

e pa

ralle

l to

a gi

ven

line

thro

ugh

a po

int n

ot o

n th

e lin

e,

and

cons

truct

ing

the

follo

win

g ob

ject

s in

scrib

ed in

a c

ircle

: an

equi

late

ral t

riang

le, s

quar

e,

and

a re

gula

r hex

agon

.

F-CO

.D.1

3 13

. Con

stru

ct a

n eq

uila

tera

l tria

ngle

, a

squa

re, a

nd a

regu

lar h

exag

on in

scrib

ed

in a

circ

le.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

G-C

.A.1

1.

Pro

ve th

at a

ll ci

rcle

s ar

e si

mila

r. Th

ere

is n

o ad

ditio

nal s

cope

or

clar

ifica

tion

for t

his

stan

dard

. M

3.G

.C.A

.1 R

ecog

nize

that

all

circ

les

are

sim

ilar.

Th

ere

are

no a

sses

smen

t lim

its

for t

his

stan

dard

. Th

e en

tire

stan

dard

is a

sses

sed

in th

is

cour

se.

G-C

.A.2

2.

Iden

tify

and

desc

ribe

rela

tions

hips

am

ong

insc

ribed

ang

les,

radi

i, an

d ch

ords

. Inc

lude

the

rela

tions

hip

betw

een

cent

ral,

insc

ribed

, and

circ

umsc

ribed

an

gles

; ins

crib

ed a

ngle

s on

a d

iam

eter

ar

e rig

ht a

ngle

s; th

e ra

dius

of a

circ

le is

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.C

.A.2

Iden

tify

and

desc

ribe

rela

tions

hips

am

ong

insc

ribed

an

gles

, rad

ii, a

nd c

hord

s.

Incl

ude

the

rela

tions

hip

betw

een

cent

ral,

insc

ribed

, and

ci

rcum

scrib

ed a

ngle

s; in

scrib

ed

angl

es o

n a

diam

eter

are

righ

t an

gles

; the

radi

us o

f a c

ircle

is

perp

endi

cula

r to

the

tang

ent

113

perpendicular to the tangent where the

radius intersects the circle. w

here the radius intersects the circle, and properties of angles for a qu adrilateral inscribed in a circle.

G-C.A.3

3.Construct the inscribed and

circumscribed circles of a triangle, and

prove properties of angles for aquadrilateral inscribed in a circle.


M3.G

.C.A.3 Construct the incenter

and circumcenter of a triangle and

use their properties to solve problem

s in context.



G-C.B.5

5.Derive using sim

ilarity the fact that thelength of the arc intercepted by an angleis proportional to the radius, and defi net he radian m

easure of the angle as theconstant of proportionality; derive theform

ula for the area of a sector.


M3.G

.C.B.4 Find the area of a sector of a circle in a real w

orld context.

For example, use proportional

relationships and angles m

easured in degrees or radians.



G-G

PE.A.11.D

erive the equation of a circle of givencenter and radius using the PythagoreanTheorem

; complete the square to fi nd

t he center and radius of a circle given byan equation.


M3.G

.GPE.A.1 W

rite the equation of a circle of given center and radius using the Pythagorean Theorem

.



G-G

PE.A.22.D

erive the equation of a parabolagiven a focus and directrix.


G-G

PE.B.44.U

se coordinates to prove simple

geometric theorem

s algebraically. Forexam

ple, prove or disprove that a figuredefined by four given points in thecoordinate plane is a rectangle; prove ordisprove that the point (1, √3) lies on t hec ircle centered at the origin andcontaining the point (0, 2).


M3.G

.GPE.B.2 U

se coordinates to prove sim

ple geometric theorem

s algebraically.

For example, prove or disprove

that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √ 3 ) lies on the circle centered at the origin and containing the point (0, 2).



G-G

PE.B

.55.

Prov

e th

e sl

ope

crite

ria fo

r par

alle

lan

d pe

rpen

dicu

lar l

ines

and

use

them

toso

lve

geom

etric

pro

blem

s (e

.g.,

find

the

equa

tion

of a

line

par

alle

l or

perp

endi

cula

r to

a gi

ven

line

that

pas

ses

thro

ugh

a gi

ven

poin

t).

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.G

PE.B

.3 P

rove

the

slop

e cr

iteria

for p

aral

lel a

nd p

erpe

ndic

ular

lin

es a

nd u

se th

em to

sol

ve

geom

etric

pro

blem

s.

For e

xam

ple,

find

the

equa

tion

of a

line

par

alle

l or

perp

endi

cula

r to

a gi

ven

line

that

pas

ses

thro

ugh

a gi

ven

poin

t.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

G-G

PE.B

.66.

Find

the

poin

t on

a di

rect

ed li

nese

gmen

t bet

wee

n tw

o gi

ven

poin

ts th

atpa

rtitio

ns th

e se

gmen

t in

a gi

ven

ratio

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.G

PE.B

.4 F

ind

the

poin

t on

a di

rect

ed li

ne s

egm

ent b

etw

een

two

give

n po

ints

that

par

titio

ns th

e se

gmen

t in

a gi

ven

ratio

.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

G-G

PE.B

.77.

Use

coo

rdin

ates

to c

ompu

tepe

rimet

ers

of p

olyg

ons

and

area

s of

trian

gles

and

rect

angl

es, e

.g.,

usin

g th

edi

stan

ce fo

rmul

a.★

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.G

PE.B

.5 U

se c

oord

inat

es to

co

mpu

te p

erim

eter

s of

pol

ygon

s an

d ar

eas

of tr

iang

les

and

rect

angl

es.*

For e

xam

ple,

use

the

dist

ance

fo

rmul

a.

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

G-G

MD.

A.4

4.Id

entif

y th

e sh

apes

of t

wo-

dim

ensi

onal

cros

s-se

ctio

ns o

f thr

ee-d

imen

sion

alob

ject

s, a

nd id

entif

y th

ree-

dim

ensi

onal

obje

cts

gene

rate

d by

rota

tions

of t

wo-

dim

ensi

onal

obj

ects

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

G-M

G.A

.11.

Use

geo

met

ric s

hape

s, th

eir

mea

sure

s, a

nd th

eir p

rope

rties

tode

s crib

e ob

ject

s (e

.g.,

mod

elin

g a

tree

trunk

or a

hum

an to

rso

as a

cyl

inde

r).★

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.M

G.A

.1 U

se g

eom

etric

sh

apes

, the

ir m

easu

res,

and

thei

r pr

oper

ties

to d

escr

ibe

obje

cts.

★

Ther

e ar

e no

ass

essm

ent l

imits

fo

r thi

s st

anda

rd.

The

entir

e st

anda

rd is

ass

esse

d in

this

co

urse

.

G-M

G.A

.22.

Appl

y co

ncep

ts o

f den

sity

bas

ed o

nar

ea a

nd v

olum

e in

mod

elin

g si

tuat

ions

(e.g

., pe

rson

s pe

r squ

are

mile

, BTU

s pe

rcu

bic

foot

).★

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

G-M

G.A

.33.

Appl

y ge

omet

ric m

etho

ds to

sol

vede

sign

pro

blem

s (e

.g.,

desi

gnin

g an

obje

ct o

r stru

ctur

e to

sat

isfy

phy

sica

lco

nstra

ints

or m

inim

ize

cost

; wor

king

with

typo

grap

hic

grid

sys

tem

s ba

sed

on

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

G.M

G.A

.2 A

pply

geo

met

ric

met

hods

to s

olve

real

wor

ld

prob

lem

s.★

Geo

met

ric m

etho

ds m

ay

incl

ude

but a

re n

ot li

mite

d to

us

ing

geom

etric

sha

pes,

the

prob

abilit

y of

a s

hade

d re

gion

, de

nsity

, and

des

ign

prob

lem

s.

115

ratios). ★ There are no assessm

ent limits


S-ID.A.44.U

se the mean and standard deviation

of a data set to fit it to a normal

distribution and to estimate populati on

per centages. Recognize that there are

data sets for which such a procedure is

not appropriate. Use calculators,

spreadsheets, and tables to estimate

areas under the normal curve.


M3.S.ID.A.1 U

se the mean and

standard deviation of a data set to fit it to a norm

al distribution and to estim

ate population percentages using the Em

pirical Rule.



S-ID.B.66.R


variables on a scatter plot, and describehow

the variables are related.a.Fit a function to the data; usefunctions fitted to data to solve problem

sin the context of the data. U

se givenfunctions or choose a function suggestedby the context. E

mphasize linear,


b.Informally assess the fit of a function

by plotting and analyzing residuals.

For S-ID.6a:


ii)Tasks are limited to exponential

functions with dom

ains not in thei ntegers and trigonom

etricfunctions

M3.S.ID.B.2 R



the variables are related. a.Fit a function to the data; usef unctions fitted to data to solveproblem

s in the context of the data.U

se given functions or choos e afunction suggested by the context.b.Fit a linear function for a scatterplot that suggests a linearassociation.

Use given functions or choose

a function suggested by the context.


c ontext.

ii)T asks are limited to linear,

quadratic, and exponentialfunctions w

ith domains not in

t he integers.

S-IC.A.11.U

nderstand statistics as a process form

aking inferences about populationparam

eters based on a random sam

pl ef rom

that population.


M3.S.IC.A.1 U

nderstand statistics as a process for m

aking inferences about population param

eters based on a random

sample from

that population.



S-IC.A.22.D

ecide if a specified model is

consistent with results from

a given data-generating process, e.g., usi ngs im

ulation. For example, a m

odel says aspinning coin falls heads up w

ithprobability 0.5. W

ould a result of 5 tailsin a row

cause you to question them

odel?


M3.S.IC.A.2 D

ecide if a specified m

odel is consistent with results from

a given data-generating process, e.g., using sim

ulation.

For example, a m

odel says a spinning coin falls heads up w

ith probability 0.5. Would a

result of 10 heads in a row

cause you to question the m

odel? There are no assessm

ent limits


★* =

Mod

elin

g St

anda

rd

S-IC

.B.3

3.R

ecog

nize

the

purp

oses

of a

nddi

ffere

nces

am

ong

sam

ple

surv

eys,

expe

rimen

ts, a

nd o

bser

vatio

nal s

tudi

es;

expl

ain

how

rand

omiz

atio

n re

late

s to

each

.

Ther

e is

no

addi

tiona

l sco

pe o

r cl

arifi

catio

n fo

r thi

s st

anda

rd.

M3.

S.IC

.B.3

Rec

ogni

ze th

e pu

rpos

es o

f and

diff

eren

ces

amon

g sa

mpl

e su

rvey

s, e

xper

imen

ts, a

nd

obse

rvat

iona

l stu

dies

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117

Standards Comparison ChartStandard

CodingDropped

from CourseAdded to

Course No change

118

Revised


Standards Comparison Activity

1. If you had to summarize the revisions to these selected standards in twenty words or less, what would you say?

Notes:

Small Group Consensus:

Whole Group Consensus:

119

Notes:

Appointment with Peers

Please meet with your first partner to discuss the following:

• How will these changes impact your classroom?

• What are your takeaways from modules 1–3?

• How does this align to your observation rubric?

120

Module 3 Review

Strong Standards


• The instructional expectations remain the same and are still the focus of the standards.

• The revised standards represent a stronger foundation that will support the progression of rigorous standards throughout the grade levels.

• The revised standards improve connections:

– within a single grade level, and

– between multiple grade levels.

121

Strong StandardsHigh

Expectations



Part 2: Developing a Deeper UnderstandingModule 4: Diving Into 9–12 Math

123

TAB PAGE

Goals• Concisely describe a course based on its introduction.

emphasis of the standard—its intent and purpose.

• Use the KUD approach to guide planning, instruction, and assessment.

High Expectations


Strong Standards






125

• Develop a means for deconstructing standards to determine the mathematical

Closer Look

Take a few minutes to read the overview page for your grade level and think about how this relates to the overarching revisions we have just seen.

Notes:

Now summarize your course in 140 characters. Write your tweet to inform others regarding what is included in your grade.

My Tweet:

126

“With my ears to the ground, listening to my students, my eyes are focused on the mathematical horizon.”

—Ball, 1993

Intent and Purpose of the Standards

Analyzing Standards

A1.F.IF.C.7 (Note also: A2.F.IF.B.4 and M2.F.IF.B.5)

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.

Notes:

127

We are going to look closely at A1.F.IF.C.7.

Know (facts, vocabulary)

Understand(concepts,

generalizations)

Do(verbs, skills)

Essential Questions:

Instruction & Assessment (What does the math look like?)

128

• What is it that the standard wants the student to know, understand, and do?

• KUD – helps to maintain focus in differentiated instruction

– Know: facts, vocabulary, properties, procedures, etc.

– Understand: concepts, ideas, etc.

– Do: tasks, approaches, assessment problems, etc.

• The two go together: What is the intent and purpose of the standard and how do I put this into instructional form?

From Standard to Instruction: KUD

Know, Understand, and Do

Know

Understand

Do

129

You Try One!

Know (facts, vocabulary)

Understand(concepts,

generalizations)

Do(verbs, skills)

Essential Questions:

Instruction & Assessment (What does the math look like?)

130

Module 4 Review

High Expectations


• Concisely describe a course based on its introduction.

• Develop a means for deconstructing standards to determine the mathematical emphasis, intent, and purpose of the standard.

• Use the KUD approach to guide planning, instruction, and assessment.

131


Expectations



Part 3: Instructional ShiftsModule 5: Revisiting the SMP’s and Instructional Expectations

133

TAB PAGE

Goals

• Revisit the concepts of focus, coherence, and rigor and how they play out in instruction.

• Discuss the purpose and place of the content and practice standards.

• Share instructional strategies related to the Standards for Mathematical Practice.

• Discuss research on the influence of mindsets in the math classroom.

High Expectations


Strong Standards






135

• Explore students’ mathematical mindsets.

Why Standards for Mathematical Practice?

“Beginning to experiment with small changes to one’s teaching practice and collaborating with colleagues can help move students toward the vision of mathematical proficiency described in the Standards for Mathematical Practice”

—Mateas, 2016

• Tell us what students should know and be able to do

• So, what should students know and do?

• Content Standards

• Standards for Mathematical Practice

• Literacy Skills

• Knowing that these are what students need to learn, teachers determine howto teach these.

136

Mindset

Notes:

137

• The TN Academic Standards for Mathematics may seem challenging for students whose mindsets have been fixed by their past experiences in mathematics classrooms.

• As teachers, we are best positioned to influence students’ mathematical mindsets through our actions and practices in the mathematics classroom.

_____________________________ -Intelligence is a fixed trait. You cannot change it.

_____________________________ -You can grow your intelligence through effort.

Notes:

Why Address Mindsets?“If there’s a threat of being wrong every time I raise my hand, and being wrong is a bad thing, then very quickly I decide math isn’t for me, I don’t like this, I’m not a smart person.”

—Noah Heller, Harvard Graduate School of Education, 2016

Everyone can learn math to

the highest levels

Mistakes are valuable

Questions are important

Math is about creativity and making sense

Math is about connections

and communicating

Math class is about learning not preforming

Depth is more important than

speed

138

Instructional Expectations

Focus1. In your grade-level groups, discuss ways you could respond if someone asks you

the following question, “Why focus? There’s so much math that students could be learning. Why limit them?”

2. Review the table below and answer the questions, “Which two of the following represent areas of major focus for the indicated grade?”

8 Standard form of a linear equation.

Define, evaluate, and compare functions.

Understand and apply the Pythagorean Theorem.

Alg. 1 Zeros of polynomials Linear and quadratic functions.

Creating equations to model situations.

Alg. 2 Exponential andlogarithmic functions. Polar coordinates Using fractions to

model situations.

139


Coherence

In the space below, copy all of the standards for your assigned domain and note how coherence is evident in the vertical progression of these standards.

Grade Standard Summary of the Standard (If the standard has sub-parts,summarize each sub-part.)

140


Rigor 1. Make a true statement: Rigor = _______________ + _______________ + _______________

2. In your groups, discuss ways to respond to one of the following comments: “These standards are expecting that we just teach rote memorization. Seems like a step backwards to me.” Or “I’m not going to spend time on fluency—it should just be a natural outcome of conceptual understanding.”

3. The shift towards rigor is required by the standards. Find and copy in the space below standards which specifically set expectations for each component of rigor.

141

EvidenceStandard


The instructional expectations are an essential component of the standards and provide guidance for how the standards should be taught and implemented.

Module 5 Review

• We connected the instructional shifts to the standards and our classroom practices.

• We explored students’ mathematical mindsets.

• We shared instructional strategies related to the Standards for Mathematical Practice.

142

What do the instructional shifts look like in the classroom?


Expectations



Part 3: Instructional ShiftsModule 6: Literacy Skills for Mathematical Proficiency

143

TAB PAGE

Goal

• Develop a better understanding of the Literacy Skills for Mathematical Proficiency.

High Expectations


Strong Standards






144

Reflect on ways literacy skills are already present in your mathematics classroom.

Literacy in your Math Classroom

Literacy Skills for Math ProficiencyCommunication in mathematics requires literacy skills in reading, vocabulary, speaking, listening, and writing.

145

Reading

Vocabulary

Speaking & Listening

Writing

Literacy Skills for Math ProficiencyCategorize the strategies you listed and discussed with your table partners in the chart below.

146

Literacy Skills for Mathematical Proficiency

1. Read and annotate your assigned section from pages 13–14 of the TN Math Standards. Work with your group to present this information to your colleagues.

2. Use the chart below to take notes and highlight the main ideas of each section.

Reading

Vocabulary

Speaking & Listening

Writing

147

Strategies for Incorporating LiteracyText Features

Highlight key symbols

Color code steps or circle action steps

Place a box around key terms in vocabulary

Notes:

148

Brandon and Allison participate in an annual community 5k. Brandon can run at a rate

of 3 miles in 24 minutes. Allison can run at a rate of 1 mile in 9 minutes. Who has the

faster rate? Who finished the race first?

Let’s Try a Problem

Strategies for Incorporating LiteracyGraphic Organizers

Graphs

Tables

Four Corner’s and a Diamond

Frayer’s Model

Four Square Graphic Organizer

Semantic Grid Analysis

Originally from Teaching Children Mathematics, © November 2009, Mathematical graphic organizers, p. 222.

May be adapted for personal use with students.

in

What do you

need to find?

Four-Corners-and-a-Diamond Math Graphic Organizer

What do you already know? Brainstorm ways to solve this problem.

Try it here. Explanations you need to include in your

extended-response write up

149

Notes:

Four Stages of Word Knowledge

Mathematics Vocabulary

150

2. Students have seen/heard the word but do not know the definition.

1. Students have never encountered the word before.

3. Students know the word but rely on context to define it.

4. Students know the word and can use it comfortably.

Mathematical Vocabulary• Student achievement is dependent upon students’ reading comprehension and

content area learning.

• Math vocabulary is decontextualized because they are not in everyday conversations.

• Terms in math can have multiple meanings– i.e. table, origin, and leg.

• Mathematical terms can have specific meanings – i.e. average, reflection.

• Students need to develop a conceptual meaning and read use the words accurately.

Frayer’s Model

151

Module 6 Review



• Literacy skills in the math classroom will support students’ understanding of the content standards.

• When students can read, write, and speak about math ideas, connections are made between concepts.

152

Making Connections

153

Notes:


Please meet with your second partner to discuss the following:

• What are your key takeaways from today?

• How does the align to your observation rubric?

154

Strong Standards

High Expectations



Part 4: Assessment and MaterialsModule 7: Connecting Standards and Assessment

155

TAB PAGE

Goals

• Discuss the role assessment plays in the integrated system of learning.

• Discuss the cycle of assessment.

• Discuss the areas of focus for standards-aligned assessments.

• Review and write mathematics assessment items.

High Expectations


Strong Standards






156

Connecting Standards and Assessment

Assessment

Curriculum

Instruction

Teacher Development

Standards

Assessment is_____________________________________________________________________________.

Considering this definition of assessment, what are educators “making a judgement about” when assessing students?

157

The Cycle of Assessment

“The good news is that research has shown for years that consistently applying principles of assessment for learning has yielded remarkable, if not unprecedented, gains in student achievement, especially for low achievers.”

—Black & Wiliam, 1998

Teach

Assess

Analyze

Action

158


Areas of Focus1. Intent of the Assessment

• Summative

• Formative

2. Content and structure of Assessments

3. Analysis of Assessments

Teach

Assess

Analyze

Action

Standards Aligned Assessment

159

“Benchmark assessments, either purchased by the district or from commercial vendors or developed locally, are generally meant to measure progress toward state or district content standards and to predict performance on large-scale summative tests. A common misconception is that this level of assessment is automatically formative.”

—Stephen and Jan Chappuis 2012

Intent of Assessments


• Summative• Formative

2. Content and Structure of Assessments


How are the results used?

Formative Summative

160

Intent of Assessments


• Summative

• Formative



Things to think about…Universal Design Principles:

• No barriers

• Accessible for all students

• Upholds the expectations of our state standards

Notes:

161

Developing a Classroom Assessment

Identify targeted

standards

What essential understandings do I want my students to display mastery

of now?

Deconstruct standards

What types of questions

should I ask?

Will this generate the

data that I really need?

Identify essential understandings

Notes:

162

Inventory for a Classroom Assessment

Purpose of Assessment

Formative

What items do I have? Assess Items

What items do I still need? Create Items

Summative

What items do I have? Assess Items

What items do I still need? Create Items

Notes:

163

Item ReviewStandard:4.OA.A.3: Solve multi-step contextual 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.

Item 2: Samantha bought stickers.

• She bought 6 packs of stickers.• Each pack has 12 stickers.• She got 8 more stickers from a

friend.

How many stickers does Samantha have in all?A. 26B. 64C. 72D. 80

Item 1: Samantha bought stickers.

• She bought 6 packs of stickers.• Each pack has 12 stickers.• She got 8 more stickers from a

friend.

How many stickers does Samantha have in all?A. 76B. 78C. 80D. 82

Which item is better?

Notes:

164

Item Review

Assessment Terminology

Item Type

Selected response

Open response

Verbal

Extended writing

Item Components

Stimulus

Stem

Key

Distractor

165

Rationale

Examining Items: Formative vs. Summative

• What is the question actually asking?

• Is the question aligned to the depth of the standard?

• Are the answers precise?

• Is the wording grade appropriate?

• Is the question aligned to the standard?

• Do the distractors give insight into student thinking?

• Is the entire standard assessed?

• Is the question precise?

• Is there a better way to assess the standard?

166

You will be looking at five items. Decide if you would keep them, revise them in some way, or throw the item out all together. Look first at the items independently. Then you may work with a partner to complete the activity.

Item Assessment Activity

Item #2

G.SRT.B.4 (M2.G.SRT.B.4)

Know and write the equation of a circle of given center and radius using the Pythagorean Theorem.

Item #1

M3.G.GPE.A.1 (G.GPE.A.1)

Prove theorems about similar triangles.

Right Triangle ABC has side lengths 5, 12 and 13. If Triangle DEF is similar to ABC and has one side length 60, what are the possible missing side lengths of DEF?

A. 10 and 24B. 25 and 65C. 30 and 78D. 36 and 96E. 100 and 125F. 144 and 156

The equation for a given circle is 2x2 + 2y2 – 8x – 12y + 8 = 0.

What is the radius of the circle?

A. 2B. 3C. 4D. 12

167

For each of the provided formative assessment items, think about the things we just discussed. Would you include it on a formative assessment when paired with the provided standard?

Item Assessment Activity

Item #3

A2.A.REI.D.6 (M1.A.REI.C.4)

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 approximate solutions using technology.

!(")=|"|− 2 and #(")= −|"|+2

Graph f(x) and g(x). Identify all solutions to the equation f(x)=g(x) on the graph.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

Item #4

M1.F.IF.C.7 (A1.F.IF.C.8)

Two savings plans are represented by the functions f(x) and g(x) respectively, where x is the number of months and f(x) and g(x) represent the value of each account in dollars.

What is the average rate of change of the higher earning account after 6 months?

x g(x)

1 50

2 200

3 450Write your answer in the space provided.

168

Share one or two “ah-ha” moments from this activity with your neighbor.

Item Assessment ActivityItem #5

A2.A.APR.A.2 (M3.A.APR.A.2)

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Identify all zeroes for the following:

$ = 2"' − 5"* − 3", − 2"- + 5" + 3

169

Creating Formative Items

Before you actually start writing items:

• Read the standards carefully with the assessment purpose in mind. Ask yourself: “What skills/knowledge are the standards asking the student to display?”

• Revisit the “I can” statements or “essential questions” you wrote for the standard(s). They may provide guidance as you write items.

• Brainstorm.

Revisiting Standard A1.F.IF.C.7

Determine if each equation will have a minimum or a maximum value. You do not have to provide any coordinates. Match the equations on the left with the correct choice on the top.

FORMATIVE Assessment

A1.F.IF.C.7 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.

Maximum Minimum

$ = (" − 2)-+4

$ = −1(" − 2)-+4

$ = −2"- + 4" − 2

$ = 2"- − 8" + 1

170

• Think about the purpose of the assessment as a whole. Is it formative or summative?

Creating Formative Items

Revisiting Standard A1.F.IF.C.7

Did we cover all aspects of the standard with these items?

Identify the zeroes for the following quadratic equation:

$ = 2"- − 4x + 1

For a quadratic function, complete the square in order to identify the zeroes.

Interpret zeros and extreme values for quadratic equations in terms of a context.

A ball is thrown straight up, from 4 m above the ground with a velocity of 20 m/s. Itsheight is modeled by the following equation:

ℎ = 4 + 208 −58-

How long will it take the ball to hit the ground?

A. 5 secondsB. 4.2 secondsC. 2.2 seconds D. 1 second

Item #2

Item #3

171

Review

• Formative Assessments may need items that scaffold in order for the teacher to diagnose what a student does/does not understand.

• It is very difficult to formatively assess student understanding through a single item.

• Don’t forget the principles of universal design.

• You will be provided a set of standards.

• You and a partner will be writing items to post for our gallery walk.

– On selected response items you do not have to post the rationale for the distractors.

– Please post the coding for the standard(s) to which your items are written.

Item Writing-Your Turn

Selected Response

Multiple Choice Multiple Select

• Effectively writing “I can” or “essential questions” helps target assessment items specifically to standards.

• It’s important to ask yourself the nine essential questions during item review or item writing.

172

Option 1 Option 2

1. Choose three of the standards.2. Write an item to assess each

standard that you would use on a formative assessment.

3. Try to write at least one multiple choice or multiple select item. Focus on writing distractors that provide instructional information.

1. Choose one standard.2. Write three formative assessment

items to the single standard that you selected. Make sure that each item requires students to demonstrate a different level of understanding of the standard.

3. Try to write at least one multiple choice or multiple select item. Focus on writing distractors that provide instructional information.

Item Writing: Your Turn

Use this space to write out your standard(s) and assessment item(s).

173

Gallery WalkAs you review your colleagues’ items, look for similarities and differences in the items created.

• What was challenging about this experience?

• What did you learn from this experience?

• What supports do you need to better understand the relationship between standards and assessments in this way?

Reflection

Reflect on your experience evaluating and creating assessment items and discuss the following:

Notes:

174

Analyzing Assessments


• Summative

• Formative



• Is the data ______________________________________________________________ ?

• How is it analyzed?

• On which questions ________________________________________________________ ? Why?

• On which questions ________________________________________________________ ? Why?

• Were there issues with…

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________ ?

Analysis of Assessment

175

Taking Action

• How is instruction changing/adapting as a result of student data?

• Are results shared with all stakeholders (including students)?

• Are assessments adapted to address weaknesses found?

Assessment Instruction

“The assessments will produce no formative benefit if teachers administer them, report the results, and then continue with instruction as previously planned.”

—Stephen and Jan Chappuis, 2012

Notes:

176

Summary


Teach

Assess

Analyze

Action



177

Notes:


Please meet with your third partner to discuss the following:

• What are your takeaways from module 7?

• How does this align to your observation rubric?

178

Strong Standards

High Expectations



Part 4: Assessment and MaterialsModule 8: Evaluating Instructional Materials

179

TAB PAGE

Goals

• Know which key criteria to use for reviewing materials, lessons, and/or units for alignment and quality.

How do we know that our instructional materials address the depth of the content and the instructional shifts of focus, coherence, and rigor of the TN State Standards?

Key Question

High Expectations


Strong Standards






180

• Examine the TEAM rubric to define what is meant by standards based materials.

Standards Based Materials and Practice

“…teachers have a responsibility to make day-to-day instructional choices that ensure that students work with problems that engage their interest and their intellect.”

—Cathy L. Seeley, 2014

Reflect on Our Practice

When your students’ work is on public display, in the hallway or shared with families, can anyone see the math?

If anyone looked at your students’ work, would they be able to see the math or would they be left asking “where’s the math?”

Notes:

181

Are the materials and the instructional practices you are using focused on the mathematics?

Standards Based Materials and PracticeTEAM ConnectionActivities & Materials• Support the lesson objective

• Are challenging

• Sustain students’ attention

• Elicit a variety of thinking

• Provide time for reflection

• Provide opportunities for student-to-student interaction

• Provide students with choices

• Incorporate technology

• Induce curiosity & suspense

• In addition sometimes activities are game-like, involve simulations, require creating products, and demand self-direction and self-monitoring.

• The preponderance of activities demand complex thinking and analysis

• Texts & task are appropriately complex

TEAM ConnectionProblem Solving

• Abstraction

• Categorization

• Predicting Outcomes

• Improving Solutions

• Generating Ideas

• Creating & Designing

• Observing & Experimenting

• Drawing Conclusions/Justifying Solutions

• Identify Relevant/Irrelevant Information

182


Effective Mathematics Teaching Practices1. Establish mathematics goals to focus learning.

2. Implement tasks that promote reasoning and problem solving.

3. Use and connect mathematical representations.

4. Facilitate meaningful mathematical discourse.

5. Pose purposeful questions.

6. Build procedural fluency from conceptual understanding.

7. Support productive struggle in learning mathematics.

8. Elicit and use evidence of student thinking.

• What content standard do you think these activities address?

• Where is the evidence of student understanding of the mathematical content?

Notes:

183

Missing Angle Activity


Notes:

Missing Angle Activity

If a teacher was trying to address the depth of the content standard 8.G.A.3 , does the Missing Angle Activity accomplish this goal?

8.G.A.3. 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.

184

Criteria for Alignment and Quality

Research

Notes:

“A curriculum is more than a collection of activities.”

—from the Curriculum Principle in Principles and Standards for School Mathematics

A well-articulated curriculum will:

• Make clear the most important mathematics of the grade level.

• Specify when concepts and skills are introduced and when they should be mastered.

• Detail how student conceptual understanding of big ideas develops across units and across multiple grade levels.

When choosing instructional materials, what should a teacher consider?

185

Criteria for Alignment and Quality

Course Standards Instructional Shifts

Clear Progressions Access for All Students

Quality

Materials Review InstrumentWhen reviewing materials, it is important to have a deep understanding of the standards and a deep understanding of the review instrument before looking at the materials.

• Section I: Non-Negotiable Alignment Criteria

– Part A: Standards

– Part B: Shifts

• Focus

• Rigor

• Coherence

• Section II: Additional Alignment Criteria and Indicators of Quality

– Part A: Key areas of focus

– Part B: Student engagement and instructional focus

– Part C: Monitoring student progress

186

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: Do

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ify

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s: Use

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ta to

info

rm

instru

ction

.

Evaluating Instructional Materials: Best Practices

• It’s important to review instructional materials you use to determine where you have strong alignment to standards and where you may have gaps to fill.

• School leaders and teachers should engage in reviewing instructional materials on an ongoing basis to develop pedagogy and capacity.

• There is a new adoption.

• Current materials have gaps that may require supplemental materials.

• They are looking for supplemental instructional materials.

Notes:

188

Teachers need to review materials when:

Supplemental Materials

Let’s Discuss:

• What resources do you have on hand?

• Where do you find supplemental materials?

• How can you use this process to evaluate supplemental materials?

Notes:

Reviewing Materials: A RecapAs you look for materials…

• Is it aligned to the standards?

• Does it reflect high leverage best practices?

• Is it accessible for ALL students?

• Does it lead to students being able to demonstrate mastery of the standard?

189

Potential Gaps in Materials

Grades 6-8:

• Shifted Compound Probability standard

• Moved from seventh to eighth grade

• Revised Geometry standards

• Removed from seventh grade: slice of 3-dimensional objects

• Removed from eighth grade: congruency and similarity of 2-dimensional objects

Grades 9-12:

• Shifted a number of standards from Algebra II and Integrated Math III to the Additional Math Courses

Notes:

190

Module 8 Review



The review process of instructional materials will:

alignment or gaps, and

§ Result in wise decisions about how best to use the materials already on-site to teach the new standards to mastery OR effectively fill any gaps uncovered in the review process.

191

§ Deepen understanding of the standards,

§ Make use of review instruments to analyze materials to determine

Notes:


Please meet with your fourth partner to discuss the following:

• How can the materials review instrument lead to improved student outcomes in your classroom?

192

Strong Standards

High Expectations



Part 5: Putting It All TogetherModule 9: Instructional Planning

193

TAB PAGE

Goals

• Develop lesson planning techniques to strengthen the understanding of the relationship between standards and practice.

• Create lessons based on the revised standards to be used for instruction.

High Expectations




Strong Standards




194

• Understand intentional instruction as a bridge between strong standards and assessment.

Designing Effective Learning Experiences

“…teachers have a responsibility to make day-to-day instructional choices that ensure that students work with problems that engage their interest and their intellect.”

—Cathy L. Seeley, 2014

Notes:

195

Designing Effective Learning Experiences

What is intentional instruction?

Notes:

Know your standard

Essential Questions or I

Can’s

KUD

196

• Standards are driving your instruction.• What does "intentional" mean?• Gather evidence of learning (assessments)• What are your end goals?

Designing Effective Learning ExperiencesPutting It All Together

Now it’s your turn! Use this space for your small group planning.

197

• We brought together concepts studied in the previous modules to plan for instruction.

• We were intentional in our instruction, using instruction to bridge the gap between standards and assessment.

Intentionally Planned

Instruction

Launch, Explore, Summarize

Understanding by Design

Your Method

198

There are many ways to intentionally plan instruction.

High Expectations


Strong Standards


Shifts in Instructional Practice


Aligned Materials and Assessment


199

Module 9 Review

• There are many ways to “do” intentional instruction.• Intentional instruction is the bridge between standards and assessment.• Start with the standard: determine what students need to know, understand, and

do.• Create learning experiences that connect to students’ experiences and give them a

chance to explore the concept.

High Expectations


Strong Standards


Shifts in Instructional Practice


Aligned Materials and Assessment


145

• Assessment plays a critical role in instruction, should be standards based, and should be used to determine student mastery of the standard(s).

“Districts and schools in Tennessee will exemplify excellence and equity such that

all students are equipped with the knowledge and skills to successfully

embark upon their chosen path in life.”

200

Teacher Training Revised ELA and Math Standards Math 9–12...• K–12 learning progressions •...

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