Common Core Essential Elements Alternate Achievement ...

&Common Core Essential ElementsAlternate Achievement Descriptors

6-8 GradeInstructional Guide

Mathematics

West Virginia Board of education

2012-2013

L. Wade Linger Jr., PresidentGayle C. Manchin, Vice President

Robert W. Dunlevy, Secretary

Michael I. Green, MemberPriscilla M. Haden, MemberLloyd G. Jackson II, MemberLowell E. Johnson, MemberJenny N. Phillips, MemberWilliam M. White, Member

Paul L. Hill, Ex OfficioChancellor

West Virginia Higher Education Policy Commission

James L. Skidmore, Ex OfficioChancellor

West Virginia Council for Community and Technical College Education

Jorea M. Marple, Ex OfficioState Superintendent of Schools

Math | 6-8 Grade i

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ......................................................................................................... iv

INTRODUCTION ...................................................................................................................... 1

NCLB GUIDANCE ..................................................................................................................... 2

ACCESS TO INSTRUCTION AND ASSESSMENT ................................................................ 3

ACCESSING THE GENERAL CURRICULUM ........................................................................ 4Model Symbol Use Throughout Instruction ............................................................................. 4Use Partner-Assisted Scanning Across the Day......................................................................... 4Use First-Letter Cueing as a Communication Strategy Whenever Possible .............................. 4

GUIDANCE AND SUPPORT .................................................................................................... 5

RELATIONSHIP TO THE DYNAMIC LEARNING MAPS ASSESSMENT ........................... 6

SYSTEM ALIGNMENT .............................................................................................................. 6Levels of Performance .............................................................................................................. 7

DOCUMENT ORGANIZATION ............................................................................................... 8Directions for Interpreting Essential Elements ......................................................................... 9

COMMON CORE ESSENTIAL ELEMENTS AND ACHIEVEMENT DESCRIPTORS

Sixth Grade Mathematics StandardsRatios and Proportional Relationships ............................................................................... 10The Number System ........................................................................................................... 11Expressions and Equations ................................................................................................. 16Geometry ........................................................................................................................... 20Statistics and Probability .................................................................................................... 22

Seventh Grade Mathematics StandardsRatios and Proportional Relationships ............................................................................... 24The Number System ........................................................................................................... 25Expressions and Equations ................................................................................................. 30Geometry ........................................................................................................................... 32 Statistics and Probability .................................................................................................... 36

Eighth Grade Mathematics StandardsThe Number System ........................................................................................................... 39Expressions and Equations ................................................................................................. 41Functions ............................................................................................................................ 44Geometry ........................................................................................................................... 47 Statistics and Probability .................................................................................................... 50

ii Common Core Essential Elements

GLOSSARY AND EXAMPLES OF MATHEMATICS TERMS................................................ 53

GLOSSARY OF SPECIAL EDUCATION TERMS .................................................................. 59

BIBLIOGRAPHY OF DEVELOPMENT PROCESS ................................................................ 63

BIBLIOGRAPHY FOR MATHEMATICS CONTENT ............................................................ 65

APPENDIX A: SEA/STAKEHOLDER DEMOGRAPHICS

Math | 6-8 Grade iii

ACKNOWLEDGEMENTS*For stakeholder demographics, See Appendix A.

Edvantia FacilitatorsJan Sheinker, Sheinker Educational Services, Inc.

Beth Judy, Director, Assessment, Alignment, and Accountability ServicesNathan Davis, Information Technology Specialist

Kristen Deitrick, Corporate Communications SpecialistLinda Jones, Executive Assistant

Dynamic Learning Maps (DLM) Staff and ConsultantsNeal Kingston, Project Director

Alan Sheinker, Associate Project DirectorLaura Kramer, Test Development Lead

Karthick Palaniswamy, Technology Development LeadKelli Thomas, Mathematics Learning Map Team Lead

Carrie Mark, English Language Arts Learning Map Team LeadPatti Whetstone, Research Associate

Sue Bechard, ConsultantKaren Erickson, Consultant

Chris Cain, Consultant

Dynamic Learning Maps (DLM) Consortia StatesIowa

KansasMichigan

MississippiMissouri

New JerseyNorth Carolina

OklahomaUtah

VirginiaWashington

West VirginiaWisconsin

iv Common Core Essential Elements

Mathematics State Education Agency (SEA)/Stakeholder Representatives

IOWASEA Representatives: Tom Deeter, Emily Thatcher

Stakeholders: Barbara Adams, John Butz, Laurel Cakinberk, Dagny FidlerKANSAS

SEA Representatives: Sidney Cooley, Debbie MatthewsStakeholders: DiRae Boyd, Teresa Kraft, Michele Luksa, Mona Tjaden

MICHIGANSEA Representatives: Linda Howley, Joanne Winkelman

Stakeholders: Tamara Barrientos, Roula AlMouabbi, Brian Pianosi, Larry TimmMISSOURI

SEA Representatives: Lin Everett, Sara King, Jane VanDeZandeStakeholders: Sharon Campione, Emily Combs, Karen Pace

NEW JERSEYSEA Representatives: Shirley Cooper, MaryAnn Joseph

Stakeholders: Sue Burger, Tracey Lank, Katie SlaneNORTH CAROLINA

SEA Representative: Robin BarbourStakeholders: Ronda Layman, Janet Sockwell

OKLAHOMASEA Representatives: Jennifer Burnes, Amy Daugherty

Stakeholder: Christie StephensonUTAH

SEA Representatives: Wendy Carver, Jennie DeFriezStakeholders: Lynda Brown, Kim Fratto, Lisa Seipert, Nicole Warren

VIRGINIASEA Representatives: John Eisenberg, Deborah Wickham

Stakeholders: Diane Lucas, Laura Scearce, Joyce Viscomi, Roslynn WebbWASHINGTON

SEA Representatives: Debra Hawkins, Janice TornowStakeholders: Jeff Crawford, John DeBenedetti, Kirsten Dlugo, Angelita Jagla

WEST VIRGINIASEA Representatives: Melissa Gholson, Beth Cipoletti

Stakeholders: Wes Lilly, Melissa Mobley, Lisa New, Deena SwainWISCONSIN

SEA Representative: Brian JohnsonStakeholders: Amber Eckes, Rosemary Gardner, Mary Richards, Jeff Ziegler

Math | 6-8 Grade 1

The Common Core Essential Elements (EEs) are linked to the Common Core State Standards (CCSS) for Mathematics. A group of general educators, special educators, and content specialists from member states in the Dynamic Learning Maps (DLM) Consortium gathered to determine the essence of the CCSS.

This document provides a high-level view of the relationship between the CCSS and the links to performance for students with significant cognitive disabilities. It is intended to provide a beginning structure for the design of a summative alternate assessment. The document is not intended as a stand-alone guide to instruction, nor is it intended to contain all the steps in a complete learning progression or detailed curriculum. The DLM and associated professional development will provide greater detail than described in this document.

Beginning with the Mathematics CCSS, stakeholders defined links to illuminate the precursors for the essential content and skills contained in the grade level CCSS clusters and indicators. These EEs are not intended as a redefinition of the standards. Rather, they are intended to describe challenging expectations for students with significant cognitive disabilities in relation to the CCSS. The EEs clarify the bridge between grade level achievement expectations for students with significant cognitive disabilities who participate in alternate assessments and the CCSS.

Neither are the EEs intended to prescribe the beginning or end of instruction on the content and skills they represent; rather, they indicate the grade level at which initial mastery would be the target to be assessed. Students should begin instruction in content and skills at the earliest point possible and continue instruction until mastery is attained.

The stakeholder group, consisting of state education agency (SEA) representatives and SEA-selected content teachers of students with significant cognitive disabilities, developed instructional achievement level descriptors (IALDs) for each of the EEs. IALDs were defined for four performance levels: I, II, III, and IV. Level III IALDs are aligned with the EEs. The target content and skills for each level of achievement, from Level I to Level IV, were then defined. For each target skill, the stakeholder group developed examples to illustrate how students might demonstrate achievement of the performance level. The IALDs are intended to provide an achievement ladder for students working toward achievement (Level III) of the EEs and onward (Level IV) and toward greater participation in the grade level CCSS to which the EEs are linked. The provided examples are intended to assist teachers to envision how the broad range of students with significant cognitive disabilities might perform the same content, despite the different challenges their disabilities might present. The examples are not exhaustive and do not represent the full range of possibilities in which the highly diverse population of students with significant cognitive disabilities might access the EEs or demonstrate the achievement of those elements. However, the examples do provide some of the ways that performance might be elicited and demonstrated across the spectrum of students with significant cognitive disabilities.

Finally, the stakeholder group developed alternate assessment achievement descriptors for each grade level -- from third grade through high school -- where summative assessments might be required. The alternate assessment achievement descriptors will provide a bridge between the EEs and a summative alternate assessment aligned to them. The descriptors are intended to provide one element to guide development of the test blueprint, development of items and

INTRODUCTION

2 Common Core Essential Elements

tasks that measure the full range of achievement, and the setting of cut scores during standard setting for the assessment. The focus of an alternate assessment in a standards-based system is based on the achievement that aligns with EEs linked to grade level content.

Together, the system of standards and descriptors is designed to allow students with significant cognitive disabilities to progress toward the achievement of state standards linked to grade level expectations. The relationship of standards and assessment to teaching and learning are depicted for use by teachers, assessment designers, and users of alternate assessment results.

NCLB GUIDANCEThe stakeholder group’s work was guided by the U. S. Department of Education’s Peer Review Guidance (Standards and Assessments Peer Review Guidance: Information and Examples for Meeting Requirements of the No Child Left Behind Act of 2001 [NCLB]), which requires that alternate academic achievement standards align with the alternate assessment. They must• include knowledge and skills that link to grade level expectations,• promote access to the general curriculum, and• reflect professional judgment of the highest learning standards possible for the group of

students with the most significant cognitive disabilities.

Although the grade-level content may be reduced in complexity or adjusted to reflect prerequisite skills, the link to grade-level standards must be clear. The Peer Review Guidance notes that the concept of alternate achievement standards related to grade level may be ambiguous. According to the Guidance, the descriptors• should be defined in a way that supports individual growth because of their linkage to

different content across grades;• are not likely to show the same clearly defined advances in cognitive complexity as the

general education standards when examined across grade levels;• should rely on the judgment of experienced special educators and administrators, higher

education representatives, and parents of students with disabilities as they define alternate achievement standards; and

• should provide an appropriate challenge for students with the most significant cognitive disabilities as they move through their schooling.

The Guidance requires links to grade-level standards. The EEs were developed by DLM consortium states to differentiate knowledge and skills by grade level. This differentiation is intended to clarify the link between the grade-level EEs and the grade-level CCSS and to show a forward progression across grades. The progression of content and skills across years of instruction reflect the changing priorities for instruction and learning as students move from grade to grade. The differences from grade level to grade level are often subtle and progression is sometimes more horizontal than vertical. For example, the grade-to-grade level differences may consist of added skills that are not of obvious increasing rigor compared to the differences found in the CCSS across grade levels. To the degree possible, skills escalate in complexity or rigor at Levels III and IV across the grades, with clear links to the shifting emphasis at each grade level in the CCSS.

Math | 6-8 Grade 3

The EEs and Achievement Descriptors developed by the DLM consortium states are intended to create the maximum possible access to the CCSS for students with significant cognitive disabilities. The way in which information is presented for instruction and assessment and the manner in which students demonstrate achievement is in no way intended to be limited by statements of EEs or Achievement Descriptors. To that end, modes of communication, both for presentation or response, are not stated in either the EEs or Achievement Descriptors unless a specific mode is an expectation. Where no limitation has been stated, no limitation should be inferred. Students’ opportunities to learn and to demonstrate learning should be maximized by providing whatever communication, assistive technologies, augmentative and alternative communication (AAC) devices, or other access tools that are necessary and routinely used by the student during instruction.

Students with significant cognitive disabilities include a broad range of students with diverse disabilities and communication needs. For some students with significant cognitive disabilities, graphic organizers similar to those used by students without disabilities provide useful access to content and are adequate to maximize opportunities to learn and demonstrate achievement. Other students require a range of assistive technologies to access content and demonstrate achievement. For some students, AAC devices and accommodations for hearing and visual impairments will be needed. As with other physical disabilities, students with visual impairments may perform some expectations using modified items, presentations, or response formats. A few items may not lend themselves to such modifications. Decisions about the appropriate modifications for visual impairments are accounted for in the design of the assessments.

The access challenge for some is compounded by the presence of multiple disabilities. All of these needs, as well as the impact of levels of alertness due to medication and other physical disabilities which may affect opportunities to respond appropriately, need to be considered.

Most presentation and response access conditions do not constitute accommodations as they are understood for students who take the general assessment. Methods of presentation that do not violate the intended construct by aiding or directing the students’ response allow the student to perceive what knowledge or skill is expected. Aids to responding that do not constitute a violation of the intended construct allow the student to demonstrate the expected knowledge and skills. Examples of acceptable access technologies include the following:• communication devices that compensate for a students’ physical inability to produce

independent speech.• devices that compensate for a students’ physical inability to manipulate objects or

materials, point to responses, turn pages in a book, or use a pencil or keyboard to answer questions or produce writing.

• tools that maximize a students’ ability to acquire knowledge and skills and to demonstrate the products of their learning.

ACCESS TO INSTRUCTION AND ASSESSMENT


Technology is also of particular importance to students with significant cognitive disabilities to access the general curriculum and achieve the EEs. Although educators have traditionally viewed technology as hardware and software, assistive technology tenets provide a broader view of the applications of low, medium, and high levels of technology use. Assistive technology tools can be vital to a student in acquiring and demonstrating learning unimpeded by the barriers that the disability presents.

Model Symbol Use Throughout Instruction

Many students with significant cognitive disabilities have difficulty with or cannot use speech to communicate and/or are supported by the use of communication symbols (e.g., communication boards, speech generating devices, voice output communication devices) and supports to augment their speech and other means of communication. Students who require symbols and other AAC supports require frequent modeling in the use of those symbols to interact and respond during instruction. Students who use symbols and other communication supports need as much modeling as children who use speech to communicate. Modeling in this way is not viewed as a means of prompting, guidance, or support, just as having a teacher talk serves those purposes for a student who communicates using speech.

When modeling the use of symbols and other communication supports, teachers use the symbols and supports themselves, hand them to students without communication impairments to use, and involve the students who need to use them every day. Each of these steps can play an important role in validating the use of symbols and communication supports and demonstrating multiple levels of expertise in their use.

Use Partner-Assisted Scanning Across the Day

Making a choice from the items on a list, symbols, tactuals, or a communication board can be difficult for some students because they lack the ability to point, cannot see or read the choices, or are positioned too far away (as in group activities). Partner-assisted scanning addresses these issues by asking the communication partner (a teacher, paraprofessional, or peer) to point to each of the options pausing long enough at each for the students with physical and communication impairments to respond “yes” if the item is their desired choice. Depending on the needs of an individual child, the partner can name each option when pointing or simply point.

Throughout the IALDS, examples are provided that require students to select, identify, recognize, and so forth from a number of options. It is suggested that teachers use partner-assisted scanning to support these modes of responding and communicating whenever it appears that the act of directly pointing to a response is too difficult for a particular student.

Use First-Letter Cueing as a Communication Strategy Whenever Possible

Students with communication impairments who are beginning to read, write, and communicate regularly face the challenge of not having access to the words or symbols they want or need to communicate effectively. When attempting to provide them with every possible word they might need, the result is an unmanageable communication system. When guessing what will be most important, it is inevitable that some guesses will be wrong. Until students can spell well enough to communicate their own thoughts, it is important to rely on cueing strategies. First-letter cueing is one such strategy. Students can use an alphabet display to point to the

ACCESSING THE GENERAL CURRICULUM

Math | 6-8 Grade 5

first letter (or try to spell more) of the word they are trying to communicate. Teachers can use this strategy to help students respond efficiently to questions that involve known choices. Teachers can also model the use of first-letter cueing in their day-to-day interactions with the class. Natural opportunities to incorporate this strategy occur when the teacher is prompting students to recall a specific word (e.g., “I am thinking of a new word we learned yesterday that started with the letter ‘t’”.) or concept (e.g., “Who remembers the big word we learned to describe when we put things together to find out how many we have in all? It begins with the letter ‘a’”.). There are times every school day when the adults in the class can model the use of first-letter cueing.

GUIDANCE AND SUPPORTThe authors of the CCSS use the words, “prompting and support” at the earliest grade levels to indicate when students were not expected to achieve standards completely independently. Generally, “prompting” refers to “the action of saying something to persuade, encourage, or remind someone to do or say something” (McKean, 2005). However, in special education, prompting is often used to mean a system of structured cues to elicit desired behaviors that otherwise would not occur. In order to communicate clearly that teacher assistance is permitted during instruction of the EEs, and is not limited to structured prompting procedures, the decision was made by the stakeholder group to use the more general term guidance throughout the EEs and alternate achievement descriptors.

Guidance and support during instruction should be interpreted as teacher encouragement, general assistance, and informative feedback to support the student in learning. Some examples of the kinds of teacher behaviors that would be considered guidance and support include• getting the student started (e.g., “Tell me what to do first”),• providing a hint in the right direction without revealing the answer (e.g., Student wants to

write dog but is unsure how, the teacher might say, “See if you can write the first letter in the word, /d/og.”),

• narrowing the field of choices as a student provides an inaccurate response,• using structured technologies such as task specific word banks, or• providing the structured cues such as those found in prompting procedures (e.g., least-to-

most prompts, simultaneous prompting, and graduated guidance).

Guidance and support as described above apply to instruction per the examples provided in the IALDS. The IALDs are intended to provide an idea of how students might perform the EEs at the threshold to various achievement levels as they work toward independent mastery.

Alternate assessments measure the degree to which students with significant cognitive disabilities have mastered the EEs. During any assessment, accommodation(s) allowed on the assessment must have been used and practiced during instruction; however, some accommodations that are permissible during instruction would compromise the integrity of the assessments, thereby yielding invalid and unreliable results and cannot be used for assessment purposes. Some guidance and support strategies may not be allowed for assessment purposes when variance in teacher assistance, cues, and prompts could compromise judgments about mastery of the EEs and comparability of administration.


The EEs and Achievement Descriptors developed by the DLM consortium states and their stakeholder representatives serve two functions. Instructionally, they provide teachers with information about the level of knowledge and skills expected of their students. Second, they provide elaboration that teachers can use to help guide instruction toward achievement expectations. IALDs were developed for each of the EEs. Each IALD is further clarified by a range of examples. Teachers may find these examples useful for envisioning how their students might perform as they progress toward the expected achievement, as long as they keep in mind that they are examples only and cannot represent the full range of ways in which students might demonstrate their achievement.

Assessment Achievement Level Descriptors (AALDs) will emerge as drafts from the IALDS. The AALDs are content and grade specific, but summarize across the EEs the key performance differences across levels of achievement and across grade levels. While draft AALDs will be used in the initial stages of standard setting to help guide that process, final AALDs will emerge from the standard setting process. Standard setting will take into account the overall degree of accuracy with which a student would need to perform in order to achieve at a particular level. Just as on a general education assessment, no individual student will be expected to perform proficiently on every EE in order to be considered Level III.

For purposes of the DLM assessments under development, the achievement descriptors provide a useful link between the EEs and the DLM assessments. The descriptors, along with DLM developed from the CCSS, provide guidance to the development of the alternate assessment so that a full range of performance is measured and the setting of score ranges within each level rests on a defined frame of reference. The grade level EEs and alternate achievement standards• standardize meaning for the content and skill expectations,• create consistency in expected performance,• emphasize skill similarities for all students participating in the alternate assessments,• accommodate diverse disabilities, and• ground alternate assessments in a consistent set of expectations.

Achievement descriptors are used to categorize and explain student performance both in the course of instruction and on the alternate assessment.

SYSTEM ALIGNMENTThe EEs and alternate achievement descriptors are intended to contribute to a fully aligned system of standards, curriculum, teaching, learning, technology, and assessment that optimize equity of opportunity for all students in each classroom, school, and local education agency to access and learn the standards. To the degree possible, the grade level EEs are vertically aligned and linked to the grade level CCSS.

The linkages provided by the EEs to the CCSS are intended to increase access to the general curriculum for all students with disabilities. Examples provided for IALDs at each level of achievement are designed for special education and general education classroom teachers to

RELATIONSHIP TO THE DYNAMIC LEARNING MAPS ASSESSMENT

Math | 6-8 Grade 7

use in working with special education students who have significant cognitive disabilities. The examples are designed to help teachers evaluate students’ progress toward achievement of the EEs as well as illuminate the kinds of performances that indicate various levels of achievement.

Just as the EEs and IALDS are designed to guide teaching practices toward achievement in academic content areas, the standards reframe the expectations for foundational skills in pre-academic and academic areas. Precursor/prerequisite and the unique enabling skills related to mathematics content is specified in the context of their roles as a foundation for students with significant cognitive disabilities to achieve skills related to academic content.

Levels of Performance

Within this document, each grade level EE is cross-referenced to one or several CCSS.

Four performance levels have been proposed for the DLM’s alternate academic achievement standards: I, II, III, and IV. Mastery is considered to be demonstrated at Level III and Level IV and is identified as meeting the Level III level on an alternate assessment as specified in the NCLB. A general description of each of these levels is included below:

Level I - A student at this level attempts to perform tasks with support.

Level II - A student at this level demonstrates some content knowledge and skills from the extended grade level standards.

Level III - A student at this level demonstrates content knowledge and skills at a level aligned with the complexity of the EEs.

Level IV - A student at this level demonstrates content knowledge and skills at a higher level of complexity than those described for Level III. Typically, this complexity includes the routine use of symbol systems as applied to mathematics.

For each performance level, specific descriptions of content and skills are bulleted and examples of each level of performance are provided. The EEs, IALDS, and examples are intended as a resource for developing individualized education plan (IEP) goals, benchmarks, and curricular materials in reading, language arts, and mathematics. Students may need goals and benchmarks in areas other than academic content domains (e.g., self-care/living skills, mobility). As always, IEPs address the individual needs of each student to make progress toward the standards.


Common Core Grade-Level Clusters are the Cluster titles and Grade-Level Indicators as they appear in the CCSS for Mathematics (Common Core State Standards Initiative, 2010).

Common Core Essential Elements (EEs) describe links to the CCSS for access by students with significant cognitive disabilities.

Instructional Achievement Level Descriptors (IALDs) describe performance at four achievement levels based on the EEs and are accompanied by examples at each achievement level.

CCSS Grade-Level Clusters

Common Core Essential Elements

Instructional Achievement Level Descriptor

Represent and solve problems involving addition and subtraction.

1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

EE1.OA.1.a. Use language to describe putting together and taking apart, aspects of addition and subtraction.

Level IV AA Students will:EE1.OA.1.a. Use words like take away, subtract, give, add, more, and same quantity, when putting together and taking apart.Ex. When gathering and distributing classroom supplies, appropriately use words like “more” and “take away” (handing out paper, pencils, or other tools used in a lesson).Ex. When picking teams for P.E., use the language of “I need one more student” or “I need to take away one more from my team.”Ex. Request “one more” or “take away” one or more when the teacher has set up an activity where there is an uneven number of supplies.Ex. During an activity, use “add,” “more,” “less,” etc. to indicate when a different amount is needed.

Level III AA Students will:EE1.OA.1.a. Use language to describe putting together and taking apart, aspects of addition and subtraction.Ex. After the teacher shows six blocks and removes two, label the action as “take away” or informal language with the same meaning.Ex. Appropriately use “more” and “give” to express desire for more snacks or blocks.Ex. Use one-to-one correspondence to line up two sets of objects and ask which group has more/less.Ex. During practice of adding __ more to a numeral, show correct flashcard when asked, “I have two; who has two more (4)?”

Level II AA Students will:EE1.OA.1.a. Put together or take away.Ex. Take away one crayon from the box.Ex. Put together red blocks and green blocks when asked.Ex. Give coins to purchase an item or take change at end of purchase.Ex. Give the teacher two blocks and then two more blocks.

Level I AA Students will:EE1.OA.1.a. Follow directions to put together or take away an object with a verbal prompt.Ex. In a classroom routine and when presented with a component needed for the routine, give component(s) when asked to put together for the activity.Ex. Take a paper or object from peer when passed out.Ex. Offer paper or object to peer to put together with group’s work when collected at the end of the lesson.

DOCUMENT ORGANIZATION

Math | 6-8 Grade 9

Directions for Interpreting Essential Elements

Essential Elements (EEs). The EEs are statements that provide links for students with significant cognitive disabilities to the essential content and skills defined in the grade-level clusters of the CCSS. The EEs provide a bridge for students with significant cognitive disabilities to the CCSS. The EEs are not intended as a reinterpretation of the CCSS; rather, they were developed to create a bridge between the CCSS and challenging achievement expectations for students with significant cognitive disabilities. The order in which the EEs are listed is a direct reflection of the order in which the CCSS are listed. The order is not intended to convey a sequence for instruction; rather, it illustrates progress across years. In the tables, the left column contains the CCSS grade-level clusters and indicators, the middle column contains the EE linked to them, and the right column contains the IALDs for each EE and examples for each IALD (as demonstrated by the example provided on the previous page). Each EE and IALD completes the phrase “Students will . . . .”

CCSS marked with an (+) are advanced standards and are not included in this document as it was determined by the stakeholder group that students of this population would not be accessing the curriculum at this advanced level and writing Essential Elements to this level would be unnecessary. Also, if it appears that a standard has been omitted in the high school grades, it is an advanced standard.

NOTE: N/A is used instead of a descriptor under Level IV, if it was determined by the stakeholder group that the content of the CCSS could not be addressed.

Bullets under instructional achievement levels denote descriptions of achievement at that level for the content related to the essential element.

Examples clarify certain components of EEs. The provided examples are illustrative, not exhaustive. They are intended to provide a range of ways in which a student may demonstrate progress toward the essential element and beyond.


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spe

ed.

For e

xam

ple,

if it

took

7

hour

s to

mow

4 la

wns

, the

n at

that

rate

, ho

w m

any

law

ns c

ould

be

mow

ed in

35

hour

s? A

t wha

t rat

e w

ere

law

ns b

eing

m

owed

?•

Find

a p

erce

nt o

f a q

uanti

ty

as a

rate

per

100

(e.g

., 30

% o

f a q

uanti

ty

mea

ns 3

0/10

0 tim

es th

e qu

antit

y); s

olve

pr

oble

ms

invo

lvin

g fin

ding

the

who

le,

give

n a

part

and

the

perc

ent.

•U

se ra

tio re

ason

ing

to c

onve

rt

mea

sure

men

t uni

ts; m

anip

ulat

e an

d tr

ansf

orm

uni

ts a

ppro

pria

tely

whe

n m

ultip

lyin

g or

div

idin

g qu

antiti

es.

EE6.

RP.1

. Dem

onst

rate

a s

impl

e ra

tio

rela

tions

hip.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

RP.1

. Use

a ra

tio to

des

crib

e a

rela

tions

hip

usin

g nu

mbe

rs a

nd o

bjec

ts.

Ex. G

iven

an

even

num

ber

of re

d an

d tw

ice

as m

any

gree

n be

ads,

iden

tify

the

ratio

of g

reen

bea

ds c

ompa

red

to

red

bead

s.Ex

. Whi

le p

repa

ring

a re

cipe

, fill

in a

ratio

of fl

our

to s

ugar

(e.g

., on

e cu

p of

sug

ar to

four

cup

s of

flou

r.)Ex

. Com

pare

the

num

ber

of m

ale

stud

ents

to fe

mal

e st

uden

ts.

Ex. G

iven

the

quan

tity

of m

ater

ials

ava

ilabl

e an

d th

e nu

mbe

r of

gro

ups

who

will

con

duct

a s

cien

ce e

xper

imen

t,

use

a ra

tio re

latio

nshi

p to

des

crib

e ho

w m

uch

each

gro

up w

ill re

ceiv

e.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

RP.1

. Dem

onst

rate

a s

impl

e ra

tio re

latio

nshi

p.Ex

. Giv

e a

pen

and

a pe

ncil

to e

ach

clas

smat

e.Ex

. Aft

er th

e te

ache

r ex

plai

ns w

hat m

ater

ials

eac

h gr

oup

need

s, u

se a

n A

AC to

tell

anot

her

stud

ent t

o ge

t tw

o cu

ps fo

r on

e ta

ble.

Leve

l II A

A S

tude

nts

will

:EE

6.RP

.1. C

ompl

ete

a pa

tter

n gi

ven

a si

mpl

e ra

tio.

Ex. T

ake

two

step

s on

a n

umbe

r lin

e ea

ch ti

me

the

teac

her

says

“st

ep.”

Ex. G

ive

a ra

tio o

f tw

o-to

-one

, com

plet

e a

AA

BAA

BAA

B pa

tter

n (e

.g.,

jum

p, ju

mp,

cla

p; ju

mp,

jum

p, c

lap)

.

Leve

l I A

A S

tude

nts

will

:EE

6.RP

.1. I

denti

fy a

one

-to-

one

rela

tions

hip.

Ex. G

iven

a s

tack

of n

apki

ns, g

ive

a na

pkin

to e

ach

clas

smat

e.Ex

. Whe

n so

rting

mai

l in

the

mai

n offi

ce, p

lace

one

cop

y of

the

scho

ol n

ewsl

etter

in e

ach

teac

her’

s m

ailb

ox.

Ex. T

ouch

eac

h ob

ject

as

teac

her

coun

ts.

Math | 6-8 Grade 11

Sixt

h G

rad

e M

ath

emat

ics

Stan

dar

ds:

Th

e N

um

ber

Sys

tem

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

App

ly a

nd e

xten

d pr

evio

us

unde

rsta

ndin

gs o

f mul

tipl

icati

on

and

divi

sion

to d

ivid

e fr

acti

ons

by

frac

tion

s.

6.N

S.1.

Inte

rpre

t and

com

pute

qu

otien

ts o

f fra

ction

s, a

nd s

olve

w

ord

prob

lem

s in

volv

ing

divi

sion

of

frac

tions

by

frac

tions

, e.g

., by

usi

ng v

isua

l fra

ction

mod

els

and

equa

tions

to re

pres

ent t

he

prob

lem

. Fo

r exa

mpl

e, c

reat

e a

stor

y co

ntex

t for

(2/3

) ÷ (3

/4)

and

use

a vi

sual

frac

tion

mod

el

to s

how

the

quoti

ent;

use

the

rela

tions

hip

betw

een

mul

tiplic

ation

an

d di

visi

on to

exp

lain

that

(2/3

) ÷

(3/4

) = 8

/9 b

ecau

se 3

/4 o

f 8/9

is

2/3

. (In

gen

eral

, (a/

b) ÷

(c/d

) =

ad/b

c.)

How

muc

h ch

ocol

ate

will

ea

ch p

erso

n ge

t if 3

peo

ple

shar

e 1/

2 lb

. of c

hoco

late

equ

ally

? H

ow

man

y 3/

4-cu

p se

rvin

gs a

re in

2/3

of

a c

up o

f yog

urt?

How

wid

e is

a

rect

angu

lar s

trip

of l

and

with

leng

th

3/4

mi a

nd a

rea

1/2

squa

re m

i?

Com

pute

flue

ntly

with

mul

ti-di

git

num

bers

and

find

com

mon

fact

ors

and

mul

tiple

s.

EE6.

NS.

1. C

ompa

re th

e re

latio

nshi

ps b

etw

een

two

unit

frac

tions

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

NS.

1. C

ompa

re th

e re

latio

nshi

ps b

etw

een

the

thre

e un

it fr

actio

ns (1

/2, 1

/4, 1

/8).

Ex. G

iven

thre

e m

easu

ring

cup

s fil

led

to 1

/2, 1

/4, a

nd 1

/8 w

ith w

ater

, com

pare

frac

tiona

l am

ount

s to

det

erm

ine

whi

ch is

gre

ater

.Ex

. Giv

en p

icto

rial

repr

esen

tatio

ns o

f sha

ded

pict

ures

and

/or

frac

tion

bars

, com

pare

fr

actio

ns to

det

erm

ine

whi

ch is

a s

mal

ler

or le

sser

am

ount

.Ex

. Usi

ng c

ircle

sha

ped

frac

tion

puzz

les,

com

pare

a 1

/2, 1

/4, a

nd 1

/8 to

det

erm

ine

whi

ch

is g

reat

er.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

NS.

1. C

ompa

re th

e re

latio

nshi

ps b

etw

een

two

unit

frac

tions

.Ex

. Giv

en tw

o m

easu

ring

cup

s of

1/2

and

1/4

full

of s

and,

com

pare

the

amou

nts

in e

ach

of

the

mea

suri

ng c

ups

to a

who

le c

up.

Whi

ch is

mor

e?Ex

. Giv

en tw

o m

easu

ring

cup

s of

1/4

and

1/8

full

of w

ater

, com

pare

the

amou

nts

in e

ach

of th

e m

easu

ring

cup

s to

a w

hole

cup

. W

hich

is m

ore?

Ex. W

hen

give

n a

grou

p of

eve

n-nu

mbe

red

obje

cts

that

repr

esen

ts 1

/2 a

nd 1

/4,

dete

rmin

e w

hich

set

is m

ore

or le

ss.

Ex. S

plit

an e

ven-

num

bere

d gr

oup

of o

bjec

ts in

to tw

o eq

ual g

roup

s to

sho

w o

ne h

alf o

f th

e gr

oup;

then

spl

it ea

ch g

roup

aga

in to

sho

w fo

urth

s of

the

who

le; a

nd s

plit

each

gro

up

agai

n to

sho

w e

ight

hs o

f the

who

le.

Leve

l II A

A S

tude

nts

will

:EE

6.N

S.1.

Dem

onst

rate

an

amou

nt o

f 1/2

.Ex

. Fol

d on

e pi

ece

of p

aper

in h

alf t

o sh

ow tw

o ha

lves

in e

very

one

who

le.

Ex. S

hade

a s

hape

to s

how

1/2

.Ex

. Giv

en a

who

le a

nd a

hal

f, id

entif

y th

e ha

lf (e

.g.,

a w

hole

or

half

sand

wic

h).

Ex. S

how

n a

glas

s th

at is

full

and

a gl

ass

that

is 1

/2 (h

alf)

full,

sel

ect t

he h

alf-

full

glas

s.

Leve

l I A

A S

tude

nts

will

:EE

6.N

S.1.

Dis

tingu

ish

betw

een

mor

e or

less

.Ex

. Giv

en tw

o gr

oups

of o

bjec

ts w

ith s

igni

fican

tly d

iffer

ent a

mou

nts

(thr

ee v

s. 1

0),

dete

rmin

e w

hich

gro

up h

as m

ore

or le

ss.

Ex. G

iven

a p

ictu

re o

f a fa

mili

ar s

ymm

etri

cal o

bjec

t cut

in h

alf,

com

bine

bot

h ha

lves

to

mak

e a

who

le.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Com

pute

flue

ntly

wit

h m

ulti

-dig

it

num

bers

and

find

com

mon

fact

ors

and

mul

tipl

es.

6.N

S.2.

Flu

ently

div

ide

mul

ti-di

git n

umbe

rs u

sing

the

stan

dard

al

gori

thm

.

EE6.

NS.

2. A

pply

the

conc

ept o

f fa

ir s

hare

and

equ

al s

hare

s to

di

vide

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

NS.

2. S

olve

a d

ivis

ion

prob

lem

usi

ng th

e co

ncep

t of e

qual

sha

res.

Ex. G

iven

a re

al-li

fe d

ivis

ion

prob

lem

, sol

ve th

e pr

oble

m u

sing

man

ipul

ative

s.Ex

. Giv

en a

gro

up o

f obj

ects

, det

erm

ine

wha

t num

ber

to g

ive

each

cla

ssm

ate

to c

reat

e eq

ual s

hare

s.Ex

. Div

ide

stud

ents

into

four

equ

al g

roup

s fo

r a

spor

ts to

urna

men

t.Ex

. Whe

n pl

antin

g se

eds

for

a sc

ienc

e ex

peri

men

t, d

ivid

e th

e se

eds

into

equ

al s

hare

s.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

NS.

2. A

pply

the

conc

ept o

f fai

r sh

are

and

equa

l sha

res

to d

ivid

e.Ex

. Whe

n pl

antin

g se

eds

for

a sc

ienc

e ex

peri

men

t, d

ivid

e th

e se

eds

into

10

equa

l sha

res.

Ex. D

ivid

e co

nstr

uctio

n pa

per

equa

lly a

mon

g cl

assm

ates

.Ex

. Div

ide

stud

ents

in th

e cl

assr

oom

into

two

equa

l tea

ms.

Ex. D

ivid

e 10

one

dol

lar

bills

into

two

fair

sha

res

(e.g

., “I

f I fi

nd 1

0 do

llars

and

I di

vide

it

equa

lly w

ith s

omeo

ne, h

ow m

uch

do w

e ea

ch g

et?”

).

Leve

l II A

A S

tude

nts

will

:EE

6.N

S.2.

Iden

tify

the

conc

ept o

f div

isio

n us

ing

fair

and

equ

al s

hare

s.Ex

. Giv

en a

pap

er fo

lded

in h

alf,

iden

tify

whe

ther

they

are

equ

al s

hare

s.

Ex. D

istr

ibut

e ca

rds

in a

car

d ga

me

givi

ng e

ach

stud

ent a

fair

sha

re.

Ex. G

iven

a s

et o

f boo

ks, d

ivid

e th

em in

to tw

o bu

cket

s.Ex

. Giv

en Z

iplo

c ba

ggie

s w

ith a

n eq

ual n

umbe

r of

pen

cils

in th

em, s

ay th

e nu

mbe

r of

ba

ggie

s an

d th

e nu

mbe

r of

pen

cils

in e

ach

bag.

Leve

l I A

A S

tude

nts

will

:EE

6.N

S.2.

Rep

licat

e eq

ual s

ets.

Ex. G

iven

a m

odel

, rep

licat

e eq

ual s

ets

usin

g ri

ngs

and

patt

ern

bloc

ks.

Ex. G

iven

a m

odel

, pla

ce fi

ve d

iffer

ent c

olor

s in

equ

al s

ets.

Math | 6-8 Grade 13

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

6.N

S.3.

Flu

ently

add

, sub

trac

t,

mul

tiply

, and

div

ide

mul

ti-di

git

deci

mal

s us

ing

the

stan

dard

al

gori

thm

for

each

ope

ratio

n.

EE6.

NS.

3. S

olve

two

fact

or

mul

tiplic

ation

pro

blem

s w

ith

prod

ucts

up

to 5

0 us

ing

conc

rete

ob

ject

s an

d/or

cal

cula

tors

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

NS.

3. S

olve

mul

tiplic

ation

pro

blem

s w

ith w

hole

num

ber

prod

ucts

to 5

0 us

ing

num

eric

al re

pres

enta

tions

.Ex

. Giv

en a

set

of m

ultip

licati

on p

robl

ems

in n

umer

ical

form

, find

the

prod

uct.

Ex. G

iven

a c

ompu

ter

prog

ram

with

mul

tiplic

ation

pro

blem

s, fi

nd th

e pr

oduc

t.Ex

. Fin

d th

e pr

oduc

t of w

hole

num

bers

to 2

0 vi

a m

ultip

le a

lgor

ithm

s (e

.g.,

diffe

rent

way

s to

get

to 2

0 =

10 x

2, 2

x 1

0, 1

0 +

10 o

r 5

+ 5

+ 5

+ 5)

.Ex

. Giv

en a

sto

ry p

robl

em, fi

nd th

e pr

oduc

t and

repr

esen

t it n

umer

ical

ly (e

.g.,

If I h

ave

thre

e sh

irts

and

two

pair

of p

aint

s ho

w m

any

outf

its c

an o

ne m

ake?

If I

hav

e fiv

e ro

ws

of

desk

s an

d 10

des

ks in

eac

h ro

w, h

ow m

any

desk

s w

ill I

have

? If

I ba

bysa

t for

five

day

s an

d ea

rned

10

dolla

rs e

ach

day

how

muc

h m

oney

wou

ld I

mak

e?).

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

NS.

2. S

olve

two

fact

or m

ultip

licati

on p

robl

ems

with

pro

duct

s up

to 5

0 us

ing

conc

rete

ob

ject

s an

d/or

cal

cula

tors

.Ex

. Giv

en a

set

of m

anip

ulati

ves,

mak

e th

ree

grou

ps o

f thr

ee a

nd th

en fi

nd th

e pr

oduc

t.Ex

. Giv

en a

100

s bo

ard,

sho

w 3

x 1

0, th

ree

sets

of 1

0, a

nd s

tate

the

prod

uct.

Ex. G

iven

num

bers

pai

red

with

con

cret

e re

pres

enta

tions

, sel

ect t

he c

orre

ct a

nsw

er.

Leve

l II A

A S

tude

nts

will

:EE

6.N

S.2.

Sol

ve re

peat

ed a

dditi

on p

robl

ems

whe

re th

e ad

dend

s ar

e th

e sa

me

(i.e.

, 5 +

5 +

5

= 15

is e

qual

to th

ree

grou

ps o

f five

) usi

ng c

oncr

ete

man

ipul

ative

s an

d/or

a c

alcu

lato

r.Ex

. Giv

en a

sto

ry p

robl

em, fi

nd th

e su

m o

f a re

peat

ed a

dditi

on p

robl

em u

sing

obj

ects

or

thei

r re

pres

enta

tions

(e.g

., If

I hav

e tw

o ro

ws

of d

esks

and

thre

e de

sks

in e

ach

row

how

m

any

desk

s w

ill I

have

? If

I ba

bysa

t for

thre

e da

ys a

nd e

arne

d fo

ur d

olla

rs e

ach

day

how

m

uch

mon

ey w

ould

I m

ake?

[Giv

en p

lay

mon

ey a

s a

man

ipul

ative

]).

Ex. G

iven

a p

ictu

re o

f thr

ee g

roup

s of

thre

e pe

ncils

, rep

rese

nt a

nd s

olve

the

repe

ated

ad

ditio

n pr

oble

m.

Ex. B

efor

e st

artin

g an

art

pro

ject

, gat

her

two

piec

es e

ach

of fi

ve d

iffer

ent c

olor

ed p

aper

s an

d de

scri

be h

ow m

any

tota

l pie

ces

of p

aper

are

requ

ired.

Leve

l I A

A S

tude

nts

will

:EE

6.N

S.2.

Iden

tify

a gr

oup

of a

giv

en q

uanti

ty.

Ex. G

iven

a g

roup

of o

bjec

ts w

ith n

o gr

eate

r th

an th

ree

item

s, id

entif

y ho

w m

any

are

in

the

grou

p th

at m

atch

es th

e te

ache

r’s

hand

held

num

eric

sym

bol (

e.g.

, gro

up o

f tw

o, g

roup

of

one

, gro

up o

f thr

ee -

mat

ch to

the

num

bers

two,

one

, and

thre

e).

Ex. S

ubiti

ze s

ets

of fo

ur (e

.g.,

usin

g a

die)

.Ex

. Giv

en a

set

num

ber

of s

ound

s, n

o gr

eate

r th

an th

ree,

iden

tify

the

quan

tity

of s

ound

s he

ard

(e.g

., in

dica

ting

thre

e do

ts o

r th

e nu

mbe

r th

ree)

. D

o th

is tw

ice

and

iden

tify

if th

e nu

mbe

r of

sou

nds

are

the

sam

e or

diff

eren

t as

the

first

roun

d.Ex

. Whe

n sh

own

a re

peati

ng p

atter

n of

thre

e ob

ject

s, th

ree

obje

cts,

thre

e ob

ject

s, te

ll th

e te

ache

r ho

w m

any

obje

cts

are

in th

e re

peat

ed p

atter

n.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

6.N

S.4.

Fin

d th

e gr

eate

st c

omm

on

fact

or o

f tw

o w

hole

num

bers

less

th

an o

r eq

ual t

o 10

0 an

d th

e le

ast

com

mon

mul

tiple

of t

wo

who

le

num

bers

less

than

or

equa

l to

12.

Use

the

dist

ribu

tive

prop

erty

to

exp

ress

a s

um o

f tw

o w

hole

nu

mbe

rs 1

–100

with

a c

omm

on

fact

or a

s a

mul

tiple

of a

sum

of t

wo

who

le n

umbe

rs w

ith n

o co

mm

on

fact

or.

For e

xam

ple,

exp

ress

36

+ 8

as 4

(9 +

2).

App

ly a

nd e

xten

d pr

evio

us u

nder

stan

ding

s of

nu

mbe

rs to

the

syst

em o

f rati

onal

nu

mbe

rs.

EE6.

NS.

4. N

/A

Math | 6-8 Grade 15

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

App

ly a

nd e

xten

d pr

evio

us u

nder

stan

ding

s of

nu

mbe

rs to

the

sys

tem

of r

ation

al n

umbe

rs.

6.N

S.5.

Und

erst

and

that

pos

itive

and

neg

ative

nu

mbe

rs a

re u

sed

toge

ther

to d

escr

ibe

quan

tities

ha

ving

opp

osite

dire

ction

s or

val

ues

(e.g

., te

mpe

ratu

re a

bove

/bel

ow z

ero,

ele

vatio

n ab

ove/

belo

w s

ea le

vel,

cred

its/d

ebits

, pos

itive

/neg

ative

el

ectr

ic c

harg

e); u

se p

ositi

ve a

nd n

egati

ve n

umbe

rs

to re

pres

ent q

uanti

ties

in re

al-w

orld

con

text

s,

expl

aini

ng th

e m

eani

ng o

f 0 in

eac

h si

tuati

on.

6.N

S.6.

Und

erst

and

a ra

tiona

l num

ber

as a

poi

nt o

n th

e nu

mbe

r lin

e. E

xten

d nu

mbe

r lin

e di

agra

ms

and

coor

dina

te a

xes

fam

iliar

from

pre

viou

s gr

ades

to

repr

esen

t poi

nts

on th

e lin

e an

d in

the

plan

e w

ith

nega

tive

num

ber

coor

dina

tes.

•Re

cogn

ize

oppo

site

sig

ns o

f num

bers

as

indi

catin

g lo

catio

ns o

n op

posi

te s

ides

of 0

on

the

num

ber

line;

reco

gniz

e th

at th

e op

posi

te o

f th

e op

posi

te o

f a n

umbe

r is

the

num

ber

itsel

f, e.

g.,

–(–3

) = 3

, and

that

0 is

its

own

oppo

site

.•

Und

erst

and

sign

s of

num

bers

in

orde

red

pairs

as

indi

catin

g lo

catio

ns in

qua

dran

ts

of th

e co

ordi

nate

pla

ne; r

ecog

nize

that

whe

n tw

o or

dere

d pa

irs d

iffer

onl

y by

sig

ns, t

he lo

catio

ns o

f th

e po

ints

are

rela

ted

by re

flecti

ons

acro

ss o

ne o

r bo

th a

xes.

•Fi

nd a

nd p

ositi

on in

tege

rs a

nd o

ther

ra

tiona

l num

bers

on

a ho

rizo

ntal

or

verti

cal

num

ber

line

diag

ram

; find

and

pos

ition

pai

rs

of in

tege

rs a

nd o

ther

ratio

nal n

umbe

rs o

n a

coor

dina

te p

lane

.

6.N

S.7.

Und

erst

and

orde

ring

and

abs

olut

e va

lue

of

ratio

nal n

umbe

rs.

•In

terp

ret s

tate

men

ts o

f ine

qual

ity

as s

tate

men

ts a

bout

the

rela

tive

posi

tion

of tw

o nu

mbe

rs o

n a

num

ber

line

diag

ram

. Fo

r exa

mpl

e,

inte

rpre

t -3

> -7

as

a st

atem

ent t

hat -

3 is

loca

ted

to th

e rig

ht o

f -7

on a

num

ber l

ine

orie

nted

from

le

ft to

righ

t.•

Wri

te, i

nter

pret

, and

exp

lain

st

atem

ents

of o

rder

for

ratio

nal n

umbe

rs in

real

-w

orld

con

text

s. F

or e

xam

ple,

writ

e -3

o C

> -7

o C

to

expr

ess

the

fact

that

-3

o C is

war

mer

than

-7

o C.•

Und

erst

and

the

abso

lute

val

ue o

f a

ratio

nal n

umbe

r as

its

dist

ance

from

0 o

n th

e nu

mbe

r lin

e; in

terp

ret a

bsol

ute

valu

e as

mag

nitu

de

for

a po

sitiv

e or

neg

ative

qua

ntity

in a

real

-wor

ld

situ

ation

. Fo

r exa

mpl

e, fo

r an

acco

unt b

alan

ce o

f -3

0 do

llars

, writ

e |-

30|

= 30

to d

escr

ibe

the

size

of

the

debt

in d

olla

rs.

•D

istin

guis

h co

mpa

riso

ns o

f abs

olut

e va

lue

from

sta

tem

ents

abo

ut o

rder

. Fo

r exa

mpl

e,

reco

gniz

e th

at a

n ac

coun

t bal

ance

less

than

-30

dolla

rs re

pres

ents

a d

ebt g

reat

er th

an 3

0 do

llars

.

6.N

S.8.

Sol

ve re

al-w

orld

and

mat

hem

atica

l pr

oble

ms

by g

raph

ing

poin

ts in

all

four

qua

dran

ts

of th

e co

ordi

nate

pla

ne.

Incl

ude

use

of c

oord

inat

es

and

abso

lute

val

ue to

find

dis

tanc

es b

etw

een

poin

ts w

ith th

e sa

me

first

coo

rdin

ate

or th

e sa

me

seco

nd c

oord

inat

e.

EE6.

NS.

5-8.

Und

erst

and

that

pos

itive

and

ne

gativ

e nu

mbe

rs a

re u

sed

toge

ther

to

desc

ribe

qua

ntitie

s ha

ving

opp

osite

dire

ction

s or

val

ues

(e.g

., te

mpe

ratu

re a

bove

/bel

ow

zero

).

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

NS.

5-8.

App

ly p

ositi

ve a

nd n

egati

ve n

umbe

rs to

a re

al-w

orld

con

text

from

gre

ater

than

pos

itive

10

and

less

than

neg

ative

10.

Ex. G

iven

thre

e ne

gativ

e an

d po

sitiv

e te

mpe

ratu

res

on th

ree

ther

mom

eter

s, o

rder

the

tem

pera

ture

s fr

om le

ast t

o gr

eate

st (e

.g.,-

15, 0

, 15)

.Ex

. Whe

n gi

ven

a th

erm

omet

er re

adin

g -5

deg

rees

, tel

l how

muc

h th

e te

mpe

ratu

re w

ill h

ave

to r

ise

to g

et to

15

degr

ees?

Ex. G

iven

thre

e ba

nk s

tate

men

ts, o

rder

the

stat

emen

t bal

ance

s fr

om le

ast t

o gr

eate

st.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

NS.

5-8.

Und

erst

and

that

pos

itive

and

neg

ative

num

bers

are

use

d to

geth

er to

des

crib

e qu

antiti

es h

avin

g op

posi

te d

irecti

ons

or v

alue

s (e

.g.,

tem

pera

ture

abo

ve/b

elow

zer

o).

Ex. G

iven

a n

umbe

r lin

e an

d as

ked

to s

how

the

num

ber

that

is o

ppos

ite o

f 5, s

elec

t -5.

Ex. G

iven

two

tem

pera

ture

s on

two

ther

mom

eter

s, o

ne p

ositi

ve a

nd o

ne n

egati

ve, d

eter

min

e w

hich

tem

pera

ture

is th

e co

ldes

t.Ex

. Loo

k at

the

reco

rds

(win

s/lo

sses

) of t

hree

bas

ebal

l tea

ms

(pos

itive

num

bers

to in

dica

te n

umbe

r of

win

s an

d ne

gativ

e nu

mbe

rs to

indi

cate

num

ber

of lo

sses

) and

then

rank

the

team

s in

ord

er fr

om th

e gr

eate

st n

umbe

r of

win

s/le

ast a

mou

nt o

f lo

sses

.Ex

. Loo

k at

a b

ank

stat

emen

t/ch

eckb

ook

regi

ster

and

tell

if th

ere

is a

pos

itive

or

nega

tive

bala

nce

(do

you

have

any

mon

ey o

r do

yo

u ow

e th

e ba

nk m

oney

?).

Leve

l II A

A S

tude

nts

will

:EE

6.N

S.5-

8. O

rder

pos

itive

num

bers

from

leas

t to

grea

test

.Ex

. Giv

en th

ree

tem

pera

ture

s ab

ove

zero

, put

them

in o

rder

from

col

dest

to h

ottes

t.Ex

. Seq

uenc

e po

sitiv

e nu

mbe

rs c

orre

ctly

on

a nu

mbe

r lin

e (e

.g.,

tem

pera

ture

s).

Ex. L

ook

at th

ree

chec

kboo

k re

gist

ers

with

pos

itive

bal

ance

s an

d or

der

the

bala

nces

from

leas

t to

grea

test

.Ex

. Giv

en te

mpe

ratu

res

from

thre

e se

ason

s pu

t the

m in

ord

er fr

om c

olde

st to

hott

est.

Leve

l I A

A S

tude

nts

will

:EE

6.N

S.5-

8. Id

entif

y w

hich

is g

reat

er th

an a

nd le

ss th

an u

sing

few

er th

an 1

0.Ex

. Giv

en tw

o se

ts o

f man

ipul

ative

s, id

entif

y w

hich

has

the

grea

ter

amou

nt o

r w

hich

has

the

less

er a

mou

nt.

Ex. I

n a

scie

nce

expe

rim

ent g

row

ing

plan

ts, d

eter

min

e ho

w m

any

plan

ts h

ave

lived

and

how

man

y ha

ve d

ied

to d

eter

min

e if

mor

e liv

ed o

r di

ed.

Ex. J

oe h

as th

ree

mar

bles

, Fra

nk h

as s

ix.

Who

has

mor

e?Ex

. Far

mer

John

has

five

cow

s an

d ni

ne p

igs.

Are

ther

e m

ore

cow

s or

pig

s?Ex

. Giv

en a

repr

esen

tatio

n of

a th

erm

omet

er, i

ndic

ate

whi

ch d

irecti

on im

plie

s a

grea

ter

tem

pera

ture

.Ex

. On

a nu

mbe

r lin

e, w

hich

num

ber

is c

lose

r to

zer

o: th

ree

or fi

ve?

Ex. G

iven

two

tem

pera

ture

s ab

ove

zero

, ind

icat

e w

hich

is g

reat

er.


Sixt

h G

rad

e M

ath

emat

ics

Stan

dar

ds:

Exp

ress

ion

s an

d E

qu

atio

ns

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

App

ly a

nd e

xten

d pr

evio

us

unde

rsta

ndin

gs o

f ari

thm

etic

to

alge

brai

c ex

pres

sion

s.

6.EE

.1. W

rite

and

eva

luat

e nu

mer

ical

exp

ress

ions

invo

lvin

g w

hole

-num

ber

expo

nent

s.

6.EE

.2. W

rite

, rea

d, a

nd e

valu

ate

expr

essi

ons

in w

hich

lett

ers

stan

d fo

r nu

mbe

rs.

•W

rite

exp

ress

ions

that

re

cord

ope

ratio

ns w

ith n

umbe

rs

and

with

lett

ers

stan

ding

for

num

bers

. Fo

r exa

mpl

e, e

xpre

ss

the

calc

ulati

on “

Subt

ract

y fr

om

5” a

s 5

– y.

•Id

entif

y pa

rts

of a

n ex

pres

sion

usi

ng m

athe

mati

cal

term

s (s

um, t

erm

, pro

duct

, fa

ctor

, quo

tient

, coe

ffici

ent)

; vi

ew o

ne o

r m

ore

part

s of

an

expr

essi

on a

s a

sing

le e

ntity

. Fo

r ex

ampl

e, d

escr

ibe

the

expr

essi

on

2 (8

+ 7

) as

a pr

oduc

t of t

wo

fact

ors;

vie

w (8

+ 7

) as

both

a

sing

le e

ntity

and

a s

um o

f tw

o te

rms.

•Ev

alua

te e

xpre

ssio

ns

at s

peci

fic v

alue

s of

thei

r va

riab

les.

Inc

lude

exp

ress

ions

th

at a

rise

from

form

ulas

us

ed in

real

-wor

ld p

robl

ems.

Pe

rfor

m a

rith

meti

c op

erati

ons,

in

clud

ing

thos

e in

volv

ing

who

le-

num

ber

expo

nent

s, in

the

conv

entio

nal o

rder

whe

n th

ere

are

no p

aren

thes

es to

spe

cify

a

parti

cula

r or

der

(Ord

er o

f O

pera

tions

). F

or e

xam

ple,

use

th

e fo

rmul

as V

= s

3 and

A =

6 s

2 to

find

the

volu

me

and

surf

ace

area

of a

cub

e w

ith s

ides

of

leng

th s

= 1

/2.

EE6.

EE.1

-2. I

denti

fy e

quiv

alen

t nu

mbe

r se

nten

ces.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

EE.1

. Gen

erat

e a

two-

step

mat

h se

nten

ce u

sing

app

ropr

iate

num

bers

and

sym

bols

.Ex

. Giv

en a

two-

step

wor

d pr

oble

m, i

denti

fy th

e nu

mer

ical

equ

ival

ent (

e.g.

, “Jo

hn h

as

two

appl

es, M

ary

has

thre

e. J

ohn

ate

one

appl

e. H

ow m

any

appl

es a

re le

ft?”

Stu

dent

pr

oduc

es th

e m

ath

sent

ence

(2 +

3 –

1 =

) or

(2 –

1 +

3 =

).Ex

. Giv

en a

two-

step

wor

d pr

oble

m, i

denti

fy th

e nu

mer

ical

equ

ival

ent (

e.g.

“Tr

udy

has

thre

e ca

kes.

She

was

giv

en o

ne m

ore.

Fra

nk h

as tw

o ca

kes.

Sho

w w

ho h

as th

e gr

eate

r nu

mbe

r of

cak

es.”

(3

+ 1

> 2)

, (3

+ 1

= 4,

4 >

2).

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

EE.1

. Ide

ntify

equ

ival

ent n

umbe

r se

nten

ces.

Ex. G

iven

a w

ord

prob

lem

, ide

ntify

the

num

eric

al e

quiv

alen

t (e.

g. “

John

has

one

pen

cil.

H

e is

giv

en fi

ve m

ore.

How

man

y pe

ncils

doe

s he

hav

e?”

Stu

dent

iden

tifies

1 +

5 =

as

an

equi

vale

nt to

the

stat

emen

t.).

Ex. G

iven

a w

ord

prob

lem

, ide

ntify

the

num

eric

al e

quiv

alen

t (e.

g. “

Teac

her

plac

es g

roup

of

thre

e pe

ncils

and

a g

roup

of f

our

penc

ils to

the

left

of s

tude

nt.

Teac

her

then

pla

ces

a se

cond

gro

up o

f five

pen

cils

and

two

penc

ils to

the

righ

t of t

he s

tude

nt a

nd a

sks,

“do

es

this

gro

up o

f pen

cils

hav

e th

e sa

me

amou

nt a

s th

e ot

her

grou

p of

pen

cils

?” (

3 +

4 =

5 +

2).

Ex. G

iven

a n

umbe

r pr

oble

m, s

elec

t fro

m c

hoic

es a

n eq

uiva

lent

pro

blem

(e.g

., 1

+ 3

has

the

sam

e re

sult

as 2

+ 2

).

Leve

l II A

A S

tude

nts

will

:EE

6.EE

.1. M

atch

num

ber

sent

ence

with

the

corr

ect p

ictu

re re

pres

enta

tion.

Ex. G

iven

a p

ictu

re s

how

ing

sing

le a

dditi

on, i

denti

fy c

orre

ct n

umbe

r se

nten

ce.

Ex. G

iven

a p

ictu

re a

nd a

cor

rect

and

inco

rrec

t num

ber

sent

ence

, cho

ose

one

that

is

corr

ect.

Leve

l I A

A S

tude

nts

will

:EE

6.EE

.1. I

denti

fy m

ath

sym

bol “

=” a

s m

eani

ng e

qual

to.

Ex. I

ndic

ate

the

sym

bol i

n a

mat

h se

nten

ce.

Ex. G

iven

pic

ture

repr

esen

tatio

ns o

f tw

o eq

ual g

roup

s of

obj

ects

with

an

equa

l sig

n be

twee

n, re

spon

ds th

at th

ey a

re th

e sa

me.

Math | 6-8 Grade 17

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

6.EE

.3. A

pply

the

prop

ertie

s of

ope

ratio

ns to

gen

erat

e eq

uiva

lent

exp

ress

ions

. Fo

r ex

ampl

e, a

pply

the

dist

ributi

ve

prop

erty

to th

e ex

pres

sion

3 (2

+

x) to

pro

duce

the

equi

vale

nt

expr

essi

on 6

+ 3

x; a

pply

the

dist

ributi

ve p

rope

rty

to th

e ex

pres

sion

24x

+ 1

8y to

pro

duce

th

e eq

uiva

lent

exp

ress

ion

6 (4

x +

3y);

appl

y pr

oper

ties

of

oper

ation

s to

y +

y +

y to

pro

duce

th

e eq

uiva

lent

exp

ress

ion

3y.

6.EE

.4. I

denti

fy w

hen

two

expr

essi

ons

are

equi

vale

nt (i

.e.,

whe

n th

e tw

o ex

pres

sion

s na

me

the

sam

e nu

mbe

r re

gard

less

of

whi

ch v

alue

is s

ubsti

tute

d in

to th

em).

For

exa

mpl

e, th

e ex

pres

sion

s y

+ y

+ y

and

3y a

re

equi

vale

nt b

ecau

se th

ey n

ame

the

sam

e nu

mbe

r reg

ardl

ess

of w

hich

num

ber y

sta

nds

for.

Rea

son

abou

t and

sol

ve

one-

varia

ble

equa

tions

and

in

equa

lities

.

EE6.

EE.3

-4. D

emon

stra

te

unde

rsta

ndin

g of

equ

ival

ent

expr

essi

ons.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

EE.3

-4. S

olve

equ

ival

ent e

xpre

ssio

ns to

illu

stra

te th

at th

ey a

re e

quiv

alen

t.Ex

. Fill

in th

e bl

ank

to m

ake

a tr

ue s

tate

men

t: 2

+ 6

= 6

+ _

__.

Ex. F

ill in

the

blan

k to

mak

e a

true

sta

tem

ent:

3 +

5 =

___

+ 3

.Ex

. Fill

in th

e bl

ank

to m

ake

a tr

ue s

tate

men

t: 4

+ _

__ =

3 +

4.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

EE.3

-4. D

emon

stra

te u

nder

stan

ding

of e

quiv

alen

t exp

ress

ions

.Ex

. Ind

icat

e th

at 2

+ 3

is th

e sa

me

as 3

+ 2

.Ex

. Ans

wer

yes

or

no w

hen

aske

d, “

Is 2

+ 3

equ

al to

3 +

2?”

Ex. A

nsw

er y

es o

r no

whe

n as

ked,

“Is

2 +

3 e

qual

to 4

+ 2

?”

Leve

l II A

A S

tude

nts

will

:EE

6.EE

.3-4

. Rec

ogni

ze d

iffer

ent d

ispl

ays

of th

e eq

ual q

uanti

ties.

Ex. G

iven

a m

odel

, cre

ate

an e

xpre

ssio

n us

ing

man

ipul

ative

s (e

.g.,

thre

e bl

ocks

plu

s tw

o bl

ocks

equ

als

five

bloc

ks).

Ex. G

iven

a g

roup

of t

hree

obj

ects

, a g

roup

of f

our

obje

cts,

and

a g

roup

of s

even

obj

ects

, m

atch

to 3

+ 4

= 7

.

Leve

l I A

A S

tude

nts

will

:EE

6.EE

.3-4

. Mat

ch d

iffer

ent d

ispl

ays

of th

e sa

me

quan

tity.

Ex. M

atch

pic

ture

s of

qua

ntitie

s of

obj

ects

to th

eir

num

eric

al e

quiv

alen

t (e.

g., f

our

balls

m

atch

es to

the

num

ber

4).


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Reas

on a

bout

and

sol

ve

one-

vari

able

equ

ation

s an

d in

equa

lities

.

6.EE

.5. U

nder

stan

d so

lvin

g an

equ

ation

or

ineq

ualit

y as

a

proc

ess

of a

nsw

erin

g a

ques

tion:

w

hich

val

ues

from

a s

peci

fied

set,

if a

ny, m

ake

the

equa

tion

or

ineq

ualit

y tr

ue?

Use

sub

stitu

tion

to d

eter

min

e w

heth

er a

giv

en

num

ber

in a

spe

cifie

d se

t mak

es

an e

quati

on o

r in

equa

lity

true

.

6.EE

.6. U

se v

aria

bles

to

repr

esen

t num

bers

and

wri

te

expr

essi

ons

whe

n so

lvin

g a

real

-w

orld

or

mat

hem

atica

l pro

blem

; un

ders

tand

that

a v

aria

ble

can

repr

esen

t an

unkn

own

num

ber,

or, d

epen

ding

on

the

purp

ose

at

hand

, any

num

ber

in a

spe

cifie

d se

t.

6.EE

.7. S

olve

real

-wor

ld a

nd

mat

hem

atica

l pro

blem

s by

w

ritin

g an

d so

lvin

g eq

uatio

ns o

f th

e fo

rm x

+ p

= q

and

px

= q

for

case

s in

whi

ch p

, q a

nd x

are

all

nonn

egati

ve ra

tiona

l num

bers

.

6.EE

.8. W

rite

an

ineq

ualit

y of

the

form

x >

c o

r x

< c

to re

pres

ent a

co

nstr

aint

or

cond

ition

in a

real

w

orld

or

mat

hem

atica

l pro

blem

. Re

cogn

ize

that

ineq

ualiti

es o

f the

fo

rm x

> c

or

x <

c ha

ve in

finite

ly

man

y so

lutio

ns; r

epre

sent

so

lutio

ns o

f suc

h in

equa

lities

on

num

ber

line

diag

ram

s.

EE6.

EE.5

-7. M

atch

an

equa

tion

to a

real

-wor

ld p

robl

em in

whi

ch

vari

able

s ar

e us

ed to

repr

esen

t nu

mbe

rs.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

EE.2

. Usi

ng a

var

iabl

e, g

ener

ate

an e

quiv

alen

t equ

ation

that

repr

esen

ts a

real

-wor

ld

prob

lem

.Ex

. Arr

ange

sym

bols

and

num

bers

to s

how

this

equ

ation

: Joe

has

thre

e cu

ps a

nd S

ue h

as

som

e m

ore

cups

. If

they

hav

e ei

ght c

ups

toge

ther

, how

wou

ld w

e w

rite

this

? A

nsw

er: 3

+

X =

8.Ex

. Sho

w h

ow to

wri

te th

is e

quati

on: t

wo

stud

ents

hav

e ap

ples

, one

stu

dent

has

five

ap

ples

, the

oth

er s

tude

nt h

as m

ore

appl

es, a

nd th

ere

are

12 a

pple

s al

toge

ther

. H

ow

wou

ld y

ou w

rite

this

? A

nsw

er 5

+ X

= 1

2.Ex

. Tog

ethe

r Pe

te a

nd Jo

e ha

ve fi

ve c

andi

es.

Pete

has

two.

How

man

y do

es Jo

e ha

ve?

Sh

ow th

e pr

oble

m w

ith m

anip

ulati

ves

usin

g X

to re

pres

ent t

he u

nkno

wn,

how

wou

ld y

ou

wri

te th

e eq

uatio

n us

ing

X. A

nsw

er: 2

+ X

= 7

.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

EE.2

. Mat

ch a

n eq

uatio

n to

a re

al-w

orld

pro

blem

in w

hich

var

iabl

es a

re u

sed

to

repr

esen

t num

bers

.Ex

. Mat

ch a

n eq

uatio

n us

ing

X to

repr

esen

t how

man

y Fr

ed h

as: F

red

and

June

hav

e fiv

e ap

ples

. Ju

ne h

as tw

o. S

how

me

this

pro

blem

. A

nsw

er: 2

+ X

= 5

.Ex

. Tel

l tha

t X m

eans

“ho

w m

any”

in 2

+

=5 a

nd in

sert

X in

the

box.

Ex. M

atch

an

equa

tion

to th

is w

ord

prob

lem

: I k

now

Tom

my

has

thre

e tic

kets

. H

ow m

any

mor

e tic

kets

will

he

need

if h

e w

ants

to ta

ke fi

ve fr

iend

s to

a m

ovie

? A

nsw

er: 3

+ X

= 5

.

Leve

l II A

A S

tude

nts

will

:EE

6.EE

.2. D

eter

min

e w

hat i

s un

know

n in

an

equa

tion.

Ex. A

fter

hea

ring

a s

tory

pro

blem

, ind

icat

e w

hat i

s un

know

n (t

he te

ache

r la

bels

that

as

X).

Ex. T

ell t

hat X

mea

ns “

how

man

y” in

2 +

=5

and

inse

rt X

in th

e bo

x.Ex

. Ind

icat

e th

e X

whe

n as

ked,

“W

hat n

umbe

r do

I no

t kno

w in

this

equ

ation

?

Leve

l I A

A S

tude

nts

will

:EE

6.EE

.2. I

denti

fy th

e le

tter

in a

mat

hem

atica

l sen

tenc

e.Ex

. Poi

nt to

or

indi

cate

the

lett

er/fi

xed/

vari

able

.Ex

. Ind

icat

e “X

” in

the

equa

tion

whe

n as

ked.

Math | 6-8 Grade 19

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Repr

esen

t an

d an

alyz

e qu

anti

tati

ve re

lati

onsh

ips

betw

een

depe

nden

t and

in

depe

nden

t va

riab

les.

6.EE

.9.U

se v

aria

bles

to

repr

esen

t tw

o qu

antiti

es in

a

real

-wor

ld p

robl

em th

at c

hang

e in

rela

tions

hip

to o

ne a

noth

er;

wri

te a

n eq

uatio

n to

exp

ress

on

e qu

antit

y, th

ough

t of a

s th

e de

pend

ent v

aria

ble,

in te

rms

of th

e ot

her

quan

tity,

thou

ght

of a

s th

e in

depe

nden

t var

iabl

e.

Ana

lyze

the

rela

tions

hip

betw

een

the

depe

nden

t and

in

depe

nden

t var

iabl

es u

sing

gr

aphs

and

tabl

es, a

nd re

late

th

ese

to th

e eq

uatio

n. F

or

exam

ple,

in a

pro

blem

invo

lvin

g m

otion

at c

onst

ant s

peed

, lis

t and

gra

ph o

rder

ed p

airs

of

dis

tanc

es a

nd ti

mes

, and

w

rite

the

equa

tion

d =

65t

to re

pres

ent t

he re

latio

nshi

p be

twee

n di

stan

ce a

nd ti

me.

EE6.

EE.9

. N/A


Sixt

h G

rad

e M

ath

emat

ics

Stan

dar

ds:

Geo

met

ry

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Solv

e re

al-w

orld

and

mat

hem

atica

l pr

oble

ms

invo

lvin

g ar

ea, s

urfa

ce

area

, and

vol

ume.

6.G

.1. F

ind

the

area

of r

ight

tr

iang

les,

oth

er tr

iang

les,

spe

cial

qu

adri

late

rals

, and

pol

ygon

s by

com

posi

ng in

to re

ctan

gles

or

dec

otm

posi

ng in

to tr

iang

les

and

othe

r sh

apes

; app

ly th

ese

tech

niqu

es in

the

cont

ext o

f sol

ving

re

al w

orld

and

mat

hem

atica

l pr

oble

ms.

6.G

.2. F

ind

the

volu

me

of a

rig

ht

rect

angu

lar

pris

m w

ith fr

actio

nal

edge

leng

ths

by p

acki

ng it

with

un

it cu

bes

of th

e ap

prop

riat

e un

it fr

actio

n ed

ge le

ngth

s, a

nd s

how

that

th

e vo

lum

e is

the

sam

e as

wou

ld

be fo

und

by m

ultip

lyin

g th

e ed

ge

leng

ths

of th

e pr

ism

. A

pply

the

form

ulas

V =

l w

h a

nd V

= b

h to

find

vo

lum

es o

f rig

ht re

ctan

gula

r pr

ism

s w

ith fr

actio

nal e

dge

leng

ths

in th

e co

ntex

t of s

olvi

ng re

al w

orld

and

m

athe

mati

cal p

robl

ems.

EE6.

G.1

-2. D

emon

stra

te a

rea.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

G.1

-2. F

ind

area

.

Ex. D

eter

min

e ho

w m

any

tiles

in a

sin

gle

laye

r ar

e re

quire

d to

cov

er a

rect

angl

e.

11

1

11

1

11

13

+ 3

+ 3

= 9

tiles

Ex. D

eter

min

e ho

w m

any

cube

s in

a s

ingl

e la

yer

are

requ

ired

to c

over

the

bott

om o

f a b

ox a

nd s

tate

th

e nu

mbe

r re

quire

d.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

G.1

-2. D

emon

stra

te a

rea.

Ex. G

iven

two

repr

esen

tatio

ns, i

denti

fy w

hich

has

are

a (e

.g. l

ine

segm

ent,

ang

le, s

quar

e).

Ex. U

se s

quar

es o

f col

ored

pap

er to

cov

er th

eir

desk

or

tray

on

a w

heel

chai

r.

Ex. T

ell w

hich

figu

re is

larg

er in

side

.

Leve

l II A

A S

tude

nts

will

:

EE6.

G.1

-2. D

eter

min

e w

hat i

s th

e la

rger

are

a.

Leve

l I A

A S

tude

nts

will

:

EE6.

G.1

-2. I

ndic

ate

the

insi

de o

f a s

pace

.

Ex. F

ill in

the

insi

de o

f a fi

gure

whe

n th

e di

ffere

nce

betw

een

the

insi

de a

nd o

utsi

de is

cle

ar.

Ex. A

nsw

er y

es o

r no

whe

n as

ked,

“H

ere

is a

bas

ket.

Her

e is

a b

all.

Put

the

ball

insi

de th

e ba

sket

. Is

th

e ba

ll in

side

or

outs

ide

the

bask

et?”

Ex. P

oint

aro

und

the

room

or

spre

ad a

rms

whe

n as

ked

“Are

we

insi

de o

r ou

tsid

e ou

r cl

assr

oom

?”

Ex. P

oint

to th

e in

side

of a

box

or

fram

e w

hen

aske

d, “

Whe

re is

the

insi

de?”

Math | 6-8 Grade 21

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

6.G

.3. D

raw

pol

ygon

s in

the

coor

dina

te p

lane

giv

en c

oord

inat

es

for

the

verti

ces;

use

coo

rdin

ates

to

find

the

leng

th o

f a s

ide

join

ing

poin

ts w

ith th

e sa

me

first

coo

rdin

ate

or th

e sa

me

seco

nd c

oord

inat

e.

App

ly th

ese

tech

niqu

es in

the

cont

ext o

f sol

ving

real

-wor

ld a

nd

mat

hem

atica

l pro

blem

s.6.

G.4

. Rep

rese

nt th

ree-

dim

ensi

onal

fig

ures

usi

ng n

ets

mad

e up

of

rect

angl

es a

nd tr

iang

les,

and

use

the

nets

to fi

nd th

e su

rfac

e ar

ea o

f the

se

figur

es.

App

ly th

ese

tech

niqu

es in

th

e co

ntex

t of s

olvi

ng re

al-w

orld

and

m

athe

mati

cal p

robl

ems.

EE6.

G.4

. Ide

ntify

com

mon

thre

e-di

men

sion

al s

hape

s.Le

vel I

V A

A S

tude

nts

will

:

EE6.

G.4

. Rel

ate

real

-wor

ld it

ems

as th

ree-

dim

ensi

onal

sha

pes

to th

eir

two-

dim

ensi

onal

re

pres

enta

tions

.

Ex. M

atch

the

pict

ure

of th

e so

da c

an to

the

pict

ure

of th

e cy

linde

r, et

c.

Ex. I

denti

fy in

the

envi

ronm

ent i

tem

s th

at a

re th

ree-

dim

ensi

onal

whe

n pr

esen

ted

with

in th

e tw

o-di

men

sion

al fo

rmat

.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

G.4

. Ide

ntify

com

mon

thre

e-di

men

sion

al s

hape

s.

Ex. W

hen

pres

ente

d w

ith a

sph

ere

and

a cu

be, n

ame

the

thre

e-di

men

sion

al s

hape

.

Ex. I

denti

fy s

pher

es a

nd c

ubes

in th

e cl

assr

oom

.

Leve

l II A

A S

tude

nts

will

:

EE6.

G.4

. Sor

t thr

ee-d

imen

sion

al s

hape

s an

d tw

o-di

men

sion

al s

hape

s.

Ex. W

hen

give

n a

bag

of th

ree-

dim

ensi

onal

sha

pes

and

thei

r tw

o-di

men

sion

al p

ictu

res,

sor

t int

o th

e ap

prop

riat

e th

ree-

dim

ensi

onal

or

two-

dim

ensi

onal

sha

pe.

Ex. L

abel

obj

ects

as

thre

e-di

men

sion

al a

nd tw

o-di

men

sion

al s

hape

s in

the

clas

sroo

m.

Leve

l I A

A S

tude

nts

will

:

EE6.

G.4

. Mat

ch s

hape

s.

Ex. W

hen

give

n a

pict

ure

of a

sha

pe, fi

nd li

ke s

hape

s in

the

clas

sroo

m.

Ex. S

hape

BIN

GO

.


Sixt

h G

rad

e M

ath

emat

ics

Stan

dar

ds:

Sta

tist

ics

and

Pro

bab

ilit

y

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Dev

elop

und

erst

andi

ng o

f st

atisti

cal v

aria

bilit

y.

6.SP

.1. R

ecog

nize

a s

tatis

tical

qu

estio

n as

one

that

anti

cipa

tes

vari

abili

ty in

the

data

rela

ted

to

the

ques

tion

and

acco

unts

for

it in

the

answ

ers.

For

exa

mpl

e,

“How

old

am

I?”

is n

ot a

sta

tistic

al

ques

tion,

but

“H

ow o

ld a

re th

e st

uden

ts in

my

scho

ol?”

is a

st

atisti

cal q

uesti

on b

ecau

se o

ne

antic

ipat

es v

aria

bilit

y in

stu

dent

s’

ages

.

6.SP

.2. U

nder

stan

d th

at a

set

of

dat

a co

llect

ed to

ans

wer

a

stati

stica

l que

stion

has

a

dist

ribu

tion,

whi

ch c

an b

e de

scri

bed

by it

s ce

nter

, spr

ead,

an

d ov

eral

l sha

pe.

EE6.

SP.1

-2. D

ispl

ay d

ata

on a

gra

ph

or ta

ble

that

sho

ws

vari

abili

ty in

th

e da

ta.

Leve

l IV

AA

Stu

dent

s w

ill:

EE6.

SP.1

-2. C

olle

ct, d

ispl

ay, a

nd d

escr

ibe

data

on

a gr

aph

or ta

ble.

Ex. C

olle

ct d

ata

for

a cl

assr

oom

exp

erim

ent a

nd c

hart

hei

ght o

f pla

nts,

tem

pera

ture

of s

oil,

etc.

Ex. C

olle

ct d

ata

from

a c

lass

sur

vey

of h

eigh

t and

cre

ate

a ta

ble

show

ing

the

vari

ance

in h

eigh

t (e

.g.,

shor

test

per

son

is 4

’6”,

the

talle

st p

erso

n is

5’4

”).

Ex. C

olle

ct w

eath

er d

ata

and

grap

h to

sho

w v

aria

nce

(e.g

., fiv

e su

nny

days

, thr

ee c

loud

y, tw

o ra

iny)

.Ex

. Des

crib

e da

ta la

id o

ut o

n a

grap

h sh

owin

g a

dist

ribu

tion

of re

spon

ses.

For

exa

mpl

e,

stud

ents

hav

e di

ffere

nt h

eigh

ts, b

ut th

ere

are

man

y w

ith s

imila

r he

ight

s, w

hile

som

e ar

e m

uch

talle

r or

sho

rter

.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

SP.1

-2. D

ispl

ay d

ata

on a

gra

ph o

r ta

ble

that

sho

ws

vari

abili

ty o

f dat

a.Ex

. Giv

en w

eath

er d

ata

for

the

wee

k, d

ispl

ay it

on

a gr

aph

to s

how

var

ianc

e (e

.g.,

five

sunn

y da

ys, t

hree

clo

udy,

two

rain

y).

Ex. G

iven

dat

a ab

out t

he a

ges

of s

tude

nts

in th

e cl

ass

(e.g

., 12

, 13,

and

14)

, dis

play

dat

a in

a

tabl

e sh

owin

g th

e va

rian

ce in

age

(e.g

., fe

wes

t are

12

year

s ol

d, m

ost a

re 1

3 ye

ars

old)

.

Leve

l II A

A S

tude

nts

will

:EE

6.SP

.1-2

. Org

aniz

e da

ta.

Ex. S

urve

y st

uden

ts in

the

clas

sroo

m c

once

rnin

g fa

vori

tes

amon

g th

ree

choi

ces

and

repr

esen

t re

spon

ses

(e.g

., ho

w m

any

pick

eac

h of

thre

e st

orie

s or

eac

h of

thre

e su

bjec

ts).

Ex. G

iven

dat

a, s

ort t

o de

term

ine

how

man

y (e

.g.,

how

man

y st

uden

ts h

ave

cert

ain

num

ber

of

sibl

ings

).Le

vel I

AA

Stu

dent

s w

ill:

EE6.

SP.1

-2. S

ort i

nfor

mati

on in

to c

ateg

orie

s of

sam

e an

d di

ffere

nt.

Ex. A

fter

cha

rting

the

wea

ther

for

a w

eek,

iden

tify

if to

day’

s w

eath

er w

as th

e sa

me

or d

iffer

ent

than

yes

terd

ay.

Ex. G

iven

a g

raph

ic o

rgan

izer

with

thre

e ca

tego

ries

of c

olor

s id

entifi

ed, s

ort s

even

dis

cs o

f th

ree

diffe

rent

col

ors

into

the

cate

gori

es a

nd p

lace

them

in th

e ap

prop

riat

e pl

ace

on th

e gr

aphi

c or

gani

zer.

6.SP

.3. R

ecog

nize

that

a m

easu

re

of c

ente

r fo

r a

num

eric

al d

ata

set

sum

mar

izes

all

of it

s va

lues

with

a

sing

le n

umbe

r, w

hile

a m

easu

re o

f va

riati

on d

escr

ibes

how

its

valu

es

vary

with

a s

ingl

e nu

mbe

r.

EE6.

SP.3

. N/A

Sum

mar

ize

and

desc

ribe

di

stri

buti

ons.

6.SP

.4. D

ispl

ay n

umer

ical

dat

a in

pl

ots

on a

num

ber

line,

incl

udin

g do

t plo

ts, h

isto

gram

s, a

nd b

ox

plot

s.

EE6.

SP.4

. N/A

(See

EE6

.SP.

1-2)

Math | 6-8 Grade 23

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

6.SP

.5. S

umm

ariz

e nu

mer

ical

da

ta s

ets

in re

latio

n to

thei

r co

ntex

t, s

uch

as b

y:•

Repo

rting

the

num

ber

of o

bser

vatio

ns.

•D

escr

ibin

g th

e na

ture

of

the

attri

bute

und

er in

vesti

gatio

n,

incl

udin

g ho

w it

was

mea

sure

d an

d its

uni

ts o

f mea

sure

men

t.•

Giv

ing

quan

titati

ve

mea

sure

s of

cen

ter

(med

ian

and/

or m

ean)

and

var

iabi

lity

(inte

rqua

rtile

rang

e an

d/or

mea

n ab

solu

te d

evia

tion)

, as

wel

l as

desc

ribi

ng a

ny o

vera

ll pa

tter

n an

d an

y st

riki

ng d

evia

tions

fr

om th

e ov

eral

l patt

ern

with

re

fere

nce

to th

e co

ntex

t in

whi

ch

the

data

wer

e ga

ther

ed.

•Re

latin

g th

e ch

oice

of

mea

sure

s of

cen

ter

and

vari

abili

ty to

the

shap

e of

the

data

dis

trib

ution

and

the

cont

ext

in w

hich

the

data

wer

e ga

ther

ed.

EE6.

SP.5

. Sum

mar

ize

data

di

stri

butio

ns o

n a

grap

h or

tabl

e.Le

vel I

V A

A S

tude

nts

will

:EE

6.SP

.5. S

umm

ariz

e th

e da

ta o

n a

grap

h or

tabl

e.Ex

. Whe

n lo

okin

g at

a ta

ble

of w

hat s

tude

nts

like

to e

at fo

r lu

nch,

sum

mar

ize

the

data

in

mul

tiple

way

s (i.

e., c

hick

en n

ugge

ts h

as th

e m

ost,

piz

za h

as th

e le

ast)

.Ex

. Whe

n lo

okin

g at

a g

raph

of t

empe

ratu

res

from

the

wee

k, s

umm

ariz

e th

e da

ta in

m

ultip

le w

ays

(i.e.

, thr

ee d

ays

wer

e ab

ove

70 d

egre

es, s

ix d

ays

wer

e be

twee

n 60

-70

degr

ees,

and

two

days

wer

e 50

-60

degr

ees)

.

Leve

l III

AA

Stu

dent

s w

ill:

EE6.

SP.5

. Sum

mar

ize

data

dis

trib

ution

s on

a g

raph

or

tabl

e.Ex

. Whe

n lo

okin

g at

a g

raph

of t

empe

ratu

res

from

the

wee

k, s

umm

ariz

e th

e da

ta in

one

w

ay (i

.e.,

thre

e da

ys w

ere

abov

e 70

deg

rees

).Ex

. Whe

n lo

okin

g at

a ta

ble

of w

hat s

tude

nts

like

to e

at fo

r lu

nch,

sum

mar

ize

the

data

in

one

way

(e.g

., ch

icke

n nu

gget

s ha

s th

e m

ost;

piz

za h

as th

e le

ast)

.

Leve

l II A

A S

tude

nts

will

:EE

6.SP

.5. U

se a

gra

ph to

det

erm

ine

whi

ch c

ateg

ory

has

the

mos

t.Ex

. Loo

king

at a

bar

gra

ph o

n th

e st

uden

ts’ f

avor

ite s

ubje

ct in

sch

ool,

iden

tify

whi

ch is

the

mos

t pre

ferr

ed s

ubje

ct.

Ex. L

ooki

ng a

t a p

icto

grap

h of

the

stud

ents

’ fav

orite

spo

rts

team

s, id

entif

y w

hich

is th

e m

ost p

refe

rred

team

.

Leve

l I A

A S

tude

nts

will

:EE

6.SP

.5. I

denti

fy w

hich

has

mor

e or

less

.Ex

. Giv

en tw

o ite

ms

on a

bar

gra

ph, i

denti

fy w

hich

has

mor

e or

less

.Ex

. Giv

en tw

o to

wer

s of

inte

rloc

king

cub

es, i

denti

fy w

hich

has

mor

e or

less

.


CO

MM

ON

CO

RE

ESS

EN

TIA

L E

LEM

EN

TS

AN

D A

CH

IEV

EM

EN

T

DE

SCR

IPT

OR

S FO

R S

EVE

NT

H G

RA

DE

Seve

nth

Gra

de

Mat

hem

atic

s St

and

ard

s: R

atio

s an

d P

rop

ort

ion

al R

elat

ion

ship

s

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Ana

lyze

pro

porti

onal

rela

tion

ship

s an

d us

e th

em to

sol

ve re

al-w

orld

and

m

athe

mati

cal p

robl

ems.

7.RP

.1. C

ompu

te u

nit r

ates

ass

ocia

ted

with

ratio

s of

frac

tions

, inc

ludi

ng ra

tios

of le

ngth

s, a

reas

and

oth

er q

uanti

ties

mea

sure

d in

like

or

diffe

rent

uni

ts.

For

exam

ple,

if a

per

son

wal

ks 1

/2 m

ile in

ea

ch 1

/4 h

our,

com

pute

the

unit

rate

as

the

com

plex

frac

tion

1/2 / 1/

4 mile

s pe

r hou

r, eq

uiva

lent

ly 2

mile

s pe

r hou

r.

7.RP

.2. R

ecog

nize

and

repr

esen

t pr

opor

tiona

l rel

ation

ship

s be

twee

n qu

antiti

es.

•D

ecid

e w

heth

er tw

o qu

antiti

es a

re in

a p

ropo

rtion

al

rela

tions

hip,

e.g

., by

testi

ng fo

r eq

uiva

lent

ratio

s in

a ta

ble

or g

raph

ing

on a

coo

rdin

ate

plan

e an

d ob

serv

ing

whe

ther

the

grap

h is

a s

trai

ght l

ine

thro

ugh

the

orig

in.

•Id

entif

y th

e co

nsta

nt o

f pr

opor

tiona

lity

(uni

t rat

e) in

tabl

es,

grap

hs, e

quati

ons,

dia

gram

s, a

nd

verb

al d

escr

iptio

ns o

f pro

porti

onal

re

latio

nshi

ps.

•Re

pres

ent p

ropo

rtion

al

rela

tions

hips

by

equa

tions

. Fo

r ex

ampl

e, if

tota

l cos

t t is

pro

porti

onal

to

the

num

ber n

of i

tem

s pu

rcha

sed

at a

con

stan

t pric

e p,

the

rela

tions

hip

betw

een

the

tota

l cos

t and

the

num

ber o

f ite

ms

can

be e

xpre

ssed

as

t = p

n.•

Expl

ain

wha

t a p

oint

(x, y

) on

the

grap

h of

a p

ropo

rtion

al re

latio

nshi

p m

eans

in te

rms

of th

e si

tuati

on, w

ith

spec

ial a

tten

tion

to th

e po

ints

(0, 0

) and

(1

, r) w

here

r is

the

unit

rate

.

7.RP

.3. U

se p

ropo

rtion

al re

latio

nshi

ps

to s

olve

mul

tiste

p ra

tio a

nd p

erce

nt

prob

lem

s. E

xam

ples

: sim

ple

inte

rest

, tax

, m

arku

ps a

nd m

arkd

owns

, gra

tuiti

es a

nd

com

mis

sion

s, fe

es, p

erce

nt in

crea

se a

nd

decr

ease

, per

cent

err

or.

EE7.

RP.1

-3. U

se a

ratio

to m

odel

or

desc

ribe

a re

latio

nshi

p.Le

vel I

V A

A S

tude

nts

will

:EE

7.RP

.1-3

. Com

plet

e th

e ra

tio u

sing

num

bers

to s

how

rela

tions

hips

.Ex

. Giv

en o

ne c

ompo

nent

of a

ratio

in s

tand

ard

form

(1:_

) com

plet

e th

e ra

tio.

Ex. G

iven

a fa

mily

pic

ture

, wha

t is

the

ratio

of p

eopl

e w

eari

ng h

ats

com

pare

d to

the

tota

l num

ber

of p

eopl

e in

th

e pi

ctur

e?Ex

. Des

crib

e th

e re

latio

nshi

p be

twee

n m

iles

driv

en a

nd th

e tim

e ta

ken

by c

reati

ng a

ratio

(e.g

., Ka

tie k

now

s sh

e ca

n dr

ive

one

mile

in tw

o m

inut

es is

1:2

.)

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

RP.1

-3. U

se a

ratio

to m

odel

or

desc

ribe

a re

latio

nshi

p.Ex

. Giv

en a

bag

of g

reen

and

red

chip

s, id

entif

y th

e ra

tio o

f gre

en c

hips

com

pare

d to

red

chip

s.Ex

. Use

a p

icto

rial

repr

esen

tatio

n to

sho

w p

art-

who

le re

latio

nshi

p (e

.g.,

Wha

t par

t of t

he p

ictu

re is

sha

ded?

Th

ree

part

s ar

e sh

aded

and

one

par

t is

not.

).

Leve

l II A

A S

tude

nts

will

:EE

7.RP

.1-3

. Dem

onst

rate

a s

impl

e ra

tio re

latio

nshi

p.Ex

. Usi

ng a

dry

eas

e bo

ard

dem

onst

rate

a ra

tio re

latio

nshi

p of

squ

ares

to c

ircle

s.Ex

. Whe

n pl

ayin

g a

boar

d ga

me,

mov

e on

e sp

ace

for

ever

y do

t on

the

die.

Ex. C

ompl

ete

a pa

tter

n gi

ven

a si

mpl

e ra

tio.

Leve

l I A

A S

tude

nts

will

:EE

7.RP

.1-3

. Ide

ntify

one

item

as

it re

late

s to

ano

ther

.Ex

. Whe

n gi

ven

two

bask

ets

with

mar

kers

, cou

nt th

e nu

mbe

r in

eac

h ba

sket

and

com

pare

.Ex

. Giv

en tw

o ca

rds

with

att

enda

nce

card

s, c

ompa

re th

e nu

mbe

r he

re a

nd a

bsen

t.Ex

. Giv

en a

hal

f an

appl

e an

d a

who

le a

pple

, ide

ntify

“th

e w

hole

” ap

ple.

Math | 6-8 Grade 25

Seve

nth

Gra

de

Mat

hem

atic

s St

and

ard

s: T

he

Nu

mb

er S

yste

m

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

App

ly a

nd e

xten

d pr

evio

us

unde

rsta

ndin

gs o

f ope

rati

ons

wit

h fr

acti

ons

to a

dd, s

ubtr

act,

mul

tipl

y, a

nd

divi

de ra

tion

al n

umbe

rs.

7.N

S.1.

App

ly a

nd e

xten

d pr

evio

us

unde

rsta

ndin

gs o

f add

ition

and

su

btra

ction

to a

dd a

nd s

ubtr

act r

ation

al

num

bers

; rep

rese

nt a

dditi

on a

nd

subt

racti

on o

n a

hori

zont

al o

r ve

rtica

l nu

mbe

r lin

e di

agra

m.

•D

escr

ibe

situ

ation

s in

whi

ch

oppo

site

qua

ntitie

s co

mbi

ne to

mak

e 0.

For

exa

mpl

e, a

hyd

roge

n at

om h

as 0

ch

arge

bec

ause

its

two

cons

titue

nts

are

oppo

site

ly c

harg

ed.

•U

nder

stan

d p

+ q

as th

e nu

mbe

r lo

cate

d a

dist

ance

|q|

from

p,

in th

e po

sitiv

e or

neg

ative

dire

ction

de

pend

ing

on w

heth

er q

is p

ositi

ve o

r ne

gativ

e. S

how

that

a n

umbe

r an

d its

op

posi

te h

ave

a su

m o

f 0 (a

re a

dditi

ve

inve

rses

). In

terp

ret s

ums

of ra

tiona

l nu

mbe

rs b

y de

scri

bing

real

-wor

ld

cont

exts

.

•U

nder

stan

d su

btra

ction

of

ratio

nal n

umbe

rs a

s ad

ding

the

addi

tive

inve

rse,

p –

q =

p +

(–q)

. Sh

ow th

at th

e di

stan

ce b

etw

een

two

ratio

nal n

umbe

rs

on th

e nu

mbe

r lin

e is

the

abso

lute

va

lue

of th

eir

diffe

renc

e, a

nd a

pply

this

pr

inci

ple

in re

al-w

orld

con

text

s.

EE7.

NS.

1. A

dd fr

actio

ns w

ith li

ke

deno

min

ator

s (h

alve

s, th

irds,

four

ths,

an

d te

nths

) so

the

solu

tion

is le

ss th

an o

r eq

ual t

o on

e.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

NS.

1. S

ame

as L

evel

III A

A S

tude

nts.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

NS.

1. A

dd fr

actio

ns w

ith li

ke d

enom

inat

ors

(hal

ves,

third

s fo

urth

s, a

nd te

nths

) so

the

solu

tion

is le

ss th

an o

r eq

ual t

o on

e.

Ex. U

se fr

actio

n ba

rs o

r fr

actio

n ci

rcle

s to

add

so

that

ans

wer

is le

ss th

an o

r eq

ual t

o on

e. M

atch

a n

umer

ical

re

pres

enta

tion

to th

e m

odel

.

Ex. G

iven

tent

hs, c

onst

ruct

the

who

le a

nd re

cogn

ize

that

10

tent

hs a

re n

eede

d to

mak

e a

who

le.

(Con

nect

to

mon

ey --

10

dim

es =

one

who

le d

olla

r).

Leve

l II A

A S

tude

nts

will

:

EE7.

NS.

1. U

se m

odel

s to

add

hal

ves,

third

s, a

nd fo

urth

s.

Ex. G

iven

third

s, c

onst

ruct

the

who

le a

nd a

dd th

e nu

mbe

r of

third

s ne

eded

to m

ake

a w

hole

.

Ex. G

iven

four

ths,

con

stru

ct th

e w

hole

and

add

the

num

ber

of fo

urth

s ne

eded

to m

ake

a w

hole

.

Ex. G

iven

a re

cipe

that

cal

ls fo

r a

1/4

cup

of s

ugar

, sha

de a

pic

ture

of a

mea

suri

ng c

up m

arke

d in

to fo

urth

s to

sh

ow h

ow m

uch

suga

r is

nee

ded

to d

oubl

e th

e re

cipe

(1/4

+ 1

/4 =

2/4

or

1/2)

.

Ex. D

emon

stra

te th

at a

who

le c

an b

e di

vide

d in

to e

qual

par

ts, a

nd w

hen

reas

sem

bled

, rec

reat

es th

e w

hole

usi

ng

a m

odel

.

Leve

l I A

A S

tude

nts

will

:

EE7.

NS.

1. U

se m

odel

s to

iden

tify

the

who

le a

nd fi

nd th

e m

issi

ng p

iece

s of

a w

hole

.

Ex. G

iven

thre

e ch

oice

s, id

entif

y w

hich

is m

ore,

a w

hole

or

a ha

lf.

Ex. P

rese

nted

with

a w

hole

obj

ect a

nd th

e sa

me

obje

ct w

ith a

pie

ce m

issi

ng, i

denti

fy th

e w

hole

.

Ex. G

iven

1/2

a p

izza

, ide

ntify

the

mis

sing

par

t (co

ncre

te m

odel

or

touc

h bo

ard)

.

Ex. S

how

n pa

pers

cut

in h

alve

s, th

irds,

etc

., ch

oose

the

obje

ct c

ut in

hal

ves.

Ex. G

iven

box

es w

ith o

ne-t

hird

sha

ded,

one

-hal

f sha

ded,

and

the

who

le s

hade

d, c

hoos

e th

e on

e w

ith th

e w

hole

sh

aded

.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

App

ly p

rope

rties

of o

pera

tion

s as

st

rate

gies

to a

dd a

nd s

ubtr

act r

ation

al

num

bers

.

7.N

S.2.

App

ly a

nd e

xten

d pr

evio

us

unde

rsta

ndin

gs o

f mul

tiplic

ation

and

di

visi

on a

nd o

f fra

ction

s to

mul

tiply

and

di

vide

ratio

nal n

umbe

rs.

U

nder

stan

d th

at

mul

tiplic

ation

is e

xten

ded

from

frac

tions

to

ratio

nal n

umbe

rs b

y re

quir

ing

that

op

erati

ons

conti

nue

to s

atisf

y th

e pr

oper

ties

of o

pera

tions

, par

ticul

arly

th

e di

stri

butiv

e pr

oper

ty, l

eadi

ng to

pr

oduc

ts s

uch

as (–

1)(–

1) =

1 a

nd th

e ru

les

for

mul

tiply

ing

sign

ed n

umbe

rs.

Inte

rpre

t pro

duct

s of

ratio

nal n

umbe

rs

by d

escr

ibin

g re

al-w

orld

con

text

s.

EE7.

NS.

2.a.

Sol

ve m

ultip

licati

on

prob

lem

s w

ith p

rodu

cts

to 1

00.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

NS.

2.a.

Sol

ve m

ultip

licati

on p

robl

ems

with

pro

duct

s to

144

.

Ex. G

iven

a m

ultip

licati

on p

robl

em, s

olve

inde

pend

ently

usi

ng a

var

iety

of m

etho

ds.

Ex. G

iven

the

prod

uct a

nd th

ree

poss

ible

mul

tiplic

ation

pro

blem

s, id

entif

y th

e co

rrec

t mul

tiplic

ation

pro

blem

for

the

answ

er.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

NS.

2.a.

Sol

ve m

ultip

licati

on p

robl

ems

with

pro

duct

s to

100

.

Ex. G

iven

the

mod

el o

f a m

ultip

licati

on p

robl

em, i

denti

fy th

e m

ultip

licati

on p

robl

em a

nd th

e co

rres

pond

ing

answ

er.

Ex. G

iven

a m

ultip

licati

on p

robl

em (4

x 3

) and

thre

e an

swer

cho

ices

, use

a c

alcu

lato

r to

sol

ve th

e pr

oble

m a

nd

choo

se th

e co

rrec

t ans

wer

.

Ex. G

iven

an

arra

y of

mod

els,

sho

w w

hich

arr

ay d

epic

ts a

pro

blem

(e.g

., 5

x 7

= 35

).

Ex. S

olve

wor

d pr

oble

ms

usin

g m

ultip

licati

on (e

.g.,

I wan

t bri

ng 1

0 pe

ople

to m

y pa

rty

and

I hav

e tw

o pa

rty

hats

fo

r ea

ch p

erso

n. H

ow m

any

part

y ha

ts d

o I h

ave?

).

Leve

l II A

A S

tude

nts

will

:

EE7.

NS.

2.a.

Sol

ve m

ultip

licati

on p

robl

ems

usin

g fa

ctor

s 1

– 10

.

Ex. U

se re

peat

ed a

dditi

on to

sol

ve m

ultip

licati

on p

robl

ems.

Ex. U

sing

a m

ultip

licati

on c

hart

, ide

ntify

the

answ

er to

mul

tiplic

ation

pro

blem

s.

Ex. C

reat

e ar

rays

to m

odel

mul

tiplic

ation

fact

s.

Ex. U

se 1

00s

boar

d or

touc

h bo

ard

to m

odel

ski

p co

untin

g (i.

e., 2

, 4, 6

, 8

. . .

).

Ex. G

roup

item

s to

mod

el m

ultip

licati

on (e

.g.,

3 x

5 co

uld

be m

odel

ed b

y th

ree

grou

ps w

ith fi

ve in

eac

h gr

oup)

.

Leve

l I A

A S

tude

nts

will

:

EE7.

NS.

2.a.

Ski

p co

unt b

y tw

os a

nd te

ns.

Ex. M

odel

repe

ated

add

ition

.

Ex. U

se a

100

s bo

ard

or to

uch

boar

d to

ski

p co

unt (

i.e.,

2, 4

, 6, 8

, . .

. ).

Ex. G

iven

bun

dles

of p

ipe

clea

ners

(10

in e

ach

bund

le),

skip

cou

nt to

find

the

tota

l.

Math | 6-8 Grade 27

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

U

nder

stan

d th

at in

tege

rs

can

be d

ivid

ed, p

rovi

ded

that

the

divi

sor

is n

ot z

ero,

and

eve

ry q

uotie

nt

of in

tege

rs (w

ith n

on-z

ero

divi

sor)

is a

ra

tiona

l num

ber.

If p

and

q a

re in

tege

rs,

then

–(p

/q) =

(–p)

/q =

p/(

–q).

Inte

rpre

t qu

otien

ts o

f rati

onal

num

bers

by

desc

ribi

ng re

al-w

orld

con

text

s.

EE7.

NS.

2.b.

Sol

ve d

ivis

ion

prob

lem

s w

ith

divi

sors

up

to fi

ve a

nd a

lso

with

a d

ivis

or

of 1

0 w

ithou

t rem

aind

ers.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

NS.

2.b.

Sol

ve d

ivis

ion

prob

lem

s w

ith d

ivis

ors

up to

10

usin

g nu

mbe

rs.

Ex. G

iven

a re

al-w

orld

pro

blem

, find

the

solu

tion

usin

g di

visi

on (e

.g.,

“If I

hav

e th

e ar

ea o

f a h

all t

hat i

s 50

feet

an

d on

e si

de h

as a

leng

th o

f 5 fe

et, h

ow lo

ng is

the

othe

r si

de?)

.

Ex. G

iven

a p

robl

em in

volv

ing

mon

ey, fi

nd th

e so

lutio

n us

ing

divi

sion

(e.g

., “I

f a fr

iend

and

I fin

d 20

dol

lars

, how

w

ill w

e sp

lit it

up

so th

at w

e ea

ch g

et th

e sa

me

amou

nt?”

).

Ex. I

f I h

ave

a la

rge

bow

l with

eig

ht c

ups

of b

eans

, how

man

y tw

o-cu

p se

rvin

gs c

an I

get o

ut o

f tha

t bow

l?

Ex. G

iven

a c

ompu

ter

prog

ram

with

div

isio

n pr

oble

ms,

find

the

quoti

ent.

Ex. W

hen

plan

ting

seed

s fo

r a

scie

nce

expe

rim

ent,

div

ide

the

seed

s in

to 1

0 eq

ual s

hare

s an

d re

pres

ent t

he

prob

lem

in n

umer

als.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

NS.

2.b.

Sol

ve d

ivis

ion

prob

lem

s w

ith d

ivis

ors

up to

five

and

als

o w

ith a

div

isor

of 1

0 w

ithou

t rem

aind

ers.

Ex. U

se m

oney

to s

olve

div

isio

n pr

oble

ms

(e.g

., If

a fr

iend

and

I fin

d 10

dol

lars

, how

will

we

split

it u

p so

that

we

each

get

the

sam

e am

ount

? D

ivid

e th

e pa

per

mon

ey to

find

the

answ

er.).

Ex. G

iven

10

man

ipul

ative

s, d

ivid

e in

to tw

o eq

ual g

roup

s of

five

. Sh

ow th

at 1

0 /

2 =

5.

Ex. D

ivid

e th

e cl

assr

oom

into

four

equ

al g

roup

s fo

r a

spor

ts to

urna

men

t.

Ex. U

se th

e nu

mbe

r lin

e to

sho

w h

ow m

any

times

you

can

sub

trac

t five

out

of 1

5.

Ex. I

f you

giv

e ea

ch p

erso

n tw

o cu

ps o

f sou

p an

d yo

u ha

ve 1

0 cu

ps o

f sou

p, h

ow m

any

peop

le c

ould

com

e to

yo

ur s

oup

part

y?

Leve

l II A

A S

tude

nts

will

:

EE7.

NS.

2.b.

Det

erm

ine

how

man

y tim

es a

num

ber

can

be s

ubtr

acte

d fr

om a

n eq

ually

div

isib

le n

umbe

r.

Ex. G

iven

a n

umbe

r di

visi

ble

by fi

ve o

r 10

, sub

trac

t out

five

or

10, s

how

the

num

ber

of ti

mes

this

num

ber

can

be

subt

ract

ed (e

.g.,

“Sho

w m

e ho

w m

any

sets

of fi

ve p

ipe

clea

ners

you

can

div

ide

20 p

ipe

clea

ners

into

”).

Ex. G

iven

a n

umbe

r lin

e, d

emon

stra

te h

ow m

any

times

a n

umbe

r ca

n be

sub

trac

ted

from

an

equa

lly d

ivis

ible

nu

mbe

r (e

.g.,

“Sho

w m

e ho

w m

any

times

can

you

sub

trac

t five

from

25

usin

g th

e nu

mbe

r lin

e”).

Ex. G

iven

pic

ture

s of

pai

rs o

f sho

es, s

ubtr

act p

airs

to d

eter

min

e ho

w m

any

peop

le (e

.g.,

“If t

here

are

10

shoe

s in

th

e ro

om, h

ow m

any

peop

le a

re th

ere?

”).

Leve

l I A

A S

tude

nts

will

:

EE7.

NS.

2.b.

Ass

ocia

te v

alue

with

the

num

ber

one

by re

cogn

izin

g th

e gr

oup/

set t

hat h

as m

ore

than

one

.

Ex. G

iven

a s

tack

of l

ibra

ry b

ooks

and

a s

ingl

e bo

ok, i

denti

fy w

hich

set

has

mor

e th

an o

ne.

Ex. C

ompo

se a

set

with

mor

e th

an o

ne m

anip

ulati

ve.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

A

pply

pro

perti

es o

f op

erati

ons

as s

trat

egie

s to

mul

tiply

and

di

vide

ratio

nal n

umbe

rs.

Co

nver

t a ra

tiona

l num

ber

to

a de

cim

al u

sing

long

div

isio

n; k

now

that

th

e de

cim

al fo

rm o

f a ra

tiona

l num

ber

term

inat

es in

0s

or e

vent

ually

repe

ats.

EE7.

NS.

2.c-

d. C

ompa

re fr

actio

ns to

fr

actio

ns a

nd d

ecim

als

to d

ecim

als

usin

g ra

tiona

l num

bers

less

than

one

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

NS.

2.c-

d. C

ompa

re a

nd o

rder

frac

tions

and

dec

imal

s w

hen

all n

umbe

rs a

re fr

actio

ns o

r w

hen

all n

umbe

rs

are

deci

mal

s or

whe

n fr

actio

ns a

nd d

ecim

als

are

mix

ed.

Ex. D

ivid

e a

who

le p

izza

into

diff

eren

t fra

ction

s (1

/4 a

nd 1

/2).

Ex. O

rder

frac

tions

or

deci

mal

s fr

om le

ast t

o gr

eate

st (1

/4, 1

/2, a

nd 3

/4) o

n a

num

ber

line.

Ex. S

ort f

racti

ons

and

deci

mal

s an

d m

atch

mon

etar

y am

ount

s (1

/4 o

f a d

olla

r =

25¢,

1/2

of a

dol

lar

= $0

.50)

.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

NS.

2.c-

d. C

ompa

re fr

actio

ns to

frac

tions

and

dec

imal

s to

dec

imal

s us

ing

ratio

nale

num

bers

less

than

one

.

Ex. C

ompa

re tw

o fr

actio

ns a

nd lo

cate

them

on

a nu

mbe

r lin

e.

Ex. U

se p

icto

rial

repr

esen

tatio

ns to

com

pare

frac

tions

to fr

actio

ns a

nd d

ecim

als

to d

ecim

als.

Ex. P

oint

to th

e m

easu

ring

cup

that

sho

ws

1/2.

Ex. G

iven

a q

uart

er a

nd a

dim

e, s

how

whi

ch h

as a

sm

alle

r va

lue.

Ex. G

iven

two

cloc

ks, o

ne o

n th

e ho

ur a

nd o

ne o

n th

e ha

lf ho

ur, c

hoos

e w

hich

sho

ws

a ha

lf ho

ur.

Leve

l II A

A S

tude

nts

will

:

EE8.

NS.

2.c-

d. Id

entif

y th

e lo

catio

n of

a fr

actio

n or

dec

imal

use

d in

the

real

wor

ld a

nd/o

r on

a n

umbe

r lin

e.

Ex. L

abel

the

loca

tion

of a

frac

tion

or d

ecim

al o

n a

num

ber

line.

Ex. G

iven

a n

umbe

r 2

1/2,

poi

nt to

the

num

ber

on a

num

ber

line.

Ex. L

ocat

e a

deci

mal

use

d in

the

real

wor

ld o

n a

num

ber

line

to te

ll w

hich

is m

ore

(e.g

., “I

f an

item

cos

t $0.

58 a

nd

anot

her

item

cos

t $0.

59 c

ents

, find

bot

h am

ount

s on

the

num

ber

line

and

tell

whi

ch c

osts

mor

e.”)

.

Ex. L

ocat

e a

frac

tion

used

in th

e re

al w

orld

on

a nu

mbe

r lin

e to

tell

whi

ch is

mor

e (e

.g.,

If I h

ave

3/4

of a

pie

and

yo

u ha

ve 1

/2 o

f a p

ie u

sing

the

num

ber

line,

sho

w w

ho h

as m

ore

pie.

Fin

d th

e lo

catio

n of

the

num

ber

0.5

on a

nu

mbe

r lin

e.).

Leve

l I A

A S

tude

nts

will

:

EE8.

NS.

2.c-

d. Id

entif

y de

cim

als

or fr

actio

ns.

Ex. G

iven

a w

hole

num

ber

and

a de

cim

al, c

hoos

e th

e de

cim

al.

Ex. G

iven

a b

all,

a bl

ock,

and

a d

ecim

al, p

oint

to th

e de

cim

al.

Ex. S

elec

t 1/2

of a

n ob

ject

whe

n as

ked

to s

how

1/2

(i.e

., 1/

2 of

an

appl

e).

Math | 6-8 Grade 29

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

7.N

S.3.

Sol

ve re

al-w

orld

and

m

athe

mati

cal p

robl

ems

invo

lvin

g th

e fo

ur o

pera

tions

with

ratio

nal

num

bers

.2

EE7.

NS.

3. D

emon

stra

te th

e va

lue

of v

ario

us m

oney

am

ount

s us

ing

deci

mal

s.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

NS.

3. D

eter

min

e th

e to

tal v

alue

of m

oney

wri

tten

as

a de

cim

al g

iven

real

-wor

ld

situ

ation

s.

Ex. U

se a

cal

cula

tor

to d

eter

min

e ho

w m

uch

mon

ey th

ey h

ave

tota

l in

deci

mal

form

.

Ex. C

ount

mon

ey u

sing

dec

imal

s/ca

lcul

ator

to “

shop

” fo

r ite

ms

and

dete

rmin

e ho

w m

uch

mon

ey to

pay

the

cash

ier

whe

n gi

ven

the

tota

l of t

he p

urch

ase.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

NS.

3. D

emon

stra

te th

e va

lue

of v

ario

us m

oney

am

ount

s us

ing

deci

mal

s.

Ex. G

iven

a v

arie

ty o

f coi

ns a

nd b

ills,

wri

te th

e va

lue

of th

e gi

ven

mon

ey u

sing

a d

ecim

al.

Ex. G

iven

a v

arie

ty o

f coi

ns, b

ills,

and

car

ds w

ith a

mou

nts

wri

tten

with

dec

imal

s, m

atch

th

e ca

rds

to th

e va

lue

of th

e co

ins.

Ex. U

se a

cal

cula

tor

to s

how

the

valu

e of

coi

ns in

dec

imal

s (e

.g.,

quar

ters

($0.

25),

dim

es

($0.

10) n

icke

ls ($

0.05

), an

d pe

nnie

s ($

0.01

).

Leve

l II A

A S

tude

nts

will

:

EE7.

NS.

3. Id

entif

y th

e de

cim

al v

alue

of v

ario

us c

oins

.

Ex. G

iven

pic

ture

s of

coi

ns, i

denti

fy th

e va

lue

of e

ach

coin

in c

ents

.

Ex. G

iven

car

ds w

ith d

iffer

ent c

oin

amou

nts

wri

tten

in d

ecim

als

($0.

05, $

0.10

, $0.

20, e

tc.),

m

atch

the

amou

nt w

ith th

e co

rrec

t coi

n.

Ex. G

iven

mor

e th

an o

ne o

f the

sam

e co

in, i

denti

fy th

e to

tal v

alue

of t

he g

iven

coi

ns.

Leve

l I A

A S

tude

nts

will

:

EE7.

NS.

3. Id

entif

y m

oney

.

Ex. G

iven

a g

roup

of c

oins

repr

esen

ting

diffe

rent

val

ues,

sor

t coi

ns b

y lik

e am

ount

s.

Ex. G

iven

a p

ictu

re o

f a c

oin,

mat

ch re

al c

oins

to th

e pi

ctur

e.

Ex. D

iffer

entia

te b

etw

een

dolla

r m

oney

and

cha

nge

(coi

ns).

Ex. C

hoos

e m

oney

ver

sus

non-

mon

ey (e

.g.,

colo

red

chip

s, e

tc.)

to p

ay fo

r pu

rcha

ses.


Seve

nth

Gra

de

Mat

hem

atic

s St

and

ard

s: E

xpre

ssio

ns

and

Eq

uat

ion

s

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Use

pro

perti

es o

f ope

rati

ons

to

gene

rate

equ

ival

ent

expr

essi

ons.

7.EE

.1. A

pply

pro

perti

es o

f op

erati

ons

as s

trat

egie

s to

add

, su

btra

ct, f

acto

r, an

d ex

pand

lin

ear

expr

essi

ons

with

ratio

nal

coeffi

cien

ts.

7.EE

.2. U

nder

stan

d th

at re

wri

ting

an e

xpre

ssio

n in

diff

eren

t for

ms

in

a pr

oble

m c

onte

xt c

an s

hed

light

on

the

prob

lem

and

how

the

quan

tities

in

it a

re re

late

d. F

or e

xam

ple,

a +

0.

05a

= 1.

05a

mea

ns th

at “

incr

ease

by

5%

” is

the

sam

e as

“m

ultip

ly b

y 1.

05.”

EE7.

EE.1

-2. U

se th

e re

latio

nshi

p w

ithin

add

ition

and

/or m

ultip

licati

on

to il

lust

rate

that

two

expr

essi

ons

are

equi

vale

nt.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

EE.1

-2. A

pply

the

com

mut

ative

pro

pert

y to

com

plet

e an

equ

ation

.

Ex. G

iven

12

obje

cts

and

an e

quati

on w

ith th

ree

grou

ps o

n on

e si

de o

f the

equ

als

sign

and

two

grou

ps o

n ot

her

side

, cre

ate

a ba

lanc

ed e

quati

on b

y re

cogn

izin

g th

at th

e si

de w

ith th

ree

grou

ps w

ill

have

two

obje

cts

in e

ach

grou

p, a

nd th

e si

de w

ith tw

o gr

oups

will

hav

e th

ree

obje

cts

in e

ach

grou

p.

Ex. 5

x 7

= _

___

x __

___

(7 x

5)

Ex. _

___

+ __

__ =

4 +

8

(8 +

4)

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

EE.1

-2. U

se th

e re

latio

nshi

p w

ithin

add

ition

and

/or

mul

tiplic

ation

to il

lust

rate

that

two

expr

essi

ons

are

equi

vale

nt.

Ex. 4

+ 7

= 7

+ _

___

Ex. 2

x 4

= _

___

x 2

Ex. 3

+ _

___

= 5

+ 3

Leve

l II A

A S

tude

nts

will

:

EE7.

EE.1

-2. U

se th

e re

latio

nshi

p w

ithin

add

ition

to il

lust

rate

that

two

expr

essi

ons

are

equi

vale

nt.

Ex. G

iven

a m

odel

sho

win

g fiv

e ob

ject

s pl

us tw

o ob

ject

s on

one

sid

e of

an

equa

ls s

ign

and

two

obje

cts

on th

e ot

her

side

, rec

ogni

ze th

at fi

ve o

bjec

ts a

re n

eede

d to

get

the

sam

e am

ount

.

Ex. I

s 2

+ 3

= to

3 +

2?

Ans

wer

yes

/no.

Ex. I

s 2

+ 3

= to

4 +

2?

Ans

wer

yes

/no.

Leve

l I A

A S

tude

nts

will

:

EE7.

EE.1

-2. U

nder

stan

d th

at d

iffer

ent d

ispl

ays

of th

e sa

me

quan

tity

are

equa

l.

Ex. R

ecog

nize

that

thre

e di

scs

and

thre

e sq

uare

s ar

e th

e sa

me

quan

tity.

Ex. R

ecog

nize

that

diff

eren

t arr

ange

men

ts o

f the

sam

e am

ount

are

equ

al (e

.g.,

diffe

rent

ar

rang

emen

ts o

f 4 d

ots

– co

nnec

tion

to s

ubiti

zing

).

Math | 6-8 Grade 31

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Solv

e re

al-li

fe a

nd m

athe

mati

cal

prob

lem

s us

ing

num

eric

al a

nd a

lgeb

raic

ex

pres

sion

s an

d eq

uati

ons.

7.EE

.3. S

olve

mul

ti-st

ep re

al-li

fe a

nd

mat

hem

atica

l pro

blem

s po

sed

with

po

sitiv

e an

d ne

gativ

e ra

tiona

l num

bers

in

any

form

(who

le n

umbe

rs, f

racti

ons,

and

de

cim

als)

, usi

ng to

ols

stra

tegi

cally

. A

pply

pr

oper

ties

of o

pera

tions

to c

alcu

late

with

nu

mbe

rs in

any

form

; con

vert

bet

wee

n fo

rms

as a

ppro

pria

te; a

nd a

sses

s th

e re

ason

able

ness

of a

nsw

ers

usin

g m

enta

l co

mpu

tatio

n an

d es

timati

on s

trat

egie

s.

For e

xam

ple:

If a

wom

an m

akin

g $2

5 an

ho

ur g

ets

a 10

% ra

ise,

she

will

mak

e an

ad

ditio

nal 1

/10

of h

er s

alar

y an

hou

r, or

$2

.50,

for a

new

sal

ary

of $

27.5

0. I

f you

w

ant t

o pl

ace

a to

wel

bar

9 3

/4 in

ches

lo

ng in

the

cent

er o

f a d

oor t

hat i

s 27

1/2

in

ches

wid

e, y

ou w

ill n

eed

to p

lace

the

bar a

bout

9 in

ches

from

eac

h ed

ge; t

his

estim

ate

can

be u

sed

as a

che

ck o

n th

e ex

act c

ompu

tatio

n.

7.EE

.4. U

se v

aria

bles

to re

pres

ent

quan

tities

in a

real

-wor

ld o

r m

athe

mati

cal p

robl

em, a

nd c

onst

ruct

si

mpl

e eq

uatio

ns a

nd in

equa

lities

to

solv

e pr

oble

ms

by re

ason

ing

abou

t the

qu

antiti

es.

•So

lve

wor

d pr

oble

ms

lead

ing

to e

quati

ons

of th

e fo

rm p

x +

q =

r and

p(

x +

q) =

r, w

here

p, q

, and

r ar

e sp

ecifi

c ra

tiona

l num

bers

. So

lve

equa

tions

of

thes

e fo

rms

fluen

tly.

Com

pare

an

alge

brai

c so

lutio

n to

an

arith

meti

c so

lutio

n, id

entif

ying

the

sequ

ence

of t

he

oper

ation

s us

ed in

eac

h ap

proa

ch.

For

exam

ple,

the

perim

eter

of a

rect

angl

e is

54

cm.

Its

leng

th is

6 c

m.

Wha

t is

its

wid

th?

•So

lve

wor

d pr

oble

ms

lead

ing

to in

equa

lities

of t

he fo

rm p

x +

q >

r or

px +

q <

r, w

here

p, q

, and

r ar

e sp

ecifi

c ra

tiona

l num

bers

. G

raph

the

solu

tion

set

of th

e in

equa

lity

and

inte

rpre

t it i

n th

e co

ntex

t of t

he p

robl

em.

For e

xam

ple:

A

s a

sale

sper

son,

you

are

pai

d $5

0 pe

r w

eek

plus

$3

per s

ale.

Thi

s w

eek

you

wan

t you

r pay

to b

e at

leas

t $10

0. W

rite

an in

equa

lity

for t

he n

umbe

r of s

ales

you

ne

ed to

mak

e, a

nd d

escr

ibe

the

solu

tions

.

EE7.

EE.3

-4. U

se th

e co

ncep

t of e

qual

ity

with

mod

els

to s

olve

one

-ste

p ad

ditio

n an

d su

btra

ction

equ

ation

s.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

EE.3

-4. S

olve

two-

step

add

ition

and

sub

trac

tion

equa

tions

.

Ex. A

fter

det

erm

inin

g th

at 5

+ 5

= 1

0, d

ecom

pose

10

into

thre

e an

d se

ven.

Ex. A

fter

det

erm

inin

g th

at 9

- 6

= 3,

det

erm

ine

that

thre

e is

com

pose

d of

3 +

1).

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

EE.3

-4. U

se th

e co

ncep

t of e

qual

ity w

ith m

odel

s to

sol

ve o

ne-s

tep

addi

tion

and

subt

racti

on e

quati

ons.

Ex. I

f the

re is

a q

uanti

ty o

f five

on

one

side

of t

he e

quati

on a

nd a

qua

ntity

of t

wo

on th

e ot

her

side

, wha

t qu

antit

y is

add

ed to

mak

e it

equa

l?

Ex. I

f I h

ave

thre

e ba

lls a

nd I

get s

ome

mor

e ba

lls –

how

man

y di

d I g

et if

I no

w h

ave

seve

n?

Ex. G

iven

4 +

___

= 1

2, id

entif

y th

e m

issi

ng a

mou

nt u

sing

mod

els.

Ex. G

iven

12

- ___

= 5

, ide

ntify

the

mis

sing

am

ount

usi

ng m

odel

s.

Ex. G

iven

10

= 2

+ __

__, i

denti

fy th

e m

issi

ng a

mou

nt u

sing

mod

els.

Leve

l II A

A S

tude

nts

will

:

EE7.

EE.3

-4. I

denti

fy th

e am

ount

nee

ded

to e

qual

the

valu

e on

the

give

n si

de o

f an

equa

tion.

Ex. T

hree

obj

ects

+ tw

o ob

ject

s w

ill e

qual

five

obj

ects

.

Ex. G

iven

a n

umbe

r fr

om 2

to 1

0, d

ecom

pose

the

num

ber

to c

reat

e a

bala

nced

equ

ation

(con

necti

on to

de

com

posi

tion

of n

umbe

rs).

Leve

l I A

A S

tude

nts

will

:

EE7.

EE.3

-4. R

ecog

nize

equ

al q

uanti

ties

on b

oth

side

s of

an

equa

tion.

Ex. M

atch

equ

al q

uanti

ties:

thre

e tr

iang

les

is th

e sa

me

quan

tity

as th

ree

circ

les.

Ex. G

ive

the

digi

t 5, c

ount

out

five

obj

ects

as

an e

qual

qua

ntity

.


Seve

nth

Gra

de

Mat

hem

atic

s St

and

ard

s: G

eom

etry

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Dra

w c

onst

ruct

, and

des

crib

e ge

omet

rica

l figu

res

and

desc

ribe

th

e re

lati

onsh

ips

betw

een

them

.

7.G

.1. S

olve

pro

blem

s in

volv

ing

scal

e dr

awin

gs o

f geo

met

ric

figur

es, i

nclu

ding

com

putin

g ac

tual

leng

ths

and

area

s fr

om a

sc

ale

draw

ing

and

repr

oduc

ing

a sc

ale

draw

ing

at a

diff

eren

t sca

le.

7.G

.2. D

raw

(fre

ehan

d, w

ith

rule

r an

d pr

otra

ctor

, and

with

te

chno

logy

) geo

met

ric

shap

es

with

giv

en c

ondi

tions

. Fo

cus

on

cons

truc

ting

tria

ngle

s fr

om th

ree

mea

sure

s of

ang

les

or s

ides

, no

ticin

g w

hen

the

cond

ition

s de

term

ine

a un

ique

tria

ngle

, m

ore

than

one

tria

ngle

, or

no

tria

ngle

.

EE7.

G.1

-2. D

raw

or

clas

sify

and

re

cogn

ize

basi

c tw

o-di

men

sion

al

geom

etri

c sh

apes

with

out a

m

odel

(circ

le, t

rian

gle,

rect

angl

e/sq

uare

).

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

G.1

-2. D

raw

or

mod

el tw

o-di

men

sion

al s

hape

s in

clud

ing

a tr

apez

oid

and

rhom

bus

with

out a

mod

el.

Ex. D

raw

/cre

ate

a tr

apez

oid.

Ex. D

raw

/cre

ate

a rh

ombu

s.Ex

. Rep

licat

e a

geom

etri

c sh

ape

with

giv

en d

imen

sion

s.Ex

. Dra

w a

sha

pe th

at is

twic

e as

big

in o

ne d

imen

sion

(len

gth

or w

idth

) as

a gi

ven

shap

e (e

.g.,

give

n a

coor

dina

te g

rid,

hav

e th

e st

uden

t dra

w a

rect

angl

e th

at is

twic

e as

long

and

tw

ice

as h

igh

as th

e on

e he

/she

is g

iven

).

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

G.1

-3. D

raw

or

clas

sify

and

reco

gniz

e ba

sic

two-

dim

ensi

onal

geo

met

ric

shap

es

with

out a

mod

el (c

ircle

, tri

angl

e, re

ctan

gle/

squa

re).

Ex. R

ecog

nize

and

gro

up to

geth

er d

iffer

ent t

ypes

of r

ecta

ngle

s an

d ci

rcle

sEx

. Sta

te th

e na

me

of c

ircle

, tri

angl

e, re

ctan

gle,

and

squ

are.

Ex. D

raw

a re

ctan

gle

and

circ

le.

Leve

l II A

A S

tude

nts

will

:EE

7.G

.1-2

. Dem

onst

rate

the

abili

ty to

com

plet

e a

two-

dim

ensi

onal

sha

pe (c

ircle

, tri

angl

e,

rect

angl

e, s

quar

e).

Ex. C

ompa

re s

hape

s w

hen

give

n m

anip

ulati

ves/

pict

ures

and

ask

ed to

tell

wha

t sha

pes

are

the

sam

e an

d w

hat s

hape

s ar

e is

diff

eren

t.Ex

. Giv

en a

n ar

c, c

ompl

ete

the

draw

ing

of a

circ

le.

Ex. G

iven

con

cret

e pi

eces

, com

plet

e a

spec

ified

sha

pe (i

.e.,

four

equ

al le

ngth

pop

sicl

e sti

cks

to c

reat

e a

squa

re).

Leve

l I A

A S

tude

nts

will

:EE

7.G

.1-2

. Dem

onst

rate

the

abili

ty to

reco

gniz

e a

two-

dim

ensi

onal

sha

pe (c

ircle

, tri

angl

e,

rect

angl

e, s

quar

e) w

hen

give

n a

com

plet

e sh

ape.

Ex. R

ecog

nize

a s

hape

.Ex

. Whe

n gi

ven

a sh

ape,

find

ano

ther

sha

pe li

ke th

e on

e ju

st g

iven

.Ex

. Com

pare

sha

pes

whe

n gi

ven

man

ipul

ative

s –

to s

ay tw

o sh

apes

are

the

sam

e (c

ongr

uent

) aft

er m

atch

ing

the

side

s on

eac

h.Ex

. Use

var

ious

med

ia fo

r st

uden

ts to

form

a s

impl

e ge

omet

ric

shap

e (i.

e. s

and,

sha

ving

cr

eam

)Ex

. Giv

en a

sam

ple

shap

e, tr

ace

the

shap

e (t

ouch

boa

rd, r

aise

d pa

per,

wik

i stic

ks, e

tc.)

Math | 6-8 Grade 33

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

7.G

.3. D

escr

ibe

the

two-

dim

ensi

onal

figu

res

that

resu

lt fr

om s

licin

g th

ree-

dim

ensi

onal

fig

ures

, as

in p

lane

sec

tions

of

righ

t rec

tang

ular

pri

sms

and

righ

t rec

tang

ular

pyr

amid

s.

EE7.

G.3

. Mat

ch a

two-

dim

ensi

onal

sha

pe w

ith a

thre

e-di

men

sion

al s

hape

that

sha

res

an a

ttri

bute

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

G.3

. Pai

r tw

o- a

nd th

ree-

dim

ensi

onal

sha

pes

to c

ompl

ete

a re

al-w

orld

task

.Ex

. Giv

en a

thre

e-di

men

sion

al s

hape

and

sev

eral

diff

eren

t tw

o-di

men

sion

al s

hape

s (e

.g.,

cube

, cyl

inde

rs),

sele

ct th

e tw

o-di

men

sion

al s

hape

that

repr

esen

ts o

ne fa

ce o

f the

thre

e-di

men

sion

al s

hape

(e.g

., sq

uare

, circ

le).

Ex. G

iven

a d

iagr

am to

sho

w th

e pl

acem

ent o

f diff

eren

t sha

ped

obje

cts

in a

sto

rero

om,

use

the

two-

dim

ensi

onal

sha

pe in

the

diag

ram

to p

lace

thre

e-di

men

sion

al o

bjec

ts

appr

opri

atel

y on

the

shel

f (e.

g., s

quar

e bo

xes

on s

quar

es, r

ecta

ngul

ar b

oxes

on

rect

angl

es,

and

bott

les

on c

ircle

s).

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

G.3

. Mat

ch a

two-

dim

ensi

onal

sha

pe w

ith a

thre

e- d

imen

sion

al s

hape

that

sha

res

an

attri

bute

.Ex

. Giv

en a

circ

le, fi

nd o

bjec

ts th

at a

re th

ree-

dim

ensi

onal

cou

nter

part

s (e

.g.,

ball,

glo

be,

sphe

re).

Ex. G

iven

a s

quar

e, fi

nd o

bjec

ts th

at a

re th

ree-

dim

ensi

onal

cou

nter

part

s (e

.g.,

box,

lo

cker

).Ex

. Giv

en a

squ

are,

find

thre

e-di

men

sion

al o

bjec

ts th

at s

hare

one

att

ribu

te (e

.g.,

squa

re

with

cub

e, c

ircle

with

cyl

inde

r).

Leve

l II A

A S

tude

nts

will

:EE

7.G

.3. I

denti

fy th

e att

ribu

tes

of a

thre

e-di

men

sion

al s

hape

(col

or, n

umbe

r of

sid

es,

face

s, s

ize,

text

ures

, sha

pe, e

tc.).

Ex. G

iven

a re

d ba

ll an

d co

mm

unic

ation

dev

ice,

iden

tify

wor

ds th

at d

escr

ibe

the

attri

bute

s of

the

ball.

Ex. G

iven

a g

roup

of s

hape

s, d

escr

ibe

com

mon

att

ribu

tes.

Ex. G

iven

a c

lass

of o

bjec

ts, i

denti

fy c

omm

on a

ttri

bute

s an

d ch

oose

one

to s

ort b

y.

Leve

l I A

A S

tude

nts

will

:EE

7.G

.3. R

eplic

ate

the

two-

dim

ensi

onal

cro

ss-s

ectio

n of

a th

ree-

dim

ensi

onal

sha

pe (c

ube,

sp

here

, cyl

inde

r) w

hen

give

n a

com

plet

e sh

ape.

Ex. G

iven

a c

ube,

out

line

the

base

to fo

rm a

squ

are.

Ex. G

iven

a s

oda

can,

out

line

the

base

to fo

rm a

circ

le.

Solv

e re

al-li

fe a

nd m

athe

mati

cal

prob

lem

s in

volv

ing

angl

e m

easu

re, a

rea,

sur

face

are

a,

and

volu

me.

7.G

.4. K

now

the

form

ulas

for

the

area

and

circ

umfe

renc

e of

a

circ

le a

nd u

se th

em to

sol

ve

prob

lem

s; g

ive

an in

form

al

deri

vatio

n of

the

rela

tions

hip

betw

een

the

circ

umfe

renc

e an

d ar

ea o

f a c

ircle

.

EE7.

G.4

. N/A


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

7.G

.5. U

se fa

cts

abou

t su

pple

men

tary

, com

plem

enta

ry,

verti

cal,

and

adja

cent

ang

les

in a

m

ulti-

step

pro

blem

to w

rite

and

so

lve

sim

ple

equa

tions

for

an

unkn

own

angl

e in

a fi

gure

.

EE7.

G.5

. Fin

d th

e pe

rim

eter

of

a re

ctan

gle

give

n th

e le

ngth

and

w

idth

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

G.5

. Sol

ve s

impl

e pe

rim

eter

pro

blem

s w

ith re

ctan

gles

.Ex

. Giv

en a

rect

angl

e w

ith id

entifi

ed d

imen

sion

s, d

eter

min

e th

e pe

rim

eter

.Ex

. A b

ulle

tin b

oard

is 5

’ by

5’.

How

muc

h bo

rder

pap

er is

nee

ded

for

the

peri

met

er?

Ex. W

hen

give

n a

pict

ure

of a

gar

den

with

onl

y th

e le

ngth

and

wid

th id

entifi

ed, s

olve

for

peri

met

er.

3 +

8 +

3 +

8 =

22 y

ards

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

G.5

. Fin

d th

e pe

rim

eter

of a

rect

angl

e gi

ven

the

leng

th a

nd w

idth

.Ex

. Det

erm

ine

the

peri

met

er o

f a re

ctan

gle

give

n a

visu

al m

odel

and

a c

alcu

lato

r.Ex

. Giv

en a

rect

angl

e w

ith ti

c m

arks

indi

catin

g a

leng

th o

f six

and

a w

idth

of f

our,

dete

rmin

e th

e pe

rim

eter

by

coun

ting

(6 +

4 +

6 +

4).

Ex. S

how

n a

tape

d re

ctan

gle

on th

e flo

or w

ith ti

c m

arks

or

floor

tile

s de

notin

g sq

uare

s w

ithin

the

rect

angl

e, w

alk

arou

nd th

e re

ctan

gle,

cou

nting

ste

ps/ti

les/

tic m

arks

, to

dete

rmin

e th

e pe

rim

eter

.Ex

. Mea

sure

the

leng

th a

nd w

idth

of a

des

k an

d ot

her

rect

angu

lar

obje

cts

in th

e cl

assr

oom

(i.e

., bo

oks,

pic

ture

fram

es).

Leve

l II A

A S

tude

nts

will

:EE

7.G

.5. I

denti

fy th

e le

ngth

and

wid

th o

f a re

ctan

gle.

Ex. C

over

a re

ctan

gle

with

squ

ares

(i.e

., co

lor

tiles

) and

iden

tify

the

sum

of n

umbe

rs o

f til

es o

f the

top/

bott

om a

nd th

e si

des.

Ex. G

iven

a c

ircle

, mea

sure

the

dist

ance

aro

und

the

circ

le (c

ircum

fere

nce

– pe

rim

eter

of a

ci

rcle

).Ex

. Pla

ce a

str

ing

arou

nd th

e pe

rim

eter

of a

n ob

ject

and

then

mea

sure

the

leng

th o

f the

st

ring

to te

ll th

e di

stan

ce a

roun

d th

e ob

ject

.Ex

. Giv

en a

gri

dded

rect

angl

e, id

entif

y th

e le

ngth

of t

he to

p/bo

ttom

and

the

side

s.

Leve

l I A

A S

tude

nts

will

:EE

7.G

.5. O

utlin

e th

e pe

rim

eter

of a

n ob

ject

.Ex

. Use

wik

i stic

ks to

out

line

the

bord

er o

f a s

quar

e/re

ctan

gle.

Ex. O

utlin

e th

e pe

rim

eter

of a

rect

angu

lar

pan

by tr

acin

g th

e ed

ge w

ith a

fing

er.

Ex. O

utlin

e th

e pe

rim

eter

of a

tabl

et b

y la

ying

str

ing

arou

nd th

e ed

ge.

Ex. C

ount

the

num

ber

of s

quar

es a

roun

d th

e ou

tsid

e of

a g

ridd

ed re

ctan

gle.

12

34

5

126

1110

98

7

Math | 6-8 Grade 35

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

7.G

.6. S

olve

real

-wor

ld a

nd

mat

hem

atica

l pro

blem

s in

volv

ing

area

, vol

ume

and

surf

ace

area

of

two-

and

thre

e-di

men

sion

al

obje

cts

com

pose

d of

tria

ngle

s,

quad

rila

tera

ls, p

olyg

ons,

cub

es,

and

righ

t pri

sms.

EE7.

G.6

. Fin

d th

e ar

ea o

f a

rect

angl

e gi

ven

the

leng

th a

nd

wid

th u

sing

a m

odel

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

G.6

. Sol

ve s

impl

e ar

ea p

robl

ems

with

rect

angl

es.

Ex. A

rect

angu

lar

rug

is 4

’ by

5’.

Wha

t is

the

area

of t

he r

ug?

Use

a c

alcu

lato

r to

app

ly to

th

e gi

ven

mod

el p

robl

em a

nd fi

nd th

e an

swer

.Ex

. Giv

en a

rect

angl

e w

ith id

entifi

ed le

ngth

and

wid

th d

imen

sion

s, d

eter

min

e th

e ar

ea.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

G.6

. Fin

d th

e ar

ea o

f a re

ctan

gle

give

n th

e le

ngth

and

wid

th u

sing

a m

odel

.Ex

. Giv

en re

ctan

gles

(inc

ludi

ng s

quar

es) w

ith g

rids

, cou

nt s

quar

es to

cal

cula

te th

e ar

ea.

Ex

. Par

tition

rect

angu

lar

figur

es in

to ro

ws

and

colu

mns

of t

he s

ame-

size

squ

ares

with

out

gaps

and

ove

rlap

s an

d co

unt t

hem

to fi

nd th

e ar

ea.

Ex. G

iven

a p

ictu

re o

f a re

ctan

gle,

hav

e st

uden

ts d

ivid

e th

e in

teri

or o

f the

figu

re in

to

equa

lly s

quar

ed u

nits

and

det

erm

ine

the

num

ber

of s

quar

ed u

nits

with

in th

e re

ctan

gle.

Leve

l II A

A S

tude

nts

will

:EE

7.G

.6. I

denti

fy th

e le

ngth

and

wid

th (d

imen

sion

s) o

f a re

ctan

gle.

Ex. C

over

a g

iven

rect

angl

e w

ith s

quar

es (i

.e.,

colo

r til

es) a

nd id

entif

y th

e nu

mer

ical

val

ue

of th

e to

tal n

umbe

r of

squ

are

units

.Ex

. Giv

en a

gri

dded

rect

angu

lar

box

plac

e sm

alle

r bo

xes

side

-by-

side

(in

one

laye

r) to

co

unt h

ow m

any

smal

l box

es th

e la

rge

box

hold

s an

d id

entif

y th

e nu

mer

ical

val

ue (s

um) o

f th

e gr

ids

insi

de th

e re

ctan

gle.

Leve

l I A

A S

tude

nts

will

:EE

7.G

.6. D

uplic

ate

the

area

of a

rect

angl

e (s

quar

e).

Ex. C

over

a s

quar

e pa

n w

ith p

iece

s of

toas

t, s

quar

e cr

acke

rs, e

tc. i

n a

sing

le la

yer.

Ex. U

se s

quar

es o

f col

ored

pap

er to

cov

er th

eir

desk

or

tray

on

a w

heel

chai

r.


Seve

nth

Gra

de

Mat

hem

atic

s St

and

ard

s: S

tati

stic

s an

d P

rob

abil

ity

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Use

rand

om s

ampl

ing

to d

raw

in

fere

nces

abo

ut a

pop

ulati

on.

7.SP

.1. U

nder

stan

d th

at s

tatis

tics

can

be u

sed

to g

ain

info

rmati

on

abou

t a p

opul

ation

by

exam

in-

ing

a sa

mpl

e of

the

popu

latio

n;

gene

raliz

ation

s ab

out a

pop

ula-

tion

from

a s

ampl

e ar

e va

lid o

nly

if th

e sa

mpl

e is

repr

esen

tativ

e of

that

pop

ulati

on.

Und

erst

and

that

rand

om s

ampl

ing

tend

s to

pr

oduc

e re

pres

enta

tive

sam

ples

an

d su

ppor

t val

id in

fere

nces

.

7.SP

.2. U

se d

ata

from

a ra

ndom

sa

mpl

e to

dra

w in

fere

nces

abo

ut

a po

pula

tion

with

an

unkn

own

char

acte

risti

c of

inte

rest

. G

ener

-at

e m

ultip

le s

ampl

es (o

r si

mul

at-

ed s

ampl

es) o

f the

sam

e si

ze to

ga

uge

the

vari

ation

in e

stim

ates

or

pre

dicti

ons.

For

exa

mpl

e,

estim

ate

the

mea

n w

ord

leng

th

in a

boo

k by

rand

omly

sam

plin

g w

ords

from

the

book

; pre

dict

the

win

ner o

f a s

choo

l ele

ction

bas

ed

on ra

ndom

ly s

ampl

ed s

urve

y da

ta.

Gau

ge h

ow fa

r off

the

esti-

mat

e or

pre

dicti

on m

ight

be.

EE7.

SP.1

-2. A

nsw

er a

que

stion

re

late

d to

the

colle

cted

dat

a fr

om a

n ex

peri

men

t, g

iven

a

mod

el o

f dat

a, o

r fr

om d

ata

col-

lect

ed b

y th

e st

uden

t.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

SP.1

-2. A

nsw

er a

que

stion

abo

ut d

ata

colle

cted

from

an

expe

rim

ent a

nd e

xpla

in o

r de

mon

stra

te th

e re

sults

.Ex

. Pol

l cla

ssm

ates

to d

eter

min

e w

here

to g

o on

a fi

eld

trip

and

exp

lain

resu

lts.

Ex. G

iven

dat

a on

hei

ght o

f stu

dent

s in

two

clas

ses,

iden

tify

whi

ch c

lass

has

the

talle

st

stud

ents

.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

SP.1

-2. A

nsw

er a

que

stion

rela

ted

to th

e co

llect

ed d

ata

from

an

expe

rim

ent,

giv

en a

m

odel

of d

ata,

or

from

dat

a co

llect

ed b

y th

e st

uden

t.Ex

. Giv

en d

ata

(i.e.

, a fr

eque

ncy

tabl

e) o

f fav

orite

piz

za to

ppin

gs, w

hich

type

of p

izza

w

ould

be

orde

red

mos

t oft

en.

Ex. A

sked

wha

t the

ir fa

vori

te s

easo

n is

, pla

ce th

emse

lves

in o

ne o

f the

four

gro

ups

and

answ

er a

que

stion

abo

ut th

e re

sults

. (W

hat i

s th

e gr

oup’

s fa

vori

te s

easo

n? W

hat i

s th

e gr

oup’

s le

ast f

avor

ite s

easo

n?)

Leve

l II A

A S

tude

nts

will

:EE

7.SP

.1-2

. Col

lect

dat

a to

ans

wer

a g

iven

que

stion

.Ex

. Ask

fello

w c

lass

mat

es w

hat t

heir

favo

rite

acti

vity

sub

ject

is a

nd k

eep

tally

mar

ks o

f the

re

spon

ses.

Ex. U

se a

gri

d to

reco

rd th

e nu

mbe

r of

tenn

is s

hoes

in th

e cl

assr

oom

.

Leve

l I A

A S

tude

nts

will

:EE

7.SP

.1-2

. Ans

wer

a q

uesti

on fo

r da

ta c

olle

ction

.Ex

. Ans

wer

a q

uesti

on a

bout

wha

t the

y at

e fo

r br

eakf

ast.

Ex. A

nsw

er a

que

stion

abo

ut th

eir

favo

rite

can

dy b

ar.

Math | 6-8 Grade 37

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Dra

w in

form

al c

ompa

rati

ve in

-fe

renc

es a

bout

tw

o po

pula

tion

s.

7.SP

.3. I

nfor

mal

ly a

sses

s th

e de

gree

of v

isua

l ove

rlap

of t

wo

num

eric

al d

ata

dist

ribu

tions

with

si

mila

r va

riab

ilitie

s, m

easu

ring

th

e di

ffere

nce

betw

een

the

cen-

ters

by

expr

essi

ng it

as

a m

ultip

le

of a

mea

sure

of v

aria

bilit

y. F

or

exam

ple,

the

mea

n he

ight

of

play

ers

on th

e ba

sket

ball

team

is

10

cm g

reat

er th

an th

e m

ean

heig

ht o

f pla

yers

on

the

socc

er

team

, abo

ut tw

ice

the

varia

bilit

y (m

ean

abso

lute

dev

iatio

n) o

n ei

ther

team

; on

a do

t plo

t, th

e se

para

tion

betw

een

the

two

dist

ributi

ons

of h

eigh

ts is

noti

ce-

able

.

7.SP

.4. U

se m

easu

res

of c

ente

r an

d m

easu

res

of v

aria

bilit

y fo

r nu

mer

ical

dat

a fr

om ra

ndom

sa

mpl

es to

dra

w in

form

al

com

para

tive

infe

renc

es a

bout

tw

o po

pula

tions

. Fo

r exa

mpl

e,

deci

de w

heth

er th

e w

ords

in a

ch

apte

r of a

sev

enth

-gra

de s

ci-

ence

boo

k ar

e ge

nera

lly lo

nger

th

an th

e w

ords

in a

cha

pter

of a

fo

urth

-gra

de s

cien

ce b

ook.

EE7.

SP.3

. Com

pare

two

sets

of

data

with

in a

sin

gle

data

dis

play

su

ch a

s a

pict

ure

grap

h, li

ne p

lot,

or

bar

gra

ph.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

SP.3

. Com

pare

dat

a fr

om tw

o pi

ctur

e gr

aphs

, tw

o lin

e pl

ots,

or

two

bar

grap

hs.

Ex. G

iven

two

bar

grap

hs s

how

ing

the

num

ber

of p

ets

stud

ents

from

two

diffe

rent

cla

ss-

room

s ha

ve, d

eter

min

e w

hich

cla

ssro

om o

f stu

dent

s ha

s th

e m

ost p

ets.

Ex. G

iven

two

bar

grap

hs, s

how

ing

the

num

ber

of b

oys

and

the

num

ber

of g

irls

from

two

diffe

rent

cla

ssro

oms,

det

erm

ine

whi

ch c

lass

room

has

the

leas

t num

ber

of g

irls

(or

the

leas

t num

ber

of b

oys,

or

the

grea

test

num

ber

of b

oys,

or

the

grea

test

num

ber

of g

irls

).

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

SP.3

. Com

pare

two

sets

of d

ata

with

in a

sin

gle

data

dis

play

suc

h as

a p

ictu

re g

raph

, lin

e pl

ot, o

r ba

r gr

aph.

Ex. C

ompa

re th

e ch

ange

in th

e nu

mbe

r of

day

s of

sun

light

in s

umm

er a

nd w

inte

r on

a li

ne

plot

on

a gi

ven

grap

h.Ex

. Giv

en a

bar

gra

ph, c

ompa

re th

e nu

mbe

r of

red

M&

Ms

to b

lue

M&

Ms.

Leve

l II A

A S

tude

nts

will

:EE

7.SP

.3. S

umm

ariz

e da

ta o

n a

grap

h or

tabl

e in

one

way

.Ex

. Whe

n lo

okin

g at

a g

raph

of t

empe

ratu

res

from

the

wee

k, s

umm

ariz

e th

e da

ta in

one

w

ay (i

.e.,

thre

e da

ys w

ere

abov

e 70

deg

rees

).Ex

. Whe

n lo

okin

g at

a ta

ble

that

con

tain

s da

ta a

bout

wha

t stu

dent

s lik

e to

eat

or

wha

t st

uden

ts li

ke to

do,

sum

mar

ize

the

data

in o

ne w

ay (i

.e.,

“wat

ch m

ovie

s” h

as th

e m

ost)

.

Leve

l I A

A S

tude

nts

will

:EE

7.SP

.3. R

ead

data

from

one

giv

en s

ourc

e.Ex

. Usi

ng a

pic

togr

aph,

iden

tify

the

num

ber

of s

tude

nts

who

hav

e a

dog,

are

pre

sent

, eat

br

eakf

ast,

etc

.Ex

. Usi

ng a

bar

gra

ph, i

denti

fy w

hich

is m

ore

or w

hich

is le

ss.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Inve

stiga

te c

hanc

e pr

oces

ses

and

deve

lop,

use

, and

eva

luat

e pr

obab

ility

m

odel

s.

7.SP

.5. U

nder

stan

d th

at th

e pr

obab

ility

of

a c

hanc

e ev

ent i

s a

num

ber

betw

een

0 an

d 1

that

exp

ress

es th

e lik

elih

ood

of th

e ev

ent o

ccur

ring

. La

rger

num

bers

indi

cate

gr

eate

r lik

elih

ood.

A p

roba

bilit

y ne

ar 0

in

dica

tes

an u

nlik

ely

even

t, a

pro

babi

lity

arou

nd 1

/2 in

dica

tes

an e

vent

that

is n

ei-

ther

unl

ikel

y no

r lik

ely,

and

a p

roba

bilit

y ne

ar 1

indi

cate

s a

likel

y ev

ent.

7.SP

.6. A

ppro

xim

ate

the

prob

abili

ty o

f a

chan

ce e

vent

by

colle

cting

dat

a on

th

e ch

ance

pro

cess

that

pro

duce

s it

and

obse

rvin

g its

long

-run

rela

tive

freq

uenc

y,

and

pred

ict t

he a

ppro

xim

ate

rela

tive

freq

uenc

y gi

ven

the

prob

abili

ty.

For

exam

ple,

whe

n ro

lling

a n

umbe

r cub

e 60

0 tim

es, p

redi

ct th

at a

3 o

r 6 w

ould

be

rolle

d ro

ughl

y 20

0 tim

es, b

ut p

roba

bly

not e

xact

ly 2

00 ti

mes

.

7.SP

.7. D

evel

op a

pro

babi

lity

mod

el a

nd

use

it to

find

pro

babi

lities

of e

vent

s.

Com

pare

pro

babi

lities

from

a m

odel

to

obse

rved

freq

uenc

ies;

if th

e ag

reem

ent

is n

ot g

ood,

exp

lain

pos

sibl

e so

urce

s of

th

e di

scre

panc

y.•

Dev

elop

a u

nifo

rm p

roba

bilit

y m

odel

by

assi

gnin

g eq

ual p

roba

bilit

y to

all

outc

omes

, and

use

the

mod

el

to d

eter

min

e pr

obab

ilitie

s of

eve

nts.

Fo

r exa

mpl

e, if

a s

tude

nt is

sel

ecte

d at

ra

ndom

from

a c

lass

, find

the

prob

abili

ty

that

Jane

will

be

sele

cted

and

the

prob

-ab

ility

that

a g

irl w

ill b

e se

lect

ed.

•D

evel

op a

pro

babi

lity

mod

el

(whi

ch m

ay n

ot b

e un

iform

) by

obse

rvin

g fr

eque

ncie

s in

dat

a ge

nera

ted

from

a

chan

ce p

roce

ss.

For e

xam

ple,

find

the

appr

oxim

ate

prob

abili

ty th

at a

spi

nnin

g pe

nny

will

land

hea

ds u

p or

that

a to

ssed

pa

per c

up w

ill la

nd o

pen-

end

dow

n. D

o th

e ou

tcom

es fo

r the

spi

nnin

g pe

nny

appe

ar to

be

equa

lly li

kely

bas

ed o

n th

e ob

serv

ed fr

eque

ncie

s?

EE7.

SP.5

-7. D

escr

ibe

the

prob

abili

ty o

f ev

ents

occ

urri

ng a

s po

ssib

le o

r im

pos-

sibl

e.

Leve

l IV

AA

Stu

dent

s w

ill:

EE7.

SP.5

-7. D

iffer

entia

te a

nd d

escr

ibe

exam

ples

of a

situ

ation

that

is p

ossi

ble,

a s

ituati

on th

at is

like

ly, a

nd a

situ

-ati

on th

at is

impo

ssib

le.

Ex. S

tate

a s

ituati

on th

at is

impo

ssib

le.

Ex. S

tate

a s

ituati

on th

at is

pos

sibl

e.

Leve

l III

AA

Stu

dent

s w

ill:

EE7.

SP.5

-7. D

escr

ibe

the

prob

abili

ty o

f eve

nts

occu

rrin

g as

pos

sibl

e or

impo

ssib

le.

Ex. A

nsw

er, “

Is it

pos

sibl

e th

at a

squ

irre

l att

ends

sch

ool w

ith y

ou?”

Ex. A

nsw

er, “

Is it

pos

sibl

e th

at a

cow

will

eve

r dr

ive

a ca

r?”

Ex. A

nsw

er, “

If yo

u on

ly o

wn

only

thre

e sh

irts

- a

red

one,

a b

lue

one,

and

a b

lack

one

- is

it p

ossi

ble

to p

ull a

w

hite

one

from

you

r dr

awer

?”

Leve

l II A

A S

tude

nts

will

:EE

7.SP

.5-7

. Ide

ntify

pos

sibl

e ev

ents

that

cou

ld o

ccur

in th

e na

tura

l env

ironm

ent.

Ex. G

iven

the

lunc

h m

enu

of p

izza

and

ham

burg

ers,

iden

tify

whe

ther

it is

pos

sibl

e to

get

a h

ambu

rger

for

lunc

h.Ex

. Giv

en a

wee

kly

char

t of c

lass

room

jobs

(diff

eren

t job

s ev

ery

day

of th

e w

eek)

, ans

wer

“W

hat j

ob is

pos

sibl

e fo

r M

onda

y?”

Leve

l I A

A S

tude

nts

will

:EE

7.SP

.5-7

. Ide

ntify

out

com

es b

ased

on

a po

ssib

le e

vent

.Ex

. Giv

en a

pic

ture

of a

per

son

wea

ring

a h

eavy

coa

t, s

carf

, and

hat

, ide

ntify

if th

e cl

othi

ng is

app

ropr

iate

for

a pi

ctur

e of

som

e w

eath

er c

ondi

tion.

Ex. “

We

are

goin

g on

a fi

eld

trip

in to

wn.

In

whi

ch o

f the

follo

win

g w

ould

it b

e po

ssib

le to

tran

spor

t the

enti

re

clas

s (s

how

pic

ture

s of

a ro

cket

, bic

ycle

, and

a b

us)?

”

Math | 6-8 Grade 39

CO

MM

ON

CO

RE

ESS

EN

TIA

L E

LEM

EN

TS

AN

D A

CH

IEV

EM

EN

T

DE

SCR

IPT

OR

S FO

R E

IGH

TH

GR

AD

EE

igh

th G

rad

e M

ath

emat

ics

Stan

dar

ds:

Th

e N

um

ber

Sys

tem

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Know

that

the

re a

re n

umbe

rs

that

are

not

rati

onal

, and

ap

prox

imat

e th

em b

y ra

tion

al

num

bers

.

8.N

S.1.

Kno

w th

at n

umbe

rs

that

are

not

ratio

nal a

re c

alle

d ir

ratio

nal.

Und

erst

and

info

rmal

ly

that

eve

ry n

umbe

r ha

s a

deci

mal

ex

pans

ion;

for

ratio

nal n

umbe

rs

show

that

the

deci

mal

exp

ansi

on

repe

ats

even

tual

ly, a

nd c

onve

rt

a de

cim

al e

xpan

sion

whi

ch

repe

ats

even

tual

ly in

to a

ratio

nal

num

ber.

EE8.

NS.

1. S

ubtr

act f

racti

ons

with

like

den

omin

ator

s (h

alve

s,

third

s, fo

urth

s, a

nd te

nths

) with

m

inue

nds

less

than

or

equa

l to

one.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

NS.

1. S

ubtr

act f

racti

ons

with

like

den

omin

ator

s (h

alve

s, th

irds,

four

ths,

and

tent

hs)

with

min

uend

s th

at m

ay b

e gr

eate

r th

an o

ne.

Ex. S

ubtr

act t

wo

frac

tions

with

like

den

omin

ator

s w

ith m

odel

s or

num

bers

.Ex

. If I

hav

e 1

3/4

and

I tak

e 1/

4 aw

ay, h

ow m

any

who

les

and

four

ths

are

left

?

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

NS.

1. S

ubtr

act f

racti

ons

with

like

den

omin

ator

s (h

alve

s, th

irds,

four

ths,

and

tent

hs)

with

min

uend

s le

ss th

an o

r eq

ual t

o on

e.Ex

. Use

frac

tion

bars

or

frac

tion

circ

les

to a

dd a

nd m

atch

a n

umer

ical

repr

esen

tatio

n to

th

e m

odel

so

the

answ

er is

less

than

or

equa

l to

one.

Ex. G

iven

3/4

, tak

e 1/

4 aw

ay a

nd te

ll or

sho

w h

ow m

any

four

ths

are

left

.Ex

. Giv

en 7

/10,

reco

gniz

e th

at 3

/10

are

need

ed to

mak

e a

who

le.

(Con

nect

to m

oney

– 1

0 di

mes

= o

ne w

hole

dol

lar)

Leve

l II A

A S

tude

nts

will

:EE

8.N

S.1.

Use

mod

els

to s

ubtr

act h

alve

s, th

irds,

and

four

ths.

Ex. G

iven

a w

hole

div

ided

into

third

s, te

ll m

e ho

w m

any

times

they

can

take

a th

ird o

ut o

f th

e w

hole

.Ex

. Pre

sent

ed a

rect

angl

e w

ith 1

/3 o

f the

who

le s

hade

d, te

ll ho

w m

any

third

s ar

e le

ft.

Leve

l I A

A S

tude

nts

will

:EE

8.N

S.1.

Use

mod

els

to id

entif

y th

e w

hole

and

find

the

mis

sing

pie

ces

of a

who

le u

sing

ha

lves

.Ex

. Pre

sent

ed a

n ob

ject

with

a p

iece

mis

sing

and

a w

hole

obj

ect,

iden

tify

the

who

le.

Ex. G

iven

1/2

of a

piz

za, i

denti

fy th

e m

issi

ng p

art (

conc

rete

mod

el o

r to

uch

boar

d).

Ex. G

iven

a w

hole

with

1/2

sha

ded,

iden

tify

the

mis

sing

par

t.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

8.N

S.2.

Use

ratio

nal

appr

oxim

ation

s of

irra

tiona

l nu

mbe

rs to

com

pare

the

size

of

irra

tiona

l num

bers

, loc

ate

them

ap

prox

imat

ely

on a

num

ber

line

diag

ram

, and

esti

mat

e th

e va

lue

of e

xpre

ssio

ns (e

.g.,

π2 ).

For e

xam

ple,

by

trun

catin

g th

e de

cim

al e

xpan

sion

of √

2, s

how

th

at √

2 is

bet

wee

n 1

and

2, th

en

betw

een

1.4

and

1.5,

and

exp

lain

ho

w to

con

tinue

on

to g

et b

etter

ap

prox

imati

ons

EE8.

NS.

2. R

epre

sent

diff

eren

t fo

rms

and

valu

es o

f dec

imal

nu

mbe

rs u

sing

frac

tions

with

nu

mer

ator

s th

at a

re m

ultip

les

of

five

and

a de

nom

inat

or o

f 100

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

NS.

2. R

epre

sent

diff

eren

t for

ms

and

valu

es o

f dec

imal

num

bers

to th

e hu

ndre

ds p

lace

(d

ecim

al, f

racti

on, h

undr

eds

grid

, and

mon

ey re

pres

enta

tion)

.Ex

. Giv

en a

hun

dred

s gr

id, s

hade

in a

n ap

prox

imati

on to

a g

iven

dec

imal

or

frac

tion.

Ex. G

iven

a p

ictu

re o

f a s

hade

d hu

ndre

ds g

rid,

det

erm

ine

the

deci

mal

or

frac

tiona

l par

t.Ex

. Whe

n gi

ven

coin

s re

pres

entin

g 60

cen

ts, w

rite

the

deci

mal

am

ount

as

$0.6

0.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

NS.

2. R

epre

sent

diff

eren

t for

ms

and

valu

es o

f dec

imal

num

bers

usi

ng fr

actio

ns w

ith

num

erat

ors

that

are

mul

tiple

s of

five

and

a d

enom

inat

or o

f 100

.Ex

. Giv

en a

hun

dred

s gr

id w

ith o

ne fo

urth

sha

ded-

in, i

denti

fy th

e co

rrec

t dec

imal

re

pres

enta

tion

from

cho

ices

25/

100,

10/

100,

or

100/

100.

Ex. W

hen

give

n co

ins

repr

esen

ting

50 c

ents

, wri

te th

e de

cim

al v

alue

as

$0.5

0.

Leve

l II A

A S

tude

nts

will

:EE

8.N

S.2.

Dis

tingu

ish

betw

een

a pa

rt re

pres

ente

d by

a d

ecim

al a

nd a

who

le n

umbe

r w

ithou

t dec

imal

s.Ex

. Giv

en a

dol

lar

and

two

quar

ters

, ide

ntify

whi

ch re

pres

ents

the

who

le (d

olla

r) a

nd th

e de

cim

al p

art (

two

quar

ters

).Ex

. Giv

en a

fully

sha

ded-

in h

undr

eds

grid

and

a p

artia

lly s

hade

d-in

hun

dred

s gr

id, i

denti

fy

whi

ch re

pres

ents

the

who

le a

nd w

hich

repr

esen

ts th

e de

cim

al (p

art o

f a w

hole

).

Leve

l I A

A S

tude

nts

will

:EE

8.N

S.2.

Iden

tify

a pa

rt o

f a w

hole

in c

oncr

ete

real

-wor

ld o

bjec

ts.

Ex. W

hen

show

n an

app

le w

ith a

mis

sing

pie

ce, i

denti

fy th

e pa

rt th

at is

mis

sing

.Ex

. Whe

n gi

ven

a st

uden

t’s s

ched

ule

for

the

day

with

one

acti

vity

mis

sing

, ide

ntify

wha

t ac

tivity

is m

issi

ng fr

om th

eir

sche

dule

.Ex

. Sho

w w

hich

pie

ce is

mis

sing

from

a fa

mili

ar o

bjec

t.

Math | 6-8 Grade 41

Eig

hth

Gra

de

Mat

hem

atic

s St

and

ard

s: E

xpre

ssio

ns

and

Eq

uat

ion

s

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Expr

essi

ons

and

Equa

tion

s.

Wor

k w

ith

radi

cals

and

inte

ger

expo

nent

s.

8.EE

.1. K

now

and

app

ly th

e pr

oper

ties

of in

tege

r ex

pone

nts

to g

ener

ate

equi

vale

nt n

umer

ical

ex

pres

sion

s. F

or e

xam

ple,

32 ×

3–5

=

3–3 =

1/3

3 = 1

/27.

8.EE

.2. U

se s

quar

e ro

ot a

nd c

ube

root

sym

bols

to re

pres

ent s

oluti

ons

to e

quati

ons

of th

e fo

rm x

2 = p

and

x3 =

p, w

here

p is

a p

ositi

ve ra

tiona

l nu

mbe

r. E

valu

ate

squa

re ro

ots

of

smal

l per

fect

squ

ares

and

cub

e ro

ots

of s

mal

l per

fect

cub

es.

Know

that

√2

is ir

ratio

nal.

8.EE

.3. U

se n

umbe

rs e

xpre

ssed

in

the

form

of a

sin

gle

digi

t tim

es

a w

hole

-num

ber

pow

er o

f 10

to

estim

ate

very

larg

e or

ver

y sm

all

quan

tities

, and

to e

xpre

ss h

ow

man

y tim

es a

s m

uch

one

is th

an th

e ot

her.

For

exa

mpl

e, e

stim

ate

the

popu

latio

n of

the

Uni

ted

Stat

es a

s 3

times

108 a

nd th

e po

pula

tion

of th

e w

orld

as

7 tim

es 1

09 , and

det

erm

ine

that

the

wor

ld p

opul

ation

is m

ore

than

20

times

larg

er.

8.EE

.4. P

erfo

rm o

pera

tions

with

nu

mbe

rs e

xpre

ssed

in s

cien

tific

nota

tion,

incl

udin

g pr

oble

ms

whe

re

both

dec

imal

and

sci

entifi

c no

tatio

n ar

e us

ed.

Use

sci

entifi

c no

tatio

n an

d ch

oose

uni

ts o

f app

ropr

iate

si

ze fo

r m

easu

rem

ents

of v

ery

larg

e or

ver

y sm

all q

uanti

ties

(e.g

., us

e m

illim

eter

s pe

r ye

ar fo

r se

afloo

r sp

read

ing)

. In

terp

ret s

cien

tific

nota

tion

that

has

bee

n ge

nera

ted

by

tech

nolo

gy.

EE8.

EE.1

-4. C

ompo

se a

nd

deco

mpo

se n

umbe

rs to

thre

e di

gits

.Le

vel I

V A

A S

tude

nts

will

:EE

8.EE

.1-4

. Use

pow

ers

of 1

0 to

com

pose

and

dec

ompo

se n

umbe

rs.

Ex. R

ecog

nize

3 x

102

= 30

0 as

ano

ther

way

to s

tate

3 x

100

= 3

00.

Ex. 5

x 1

01 = _

__.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

EE.1

-4. C

ompo

se a

nd d

ecom

pose

num

bers

to th

ree

digi

ts.

Ex. 3

00 +

50

+ 7

= __

___.

Ex. 5

7 =

____

_ +

____

_.Ex

. Sho

w th

at tw

elve

is o

ne 1

0 an

d tw

o on

es, o

r 12

one

s, o

r se

ven

ones

and

five

one

s, e

tc.

Leve

l II A

A S

tude

nts

will

:EE

8.EE

.1-4

. Use

mod

els

to re

pres

ent t

he c

ompo

sitio

n of

num

bers

.Ex

. Illu

stra

te a

num

ber

usin

g m

odel

s.Ex

. Sho

w th

at 1

2 is

one

10

and

two

ones

.Ex

. Com

pose

num

bers

to fi

ve.

Ex. C

ompo

se n

umbe

rs to

10.

Ex. M

odel

num

bers

usi

ng b

ase

ten

bloc

ks.

Ex. D

istin

guis

h th

e va

lue

of th

e di

gits

in 1

34 (e

.g.,

1 =

100,

3 =

30,

and

4 =

1).

Ex. G

iven

two

nick

els,

sho

w th

e co

rrec

t num

ber

to re

pres

ent t

hat v

alue

.

Leve

l I A

A S

tude

nts

will

:EE

8.EE

.1-4

. Rec

ogni

ze th

e sp

ecifi

c va

lue

a nu

mbe

r re

pres

ents

.Ex

. Rec

ogni

ze a

num

ber

usin

g pi

ctor

ial r

epre

sent

ation

s.Ex

. Mat

ch a

num

eric

al v

alue

with

a p

icto

rial

repr

esen

tatio

n or

con

cret

e ob

ject

s.Ex

. Loo

k at

a m

odel

and

det

erm

ine

the

num

eric

val

ue.

Ex. G

iven

a ji

g or

a m

odel

with

10

spac

es, p

ut o

ne o

bjec

t per

spa

ce a

nd a

ssem

ble

a gr

oup

of 1

0.Ex

. Giv

en th

ree

bear

s, s

elec

t the

num

ber

thre

e ca

rd.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Und

erst

and

the

conn

ecti

ons

betw

een

prop

orti

onal

re

lati

onsh

ips,

line

s, a

nd li

near

eq

uati

ons.

8.EE

.5. G

raph

pro

porti

onal

re

latio

nshi

ps, i

nter

preti

ng th

e un

it ra

te a

s th

e sl

ope

of th

e gr

aph.

Com

pare

two

diffe

rent

pr

opor

tiona

l rel

ation

ship

s re

pres

ente

d in

diff

eren

t w

ays.

For

exa

mpl

e, c

ompa

re

a di

stan

ce-ti

me

grap

h to

a

dist

ance

-tim

e eq

uatio

n to

de

term

ine

whi

ch o

f tw

o m

ovin

g ob

ject

s ha

s gr

eate

r sp

eed.

8.EE

.6. U

se s

imila

r tr

iang

les

to

expl

ain

why

the

slop

e m

is th

e sa

me

betw

een

any

two

disti

nct

poin

ts o

n a

non-

verti

cal l

ine

in

the

coor

dina

te p

lane

; der

ive

the

equa

tion

y =

mx

for

a lin

e th

roug

h th

e or

igin

and

the

equa

tion

y =

mx

+ b

for

a lin

e in

terc

eptin

g th

e ve

rtica

l axi

s at

b.

EE8.

EE.5

-6. G

raph

a s

impl

e ra

tio

usin

g th

e x

and

y ax

is p

oint

s w

hen

give

n th

e ra

tio in

sta

ndar

d fo

rm (2

:1) a

nd c

onve

rt to

2/1

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

EE.5

-6. G

raph

a s

impl

e ra

tio u

sing

the

x an

d y

axis

poi

nts

whe

n gi

ven

the

ratio

in

stan

dard

form

(2:1

) and

exp

and

on th

e ra

tio b

y tw

o or

mor

e po

ints

.Ex

. Giv

en a

ratio

2:1

(the

re a

re tw

o ba

lloon

s fo

r ev

ery

child

), gr

aph

the

linea

r eq

uatio

n on

a

grap

h la

bele

d x

axis

and

the

y ax

is.

This

equ

ation

wou

ld h

ave

a sl

ope

of 2

.Ex

. Giv

en th

ere

is o

ne b

oy fo

r ev

ery

one

girl

, gra

ph p

oint

s fo

r th

e ra

tio o

f 1:1

(thi

s lin

ear

equa

tion

will

hav

e a

slop

e of

1).

Ex. G

iven

two

plott

ed d

ata

poin

ts, p

lot a

third

poi

nt u

sing

pic

ture

s.Ex

. Giv

en a

ratio

of 3

:1 in

dica

ting

that

eac

h st

uden

t nee

ds th

ree

item

s, c

onve

rt th

e ra

tio

to fr

actio

n fo

rm (2

/1) a

nd p

lot o

n a

pre-

labe

led

grap

h th

is p

oint

and

two

addi

tiona

l poi

nts

that

are

func

tions

of t

he o

rigi

nal r

atio

(3:1

, 6:2

, 9:3

).

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

EE.5

-6. G

raph

a s

impl

e ra

tio u

sing

the

x an

d y

axis

poi

nts

whe

n gi

ven

the

ratio

in

stan

dard

form

(2:1

) and

con

vert

to 2

/1.

Ex. G

iven

two

piec

es o

f dat

a, p

lace

on

a gr

aph.

Ex. G

iven

a ra

tio o

f 3:1

indi

catin

g th

at e

ach

stud

ent n

eeds

thre

e ite

ms,

gui

de s

tude

nt in

co

nver

ting

ratio

to fr

actio

n fo

rm (2

/1) a

nd p

lot o

n a

pre-

labe

led

grap

h.

Leve

l II A

A S

tude

nts

will

:EE

8.EE

.5-6

. Ide

ntify

a s

peci

fic d

ata

poin

t whe

n gi

ven

the

coor

dina

tes.

Ex. R

ead

and

plot

coo

rdin

ates

on

a m

ap.

Ex. G

iven

thre

e w

ides

prea

d da

ta p

oint

s an

d co

ordi

nate

s, id

entif

y na

med

poi

nt.

Ex. G

iven

a s

tand

ard

mul

tiplic

ation

cha

rt, fi

nd th

e pr

oduc

t of t

wo

num

bers

usi

ng

coor

dina

te s

kills

.Ex

. Ind

icat

e w

ith c

oord

inat

es w

hat d

ata

poin

ts m

ean

or th

e da

ta re

veal

ed b

y th

e sp

ecify

po

int.

Leve

l I A

A S

tude

nts

will

:EE

8.EE

.5-6

. Pla

ce o

r lo

cate

dat

a on

a s

impl

e tw

o-ca

tego

ry g

raph

.Ex

. Use

dis

tanc

e la

ndm

ark

to te

ll if

som

ethi

ng is

clo

se o

r fa

r aw

ay.

Ex. F

inds

obj

ects

aft

er m

ovem

ent (

sear

ches

a s

mal

l are

a co

mpr

ehen

sive

ly).

Ex. L

ocat

e ob

ject

s on

a m

ap (w

ith o

r w

ithou

t coo

rdin

ates

).

Math | 6-8 Grade 43

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Ana

lyze

and

sol

ve li

near

equ

ation

s an

d pa

irs

of s

imul

tane

ous

linea

r eq

uati

ons.

8.EE

.7. S

olve

line

ar e

quati

ons

in o

ne

vari

able

.•

Giv

e ex

ampl

es o

f lin

ear

equa

tions

in o

ne v

aria

ble

with

one

so

lutio

n, in

finite

ly m

any

solu

tions

, or

no

solu

tions

. Sh

ow w

hich

of

thes

e po

ssib

ilitie

s is

the

case

by

succ

essi

vely

tran

sfor

min

g th

e gi

ven

equa

tion

into

sim

pler

form

s, u

ntil

an e

quiv

alen

t equ

ation

of t

he fo

rm

x =

a, a

= a

, or

a =

b re

sults

(whe

re a

an

d b

are

diffe

rent

num

bers

).•

Solv

e lin

ear

equa

tions

w

ith ra

tiona

l num

ber

coeffi

cien

ts,

incl

udin

g eq

uatio

ns w

hose

sol

ution

s re

quire

exp

andi

ng e

xpre

ssio

ns

usin

g th

e di

stri

butiv

e pr

oper

ty a

nd

colle

cting

like

term

s.

EE8.

EE.7

. Sol

ve a

lgeb

raic

exp

ress

ions

us

ing

sim

ple

addi

tion

and

subt

racti

on.

Leve

l IV

AA

Stu

dent

s w

ill:

8.EE

.7. S

olve

alg

ebra

ic e

xpre

ssio

ns u

sing

two-

digi

t add

ition

and

sub

trac

tion.

Ex. S

olve

20

+ x,

whe

n x

=25.

Ex. S

olve

35

– x,

whe

n x

= 12

.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

EE.7

. Sol

ve a

lgeb

raic

exp

ress

ions

usi

ng s

impl

e ad

ditio

n an

d su

btra

ction

.Ex

. Mar

k ha

d 10

dol

lars

and

nee

ds 1

5. H

ow m

any

mor

e do

llars

doe

s he

nee

d?Ex

. Giv

en a

set

of b

aske

tbal

ls, s

ome

in a

bag

and

five

out

side

of t

he b

ag, s

olve

for

find

the

tota

l nu

mbe

r of

bas

ketb

alls

in th

e se

t whe

n th

e ba

g co

ntai

ns tw

o ba

sket

balls

.Ex

. Fin

d th

e di

ffere

nce

whe

n gi

ven

the

tota

l and

the

solu

tion

(e.g

., A

stu

dent

has

10

choc

olat

e ch

ips

and

a ba

g of

cho

cola

te c

hips

. So

lve

for

the

amou

nt th

e ba

g co

ntai

ns w

hen

the

tota

l is

25.)

Leve

l II A

A S

tude

nts

will

:EE

8.EE

.7. S

olve

sim

ple

addi

tion

and

subt

racti

on p

robl

ems.

Ex. P

layi

ng a

gam

e, ro

ll tw

o di

ce a

nd a

dd u

p th

e do

ts (d

ice

with

dot

s or

dic

e w

ith n

umer

als)

.Ex

. Usi

ng a

pic

tori

al re

pres

enta

tion

of n

umbe

rs, s

olve

the

addi

tion

and

subt

racti

on p

robl

ems

(i.e.

th

ree

ballo

ons

min

us o

ne b

allo

on).

Leve

l I A

A S

tude

nts

will

:EE

8.EE

.7. D

istin

guis

h be

twee

n a

lett

er a

nd a

num

ber.

Ex. W

hen

aske

d to

wri

te th

eir

hom

e ad

dres

s, id

entif

y be

twee

n th

e le

tter

s an

d nu

mbe

rs in

the

addr

ess.

Ex. W

hen

a bo

ok is

read

to th

em, i

denti

fy th

e pa

ge n

umbe

r.Ex

. Whe

n lo

okin

g in

a te

leph

one

book

iden

tify

the

tele

phon

e nu

mbe

r vs

. the

nam

e.8.

EE.8

. Ana

lyze

and

sol

ve p

airs

of

sim

ulta

neou

s lin

ear

equa

tions

.•

Und

erst

and

that

so

lutio

ns to

a s

yste

m o

f tw

o lin

ear

equa

tions

in tw

o va

riab

les

corr

espo

nd to

poi

nts

of in

ters

ectio

n of

thei

r gr

aphs

, bec

ause

poi

nts

of

inte

rsec

tion

satis

fy b

oth

equa

tions

si

mul

tane

ousl

y.•

Solv

e sy

stem

s of

two

linea

r eq

uatio

ns in

two

vari

able

s al

gebr

aica

lly, a

nd e

stim

ate

solu

tions

by

gra

phin

g th

e eq

uatio

ns.

Solv

e si

mpl

e ca

ses

by in

spec

tion.

For

ex

ampl

e, 3

x +

2y =

5 a

nd 3

x +

2y =

6

have

no

solu

tion

beca

use

3x +

2y

cann

ot s

imul

tane

ousl

y be

5 a

nd 6

.•

Solv

e re

al-w

orld

and

m

athe

mati

cal p

robl

ems

lead

ing

to tw

o lin

ear

equa

tions

in tw

o va

riab

les.

For

exa

mpl

e, g

iven

co

ordi

nate

s fo

r tw

o pa

irs o

f poi

nts,

de

term

ine

whe

ther

the

line

thro

ugh

the

first

pai

r of

poi

nts

inte

rsec

ts th

e lin

e th

roug

h th

e se

cond

pai

r.

EE8.

EE.8

. N/A

(See

EE.

8.EE

.5-6

)


Eig

hth

Gra

de

Mat

hem

atic

s St

and

ard

s: F

un

ctio

ns

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Defi

ne, e

valu

ate,

and

com

pare

fu

ncti

ons.

8.F.

1. U

nder

stan

d th

at a

func

tion

is a

rul

e th

at a

ssig

ns to

eac

h in

put e

xact

ly o

ne o

utpu

t. T

he

grap

h of

a fu

nctio

n is

the

set o

f or

dere

d pa

irs c

onsi

sting

of a

n in

put a

nd th

e co

rres

pond

ing

outp

ut.3

8.F.

2. C

ompa

re p

rope

rties

of

two

func

tions

eac

h re

pres

ente

d in

a d

iffer

ent w

ay (a

lgeb

raic

ally

, gr

aphi

cally

, num

eric

ally

in ta

bles

, or

by

verb

al d

escr

iptio

ns).

For

ex

ampl

e, g

iven

a li

near

func

tion

repr

esen

ted

by a

tabl

e of

val

ues

and

a lin

ear f

uncti

on re

pres

ente

d by

an

alge

brai

c ex

pres

sion

, de

term

ine

whi

ch fu

nctio

n ha

s th

e gr

eate

r rat

e of

cha

nge.

8.F.

3. In

terp

ret t

he e

quati

on

y =

mx

+ b

as d

efini

ng a

line

ar

func

tion,

who

se g

raph

is a

st

raig

ht li

ne; g

ive

exam

ples

of

func

tions

that

are

not

line

ar.

For e

xam

ple,

the

func

tion

A =

s2

givi

ng th

e ar

ea o

f a s

quar

e as

a

func

tion

of it

s si

de le

ngth

is n

ot

linea

r bec

ause

its

grap

h co

ntai

ns

the

poin

ts (1

,1),

(2,4

) and

(3,9

), w

hich

are

not

on

a st

raig

ht li

ne.

EE8.

F.1-

3. G

iven

a fu

nctio

n ta

ble,

id

entif

y th

e m

issi

ng n

umbe

r.Le

vel I

V A

A S

tude

nts

will

:EE

8.F.

1-3.

Giv

en a

func

tion

tabl

e, id

entif

y th

e ru

le a

nd e

xpre

ss th

e ru

le fo

r th

e m

issi

ng

vari

able

(e.g

., n

times

2).

Ex. G

iven

a fu

nctio

n ta

ble,

iden

tify

the

rule

to fi

nd th

e m

issi

ng n

umbe

r.

12

34

n2

46

8X

Ex. G

iven

a fu

nctio

n ta

ble,

iden

tify

the

rule

to fi

nd th

e m

issi

ng n

umbe

r.

12

34

n5

1015

20X

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

F.1-

3. G

iven

a fu

nctio

n ta

ble,

iden

tify

the

mis

sing

num

ber.

Ex.

12

34

24

X8

Leve

l II A

A S

tude

nts

will

:EE

8.F.

1-3.

Iden

tify

the

rela

tions

hip

betw

een

two

num

bers

.Ex

. Giv

en c

hoic

es, t

ell t

he re

latio

nshi

p be

twee

n tw

o nu

mbe

rs (e

.g.,

How

muc

h m

ore

is fi

ve

than

thre

e? F

ive

is tw

o m

ore

than

thre

e.).

Ex. I

denti

fy th

e re

latio

nshi

p be

twee

n tw

o gi

ven

num

bers

(e.g

., If

you

doub

le fo

ur, y

ou

have

eig

ht).

Leve

l I A

A S

tude

nts

will

:EE

8.F.

1-3.

Giv

en a

seq

uenc

e, m

atch

the

elem

ent o

f a s

eque

nce.

Ex. G

iven

the

sequ

ence

1, 2

, 1, 2

and

a 1

, mat

ch to

num

ber

1.Ex

. Giv

en a

seq

uenc

e of

tria

ngle

, circ

le, t

rian

gle,

circ

le a

nd a

circ

le, m

atch

the

circ

le.

Math | 6-8 Grade 45

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Use

func

tion

s to

mod

el

rela

tion

ship

s be

twee

n qu

anti

ties

.

8.F.

4. C

onst

ruct

a fu

nctio

n to

m

odel

a li

near

rela

tions

hip

betw

een

two

quan

tities

. D

eter

min

e th

e ra

te o

f cha

nge

and

initi

al v

alue

of t

he fu

nctio

n fr

om a

des

crip

tion

of a

re

latio

nshi

p or

from

two

(x, y

) va

lues

, inc

ludi

ng re

adin

g th

ese

from

a ta

ble

or fr

om a

gra

ph.

Inte

rpre

t the

rate

of c

hang

e an

d in

itial

val

ue o

f a li

near

func

tion

in te

rms

of th

e si

tuati

on it

m

odel

s, a

nd in

term

s of

its

grap

h or

a ta

ble

of v

alue

s.

EE8.

F.4.

Det

erm

ine

the

valu

es o

r ru

le o

f a fu

nctio

n us

ing

a gr

aph

or a

tabl

e.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

F.4.

Giv

en th

e in

put v

alue

s an

d a

rule

, com

plet

e th

e ou

tput

.Ex

. Com

plet

e th

e ta

ble

by a

ddin

g th

ree

to e

ach

inpu

t val

ue.

xy

1 2 3 4Le

vel I

II A

A S

tude

nts

will

:EE

8.F.

4. D

eter

min

e th

e va

lues

or

rule

of a

func

tion

usin

g a

grap

h or

a ta

ble.

Ex. G

iven

a ta

ble,

det

erm

ine

rule

app

lied.

xy

11

+ __

=4

22+

__

=5

33

+ __

=6

Ex. G

iven

a ta

ble,

det

erm

ine

incr

ease

or

decr

ease

.

xy

14

25

36

Leve

l II A

A S

tude

nts

will

:EE

8.F.

4. N

avig

ate,

read

, use

, or

appl

y a

grap

h or

tabl

e.Ex

. Giv

en a

set

of c

oord

inat

es, l

ocat

e on

a g

raph

.Ex

. Giv

en a

loca

tion,

iden

tify

coor

dina

tes.

Ex. U

sing

a b

asic

map

of t

own,

iden

tify

two

stre

ets

over

.

Leve

l I A

A S

tude

nts

will

:EE

8.F.

4. Id

entif

y th

e di

ffere

nt p

arts

of a

gra

ph o

r a

tabl

e.Ex

. Rec

ogni

ze m

ore

or le

ss.

Ex. R

ecog

nize

a g

raph

.Ex

. Rec

ogni

ze a

tabl

e.Ex

. Ide

ntify

row

s/co

lum

ns.


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

8.F.

5. D

escr

ibe

qual

itativ

ely

the

func

tiona

l rel

ation

ship

bet

wee

n tw

o qu

antiti

es b

y an

alyz

ing

a gr

aph

(e.g

., w

here

the

func

tion

is

incr

easi

ng o

r de

crea

sing

, lin

ear

or n

onlin

ear)

. Sk

etch

a g

raph

th

at e

xhib

its th

e qu

alita

tive

feat

ures

of a

func

tion

that

has

be

en d

escr

ibed

ver

bally

.

EE8.

F.5.

Des

crib

e ho

w a

gra

ph

repr

esen

ts a

rela

tions

hip

betw

een

two

quan

tities

.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

F.5.

Des

crib

e ho

w a

gra

ph re

pres

ents

a re

latio

nshi

p be

twee

n tw

o qu

antiti

es a

nd u

se

the

grap

h to

ans

wer

que

stion

s us

ing

that

rela

tions

hip.

Ex. G

iven

a c

hart

sho

win

g th

e nu

mbe

rs o

f eac

h co

lore

d di

sk in

a b

ag, s

how

how

the

grap

h re

late

s co

lor

to n

umbe

r (e

.g.,

Poin

t to

the

axis

that

tells

you

the

num

ber

and

to th

e ax

is th

at te

lls y

ou th

e co

lor

and

poin

t to

the

bar

that

sho

ws

the

colo

r w

ith th

e hi

ghes

t nu

mbe

r.).

Ex. G

iven

a li

ne g

raph

sho

win

g da

ys o

f con

secu

tive

snow

fall

and

inch

es o

f acc

umul

ated

sn

ow, s

how

how

the

grap

h re

late

s nu

mbe

r of

day

s to

am

ount

of a

ccum

ulat

ed s

now

(e.g

., Sa

y th

e na

me

of th

e ax

is th

at s

how

s in

ches

of s

now

and

the

axis

that

sho

w c

onse

cutiv

e da

ys o

f sno

wfa

ll an

d th

en te

ll w

hich

poi

nt o

n th

e gr

aph

show

s th

e m

ost s

now

and

mos

t co

nsec

utive

day

s of

sno

wfa

ll.).

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

F.5.

Des

crib

e ho

w a

gra

ph re

pres

ents

a re

latio

nshi

p be

twee

n tw

o qu

antiti

es.

Ex. G

iven

a c

hart

sho

win

g th

e nu

mbe

rs o

f eac

h co

lore

d di

sk in

a b

ag, s

how

how

the

grap

h re

late

s co

lor

to n

umbe

r (e

.g.,

Poin

t to

the

axis

that

tells

you

the

num

ber

and

to th

e ax

is

that

tells

you

the

colo

r.).

Ex. G

iven

a li

ne g

raph

sho

win

g da

ys o

f con

secu

tive

snow

fall

and

inch

es o

f acc

umul

ated

sn

ow, s

how

how

the

grap

h re

late

s nu

mbe

r of

day

s to

am

ount

of a

ccum

ulat

ed s

now

(e.g

., sa

y th

e na

me

of th

e ax

is th

at s

how

s in

ches

of s

now

and

the

axis

that

sho

ws

cons

ecuti

ve

days

of s

now

fall)

.

Leve

l II A

A S

tude

nts

will

:EE

8.F.

5. A

nsw

er q

uesti

ons

abou

t dat

a fr

om a

gra

ph.

Ex. G

iven

a c

hart

of c

olor

s in

an

M&

M b

ag, a

nsw

er a

que

stion

abo

ut th

e in

form

ation

on

the

grap

h (e

.g.,

Whi

ch is

the

mos

t com

mon

col

or?)

.Ex

. Giv

en a

bar

gra

ph re

pres

entin

g nu

mbe

rs o

f col

ored

dis

ks fo

und

in a

bag

, ans

wer

a

ques

tion

abou

t the

info

rmati

on (e

.g.,

A b

ag o

f col

ored

dis

cs c

onta

ins

15 re

d, 1

2 bl

ue, e

ight

gr

een,

and

five

yel

low

. W

hich

bar

sho

ws

how

man

y re

d di

scs

are

in th

e ba

g?).

Ex. G

iven

a p

ictu

re g

raph

sho

win

g a

five-

day

fore

cast

sho

win

g sn

ow s

how

ers

for

all d

ays,

id

entif

y w

hich

poi

nt s

how

s ho

w m

uch

snow

is e

xpec

ted

to fa

ll on

the

fifth

day.

Leve

l I A

A S

tude

nts

will

:EE

8.F.

5. P

lace

dat

a in

a g

raph

.Ex

. Pla

ce s

ticke

rs o

f the

sam

e ty

pe (e

.g.,

colo

r, an

imal

) on

the

sam

e ba

r in

a g

raph

?Ex

. Gro

up d

ata

into

cat

egor

ies

and

plac

e on

a g

raph

(e.g

., ty

pes

of m

usic

, typ

es o

f foo

d).

Math | 6-8 Grade 47

Eig

hth

Gra

de

Mat

hem

atic

s St

and

ard

s: G

eom

etry

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Und

erst

and

cong

ruen

ce a

nd

sim

ilari

ty u

sing

phy

sica

l mod

els,

tr

ansp

aren

cies

, or

geom

etry

so

ftw

are.

8.G

.1. V

erify

exp

erim

enta

lly

the

prop

ertie

s of

rota

tions

, re

flecti

ons,

and

tran

slati

ons:

a.

Line

s ar

e ta

ken

to

lines

, and

line

seg

men

ts to

line

se

gmen

ts o

f the

sam

e le

ngth

.b.

A

ngle

s ar

e ta

ken

to

angl

es o

f the

sam

e m

easu

re.

c.

Para

llel l

ines

are

take

n to

par

alle

l lin

es.

8.G

.2. U

nder

stan

d th

at a

two-

dim

ensi

onal

figu

re is

con

grue

nt

to a

noth

er if

the

seco

nd c

an

be o

btai

ned

from

the

first

by

a s

eque

nce

of ro

tatio

ns,

refle

ction

s, a

nd tr

ansl

ation

s;

give

n tw

o co

ngru

ent fi

gure

s,

desc

ribe

a s

eque

nce

that

ex

hibi

ts th

e co

ngru

ence

be

twee

n th

em.

8.G

.3. D

escr

ibe

the

effec

t of

dila

tions

, tra

nsla

tions

, ro

tatio

ns, a

nd re

flecti

ons

on

two-

dim

ensi

onal

figu

res

usin

g co

ordi

nate

s.

EE8.

G.1

-3. I

denti

fy s

imila

rity

and

co

ngru

ence

(sam

e) in

obj

ects

an

d sh

apes

con

tain

ing

angl

es

with

out t

rans

latio

ns.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

G.1

-3. N

/A

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

G.1

-3. I

denti

fy s

imila

rity

and

con

grue

nce

(sam

e) in

obj

ects

and

sha

pes

cont

aini

ng

angl

es w

ithou

t tra

nsla

tions

.Ex

. Mat

ch a

n an

gle

in o

ne s

hape

with

the

sam

e an

gle

in a

noth

er s

hape

with

man

ipul

ative

s or

pic

ture

s.Ex

. Giv

en d

iffer

ent s

ize

shap

es, fi

nd th

e tw

o sh

apes

that

are

sim

ilar

and

tell

why

.Ex

. Giv

en a

pic

ture

of a

sha

pe, m

atch

that

pic

ture

to th

e co

ngru

ent o

bjec

t on

the

tabl

e.Ex

. Usi

ng a

pic

ture

of a

doo

r at

a 4

5 or

90-

degr

ee a

ngle

adj

ust t

he c

lass

room

doo

r to

the

sam

e an

gle.

Leve

l II A

A S

tude

nts

will

:EE

8.G

.1-3

. Mat

ch s

imila

r sh

apes

.Ex

. Mat

ch a

squ

are

to a

squ

are.

Ex. M

atch

a la

rge

squa

re w

ith a

larg

e sq

uare

.Ex

. Giv

en s

hape

s, fi

nd th

e tw

o sh

apes

that

are

sim

ilar

and

tell

why

.

Leve

l I A

A S

tude

nts

will

:EE

8.G

.1-3

. Mat

ch s

hape

s us

ing

a th

ree-

dim

ensi

onal

obj

ect.

Ex. O

verl

ay th

e ou

tline

of a

sha

pe w

ith a

thre

e-di

men

sion

al o

bjec

t usi

ng a

ngle

s in

the

outli

ne a

s gu

ides

(e.g

., bu

ildin

g w

ith b

lock

s).

Ex. T

ell,

whi

ch s

ocks

mat

ch in

col

or, s

hape

, and

siz

e.Ex

. If a

soc

k is

ups

ide

dow

n an

d an

othe

r so

ck is

rig

ht s

ide

up, c

an y

ou m

ake

them

mat

ch?


CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

8.G

.4. U

nder

stan

d th

at a

two-

dim

ensi

onal

figu

re is

sim

ilar

to a

noth

er if

the

seco

nd c

an

be o

btai

ned

from

the

first

by

a se

quen

ce o

f rot

ation

s, re

flecti

ons,

tr

ansl

ation

s, a

nd d

ilatio

ns; g

iven

tw

o si

mila

r tw

o-di

men

sion

al fi

gure

s,

desc

ribe

a s

eque

nce

that

exh

ibits

th

e si

mila

rity

bet

wee

n th

em.

EE8.

G.4

. Ide

ntify

sim

ilar

shap

es w

ith

and

with

out r

otati

on.

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

G.4

. Det

erm

ine

if ge

omet

ric

shap

es a

re s

imila

r w

ith ro

tatio

ns o

r re

flecti

ons.

Ex. S

ort s

hape

s in

to g

roup

s of

sim

ilar

shap

es w

ith ro

tatio

n an

d si

mila

r sh

apes

with

refle

ction

s.Ex

. Mat

ches

com

bina

tions

of s

imila

r sh

apes

to e

ach

othe

r (e

.g.,

mat

ch s

imila

r sh

apes

with

rota

tions

to

eac

h ot

her

and

mat

ch s

imila

r sh

apes

with

refle

ction

s to

eac

h ot

her)

.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

G.4

. Ide

ntify

sim

ilar

shap

es w

ith a

nd w

ithou

t rot

ation

.Ex

. Giv

en a

sha

pe fi

nd it

s si

mila

r ro

tatio

n.Ex

. Com

pare

sha

pes

in th

e en

viro

nmen

t to

find

a si

mila

r sh

ape

that

is ro

tate

d.Ex

. Whe

n gi

ven

a gr

oup

of tr

iang

les,

sel

ect t

wo

that

are

sim

ilar

whe

n on

e is

rota

ted.

Ex. S

elec

t the

sha

pe th

at is

not

sim

ilar

from

a g

roup

of t

hree

sha

pes.

Leve

l II A

A S

tude

nts

will

:EE

8.G

.4. I

denti

fy s

imila

r ge

omet

ric

shap

es.

Ex. S

ort r

egul

ar p

olyg

ons

into

gro

ups

of s

imila

r sh

apes

.Ex

. Whe

n gi

ven

a sh

ape,

sel

ect a

sim

ilar

shap

e.Ex

. Mat

ch th

e sh

ape

of o

ne s

mal

l squ

are

to th

e sh

ape

of a

larg

e sq

uare

.

Leve

l I A

A S

tude

nts

will

:EE

8.G

.4. R

ecog

nize

geo

met

ric

shap

es.

Ex. S

ame

thin

g co

mpa

rer

– co

mpa

re to

sha

pes

to s

ee if

they

are

the

sam

e.Ex

. Sel

ect t

he n

amed

sha

pe.

Ex. W

hen

show

n a

shap

e, n

ame

the

shap

e.Ex

. Poi

nt to

a tr

iang

le w

hen

show

n a

circ

le a

nd a

tria

ngle

.Ex

. Tra

ce a

roun

d a

geom

etri

c sh

ape.

8.G

.5. U

se in

form

al a

rgum

ents

to

esta

blis

h fa

cts

abou

t the

ang

le s

um

and

exte

rior

ang

le o

f tri

angl

es,

abou

t the

ang

les

crea

ted

whe

n pa

ralle

l lin

es a

re c

ut b

y a

tran

sver

sal,

and

the

angl

e-an

gle

crite

rion

for

sim

ilari

ty o

f tri

angl

es.

For e

xam

ple,

ar

rang

e th

ree

copi

es o

f the

sam

e tr

iang

le s

o th

at th

e su

m o

f the

th

ree

angl

es a

ppea

rs to

form

a li

ne,

and

give

an

argu

men

t in

term

s of

tr

ansv

ersa

ls w

hy th

is is

so.

EE8.

G.5

. Com

pare

mea

sure

s of

an

gles

to a

rig

ht a

ngle

(gre

ater

than

, le

ss th

an, o

r eq

ual t

o).

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

G.5

. Com

pare

mea

sure

s of

ang

les

form

ed b

y in

ters

ectin

g lin

es.

Ex. G

iven

inte

rsec

ting

lines

, ide

ntify

line

ar p

air

angl

es.

Ex. G

iven

a p

air

of p

aral

lel l

ines

inte

rsec

ted

by a

third

line

, ide

ntify

ang

les

that

are

the

sam

e m

easu

re.

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

G.5

. Com

pare

mea

sure

s of

ang

les

to a

rig

ht a

ngle

(gre

ater

than

, les

s th

an, o

r eq

ual t

o).

Ex. L

ocat

e an

ang

le w

ith a

mea

sure

gre

ater

than

the

mea

sure

of a

rig

ht a

ngle

.Ex

. Use

a r

ight

-ang

le to

ol (s

quar

e co

rner

- co

rner

of a

not

e ca

rd),

to fi

nd r

ight

ang

les.

Leve

l II A

A S

tude

nts

will

:EE

8.G

.5. R

ecog

nize

a r

ight

ang

le.

Ex. I

denti

fy a

rig

ht a

ngle

in th

e sc

hool

env

ironm

ent.

Ex. W

hich

of t

hese

is a

rig

ht a

ngle

?Ex

. Tea

cher

cre

ates

on

a ge

oboa

rd.

Is th

is a

rig

ht a

ngle

?

Leve

l I A

A S

tude

nts

will

:EE

8.G

.5. R

ecog

nize

an

angl

e.Ex

. Fin

d an

gles

in g

iven

sha

pes.

Ex. F

ind

a co

rner

in th

e cl

assr

oom

(e.g

., co

rner

of t

he ro

om o

r a

tabl

e).

Math | 6-8 Grade 49

CCSS

Gra

de-L

evel

Clu

ster

sCo

mm

on C

ore

Esse

ntial

Ele

men

tsIn

stru

ction

al A

chie

vem

ent L

evel

Des

crip

tor

Und

erst

and

and

appl

y th

e Py

thag

orea

n Th

eore

m.

8.G

.6. E

xpla

in a

pro

of o

f the

Py

thag

orea

n Th

eore

m a

nd it

s co

nver

se.

8.G

.7. A

pply

the

Pyth

agor

ean

Theo

rem

to d

eter

min

e un

know

n si

de le

ngth

s in

rig

ht tr

iang

les

in re

al-

wor

ld a

nd m

athe

mati

cal p

robl

ems

in

two

and

thre

e di

men

sion

s.

8.G

.8. A

pply

the

Pyth

agor

ean

Theo

rem

to fi

nd th

e di

stan

ce

betw

een

two

poin

ts in

a c

oord

inat

e sy

stem

.

EE8.

G.6

-8. N

/A

Solv

e re

al-w

orld

and

mat

hem

atica

l pr

oble

ms

invo

lvin

g vo

lum

e of

cy

linde

rs, c

ones

, and

sph

eres

.

8.G

.9. K

now

the

form

ulas

for

the

volu

mes

of c

ones

, cyl

inde

rs, a

nd

sphe

res

and

use

them

to s

olve

real

-w

orld

and

mat

hem

atica

l pro

blem

s.

EE8.

G.9

. Ide

ntify

vol

ume

of c

omm

on

mea

sure

s (c

ups,

pin

ts, q

uart

s,

gallo

ns, e

tc.).

Leve

l IV

AA

Stu

dent

s w

ill:

EE8.

G.9

. App

ly k

now

ledg

e of

vol

ume.

Ex. U

se s

impl

e un

its to

fill

a co

ntai

ner

with

acc

urat

e co

untin

g.Ex

. Use

s cu

bes

to fi

ll a

smal

l con

tain

er a

nd e

stim

ate

the

num

ber

of c

ubes

it to

ok b

y m

athe

mati

cal

reas

onin

g (a

dditi

on o

r m

ultip

licati

on o

f row

/col

umn)

.Ex

. Sel

ect a

ppro

pria

te to

ol to

fill

a pi

tche

r (e

.g.,

tsp.

, cup

, buc

ket)

.Ex

. Sel

ect a

ppro

pria

te to

ol to

mea

sure

flou

r fo

r a

cake

– c

up o

r bu

cket

.Ex

. Con

vert

– h

ow m

any

cups

in a

pin

t?

Leve

l III

AA

Stu

dent

s w

ill:

EE8.

G.9

. Ide

ntify

vol

ume

of c

omm

on m

easu

res

(cup

s, p

ints

, gal

lons

, etc

.).Ex

. Tel

l whi

ch h

olds

mor

e w

hen

usin

g cu

bes

to fi

ll tw

o bo

xes

(e.g

., co

unt t

he c

ubes

that

fit i

n on

e bo

x as

com

pare

d to

ano

ther

).Ex

. Ide

ntify

whi

ch is

a c

up w

hen

give

n a

cup,

teas

poon

, and

a g

allo

n co

ntai

ner.

Ex. S

how

whi

ch is

a g

allo

n w

hen

give

n a

teas

poon

, bal

l, an

d a

gallo

n co

ntai

ner.

Ex. G

iven

a g

allo

n, te

ll if

it w

ill ta

ke lo

nger

to fi

ll th

e ga

llon

with

cup

s or

with

pin

ts?

Leve

l II A

A S

tude

nts

will

:EE

8.G

.9. I

denti

fy w

hich

is m

ore

or le

ss?

Ex. C

ompa

res

two

cont

aine

rs u

sing

a th

ird fo

r tr

ansi

tive

reas

onin

g –

pour

s on

e co

ntai

ner

into

two

othe

rs to

see

whi

ch h

olds

mor

e be

caus

e on

e m

ay o

verfl

ow a

nd o

ne m

ay n

ot b

ecom

e fu

ll.Ex

. Whi

ch c

onta

iner

has

mor

e m

arbl

es in

it?

Ex. W

hich

con

tain

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Math | 6-8 Grade 51

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8.


Math | 6-8 Grade 53

Acute triangle. A triangle with all acute angles (acute means measuring less than 90°). See http://www.mathsisfun.com/definitions/acute-triangle.html

Angles. A shape formed by two lines or rays that diverge from a common point or vertex.

Area. The size of a region enclosed by the figure. Area is measured in square units (e.g., the area of this rectangle is six square units).

Associative property for addition. The sum of three or more numbers which are always the same when added together, no matter what order they are in. This is illustrated by a + (b + c) = (a + b) + c; 2 + (3 + 4) = (2 + 3) + 4.

Associative property for multiplication. The product of three or more numbers which are always the same when multiplied together, regardless of their grouping. This is illustrated by a(bc) = (ab)c; 2(3×4) = (2×3)4.

Attributes. For math purposes, “attributes” refer to characteristics of an object or geometric shape. These include qualities of shape, color, size, side, length, etc.

Base ten blocks. Blocks used to learn place value, addition, subtraction, multiplication, and division. Base ten blocks consist of cubes (ones place), rods (tens place), flats (hundreds place), and blocks (thousands place).

Categorical data. Types of data, which may be divided into groups such as race, sex, age group, and educational level when categorized into a small number of groups.

Commutative property of addition. The sum of numbers are always the same when added together, no matter if the order of the addends are changed. This is illustrated by a + b = b + a (2 + 1 = 1 + 2).

Commutative property of multiplication. The product of numbers are always the same when multiplied together, even if the order of factors are changed (i.e., if a and b are two real numbers, then a × b = b × a.)

Compose numbers. To combine parts/components to form a number (adding parts to obtain a number).

Congruent figures. Figures that have the same size and shape.

Congruent/congruence. The same.

Decompose numbers. The process of separating numbers into their components (to divide a number into smaller parts). Example: 456 can be decomposed as 456 = 400 + 50 + 6.

Denominator. The “bottom” number of a fraction; the number that represents the total number of parts into which one whole is divided (e.g., in 3/4, the 4 is the denominator and indicates that one whole is divided into 4 parts).

Dividend. The number that is being divided (e.g., In the problem, there are 550 pencils; each pack has 10 pencils; how many packs are there? 550 ÷ 10 = 55, 550 is the dividend because it tells how many pencils there are in all to be divided.).

GLOSSARY & EXAMPLES OF MATHEMATICS TERMS


Divisor. A number by which another number is divided (e.g., In the problem, there are 550 pencils; each pack has 10 pencils; how many packs are there? 550 ÷ 10 = 55, 10 is the divisor because it tells how many times 550 is to be divided.

Edge. The line segment where two faces of a solid figure meet (i.e., a cube has 12 edges).

ELA. English Language Arts

Equation. A mathematical sentence of equality between two expressions; equations have an equal sign (e.g., n + 50 = 75 or 75 = n + 50 means that n + 50 must have the same value as 75).

Equilateral triangle. A triangle with all three sides of equal length, corresponding to what could also be known as a “regular” triangle – an equilateral triangle is therefore a special case of an isosceles triangle having not just two but all three sides equal. An equilateral triangle also has three equal angles. See http://www.mathsisfun.com/definitions/equilateral-triangle.html

Expression. An operation between numbers that represents a single numeric quantity; expressions do not have an equal sign (e.g., 4r, x+2, y-1).

Face. A plane surface of a three-dimensional figure.

Fact families. Sets of related math facts. For example:Addition fact family: 3 + 5 = 8; 8 - 3 = 5; 5 + 3 = 8; and 8 - 5 = 3Multiplication fact family: 5 x 4 = 20; 20 ÷ 5 = 4; 4 x 5=20; and 20 ÷ 4 = 5

Fair share. In division meaning splitting into equal parts or groups with nothing left over.

Frequency table. A table that lists items and uses tally marks to record and show the number of times they occur.

Functions. A special kind of relation where each x-value has one and only one y-value.

Function table. A table that lists pairs of numbers that show a function.

Inequality. A mathematical sentence in which the value of the expressions on either side of the relationship symbol are unequal; relation symbols used in inequalities include > (greater than) and < (less than) symbols (e.g., 7 > 3, x < y).

Input/output table. A table that lists pairs of numbers that show a function.

Integers. Positive and negative whole numbers.

Interlocking cubes. Manipulatives that help students learn number and math concepts - cubes represent “units” and link in one direction. Interlocking cubes are used for patterning, grouping, sorting, counting, numbers, addition, subtraction, multiplication, division, and measurement.

Intersecting lines. Lines that cross.

Inverse operations. Opposite/reverse operations (e.g., subtraction is the inverse operation of addition, which is why 4 + 5 = 9 and 9 – 5 = 4; division is the inverse operation of multiplication, which is why 4 x 5 = 20 and 20 ÷ 5 = 4).

Math | 6-8 Grade 55

Linear equation. An equation that is made up of two expressions set equal to each other (e.g., y = 2x + 5) - A linear equation has only one or two variables and graph as a straight line. See http://www.eduplace.com/math/mathsteps/7/d/index.html

Line graph. A graphical representation using points connected by line segments to show how something changes over time.

Lines of symmetry. Any imaginary line along which a figure could be folded so that both halves match exactly.

Manipulatives. Objects that are used to explore mathematical ideas and solve mathematical problems (e.g., tools, models, blocks, tiles cubes, geoboards, colored rods, M&M’s).

Mathematical structures.

Addition – compare-total unknownEx. If Anita has 10 sheets of paper and you have 10 more sheets than Anita. How many sheets do you have?

Addition – start unknownEx. Sam gave away 10 apples and has five apples left. How many apples did he start have before he gave 10 apples?

Addition join-part/part – wholeEx. Jessie had 20 cakes and bought five more. How many does he have now?

Subtraction – classic take awayEx. If Judy had $50 and spent $10, how much does she have left?

Subtraction – difference unknownEx. Sandi has 10 cats and 20 dogs. Which does she have more of, cats or dogs? How many more?

Subtraction – deficit missing amountEx. Sandy wants to collect 35 cards and she already has 15. How many more cards does she need?

Multiplication – repeated additionEx. James got paid $5 each day for five days. How much money did he have at the end of the five days?

Multiplication – arrayEx. Carlos wanted to cover his rectangular paper with one-inch tiles. If his paper is five inches long and four inches wide, how many tiles will it take to cover the paper?

Multiplication – fundamental counting principleEx. Julie packed four shirts and four jeans for her trip. How many outfits can she make?

Division – repeated subtractionEx. James pays $5 each day to ride the bus. How many days can he ride for $20?

Division – factor/area – side lengthEx. Tim wants to know the width of a rectangular surface covered in 20 one-inch tiles. He knows the length is five inches, but what is the width?


Division – partitive/fair shareEx. Julie has 20 different outfits. She has five shirts – how many pair of jeans does she have to make 20 different outfits?

Mean. The “average” – To find the mean, add up all the numbers and then divide by the number of numbers.

Median. The “middle” value in the list of numbers - To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list.

Minuend. The number one is subtracting from (e.g., 9 in 9 – 2 = __).

Mode. The value that occurs most often - If no number is repeated, then there is no mode for the list. See http://www.purplemath.com/modules/meanmode.htm

Models. Pictorial or tactile aids used explore mathematical ideas and solve mathematical problems – Manipulatives can be used to model situations.

Non-numeric patterns. Using symbols, shapes, designs, and pictures to make patterns (e.g., □□ΔΔ◊◊□□ΔΔ◊◊).

Non-standard units of measure. Measurements that are neither metric nor English (e.g., number of footsteps used to measure distance or using a piece of yarn used to measure length).

Number line. A diagram that represents numbers as points on a line; a number line must have the arrows at the end.

Number sentence. An equation or inequality using numbers and symbols that is written horizontally (e.g., 5 < 7 or 5 +7+12).

Numerals. 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Numeric patterns. A pattern that uses skip counting, often starting with the number 1 or 2 – Counting by tens and twos may also be presented to students beginning with different numbers such as 7 or 23; this is more difficult for students but indicates a deeper understanding of skip counting (e.g., 7, 17, 27, 37, 47, . . . or 7, 9, 11, 13, 15, 17).

Numerical expression. A mathematical phrase that involves only numbers and one or more operational symbols.

Obtuse triangle. A triangle that has one obtuse angle (obtuse means measuring more than 90°). See http://www.mathsisfun.com/definitions/obtuse-triangle.html

Operations. Addition, subtraction, multiplication, and division.

Ordered pair. In the ordered pair (1, 3), the first number is called the x-coordinate; the second number is called the y-coordinate; this ordered pair represents the coordinates of point A.• The x-coordinate tells the distance right (positive) or left (negative).• The y-coordinate tells the distance up (positive) or down (negative).

Math | 6-8 Grade 57

Parallel Lines. Lines that are the same distance apart and that never intersect – Lines that have the same slope are parallel.

Pattern. Patterns with a minimum of three terms• using numbers by repeatedly adding or subtracting (i.e., 2, 4, 6, 8, 10, 12; 0, 3, 6, 9, 12, 15;

or 50, 45, 40, 35, 30, 25).• using objects, figures, colors, sound, etc. - a repeated pattern needs to be at least six terms.

Extend a pattern - When a student is asked to continue a pattern, the pattern is presented, and the student is asked, “What comes next?” before a student can extend or describe a pattern, the given pattern must be comprised of a minimum of three terms so that the student can see the regularities of the situation and extend or describe the pattern based on those regularities.

Percent. A way of expressing a fraction as “out of 100” (e.g., 50% means 50 out of 100 or 50/100).

Perpendicular lines. Lines that intersect, forming right angles.

Polygon. A closed plane figure made by line segments.

Prediction. A guess based on available information.

Quadrilateral. A four-sided polygon.

Rational numbers. Any number that can be expressed as a/b (b≠0) where a and b are integers; also, in decimal form, any terminating or ultimately repeating decimal.

Ratios. A comparison between two things. For instance, someone can look at a group of people and refer to the “ratio of boys to girls” in the class. Suppose there are 35 students, 15 of whom are boys; the ratio of boys to girls is 15 to 20. See http://www.purplemath.com/modules/ratio.htm

Real-life situations. Ways in which mathematical concepts are used in real life.

Real numbers. All numbers on a number line, including negative and positive integers, fractions, and irrational numbers.

Real-world applications. Ways in which mathematical concepts are used in real-life situations.

Rectangle. A four-sided polygon (a flat shape with straight sides) where every angle is a right angle (90°); opposite sides are parallel and of equal length.

Right triangle. A triangle that has one right angle (a right angle measures exactly 90°) – Only a single angle in a triangle can be a right angle or it would not be a triangle. A small square is used to mark which angle in the figure is the right angle.

Sets. A group or collection of things that go together (e.g., a group of four stars).


Side. In most general terms, a line segment that is part of the figure - it is connected at either end to another line segment, which, in turn, may or may not be connected to still other line segments.

Similar figures. Figures that have the same shape but different sizes.

Similar shapes. Objects of the same shape but different sizes in which the corresponding angles are the same.

Slope. The steepness/incline/grade of a line.

Positive slope – the condition in which a line inclines from left to right.

Negative slope – the condition in which a line declines from left to right.

Square. A four-sided polygon (a flat shape with straight sides) where all sides have equal length and every angle is a right angle (90°).

Square root. A value that can be multiplied by itself to give the original number (e.g., the square root of 25 is 5 because 5 x 5 = 25).

Square root notation. Numbers written using a radical √.

Subitize. To judge the number of objects in a group accurately without counting.

Three-dimensional geometric figures. The study of solid figures in three-dimensional space: cube, rectangular prism, sphere, cone, cylinder, and pyramid.

Two-dimensional figures. The study of two-dimensional figures in a plane; drawings of square, rectangle, circle, triangle, pentagon, hexagon, and octagon.

Unknown fixed quantities. A constant that is a quantity; a value that does not change.

Variable. A symbol for an unknown number to be solved; it is usually a letter like x or y (e.g., in x + 3 = 7, x is the variable).

Venn diagram. Made up of two or more overlapping circles. It is often used in mathematics to show relationships between sets. A Venn diagram enables students to organize similarities and differences visually.

Vertex (vertices, pl.). The point(s) where two or more edges meet (corners).

Volume. The amount of three-dimensional space an object occupies; capacity.

Math | 6-8 Grade 59

Accommodations. Changes in the administration of an assessment, such as setting, scheduling, timing, presentation format, response mode, or others, including any combination of these that does not change the construct intended to be measured by the assessment or the meaning of the resulting scores. Accommodations are used for equity, not advantage, and serve to level the playing field. To be appropriate, assessment accommodations must be identified in the student’s Individualized Education Plan (IEP) or Section 504 plan and used regularly during instruction and classroom assessment.

Achievement descriptors. Narrative descriptions of performance levels that convey student performance at each achievement level and further defines content standards by connecting them to information that describes how well students are doing in learning the knowledge and skills contained in the content standards. (See also “performance descriptors.”)

Achievement levels. A measurement that distinguishes an adequate performance from a Level I or expert performance. Achievement levels provide a determination of the extent to which a student has met the content standards. (See also Performance levels.)

Achievement standard .A system that includes performance levels (e.g., unsatisfactory, Level III, advanced), descriptions of student performance for each level, examples of student work representing the entire range of performance for each level, and cut scores. A system of performance standards operationalizes and further defines content standards by connecting them to information that describes how well students are doing in learning the knowledge and skills contained in the content standards. (See also “performance standards.”)

Achievement test. An instrument designed to efficiently measure the amount of academic knowledge and/or skill a student has acquired from instruction. Such tests provide information that can be compared to either a norm group or a measure of performance, such as a standard.

Age appropriate. The characteristics of the skills taught, the activities and materials selected, and the language level employed that reflect the chronological age of the student.

Alignment. The similarity or match between or among content standards, achievement (performance) standards, curriculum, instruction, and assessments in terms of equal breadth, depth, and complexity of knowledge and skill expectations.

Alternate assessment. An instrument used in gathering information on the standards-based performance and progress of students whose disabilities preclude their valid and reliable participation in general assessments. Alternate assessments measure the performance of a relatively small population of students who are unable to participate in the general assessment system, even with accommodations, as determined by the IEP team.

Assessment. The process of collecting information about individuals, groups, or systems that relies upon a number of instruments, one of which may be a test. Therefore, assessment is a more comprehensive term than test.

Assessment literacy. The knowledge of the basic principles of sound assessment practice including terminology, development, administration, analysis, and standards of quality.

GLOSSARY OF SPECIAL EDUCATION TERMS


Assistance (vs. support). The degree to which the teacher provides aid to the student’s performance that provides direct assistance in the content or skill being demonstrated by the student. That is, the assistance involves the teacher performing the cognitive work required. Assistance results in an invalidation of the item or score. (See also “support.”)

Assistive technology. A device, piece of equipment, product system, or service that is used to increase, maintain, or improve the functional capabilities of a student with a disability. (See 34 CFR §300.5 and 300.6.)

Cues. Assistance, words, or actions provided to a student to increase the likelihood that the student will give the desired response.

Curriculum. A document that describes what teachers do in order to convey grade-level knowledge and skills to a student.

Depth. The level of cognitive processing (e.g., recognition, recall, problem solving, analysis, synthesis, and evaluation) required for success relative to the performance standards.

Disaggregation. The collection and reporting of student achievement results by particular subgroups (e.g., students with disabilities, limited English Level III students) to ascertain the subgroup’s academic progress. Disaggregation makes it possible to compare subgroups or cohorts.

Essence of the standard. That which conveys the same ideas, skills, and content of the standard, expressed in simpler terms.

Essential Elements (EEs or CCEEs). The Common Core Essential Elements are specific statements of the content and skills that are linked to the Common Core State Standards (CCSS) grade level specific expectations for students with significant cognitive disabilities.

Grade Band Essential Element. A statement of essential precursor content and skills linked to the Common Core State Standards (CCSS) grade level clusters and indicators that maintain the essence of that standard, thereby identifying the grade-level expectations for students with significant cognitive disabilities to access and make progress in the general curriculum.

Grade level. The grade in which a student is enrolled.

Instructional Achievement Level Descriptors (IALDs). Describes student achievement and illustrates student performance. IALDs operationalize and further define Essential Elements by connecting them to information that describes how well students are doing in learning the knowledge and skills contained in the Essential Elements.

Individualized Education Program (IEP). An IEP is a written plan, developed by a team of regular and special educators, parents, related service personnel, and the student, as appropriate, describing the specially designed instruction needed for an eligible exceptional student to progress in the content standards and objectives and to meet other educational needs.

Linked. A relationship between a grade level indicator for Common Core State Standards (CCSS) and Common Core Essential Elements (EEs or CCEEs) that reflects similar content and skills but does not match the breadth, depth, and complexity of the standards.

Math | 6-8 Grade 61

Multiple measures. Measurement of student or school performance through more than one form or test.• For students, these might include teacher observations, performance assessments or

portfolios.• For schools, these might include dropout rates, absenteeism, college attendance or

documented behavior problems

Natural cue. Assistance given to a student that provides a flow among the expectations presented by the educator, opportunities to learn, and the desired outcome exhibited by the student.

Opportunity to learn. The provision of learning conditions, including suitable adjustments, to maximize a student’s chances of attaining the desired learning outcomes, such as the mastery of content standards.

Readability. The formatting of presented material that considers the organization of text; syntactic complexity of sentences; use of abstractions; density of concepts; sequence and organization of ideas; page format; sentence length; paragraph length; variety of punctuation; student background knowledge or interest; and use of illustrations or graphics in determining the appropriate level of difficulty of instructional or assessment materials.

Real-world application. The opportunity for a student to exhibit a behavior or complete a task that he or she would normally be expected to perform outside of the school environment.

Response requirements. The type, kind, or method of action required of a student to answer a question or testing item. The response may include, but is not limited to, reading, writing, speaking, creating, and drawing.

Stakeholders. A group of individuals perceived to be vested in a particular decision (e.g., a policy decision).

Standardized. An established procedure that assures that a test is administered with the same directions, and under the same conditions and is scored in the same manner for all students to ensure the comparability of scores. Standardization allows reliable and valid comparison to be made among students taking the test. The two major types of standardized tests are norm-referenced and criterion-referenced.

Standards. There are two types of standards, content and achievement (performance).• Content standards. Statements of the subject-specific knowledge and skills that schools are

expected to teach students, indicating what students should know and be able to do.• Achievement (Performance) standards. Indices of qualities that specify how adept or

competent a student demonstration must be and consist of the following four components:levels that provide descriptive labels or narratives for student performance (i.e.,

advanced, Level III, etc.);descriptions of what students at each particular level must demonstrate relative to the

task;examples of student work at each level illustrating the range of performance within each

level; andcut scores clearly separating each performance level.


Standards-based assessments. Assessments constructed to measure how well students have mastered specific content standards or skills.

Test. A measuring device or procedure. Educational tests are typically composed of questions or tasks designed to elicit predetermined behavioral responses or to measure specific academic content standards.

Test presentation. The method, manner, or structure in which test items or assessments are administered to the student.

Universal design of assessment. A method for developing an assessment to ensure accessibility by all students regardless of ability or disability. Universal design of assessment is based on principles used in the field of architecture in which user diversity is considered during the conceptual stage of development.

*Adapted from the Glossary of Assessment Terms and Acronyms Used in Assessing Special Education Students: A Report from the Assessing Special Education Students (ASES) State Collaborative on Assessment and Student Standards (SCASS)

Math | 6-8 Grade 63

Council of Chief State School Officers (CCSSO). (2003). Glossary of assessment terms and acronyms used in assessing special education students: A report from the Assessing Special Education Students (ASES) State Collaborative on Assessment and Student Standards (SCASS). Washington, DC: Author. Retrieved from http://www.ccsso.org/Documents/2006/Assessing_Students_with_Disabilities_Glossary_2006.pdf

Education Commission of the States. (1998). Designing and implementing standards-based accountability systems. Denver, CO: Author. Retrieved from http://www.eric.ed.gov/PDFS/ED419275.pdf

Hansche, L. (1998). Handbook for the development of performance standards: Meeting the requirements of Title I. Washington, DC: U.S. Department of Education (ED) and CCSSO. Retrieved from http://www.eric.ed.gov/PDFS/ED427027.pdf

Jaeger, R. M., & Tucker, C. G. (1998). Analyzing, disaggregating, reporting, and interpreting students’ achievement test results: A guide to practice for Title I and beyond. Washington, DC: CCSSO.

Johnstone, C. J. (2003). Improving validity of large-scale tests: Universal design and student performance (Technical Report 37). Minneapolis, MN: University of Minnesota, National Center on Educational Outcomes (NCEO). Retrieved from http://www.cehd.umn.edu/nceo/onlinepubs/technical37.htm.

Lehr, C.,& Thurlow, M. (2003). Putting it all together: Including students with disabilities in assessment and accountability systems (Policy Directions No. 16). Minneapolis, MN: University of Minnesota, NCEO. Retrieved from http://www.cehd.umn.edu/NCEO/onlinepubs/Policy16.htm

Linn, R. L., & Herman, J. L. (1997). A policymaker’s guide to standards-led assessment. Denver, CO: National Center for Research on Evaluation, Standards, and Student Testing (CRESST) & Education Commission of the States (ECS) Distribution Center. Retrieved from http://www.eric.ed.gov/PDFS/ED408680.pdf

McKean, E. (Ed.). (2005). The New Oxford American Dictionary (2nd ed.). New York, NY: Oxford University Press.

Quenemoen, R., Thompson, S., & Thurlow, M. (2003). Measuring academic achievement of students with significant cognitive disabilities: Building understanding of alternate assessment scoring criteria (Synthesis Report 50). Minneapolis, MN: University of Minnesota, NCEO. Retrieved from http://www.cehd.umn.edu/NCEO/onlinepubs/Synthesis50.html

Rabinowitz, S., Roeber, E., Schroeder, C., & Sheinker, J. (2006). Creating aligned standards and assessment systems. Washington, DC: CCSSO. Retrieved from http://www.ccsso.org/Documents/2006/Creating_Aligned_Standards_2006.pdf

Roeber, E. (2002). Setting standards on alternate assessments (Synthesis Report 42). Minneapolis, MN: University of Minnesota, NCEO. Retrieved from http://www.cehd.umn.edu/nceo/OnlinePubs/Synthesis42.html

BIBLIOGRAPHY OF DEVELOPMENT PROCESS


Sheinker, J. M. (2004, April 26). Achievement standards for alternate assessments: What is standard setting? Teleconference presentation for the National Center for Educational Outcomes to 38 State Departments of Education, Minneapolis, MN. Retrieved from http://www.cehd.umn.edu/nceo/teleconferences/tele08/default.html

Sheinker, J. M., & Redfield, D. L. (2001). Handbook for professional development on assessment literacy. Washington, DC: CCSSO.

Thompson, S. J., Johnstone, C. J., & Thurlow, M. L. (2002). Universal design applied to large scale assessments (Synthesis Report 44). Minneapolis, MN: University of Minnesota, NCEO. Retrieved from http://www.cehd.umn.edu/nceo/onlinepubs/Synthesis44.html

Ysseldyke, J., Krentz, J., Elliott, J., Thurlow, M. L., Erickson, R., & Moore, M. L. (1998). NCEO framework For educational accountability. Minneapolis, MN: University of Minnesota, NCEO. Retrieved from http://www.cehd.umn.edu/NCEO/onlinepubs/archive/Framework/FrameworkText.html

Math | 6-8 Grade 65

Blaha, R., & Cooper, H. (2009, February 12-14). Academic learners with deafblindness: Providing access to the general curriculum. Paper presented at the Purpose, Satisfaction, and Joy in the Lives of Students with Deafblindness and the People Who Care Conference, Austin, TX. Retrieved from http://www.tsbvi.edu/attachments/handouts/feb09/BlahaCooperAcademAccessGenEd_handout.doc

Browder, D. M. & Spooner, F. (Eds.). (2006). Teaching language arts, math, and science to students with significant cognitive disabilities. Baltimore, MD: Brookes Publishing Co. Retrieved from http://www.brookespublishing.com/store/books/browder-7985/index.htm

Burris, C., Heubert, J., & Levin, H. (2004). Math acceleration for all. Educational Leadership: Improving Achievement in Math and Science, 61(5), 68-71. Alexandria, VA: Association for Supervision and Curriculum Development (ASCD). Retrieved from http://www.ascd.org/publications/educational-leadership/feb04/vol61/num05/Math-Acceleration-for-All.aspx

Clements, D. H. (1999, March). Subitizing: What is it? Why teach it? Teaching Children Mathematics, 5(7), 400-405. Reston, VA: National Council of Teachers of Mathematics. Retrieved from http://www.nctm.org/publications/article.aspx?id=20890

Clements, D. H. (1999, January). Teaching length measurement: Research challenges. School Science and Mathematics, 99(1), 5-11. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/ssm.1999.99.issue-1/issuetoc

Clements, D. H., & Sarama, J. (2010). Technology. In V. Washington & J. Andrews (Eds.), Children of 2020: Creating a better tomorrow (pp. 119-123). Washington, DC: Council for Professional Recognition/National Association for the Education of Young Children.

Clements, D. H., Sarama, J., & Wolfe, C. B. (2011). Tools for early assessment in mathematics (TEAM). Columbus, OH: McGraw-Hill Education. Retrieved from https://www.mheonline.com/program/view/4/4/335/007TEAM

Cooney, S., & Bottoms, G. (2002). Middle grades to high school: Mending a weak link (Research brief). Atlanta, GA: Southern Regional Education Board. Retrieved from http://publications.sreb.org/2002/02V08_Middle_Grades_To_HS.pdf

Daro, P. (2011, February). Unlocking the common core: Common core state standards. Webinar for the Common Core Virtual Conference, sponsored by Pearson Education. Retrieved from http://commoncore.pearsoned.com/index.cfm?locator=PS1324

Fletcher, J. M., Lyon, G. R., Fuchs, L. S., & Barnes, M. A. (2006). Learning disabilities: From identification to intervention. New York, NY: Guilford Press.

Ford, R. (2006, January). High school profiles of mathematically unprepared college freshmen. Paper presented at the fourth annual International Conference on Education, Honolulu, HI.

Fuchs, L. S., Compton, D. L., Fuchs, D., Paulsen, K., Bryant, J. D., & Hamlett, C. L. (2005). The prevention, identification, and cognitive determinants of math difficult. Journal of Educational Psychology, 97(3), 493-513.

Fuchs, L. S., Fuchs, D., Compton, D. L., Powell, S. R., Seethaler, P. M., Capizzi, A. M., Schatschneider, C., & Fletcher, J. M. (2006). The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems. Journal of Educational Psychology, 98(1), 29-43.

BIBLIOGRAPHY FOR MATHEMATICS CONTENT


Fuchs, L. S., Fuchs, D., Powell, S. R., Seethaler, P. M., Cirino, P. T., & Fletcher, J. M. (2008). Intensive intervention for students with mathematics disabilities: seven principles of effective practice. Learning Disability Quarterly, 31(2), 79-92. Retrieved from http://www.mendeley.com/research/intensive-intervention-students-mathematics-disabilities-seven-principles-effective-practice-4/

Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P., & Fletcher, J. M. (2007). Intensive intervention on number combination and story problem deficits in third graders with math difficulties, with and without concurrent reading difficulties. Manuscript submitted for publication.

Fuson, K. C., Clements, D. H., & Beckmann, S. (2010). Focus in Grade 1: Teaching with the curriculum focal points. Reston, VA: National Council of Teachers of Mathematics. Retrieved from http://www.nctm.org/catalog/product.aspx?id=13628

Ginsburg, A., & Leinwand, S. (2009). Informing Grades 1-6 mathematics standards development: What can be learned from high-performing Hong Kong, Korea, and Singapore? Washington, DC: American Institutes for Research. Retrieved from http://www.air.org/files/MathStandards.pdf

James B. Hunt Jr. Institute for Educational Leadership and Policy. (n.d.). Common core state standards – General brief. Retrieved from http://www.edweek.org/media/fordham_event.pdf

Kroesbergen, E. H., & Van Luit, J.E.H. (2003). Mathematics interventions for children with special needs: A meta-analysis. Remedial and Special Education, 24(2), 97-114. Retrieved from http://rse.sagepub.com/content/24/2/97.abstract

Maccini, P., Mulcahy, C. A., & Wilson, M. G. (2007). A follow-up of mathematics interventions for secdonary students with learning disabilities. Learning Disabilities Research and Practice, 22(1), 58-74. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5826.2007.00231.x/abstract

Miller, S. P., & Hudson, P.J. (2007). Using evidence-based practices to build mathematics competence related to conceptual, procedural, and declarative knowledge. Learning Disabilities Research and Practice, 22(1), 47-57. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5826.2007.00230.x/abstract

Montague, M. (2007). Self-regulation and mathematics instruction. Learning Disabilities Research and Practice, 22(1), 75-83. Retrieved from http://onlinelibrary.wiley.com/doi/10.1111/j.1540-5826.2007.00232.x/abstract

National Governors Association for Best Practices (NGA Center) & Council of Chief State School Officers (CCSSO). (2011). The standards: Mathematics. Retrieved from http://www.corestandards.org/the-standards/mathematics.

NGA Center & CCSSO. (2010). Common core state standards for mathematics. Appendix A: Designing high school mathematics courses based on the common core state standards. Retrieved from http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf

Math | 6-8 Grade 67

Sarama, J., & Clements, D. H. (2010). The mathematical lives of young children. In V. Washington & J. Andrews (Eds.), Children of 2020: Creating a better tomorrow (pp. 81-84). Washington, DC: Council for Professional Recognition/National Association for the Education of Young Children.

Sarama, J., & Clements, D. H. (2009). Teaching math in the primary grades: The learning trajectories approach. Young Children, 64(2), 63-65. Retrieved from http://www.naeyc.org/files/yc/file/Primary_Interest_BTJ.pdf

Schmidt, B., Houang, R., & Cogan, L. (2002). A coherent curriculum: The case of mathematics. American Educator, 26(2), 1-18. Retrieved from http://www.aft.org/pdfs/americaneducator/summer2002/curriculum.pdf

Swanson, H. L. (2006). Cross-sectional and incremental changes in working memory and mathematical problem solving. Journal of Educational Psychology, 98, 265-281.

Thompson, S. J., Morse, A. B., Sharpe, M., & Hall, S. (2005). Accommodations manual: How to select, administer and evaluate use of accommodations and assessment for students with disabilities (2nd ed.). Washington, DC: CCSSO. Retrieved from http://www.ccsso.org/Documents/2005/Accommodations_Manual_How_2005.pdf

Washington Office of the Superintendent of Public Instruction. (2008). Guidelines for accelerating students into high school mathematics in grade 8. Olympia, WA: Author. Retrieved from http://www.k12.wa.us/Mathematics/Standards/Compression.pdf

Wiley, A., Wyatt, J. & Camara, W. J. (2010). The development of a multidimensional college readiness index (Research Report No. 2010-3). New York, NY: The College Board. Retrieved from http://professionals.collegeboard.com/profdownload/pdf/10b_2084_DevMultiDimenRR_WEB_100618.pdf


Math | 6-8 Grade 69

APPENDIX A

SEA/STAKEHOLDER DEMOGRAPHICS


Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Barb

ara

Ada

ms

IAN

o re

spon

seK-

12 M

athe

mati

cs

Curr

icul

um

Coor

dina

tor

No

resp

onse

No

resp

onse

Cauc

asia

n20

-25

PhD

Roul

a A

lMou

abbi

MI

Seco

ndar

y M

ath

6-12

; Bili

ngua

l A

rabi

c/Fr

ench

6-1

2

HS

Bilin

gual

A

lgeb

ra/G

eom

etry

. Co

llege

Alg

ebra

9-11

and

Co

llege

Ara

bic,

Fre

nch,

Afr

ican

Cauc

asia

n20

-25

MA

Robi

n Ba

rbou

rN

CA

ll Su

bjec

ts 4

-6; 6

-9

Mat

h an

d Sc

ienc

e;

AIG

cer

tifica

tion

Seco

ndar

y M

ath

Cons

ulta

nt fo

r N

C D

ept.

of P

ublic

In

stru

ction

7-8

Mat

h;

9th

Phys

ical

Sc

ienc

e; A

lgeb

ra

1; In

tegr

ated

M

ath

Gen

eral

Edu

catio

n w

ith

incl

usio

n ex

peri

ence

Cauc

asia

n20

-25

MA

Tam

ara

Barr

ient

osM

IK-

5 El

emen

tary

; 6-8

M

ath/

Scie

nce

Dire

ctor

, Sag

inaw

Va

lley

Stat

e U

nive

rsity

Reg

iona

l M

athe

mati

cs a

nd

Scie

nce

Cent

er

6-8

Mat

hN

/AH

ispa

nic

11-1

5M

A

DiR

ae B

oyd

KSCo

re C

onte

nt M

esh

K-6;

Ele

men

tary

K-

9; L

D K

-9; M

R K-

9; S

PED

ELA

K-9

; SP

ED H

isto

ry a

nd

Gov

ernm

ent K

-9;

SPED

Mat

h K-

9; S

PED

Sc

ienc

e K-

9

Func

tiona

l 6-8

inte

r-re

late

d te

ache

r Sp

ecia

l Ed

ucati

on 6

-8;

Sum

mer

Sch

ool

to K

-12

Spec

ial

Educ

ation

MR;

S/P

; Auti

sm; E

D;

DB;

MD

: HI;

OH

I; TB

I; LD

Cauc

asia

n16

-20

BA

Lynd

a Br

own

UT

ESL/

Elem

Mat

h/Ea

rly

Child

hood

En

dors

emen

t

Mat

h Co

ach

K-6

(4

scho

ols,

gen

eral

and

sp

ecia

l ed.

)

2-6

Gen

eral

Ed

ucati

onSp

ecia

l Edu

catio

n an

d In

clus

ion

Cauc

asia

n30

+M

ED

Sue

Burg

erN

JEl

emen

tary

/ Te

ache

r of

H

andi

capp

ed

Spec

ial E

duca

tion/

Cu

rric

ulum

Sp

ecia

list

HS

Reso

urce

HS

Reso

urce

; Auti

sm;

OH

I; M

LD; B

D;

Pres

choo

l Dis

able

d

Cauc

asia

n30

+BA

Jenn

ifer

Burn

sO

KSp

ecia

l Edu

catio

n –

all c

onte

nts

Ass

essm

ent

Coor

dina

tor

for

Spec

ial E

duca

tion

Serv

ices

for

Stat

e D

ept.

of E

d.

Spec

ial

Educ

ation

Pre

-K

and

6-8

S/P;

MI/

MO

Cauc

asia

n6-

10M

ED/

MS

John

But

zIA

Mat

h K-

8; K

-6

Elem

enta

ry

Educ

ation

2nd

grad

e te

ache

r5t

h gr

ade

Inst

ructi

on o

f Spe

cial

Ed

ucati

on in

Gen

eral

Ed

ucati

on c

lass

room

Cauc

asia

n16

-20

BA

Math | 6-8 Grade 71

Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Laur

el C

akin

berk

IASp

ecia

l Edu

catio

n St

rate

gist

IISp

ecia

l Edu

catio

nM

iddl

e/H

SM

O/S

/PCa

ucas

ian

11-1

5M

A

Shar

on C

ampi

one

MO

LD 1

-8; M

H/B

D K

-9;

Spec

Ed

Adm

in K

-12;

Pr

inci

pal K

-12

Func

tiona

l, Li

fe

Skill

s, S

elf-

cont

aine

d 4-

6

Mid

dle

Scho

ol

7-8/

Spec

ial

Educ

ation

SSD

Coo

rdin

ator

; Te

ache

r A

ssis

t sev

ere

popu

latio

n

Cauc

asia

n16

-20

MS

Wen

dy C

arve

rU

TCo

mm

unic

ation

D

isor

ders

/Spe

cial

Ed

ucati

on K

-12+

; Sp

eech

Lan

guag

e Pa

thol

ogy,

Ps

ycho

logy

, Mild

/M

od D

is, E

LA

Spec

ial E

duca

tion

Ass

essm

ent

Spec

ialis

t

Spec

ial

Educ

ation

K-1

2+

MI/

MO

/SCa

ucas

ian

30+

MS

Beth

Cip

oletti

WV

Mat

h 7-

12A

ssis

tant

D

irect

or, O

ffice

of

Ass

essm

ent a

nd

Acc

ount

abili

ty

Mat

h 7-

12

and

colle

ge;

taug

ht te

ache

r pr

epar

ation

co

urse

s (m

athe

mati

cs)

Incl

usio

n Cl

asse

sCa

ucas

ian

30+

EdD

Emily

Com

bsM

OM

ath

5-9/

ELA

5-9

Mat

h 7t

h gr

ade

Gen

eral

Ed

ucati

on G

rade

6

Incl

usio

n; s

peci

al

serv

ice,

IEP

Cauc

asia

n11

-15

MS

Sidn

ey C

oole

yKS

Mat

h; S

peci

al

Educ

ation

Stat

e M

athe

mati

cs

Cons

ulta

ntG

ener

al

Educ

ation

7-1

2In

tegr

ated

Mat

h gr

ades

7-

9; S

tate

LD

con

sulta

ntCa

ucas

ian

30+

PhD

Shir

ley

Coop

erN

JM

ath

Stat

e M

athe

mati

cs

Coor

dina

tor

Gen

eral

Ed

ucati

onIn

clus

ion

Afr

ican

A

mer

ican

30+

MS

Jeff

Cra

wfo

rdW

AM

ath

HS

Mat

h, 9

-12

Colle

ge

Mat

hem

atics

Low

SES

Cauc

asia

n16

-20

MS

Am

y D

augh

erty

OK

Spec

ial E

duca

tion

– A

ll co

nten

tsA

ssoc

iate

Sta

te

Dire

ctor

for

Spec

ial

Educ

ation

Ser

vice

s,

Stat

e D

ept.

of E

d.

Spec

ial

Educ

ation

K-1

2S/

P; E

moti

onal

D

istu

rbed

Cauc

asia

n6-

10BS

John

DeB

ened

ettiW

ASp

ecia

l Edu

catio

n4-

5 Ex

tend

ed

Reso

urce

N/A

Spec

ial E

duca

tion

teac

her

Cauc

asia

n6-

10BS


Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Thom

as D

eete

rIA

NA

Lead

Con

sulta

nt

(Gen

eral

Edu

catio

n)

Ass

essm

ent,

A

ccou

ntab

ility

, Pr

ogra

m E

valu

ation

Gen

eral

Ed

ucati

onA

sian

-Ca

ucas

ian

20-2

5Ph

D

Jenn

ie D

eFri

ezU

TA

dmin

istr

ative

/ Su

perv

isor

y Ce

rtific

ation

; Lev

el 2

M

ath

endo

rsem

ent;

Le

vel 2

Ele

men

tary

Ed

ucati

on L

icen

se,

mid

dle

leve

l ed

ucati

on

Uta

h St

ate

Offi

ce

of E

duca

tion

Elem

enta

ry M

ath

Ass

essm

ent

Spec

ialis

t/A

ssis

tant

Sp

ecia

l Edu

catio

n A

sses

smen

t Sp

ecia

list

Gen

eral

Ed

ucati

on

Gra

des

4-7;

M

ath/

Scie

nce

Ass

ista

nt to

Sta

te

Spec

ial E

duca

tion

Ass

essm

ent S

peci

alis

t

Cauc

asia

n11

-15

MED

Kirs

ten

Dlu

goW

A6-

8 EL

A, M

ath,

Re

adin

g an

d Sp

ecia

l Ed

ucati

on

Spec

ial E

duca

tion

Teac

her

6-8,

Life

Sk

ills

Clas

sroo

m

Ung

rade

d cl

assr

oom

for

blin

d ag

es 1

2-16

VI; D

B; A

ut; M

D; L

D;

BD, I

DCa

ucas

ian

6-10

MED

Am

ber

Ecke

sW

IEl

emen

tary

Ed

ucati

on a

nd L

D;

Read

ing

Teac

her

Spec

ial E

duca

tion

Man

ager

Gra

des

6-8

Read

ing

6-8;

M

ath

6-8

and

sum

mer

cla

sses

K-

3

Spec

ial E

duca

tion

man

ager

/tea

cher

Cauc

asia

n6-

10BS

John

Eis

enbe

rgVA

Spec

ial E

duca

tion

Virg

inia

Dep

artm

ent

of E

duca

tion

Dire

ctor

of

Inst

ructi

onal

Su

ppor

t and

Rel

ated

Se

rvic

es

Spec

ial

Educ

ation

ASD

; SD

; ID

Cauc

asia

n11

-15

MS

Lin

Ever

ettM

OK-

5 A

dmin

istr

ator

/Pr

inci

pal;

4-8

SS; K

-8

Gen

eral

Edu

catio

n:

Life

time

Certi

ficat

e;

4-8

Mid

dle

Scho

ol

Adm

in/P

rinc

ipal

; Su

peri

nten

dent

’s

certi

ficati

on, K

-12

MO

Dep

t. o

f Ed

ucati

on A

ssis

tant

D

irect

or o

f A

sses

smen

t/O

ffice

of

CCR

Self-

cont

aine

d 1-

4; E

LA M

iddl

e;

Prin

cipa

l K-8

, M

etho

ds fo

r pr

e-se

rvic

e te

ache

rs/

Uni

vers

ity

Spec

ial E

d Co

ordi

nato

rCa

ucas

ian

30+

EdS

Math | 6-8 Grade 73

Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Dag

ny F

idle

rIA

Dire

ctor

of S

peci

al

Educ

ation

; PK-

12

Prin

cipa

l; PK

-12

Spec

ial E

duca

tion

Supe

rvis

or

Vice

-Pri

ncip

al/

Spec

ial E

duca

tion

Supe

rvis

or (f

ocus

on

stud

ents

with

SCD

)

Spec

ial

Educ

ation

K-

12, C

olle

ge

inst

ructi

on

Focu

s on

stu

dent

s w

ith

sign

ifica

nt d

isab

ilitie

sCa

ucas

ian

30+

PhD

Kim

Fra

tto

UT

Und

er re

view

Dis

tric

t Lev

el

Teac

her

Spec

ialis

t fo

r St

uden

ts w

/Si

gnifi

cant

Cog

nitiv

e D

isab

ilitie

s

K-6

Spec

ial

Educ

ation

K-6

Reso

urce

Tea

cher

; In

clus

ion

Spec

ialis

t;

Spec

ial E

duca

tion

Coor

dina

tor;

Tea

cher

sp

ecia

list K

-12+

, Te

ache

r Sp

ecia

list,

st

uden

ts w

ith S

CD

Cauc

asia

n11

-15

MS

Rose

mar

y G

ardn

erW

IEl

emen

tary

Ed

ucati

on 1

-8; S

SLD

Pr

eK-1

2; P

rinc

ipal

; D

irect

or o

f Spe

cial

Ed

ucati

on; P

upil

Serv

ices

Spec

ial E

duca

tion;

Ed

ucati

onal

Pr

ogra

mm

er

Gen

eral

Ed

ucati

on 1

&

2, a

nd S

peci

al

Educ

ation

in

term

edia

te

and

mid

dle

scho

ol

Spec

ial E

duca

tion

Teac

her/

Supp

ort A

dmin

Cauc

asia

n26

-30

MS

Mel

issa

Gho

lson

WV

Mul

ti-Su

bjec

ts K

-8;

Men

tal I

mpa

irm

ents

, A

utism

, Beh

avio

r D

isor

ders

, Spe

cific

LD

K-2

1; P

rinc

ipal

an

d Su

peri

nten

dent

WV

Dep

t. o

f Ed

ucati

on, O

ffice

of

Ass

essm

ent a

nd

Acc

ount

abili

ty,

Alte

rnat

e A

sses

smen

t and

A

ccom

mod

ation

s

Elem

enta

ry

(gen

eral

an

d sp

ecia

l ed

ucati

on),

Mid

dle

Scho

ol

(spe

cial

ed

ucati

on);

Hig

h Sc

hool

(gen

eral

an

d sp

ecia

l ed

ucati

on),

, Co

llege

(tea

cher

pr

epar

ation

co

urse

s)

Supe

rvis

or o

f Spe

cial

Ed

ucati

on; S

peci

al

educ

ation

teac

hing

ex

peri

ence

with

auti

sm,

mild

, mod

erat

e, s

ever

e an

d pr

ofou

nd, m

enta

l im

pair

men

ts, b

ehav

ior

diso

rder

s, g

ifted

and

le

arni

ng d

isab

ilitie

s

Cauc

asia

n16

-20

MA

Deb

ra H

awki

nsW

AES

EA S

choo

l Ps

ycho

logy

Dire

ctor

Cla

ssro

om

Ass

essm

ent

Inte

grati

on

Gen

eral

Ed

ucati

on P

ost-

Seco

ndar

y Le

vel

Prof

ound

ly M

enta

lly

Han

dica

pped

Cauc

asia

n20

-25

EdD


Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Lind

a H

owle

yM

ISt

ate

Educ

ation

A

sses

smen

t Re

pres

enta

tive

Stat

e Ed

ucati

on

Ass

essm

ent

Repr

esen

tativ

e

Cauc

asia

n11

-15

MS

Ang

elita

Jagl

aW

AEl

emen

tary

K-8

; Te

ache

r of

Eng

lish

as

a Se

cond

Lan

guag

e;

Read

ing

and

Mat

h M

.S. E

d; N

BCT

Gen

eral

Edu

catio

n–

4th

grad

eSp

ecia

l Edu

catio

n, lo

w

SES,

ELL

Mex

ican

-A

mer

ican

6-10

MS

Bria

n Jo

hnso

nW

ISp

ecia

l Edu

catio

nSp

ecia

l Edu

catio

nCD

; Auti

sm; E

BDCa

ucas

ian

6-10

MS

Mar

yAnn

Jose

phN

JN

BCT;

Mid

dle

Child

hood

G

ener

alis

t; S

peci

al

Educ

ation

K-1

2

Spec

ial E

duca

tion

Cons

ulta

nt N

JDO

E/O

SEP

Spec

ial

Educ

ation

Se

vere

/Pr

ofou

nd,

Mid

dle

Scho

ol;

5-6

In C

lass

Re

sour

ce

Plan

ning

(s

peci

al e

d),

self-

cont

aine

d cl

assr

oom

age

s 7-

11; G

ener

al

and

Spec

ial

Educ

ation

Pr

e-K-

1

Seve

re/P

rofo

und;

Le

arni

ng D

isab

led

K-8

Cauc

asia

n30

+M

ED

Sara

Kin

gM

ON

o re

spon

seSp

ecia

l Edu

catio

n ag

es 1

8-20

Spec

ial

Educ

ation

age

s 14

-20

Spec

ial E

duca

tion

Cauc

asia

n6-

10M

A

Tere

sa K

raft

KSEd

ucati

on o

f the

D

eaf

Curr

icul

um a

nd

Ass

essm

ent

Coor

dina

tor,

KS

Scho

ol fo

r th

e D

eaf

Dea

f/H

OH

/Mul

ti-ha

ndic

appe

d; V

isua

l Im

pair

men

ts

Cauc

asia

n30

+M

ED

Trac

ey L

ank

NJ

Spec

ial E

duca

tion

Spec

ial E

duca

tion

3-5

grad

esSp

ecia

l Ed

ucati

on 1

, 2,

and

6th

grad

es

Mul

tiple

Dis

abili

ties

Cauc

asia

n1-

5

Rond

a La

yman

NC

Spee

ch L

angu

age;

EC

Adm

inis

trati

onEC

Lea

d Te

ache

r/SL

P-A

utism

and

low

in

cide

nce

Auti

sm; S

ever

e/Pr

ofou

ndCa

ucas

ian

20-2

5M

ED

Math | 6-8 Grade 75

Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Wes

ley

Lilly

WV

Spec

ial E

duca

tion

K-A

dult

(MI,

LD,

BD, A

utism

, Sev

ere

Men

tal D

isab

ilitie

s;

Seco

ndar

y Ed

ucati

on; K

-12

(Phy

sica

l Edu

catio

n)

Seco

ndar

y Sp

ecia

l Ed

ucati

on M

I/Se

vere

/Auti

sm

Spec

ial

Educ

ation

K-8

M

I/Se

vere

/A

utism

/LD

/BD

MI/

Seve

re/

Auti

sm/L

D/B

D; w

orke

d w

ith d

esig

ning

alte

rnat

e as

sess

men

t

Cauc

asia

n6-

10M

A

Dia

ne L

ucas

VAEl

emen

tary

Rea

ding

, M

ath,

Soc

ial S

tudi

es,

and

Scie

nce

Spec

ial E

duca

tion

Clas

sroo

m R

esou

rce

Teac

her

(AT

Team

Le

ader

)

Earl

y Ch

ildho

od

Spec

ial

Educ

ation

Spec

ial E

duca

tion

pre

K-12

, ID

, SD

, Auti

sm, L

DCa

ucas

ian

30+

MS

Mic

hele

Luk

saKS

Seve

re D

isab

ilitie

sSp

ecia

l Edu

catio

n Co

nsul

ting

Teac

her

for

Elem

enta

ry

Spec

ial

Educ

ation

Co

nsul

ting

Teac

her

5-12

Seve

re D

isab

ilitie

s;

Dea

f-Bl

ind,

Auti

smCa

ucas

ian

26-3

0M

A

Deb

orah

M

atthe

ws

KSSt

uden

ts w

ith

Sign

ifica

nt C

ogni

tive

Dis

abili

ties

and

Earl

y Ch

ildho

od

Kans

as S

tate

D

epar

tmen

t of

Educ

ation

Earl

y ch

ildho

od-

high

sch

ool

Earl

y Ch

ildho

od;

Stud

ents

with

Si

gnifi

cant

Cog

nitiv

e D

isab

ilitie

s

Cauc

asia

n20

-25

MS

Mel

issa

Mob

ley

WV

Auti

sm/M

enta

l Im

pair

men

tSu

perv

isor

of

Spec

ial E

duca

tion

– A

utism

and

all

leve

ls o

f men

tal

impa

irm

ent

Auti

sm K

-8A

utism

; Men

tal

Impa

irm

ents

Pre

K-A

dult

Cauc

asia

n6-

10M

A

Lisa

New

WV

Mat

h 7-

12; B

usin

ess

Prin

cipl

es 7

-12

HS

Alg

ebra

I,

Alg

ebra

sup

port

te

ache

r

Gen

eral

Ed

ucati

on

Gra

des

5-12

Team

teac

her;

incl

usio

n;

item

wri

ting

for

alte

rnat

e as

sess

men

t

Cauc

asia

nN

ative

A

mer

ican

20-2

5M

S

Kare

n Pa

ceM

OM

ath

7-12

HS

Mat

h Te

ache

rG

ener

al

Educ

ation

Mat

h 7-

9

LD, B

D, E

LL, l

ow S

ESCa

ucas

ian

30+

MED


Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Brai

n Pi

anos

iM

ISe

lf-co

ntai

ned

Elem

enta

ry 6

-8

Mat

h/Sc

ienc

e; K

-12

Spec

ial E

d.; C

ogni

tive

Impa

irm

ent

Adm

inis

trati

on –

ce

rtifie

d el

emen

tary

pr

inci

pal,

supe

rvis

or

and

dire

ctor

ce

rtific

ation

s in

sp

ecia

l ed.

Dire

ctor

of a

Ce

nter

-bas

ed s

choo

l se

rvin

g st

uden

ts

with

Mod

erat

e to

Se

vere

Cog

nitiv

e,

seve

re m

ultip

le

impa

irm

ents

, au

tism

; beh

avio

r ne

eds

Gen

eral

Ed

ucati

on 3

rd

grad

e; S

peci

al

Educ

ation

H

S Cr

oss

Cate

gori

cal

Dea

f son

; Dau

ghte

r w

ith

LD; S

peci

al O

lym

pics

vo

lunt

eer

Cauc

asia

n20

-25

MA

Mar

y Ri

char

dsW

IW

I Edu

cato

r G

rade

s 1-

8M

ath

Coac

h PK

-8G

ener

al

Educ

ation

K-6

; Ti

tle I

Mat

h 1-

4; G

ifted

and

Ta

lent

ed G

rade

s 1-

5

Incl

usio

nCa

ucas

ian

30+

MS

Laur

a Sc

earc

eVA

Mat

h Sp

ecia

list K

-8M

ath

Coac

h K-

5In

clus

ion

Gra

des

3 an

d 5

Incl

usio

n; G

ifted

and

Ta

lent

edCa

ucas

ian

11-1

5M

ED

Lisa

Sei

pert

UT

MI/

MO

D/S

ever

e Sp

ecia

l Edu

catio

nID

/SID

sel

f-co

ntai

ned

Gra

des

7-9

LD/C

D S

elf-

cont

aine

d G

rade

s 7-

9

LD/I

D/S

IDCa

ucas

ian

11-1

5BS

Katie

Sla

neN

JM

ath

and

LA7t

h G

rade

Spe

cial

Ed

ucati

on, s

elf-

cont

aine

d an

d in

clus

ive

Spec

ial

Educ

ation

2-5

se

lf-co

ntai

ned

LD a

nd A

utism

Cauc

asia

n1-

5BA

Jane

t Soc

kwel

lN

CSe

vere

/Pro

foun

d K-

12; M

enta

lly

hand

icap

ped

K-12

; B/

E H

andi

capp

ed

K-12

; LD

K-1

2; B

irth

- Ki

nder

gart

en

Spec

ial E

duca

tion

Pres

choo

l Co

ordi

nato

r an

d Su

ppor

t for

ID-

Mod

/Sev

ere

Spec

ial

Educ

ation

K-1

2 m

oder

ate

to

prof

ound

Mod

erat

e/se

vere

/pr

ofou

nd, b

ehav

ior-

emoti

onal

dis

turb

ed,

pre-

scho

ol

Cauc

asia

n21

-25

BS

Chri

stie

Step

hens

onO

KM

I/M

od; S

ever

e/Pr

ofou

ndEl

emen

tary

Sp

ecia

l Edu

catio

n Su

perv

isor

K-12

LD. I

D. M

D A

utism

, OH

ICa

ucas

ian

6-10

BS

Math | 6-8 Grade 77

Nam

eSt

ate

Are

a of

Certi

ficati

onCu

rren

t A

ssig

nmen

tO

ther

Gra

des

Taug

htSp

ecia

l Pop

ulati

on

Expe

rien

ceEt

hnic

ity

Year

s of

Ex

peri

ence

Hig

hest

D

egre

e

Dee

na S

wai

nW

VM

ulti-

subj

ects

K-8

; BD

; auti

sm/a

dmin

RESA

Dire

ctor

of

Spec

ial E

duca

tion

Gen

eral

Ed

ucati

on

K-8;

Mat

h an

d Sc

ienc

e at

Alt.

Sc

hool

/Juv

enile

D

eten

tion

Cent

er G

rade

s 7-

9; A

utism

K-1

2

Expe

rien

ce te

achi

ng

stud

ents

with

ASD

, Tr

aine

r of

teac

hers

and

ad

min

istr

ator

s on

SE

issu

es

Cauc

asia

n16

-20

MA

Emily

Tha

tche

rIA

K-12

Str

at I

MD

; K-1

2 St

rat I

I MD

. Mul

ti-ca

t 6-

12; B

D K

-6; S

ever

e an

d Pr

ofou

nd K

-12;

Sp

ecia

l Edu

catio

n Co

nsul

tant

Iow

a D

ept.

of E

d.,

Bure

au o

f Stu

dent

an

d Fa

mily

Sup

port

Se

rvic

es (S

PED

), In

stru

ction

al

Cont

ent R

esou

rce

and

Alte

rnat

e A

sses

smen

t Co

nsul

tant

Spec

ial

Educ

ation

and

A

rt K

-12

22 y

ears

var

ied

expe

rien

ceCa

ucas

ian

21-2

5M

ED

Larr

y Ti

mm

MI

Spec

ial E

duca

tion

CI;

Indu

stri

al E

duca

tion

Mid

dle

Scho

ol C

I M

ath

6-8

Gen

eral

Ed

ucati

on 6

-8

Tech

Ed.

Mod

to M

ild C

.I.Ca

ucas

ian

16-2

0M

A

Mon

a Tj

aden

KSEl

emen

tary

K-9

; EM

R an

d TM

R Sp

ecia

l Ed

ucati

on K

-9;

Spec

ial E

duca

tion

Supe

rvis

or K

-12;

Li

brar

y M

edia

K-1

2

Spec

ial E

duca

tion

Prog

ram

Co

ordi

nato

r

Spec

ial

Educ

ation

Pr

ogra

m

Coor

dina

tor

Spec

ial E

duca

tion

Teac

her

and

Coor

dina

tor

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MS

Math | 6-8 Grade 79

Jorea M. Marple, Ed.D.State Superintendent of Schools