everyday mathematics and the nctm curriculum focal … · mathematics project in order to enable...

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Frequently Asked Questions 1

Focused and Coherent


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About the Everyday Mathematics ® Program The Everyday Mathematics program was developed by the University of Chicago School Mathematics Project in order to enable children in elementary grades to learn more mathematical content and become life-long mathematical thinkers. Its instructional design was carefully crafted to capitalize on student interest and maximize student learning by developing concepts and skills over time and in a wide variety of contexts. Its core author team at the University of Chicago School Mathematics Project collaborated on all grade levels to provide a focused, coherent, well-articulated Pre-K through Grade 6 curriculum.

As the Everyday Mathematics program enters its third decade of success, we at the University of Chicago School Mathematics Project and Wright Group/McGraw-Hill have been asked a number of questions about how the program aligns with Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence published by the National Council of Teachers of Mathematics (NCTM). This brief session of frequently asked questions, in conjunction with the Grade-Level Goals and correlations that follow, is an attempt to address these concerns.

At the heart of the Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence (referred to here as simply “Focal Points”) is NCTM’s Curriculum Principle, which states, “A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades” (p. 3*). As shown by three decades of

proven performance, the Everyday Mathematics curriculum embodies a well-articulated, coherent curriculum. In fact, no other mathematics program available has a set of goals as explicit and articulate as does the Everyday Mathematics program. The program’s Grade-Level Goals clearly indicate the focus of the curriculum at every point and the point at which closure is expected on each skill and concept. At the same time, no other program shows as clearly as the Everyday Mathematics program how mathematical content builds coherently across the grades. As we shall see in this booklet, that mathematical content overwhelmingly agrees with the topics suggested by the Focal Points.

The Everyday Mathematics

curriculum embodies

a well-articulated,

coherent curriculum.

*All page citations contained herein pertain to quotations reprinted with permission from Curriculum Focal Points for

Prekindergarten through Grade 8 Mathematics: A Quest for Coherence, copyright 2006 by the National Council of

Teachers of Mathematics. All rights reserved.

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2 Everyday Mathematics and Curriculum Focal Points 2 Everyday Mathematics and Curriculum Focal Points

About the Focal PointsIn 2006, NCTM published the Focal Points in an attempt to offer “a starting point in a dialogue on what is important at particular levels of instruction” (p. vii). The Focal Points are meant to “provide ideas that may kindle fruitful discussions among teacher leaders and teachers about areas to emphasize as they consider the developmental needs of their students and examine a year’s program of instruction” (p. 1).

By proposing a broad grade-by-grade outline of important mathematical concepts and skills, NCTM intended to spark conversation but did not aim to answer all questions about curriculum. Within the general framework of the Focal Points, there remain many issues to be resolved, including instructional approaches, the sequencing of topics within grades, teaching tools, and professional development. For many of these and other issues, the Focal Points defer to NCTM’s Principles and Standards for School Mathematics. Indeed, as the introduction to the Focal Points notes, Principles and Standards “remains the comprehensive reference on developing mathematical knowledge across the grades” (p. 1). Many of the remaining details of determining what makes a focused and coherent curriculum are left to the state and local level.

The following are some common questions about the Focal Points, the Everyday Mathematics program, and how the two relate, complete with our responses to them.

Q Are the Focal Points a move to get “back to the basics”?A No. Basics (e.g., basic skills and basic facts) have always been important and always

will be. The Focal Points provide educators with guidance regarding the focus of instruction at each grade level. The Focal Points are intended to take us forward into the future, not backward (or back into the past), with an eye on tackling the perceived “mile-wide, inch-deep” problem in U.S. schools.

The authors of the Everyday Mathematics program agree with the importance of coherence in curriculum design and the focus on the topics suggested in the Focal Points. The Everyday Mathematics program clearly articulates the topics of focus and places emphasis on these topics throughout the curriculum in a way that both engages students and is paced to fit how they learn and retain knowledge.


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Q What does it mean for a curriculum to be “focused”?

A The Focal Points are “important mathematical topics for each grade level” and “areas of instructional emphasis” (p. 5) Since the Focal Points are the mathematical concepts and skills to be emphasized, one must first take a holistic look at a curriculum to gauge if the Focal Points are truly being emphasized for each grade. That is, one must ask, “Does the curriculum spend a significant portion of time on these key topics?” Former NCTM president Skip Fennell specifically addressed this question in his 2007 talk “Focal Points—Where We Are, and What’s Next?” when he said,

“The purpose of the focal points is to ensure direct attention, or focus, so that

what is taught can be covered thoroughly and understood deeply, with continuous

engagement in problem solving, reasoning and proof, communication, connections,

and related representations. In all candor, using the focal points calls for more than

completing the match game of checking off topics in an existing curriculum.”

Being focused requires emphasizing topics enough so they are treated in-depth in such a way that gives continuous attention to the five Process Standards in Principles and Standards: problem solving, communication, representations, reasoning, and making connections. In the Everyday Mathematics program, topics of focus are outlined in the Program Goals and the Grade-Level Goals; these mathematical ideas are taught in-depth and revisited periodically over time with constant awareness of the Process Standards to help ensure long-term retention and, ultimately, mastery.

Q Do the Focal Points call for the mastery of specific topicsin order to move on?

A No, they do not. Some people think “focus” means that students stay on one topic for a large number of consecutive class periods and then move to another topic assuming mastery has been achieved. You will not find this approach advocated in the Focal Points. In fact, such an approach, though common in many textbooks, is not supported by research on people’s learning methods. That research has shown that “massing” instruction and practice is less effective than spreading it out over time.

NCTM explicitly states that the Focal Points are not mastery objectives, and no instructional approach or topic sequencing is recommended (p. 5). The Focal Points are suggestions regarding what to stress in each grade; they are “not . . . a list of objectives for students to master” (p. 10). An effective curriculum should contain mastery objectives—as the Everyday Mathematics curriculum does—but the Focal Points are not meant to be used as such.

“Using the Focal Points

calls for more than

completing the match game

of checking off topics”

–Skip Fennell; former NCTM President


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4 Everyday Mathematics and Curriculum Focal Points

Q There are only a few Focal Points per grade. Are these supposed to make up the whole curriculum?

A No. There is a common misconception that the Focal Points provide a complete list of mathematical topics for each grade. That is not true. As former NCTM President, Skip Fennell, stated:

“The next issue is one I often mention during focal points sessions. Are the focal points

within a grade the entire curriculum for that particular grade? Absolutely not. You may decide

to establish work with data charts and data analysis as expectations. And understandings

of time and money can certainly be included as expectations in the primary grades. These

are perfectly appropriate as expectations, but we wouldn’t define them as focal points.”

Q Should I cover only the Focal Points during the year and leave out everything else?

A No. The notion that mathematics not specifically mentioned in the Focal Points can simply be omitted from curricula is false and provides a disservice for students. The complete removal of topics based solely on their omission from the Focal Points deprives curricula of valuable opportunities and can result in problems with cross-grade continuity of topics. Indeed, there are other important parts to a curriculum than the focal points. For example, the topic of money, which is not mentioned in the Focal Points, can provide a useful context in which to introduce and learn many number and computation concepts.

“Are the focal points within

a grade the entire

curriculum for that

particular grade?

Absolutely not.”

–Skip Fennell, former NCTM President


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Q If I cover topics other than the Focal Points at a certain grade level, is my curriculum guilty of being “a mile wide and an inch deep”?

A Not necessarily. The “mile-wide, inch-deep” charge pertains to the overall scope of the curriculum and has little relevance when the view is restricted to individual grade levels. If your curriculum focuses on important math topics—like the ones laid out by the Focal Points—in a way that allows for in-depth coverage, then the “mile-wide, inch-deep” label should not apply. Obviously, to determine if your curriculum achieves focus, grade-level mastery objectives are needed, but these need not completely agree with the Focal Points at every grade level. Bear in mind that the Focal Points do not recommend how topics should be presented. So, at a particular grade level, your curriculum could have additional objectives that lay the foundation for deeper concepts in future grades or objectives for reviewing and enriching topics from previous grades.

Q Are the Focal Points final or could they possibly change?A While considerable thought has been given to the Focal Points, the document was

meant as a starting point to a larger discussion. This means topics are subject to possible change. NCTM has stated this numerous times, including in the Focal Points Q&A section of their Web site:

“Curriculum focal points are designed to promote discussion and dialogue about

important mathematics. It is hoped that the focal points impact curriculum revision at

the state and local level. However, that discussion and dialogue certainly may push

focal point topics or elements of the focal points to a lower or higher grade level.”


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The Everyday Mathematics Program’s Alignment with the Focal Points

Q Does the Everyday Mathematics curriculum treat the Focal Points topics in-depth at each grade?

A Yes. The Everyday Mathematics curriculum is research-based and has been developed and refined over more than two decades. Even though the core of the Everyday Mathematics curriculum predates the Focal Points, the two match well. (See the correlation of the Focal Points to Grade-level Goals on pages 25–63.) That the key topics identified in the Focal Points should align well with the Everyday Mathematics Grade-Level Goals is not surprising for two reasons. First, both the Focal Points and the Everyday Mathematics curriculum rely upon the same body of research in mathematics education; and second, both also rely on NCTM’s Principles and Standards:

“Principles and Standards remains the definitive reference on the development

of mathematical content and processes across the grades.” (p. 8)

Q How did the authors ensure that the Everyday Mathematics curriculum is “focused” and “coherent” ?

A The key to the focus and coherence of the Everyday Mathematics curriculum is its goal structure. The Everyday Mathematics curriculum has 15 Program Goals, which are the same for all grades. These Program Goals are the overarching aims that weave the curriculum together across grade levels. The grade-level benchmarks for achieving the Program Goals are articulated in the Grade-Level Goals. These are the big ideas of each grade, covered in-depth, which all students are expected to master. The Grade-Level Goals progress logically across grades to create a coherent curriculum focused on achieving the 15 goals of the program. Refer to the tables on pages 10–24 for details on how the Everyday Mathematics Grade-Level Goals progress across grades.

The key to the focus

and coherence of the

Everyday Mathematics

curriculum is its

goal structure.


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Q The Focal Points say to avoid “inefficient reteaching.” How is reteaching in the Everyday Mathematics curriculum different?

A Here is what the Focal Points say about reteaching:

“The decision to organize instruction around focal points assumes that the learning of

mathematics is cumulative, with work in the later grades building on and deepening what students

have learned in the earlier grades, without repetitious and inefficient reteaching.” (p. 5)

The Everyday Mathematics curriculum avoids “inefficient reteaching.” Instead, Everyday Mathematics students are given cumulative practice and review distributed throughout the curriculum. This method of cumulative distributed practice is based on a wealth of research on how people learn and retain knowledge.

The Everyday Mathematics Grade-Level Goals clearly articulate this cumulative learning. Distinctly defined benchmarks are met every year. The progression of content is effectively paced in the Everyday Mathematics curriculum to meet the Grade-Level Goals—which are, in turn, paced to meet the Program Goals.

Q There are only a few Focal Points for each grade. Why does each grade of the Everyday Mathematics program have many more Grade-Level Goals?

A There are several reasons for the difference:

◆ The first is discrepancy in grain size. The language of the Focal Points is often broad and inclusive. In contrast, the language of the Grade-Level Goals is quite narrow and precise. As a result, multiple Grade-Level Goals can match to one Focal Point. The correlation on pages 25–63 shows the details of how the two match up.

◆ The second is pacing. In some grade levels of the Everyday Mathematics program, there is foundational work of a few topics that do not appear as Focal Points until later (see the next question on fractions). There are also Grade-Level Goals detailing review work from an earlier grade’s Focal Point. This difference is not in the focus of the curriculum, but in the pacing. The Everyday Mathematics curriculum teaches the Focal Point topics in-depth and to mastery. However, the program is paced so that new ideas are introduced and developed gradually, in-depth, in a coherent way over time.

The progression

of content is effec-

tively paced in the

Everyday Mathematics

curriculum to meet

the Grade-Level

Goals—which are, in

turn, paced to meet the

Program Goals.


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Q Fractions appear as early as Kindergarten in the Everyday Mathematics curriculum but first appear in the Focal Points in Grade 3. Why are fractions introduced so early?

A In both the Everyday Mathematics curriculum and in the Focal Points, fractions are a focus of instruction in grade 3. In the Everyday Mathematics curriculum, the concept of “fraction” is introduced informally at first in kindergarten; then, students develop deeper understanding of progressively more sophisticated fraction concepts in first and second grades, and are able to read, write, and model fractions by third grade. So, the third-grade Everyday Mathematics mastery objective (Grade-Level Goal) matches the focal point at grade 3.

Part of this pacing is built on research findings that say children can comprehend fraction concepts at a much younger age than previously expected, a finding that Principles and Standards recognized by including significant work with fractions in its recommendations for prekindergarten through grade 2. Topics in the Everyday Mathematics curriculum are carefully introduced in earlier grades in preparation to build a strong foundation for mastery at a later grade. This introductory work is woven into the curriculum in a way that doesn’t hinder development of on-grade mastery of other topics. This is a hallmark of the Everyday Mathematics program, and it is one of the reasons why Everyday Mathematics students retain what they learn and consistently earn high test scores.

Q The Everyday Mathematics curriculum has lessons on probability throughout grades K–6, but the Focal Points don’t mention the topic until grade 7. Why is there such a difference in the two?

A Despite their absence from the Focal Points for elementary grades, topics in probability are reasonable expectations for earlier grades. In fact, NCTM treats probability extensively in the Prekindergarten–2 and 3–5 grade bands in their Principles and Standards—which is the definitive reference on the development of mathematical content. It was previously mentioned that the Focal Points are in no way final, so probability may well be a topic that needs to be discussed further by the mathematics education community.

However, even within the context of the Focal Points, problems with probability provide an excellent application of fraction concepts. By teaching probability, you are providing students not only with fraction practice but also with exposure to a higher level of analytic reasoning gained by meeting the Process Standards from Principles and Standards.


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Q The Focal Points require exposure to the standard algorithms for the four arithmetic operations. What does the Everyday Mathematics curriculum do with the standard algorithms?

A Regarding exposure to the standard algorithms, NCTM has further clarified their intent in the Focal Points Q&A section of their Web site:

“Note that the focal point suggests efficient procedures, including the standard algorithm—meaning

including, not exclusively, access to the standard algorithm. Students need to understand how

and why algorithms work. The standard algorithms are historically validated mathematically

efficient procedures. But we want children to understand how these procedures work,

which means, in this case, having classroom experience with place value, and the important

properties (e.g. commutative and associative) which influence how we add and subtract.”

Based on three decades of validated student achievement†, students using the Everyday Mathematics program demonstrate superior proficiency using many algorithmic methods, including traditional forms. We have found that students are most successful when they understand how an algorithm works (as opposed to merely memorizing rote steps without understanding them). However, given the wide consensus that proficiency with U.S. traditional algorithms is an important expectation in U.S. culture, effective fall 2008, they will become more prominent in the Everyday Mathematics program.

We hope these questions and answers have been helpful in navigating the dynamic waters of mathematics education. Over the three decades of its existence, the Everyday Mathematics program has continuously sought feedback from teachers, administrators, and parents to find ways to improve the curriculum based on what works in the actual classroom, and we certainly look forward to continuing that dialog as the discussion turns to the Focal Points.

†See www.everydaymathsuccess.com for various student achievement data.


The Everyday Mathematics ® Goals

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8 Curriculum

Focal Points and Everyday Mathem

atics

Everyday MathematicsContent Strand: GEOMETRY

Program Goal: Investigate Characteristics and Properties

of Two- and Three-Dimensional Geometric ShapesContent Kindergarten First Grade Second Grade Third Grade Fourth Grade Fifth Grade Sixth Grade

Lines and angles Goal 1. Draw line segments and identify parallel line segments.

Goal 1. Identifyand draw points, intersecting and parallel line segments and lines, rays and right angles.

Goal 1. Identify,draw, and describe points, intersecting and parallel line segments and lines, rays, and right, acute, and obtuse angles.

Goal 1. Identify, describe, compare, name, and draw right, acute, obtuse, straight, and reflex angles; determine angle measures in vertical and supple-mentary angles and by applying properties of sums of angle measures in triangles and quadrangles.

Goal 1. Identify, describe, classify, name, and draw angles; determine angle measures by applying properties of orientations of angles and of sums of angle measures in triangles and quadrangles.

Plane and solid figures

Goal 1. Identify and describe plane and solid figures including circles, triangles, squares, rectangles,spheres, and cubes.

Goal 1. Identify and describe plane and solid figures including circles, triangles, squares, rectangles,spheres, cylinders, rectangular prisms, pyramids, cones, and cubes.

Goal 2. Identify, describe, and model plane and solid figures including circles, triangles, squares, rectangles,hexagons,trapezoids,rhombuses,spheres, cylinders, rectangular prisms, pyramids, cones, and cubes.

Goal 2. Identify, describe, model, and compare plane and solid figures including circles, polygons, spheres, cylinders, rectangular prisms, pyramids, cones, and cubes using appropriategeometric terms including the terms face, edge, vertex, and base.

Goal 2. Describe, compare, and classify plane and solid figures, including polygons, circles, spheres, cylinders, rect-angular prisms, cones, cubes, and pyramids, using appropriate geometric terms including vertex, base, face, edge, and congruent.

Goal 2. Describe, compare, and classify plane and solid figures using appropriate geometric terms; identify congruent figures and describe their properties.

Goal 2. Identify and describe similar and congruent figures and describe their properties; construct a figure that is congruent to another figure using a compass and straightedge.

Program Goal: Apply Transformations and Symmetry in Geometric SituationsTransformationsand symmetry

Goal 2. Identify shapes having line symmetry.

Goal 2. Identify shapes having line symmetry; complete line-symmetric shapes or designs.

Goal 3. Create and complete two-dimensionalsymmetric shapes or designs.

Goal 3. Createand complete two-dimensionalsymmetric shapes or designs; locate multiple lines of symmetry in a two-dimensional shape.

Goal 3. Identify,describe, and sketch examples of reflections; identify and describe examples of translations and rotations.

Goal 3. Identify, describe, and sketch examples of reflections, translations, and rotations.

Goal 3. Identify,describe, and sketch (including plotting on the coordinateplane) instances of reflections, translations, and rotations.

The Everyday Mathematics program builds understanding over a period of time, first through informal exposure and later through more formal and directed instruction. The following table shows the program’s Grade-Level Goals for Kindergarten through grade 6 sorted by content strand and how the core of the curriculum progresses across the grade levels to meet the 15 Program Goals.

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odel

fractions;

solv

e

pro

ble

ms involv

ing

fractional part

s o

f a

regio

n o

r a c

olle

ctio

n;

describe s

trate

gie

s

used.

Go

al

2.

Read,

write

, and m

odel

fractions;

solv

e

pro

ble

ms involv

ing

fractional part

s o

f a

regio

n o

r a c

olle

ctio

n;

describe a

nd e

xpla

in

str

ate

gie

s u

sed;

giv

en a

fra

ctional

part

of

a r

egio

n o

r a c

olle

ction,

identify

th

e u

nit w

hole

.

Go

al

2.

Solv

e

pro

ble

ms involv

ing

perc

ents

and

dis

counts

; describe

and e

xpla

in

str

ate

gie

s u

sed;

identify

the u

nit

whole

in s

ituations

involv

ing f

ractions.

Go

al

2.

Solv

e

pro

ble

ms involv

ing

perc

ents

and

dis

counts

; expla

in

str

ate

gie

s u

sed;

identify

the u

nit

whole

in s

ituations

involv

ing f

ractions,

decim

als

, and

perc

ents

.

Num

ber

theo

ryG

oal

5.

Use

maniu

latives t

o

identify

and m

odel

odd a

nd e

ven

num

bers

.

Go

al

4.

Recogniz

e

num

bers

as o

dd o

r even.

Go

al

3.

Fin

d

multip

les o

f 2,

5,

and 1

0.

Go

al

3.

Fin

d

multip

les o

f w

hole

num

bers

less t

han

10;

find w

hole

-num

ber

facto

rs o

f num

bers

.

Go

al

3.

Identify

prim

e a

nd c

om

posite

num

bers

; fa

cto

r num

bers

; find p

rim

e

facto

rizations.

Go

al

3.

Use G

CF

s,

LC

Ms,

and

div

isib

ility

rule

s t

o

manip

ula

te f

ractions.

Pro

gra

m G

oal:

Un

ders

tan

d E

qu

ivale

nt

Nam

es

for

Nu

mb

ers

Equi

vale

nt

nam

es f

or w

hole

nu

mbe

rs

Go

al

5.

Use

manip

ula

tives,

dra

win

gs,

and

num

erical

expre

ssio

ns involv

ing

additio

n a

nd

subtr

action o

f 1-d

igit

num

bers

to g

ive

equiv

ale

nt

nam

es f

or

whole

num

bers

up

to 2

0.

Go

al

6.

Use

manip

ula

tives,

dra

win

gs,

tally

m

ark

s,

and n

um

erical

expre

ssio

ns involv

ing

additio

n a

nd

subtr

action o

f 1-

or

2-d

igit n

um

bers

to

giv

e e

quiv

ale

nt

nam

es f

or

whole

num

bers

up t

o 1

00.

Go

al

5.

Use t

ally

m

ark

s,

arr

ays,

and n

um

erical

expre

ssio

ns involv

ing

additio

n a

nd

subtr

action t

o g

ive

equiv

ale

nt

nam

es f

or

whole

num

bers

.

Go

al

4.

Use

num

erical

expre

ssio

ns involv

ing

one o

r m

ore

of

the

basic

four

arith

metic

opera

tions t

o g

ive

equiv

ale

nt

nam

es f

or

whole

num

bers

.

Go

al

4.

Use

num

erical

expre

ssio

ns involv

ing

one o

r m

ore

of

the

basic

four

arith

metic

opera

tions a

nd

gro

upin

g s

ym

bols

to

giv

e e

quiv

ale

nt

nam

es f

or

whole

num

bers

.

Go

al

4.

Use

num

erical

expre

ssio

ns involv

ing

one o

r m

ore

of

the

basic

four

arith

metic

opera

tions,

gro

upin

g

sym

bols

, and

exponents

to g

ive

equiv

ale

nt

nam

es

for

whole

num

bers

; convert

betw

een

base-1

0,

exponential,

and r

epeate

d-f

acto

r nota

tions.

Go

al

4.

Apply

the

ord

er

of

opera

tions

to n

um

erical

expre

ssio

ns t

o g

ive

equiv

ale

nt

nam

es f

or

rational num

bers

.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: N

UM

BER

AN

D N

UM

ER

ATIO

N c

on

t.

Pro

gra

m G

oal:

Un

ders

tan

d E

qu

ivale

nt

Nam

es

for

Nu

mb

ers

co

nt.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Equi

vale

nt n

ames

fo

r fr

acti

ons,

de

cim

als,

and

pe

rcen

ts

Go

al

6.

Use

maniu

latives a

nd

dra

win

gs t

o m

odel

equiv

ale

nt

nam

es

for

1

_

2 .

Go

al

5.

Use

manip

ula

tives a

nd

dra

win

gs t

o f

ind a

nd

repre

sent

equiv

ale

nt

nam

es f

or

fractions;

use m

anip

ula

tives t

o

genera

te e

quiv

ale

nt

fractions.

Go

al

5.

Use

num

erical

expre

ssio

ns t

o

find a

nd r

epre

sent

equiv

ale

nt

nam

es

for

fractions a

nd

decim

als

; use a

nd

expla

in a

m

ultip

lication r

ule

to

fin

d e

quiv

ale

nt

fractions;

renam

e

fourt

hs,

fifths,

tenth

s,

and h

undre

dth

s

as d

ecim

als

and

perc

ents

.

Go

al

5.

Use

num

erical

expre

ssio

ns t

o

find a

nd r

epre

sent

equiv

ale

nt

nam

es f

or

fractions,

decim

als

, and p

erc

ents

; use

and e

xpla

in

multip

lication a

nd

div

isio

n r

ule

s t

o f

ind

equiv

ale

nt

fractions

and f

ractions in

sim

ple

st

form

; convert

betw

een

fractions a

nd m

ixed

num

bers

; convert

betw

een f

ractions,

decim

als

, and

perc

ents

.

Go

al

5.

Fin

d

equiv

ale

nt

fractions

and f

ractions in

sim

ple

st

form

by

apply

ing

multip

lication a

nd

div

isio

n r

ule

s a

nd

concepts

fro

m

num

ber

theory

; convert

betw

een

fractions,

mix

ed

num

bers

, decim

als

, and p

erc

ents

.

Pro

gra

m G

oal:

Un

ders

tan

d C

om

mo

n N

um

eri

cal

Rela

tio

ns

Com

pari

ng

and

orde

ring

nu

mbe

rs

Go

al

6.

Com

pare

and o

rder

whole

num

bers

up t

o 2

0.

Go

al

7.

Com

pare

and o

rder

whole

num

bers

up t

o 1

,000.

Go

al

7.

Com

pare

and o

rder

whole

num

bers

up t

o

10,0

00;

use a

rea

models

to c

om

pare

fr

actions.

Go

al

6.

Com

pare

and o

rder

whole

num

bers

up t

o

1,0

00,0

00;

use

manip

ula

tives t

o

ord

er

decim

als

th

rough h

undre

dth

s;

use a

rea m

odels

and

benchm

ark

fra

ctions

to c

om

pare

and

ord

er

fractions.

Go

al 6

. C

om

pare

and o

rder

whole

num

bers

up t

o

1,0

00,0

00,0

00 a

nd

decim

als

thro

ugh

thosandth

s;

com

pare

and o

rder

inte

gers

betw

een 2

100 a

nd

0; use a

rea m

odels

, benchm

ark

fra

ctio

ns,

and a

naly

ses o

f num

era

tors

and

denom

inato

rs to

com

pare

and o

rder

fractio

ns.

Go

al

6.

Com

pare

and o

rder

rational

num

bers

; use

are

a m

odels

, benchm

ark

fra

ctions,

and a

naly

ses o

f num

era

tors

and

denom

inato

rs t

o

com

pare

and o

rder

fractions a

nd m

ixed

num

bers

; describe

str

ate

gie

s u

sed t

o

com

pare

fra

ctions

and m

ixed n

um

bers

.

Go

al

6.

Choose

and a

pply

str

ate

gie

s

for

com

paring a

nd

ord

ering r

ational

num

bers

; expla

in

those c

hoic

es a

nd

str

ate

gie

s.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: O

PER

ATIO

NS

AN

D C

OM

PU

TATIO

N

Pro

gram

Goa

l: C

omp

ute

Acc

ura

tely

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Add

itio

n an

d su

btra

ctio

n fa

cts

Go

al

1.

Dem

onstr

ate

pro

ficie

ncy w

ith 1

/2

0,

1/2

1,

double

s,

and s

um

-equals

-te

n a

dditio

n a

nd

subtr

action f

acts

such a

s

6 1

4 5

10 a

nd

10 2

7 5

3.

Go

al

1.

Dem

onstr

ate

auto

maticity w

ith

1/2

0,

1/2

1,

double

s,

and s

um

-equals

-ten f

acts

, and p

roficie

ncy w

ith

all

additio

n a

nd

subtr

action f

acts

th

rough 1

0 1

10

.

Go

al

1.

Dem

onstr

ate

auto

maticity w

ith

all

additio

n a

nd

subtr

action f

acts

th

rough 1

0 1

10;

use b

asic

facts

to

com

pute

fact

exte

nsio

ns s

uch

as 8

0 1

70.

Go

al

1.

Dem

onstr

ate

auto

maticity w

ith

basic

additio

n a

nd

subtr

action f

acts

and

fact

exte

nsio

ns.

Add

itio

n an

d su

btra

ctio

n pr

oced

ures

Go

al

1.

Use

manip

ula

tives,

num

ber

lines,

and

menta

l arith

metic

to s

olv

e p

roble

ms

involv

ing t

he a

dditio

n

and s

ubtr

action o

f sin

gle

-dig

it w

hole

num

bers

.

Go

al

2.

Use

manip

ula

tives,

num

ber

grids,

tally

m

ark

s,

menta

l arith

metic,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f 1-d

igit

whole

num

bers

with

1-

or

2-d

igit w

hole

num

bers

; calc

ula

te

and c

om

pare

the

valu

es o

f com

bin

ations

of

coin

s.

Go

al

2.

Use

manip

ula

tives,

num

ber

grids,

tally

m

ark

s,

menta

l arith

metic,

paper

& p

encil,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f 2-d

igit w

hole

num

bers

; describe

the s

trate

gie

s

used;

calc

ula

te a

nd

com

pare

valu

es

of

coin

and b

ill

com

bin

ations.

Go

al

2.

Use

manip

ula

tives,

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f w

hole

num

bers

and

decim

als

in a

money

conte

xt; d

escribe t

he

str

ate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Go

al

2.

Use

manip

ula

tives,

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f w

hole

num

bers

and

decim

als

thro

ugh

hundre

dth

s;

describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Go

al

1.

Use

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f w

hole

num

bers

, decim

als

, and s

igned

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Go

al

1.

Use

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f w

hole

num

bers

, decim

als

, and s

igned

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: O

PER

ATIO

NS

AN

D C

OM

PU

TATIO

N c

on

t.

Pro

gram

Goa

l: C

omp

ute

Acc

ura

tely

con

t.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Mul

tipl

icat

ion

and

divi

sion

fac

tsG

oal

3.

Dem

onstr

ate

auto

maticity w

ith 3

0,

31,

32,

35,

and

310 m

ultip

lication

facts

; use s

trate

gie

s

to c

om

pute

re

main

ing f

acts

up t

o

10 3

10.

Go

al

3.

Dem

onstr

ate

auto

maticity w

ith

multip

lication f

acts

th

rough 1

0 º

10

and p

roficie

ncy

with r

ela

ted d

ivis

ion

facts

; use b

asic

fa

cts

to c

om

pute

fact

exte

nsio

ns s

uch a

s

30 º

60.

Go

al

2.

Dem

onstr

ate

auto

maticity w

ith

multip

lication f

acts

and p

roficie

ncy w

ith

div

isio

n f

acts

and

fact

exte

nsio

ns.

Mul

tipl

icat

ion

and

divi

sion

pr

oced

ures

Go

al

4.

Use a

rrays,

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication o

f 2-

and 3

-dig

it w

hole

num

bers

by 1

-dig

it

whole

num

bers

; describe t

he

str

ate

gie

s u

sed.

Go

al

4.

Use

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication

of

multid

igit w

hole

num

bers

by 2

-dig

it

whole

num

bers

and t

he d

ivis

ion

of

multid

igit w

hole

num

bers

by 1

-dig

it

whole

num

bers

; describe t

he

str

ate

gie

s u

sed a

nd

expla

in h

ow

they

work

.

Go

al

3.

Use

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication o

f w

hole

num

bers

and

decim

als

and t

he

div

isio

n o

f m

ultid

igit

whole

num

bers

and d

ecim

als

by

whole

num

bers

; expre

ss r

em

ain

ders

as w

hole

num

bers

or

fractions a

s

appro

priate

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Go

al

2.

Use

menta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication

and d

ivis

ion o

f w

hole

num

bers

, decim

als

, and s

igned

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: O

PER

ATIO

NS

AN

D C

OM

PU

TATIO

N c

on

t.

Pro

gram

Goa

l: C

omp

ute

Acc

ura

tely

con

t.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Pro

cedu

res

for

addi

tion

and

su

btra

ctio

n of

fr

acti

ons

Go

al

5.

Use

manip

ula

tives,

menta

l arith

metic,

and c

alc

ula

tors

to

solv

e p

roble

ms

involv

ing t

he a

dditio

n

and s

ubtr

action o

f fr

actions w

ith lik

e a

nd

unlik

e d

enom

inato

rs;

describe t

he

str

ate

gie

s u

sed.

Go

al

4.

Use m

enta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f fr

actions a

nd m

ixed

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Go

al

3.

Use m

enta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the a

dditio

n a

nd

subtr

action o

f fr

actions a

nd m

ixed

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.

Pro

cedu

res

for

mul

tipl

icat

ion

and

divi

sion

of

frac

tion

s

Go

al

5.

Use a

rea

models

, m

enta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication

of

fractions a

nd

mix

ed n

um

bers

; use d

iagra

ms,

a

com

mon-d

enom

inato

r m

eth

od,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the d

ivis

ion o

f fr

actions;

describe

the s

trate

gie

s u

sed.

Go

al

4.

Use m

enta

l arith

metic,

paper-

and-p

encil

alg

orith

ms,

and

calc

ula

tors

to s

olv

e

pro

ble

ms involv

ing

the m

ultip

lication

and d

ivis

ion o

f fr

actions a

nd m

ixed

num

bers

; describe

the s

trate

gie

s u

sed

and e

xpla

in h

ow

they

work

.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: O

PER

ATIO

NS

AN

D C

OM

PU

TATIO

N c

on

t.

Pro

gram

Goa

l: M

ake

Rea

son

able

Est

imat

esCo

nten

tKi

nder

gart

enFi

rst

Grad

eSe

cond

Gra

deTh

ird

Grad

eFo

urth

Gra

deFi

fth

Grad

eSi

xth

Grad

e

Com

puta

tion

al

esti

mat

ion

Go

al

3.

Estim

ate

re

asonable

ness o

f answ

ers

to b

asic

fact

pro

ble

ms (

e.g

., W

ill

7 1

8 b

e m

ore

or

less t

han 1

0?).

Go

al

3.

Make

reasonable

estim

ate

s

for

whole

num

ber

additio

n a

nd

subtr

action p

roble

ms;

expla

in h

ow

the

estim

ate

s w

ere

obta

ined.

Go

al

5.

Make

reasonable

estim

ate

s

for

whole

num

ber

additio

n a

nd

subtr

action p

roble

ms;

expla

in h

ow

the

estim

ate

s w

ere

obta

ined.

Go

al

6.

Make

reasonable

estim

ate

s

for

whole

num

ber

and d

ecim

al additio

n

and s

ubtr

action

pro

ble

ms a

nd

whole

num

ber

multip

lication a

nd

div

isio

n p

roble

ms;

expla

in h

ow

the

estim

ate

s w

ere

obta

ined.

Go

al

6.

Make

reasonable

estim

ate

s

for

whole

num

ber

and d

ecim

al

additio

n,

subtr

action,

multip

lication,

and

div

isio

n p

roble

ms

and f

raction a

nd

mix

ed n

um

ber

additio

n a

nd

subtr

action p

roble

ms;

expla

in h

ow

the

estim

ate

s w

ere

obta

ined.

Go

al

5.

Make

reasonable

estim

ate

s

for

whole

num

ber,

decim

al, f

raction,

and m

ixed n

um

ber

additio

n,

subtr

action,

multip

lication,

and

div

isio

n p

roble

ms;

expla

in h

ow

the

estim

ate

s w

ere

obta

ined.

Pro

gram

Goa

l: U

nd

erst

and

Mea

nin

gs o

f O

per

atio

ns

Mod

els

for

the

oper

atio

nsG

oal

2.

Identify

jo

in a

nd t

ake-a

way

situations.

Go

al

4.

Identify

change t

o m

ore

, change-t

o-less,

com

parison,

and

part

s-a

nd-t

ota

l situations.

Go

al

4.

Identify

and

describe c

hange,

com

parison,

and

part

s-a

nd-t

ota

l situations;

use

repeate

d a

dditio

n,

arr

ays,

and s

kip

counting t

o m

odel

multip

lication;

use

equal sharing a

nd

equal gro

upin

g t

o

model div

isio

n.

Go

al

6.

Recogniz

e

and d

escribe

change,

com

parison,

and p

art

s-a

nd-t

ota

l situations;

use

repeate

d a

dditio

n,

arr

ays,

and s

kip

counting t

o m

odel

multip

lication;

use

equal sharing a

nd

equal gro

upin

g t

o

model div

isio

n.

Go

al

7.

Use

repeate

d a

dditio

n,

skip

counting,

arr

ays,

are

a,

and s

calin

g t

o

model m

ultip

lication

and d

ivis

ion.

Go

al

7.

Use

repeate

d a

dditio

n,

arr

ays,

are

a,

and

scalin

g t

o m

odel

multip

lication a

nd

div

isio

n;

use r

atios

expre

ssed a

s w

ord

s,

fractions,

perc

ents

, and w

ith c

olo

ns;

solv

e p

roble

ms

involv

ing r

atios o

f part

s o

f a s

et

to t

he

whole

set.

Go

al

6.

Use r

atios

and s

calin

g t

o

model siz

e c

hanges

and t

o s

olv

e s

ize-

change p

roble

ms;

repre

sent

ratios a

s

fractions,

perc

ents

, and d

ecim

als

, and

usin

g a

colo

n;

model and s

olv

e

pro

ble

ms involv

ing

part

-to-w

hole

and

part

-to-p

art

ratios;

model ra

te a

nd r

atio

num

ber

sto

ries w

ith

pro

port

ions;

use

and e

xpla

in c

ross

multip

lication a

nd

oth

er

str

ate

gie

s t

o

solv

e p

roport

ions.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: D

ATA

AN

D C

HA

NCE

Pro

gram

Goa

l: S

elec

t an

d C

reat

e A

pp

rop

riat

e G

rap

hic

al R

epre

sen

tati

ons

of C

olle

cted

or

Giv

en D

ata

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Dat

a co

llec

tion

and

re

pres

enta

tion

Go

al

1.

Colle

ct

and o

rganiz

e d

ata

to

cre

ate

cla

ss-

constr

ucte

d t

ally

chart

s,

table

s,

and

bar

gra

phs.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

to

cre

ate

tally

chart

s,

table

s,

bar

gra

phs,

and lin

e p

lots

.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

or

use

giv

en d

ata

to c

reate

ta

lly c

hart

s,

table

s,

bar

gra

phs,

and lin

e

plo

ts.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

or

use

giv

en d

ata

to c

reate

chart

s,

table

s,

bar

gra

phs,

and lin

e

plo

ts.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

or

use

giv

en d

ata

to c

reate

chart

s,

table

s,

bar

gra

phs,

line p

lots

, and lin

e g

raphs.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

or

use g

iven d

ata

to

cre

ate

bar,

lin

e,

and

circle

gra

phs w

ith

reasonable

title

s,

labels

, keys,

and

inte

rvals

.

Go

al

1.

Colle

ct

and

org

aniz

e d

ata

or

use

giv

en d

ata

to c

reate

bar,

lin

e,

circle

, and

ste

m-a

nd-leaf

gra

phs

with r

easonable

titles,

labels

, keys,

and inte

rvals

.

Pro

gram

Goa

l: A

nal

yze

and

In

terp

ret

Dat

aD

ata

anal

ysis

Go

al

2.

Use g

raphs

to a

nsw

er

sim

ple

questions.

Go

al

2.

Use g

raphs

to a

nsw

er

sim

ple

questions a

nd d

raw

conclu

sio

ns;

find

the m

axim

um

and

min

imum

of

a d

ata

set.

Go

al

2.

Use g

raphs

to a

sk a

nd a

nsw

er

sim

ple

questio

ns a

nd

dra

w c

onclu

sio

ns;

find the m

axim

um

, m

inim

um

, m

ode, and

media

n o

f a d

ata

set.

Go

al

2.

Use g

raphs

to a

sk a

nd a

nsw

er

sim

ple

questio

ns a

nd

dra

w c

onclu

sio

ns;

find the m

axim

um

, m

inim

um

, ra

nge,

mode, and m

edia

n o

f a d

ata

set.

Go

al

2.

Use the

m

axim

um

, m

inim

um

, ra

nge, m

edia

n, m

ode,

and g

raphs

to a

sk a

nd a

nsw

er

questio

ns, dra

w

conclu

sio

ns, and

m

ake p

redic

tions.

Go

al

2.

Use t

he

maxim

um

, m

inim

um

, ra

nge,

media

n,

mode,

and m

ean a

nd

gra

phs t

o a

sk a

nd

answ

er

questions,

dra

w c

onclu

sio

ns,

and m

ake p

redic

tions.

Go

al

2.

Use t

he

min

imum

, ra

nge,

media

n,

mode,

and

mean a

nd g

raphs

to a

sk a

nd a

nsw

er

questions,

dra

w

conclu

sio

ns,

and

make p

redic

tions;

com

pare

and

contr

ast

the m

edia

n

and m

ean o

f a d

ata

set.

Pro

gram

Goa

l: U

nd

erst

and

an

d A

pp

ly B

asic

Con

cep

ts o

f P

rob

abil

ity

Qua

lita

tive

pr

obab

ilit

yG

oal

3.

Describe

events

usin

g c

ert

ain

, possib

le,

impossib

le,

and o

ther

basic

pro

babili

ty t

erm

s.

Go

al

3.

Describe

events

usin

g c

ert

ain

, lik

ely

, unlik

ely

, im

possib

le a

nd o

ther

basic

pro

babili

ty

term

s.

Go

al

3.

Describe

events

usin

g c

ert

ain

, lik

ely

, unlik

ely

, im

possib

le a

nd o

ther

basic

pro

babili

ty

term

s;

expla

in t

he

choic

e o

f la

nguage.

Go

al

3.

Describe

events

usin

g

cert

ain

, very

lik

ely

, lik

ely

, unlik

ely

, very

unlik

ely

, im

possib

le,

and o

ther

basic

pro

babili

ty t

erm

s;

expla

in t

he c

hoic

e o

f la

nguage.

Go

al

3.

Describe

events

usin

g

cert

ain

, very

lik

ely

, lik

ely

, unlik

ely

, very

unlik

ely

, im

possib

le

and o

ther

basic

pro

babili

ty t

erm

s;

use

more

lik

ely

, equally

lik

ely

, sam

e c

hance,

50–50,

less lik

ely

, and o

ther

basic

pro

babili

ty t

erm

s t

o

com

pare

events

; expla

in t

he c

hoic

e o

f la

nguage.

Go

al

3.

Describe

events

usin

g

cert

ain

, very

lik

ely

, lik

ely

, unlik

ely

, very

unlik

ely

, im

possib

le

and o

ther

basic

pro

babili

ty t

erm

s;

use

more

lik

ely

, equally

lik

ely

, sam

e c

hance,

50–50,

less lik

ely

, and o

ther

basic

pro

babili

ty t

erm

s t

o

com

pare

events

; expla

in t

he c

hoic

e o

f la

nguage.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: D

ATA

AN

D C

HA

NCE

Pro

gram

Goa

l: U

nd

erst

and

an

d A

pp

ly B

asic

Con

cep

ts o

f P

rob

abil

ity

con

t.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Qua

ntit

ativ

epr

obab

ilit

yG

oal

4.

Pre

dic

t th

e

outc

om

es o

f sim

ple

experim

ents

and

test

the p

redic

tions

usin

g m

anip

ula

tives;

expre

ss t

he

pro

babili

ty o

f an

event

by u

sin

g “

__

out

of

__”

language.

Go

al

4.

Pre

dic

t th

e o

utc

om

es o

f experim

ents

and

test

the p

redic

tions

usin

g m

anip

ula

tives;

sum

marize t

he

results a

nd u

se t

hem

to

pre

dic

t fu

ture

events

; expre

ss t

he

pro

babili

ty o

f an

event

as a

fra

ction.

Go

al

4.

Pre

dic

t th

e o

utc

om

es o

f experim

ents

, te

st

the p

redic

tions

usin

g m

anip

ula

tives,

and s

um

marize t

he

results;

com

pare

pre

dic

tions b

ased

on

theore

tical

pro

babili

ty w

ith

experim

enta

l re

sults;

use s

um

maries

and c

om

parisons

to p

redic

t fu

ture

events

; expre

ss t

he

pro

babili

ty o

f an

event

as a

fra

ction,

decim

al, o

r perc

ent.

Go

al

3.

Use t

he

Multip

lication

Counting P

rincip

le,

tree d

iagra

ms,

and o

ther

counting

str

ate

gie

s t

o

identify

all

possib

le

outc

om

es f

or

a s

ituation;

pre

dic

t re

sults o

f experim

ents

, te

st

the p

redic

tions

usin

g m

anip

ula

tives,

and s

um

marize t

he

findin

gs;

com

pare

pre

dic

tions b

ased o

n

theore

tical pro

babili

ty

with e

xperim

enta

l re

sults;

calc

ula

te

pro

babili

ties a

nd

expre

ss t

hem

as

fractions,

decim

als

, and p

erc

ents

; expla

in

how

sam

ple

siz

e

affects

results;

use

the r

esults t

o p

redic

t fu

ture

events

.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: M

easu

rem

en

t an

d R

efe

ren

ce F

ram

es

Pro

gram

Goa

l: U

nd

erst

and

th

e Sy

stem

s an

d P

roce

sses

of

Mea

sure

men

t; U

se A

pp

rop

riat

e Te

chn

iqu

es,

Tool

s, U

nit

s, a

nd

For

mu

las

in M

akin

g M

easu

rem

ents

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Leng

th,

wei

ght,

an

d an

gles

G

oal

1.

Use

nonsta

ndard

tools

and t

echniq

ues

to e

stim

ate

and

com

pare

weig

ht

and length

; id

entify

sta

ndard

measuring

tools

.

Go

al

1.

Use

nonsta

ndard

tools

and t

echniq

ues

to e

stim

ate

and

com

pare

weig

ht

and length

; m

easure

le

ngth

with s

tandard

m

easuring t

ools

.

Go

al

1.

Estim

ate

le

ngth

with a

nd

without

tools

; m

easure

length

to

the n

eare

st

inch

and c

entim

ete

r;

use s

tandard

and

nonsta

ndard

tools

to

measure

and

estim

ate

weig

ht.

Go

al

1.

Estim

ate

le

ngth

with a

nd

without

tools

; m

easure

length

to

the n

eare

st 1

_

2 inch

and 1

_

2 c

entim

ete

r;

dra

w a

nd d

escribe

angle

s a

s r

ecord

s o

f ro

tations.

Go

al

1.

Estim

ate

le

ngth

with a

nd

without

tools

; m

easure

length

to

the n

eare

st

1

_

4 i

nch

and 1

_

2 c

entim

ete

r;

estim

ate

the s

ize o

f angle

s w

ithout

tools

.

Go

al

1.

Estim

ate

le

ngth

with a

nd

without

tools

; m

easure

length

with

tools

to t

he n

eare

st

1

_

8

inch a

nd m

illim

ete

r;

estim

ate

the

measure

of

angle

s

with a

nd w

ithout

tools

; use t

ools

to

dra

w a

ngle

s w

ith

giv

en m

easure

s.

Go

al

1.

Estim

ate

le

ngth

with a

nd

without

tools

; m

easure

length

w

ith t

ools

to t

he

neare

st

1

__

16 i

nch a

nd

mill

imete

r; e

stim

ate

th

e m

easure

of

angle

s w

ith a

nd

without

tools

; use

tools

to d

raw

angle

s

with g

iven m

easure

s.

Are

a, p

erim

eter

, vo

lum

e, a

nd

capa

city

Go

al

2.

Count

unit

square

s t

o f

ind t

he

are

a o

f re

cta

ngle

s.

Go

al

2.

Describe

and u

se s

trate

gie

s

to m

easure

the

perim

ete

r of

poly

gons;

count

unit

square

s t

o f

ind t

he

are

as o

f re

cta

ngle

s.

Go

al

2.

Describe

and u

se s

trate

gie

s

to m

easure

the

perim

ete

r and a

rea

of

poly

gons,

to

estim

ate

the a

rea o

f irre

gula

r shapes,

and

to f

ind t

he v

olu

me o

f re

cta

ngula

r prism

s.

Go

al

2.

Describe

and u

se s

trate

gie

s

to f

ind t

he p

erim

ete

r of

poly

gons a

nd

the a

rea o

f circle

s;

choose a

nd u

se

appro

priate

form

ula

s

to c

alc

ula

te t

he a

reas

of

recta

ngle

s,

para

llelo

gra

ms,

and

tria

ngle

s,

and t

he

volu

me o

f a p

rism

; define p

i as t

he

ratio o

f a c

ircle

’s

circum

fere

nce t

o its

dia

mete

r.

Go

al

2.

Choose

and u

se a

ppro

priate

fo

rmula

s t

o c

alc

ula

te

the c

ircum

fere

nce o

f circle

s a

nd t

o s

olv

e

are

a,

perim

ete

r, a

nd

volu

me p

roble

ms.

Uni

ts a

nd

syst

ems

of

mea

sure

men

t

Go

al

3.

Describe

rela

tionship

s

betw

een d

ays in a

w

eek a

nd h

ours

in

a d

ay.

Go

al

3.

Describe

rela

tionship

s a

mong

in

ches,

feet, a

nd

yard

s;

describe

rela

tionship

s

betw

een m

inute

s in

an h

our,

hours

in a

day,

days in a

week.

Go

al

3.

Describe

rela

tionship

s a

mong

U.S

. custo

mary

units

of

length

and a

mong

metr

ic u

nits o

f le

ngth

.

Go

al

3.

Describe

rela

tionship

s a

mong

U.S

. custo

mary

units

of

length

; am

ong

metr

ic u

nits o

f le

ngth

; and a

mong U

.S.

custo

mary

units o

f capacity.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: M

EA

SUR

EM

EN

T A

ND

REFER

EN

CE F

RA

MES

con

t.

Pro

gram

Goa

l: U

nd

erst

and

th

e Sy

stem

s an

d P

roce

sses

of

Mea

sure

men

t; U

se A

pp

rop

riat

eTe

chn

iqu

es,

Tool

s, U

nit

s, a

nd

For

mu

las

in M

akin

g M

easu

rem

ents

con

t.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Mon

eyG

oal

2.

Identify

pennie

s,

nic

kels

, dim

es,

quart

ers

, and

dolla

r bill

s.

Go

al

2.

Know

and

com

pare

the v

alu

e

of

pennie

s,

nic

kels

, dim

es,

quart

ers

, and

dolla

r bill

s;

make

exchanges b

etw

een

coin

s.

Go

al

4.

Make

exchanges b

etw

een

coin

s a

nd b

ills.

Pro

gram

Goa

l: U

se a

nd

Un

der

stan

d R

efer

ence

Fra

mes

Tem

pera

ture

Go

al

3.

Describe

tem

pera

ture

usin

g a

ppro

priate

vocabula

ry,

such

as h

ot, w

arm

, and

cold

; id

entify

a

therm

om

ete

r as a

to

ol fo

r m

easuring

tem

pera

ture

.

Go

al

3.

Identify

a

therm

om

ete

r as a

to

ol fo

r m

easuring

tem

pera

ture

; re

ad

tem

pera

ture

s o

n

Fahre

nheit a

nd

Cels

ius therm

om

ete

rs

to t

he n

eare

st

10°.

Go

al

5.

Read

tem

pera

ture

on b

oth

th

e F

ahre

nheit a

nd

Cels

ius s

cale

s.

Tim

eG

oal

4.

Describe

and u

se m

easure

s o

f tim

e p

eriods r

ela

tive

to a

day a

nd w

eek;

identify

tools

that

measure

tim

e.

Go

al

4.

Use a

cale

ndar

to identify

days,

weeks,

month

s,

and d

ate

s;

tell

and s

how

tim

e t

o

the n

eare

st

half a

nd

quart

er

hour

on a

n

analo

g c

lock.

Go

al

6.

Tell

and

show

tim

e t

o t

he

neare

st

five m

inute

s

on a

n a

nalo

g c

lock;

tell

and w

rite

tim

e in

dig

ital nota

tion.

Go

al

4.

Tell

and

show

tim

e t

o t

he

neare

st

min

ute

on a

n

analo

g c

lock;

tell

and

write

tim

e in d

igital

nota

tion.

Coor

dina

te

syst

ems

Go

al

4.

Use o

rdere

d

pairs o

f num

bers

to

nam

e,

locate

, and

plo

t poin

ts in t

he

firs

t quadra

nt

of

a

coord

inate

grid.

Go

al

4.

Use o

rdere

d

pairs o

f num

bers

to

nam

e,

locate

, and

plo

t poin

ts in a

ll fo

ur

quadra

nts

of

a

coord

inate

grid.

Go

al

3.

Use o

rdere

d

pairs o

f num

bers

to

nam

e,

locate

, and

plo

t poin

ts in a

ll fo

ur

quadra

nts

of

a

coord

inate

grid.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: G

EO

METR

Y

Pro

gram

Goa

l: I

nve

stig

ate

Char

acte

rist

ics

and

Pro

per

ties

of

Two-

an

d T

hre

e-D

imen

sion

al

Geo

met

ric

Shap

esCo

nten

tKi

nder

gart

enFi

rst

Grad

eSe

cond

Gra

deTh

ird

Grad

eFo

urth

Gra

deFi

fth

Grad

eSi

xth

Grad

e

Line

s an

d an

gles

Go

al

1.

Dra

w lin

e

segm

ents

and

identify

para

llel lin

e

segm

ents

.

Go

al

1.

Identify

and d

raw

poin

ts,

inte

rsecting a

nd

para

llel lin

e s

egm

ents

and lin

es,

rays a

nd

right

angle

s.

Go

al

1.

Identify

, dra

w,

and d

escribe

poin

ts,

inte

rsecting

and p

ara

llel lin

e

segm

ents

and lin

es,

rays,

and r

ight,

acute

, and o

btu

se

angle

s.

Go

al

1.

Identify

, describe,

com

pare

, nam

e,

and d

raw

right, a

cute

, obtu

se,

str

aig

ht, a

nd r

eflex

angle

s;

dete

rmin

e

angle

measure

s

in v

ert

ical and

supple

menta

ry

angle

s a

nd b

y

apply

ing p

ropert

ies

of

sum

s o

f angle

m

easure

s in t

riangle

s

and q

uadra

ngle

s.

Go

al

1.

Identify

, describe,

cla

ssify,

nam

e,

and d

raw

angle

s;

dete

rmin

e

angle

measure

s b

y

apply

ing p

ropert

ies

of

orienta

tions o

f angle

s a

nd o

f sum

s

of

angle

measure

s

in t

riangle

s a

nd

quadra

ngle

s.

Pla

ne a

nd s

olid

fi

gure

sG

oal

1.

Identify

and

describe p

lane a

nd

solid

fig

ure

s inclu

din

g

circle

s,

tria

ngle

s,

square

s,

recta

ngle

s,

sphere

s,

and c

ubes.

Go

al

1.

Identify

and

describe p

lane a

nd

solid

fig

ure

s inclu

din

g

circle

s,

tria

ngle

s,

square

s,

recta

ngle

s,

sphere

s,

cylin

ders

, re

cta

ngula

r prism

s,

pyra

mid

s,

cones,

and

cubes.

Go

al

2.

Identify

, describe,

and m

odel

pla

ne a

nd s

olid

fig

ure

s inclu

din

g

circle

s, tr

iangle

s,

square

s, re

cta

ngle

s,

hexagons, tr

apezoid

s,

rhom

buses, sphere

s,

cylin

ders

, re

cta

ngula

r prism

s, pyra

mid

s,

cones, and c

ubes.

Go

al

2.

Identif

y,

describe, m

odel,

and

com

pare

pla

ne a

nd

solid

fig

ure

s inclu

din

g

circle

s, poly

gons,

sphere

s, cylin

ders

, re

cta

ngula

r prism

s,

pyra

mid

s, cones,

and c

ubes u

sin

g

appro

priate

geom

etr

ic

term

s inclu

din

g the

term

s face, edge,

vert

ex, and b

ase.

Go

al

2.

Describe,

com

pare

, and c

lassify

pla

ne a

nd s

olid

fig

ure

s, in

cludin

g

poly

gons,

circl

es,

sp

here

s, c

ylin

ders

, re

ctangula

r prism

s,

cones,

cubes,

and

pyr

am

ids,

usi

ng

appro

priate

geom

etr

ic

term

s in

cludin

g v

ert

ex,

base

, fa

ce, edge, and

congru

ent.

Go

al

2.

Describe,

com

pare

, and

cla

ssify p

lane

and s

olid

fig

ure

s

usin

g a

ppro

priate

geom

etr

ic t

erm

s;

identify

congru

ent

figure

s a

nd d

escribe

their p

ropert

ies.

Go

al

2.

Identify

and

describe s

imila

r and c

ongru

ent

figure

s a

nd d

escribe

their p

ropert

ies;

constr

uct

a f

igure

th

at

is c

ongru

ent

to

anoth

er

figure

usin

g

a c

om

pass a

nd

str

aig

hte

dge.

Pro

gram

Goa

l: A

pp

ly T

ran

sfor

mat

ion

s an

d S

ymm

etry

in

Geo

met

ric

Situ

atio

ns

Tran

sfor

mat

ions

an

d sy

mm

etry

Go

al

2.

Identify

shapes h

avin

g lin

e

sym

metr

y.

Go

al

2.

Identify

shapes h

avin

g lin

e

sym

metr

y;

com

ple

te

line-s

ym

metr

ic

shapes o

r desig

ns.

Go

al

3.

Cre

ate

and c

om

ple

te

two-d

imensio

nal

sym

metr

ic s

hapes o

r desig

ns.

Go

al

3.

Cre

ate

and c

om

ple

te

two-d

imensio

nal

sym

metr

ic s

hapes

or

desig

ns;

locate

m

ultip

le lin

es o

f sym

metr

y in a

tw

o-

dim

ensio

nal shape.

Go

al

3.

Identify

, describe,

and

sketc

h e

xam

ple

s

of

reflections;

identify

and

describe e

xam

ple

s

of

transla

tions a

nd

rota

tions.

Go

al

3.

Identify

, describe,

and

sketc

h e

xam

ple

s

of

reflections,

transla

tions,

and

rota

tions.

Go

al

3.

Identify

, describe,

and s

ketc

h

(inclu

din

g p

lottin

g

on t

he c

oord

inate

pla

ne)

insta

nces

of

reflections,

transla

tions,

and

rota

tions.


Copyright

© W

right

Gro

up/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: PA

TTER

NS,

FU

NCTIO

NS,

AN

D A

LGEB

RA

Pro

gram

Goa

l: U

nd

erst

and

Pat

tern

s an

d F

un

ctio

ns

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Pat

tern

s an

d fu

ncti

ons

Go

al

1.

Exte

nd,

describe,

and c

reate

vis

ual, r

hyth

mic

, and

movem

ent

pattern

s;

use r

ule

s,

whic

h w

ill

lead t

o f

unctions,

to

sort

, m

ake p

attern

s,

and p

lay “

What’s M

y

Rule

?”

and o

ther

gam

es.

Go

al

1.

Exte

nd,

describe,

and c

reate

num

eric,

vis

ual, a

nd

concre

te p

attern

s;

solv

e p

roble

ms

involv

ing f

unction

machin

es,

“What’s

My R

ule

?”

table

s,

and F

ram

es-a

nd-

Arr

ow

s d

iagra

ms.

Go

al

1.

Exte

nd,

describe,

and c

reate

num

eric,

vis

ual, a

nd

concre

te p

attern

s;

describe r

ule

s

for

pattern

s a

nd

use t

hem

to s

olv

e

pro

ble

ms;

use w

ord

s

and s

ym

bols

to

describe a

nd w

rite

ru

les f

or

functions

involv

ing a

dditio

n

and s

ubtr

action a

nd

use t

hose r

ule

s t

o

solv

e p

roble

ms.

Go

al

1.

Exte

nd,

describe,

and c

reate

num

eric p

attern

s;

describe r

ule

s

for

pattern

s a

nd

use t

hem

to s

olv

e

pro

ble

ms;

use w

ord

s

and s

ym

bols

to

describe a

nd w

rite

ru

les f

or

functions

involv

ing a

dditio

n,

subtr

action,

and

multip

lication a

nd

use t

hose r

ule

s t

o

solv

e p

roble

ms.

Go

al

1.

Exte

nd,

describe,

and c

reate

num

eric p

attern

s;

describe r

ule

s

for

pattern

s a

nd

use t

hem

to s

olv

e

pro

ble

ms;

use

word

s a

nd s

ym

bols

to

describe a

nd w

rite

ru

les f

or

functions

that

involv

e t

he f

our

basic

arith

metic

opera

tions a

nd

use t

hose r

ule

s t

o

solv

e p

roble

ms.

Go

al

1.

Exte

nd,

describe, and c

reate

num

eric p

attern

s;

describe r

ule

s

for

pattern

s a

nd

use them

to s

olv

e

pro

ble

ms; w

rite

ru

les for

functio

ns

involv

ing the four

basic

arith

metic

opera

tions; re

pre

sent

functio

ns u

sin

g w

ord

s,

sym

bols

, ta

ble

s, and

gra

phs a

nd u

se those

re

pre

senta

tions to

solv

e p

roble

ms.

Go

al

1.

Exte

nd,

describe,

and

cre

ate

num

eric

pattern

s;

describe

rule

s f

or

pattern

s

and u

se t

hem

to

solv

e p

roble

ms;

repre

sent

pattern

s

and r

ule

s u

sin

g

alg

ebra

ic n

ota

tion;

repre

sent

functions

usin

g w

ord

s,

alg

ebra

ic n

ota

tion,

table

s,

and g

raphs;

transla

te f

rom

one

repre

senta

tion t

o

anoth

er

and u

se

repre

senta

tions

to s

olv

e p

roble

ms

involv

ing f

unctions.

Pro

gram

Goa

l: U

se A

lgeb

raic

Not

atio

n t

o R

epre

sen

t an

d A

nal

yze

Situ

atio

ns

and

Str

uct

ure

sA

lgeb

raic

no

tati

on a

nd

solv

ing

num

ber

sent

ence

s

Go

al

2.

Read a

nd

w

rite

expre

ssio

ns a

nd

num

ber

sente

nces

usin

g the s

ym

bols

1

, 2

, and 5

.

Go

al

2.

Read,

write

, and e

xpla

in

expre

ssio

ns a

nd

num

ber

sente

nces

usin

g t

he s

ym

bols

1

, 2

, and 5

and

the s

ym

bols

. a

nd

, w

ith c

ues;

solv

e

equations involv

ing

additio

n a

nd

subtr

action.

Go

al

2.

Read,

write

, and e

xpla

in

expre

ssio

ns a

nd

num

ber

sente

nces

usin

g t

he s

ym

bols

1

, 2

, 5

, .

, and

,;

solv

e n

um

ber

sente

nces involv

ing

additio

n a

nd

subtr

action;

write

expre

ssio

ns a

nd

num

ber

sente

nces

to m

odel num

ber

sto

ries.

Go

al

2.

Read,

write

, and e

xpla

in n

um

ber

sente

nces u

sin

g t

he

sym

bols

1,

2,

3,

4,

5,

.,

and ,

; solv

e

num

ber

sente

nces;

write

expre

ssio

ns

and n

um

ber

sente

nces t

o m

odel

num

ber

sto

ries.

Go

al

2.

Use

conventional

nota

tion t

o w

rite

expre

ssio

ns a

nd

num

ber

sente

nces

usin

g t

he f

our

basic

arith

metic o

pera

tions;

dete

rmin

e w

heth

er

num

ber

sente

nces

are

tru

e o

r fa

lse;

solv

e o

pen

sente

nces a

nd

expla

in t

he s

olu

tions;

write

expre

ssio

ns

and n

um

ber

sente

nces t

o m

odel

num

ber

sto

ries.

Go

al

2.

Dete

rmin

e

wheth

er

num

ber

sente

nces a

re t

rue

or

fals

e;

solv

e o

pen

num

ber

sente

nces

and e

xpla

in t

he

solu

tions;

use

a letter

variable

to

write

an o

pen

sente

nce t

o m

odel

a n

um

ber

sto

ry;

use a

pan-b

ala

nce

model to

solv

e lin

ear

equations in o

ne

unknow

n.

Go

al

2.

Dete

rmin

e

wheth

er

equalit

ies

and inequalit

ies

are

tru

e o

r fa

lse;

solv

e o

pen n

um

ber

sente

nces a

nd

expla

in t

he s

olu

tions;

use a

pan-b

ala

nce

model to

solv

e lin

ear

equations in o

ne o

r tw

o u

nknow

ns;

use

tria

l-and-e

rror

and

equiv

ale

nt

equations

str

ate

gie

s t

o s

olv

e

linear

equations in

one u

nknow

n.


Copyrig

ht ©

Wrig

ht G

roup/M

cG

raw

-Hill


Every

day M

ath

em

ati

cs®

Co

nte

nt

Stra

nd

: PA

TTER

NS,

FU

NCTIO

NS,

AN

D A

LGEB

RA

co

nt.

Pro

gram

Goa

l: U

se A

lgeb

raic

Not

atio

n t

o R

epre

sen

t an

d A

nal

yze

Situ

atio

ns

and

Str

uct

ure

s co

nt.

Cont

ent

Kind

erga

rten

Firs

t Gr

ade

Seco

nd G

rade

Thir

d Gr

ade

Four

th G

rade

Fift

h Gr

ade

Sixt

h Gr

ade

Ord

er o

f op

erat

ions

Go

al

3.

Recogniz

e

that

num

eric

expre

ssio

ns c

an

have d

iffe

rent

valu

es d

ependin

g

on t

he o

rder

in

whic

h o

pera

tions

are

carr

ied o

ut;

unders

tand t

hat

gro

upin

g s

ym

bols

can b

e u

sed t

o

affect

the o

rder

in

whic

h o

pera

tions a

re

carr

ied o

ut.

Go

al

3.

Evalu

ate

num

eric e

xpre

ssio

ns

conta

inin

g g

roupin

g

sym

bols

; in

sert

gro

upin

g s

ym

bols

to

make n

um

ber

sente

nces t

rue.

Go

al

3.

Evalu

ate

num

eric e

xpre

ssio

ns

conta

inin

g g

roupin

g

sym

bols

and n

este

d

gro

upin

g s

ym

bols

; in

sert

gro

upin

g

sym

bols

and n

este

d

gro

upin

g s

ym

bols

to

make n

um

ber

sente

nces t

rue;

describe a

nd u

se

the p

recedence o

f m

ultip

lication a

nd

div

isio

n o

ver

additio

n

and s

ubtr

action.

Go

al

3.

Describe

and a

pply

the

conventional ord

er

of

opera

tions.

Pro

pert

ies

of

the

arit

hmet

ic

oper

atio

ns

Go

al

3.

Apply

th

e C

om

muta

tive

Pro

pert

y o

f A

dditio

n

and t

he A

dditiv

e

Identity

to b

asic

additio

n f

act

pro

ble

ms.

Go

al

3.

Describe

the C

om

muta

tive

and A

ssocia

tive

Pro

pert

ies o

f A

dditio

n

and a

pply

them

to

menta

l arith

metic

pro

ble

ms.

Go

al

4.

Describe

and a

pply

the

Com

muta

tive

and A

ssocia

tive

Pro

pert

ies o

f A

dditio

n,

the

Com

muta

tive

Pro

pert

y o

f M

ultip

lication,

and

the M

ultip

licative

Identity

.

Go

al

4.

Apply

the

Dis

trib

utive P

ropert

y

of

Multip

lication

over

Additio

n t

o

the p

art

ial-pro

ducts

m

ultip

lication

alg

orith

m.

Go

al

4.

Describe a

nd

apply

pro

pert

ies o

f arith

metic.

Go

al

4.

Describe

and a

pply

pro

pert

ies

of

arith

metic a

nd

multip

licative a

nd

additiv

e invers

es.


Correlation Overview 25

®

34Curriculum

Focal Points and Everyday Mathem

atics

NCTM Curriculum Focal Points andEveryday Mathematics, Grade 1

NCTM Curriculum Focal Point for Grade 1 Everyday MathematicsGrade 1 Teacher’s

Lesson Guide Pages

Everyday MathematicsGrade 1 Grade-Level Goals

Number and Operations and Algebra: Developing understandings of addition and subtraction and strategies for basic addition facts and

related subtraction facts

Children develop strategies for adding and subtracting wholenumbers on the basis of their earlier work with small numbers.

254, 330–334, 335–339, 402–408,409–413, 574–579, 676–681,698–702, 747–752, 758–762,816–821, 827–831

Operations and Computation Goal 1Demonstrate proficiency with +/- 0,+/- 1, doubles, and sum-equals-tenaddition and subtraction facts such as 6 + 4 = 10 and 10 - 7 = 3.

Operations and Computation Goal 3Estimate reasonableness of answers to basic fact problems (e.g., Will 7 + 8 bemore or less than 10?).

Operations and Computation Goal 4Identify change-to-more, change-to-less, comparison, and parts-and-totalsituations.

Patterns, Functions, and Algebra Goal 3Apply the Commutative Property of Addition and the Additive Identity to basicaddition fact problems.

Children use a variety of models, including discrete objects, length-based models (e.g., lengths of connecting cubes), and numberlines, to model “part-whole,” “adding to,” “taking away from,” and“comparing” situations to develop an understanding of the meaningsof addition and subtraction and strategies to solve such arithmeticproblems.

20–24, 33–36, 71–75, 95–98, 105,137, 140–144, 145–149, 150–155,156–161, 204–208, 209–213,219–223, 224–228, 251–255,330–334, 335–339, 388–392,393–397, 398–401, 536–541,548–552, 559–562, 698–702,811–815, 816–821, 872–875

Number and Numeration Goal 6Use manipulatives, drawings, tally marks, and numerical expressions involvingaddition and subtraction of 1- or 2-digit numbers to give equivalent names forwhole numbers up to 100.

Operations and Computation Goal 2Use manipulatives, number grids, tally marks, mental arithmetic, and calculatorsto solve problems involving the addition and subtraction of 1-digit wholenumbers with 1- or 2-digit whole numbers; calculate and compare the values ofcombinations and coins.

EM3_FPB_Corr_0034-0038.indd

34

9/15/08

10:56:45AM

The grade-level correlation charts for Pre-kindergarten through grade 6 on the following pages show how different parts of the Focal Points and Connections to the Focal Points align to the Everday Mathematics teacher’s guides as well as to Everyday Mathematics Grade-Level Goals.

NCTM Curriculum Focal Points

Content Strands are color coded

to match Focal Points strands

with Everyday Mathematics

Grade-Level Goals.

The Curriculum Focal Points and

Connections to the Focal Points

chart is divided into portions to

describe a distinct set of skills to

help identify the mathematics.

Page numbers

in the teacher’s

guides are

correlated to the

distinct portions

of a Focal Point

and Connection

to a Focal Point.

Grade-Level Goals are also

correlated to the distinct

portions of a Focal Point and

Connection to a Focal Point.

EM3_FPB_Correclation Overview_pg25 25EM3_FPB_Correclation Overview_pg25 25 9/25/08 3:07:04 PM9/25/08 3:07:04 PM


NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, P

re-K

inde

rgar

ten

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Prek

inde

rgar

ten

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en

Teac

her’

s Gu

ide

to

Acti

viti

es P

ages

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en G

rade

-Lev

el G

oals

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

wh

ole

nu

mb

ers

, in

clu

din

g c

on

ce

pts

of

co

rre

sp

on

de

nc

e,

co

un

tin

g,

ca

rdin

ality

, a

nd

co

mp

ari

so

n

Chi

ldre

n de

velo

p an

und

erst

andi

ng o

f th

e m

eani

ngs

of w

hole

num

bers

an

d re

cogn

ize

the

num

ber

of o

bjec

ts in

sm

all g

roup

s w

ithou

t co

untin

g an

d by

cou

ntin

g—th

e fir

st a

nd m

ost

basi

c m

athe

mat

ical

alg

orith

m.

26–

27, 2

8–

29, 3

8–

39, 4

0–

41,

42–

43, 4

4–

45, 4

6–

47, 4

8–

49,

50–

51,

52–

53, 5

4–

55, 5

6–

57,

58–

59, 6

0–

61, 6

2–

63, 7

2–

73,

74–

75, 7

6–

77, 8

0–

81, 8

2–

83,

84–

85, 9

4–

95, 9

6–

97, 9

8–

99,

100–

101, 1

02–

103, 1

04–

105,

106–

107, 1

08–

109, 1

00–

111,

112–

113, 1

14–

115, 1

16–

117,

118–

119, 1

20–

121, 1

22–

123,

124–

125, 1

26–

127, 1

28–

129,

130–

131, 1

40–

141, 1

42–

143,

144–

145, 1

46–

147, 1

48–

149,

150–

151, 1

52–

153, 1

54–

155,

156–

157, 1

58–

159, 1

60–

161,

162–

163

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt o

bjec

ts w

ith o

ne-t

o-on

e co

rres

pond

ence

and

kno

w t

he la

st c

ount

ing

wor

d te

lls “

how

man

y.”

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Dev

elop

an

awar

enes

s of

num

bers

and

the

ir u

ses;

ass

ocia

tes

num

ber

nam

es,

quan

titie

s, a

nd w

ritt

en n

umer

als;

rec

ogni

ze a

nd u

se d

iffer

ent

way

s to

rep

rese

nt

num

bers

(fo

r ex

ampl

e, g

roup

s of

obj

ects

or

dots

).

Chi

ldre

n un

ders

tand

tha

t nu

mbe

r w

ords

ref

er t

o qu

antit

y.26–

27, 2

8–

29, 4

2–

43, 4

6–

47,

48–

49, 5

2–

53, 5

6–

57, 6

6–

67,

68–

69, 7

0–

71, 7

2–

73, 7

4–

75,

78–

79, 9

8–

99, 1

02–

103, 1

04–

105,

106–

107, 1

08–

109, 1

10–

111,

112–

113, 1

14–

115, 1

16–

117,

122–

123, 1

28–

129, 1

40–

141,

142–

143, 1

46–

147, 1

48–

149,

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Dev

elop

an

awar

enes

s of

num

bers

and

the

ir u

ses;

ass

ocia

tes

num

ber

nam

es,

quan

titie

s, a

nd w

ritt

en n

umer

als;

rec

ogni

ze a

nd u

se d

iffer

ent

way

s to

rep

rese

nt

num

bers

(fo

r ex

ampl

e, g

roup

s of

obj

ects

or

dots

).

EM3_FPB_Corr_0026-0029.indd 26EM3_FPB_Corr_0026-0029.indd 26 9/24/08 4:48:13 PM9/24/08 4:48:13 PM

Pre-Kindergarten Correlation 27

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Prek

inde

rgar

ten

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en

Teac

her’

s Gu

ide

to

Acti

viti

es P

ages

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en G

rade

-Lev

el G

oals

Chi

ldre

n un

ders

tand

tha

t nu

mbe

r w

ords

ref

er t

o qu

antit

y.150–

151, 1

52–

153, 1

54–

155,

156–

157, 1

58–

159, 1

60–

161,

172–

173, 1

74–

175, 1

76–

177,

178–

179, 1

80–

181, 1

84–

185,

186–

187, 1

88–

189, 1

90–

191,

224–

225, 2

32–

233, 2

34–

235,

262–

263

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Dev

elop

an

awar

enes

s of

num

bers

and

the

ir u

ses;

ass

ocia

tes

num

ber

nam

es,

quan

titie

s, a

nd w

ritt

en n

umer

als;

rec

ogni

ze a

nd u

se d

iffer

ent

way

s to

rep

rese

nt

num

bers

(fo

r ex

ampl

e, g

roup

s of

obj

ects

or

dots

).

Chi

ldre

n us

e on

e-to

-one

cor

resp

onde

nce

to s

olve

pro

blem

s by

m

atch

ing

sets

and

com

pari

ng n

umbe

r am

ount

s an

d in

cou

ntin

g ob

ject

s to

10 a

nd b

eyon

d.

26–

27, 2

8–

29, 3

8–

39, 4

2–

43,

44–

45, 4

6–

47, 4

8–

49, 5

2–

53,

54–

55, 5

8–

59, 6

0–

61, 6

2–

63,

64–

65, 6

6–

67, 6

8–

69, 7

0–

71,

72–

73, 7

4–

75, 7

6–

77, 7

8–

79,

80–

81, 8

2–

83, 8

4–

85, 9

8–

99,

102–

103, 1

04–

105, 1

06–

107,

108–

109, 1

10–

111, 1

12–

113,

114–

115, 1

16–

117, 1

18–

119,

122–

123, 1

28–

129, 1

40–

141,

144–

145, 1

46–

147, 1

48–

149,

150–

151, 1

52–

153, 1

54–

155,

156–

157, 1

58–

159, 1

60–

161,

162–

163, 1

72–

173, 1

74–

175,

176–

177, 1

78–

179, 1

80–

181,

182–

183, 1

84–

185, 1

86–

187,

188–

189, 1

90–

191, 2

24–

225,

232–

233, 2

34–

235, 2

62–

263

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt o

bjec

ts w

ith o

ne-t

o-on

e co

rres

pond

ence

and

kno

w t

he la

st c

ount

ing

wor

d te

lls “

how

man

y.”

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Com

pare

and

ord

er g

roup

s of

obj

ects

usi

ng w

ords

suc

h as

mor

e, fe

wer

, les

s,

sam

e.

Chi

ldre

n un

ders

tand

tha

t th

e la

st w

ord

that

the

y st

ate

in c

ount

ing

tells

“h

ow m

any,

” th

ey c

ount

to

dete

rmin

e nu

mbe

r am

ount

s an

d co

mpa

re

quan

titie

s (u

sing

lang

uage

suc

h as

“m

ore

than

” an

d “l

ess

than

”), a

nd

they

ord

er s

ets

by t

he n

umbe

r of

obj

ects

in t

hem

.

26–

27, 2

8–

29, 4

6–

47, 4

8–

49,

52–

53, 5

4–

55, 6

6–

67, 7

0–

71,

72–

73, 7

4–

75, 7

6–

77, 7

8–

79,

84–

85, 9

8–

99, 1

02–

103, 1

04–

105,

106–

107, 1

08–

109, 1

10–

111,

112–

113, 1

14–

115, 1

16–

117,

118–

119, 1

22–

123, 1

40–

141,

144–

145, 1

46–

147, 1

48–

149,

150–

151, 1

52–

153, 1

54–

155,

156–

157, 1

58–

159, 1

60–

161,

162–

163, 1

72–

173, 1

74–

175,

176–

177, 1

78–

179, 1

80–

181,

182–

183, 1

84–

185, 1

86–

187,

188–

189, 1

90–

191, 2

32–

233,

234–

235, 2

62–

263

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt o

bjec

ts w

ith o

ne-t

o-on

e co

rres

pond

ence

and

kno

w t

he la

st c

ount

ing

wor

d te

lls “

how

man

y.”

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Com

pare

and

ord

er g

roup

s of

obj

ects

usi

ng w

ords

suc

h as

mor

e, fe

wer

, les

s,

sam

e.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Prek

inde

rgar

ten

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en

Teac

her’

s Gu

ide

to

Acti

viti

es P

ages

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en G

rade

-Lev

el G

oals

Geom

etry

: Id

en

tify

ing

sh

ap

es a

nd

de

sc

rib

ing

sp

ati

al

rela

tio

nsh

ips

Chi

ldre

n de

velo

p sp

atia

l rea

soni

ng b

y w

orki

ng f

rom

tw

o pe

rspe

ctiv

es

on s

pace

as

they

exa

min

e th

e sh

apes

of

obje

cts

and

insp

ect

thei

r re

lativ

e po

sitio

ns.

244–

245, 2

50–

251, 2

56–

257,

260–

261, 2

64–

265, 2

76–

277,

280–

281, 2

86–

287, 3

16–

317,

324–

325

Geo

metr

y G

oa

l 2

Rec

ogni

ze a

nd d

escr

ibe

the

posi

tion

and

loca

tion

of o

bjec

ts;

use

spat

ial

reas

onin

g in

con

cret

e ta

sks,

suc

h as

put

ting

toge

ther

puz

zles

and

cre

atin

g co

llage

s an

d bl

ock

stru

ctur

es.

Chi

ldre

n fin

d sh

apes

in t

heir

env

iron

men

ts a

nd d

escr

ibe

them

in t

heir

ow

n w

ords

.244–

245, 2

50–

251, 2

60–

261

Geo

metr

y G

oa

l 1

Rec

ogni

ze a

nd d

escr

ibe

basi

c 2-d

imen

sion

al g

eom

etri

c sh

apes

; ex

plor

e th

e re

latio

nshi

ps b

etw

een

basi

c 2-d

imen

sion

al a

nd 3

–di

men

sion

al s

hape

s.

Chi

ldre

n bu

ild p

ictu

res

and

desi

gns

by c

ombi

ning

tw

o- a

nd t

hree

di

men

sion

al s

hape

s, a

nd t

hey

solv

e su

ch p

robl

ems

as d

ecid

ing

whi

ch

piec

e w

ill f

it in

to a

spa

ce in

a p

uzzl

e.

68–

69, 2

48–

249, 2

54–

255,

262–

263, 2

78–

279, 2

84–

285,

308–

309, 3

18–

319

Geo

metr

y G

oa

l 1

Rec

ogni

ze a

nd d

escr

ibe

basi

c 2-d

imen

sion

al g

eom

etri

c sh

apes

; ex

plor

e th

e re

latio

nshi

ps b

etw

een

basi

c 2-d

imen

sion

al a

nd 3

-dim

ensi

onal

sha

pes.

Geo

metr

y G

oa

l 2

Rec

ogni

ze a

nd d

escr

ibe

the

posi

tion

and

loca

tion

of o

bjec

ts;

use

spat

ial

reas

onin

g in

con

cret

e ta

sks,

suc

h as

put

ting

toge

ther

puz

zles

and

cre

atin

g co

llage

s an

d bl

ock

stru

ctur

es.

Chi

ldre

n di

scus

s th

e re

lativ

e po

sitio

ns o

f ob

ject

s w

ith v

ocab

ular

y su

ch

as “

abov

e,”

“bel

ow,”

and

“ne

xt t

o.”

252–

253, 2

74–

275, 2

76–

277,

280–

281, 2

82–

283, 2

86–

287,

316–

317, 3

24–

325

Geo

metr

y G

oa

l 2

Rec

ogni

ze a

nd d

escr

ibe

the

posi

tion

and

loca

tion

of o

bjec

ts;

use

spat

ial

reas

onin

g in

con

cret

e ta

sks,

suc

h as

put

ting

toge

ther

puz

zles

and

cre

atin

g co

llage

s an

d bl

ock

stru

ctur

es.

Mea

sure

men

t: I

de

nti

fyin

g m

ea

su

rab

le a

ttri

bu

tes a

nd

co

mp

ari

ng

ob

jec

ts b

y u

sin

g t

he

se

att

rib

ute

s

Chi

ldre

n id

entif

y ob

ject

s as

“th

e sa

me”

or

“diff

eren

t,” a

nd t

hen

“mor

e” o

r “l

ess,

” on

the

bas

is o

f at

trib

utes

tha

t th

ey c

an m

easu

re.

They

iden

tify

mea

sura

ble

attr

ibut

es s

uch

as le

ngth

and

wei

ght

and

solv

e pr

oble

ms

by m

akin

g di

rect

com

pari

sons

of

obje

cts

on t

he b

asis

of

tho

se a

ttri

bute

s.

72–

73, 8

4–

85, 2

00–

201, 2

02–

203,

204–

205, 2

06–

207, 2

08–

209,

210–

211, 2

12–

213, 2

18–

219,

220–

221, 2

22–

223, 2

24–

225,

228–

229, 2

30–

231, 2

32–

233,

234–

235, 2

96–

297

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Dis

tingu

ish

and

desc

ribe

siz

e at

trib

utes

, inc

ludi

ng le

ngth

, wei

ght,

and

capa

city

or

vol

ume;

com

pare

obj

ects

acc

ordi

ng t

o va

riou

s si

ze a

ttri

bute

s.


Pre-Kindergarten Correlation 29

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or P

reki

nder

gart

en

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en

Teac

her’

s Gu

ide

to

Acti

viti

es P

ages

Ever

yday

Mat

hem

atic

s Pr

e-Ki

nder

gart

en G

rade

-Lev

el G

oals

Data

Ana

lysi

sC

hild

ren

lear

n th

e fo

unda

tions

of

data

ana

lysi

s by

usi

ng o

bjec

ts’

attr

ibut

es t

hat

they

hav

e id

entif

ied

in r

elat

ion

to g

eom

etry

and

m

easu

rem

ent

(e.g

., si

ze, q

uant

ity, o

rien

tatio

n, n

umbe

r of

sid

es o

r ve

rtic

es, c

olor

) fo

r va

riou

s pu

rpos

es, s

uch

as d

escr

ibin

g, s

ortin

g, o

r co

mpa

ring

. For

exa

mpl

e, c

hild

ren

sort

geo

met

ric

figur

es b

y sh

ape,

co

mpa

re o

bjec

ts b

y w

eigh

t (“

heav

ier,”

“lig

hter

”), o

r de

scri

be s

ets

of

obje

cts

by t

he n

umbe

r of

obj

ects

in e

ach

set.

72–

73, 8

4–

85, 2

00–

201, 2

02–

203,

204–

205, 2

06–

207, 2

08–

209,

210–

211, 2

12–

213, 2

18–

219,

220–

221, 2

28–

229, 2

30–

231,

232–

233, 2

34–

235, 2

44–

245,

246–

247, 2

48–

249, 2

56–

257,

258–

259, 2

60–

261, 2

96–

297,

300–

301, 3

04–

305, 3

06–

307,

308–

309, 3

10–

311, 3

20–

321,

322–

323, 3

26–

327

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Rec

ogni

ze a

nd m

atch

att

ribu

tes

of o

bjec

ts, s

uch

as s

ize,

sha

pe, a

nd c

olor

; us

e ru

les

to s

ort

obje

cts;

use

rul

es t

o cr

eate

and

ext

end

repe

atin

g pa

tter

ns.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Dis

tingu

ish

and

desc

ribe

siz

e at

trib

utes

, inc

ludi

ng le

ngth

, wei

ght,

and

capa

city

or

vol

ume;

com

pare

obj

ects

acc

ordi

ng t

o va

riou

s si

ze a

ttri

bute

s.

Geo

metr

y G

oa

l 1

Rec

ogni

ze a

nd d

escr

ibe

basi

c 2-d

imen

sion

al g

eom

etri

c sh

apes

; ex

plor

e th

e re

latio

nshi

ps b

etw

een

basi

c 2-d

imen

sion

al a

nd 3

-dim

ensi

onal

sha

pes.

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Dev

elop

an

awar

enes

s of

num

bers

and

the

ir u

ses;

ass

ocia

tes

num

ber

nam

es,

quan

titie

s, a

nd w

ritt

en n

umer

als;

rec

ogni

ze a

nd u

se d

iffer

ent

way

s to

rep

rese

nt

num

bers

(fo

r ex

ampl

e, g

roup

s of

obj

ects

or

dots

).

Num

ber

and

Oper

atio

nsC

hild

ren

use

mea

ning

s of

num

bers

to

crea

te s

trat

egie

s fo

r so

lvin

g pr

oble

ms

and

resp

ondi

ng t

o pr

actic

al s

ituat

ions

, suc

h as

get

ting

just

en

ough

nap

kins

for

a g

roup

, or

mat

hem

atic

al s

ituat

ions

, suc

h as

de

term

inin

g th

at a

ny s

hape

is a

tri

angl

e if

it ha

s ex

actly

thr

ee s

trai

ght

side

s an

d is

clo

sed.

94–

95, 1

20–

121, 1

44–

145,

146–

147, 1

48–

149, 1

50–

151,

154–

155, 1

56–

157, 1

58–

159,

162–

163

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Sol

ve a

nd c

reat

e nu

mbe

r st

orie

s us

ing

conc

rete

mod

elin

g; e

xplo

re p

art-

who

le

rela

tions

hips

(fo

r ex

ampl

e, 5

is m

ade

of 2

and

3).

Alge

bra

Chi

ldre

n re

cogn

ize

and

dupl

icat

e si

mpl

e se

quen

tial p

atte

rns

(e.g

., sq

uare

, cir

cle,

squ

are,

cir

cle,

squ

are,

cir

cle,

. . .

).

298–

299, 3

02–

303, 3

08–

309,

312–

313, 3

14–

315, 3

16–

317,

318–

319, 3

24–

325, 3

28–

329,

330–

331

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Rec

ogni

ze a

nd m

atch

att

ribu

tes

of o

bjec

ts, s

uch

as s

ize,

sha

pe, a

nd c

olor

; us

e ru

les

to s

ort

obje

cts;

use

rul

es t

o cr

eate

and

ext

end

repe

atin

g pa

tter

ns.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, K

inde

rgar

ten

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Kind

erga

rten

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en T

each

er’s

Gu

ide

to A

ctiv

ities

Pag

es

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en G

rade

-Lev

el G

oals

Num

ber

and

Oper

atio

ns: R

ep

rese

nti

ng

, c

om

pa

rin

g,

an

d o

rde

rin

g w

ho

le n

um

be

rs a

nd

jo

inin

g a

nd

se

pa

rati

ng

se

ts.

Chi

ldre

n us

e nu

mbe

rs, i

nclu

ding

wri

tten

num

eral

s, t

o re

pres

ent

quan

titie

s an

d to

sol

ve q

uant

itativ

e pr

oble

ms,

suc

h as

cou

ntin

g ob

ject

s in

a s

et, c

reat

ing

a se

t w

ith a

giv

en n

umbe

r of

obj

ects

, co

mpa

ring

and

ord

erin

g se

ts o

r nu

mer

als

by u

sing

bot

h ca

rdin

al

and

ordi

nal m

eani

ngs,

and

mod

elin

g si

mpl

e jo

inin

g an

d se

para

ting

situ

atio

ns w

ith o

bjec

ts.

8–

11, 1

2–

15, 1

8–

21, 2

4–

27,

28–

31, 3

2–

35, 5

0–

51, 5

2–

53,

54–

55, 6

8–

69, 7

2–

73, 9

4–

95,

98–

99, 1

04–

105, 1

06–

107,

108–

109, 1

10–

111, 1

12–

113,

114–

115, 1

16–

119, 1

24–

127,

138–

139, 1

42–

143, 1

48–

149,

150–

151, 1

56–

157, 1

58–

159,

166–

167, 1

70–

171, 1

72–

173,

188–

189, 1

90–

191, 1

94–

195,

198–

199, 2

00–

203, 2

04–

205,

210–

211, 2

12–

213, 2

18–

219,

220–

221, 2

22–

225, 2

36–

237,

242–

243, 2

44–

245, 2

50–

251,

252–

253, 2

54–

255, 2

68–

271,

282–

283, 2

84–

285, 2

90–

291,

296–

297, 2

98–

299, 3

00–

301,

302–

303, 3

14–

315

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s

to 1

00;

coun

t on

by

2s,

5s,

and

10s

and

coun

t ba

ck b

y 1s

with

nu

mbe

r gr

ids,

num

ber

lines

, and

cal

cula

tors

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt 2

0 o

r m

ore

obje

cts;

est

imat

e th

e nu

mbe

r of

obj

ects

in a

col

lect

ion.

Nu

mb

er

an

d

Nu

mera

tion

Goa

l 3

Mod

el n

umbe

rs w

ith m

anip

ulat

ives

; us

e m

anip

ulat

ives

to

exch

ange

1s

for

10s

and

10s

for

100s;

rec

ogni

ze t

hat

digi

ts c

an b

e us

ed a

nd c

ombi

ned

to r

ead

and

wri

te n

umbe

rs;

read

num

bers

up

to 3

0.

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Use

man

ipul

ativ

es t

o m

odel

hal

f of

a r

egio

n or

a c

olle

ctio

n; d

escr

ibe

the

mod

el.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

man

ipul

ativ

es, d

raw

ings

, and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

of 1

-dig

it nu

mbe

rs t

o gi

ve e

quiv

alen

t na

mes

for

who

le n

umbe

rs u

p to

20.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Com

pare

and

ord

er w

hole

num

bers

up

to 2

0.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Use

man

ipul

ativ

es, n

umbe

r lin

es, a

nd m

enta

l ari

thm

etic

to

solv

e pr

oble

ms

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

sin

gle-

digi

t w

hole

num

bers

.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Iden

tify

join

and

tak

e-aw

ay s

ituat

ions

.


Kindergarten Correlation 31

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Kind

erga

rten

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en T

each

er’s

Gu

ide

to A

ctiv

ities

Pag

es

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en G

rade

-Lev

el G

oals

Chi

ldre

n ch

oose

, com

bine

, and

app

ly e

ffec

tive

stra

tegi

es f

or

answ

erin

g qu

antit

ativ

e qu

estio

ns, i

nclu

ding

qui

ckly

rec

ogni

zing

the

nu

mbe

r in

a s

mal

l set

, cou

ntin

g an

d pr

oduc

ing

sets

of

give

n si

zes,

co

untin

g th

e nu

mbe

r in

com

bine

d se

ts, a

nd c

ount

ing

back

war

d.

12–

15, 2

4–

27, 2

8–

31, 5

0–

51,

52–

53, 7

2–

73, 1

14–

115, 1

16–

119,

156–

157, 1

70–

171, 1

72–

173,

188–

189, 1

94–

195, 2

01–

203,

204–

205, 2

10–

211, 2

18–

219,

220–

221, 2

42–

243, 3

00–

301,

332–

335, 3

88–

389, 4

06–

407,

408–

409

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s

to 1

00;

coun

t on

by

2s,

5s,

and

10s

and

coun

t ba

ck b

y 1s

with

nu

mbe

r gr

ids,

num

ber

lines

, and

cal

cula

tors

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt 2

0 o

r m

ore

obje

cts;

est

imat

e th

e nu

mbe

r of

obj

ects

in a

col

lect

ion.

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Mod

el n

umbe

rs w

ith m

anip

ulat

ives

; us

e m

anip

ulat

ives

to

exch

ange

1s

for

10s

and

10s

for

100s;

rec

ogni

ze t

hat

digi

ts c

an b

e us

ed a

nd c

ombi

ned

to r

ead

and

wri

te n

umbe

rs;

read

num

bers

up

to 3

0.

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Use

man

ipul

ativ

es t

o m

odel

hal

f of

a r

egio

n or

a c

olle

ctio

n; d

escr

ibe

the

mod

el.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

man

ipul

ativ

es, d

raw

ings

, and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

of 1

-dig

it nu

mbe

rs t

o gi

ve e

quiv

alen

t na

mes

for

who

le n

umbe

rs u

p to

20.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Use

man

ipul

ativ

es, n

umbe

r lin

es, a

nd m

enta

l ari

thm

etic

to

solv

e pr

oble

ms

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

sin

gle-

digi

t w

hole

num

bers

.

Geom

etry

: De

sc

rib

ing

sh

ap

es a

nd

sp

ac

e

Chi

ldre

n in

terp

ret

the

phys

ical

wor

ld w

ith g

eom

etri

c id

eas

(e.g

., sh

ape,

ori

enta

tion,

spa

tial r

elat

ions

) an

d de

scri

be it

with

co

rres

pond

ing

voca

bula

ry.

92–

93, 1

20–

121, 1

22–

123, 1

27,

286–

289, 2

94–

295, 3

16–

319,

416–

419

Geo

metr

y G

oa

l 1

Iden

tify

and

desc

ribe

pla

ne a

nd s

olid

fig

ures

incl

udin

g ci

rcle

s, t

rian

gles

, sq

uare

s, r

ecta

ngle

s, s

pher

es, a

nd c

ubes

.

Geo

metr

y G

oa

l 2

Iden

tify

shap

es h

avin

g lin

e sy

mm

etry

.

Chi

ldre

n id

entif

y, n

ame,

and

des

crib

e a

vari

ety

of s

hape

s, s

uch

as s

quar

es, t

rian

gles

, cir

cles

, rec

tang

les,

(re

gula

r) h

exag

ons,

and

(is

osce

les)

tra

pezo

ids

pres

ente

d in

a v

arie

ty o

f w

ays

(e.g

., w

ith

diff

eren

t si

zes

or o

rien

tatio

ns),

as w

ell a

s su

ch t

hree

-dim

ensi

onal

sh

apes

as

sphe

res,

cub

es, a

nd c

ylin

ders

. The

y us

e ba

sic

shap

es

and

spat

ial r

easo

ning

to

mod

el o

bjec

ts in

the

ir e

nvir

onm

ent

and

to

cons

truc

t m

ore

com

plex

sha

pes.

48–

49, 8

8–

89, 9

0–

91, 1

96–

197,

206–

207, 2

08–

209, 2

14–

215,

240–

241, 2

86–

289, 2

94–

295,

340–

341

Geo

metr

y G

oa

l 1

Iden

tify

and

desc

ribe

pla

ne a

nd s

olid

fig

ures

incl

udin

g ci

rcle

s, t

rian

gles

, sq

uare

s, r

ecta

ngle

s, s

pher

es, a

nd c

ubes

.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Kind

erga

rten

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en T

each

er’s

Gu

ide

to A

ctiv

ities

Pag

es

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en G

rade

-Lev

el G

oals

Mea

sure

men

t: O

rde

rin

g o

bje

cts

by m

ea

su

rab

le a

ttri

bu

tes

Chi

ldre

n us

e m

easu

rabl

e at

trib

utes

, suc

h as

leng

th o

r w

eigh

t, to

sol

ve

prob

lem

s by

com

pari

ng a

nd o

rder

ing

obje

cts.

46–

47, 7

0–

71, 1

26–

127, 1

44–

147,

152–

153, 1

64–

165, 2

48–

249,

410–

413

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Use

non

stan

dard

too

ls a

nd t

echn

ique

s to

est

imat

e an

d co

mpa

re w

eigh

t an

d le

ngth

; id

entif

y st

anda

rd m

easu

ring

too

ls.

Chi

ldre

n co

mpa

re t

he le

ngth

s of

tw

o ob

ject

s bo

th d

irec

tly (

by

com

pari

ng t

hem

with

eac

h ot

her)

and

indi

rect

ly (

by c

ompa

ring

bot

h w

ith a

thi

rd o

bjec

t), a

nd t

hey

orde

r se

vera

l obj

ects

acc

ordi

ng t

o le

ngth

.

46–

47, 7

0–

71, 1

26–

127, 1

52–

155,

246–

247, 2

48–

249, 2

56–

257

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Use

non

stan

dard

too

ls a

nd t

echn

ique

s to

est

imat

e an

d co

mpa

re w

eigh

t an

d le

ngth

; id

entif

y st

anda

rd m

easu

ring

too

ls.

NCT

M C

onne

ctio

n to

the

Foc

al

Poin

ts f

or K

inde

rgar

ten

Ev

eryd

ay M

athe

mat

ics

Kind

erga

rten

Tea

cher

’s

Guid

e to

Act

iviti

es P

ages

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en G

rade

-Lev

el G

oals

Data

Ana

lysi

sC

hild

ren

sort

obj

ects

and

use

one

or

mor

e at

trib

utes

to

solv

e pr

oble

ms.

For

exa

mpl

e, t

hey

mig

ht s

ort

solid

s th

at r

oll e

asily

fro

m

thos

e th

at d

o no

t. O

r th

ey m

ight

col

lect

dat

a an

d us

e co

untin

g to

an

swer

suc

h qu

estio

ns a

s, “

Wha

t is

our

fav

orite

sna

ck?”

12–

15, 2

4–

27, 2

8–

31, 3

2–

35,

56–

57, 6

0–

61, 6

6–

67, 7

0–

71,

88–

89, 1

04–

105, 1

68–

169,

214–

215, 2

16–

217, 2

40–

241,

252–

253, 2

60–

261, 2

62–

263,

292–

293, 3

06–

307

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

visu

al, r

hyth

mic

, and

mov

emen

t pa

tter

ns;

use

rule

s, w

hich

will

lead

to

func

tions

, to

sort

, mak

e pa

tter

ns, a

nd p

lay

“W

hat’s

My

Rul

e?”

and

othe

r ga

mes

.

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta t

o cr

eate

cla

ss-c

onst

ruct

ed t

ally

cha

rts,

tab

les,

and

ba

r gr

aphs

.

Da

ta a

nd

Ch

an

ce G

oa

l 2

Use

gra

phs

to a

nsw

er s

impl

e qu

estio

ns.

Chi

ldre

n re

-sor

t ob

ject

s by

usi

ng n

ew a

ttri

bute

s (e

.g.,

afte

r so

rtin

g so

lids

acco

rdin

g to

whi

ch o

nes

roll,

the

y m

ight

re-

sort

the

sol

ids

acco

rdin

g to

whi

ch o

nes

stac

k ea

sily

).

56–

57, 2

14–

215, 2

16–

217

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

visu

al, r

hyth

mic

, and

mov

emen

t pa

tter

ns;

use

rule

s, w

hich

will

lead

to

func

tions

, to

sort

, mak

e pa

tter

ns, a

nd p

lay

“W

hat’s

My

Rul

e?”

and

othe

r ga

mes

.


Kindergarten Correlation 33

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or K

inde

rgar

ten

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en T

each

er’s

Gu

ide

to A

ctiv

ities

Pag

es

Ever

yday

Mat

hem

atic

s Ki

nder

gart

en G

rade

-Lev

el G

oals

Geom

etry

Chi

ldre

n in

tegr

ate

thei

r un

ders

tand

ings

of

geom

etry

, mea

sure

men

t, an

d nu

mbe

r. Fo

r ex

ampl

e, t

hey

unde

rsta

nd, d

iscu

ss, a

nd c

reat

e si

mpl

e na

viga

tiona

l dir

ectio

ns (

e.g.

, “W

alk

forw

ard

10 s

teps

, tur

n ri

ght,

and

wal

k fo

rwar

d 5 s

teps

”).

92

–9

3,

24

8–2

49

, 3

18

–3

19

, 4

16

–4

17

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Use

non

stan

dard

too

ls a

nd t

echn

ique

s to

est

imat

e an

d co

mpa

re w

eigh

t an

d le

ngth

; id

entif

y st

anda

rd m

easu

ring

too

ls.

Alge

bra

Chi

ldre

n id

entif

y, d

uplic

ate,

and

ext

end

sim

ple

num

ber

patt

erns

and

se

quen

tial a

nd g

row

ing

patt

erns

(e.

g., p

atte

rns

mad

e w

ith s

hape

s) a

s pr

epar

atio

n fo

r cr

eatin

g ru

les

that

des

crib

e re

latio

nshi

ps.

16–

17, 6

2–

63, 6

4–

65, 7

4–

77,

96–

97, 1

40–

141, 1

70–

171,

192–

193, 1

96–

197, 2

38–

239,

250–

251, 2

64–

265, 2

67–

268,

302–

303, 3

10–

311, 3

12–

313,

362–

363, 3

66–

369, 4

00–

401

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

visu

al, r

hyth

mic

, and

mov

emen

t pa

tter

ns;

use

rule

s, w

hich

will

lead

to

func

tions

, to

sort

, mak

e pa

tter

ns, a

nd p

lay

“Wha

t’s M

y R

ule?

” an

d ot

her

gam

es.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

1

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns a

nd A

lgeb

ra: D

eve

lop

ing

un

de

rsta

nd

ing

s o

f a

dd

itio

n a

nd

su

btr

ac

tio

n a

nd

str

ate

gie

s f

or

ba

sic

ad

dit

ion

fa

cts

an

d

rela

ted

su

btr

ac

tio

n f

ac

ts

Chi

ldre

n de

velo

p st

rate

gies

for

add

ing

and

subt

ract

ing

who

le

num

bers

on

the

basi

s of

the

ir e

arlie

r w

ork

with

sm

all n

umbe

rs.

254, 3

30–

334, 3

35–

339, 4

02–

408,

409–

413, 5

74–

579, 6

76–

681,

698–

702, 7

47–

752, 7

58–

762,

816–

821, 8

27–

831

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Dem

onst

rate

pro

ficie

ncy

with

+/-

0, +

/- 1

, dou

bles

, and

sum

-equ

als-

ten

addi

tion

and

subt

ract

ion

fact

s su

ch a

s 6 +

4 =

10 a

nd 1

0 -

7 =

3.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Estim

ate

reas

onab

lene

ss o

f an

swer

s to

bas

ic f

act

prob

lem

s (e

.g.,

Will

7 +

8 b

e m

ore

or le

ss t

han

10?)

.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Iden

tify

chan

ge-t

o-m

ore,

cha

nge-

to-l

ess,

com

pari

son,

and

par

ts-a

nd-t

otal

si

tuat

ions

.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 3

App

ly t

he C

omm

utat

ive

Pro

pert

y of

Add

ition

and

the

Add

itive

Iden

tity

to b

asic

ad

ditio

n fa

ct p

robl

ems.

Chi

ldre

n us

e a

vari

ety

of m

odel

s, in

clud

ing

disc

rete

obj

ects

, len

gth-

base

d m

odel

s (e

.g.,

leng

ths

of c

onne

ctin

g cu

bes)

, and

num

ber

lines

, to

mod

el “

part

-who

le,”

“ad

ding

to,

” “t

akin

g aw

ay f

rom

,” a

nd

“com

pari

ng”

situ

atio

ns t

o de

velo

p an

und

erst

andi

ng o

f th

e m

eani

ngs

of a

dditi

on a

nd s

ubtr

actio

n an

d st

rate

gies

to

solv

e su

ch a

rith

met

ic

prob

lem

s.

20–

24, 3

3–

36, 7

1–

75, 9

5–

98, 1

05,

137, 1

40–

144, 1

45–

149, 1

50–

155,

156–

161, 2

04–

208, 2

09–

213,

219–

223, 2

24–

228, 2

51–

255,

330–

334, 3

35–

339, 3

88–

392,

393–

397, 3

98–

401, 5

36–

541,

548–

552, 5

59–

562, 6

98–

702,

811–

815, 8

16–

821, 8

72–

875

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Use

man

ipul

ativ

es, d

raw

ings

, tal

ly m

arks

, and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

of 1

- or

2-d

igit

num

bers

to

give

equ

ival

ent

nam

es f

or

who

le n

umbe

rs u

p to

100.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, a

nd c

alcu

lato

rs

to s

olve

pro

blem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of 1

-dig

it w

hole

nu

mbe

rs w

ith 1

- or

2-d

igit

who

le n

umbe

rs;

calc

ulat

e an

d co

mpa

re t

he v

alue

s of

co

mbi

natio

ns o

f co

ins.


Grade 1 Correlation 35

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Chi

ldre

n us

e a

vari

ety

of m

odel

s, in

clud

ing

disc

rete

obj

ects

, len

gth-

base

d m

odel

s (e

.g.,

leng

ths

of c

onne

ctin

g cu

bes)

, and

num

ber

lines

, to

mod

el “

part

-who

le,”

“ad

ding

to,

” “t

akin

g aw

ay f

rom

,” a

nd

“com

pari

ng”

situ

atio

ns t

o de

velo

p an

und

erst

andi

ng o

f th

e m

eani

ngs

of a

dditi

on a

nd s

ubtr

actio

n an

d st

rate

gies

to

solv

e su

ch a

rith

met

ic

prob

lem

s.

20–

24, 3

3–

36, 7

1–

75, 9

5–

98, 1

05,

137, 1

40–

144, 1

45–

149, 1

50–

155,

156–

161, 2

04–

208, 2

09–

213,

219–

223, 2

24–

228, 2

51–

255,

330–

334, 3

35–

339, 3

88–

392,

393–

397, 3

98–

401, 5

36–

541,

548–

552, 5

59–

562, 6

98–

702,

811–

815, 8

16–

821, 8

72–

875

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Iden

tify

chan

ge-t

o-m

ore,

cha

nge-

to-l

ess,

com

pari

son,

and

par

ts-a

nd-t

otal

si

tuat

ions

.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Rea

d, w

rite

, and

exp

lain

exp

ress

ions

and

num

ber

sent

ence

s us

ing

the

sym

bols

+

, - ,

and

= a

nd t

he s

ymbo

ls >

and

< w

ith c

ues;

sol

ve e

quat

ions

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n.

Chi

ldre

n un

ders

tand

the

con

nect

ions

bet

wee

n co

untin

g an

d th

e op

erat

ions

of

addi

tion

and

subt

ract

ion

(e.g

., ad

ding

tw

o is

the

sam

e as

“co

untin

g on

” tw

o).

145–

149, 2

09–

213, 2

19–

223,

224–

228, 2

29–

234, 6

98–

702

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s,

2s,

5s,

and

10s

past

100 a

nd b

ack

by 1

s fr

om a

ny n

umbe

r le

ss

than

100 w

ith a

nd w

ithou

t nu

mbe

r gr

ids,

num

ber

lines

, and

cal

cula

tors

.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, a

nd c

alcu

lato

rs

to s

olve

pro

blem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of 1

-dig

it w

hole

nu

mbe

rs w

ith 1

- or

2-d

igit

who

le n

umbe

rs;

calc

ulat

e an

d co

mpa

re t

he v

alue

s of

co

mbi

natio

ns o

f co

ins.

Chi

ldre

n us

e pr

oper

ties

of a

dditi

on (

com

mut

ativ

ity a

nd a

ssoc

iativ

ity)

to a

dd w

hole

num

bers

, and

the

y cr

eate

and

use

incr

easi

ngly

so

phis

ticat

ed s

trat

egie

s ba

sed

on t

hese

pro

pert

ies

(e.g

., “m

akin

g te

ns”)

to

solv

e ad

ditio

n an

d su

btra

ctio

n pr

oble

ms

invo

lvin

g ba

sic

fact

s.

402–

408, 5

48–

552, 5

57, 5

59–

562

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 3

App

ly t

he C

omm

utat

ive

Pro

pert

y of

Add

ition

and

the

Add

itive

Iden

tity

to b

asic

ad

ditio

n fa

ct p

robl

ems.

By

com

pari

ng a

var

iety

of

solu

tion

stra

tegi

es, c

hild

ren

rela

te a

dditi

on

and

subt

ract

ion

as in

vers

e op

erat

ions

.548–

552, 5

53–

558, 5

59–

562,

574–

579, 6

93–

697, 7

58–

762, 8

29

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Dem

onst

rate

pro

ficie

ncy

with

+/-

0, +

/- 1

, dou

bles

, and

sum

-equ

als-

ten

addi

tion

and

subt

ract

ion

fact

s su

ch a

s 6 +

4 =

10 a

nd 1

0 -

7 =

3.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, a

nd c

alcu

lato

rs

to s

olve

pro

blem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of 1

-dig

it w

hole

nu

mbe

rs w

ith 1

- or

2-d

igit

who

le n

umbe

rs;

calc

ulat

e an

d co

mpa

re t

he v

alue

s of

co

mbi

natio

ns o

f co

ins.

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

wh

ole

nu

mb

er

rela

tio

nsh

ips,

inc

lud

ing

gro

up

ing

in

te

ns a

nd

on

es

Chi

ldre

n co

mpa

re a

nd o

rder

who

le n

umbe

rs (

at le

ast

to 1

00)

to

deve

lop

an u

nder

stan

ding

of

and

solv

e pr

oble

ms

invo

lvin

g th

e re

lativ

e si

zes

of t

hese

num

bers

.

20–

24, 3

7–

41, 5

2–

55, 5

6–

59,

324–

329, 3

68–

373, 3

84–

387,

388–

392, 8

16–

821

Nu

mb

er

an

d N

um

era

tion

Goa

l 7

Com

pare

and

ord

er w

hole

num

bers

up

to 1

,000.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Rea

d, w

rite

, and

exp

lain

exp

ress

ions

and

num

ber

sent

ence

s us

ing

the

sym

bols

+

, - ,

and

= a

nd t

he s

ymbo

ls >

and

< w

ith c

ues;

sol

ve e

quat

ions

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Chi

ldre

n th

ink

of w

hole

num

bers

bet

wee

n 10 a

nd 1

00 in

ter

ms

of

grou

ps o

f te

ns a

nd o

nes

(esp

ecia

lly r

ecog

nizi

ng t

he n

umbe

rs 1

1 t

o 19

as 1

gro

up o

f te

n an

d pa

rtic

ular

num

bers

of

ones

).

358–

362, 3

63–

367, 3

79–

383, 6

85,

688–

692, 7

42–

746, 7

53–

757,

832–

838

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

1,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Kno

w a

nd c

ompa

re t

he v

alue

of

penn

ies,

nic

kels

, dim

es, q

uart

ers,

and

dol

lar

bills

; m

ake

exch

ange

s be

twee

n co

ins.

Chi

ldre

n un

ders

tand

the

seq

uent

ial o

rder

of

the

coun

ting

num

bers

an

d th

eir

rela

tive

mag

nitu

des

and

repr

esen

t nu

mbe

rs o

n a

num

ber

line.

20–

24, 3

3–

36, 9

4–

98, 1

19–

123,

204–

208, 5

87–

592

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s,

2s,

5s,

and

10s

past

100 a

nd b

ack

by 1

s fr

om a

ny n

umbe

r le

ss

than

100 w

ith a

nd w

ithou

t nu

mbe

r gr

ids,

num

ber

lines

, and

cal

cula

tors

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Cou

nt c

olle

ctio

ns o

f ob

ject

s ac

cura

tely

and

rel

iabl

y; e

stim

ate

the

num

ber

of

obje

cts

in a

col

lect

ion.

Geom

etry

: Co

mp

osin

g a

nd

de

co

mp

osin

g g

eo

me

tric

sh

ap

es

Chi

ldre

n co

mpo

se a

nd d

ecom

pose

pla

ne a

nd s

olid

fig

ures

(e.

g.,

by p

uttin

g tw

o co

ngru

ent

isos

cele

s tr

iang

les

toge

ther

to

mak

e a

rhom

bus)

, thu

s bu

ildin

g an

und

erst

andi

ng o

f pa

rt-w

hole

rel

atio

nshi

ps

as w

ell a

s th

e pr

oper

ties

of t

he o

rigi

nal a

nd c

ompo

site

sha

pes.

644–

648, 6

49–

653, 7

03–

708,

709–

713, 7

68–

773, 7

74–

777,

778–

782, 8

26

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Use

man

ipul

ativ

es a

nd d

raw

ings

to

mod

el h

alve

s, t

hird

s, a

nd f

ourt

hs a

s eq

ual

part

s of

a r

egio

n or

a c

olle

ctio

n; d

escr

ibe

the

mod

el.

Geo

metr

y G

oa

l 1

Iden

tify

and

desc

ribe

pla

ne a

nd s

olid

fig

ures

incl

udin

g ci

rcle

s, t

rian

gles

, squ

ares

, re

ctan

gles

, sph

eres

, cyl

inde

rs, r

ecta

ngul

ar p

rism

s, p

yram

ids,

con

es, a

nd c

ubes

.

As

child

ren

com

bine

fig

ures

, the

y re

cogn

ize

them

fro

m d

iffer

ent

pers

pect

ives

and

ori

enta

tions

, des

crib

e th

eir

geom

etri

c at

trib

utes

an

d pr

oper

ties,

and

det

erm

ine

how

the

y ar

e al

ike

and

diff

eren

t, in

th

e pr

oces

s de

velo

ping

a b

ackg

roun

d fo

r m

easu

rem

ent

and

initi

al

unde

rsta

ndin

gs o

f su

ch p

rope

rtie

s as

con

grue

nce

and

sym

met

ry.

644–

648, 6

49–

653, 6

54–

657

Geo

metr

y G

oa

l 2

Iden

tify

shap

es h

avin

g lin

e sy

mm

etry

; co

mpl

ete

line-

sym

met

ric

shap

es o

r de

sign

s.

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns a

nd A

lgeb

raC

hild

ren

use

mat

hem

atic

al r

easo

ning

, inc

ludi

ng id

eas

such

as

com

mut

ativ

ity a

nd a

ssoc

iativ

ity a

nd b

egin

ning

idea

s of

ten

s an

d on

es,

to s

olve

tw

o-di

git

addi

tion

and

subt

ract

ion

prob

lem

s w

ith s

trat

egie

s th

at t

hey

unde

rsta

nd a

nd c

an e

xpla

in.

379–

383, 7

47–

752, 7

53–

757,

758–

762, 8

13, 8

17, 8

27–

831

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

1,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Chi

ldre

n us

e m

athe

mat

ical

rea

soni

ng, i

nclu

ding

idea

s su

ch a

s co

mm

utat

ivity

and

ass

ocia

tivity

and

beg

inni

ng id

eas

of t

ens

and

ones

, to

sol

ve t

wo-

digi

t ad

ditio

n an

d su

btra

ctio

n pr

oble

ms

with

str

ateg

ies

that

the

y un

ders

tand

and

can

exp

lain

.

379–

383, 7

47–

752, 7

53–

757,

758–

762, 8

13, 8

17, 8

27–

831

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Use

man

ipul

ativ

es, d

raw

ings

, tal

ly m

arks

, and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

of 1

- or

2-d

igit

num

bers

to

give

equ

ival

ent

nam

es f

or

who

le n

umbe

rs u

p to

100.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, a

nd c

alcu

lato

rs

to s

olve

pro

blem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of 1

-dig

it w

hole

nu

mbe

rs w

ith 1

- or

2-d

igit

who

le n

umbe

rs;

calc

ulat

e an

d co

mpa

re t

he v

alue

s of

co

mbi

natio

ns o

f co

ins.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Rea

d, w

rite

, and

exp

lain

exp

ress

ions

and

num

ber

sent

ence

s us

ing

the

sym

bols

+

, - ,

and

= a

nd t

he s

ymbo

ls >

and

< w

ith c

ues;

sol

ve e

quat

ions

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n.

Chi

ldre

n so

lve

both

rou

tine

and

nonr

outin

e pr

oble

ms.

104–

108, 1

56–

161, 3

78–

382,

414–

418, 4

19–

423, 5

74–

579,

693–

697, 6

98–

702, 7

14–

718,

758–

762, 8

11–

815, 8

16–

821,

827–

831, 8

72–

875

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

1,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Use

man

ipul

ativ

es, d

raw

ings

, tal

ly m

arks

, and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

of 1

- or

2-d

igit

num

bers

to

give

equ

ival

ent

nam

es f

or

who

le n

umbe

rs u

p to

100.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, a

nd c

alcu

lato

rs

to s

olve

pro

blem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of 1

-dig

it w

hole

nu

mbe

rs w

ith 1

- or

2-d

igit

who

le n

umbe

rs;

calc

ulat

e an

d co

mpa

re t

he v

alue

s of

co

mbi

natio

ns o

f co

ins.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 3

Iden

tify

a th

erm

omet

er a

s a

tool

for

mea

suri

ng t

empe

ratu

re;

read

tem

pera

ture

s on

Fah

renh

eit

and

Cel

sius

the

rmom

eter

s to

the

nea

rest

10°.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

, vis

ual,

and

conc

rete

pat

tern

s; s

olve

pr

oble

ms

invo

lvin

g fu

nctio

n m

achi

nes,

“W

hat’s

My

Rul

e?”

tabl

es, a

nd F

ram

es-

and-

Arr

ows

diag

ram

s.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Rea

d, w

rite

, and

exp

lain

exp

ress

ions

and

num

ber

sent

ence

s us

ing

the

sym

bols

+

, - ,

and

= a

nd t

he s

ymbo

ls >

and

< w

ith c

ues;

sol

ve e

quat

ions

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

1

Ever

yday

Mat

hem

atic

s Gr

ade

1 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

1 Gr

ade-

Leve

l Goa

ls

Mea

sure

men

t and

Dat

a An

alys

isC

hild

ren

stre

ngth

en t

heir

sen

se o

f nu

mbe

r by

sol

ving

pro

blem

s in

volv

ing

mea

sure

men

ts a

nd d

ata.

44, 6

8, 2

46–

250, 2

97–

301,

302–

306, 3

10, 3

74–

378, 5

63–

568,

597–

603, 7

63–

767, 8

00–

804,

852–

855, 8

56–

859, 8

62

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta t

o cr

eate

tal

ly c

hart

s, t

able

s, b

ar g

raph

s, a

nd li

ne p

lots

.

Da

ta a

nd

Ch

an

ce G

oa

l 2

Use

gra

phs

to a

nsw

er s

impl

e qu

estio

ns a

nd d

raw

con

clus

ions

; fin

d th

e m

axim

um a

nd m

inim

um o

f a

data

set

.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Use

non

stan

dard

too

ls a

nd t

echn

ique

s to

est

imat

e an

d co

mpa

re w

eigh

t an

d le

ngth

; m

easu

re le

ngth

with

sta

ndar

d m

easu

ring

too

ls.

Mea

suri

ng b

y la

ying

mul

tiple

cop

ies

of a

uni

t en

d to

end

and

the

n co

untin

g th

e un

its b

y us

ing

grou

ps o

f te

ns a

nd o

nes

supp

orts

ch

ildre

n’s

unde

rsta

ndin

g of

num

ber

lines

and

num

ber

rela

tions

hips

.

281–

285, 2

86–

290, 2

91–

296,

563–

568, 8

65

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Use

non

stan

dard

too

ls a

nd t

echn

ique

s to

est

imat

e an

d co

mpa

re w

eigh

t an

d le

ngth

; m

easu

re le

ngth

with

sta

ndar

d m

easu

ring

too

ls.

Rep

rese

ntin

g m

easu

rem

ents

and

dis

cret

e da

ta in

pic

ture

and

bar

gr

aphs

invo

lves

cou

ntin

g an

d co

mpa

riso

ns t

hat

prov

ide

anot

her

mea

ning

ful c

onne

ctio

n to

num

ber

rela

tions

hips

.

310, 5

97–

603, 8

50, 8

61

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta t

o cr

eate

tal

ly c

hart

s, t

able

s, b

ar g

raph

s, a

nd li

ne

plot

s.

Alge

bra

Thro

ugh

iden

tifyi

ng, d

escr

ibin

g, a

nd a

pply

ing

num

ber

patt

erns

and

pr

oper

ties

in d

evel

opin

g st

rate

gies

for

bas

ic f

acts

, chi

ldre

n le

arn

abou

t ot

her

prop

ertie

s of

num

bers

and

ope

ratio

ns, s

uch

as o

dd a

nd

even

(e.

g.,“

Even

num

bers

of

obje

cts

can

be p

aire

d, w

ith n

one

left

ov

er”)

, and

0 a

s th

e id

entit

y el

emen

t fo

r ad

ditio

n.

188–

193, 1

94–

198, 2

04–

208,

219–

223, 2

24–

228, 2

29–

234,

251–

255, 4

09–

413, 5

87–

592

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s,

2s,

5s,

and

10s

past

100 a

nd b

ack

by 1

s fr

om a

ny n

umbe

r le

ss

than

100 w

ith a

nd w

ithou

t nu

mbe

r gr

ids,

num

ber

lines

, and

cal

cula

tors

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

man

ipul

ativ

es t

o id

entif

y an

d m

odel

odd

and

eve

n nu

mbe

rs.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Dem

onst

rate

pro

ficie

ncy

with

+/-

0, +

/- 1

, dou

bles

, and

sum

-equ

als-

ten

addi

tion

and

subt

ract

ion

fact

s su

ch a

s 6 +

4 =

10 a

nd 1

0 -

7 =

3.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

, vis

ual,

and

conc

rete

pat

tern

s; s

olve

pr

oble

ms

invo

lvin

g fu

nctio

n m

achi

nes,

“W

hat’s

My

Rul

e?”

tabl

es, a

nd F

ram

es-

and-

Arr

ows

diag

ram

s.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 3

App

ly t

he C

omm

utat

ive

Pro

pert

y of

Add

ition

and

the

Add

itive

Iden

tity

to b

asic

ad

ditio

n fa

ct p

robl

ems.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

2

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 2

Ever

yday

Mat

hem

atic

s Gr

ade

2 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

2 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

the

ba

se

-te

n n

um

era

tio

n s

yste

m a

nd

pla

ce

-va

lue

co

nc

ep

ts

Chi

ldre

n de

velo

p an

und

erst

andi

ng o

f th

e ba

se-t

en n

umer

atio

n sy

stem

and

pla

ce-v

alue

con

cept

s (a

t le

ast

to 1

000).

Thei

r un

ders

tand

ing

of b

ase-

ten

num

erat

ion

incl

udes

idea

s of

cou

ntin

g in

un

its a

nd m

ultip

les

of h

undr

eds,

ten

s, a

nd o

nes.

38–

41, 4

6–

50, 7

3–

75, 1

84–

189,

202–

206, 2

94–

299, 4

01–

406,

764–

768, 7

69–

774, 7

75–

779

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

10,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s;

read

and

wri

te m

oney

am

ount

s in

dol

lars

-and

-cen

ts n

otat

ion.

Chi

ldre

n’s

unde

rsta

ndin

g of

bas

e-te

n nu

mer

atio

n in

clud

es a

gra

sp o

f nu

mbe

r re

latio

nshi

ps, w

hich

the

y de

mon

stra

te in

a v

arie

ty o

f w

ays,

in

clud

ing

com

pari

ng a

nd o

rder

ing

num

bers

.

18–

22, 2

8–

32, 3

3–

37, 4

6–

50,

65–

70, 2

07–

212, 5

76–

581,

896–

901

Nu

mb

er

an

d N

um

era

tion

Goa

l 7

Com

pare

and

ord

er w

hole

num

bers

up

to 1

0,0

00;

use

area

mod

els

to c

ompa

re

frac

tions

.

Chi

ldre

n un

ders

tand

mul

tidig

it nu

mbe

rs in

ter

ms

of p

lace

val

ue,

reco

gniz

ing

that

pla

ce-v

alue

not

atio

n is

a s

hort

hand

for

the

sum

s of

m

ultip

les

of p

ower

s of

10 (

e.g.

, 853 a

s 8 h

undr

eds

+ 5

ten

s +

3 o

nes)

.

294–

299, 4

01–

406, 7

64–

768

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

10,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s;

read

and

wri

te m

oney

am

ount

s in

dol

lars

-and

-cen

ts n

otat

ion.

Num

ber

and

Oper

atio

ns a

nd A

lgeb

ra: D

eve

lop

ing

qu

ick r

ec

all o

f a

dd

itio

n f

ac

ts a

nd

re

late

d s

ub

tra

cti

on

fa

cts

an

d f

lue

nc

y w

ith

mu

ltid

igit

ad

dit

ion

an

d s

ub

tra

cti

on

Chi

ldre

n us

e th

eir

unde

rsta

ndin

g of

add

ition

to

deve

lop

quic

k re

call

of

basi

c ad

ditio

n fa

cts

and

rela

ted

subt

ract

ion

fact

s.33–

37, 9

9–

104, 1

05–

110,

111–

115, 1

16–

120, 1

21–

125,

126–

131, 1

55–

160, 1

61–

165

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Dem

onst

rate

aut

omat

icity

with

+/-

0, +

/- 1

, dou

bles

, and

sum

-equ

als-

ten

fact

s, a

nd p

rofic

ienc

y w

ith a

ll ad

ditio

n an

d su

btra

ctio

n fa

cts

thro

ugh

10 +

10.

Chi

ldre

n so

lve

arith

met

ic p

robl

ems

by a

pply

ing

thei

r un

ders

tand

ing

of

mod

els

of a

dditi

on a

nd s

ubtr

actio

n (s

uch

as c

ombi

ning

or

sepa

ratin

g se

ts o

r us

ing

num

ber

lines

), re

latio

nshi

ps a

nd p

rope

rtie

s of

num

ber

(suc

h as

pla

ce v

alue

), an

d pr

oper

ties

of a

dditi

on (

com

mut

ativ

ity a

nd

asso

ciat

ivity

).

33–

37, 9

4–

98, 1

11–

115, 1

16–

120,

121–

125, 1

26–

131, 1

55–

160,

161–

165, 2

77–

281, 2

88–

293,

294–

299, 3

78–

383, 3

84–

389,

401–

406, 5

49–

553, 5

54–

558,

807–

811, 8

12–

817

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra 3

Des

crib

e th

e C

omm

utat

ive

and

Ass

ocia

tive

Pro

pert

ies

of A

dditi

on a

nd a

pply

th

em t

o m

enta

l ari

thm

etic

pro

blem

s.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 2

Ever

yday

Mat

hem

atic

s Gr

ade

2 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

2 Gr

ade-

Leve

l Goa

ls

Chi

ldre

n so

lve

arith

met

ic p

robl

ems

by a

pply

ing

thei

r un

ders

tand

ing

of

mod

els

of a

dditi

on a

nd s

ubtr

actio

n (s

uch

as c

ombi

ning

or

sepa

ratin

g se

ts o

r us

ing

num

ber

lines

), re

latio

nshi

ps a

nd p

rope

rtie

s of

num

ber

(suc

h as

pla

ce v

alue

), an

d pr

oper

ties

of a

dditi

on (

com

mut

ativ

ity a

nd

asso

ciat

ivity

).

33–

37, 9

4–

98, 1

11–

115, 1

16–

120,

121–

125, 1

26–

131, 1

55–

160,

161–

165, 2

77–

281, 2

88–

293,

294–

299, 3

78–

383, 3

84–

389,

401–

406, 5

49–

553, 5

54–

558,

807–

811, 8

12–

817

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, p

aper

& p

enci

l, an

d ca

lcul

ator

s to

sol

ve p

robl

ems

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

2-d

igit

who

le n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

; ca

lcul

ate

and

com

pare

va

lues

of

coin

and

bill

com

bina

tions

.

Chi

ldre

n de

velo

p, d

iscu

ss, a

nd u

se e

ffic

ient

, acc

urat

e, a

nd

gene

raliz

able

met

hods

to

add

and

subt

ract

mul

tidig

it w

hole

num

bers

.254–

259, 2

66–

271, 2

77–

281,

288–

293, 2

94–

299, 3

84–

389,

396–

400, 4

01–

406, 5

54–

558,

576–

581, 5

82–

587, 6

83–

687,

896–

901, 9

02–

907

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, p

aper

& p

enci

l, an

d ca

lcul

ator

s to

sol

ve p

robl

ems

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

2-d

igit

who

le n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

; ca

lcul

ate

and

com

pare

va

lues

of

coin

and

bill

com

bina

tions

.

Chi

ldre

n se

lect

and

app

ly a

ppro

pria

te m

etho

ds t

o es

timat

e su

ms

and

diff

eren

ces

or c

alcu

late

the

m m

enta

lly, d

epen

ding

on

the

cont

ext

and

num

bers

invo

lved

.

219–

223, 2

24–

229, 2

72–

276,

277–

281, 2

88–

293, 2

94–

299,

378–

383, 3

84–

389, 4

01–

406,

554–

558, 7

48–

752, 7

53–

757,

802–

806, 8

07–

811, 8

12–

817

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

addi

tion

and

subt

ract

ion

prob

lem

s; e

xpla

in h

ow t

he e

stim

ates

wer

e ob

tain

ed.

Chi

ldre

n de

velo

p flu

ency

with

eff

icie

nt p

roce

dure

s, in

clud

ing

stan

dard

al

gori

thm

s, f

or a

ddin

g an

d su

btra

ctin

g w

hole

num

bers

, und

erst

and

why

the

pro

cedu

res

wor

k (o

n th

e ba

sis

of p

lace

val

ue a

nd p

rope

rtie

s of

ope

ratio

ns),

and

use

them

to

solv

e pr

oble

ms.

254–

259, 2

66–

271, 2

77–

281, 2

88–

293, 2

94–

299, 3

84–

389, 3

96–

400,

401–

406, 5

54–

558, 5

76–

581,

582–

587, 6

83–

687, 8

96–

901,

902–

907, A

lgor

ithm

s H

andb

ook

pp.

4–

15, 2

4–

33

See

als

o w

ww

.eve

ryd

aym

ath

on

lin

e.c

om

for

Algo

rithm

s Ha

ndbo

ok p

p. 4

–15,

24–

33, a

nd A

lgor

ithm

s in

Eve

ryda

y M

athe

mat

ics

whi

ch c

onta

in o

nlin

e an

imat

ions

and

ass

ocia

ted

prin

t re

sour

ces.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, p

aper

& p

enci

l, an

d ca

lcul

ator

s to

sol

ve p

robl

ems

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

2-d

igit

who

le n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

; ca

lcul

ate

and

com

pare

va

lues

of

coin

and

bill

com

bina

tions

.

Mea

sure

men

t: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

lin

ea

r m

ea

su

rem

en

t a

nd

fa

cilit

y i

n m

ea

su

rin

g l

en

gth

s

Chi

ldre

n de

velo

p an

und

erst

andi

ng o

f th

e m

eani

ng a

nd p

roce

sses

of

mea

sure

men

t, in

clud

ing

such

und

erly

ing

conc

epts

as

part

ition

ing

(the

m

enta

l act

ivity

of

slic

ing

the

leng

th o

f an

obj

ect

into

equ

al-s

ized

uni

ts)

and

tran

sitiv

ity (

e.g.

, if

obje

ct A

is lo

nger

tha

n ob

ject

B a

nd o

bjec

t B

is

long

er t

han

obje

ct C

, the

n ob

ject

A is

long

er t

han

obje

ct C

). C

hild

ren

unde

rsta

nd li

near

mea

sure

as

an it

erat

ion

of u

nits

and

use

rul

ers

and

othe

r m

easu

rem

ent

tool

s w

ith t

hat

unde

rsta

ndin

g.

282–

287, 5

70–

575, 6

60–

665,

666–

671, 6

72–

677, 6

78–

682

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t in

ch

and

cent

imet

er;

use

stan

dard

and

non

stan

dard

too

ls t

o m

easu

re a

nd e

stim

ate

wei

ght.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

2Ev

eryd

ay M

athe

mat

ics

Grad

e 2

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

2 Gr

ade-

Leve

l Goa

ls

Chi

ldre

n un

ders

tand

the

nee

d fo

r eq

ual-

leng

th u

nits

, the

use

of

stan

dard

uni

ts o

f m

easu

re (

cent

imet

er a

nd in

ch),

and

the

inve

rse

rela

tions

hip

betw

een

the

size

of

a un

it an

d th

e nu

mbe

r of

uni

ts u

sed

in a

par

ticul

ar m

easu

rem

ent

(i.e.

, chi

ldre

n re

cogn

ize

that

the

sm

alle

r th

e un

it, t

he m

ore

itera

tions

the

y ne

ed t

o co

ver

a gi

ven

leng

th).

282–

287, 5

70–

575, 6

60–

665,

666–

671

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t in

ch

and

cent

imet

er;

use

stan

dard

and

non

stan

dard

too

ls t

o m

easu

re a

nd e

stim

ate

wei

ght.

Num

ber

and

Oper

atio

ns

Chi

ldre

n us

e pl

ace

valu

e an

d pr

oper

ties

of o

pera

tions

to

crea

te

equi

vale

nt r

epre

sent

atio

ns o

f gi

ven

num

bers

(su

ch a

s 35 r

epre

sent

ed

by 3

5 o

nes,

3 t

ens

and

5 o

nes,

or

2 t

ens

and

15 o

nes)

55–

60, 1

38–

142, 2

02–

206,

294–

299, 4

01–

406, 5

49–

553,

726–

730, 7

64–

768, 7

69–

774,

775–

779

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

tal

ly m

arks

, arr

ays,

and

num

eric

al e

xpre

ssio

ns in

volv

ing

addi

tion

and

subt

ract

ion

to g

ive

equi

vale

nt n

ames

for

who

le n

umbe

rs.

Chi

ldre

n us

e pl

ace

valu

e an

d pr

oper

ties

of o

pera

tions

to

wri

te,

com

pare

, and

ord

er m

ultid

igit

num

bers

.18–

22, 3

3–

37, 4

6–

50, 6

5–

70,

184–

189, 5

76–

581, 7

64–

768,

896–

901, 9

26–

928

Nu

mb

er

an

d N

um

era

tion

Goa

l 7

Com

pare

and

ord

er w

hole

num

bers

up

to 1

0,0

00;

use

area

mod

els

to c

ompa

re

frac

tions

.

Chi

ldre

n us

e th

ese

idea

s to

com

pose

and

dec

ompo

se m

ultid

igit

num

bers

.184–

189, 2

02–

206, 7

64–

768,

769–

774, 7

75–

779

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el w

ith m

anip

ulat

ives

who

le n

umbe

rs u

p to

10,0

00;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

tho

se p

lace

s;

read

and

wri

te m

oney

am

ount

s in

dol

lars

-and

-cen

ts n

otat

ion.

Chi

ldre

n ad

d an

d su

btra

ct t

o so

lve

a va

riet

y of

pro

blem

s, in

clud

ing

appl

icat

ions

invo

lvin

g m

easu

rem

ent,

geom

etry

, and

dat

a, a

s w

ell a

s no

nrou

tine

prob

lem

s.

94–

98, 2

07–

212, 2

19–

223,

224–

229, 2

48–

253, 2

54–

259,

266–

271, 2

77–

281, 2

88–

293,

294–

299, 3

78–

383, 3

84–

389,

396–

400, 5

54–

558, 5

76–

581,

582–

587, 6

78–

682, 6

83–

687,

743–

747, 7

53–

757, 7

80–

783,

802–

806, 8

07–

811, 8

96–

901,

902–

907, 9

26–

928

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, n

umbe

r gr

ids,

tal

ly m

arks

, men

tal a

rith

met

ic, p

aper

& p

enci

l, an

d ca

lcul

ator

s to

sol

ve p

robl

ems

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

2-d

igit

who

le n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

; ca

lcul

ate

and

com

pare

va

lues

of

coin

and

bill

com

bina

tions

.

In p

repa

ratio

n fo

r gr

ade

3, c

hild

ren

solv

e pr

oble

ms

invo

lvin

g m

ultip

licat

ive

situ

atio

ns, d

evel

opin

g in

itial

und

erst

andi

ngs

of

mul

tiplic

atio

n as

rep

eate

d ad

ditio

n.

132–

137, 4

07–

411, 4

12–

417,

418–

423, 4

24–

428, 8

18–

823,

830–

834, 8

40–

845

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Iden

tify

and

desc

ribe

cha

nge,

com

pari

son,

and

par

ts-a

nd-t

otal

situ

atio

ns;

use

repe

ated

add

ition

, arr

ays,

and

ski

p co

untin

g to

mod

el m

ultip

licat

ion;

use

equ

al

shar

ing

and

equa

l gro

upin

g to

mod

el d

ivis

ion.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

2Ev

eryd

ay M

athe

mat

ics

Grad

e 2

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

2 Gr

ade-

Leve

l Goa

ls

Geom

etry

and

Mea

sure

men

t C

hild

ren

estim

ate,

mea

sure

, and

com

pute

leng

ths

as t

hey

solv

e pr

oble

ms

invo

lvin

g da

ta, s

pace

, and

mov

emen

t th

roug

h sp

ace.

282–

287, 5

70–

575, 6

60–

665,

666–

671, 6

72–

677, 6

78–

682,

940–

942

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t in

ch

and

cent

imet

er;

use

stan

dard

and

non

stan

dard

too

ls t

o m

easu

re a

nd e

stim

ate

wei

ght.

By

com

posi

ng a

nd d

ecom

posi

ng t

wo-

dim

ensi

onal

sha

pes

(inte

ntio

nally

sub

stitu

ting

arra

ngem

ents

of

smal

ler

shap

es f

or la

rger

sh

apes

or

subs

titut

ing

larg

er s

hape

s fo

r m

any

smal

ler

shap

es),

child

ren

use

geom

etri

c kn

owle

dge

and

spat

ial r

easo

ning

to

deve

lop

foun

datio

ns f

or u

nder

stan

ding

are

a, f

ract

ions

, and

pro

port

ions

.

284–

285, 3

38–

342, 3

55–

359, 5

68,

604–

609, 6

10–

615, 6

21–

625,

688–

693, 6

94–

698, 7

58–

763,

440–

443, 9

29–

932

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Use

man

ipul

ativ

es a

nd d

raw

ings

to

mod

el f

ract

ions

as

equa

l par

ts o

f a

regi

on o

r a

colle

ctio

n; d

escr

ibe

the

mod

els

and

nam

e th

e fr

actio

ns.

Geo

metr

y G

oa

l 3

Cre

ate

and

com

plet

e tw

o-di

men

sion

al s

ymm

etri

cal s

hape

s or

des

igns

.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Cou

nt u

nit

squa

res

to f

ind

the

area

of

rect

angl

es.

Alge

bra

Chi

ldre

n us

e nu

mbe

r pa

tter

ns t

o ex

tend

the

ir k

now

ledg

e of

pro

pert

ies

of n

umbe

rs a

nd o

pera

tions

. For

exa

mpl

e, w

hen

skip

cou

ntin

g, t

hey

build

fou

ndat

ions

for

und

erst

andi

ng m

ultip

les

and

fact

ors.

46–

50, 5

1–

54, 6

1–

64, 1

05–

110,

143–

148, 1

49–

154, 5

44–

548,

559–

564, 8

18–

823, 8

35–

839,

846–

849, 8

84–

889

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Cou

nt o

n by

1s,

2s,

5s,

10s,

25s,

and

100s

past

1,0

00 a

nd b

ack

by 1

s fr

om

any

num

ber

less

tha

n 1,0

00 w

ith a

nd w

ithou

t nu

mbe

r gr

ids,

num

ber

lines

, and

ca

lcul

ator

s.

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Rec

ogni

ze n

umbe

rs a

s od

d or

eve

n.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

, vis

ual,

and

conc

rete

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd u

se t

hem

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es f

or f

unct

ions

invo

lvin

g ad

ditio

n an

d su

btra

ctio

n an

d us

e th

ose

rule

s to

sol

ve p

robl

ems.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

3

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 3

Ever

yday

Mat

hem

atic

s Gr

ade

3 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

3 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns a

nd A

lgeb

ra: D

eve

lop

ing

un

de

rsta

nd

ing

s o

f m

ult

iplic

ati

on

an

d d

ivis

ion

an

d s

tra

teg

ies f

or

ba

sic

mu

ltip

lic

ati

on

fa

cts

an

d r

ela

ted

div

isio

n f

ac

ts

Stu

dent

s un

ders

tand

the

mea

ning

s of

mul

tiplic

atio

n an

d di

visi

on o

f w

hole

num

bers

thr

ough

the

use

of

repr

esen

tatio

ns (

e.g.

, equ

al-s

ized

gr

oups

, arr

ays,

are

a m

odel

s, a

nd e

qual

“ju

mps

” on

num

ber

lines

for

m

ultip

licat

ion,

and

suc

cess

ive

subt

ract

ion,

par

titio

ning

, and

sha

ring

fo

r di

visi

on).

They

use

pro

pert

ies

of a

dditi

on a

nd m

ultip

licat

ion

(e.g

., co

mm

utat

ivity

, ass

ocia

tivity

, and

the

dis

trib

utiv

e pr

oper

ty)

to m

ultip

ly

who

le n

umbe

rs a

nd a

pply

incr

easi

ngly

sop

hist

icat

ed s

trat

egie

s ba

sed

on t

hese

pro

pert

ies

to s

olve

mul

tiplic

atio

n an

d di

visi

on p

robl

ems

invo

lvin

g ba

sic

fact

s.

242–

247, 2

47–

253, 2

54–

259,

260–

265, 2

66–

271, 2

72–

277,

277–

282, 2

83–

286, 5

76–

581,

582–

587, 7

24–

726, 7

42–

744, 7

46,

748–

753, 7

66–

771

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Dem

onst

rate

aut

omat

icity

with

×0, ×

1, ×

2, ×

5, a

nd ×

10 m

ultip

licat

ion

fact

s;

use

stra

tegi

es t

o co

mpu

te r

emai

ning

fac

ts u

p to

10 ×

10.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Rec

ogni

ze a

nd d

escr

ibe

chan

ge, c

ompa

riso

n, a

nd p

arts

-and

-tot

al s

ituat

ions

; us

e re

peat

ed a

dditi

on, a

rray

s, a

nd s

kip

coun

ting

to m

odel

mul

tiplic

atio

n; u

se e

qual

sh

arin

g an

d eq

ual g

roup

ing

to m

odel

div

isio

n.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

the

Com

mut

ativ

e an

d A

ssoc

iativ

e P

rope

rtie

s of

Add

ition

, the

C

omm

utat

ive

Pro

pert

y of

Mul

tiplic

atio

n, a

nd t

he M

ultip

licat

ive

Iden

tity.

By

com

pari

ng a

var

iety

of

solu

tion

stra

tegi

es, s

tude

nts

rela

te

mul

tiplic

atio

n an

d di

visi

on a

s in

vers

e op

erat

ions

.254–

259, 2

60–

265, 2

72–

277,

283–

287

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

arr

ays,

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

2-

and

3-d

igit

who

le n

umbe

rs b

y 1-d

igit

who

le n

umbe

rs a

nd d

escr

ibe

the

stra

tegi

es u

sed.

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

fra

cti

on

s a

nd

fra

cti

on

eq

uiv

ale

nc

e

Stu

dent

s de

velo

p an

und

erst

andi

ng o

f th

e m

eani

ngs

and

uses

of

frac

tions

to

repr

esen

t pa

rts

of a

who

le, p

arts

of

a se

t, or

poi

nts

or

dist

ance

s on

a n

umbe

r lin

e.

358–

362, 3

68–

373, 6

48–

653, 6

60,

665–

670, 6

82–

687, 7

27

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el f

ract

ions

; so

lve

prob

lem

s in

volv

ing

frac

tiona

l par

ts o

f a

regi

on o

r a

colle

ctio

n; d

escr

ibe

stra

tegi

es u

sed.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 3

Ever

yday

Mat

hem

atic

s Gr

ade

3 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

3 Gr

ade-

Leve

l Goa

ls

Stu

dent

s un

ders

tand

tha

t th

e si

ze o

f a

frac

tiona

l par

t is

rel

ativ

e to

the

si

ze o

f th

e w

hole

, and

the

y us

e fr

actio

ns t

o re

pres

ent

num

bers

tha

t ar

e eq

ual t

o, le

ss t

han,

or

grea

ter

than

1.

368–

373, 6

48–

653, 6

60, 6

65–

670,

682–

687, 7

27

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el f

ract

ions

; so

lve

prob

lem

s in

volv

ing

frac

tiona

l par

ts o

f a

regi

on o

r a

colle

ctio

n; d

escr

ibe

stra

tegi

es u

sed.

Stu

dent

s so

lve

prob

lem

s th

at in

volv

e co

mpa

ring

and

ord

erin

g fr

actio

ns b

y us

ing

mod

els,

ben

chm

ark

frac

tions

, or

com

mon

nu

mer

ator

s or

den

omin

ator

s. T

hey

unde

rsta

nd a

nd u

se m

odel

s,

incl

udin

g th

e nu

mbe

r lin

e, t

o id

entif

y eq

uiva

lent

fra

ctio

ns.

671–

676, 6

77–

681, 6

88–

693

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

man

ipul

ativ

es a

nd d

raw

ings

to

find

and

repr

esen

t eq

uiva

lent

nam

es f

or

frac

tions

; us

e m

anip

ulat

ives

to

gene

rate

equ

ival

ent

frac

tions

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Com

pare

and

ord

er w

hole

num

bers

up

to 1

,000,0

00;

use

man

ipul

ativ

es t

o or

der

deci

mal

s th

roug

h hu

ndre

dths

; us

e ar

ea m

odel

s an

d be

nchm

ark

frac

tions

to

com

pare

and

ord

er f

ract

ions

.

Geom

etry

: De

sc

rib

ing

an

d a

na

lyzin

g p

rop

ert

ies o

f tw

o-d

ime

nsio

na

l sh

ap

es.

Stu

dent

s de

scri

be, a

naly

ze, c

ompa

re, a

nd c

lass

ify t

wo-

dim

ensi

onal

sh

apes

by

thei

r si

des

and

angl

es a

nd c

onne

ct t

hese

att

ribu

tes

to

defin

ition

s of

sha

pes.

188–

193, 3

47–

351, 4

08–

413,

420–

425, 4

26–

431, 4

32–

437

Geo

metr

y G

oa

l 1

Iden

tify

and

draw

poi

nts,

inte

rsec

ting

and

para

llel l

ine

segm

ents

and

line

s, r

ays,

an

d ri

ght

angl

es.

Geo

metr

y G

oa

l 2

Iden

tify,

des

crib

e, m

odel

, and

com

pare

pla

ne a

nd s

olid

fig

ures

incl

udin

g ci

rcle

s,

poly

gons

, sph

eres

, cyl

inde

rs, r

ecta

ngul

ar p

rism

s, p

yram

ids,

con

es, a

nd c

ubes

us

ing

appr

opri

ate

geom

etri

c te

rms

incl

udin

g th

e te

rms

face

, edg

e, v

erte

x,

and

base

.

Stu

dent

s in

vest

igat

e, d

escr

ibe,

and

rea

son

abou

t de

com

posi

ng,

com

bini

ng, a

nd t

rans

form

ing

poly

gons

to

mak

e ot

her

poly

gons

. Th

roug

h bu

ildin

g, d

raw

ing,

and

ana

lyzi

ng t

wo-

dim

ensi

onal

sha

pes,

st

uden

ts u

nder

stan

d at

trib

utes

and

pro

pert

ies

of t

wo-

dim

ensi

onal

sp

ace

and

the

use

of t

hose

att

ribu

tes

and

prop

ertie

s in

sol

ving

pr

oble

ms,

incl

udin

g ap

plic

atio

ns in

volv

ing

cong

ruen

ce a

nd s

ymm

etry

.

347–

351, 4

08–

413, 4

20–

425,

426–

431, 4

32–

437, 4

50–

455,

456–

461, 6

24–

629, 6

59–

664

Geo

metr

y G

oa

l 3

Cre

ate

and

com

plet

e tw

o-di

men

sion

al s

ymm

etri

c sh

apes

or

desi

gns;

loca

te

mul

tiple

line

s of

sym

met

ry in

a t

wo-

dim

ensi

onal

sha

pe.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

3Ev

eryd

ay M

athe

mat

ics

Grad

e 3

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

3 Gr

ade-

Leve

l Goa

ls

Alge

bra

Und

erst

andi

ng p

rope

rtie

s of

mul

tiplic

atio

n an

d th

e re

latio

nshi

p be

twee

n m

ultip

licat

ion

and

divi

sion

is a

par

t of

alg

ebra

rea

dine

ss t

hat

deve

lops

at

grad

e 3.

242–

247, 2

48–

253, 2

54–

259,

260–

265, 2

66–

271, 2

72–

277,

283–

287, 5

88–

593

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

the

Com

mut

ativ

e an

d A

ssoc

iativ

e P

rope

rtie

s of

Add

ition

, the

C

omm

utat

ive

Pro

pert

y of

Mul

tiplic

atio

n, a

nd t

he M

ultip

licat

ive

Iden

tity.

The

crea

tion

and

anal

ysis

of

patt

erns

and

rel

atio

nshi

ps in

volv

ing

mul

tiplic

atio

n an

d di

visi

on s

houl

d oc

cur

at t

his

grad

e le

vel.

254–

259, 2

60–

265, 2

72–

277,

283–

287, 5

88–

593, 6

18–

623

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es

for

func

tions

invo

lvin

g ad

ditio

n, s

ubtr

actio

n, a

nd m

ultip

licat

ion

and

use

thos

e ru

les

to s

olve

pro

blem

s.

Stu

dent

s bu

ild a

fou

ndat

ion

for

late

r un

ders

tand

ing

of f

unct

iona

l re

latio

nshi

ps b

y de

scri

bing

rel

atio

nshi

ps in

con

text

with

suc

h st

atem

ents

as,

“Th

e nu

mbe

r of

legs

is 4

tim

es t

he n

umbe

r of

cha

irs.

”

112–

116, 2

54–

259, 2

60–

265,

624–

629

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es

for

func

tions

invo

lvin

g ad

ditio

n, s

ubtr

actio

n, a

nd m

ultip

licat

ion

and

use

thos

e ru

les

to s

olve

pro

blem

s.

Mea

sure

men

tS

tude

nts

in g

rade

3 s

tren

gthe

n th

eir

unde

rsta

ndin

g of

fra

ctio

ns a

s th

ey c

onfr

ont

prob

lem

s in

line

ar m

easu

rem

ent

that

cal

l for

mor

e pr

ecis

ion

than

the

who

le u

nit

allo

wed

the

m in

the

ir w

ork

in g

rade

2.

176–

181, 2

88–

293, 4

26–

431, 4

35,

808–

813

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t 1

_

2 i

nch

and

1

_

2 c

entim

eter

; dr

aw a

nd d

escr

ibe

angl

es o

f re

cord

s of

rot

atio

ns.

Stu

dent

s de

velo

p th

eir

faci

lity

in m

easu

ring

with

fra

ctio

nal p

arts

of

linea

r un

its.

176–

181, 2

88–

293, 4

26–

431, 4

35

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t 1

_

2 i

nch

and

1

_

2 c

entim

eter

; dr

aw a

nd d

escr

ibe

angl

es o

f re

cord

s of

rot

atio

ns.

Stu

dent

s de

velo

p m

easu

rem

ent

conc

epts

and

ski

lls t

hrou

gh

expe

rien

ces

in a

naly

zing

att

ribu

tes

and

prop

ertie

s of

tw

o-di

men

sion

al

obje

cts.

188–

193, 2

00–

205, 3

47–

351,

426–

431, 4

32–

437

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

mea

sure

the

per

imet

er o

f po

lygo

ns;

coun

t un

it sq

uare

s to

fin

d th

e ar

eas

of r

ecta

ngle

s.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

3Ev

eryd

ay M

athe

mat

ics

Grad

e 3

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

3 Gr

ade-

Leve

l Goa

ls

Stu

dent

s fo

rm a

n un

ders

tand

ing

of p

erim

eter

as

a m

easu

rabl

e at

trib

ute

and

sele

ct a

ppro

pria

te u

nits

, str

ateg

ies,

and

too

ls t

o so

lve

prob

lem

s in

volv

ing

peri

met

er.

188–

193, 2

00–

205, 3

47–

351,

426–

431, 4

32–

437

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

mea

sure

the

per

imet

er o

f po

lygo

ns;

coun

t un

it sq

uare

s to

fin

d th

e ar

eas

of r

ecta

ngle

s.

Data

Ana

lysi

sA

dditi

on, s

ubtr

actio

n, m

ultip

licat

ion,

and

div

isio

n of

who

le n

umbe

rs

com

e in

to p

lay

as s

tude

nts

cons

truc

t an

d an

alyz

e fr

eque

ncy

tabl

es,

bar

grap

hs, p

ictu

re g

raph

s, a

nd li

ne p

lots

and

use

the

m t

o so

lve

prob

lem

s.

37–

41, 1

94–

199, 8

35–

839,

840–

845, 8

52–

857, 8

83–

887,

899–

904, 9

14–

916

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta o

r us

e gi

ven

data

to

crea

te c

hart

s, t

able

s, b

ar g

raph

s,

and

line

plot

s.

Da

ta a

nd

Ch

an

ce G

oa

l 2

Use

gra

phs

to a

sk a

nd a

nsw

er s

impl

e qu

estio

ns a

nd d

raw

con

clus

ions

; fin

d th

e m

axim

um, m

inim

um, r

ange

, mod

e, a

nd m

edia

n of

a d

ata

set.

Num

ber

and

Oper

atio

nsB

uild

ing

on t

heir

wor

k in

gra

de 2

, stu

dent

s ex

tend

the

ir u

nder

stan

ding

of

pla

ce v

alue

to

num

bers

up

to 1

0,0

00 in

var

ious

con

text

s.329–

322, 3

24–

329, 3

31–

335,

337–

341, 3

42–

346, 3

49

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs u

p to

1,0

00,0

00;

read

, wri

te, a

nd m

odel

w

ith m

anip

ulat

ives

dec

imal

s th

roug

h hu

ndre

dths

; id

entif

y pl

aces

in s

uch

num

bers

and

the

val

ues

of t

he d

igits

in t

hose

pla

ces;

tra

nsla

te b

etw

een

who

le

num

bers

and

dec

imal

s re

pres

ente

d in

wor

ds, i

n ba

se-1

0 n

otat

ion,

and

with

m

anip

ulat

ives

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Com

pare

and

ord

er w

hole

num

bers

up

to 1

,000,0

00;

use

man

ipul

ativ

es t

o or

der

deci

mal

s th

roug

h hu

ndre

dths

; us

e ar

ea m

odel

s an

d be

nchm

ark

frac

tions

to

com

pare

and

ord

er f

ract

ions

.

Stu

dent

s al

so a

pply

thi

s un

ders

tand

ing

to t

he t

ask

of r

epre

sent

ing

num

bers

in d

iffer

ent

equi

vale

nt f

orm

s (e

.g.,

expa

nded

not

atio

n). T

hey

deve

lop

thei

r un

ders

tand

ing

of n

umbe

rs b

y bu

ildin

g th

eir

faci

lity

with

men

tal c

ompu

tatio

n (a

dditi

on a

nd s

ubtr

actio

n in

spe

cial

cas

es,

such

as

2,5

00 +

6,0

00 a

nd 9

,000 -

5,0

00),

by u

sing

com

puta

tiona

l es

timat

ion,

and

by

perf

orm

ing

pape

r-an

d-pe

ncil

com

puta

tions

.

42–

46, 6

7–

72, 1

06–

111, 1

17–

122,

123–

128, 1

29–

134, 1

35–

140,

141–

146, 1

47–

151, 6

06–

611,

612–

617, 7

12–

717, 7

18–

723,

730–

735, 7

36–

741, 7

60–

766,

772–

777, 7

78–

783, 8

40–

845

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Dem

onst

rate

aut

omat

icity

with

all

addi

tion

and

subt

ract

ion

fact

s th

roug

h 10 +

10;

use

basi

c fa

cts

to c

ompu

te f

act

exte

nsio

ns s

uch

as 8

0 +

70.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

man

ipul

ativ

es, m

enta

l ari

thm

etic

, pap

er-a

nd-p

enci

l alg

orith

ms,

and

ca

lcul

ator

s to

sol

ve p

robl

ems

invo

lvin

g th

e ad

ditio

n an

d su

btra

ctio

n of

who

le

num

bers

and

dec

imal

s in

a m

oney

con

text

; de

scri

be t

he s

trat

egie

s us

ed a

nd

expl

ain

how

the

y w

ork.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

3Ev

eryd

ay M

athe

mat

ics

Grad

e 3

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

3 Gr

ade-

Leve

l Goa

ls

Stu

dent

s al

so a

pply

thi

s un

ders

tand

ing

to t

he t

ask

of r

epre

sent

ing

num

bers

in d

iffer

ent

equi

vale

nt f

orm

s (e

.g.,

expa

nded

not

atio

n). T

hey

deve

lop

thei

r un

ders

tand

ing

of n

umbe

rs b

y bu

ildin

g th

eir

faci

lity

with

men

tal c

ompu

tatio

n (a

dditi

on a

nd s

ubtr

actio

n in

spe

cial

cas

es,

such

as

2,5

00 +

6,0

00 a

nd 9

,000 -

5,0

00),

by u

sing

com

puta

tiona

l es

timat

ion,

and

by

perf

orm

ing

pape

r-an

d-pe

ncil

com

puta

tions

.

42–

46, 6

7–

72, 1

06–

111, 1

17–

122,

123–

128, 1

29–

134, 1

35–

140,

141–

146, 1

47–

151, 6

06–

611,

612–

617, 7

12–

717, 7

18–

723,

730–

735, 7

36–

741, 7

60–

766,

772–

777, 7

78–

783, 8

40–

845

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 5

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

addi

tion

and

subt

ract

ion

prob

lem

s; e

xpla

in h

ow t

he e

stim

ates

wer

e ob

tain

ed.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es

for

func

tions

invo

lvin

g ad

ditio

n, s

ubtr

actio

n, a

nd m

ultip

licat

ion

and

use

thos

e ru

les

to s

olve

pro

blem

s.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Rea

d, w

rite

, and

exp

lain

num

ber

sent

ence

s us

ing

the

sym

bols

+, -

, ×

, ÷, =

, >

, and

<;

solv

e nu

mbe

r se

nten

ces,

wri

te e

xpre

ssio

ns a

nd n

umbe

r se

nten

ces

to m

odel

num

ber

stor

ies.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

4

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 4

Ever

yday

Mat

hem

atic

s Gr

ade

4 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns a

nd A

lgeb

ra: D

eve

lop

ing

qu

ick r

ec

all o

f m

ult

iplic

ati

on

fa

cts

an

d r

ela

ted

div

isio

n f

ac

ts a

nd

flu

en

cy w

ith

wh

ole

nu

mb

er

mu

ltip

lic

ati

on

.

Stu

dent

s us

e un

ders

tand

ings

of

mul

tiplic

atio

n to

dev

elop

qui

ck r

ecal

l of

the

bas

ic m

ultip

licat

ion

fact

s an

d re

late

d di

visi

on f

acts

.163–

168, 1

69–

174, 1

75–

179,

180–

185, 1

92, 2

03–

207, 2

08–

213,

314–

319, 3

20–

324

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Find

mul

tiple

s of

who

le n

umbe

rs le

ss t

han

10;

find

who

le-n

umbe

r fa

ctor

s of

nu

mbe

rs.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Dem

onst

rate

aut

omat

icity

with

mul

tiplic

atio

n fa

cts

thro

ugh

10 º

10 a

nd

prof

icie

ncy

with

rel

ated

div

isio

n fa

cts;

use

bas

ic f

acts

to

com

pute

fac

t ex

tens

ions

suc

h as

30 º

60.

Stu

dent

s ap

ply

thei

r un

ders

tand

ing

of m

odel

s fo

r m

ultip

licat

ion

(i.e.

, equ

al-s

ized

gro

ups,

arr

ays,

are

a m

odel

s, e

qual

inte

rval

s on

the

nu

mbe

r lin

e), p

lace

val

ue, a

nd p

rope

rtie

s of

ope

ratio

ns (

in p

artic

ular

, th

e di

stri

butiv

e pr

oper

ty)

as t

hey

deve

lop,

dis

cuss

, and

use

eff

icie

nt,

accu

rate

, and

gen

eral

izab

le m

etho

ds t

o m

ultip

ly m

ultid

igit

who

le

num

bers

.

94–

99, 1

63–

168, 1

69–

174,

180–

185, 1

92–

197, 3

14–

319,

337–

342, 9

08–

913

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs u

p to

1,0

00,0

00,0

00 a

nd d

ecim

als

thro

ugh

thou

sand

ths;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

th

ose

plac

es;

tran

slat

e be

twee

n w

hole

num

bers

and

dec

imal

s re

pres

ente

d in

w

ords

and

in b

ase-

10 n

otat

ion.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 7

Use

rep

eate

d ad

ditio

n, s

kip

coun

ting,

arr

ays,

are

a, a

nd s

calin

g to

mod

el

mul

tiplic

atio

n an

d di

visi

on.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

App

ly t

he D

istr

ibut

ive

Pro

pert

y of

Mul

tiplic

atio

n ov

er A

dditi

on t

o th

e pa

rtia

l-pr

oduc

ts m

ultip

licat

ion

algo

rith

m.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 4

Ever

yday

Mat

hem

atic

s Gr

ade

4 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Stu

dent

s se

lect

app

ropr

iate

met

hods

and

app

ly t

hem

acc

urat

ely

to e

stim

ate

prod

ucts

or

calc

ulat

e th

em m

enta

lly, d

epen

ding

on

the

cont

ext

and

num

bers

invo

lved

.

277–

282, 2

89–

293, 3

31–

336,

762–

767, 9

08–

913,9

14–

919,

920–

925

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

mul

tidig

it w

hole

num

bers

by

2-d

igit

who

le n

umbe

rs a

nd t

he d

ivis

ion

of m

ultid

igit

who

le n

umbe

rs b

y 1-d

igit

who

le

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

and

deci

mal

add

ition

and

su

btra

ctio

n pr

oble

ms

and

who

le n

umbe

r m

ultip

licat

ion

and

divi

sion

pro

blem

s;

expl

ain

how

the

est

imat

es w

ere

obta

ined

.

Stu

dent

s de

velo

p flu

ency

with

eff

icie

nt p

roce

dure

s, in

clud

ing

the

stan

dard

alg

orith

m, f

or m

ultip

lyin

g w

hole

num

bers

, und

erst

and

why

th

e pr

oced

ures

wor

k (o

n th

e ba

sis

of p

lace

val

ue a

nd p

rope

rtie

s of

op

erat

ions

), an

d us

e th

em t

o so

lve

prob

lem

s.

320–

324, 3

37–

342, 3

43–

348,

349–

354, 7

62–

767, 9

14–

919,

956–

958, 9

72–

975

See

als

o w

ww

.eve

ryd

aym

ath

on

lin

e.c

om

for

Algo

rithm

s Ha

ndbo

ok

pp. 4

2–

49, ,

and

Alg

orith

ms

in

Ever

yday

Mat

hem

atic

s w

hich

co

ntai

n on

line

anim

atio

ns a

nd

asso

ciat

ed p

rint

res

ourc

es.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

mul

tidig

it w

hole

num

bers

by

2-d

igit

who

le n

umbe

rs a

nd t

he d

ivis

ion

of m

ultid

igit

who

le n

umbe

rs b

y 1-d

igit

who

le

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

App

ly t

he D

istr

ibut

ive

Pro

pert

y of

Mul

tiplic

atio

n ov

er A

dditi

on t

o th

e pa

rtia

l-pr

oduc

ts m

ultip

licat

ion

algo

rith

m.

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

de

cim

als

, in

clu

din

g t

he

co

nn

ec

tio

ns b

etw

ee

n f

rac

tio

ns a

nd

de

cim

als

.

Stu

dent

s un

ders

tand

dec

imal

not

atio

n as

an

exte

nsio

n of

the

bas

e-te

n sy

stem

of

wri

ting

who

le n

umbe

rs t

hat

is u

sefu

l for

rep

rese

ntin

g m

ore

num

bers

, inc

ludi

ng n

umbe

rs b

etw

een

0 a

nd 1

, bet

wee

n 1 a

nd

2, a

nd s

o on

.

238–

243, 2

44–

249, 2

55–

259,

266–

270, 2

71–

276, 4

19–

421

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs u

p to

1,0

00,0

00,0

00 a

nd d

ecim

als

thro

ugh

thou

sand

ths;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

th

ose

plac

es;

tran

slat

e be

twee

n w

hole

num

bers

and

dec

imal

s re

pres

ente

d in

w

ords

and

in b

ase-

10 n

otat

ion.

Stu

dent

s re

late

the

ir u

nder

stan

ding

of

frac

tions

to

read

ing

and

wri

ting

deci

mal

s th

at a

re g

reat

er t

han

or le

ss t

han

1, i

dent

ifyin

g eq

uiva

lent

de

cim

als,

com

pari

ng a

nd o

rder

ing

deci

mal

s, a

nd e

stim

atin

g de

cim

al

or f

ract

iona

l am

ount

s in

pro

blem

sol

ving

.

244–

249, 2

50–

254, 2

55–

259,

271–

276, 7

28–

733, 7

44–

749,

750–

755, 9

31–

935

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

num

eric

al e

xpre

ssio

ns t

o fin

d an

d re

pres

ent

equi

vale

nt n

ames

for

fra

ctio

ns

and

deci

mal

s; u

se a

nd e

xpla

in a

mul

tiplic

atio

n ru

le t

o fin

d eq

uiva

lent

fra

ctio

ns;

rena

me

four

ths,

fift

hs, t

enth

s, a

nd h

undr

edth

s as

dec

imal

s an

d pe

rcen

ts.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Com

pare

and

ord

er w

hole

num

bers

up

to 1

,000,0

00,0

00 a

nd d

ecim

als

thro

ugh

thou

sand

ths;

com

pare

and

ord

er in

tege

rs b

etw

een

-100 a

nd 0

; us

e ar

ea

mod

els,

ben

chm

ark

frac

tions

, and

ana

lyse

s of

num

erat

ors

and

deno

min

ator

s to

co

mpa

re a

nd o

rder

fra

ctio

ns.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 4

Ever

yday

Mat

hem

atic

s Gr

ade

4 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Stu

dent

s co

nnec

t eq

uiva

lent

fra

ctio

ns a

nd d

ecim

als

by c

ompa

ring

m

odel

s to

sym

bols

and

loca

ting

equi

vale

nt s

ymbo

ls o

n th

e nu

mbe

r lin

e.

244–

249, 2

50–

254, 6

09–

614,

722–

727, 7

28–

733, 7

44–

749,

750–

755

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Rea

d, w

rite

, and

mod

el f

ract

ions

; so

lve

prob

lem

s in

volv

ing

frac

tiona

l par

ts o

f a

regi

on o

r a

colle

ctio

n; d

escr

ibe

and

expl

ain

stra

tegi

es u

sed;

giv

en a

fra

ctio

nal

part

of

a re

gion

or

a co

llect

ion,

iden

tify

the

unit

who

le.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

num

eric

al e

xpre

ssio

ns t

o fin

d an

d re

pres

ent

equi

vale

nt n

ames

for

fra

ctio

ns

and

deci

mal

s; u

se a

nd e

xpla

in a

mul

tiplic

atio

n ru

le t

o fin

d eq

uiva

lent

fra

ctio

ns;

rena

me

four

ths,

fift

hs, t

enth

s, a

nd h

undr

edth

s as

dec

imal

s an

d pe

rcen

ts.

Mea

sure

men

t: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

are

a a

nd

de

term

inin

g t

he

are

as o

f tw

o-d

ime

nsio

na

l sh

ap

es.

Stu

dent

s re

cogn

ize

area

as

an a

ttri

bute

of

two-

dim

ensi

onal

reg

ions

. Th

ey le

arn

that

the

y ca

n qu

antif

y ar

ea b

y fin

ding

the

tot

al n

umbe

r of

sam

e-si

zed

units

of

area

tha

t co

ver

the

shap

e w

ithou

t ga

ps o

r ov

erla

ps. T

hey

unde

rsta

nd t

hat

a sq

uare

tha

t is

1 u

nit

on a

sid

e is

the

st

anda

rd u

nit

for

mea

suri

ng a

rea.

670–

674, 6

75–

680, 6

81–

686

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

mea

sure

the

per

imet

er a

nd a

rea

of p

olyg

ons,

to

est

imat

e th

e ar

ea o

f ir

regu

lar

shap

es, a

nd t

o fin

d th

e vo

lum

e of

rec

tang

ular

pr

ism

s.

Stu

dent

s se

lect

app

ropr

iate

uni

ts, s

trat

egie

s (e

.g.,

deco

mpo

sing

sh

apes

), an

d to

ols

for

solv

ing

prob

lem

s th

at in

volv

e es

timat

ing

or

mea

suri

ng a

rea.

670–

674, 6

75–

680, 6

81–

686,

687–

692, 6

93–

698, 6

99–

703

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

mea

sure

the

per

imet

er a

nd a

rea

of p

olyg

ons,

to

est

imat

e th

e ar

ea o

f ir

regu

lar

shap

es, a

nd t

o fin

d th

e vo

lum

e of

rec

tang

ular

pr

ism

s.

Stu

dent

s co

nnec

t ar

ea m

easu

re t

o th

e ar

ea m

odel

tha

t th

ey h

ave

used

to

repr

esen

t m

ultip

licat

ion,

and

the

y us

e th

is c

onne

ctio

n to

ju

stify

the

for

mul

a fo

r th

e ar

ea o

f a

rect

angl

e.

681–

686, 6

87–

692, 6

93–

698,

762–

767

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

mea

sure

the

per

imet

er a

nd a

rea

of p

olyg

ons,

to

est

imat

e th

e ar

ea o

f ir

regu

lar

shap

es, a

nd t

o fin

d th

e vo

lum

e of

rec

tang

ular

pr

ism

s.

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

4Ev

eryd

ay M

athe

mat

ics

Grad

e 4

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Alge

bra

Stu

dent

s co

ntin

ue id

entif

ying

, des

crib

ing,

and

ext

endi

ng n

umer

ic

patt

erns

invo

lvin

g al

l ope

ratio

ns a

nd n

onnu

mer

ic g

row

ing

or r

epea

ting

patt

erns

.

82–

88, 1

58–

162, 6

81–

686,

816–

821

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es

for

func

tions

tha

t in

volv

e th

e fo

ur b

asic

ari

thm

etic

ope

ratio

ns a

nd u

se t

hose

ru

les

to s

olve

pro

blem

s.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

4Ev

eryd

ay M

athe

mat

ics

Grad

e 4

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Thro

ugh

thes

e ex

peri

ence

s, s

tude

nts

deve

lop

an u

nder

stan

ding

of

the

use

of a

rul

e to

des

crib

e a

sequ

ence

of

num

bers

or

obje

cts.

158–

162, 8

16–

821, 9

14–

919

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

use

wor

ds a

nd s

ymbo

ls t

o de

scri

be a

nd w

rite

rul

es

for

func

tions

tha

t in

volv

e th

e fo

ur b

asic

ari

thm

etic

ope

ratio

ns a

nd u

se t

hose

ru

les

to s

olve

pro

blem

s.

Geom

etry

Stu

dent

s ex

tend

the

ir u

nder

stan

ding

of

prop

ertie

s of

tw

o-di

men

sion

al

shap

es a

s th

ey f

ind

the

area

s of

pol

ygon

s. T

hey

build

on

thei

r ea

rlie

r w

ork

with

sym

met

ry a

nd c

ongr

uenc

e in

gra

de 3

to

enco

mpa

ss

tran

sfor

mat

ions

, inc

ludi

ng t

hose

tha

t pr

oduc

e lin

e an

d ro

tatio

nal

sym

met

ry. B

y us

ing

tran

sfor

mat

ions

to

desi

gn a

nd a

naly

ze s

impl

e til

ings

and

tes

sella

tions

, stu

dent

s de

epen

the

ir u

nder

stan

ding

of

two-

dim

ensi

onal

spa

ce.

41–

46, 7

94–

798, 7

99–

804,

805–

809, 8

10–

815, 8

16–

821

Geo

metr

y G

oa

l 3

Iden

tify,

des

crib

e, a

nd s

ketc

h ex

ampl

es o

f re

flect

ions

; id

entif

y an

d de

scri

be

exam

ples

of

tran

slat

ions

and

rot

atio

ns.

Mea

sure

men

tA

s pa

rt o

f un

ders

tand

ing

two-

dim

ensi

onal

sha

pes,

stu

dent

s m

easu

re

and

clas

sify

ang

les.

425–

430, 4

31–

436, 4

37–

442,

948–

951, 9

52–

955

Geo

metr

y G

oa

l 1

Iden

tify,

dra

w, a

nd d

escr

ibe

poin

ts, i

nter

sect

ing

and

para

llel l

ine

segm

ents

and

lin

es, r

ays,

and

rig

ht, a

cute

, and

obt

use

angl

es.

Data

Ana

lysi

sS

tude

nts

cont

inue

to

use

tool

s fr

om g

rade

3, s

olvi

ng p

robl

ems

by

mak

ing

freq

uenc

y ta

bles

, bar

gra

phs,

pic

ture

gra

phs,

and

line

plo

ts.

They

app

ly t

heir

und

erst

andi

ng o

f pl

ace

valu

e to

dev

elop

and

use

st

em-a

nd-l

eaf

plot

s.

106–

111, 1

12–

118, 1

26–

129,

658–

663, 7

50–

755, 9

08–

913

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta o

r us

e gi

ven

data

to

crea

te c

hart

s, t

able

s, b

ar g

raph

s,

line

plot

s, a

nd li

ne g

raph

s.

Num

ber

and

Oper

atio

nsB

uild

ing

on t

heir

wor

k in

gra

de 3

, stu

dent

s ex

tend

the

ir u

nder

stan

ding

of

pla

ce v

alue

and

way

s of

rep

rese

ntin

g nu

mbe

rs t

o 100,0

00 in

va

riou

s co

ntex

ts.

94–

99, 1

00–

105, 1

19–

125,

132–

137, 2

60–

265, 2

71–

276,

355–

360, 3

61–

366, 3

67–

372,

373–

377

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs u

p to

1,0

00,0

00,0

00 a

nd d

ecim

als

thro

ugh

thou

sand

ths;

iden

tify

plac

es in

suc

h nu

mbe

rs a

nd t

he v

alue

s of

the

dig

its in

th

ose

plac

es;

tran

slat

e be

twee

n w

hole

num

bers

and

dec

imal

s re

pres

ente

d in

w

ords

and

in b

ase-

10 n

otat

ion.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

4Ev

eryd

ay M

athe

mat

ics

Grad

e 4

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

4 Gr

ade-

Leve

l Goa

ls

Stu

dent

s us

e es

timat

ion

in d

eter

min

ing

the

rela

tive

size

s of

am

ount

s or

dis

tanc

es.

82–

88, 1

06–

111, 1

19–

125,

132–

137, 1

86–

191, 3

25–

330,

355–

360, 3

67–

372

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

and

deci

mal

add

ition

and

su

btra

ctio

n pr

oble

ms,

and

who

le n

umbe

r m

ultip

licat

ion

and

divi

sion

pro

blem

s;

expl

ain

how

the

est

imat

es w

ere

obta

ined

.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th t

o th

e ne

ares

t 1

_

4 i

nch

and

1

_

2 c

entim

eter

; es

timat

e th

e si

ze o

f an

gles

with

out

tool

s.

Stu

dent

s de

velo

p un

ders

tand

ings

of

stra

tegi

es f

or m

ultid

igit

divi

sion

by

usin

g m

odel

s th

at r

epre

sent

div

isio

n as

the

inve

rse

of

mul

tiplic

atio

n, a

s pa

rtiti

onin

g, o

r as

suc

cess

ive

subt

ract

ion.

158–

162, 4

00–

405, 4

06–

411,

412–

418, 4

19–

421, 4

55–

459,

768–

773, 9

26–

930, 9

36–

940,

972–

975

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

mul

tidig

it w

hole

num

bers

by

2-d

igit

who

le n

umbe

rs a

nd t

he d

ivis

ion

of m

ultid

igit

who

le n

umbe

rs b

y 1-d

igit

who

le

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 7

Use

rep

eate

d ad

ditio

n, s

kip

coun

ting,

arr

ays,

are

a, a

nd s

calin

g to

mod

el

mul

tiplic

atio

n an

d di

visi

on.

By

wor

king

with

dec

imal

s, s

tude

nts

exte

nd t

heir

abi

lity

to r

ecog

nize

eq

uiva

lent

fra

ctio

ns.

609–

614, 7

22–

727, 7

28–

733

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

num

eric

al e

xpre

ssio

ns t

o fin

d an

d re

pres

ent

equi

vale

nt n

ames

for

fra

ctio

ns

and

deci

mal

s; u

se a

nd e

xpla

in a

mul

tiplic

atio

n ru

le t

o fin

d eq

uiva

lent

fra

ctio

ns;

rena

me

four

ths,

fift

hs, t

enth

s, a

nd h

undr

edth

s as

dec

imal

s an

d pe

rcen

ts.

Stu

dent

s’ e

arlie

r w

ork

in g

rade

3 w

ith m

odel

s of

fra

ctio

ns a

nd

mul

tiplic

atio

n an

d di

visi

on f

acts

sup

port

s th

eir

unde

rsta

ndin

g of

te

chni

ques

for

gen

erat

ing

equi

vale

nt f

ract

ions

and

sim

plify

ing

frac

tions

.

587–

591, 5

92–

597, 5

98–

602,

603–

608, 6

09–

614, 6

15–

620

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

num

eric

al e

xpre

ssio

ns t

o fin

d an

d re

pres

ent

equi

vale

nt n

ames

for

fra

ctio

ns

and

deci

mal

s; u

se a

nd e

xpla

in a

mul

tiplic

atio

n ru

le t

o fin

d eq

uiva

lent

fra

ctio

ns;

rena

me

four

ths,

fift

hs, t

enth

s, a

nd h

undr

edth

s as

dec

imal

s an

d pe

rcen

ts.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

5

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 5

Ever

yday

Mat

hem

atic

s Gr

ade

5 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns a

nd A

lgeb

ra: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

an

d f

lue

nc

y w

ith

div

isio

n o

f w

ho

le n

um

be

rs

Stu

dent

s ap

ply

thei

r un

ders

tand

ing

of m

odel

s fo

r di

visi

on, p

lace

va

lue,

pro

pert

ies,

and

the

rel

atio

nshi

p of

div

isio

n to

mul

tiplic

atio

n as

th

ey d

evel

op, d

iscu

ss, a

nd u

se e

ffic

ient

, acc

urat

e, a

nd g

ener

aliz

able

pr

oced

ures

to

find

quot

ient

s in

volv

ing

mul

tidig

it di

vide

nds.

32–

36, 2

30–

235, 2

36–

241,

248–

253

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Dem

onst

rate

aut

omat

icity

with

mul

tiplic

atio

n fa

cts

and

prof

icie

ncy

with

div

isio

n fa

cts

and

exte

nsio

ns.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 7

Use

rep

eate

d ad

ditio

n, a

rray

s, a

rea,

and

sca

ling

to m

odel

mul

tiplic

atio

n an

d di

visi

on;

use

ratio

s ex

pres

sed

as w

ords

, fra

ctio

ns, p

erce

nts,

and

with

col

ons;

so

lve

prob

lem

s in

volv

ing

ratio

s of

par

ts o

f a

set

to t

he w

hole

set

.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

the

prop

ertie

s of

ari

thm

etic

.

Stu

dent

s se

lect

app

ropr

iate

met

hods

and

app

ly t

hem

acc

urat

ely

to

estim

ate

quot

ient

s or

cal

cula

te t

hem

men

tally

, dep

endi

ng o

n th

e co

ntex

t an

d nu

mbe

rs in

volv

ed.

80–

84, 2

48–

252, 2

59–

264,

265–

271

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

who

le n

umbe

rs a

nd d

ecim

als

and

the

divi

sion

of

mul

tidig

it w

hole

num

bers

and

dec

imal

s by

who

le n

umbe

rs;

expr

ess

rem

aind

ers

as w

hole

num

bers

or

frac

tions

as

appr

opri

ate;

des

crib

e th

e st

rate

gies

use

d an

d ex

plai

n ho

w t

hey

wor

k.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

and

deci

mal

add

ition

, sub

trac

tion,

m

ultip

licat

ion,

and

div

isio

n pr

oble

ms

and

frac

tion

and

mix

ed n

umbe

r ad

ditio

n an

d su

btra

ctio

n pr

oble

ms;

exp

lain

how

the

est

imat

es w

ere

obta

ined

.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 5

Ever

yday

Mat

hem

atic

s Gr

ade

5 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Stu

dent

s de

velo

p flu

ency

with

eff

icie

nt p

roce

dure

s, in

clud

ing

the

stan

dard

alg

orith

m, f

or d

ivid

ing

who

le n

umbe

rs, u

nder

stan

d w

hy

the

proc

edur

es w

ork

(on

the

basi

s of

pla

ce v

alue

and

pro

pert

ies

of

oper

atio

ns),

and

use

them

to

solv

e pr

oble

ms.

37–

41, 5

2–

56, 2

30–

235,

236–

241, 2

48–

252

See

als

o w

ww

.eve

ryd

aym

ath

on

lin

e.c

om

for

Algo

rithm

s Ha

ndbo

ok

pp. 6

2–

71, a

nd A

lgor

ithm

s in

Ev

eryd

ay M

athe

mat

ics

whi

ch

cont

ain

onlin

e an

imat

ions

and

as

soci

ated

print

res

ourc

es.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Dem

onst

rate

aut

omat

icity

with

mul

tiplic

atio

n fa

cts

and

prof

icie

ncy

with

div

isio

n fa

cts

and

fact

ext

ensi

ons.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

who

le n

umbe

rs a

nd d

ecim

als

and

the

divi

sion

of

mul

tidig

it w

hole

num

bers

and

dec

imal

s by

who

le n

umbe

rs;

expr

ess

rem

aind

ers

as w

hole

num

bers

or

frac

tions

as

appr

opri

ate;

des

crib

e th

e st

rate

gies

use

d an

d ex

plai

n ho

w t

hey

wor

k.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

the

prop

ertie

s of

ari

thm

etic

.

Stu

dent

s co

nsid

er t

he c

onte

xt in

whi

ch a

pro

blem

is s

ituat

ed t

o se

lect

the

mos

t us

eful

for

m o

f th

e qu

otie

nt f

or t

he s

olut

ion,

and

the

y in

terp

ret

it ap

prop

riat

ely.

236–

241, 2

54–

258, 2

59–

264

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 3

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n of

who

le n

umbe

rs a

nd d

ecim

als

and

the

divi

sion

of

mul

tidig

it w

hole

num

bers

and

dec

imal

s by

who

le n

umbe

rs;

expr

ess

rem

aind

ers

as w

hole

num

bers

or

frac

tions

as

appr

opri

ate;

des

crib

e th

e st

rate

gies

use

d an

d ex

plai

n ho

w t

hey

wor

k.

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

an

d f

lue

nc

y w

ith

ad

dit

ion

an

d s

ub

tra

cti

on

of

fra

cti

on

s a

nd

de

cim

als

Stu

dent

s ap

ply

thei

r un

ders

tand

ings

of

frac

tions

and

fra

ctio

n m

odel

s to

rep

rese

nt t

he a

dditi

on a

nd s

ubtr

actio

n of

fra

ctio

ns w

ith u

nlik

e de

nom

inat

ors

as e

quiv

alen

t ca

lcul

atio

ns w

ith li

ke d

enom

inat

ors.

290–

295, 2

96–

301, 3

02–

307,

319–

324, 4

17–

422, 4

23–

428,

618–

623, 6

24–

629, 6

30–

635

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Sol

ve p

robl

ems

invo

lvin

g pe

rcen

ts a

nd d

isco

unts

; de

scri

be a

nd e

xpla

in

stra

tegi

es u

sed;

iden

tify

the

unit

who

le in

situ

atio

ns in

volv

ing

frac

tions

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Use

num

eric

al e

xpre

ssio

ns t

o fin

d an

d re

pres

ent

equi

vale

nt n

ames

for

fra

ctio

ns,

deci

mal

s, a

nd p

erce

nts;

use

and

exp

lain

mul

tiplic

atio

n an

d di

visi

on r

ules

to

find

equi

vale

nt f

ract

ions

and

fra

ctio

ns in

sim

ples

t fo

rm;

conv

ert

betw

een

frac

tions

an

d m

ixed

num

bers

; co

nver

t be

twee

n fr

actio

ns, d

ecim

als,

and

per

cent

s.

Nu

mb

er

an

d N

um

era

tion

Goa

l 6

Com

pare

and

ord

er r

atio

nal n

umbe

rs;

use

area

mod

els,

ben

chm

ark

frac

tions

, an

d an

alys

es o

f nu

mer

ator

s an

d de

nom

inat

ors

to c

ompa

re a

nd o

rder

fra

ctio

ns

and

mix

ed n

umbe

rs;

desc

ribe

str

ateg

ies

used

to

com

pare

fra

ctio

ns a

nd m

ixed

nu

mbe

rs.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of f

ract

ions

and

mix

ed

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 5

Ever

yday

Mat

hem

atic

s Gr

ade

5 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Stu

dent

s ap

ply

thei

r un

ders

tand

ings

of

deci

mal

mod

els,

pla

ce v

alue

, an

d pr

oper

ties

to a

dd a

nd s

ubtr

act

deci

mal

s.85–

90, 9

1–

96, 9

7–

102

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs a

nd d

ecim

als;

iden

tify

plac

es in

suc

h nu

mbe

rs

and

the

valu

es o

f th

e di

gits

in t

hose

pla

ces;

use

exp

ande

d no

tatio

n to

rep

rese

nt

who

le n

umbe

rs a

nd d

ecim

als.

Stu

dent

s de

velo

p flu

ency

with

sta

ndar

d pr

oced

ures

for

add

ing

and

subt

ract

ing

frac

tions

and

dec

imal

s.85–

90, 9

1–

96, 3

25–

330, 4

23–

428,

429–

433, 6

18–

623, 6

24–

629,

630–

635, 6

36–

641

See

als

o w

ww

.eve

ryd

aym

ath

on

lin

e.c

om

for

Algo

rithm

s Ha

ndbo

ok

pp. 1

8–

23, 3

4–

41, a

nd A

lgor

ithm

s in

Eve

ryda

y M

athe

mat

ics

whi

ch

cont

ain

onlin

e an

imat

ions

and

as

soci

ated

print

res

ourc

es.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of w

hole

num

bers

, dec

imal

s,

and

sign

ed n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of f

ract

ions

and

mix

ed

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.

Stu

dent

s m

ake

reas

onab

le e

stim

ates

of

frac

tion

and

deci

mal

sum

s an

d di

ffer

ence

s.

85–

90, 9

1–

96, 3

14–

318

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber

and

deci

mal

add

ition

, sub

trac

tion,

m

ultip

licat

ion,

and

div

isio

n pr

oble

ms

and

frac

tion

and

mix

ed n

umbe

r ad

ditio

n an

d su

btra

ctio

n pr

oble

ms;

exp

lain

how

the

est

imat

es w

ere

obta

ined

.

Stu

dent

s ad

d an

d su

btra

ct f

ract

ions

and

dec

imal

s to

sol

ve p

robl

ems,

in

clud

ing

prob

lem

s in

volv

ing

mea

sure

men

t.85–

90, 9

1–

96, 9

7–

102, 1

03–

108,

290–

295, 3

02–

307, 4

29–

433,

624–

629, 6

30–

635, 6

36–

641

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 1

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of w

hole

num

bers

, dec

imal

s,

and

sign

ed n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

addi

tion

and

subt

ract

ion

of f

ract

ions

and

mix

ed

num

bers

; de

scri

be t

he s

trat

egie

s us

ed a

nd e

xpla

in h

ow t

hey

wor

k.

Geom

etry

and

Mea

sure

men

t an

d Al

gebr

a: D

esc

rib

ing

th

ree

-dim

en

sio

na

l sh

ap

es a

nd

an

aly

zin

g t

he

ir p

rop

ert

ies,

inc

lud

ing

vo

lum

e a

nd

su

rfa

ce

are

a

Stu

dent

s re

late

tw

o-di

men

sion

al s

hape

s to

thr

ee-d

imen

sion

al s

hape

s an

d an

alyz

e pr

oper

ties

of p

olyh

edra

l sol

ids,

des

crib

ing

them

by

the

num

ber

of e

dges

, fac

es, o

r ve

rtic

es a

s w

ell a

s th

e ty

pes

of f

aces

.

189–

193, 1

99–

205, 2

06–

210,

747–

752, 7

53–

759, 8

56–

860,

861–

865, 8

72–

877

Geo

metr

y G

oa

l 1

Iden

tify,

des

crib

e, c

ompa

re, n

ame,

and

dra

w r

ight

, acu

te, o

btus

e, s

trai

ght,

and

refle

x an

gles

; de

term

ine

angl

e m

easu

res

in v

ertic

al a

nd s

uppl

emen

tary

an

gles

and

by

appl

ying

pro

pert

ies

of s

ums

of a

ngle

mea

sure

s in

tri

angl

es a

nd

quad

rang

les.

Geo

metr

y G

oa

l 2

Des

crib

e, c

ompa

re, a

nd c

lass

ify p

lane

and

sol

id f

igur

es u

sing

app

ropr

iate

ge

omet

ric

term

s; id

entif

y co

ngru

ent

figur

es a

nd d

escr

ibe

thei

r pr

oper

ties.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 5

Ever

yday

Mat

hem

atic

s Gr

ade

5 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Stu

dent

s re

cogn

ize

volu

me

as a

n at

trib

ute

of t

hree

-dim

ensi

onal

sp

ace.

747–

752, 8

66–

871, 8

72–

877

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

Stu

dent

s un

ders

tand

tha

t th

ey c

an q

uant

ify v

olum

e by

fin

ding

the

to

tal n

umbe

r of

sam

e-si

zed

units

of

volu

me

that

the

y ne

ed t

o fil

l the

sp

ace

with

out

gaps

or

over

laps

.

747–

752, 8

66–

871, 8

72–

877

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

Stu

dent

s un

ders

tand

tha

t a

cube

tha

t is

1 u

nit

on a

n ed

ge is

the

st

anda

rd u

nit

for

mea

suri

ng v

olum

e. T

hey

sele

ct a

ppro

pria

te u

nits

, st

rate

gies

, and

too

ls f

or s

olvi

ng p

robl

ems

that

invo

lve

estim

atin

g or

m

easu

ring

vol

ume.

747–

752, 7

53–

759, 8

66–

871,

872–

877, 8

78–

883

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

Stu

dent

s de

com

pose

thr

ee-d

imen

sion

al s

hape

s an

d fin

d su

rfac

e ar

eas

and

volu

mes

of

pris

ms.

747–

752, 7

53–

759, 8

56–

860,

878–

883, 8

90–

894

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

As

stud

ents

wor

k w

ith s

urfa

ce a

rea,

the

y fin

d an

d ju

stify

rel

atio

nshi

ps

amon

g th

e fo

rmul

as f

or t

he a

reas

of

diff

eren

t po

lygo

ns. T

hey

mea

sure

nec

essa

ry a

ttri

bute

s of

sha

pes

to u

se a

rea

form

ulas

to

solv

e pr

oble

ms.

722–

728, 7

29–

734, 7

35–

740,

831–

836, 8

66–

871, 8

90–

894,

980–

982, 9

83–

985

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

5Ev

eryd

ay M

athe

mat

ics,

Gr

ade

5 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Alge

bra

Stu

dent

s us

e pa

tter

ns, m

odel

s, a

nd r

elat

ions

hips

as

cont

exts

for

w

ritin

g an

d so

lvin

g si

mpl

e eq

uatio

ns a

nd in

equa

litie

s.259–

264, 2

65–

271, 7

22–

728,

729–

734, 7

35–

740, 7

47–

752,

784–

790, 7

91–

796, 7

97–

802,

831–

836, 8

66–

871, 8

90–

894

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s, d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

wri

te r

ules

for

fun

ctio

ns in

volv

ing

the

four

bas

ic

arith

met

ic o

pera

tions

; re

pres

ent

func

tions

usi

ng w

ords

, sym

bols

, tab

les,

and

gr

aphs

and

use

tho

se r

epre

sent

atio

ns t

o so

lve

prob

lem

s.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

5Ev

eryd

ay M

athe

mat

ics

Grad

e 5

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Stu

dent

s us

e pa

tter

ns, m

odel

s, a

nd r

elat

ions

hips

as

cont

exts

for

w

ritin

g an

d so

lvin

g si

mpl

e eq

uatio

ns a

nd in

equa

litie

s.259–

264, 2

65–

271, 7

22–

728,

729–

734, 7

35–

740, 7

47–

752,

784–

790, 7

91–

796, 7

97–

802,

831–

836, 8

66–

871, 8

90–

894

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Det

erm

ine

whe

ther

num

ber

sent

ence

s ar

e tr

ue o

r fa

lse;

sol

ve o

pen

num

ber

sent

ence

s an

d ex

plai

n th

e so

lutio

ns;

use

a le

tter

var

iabl

e to

wri

te a

n op

en

sent

ence

to

mod

el a

num

ber

stor

y; u

se a

pan

-bal

ance

mod

el t

o so

lve

linea

r eq

uatio

ns w

ith o

ne u

nkno

wn.

Stu

dent

s cr

eate

gra

phs

of s

impl

e eq

uatio

ns.

803–

808, 8

09–

813, 8

14–

819

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s, d

escr

ibe

rule

s fo

r pa

tter

ns a

nd

use

them

to

solv

e pr

oble

ms;

wri

te r

ules

for

fun

ctio

ns in

volv

ing

the

four

bas

ic

arith

met

ic o

pera

tions

; re

pres

ent

func

tions

usi

ng w

ords

, sym

bols

, tab

les,

and

gr

aphs

and

use

tho

se r

epre

sent

atio

ns t

o so

lve

prob

lem

s.

Stu

dent

s ex

plor

e pr

ime

and

com

posi

te n

umbe

rs a

nd d

isco

ver

conc

epts

rel

ated

to

the

addi

tion

and

subt

ract

ion

of f

ract

ions

as

they

us

e fa

ctor

s an

d m

ultip

les,

incl

udin

g ap

plic

atio

ns o

f co

mm

on f

acto

rs

and

com

mon

mul

tiple

s. T

hey

deve

lop

an u

nder

stan

ding

of

the

orde

r of

ope

ratio

ns a

nd u

se it

for

all

oper

atio

ns.

42–

46, 5

7–

61, 6

–10, 5

57–

561,

562–

567, 9

14–

919, 9

64–

966,

967–

969

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Iden

tify

prim

e an

d co

mpo

site

num

bers

; fa

ctor

num

bers

; fin

d pr

ime

fact

oriz

atio

ns.

Mea

sure

men

tS

tude

nts’

exp

erie

nces

con

nect

the

ir w

ork

with

sol

ids

and

volu

me

to

thei

r ea

rlie

r w

ork

with

cap

acity

and

wei

ght

or m

ass.

760–

765, 8

66–

871, 8

72–

877,

878–

883, 8

84–

889

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Des

crib

e an

d us

e st

rate

gies

to

find

the

peri

met

er o

f po

lygo

ns a

nd t

he a

rea

of c

ircl

es;

choo

se a

nd u

se a

ppro

pria

te f

orm

ulas

to

calc

ulat

e th

e ar

eas

of

rect

angl

es, p

aral

lelo

gram

s, a

nd t

rian

gles

, and

the

vol

ume

of a

pri

sm;

defin

e pi

as

the

rat

io o

f a

circ

le’s

cir

cum

fere

nce

to it

s di

amet

er.

Stu

dent

s so

lve

prob

lem

s th

at r

equi

re a

tten

tion

to b

oth

appr

oxim

atio

n an

d pr

ecis

ion

of m

easu

rem

ent.

80–

84, 1

03–

108, 1

65–

169,

170–

176, 1

77–

182, 1

83–

188,

199–

205, 2

06–

210, 2

42–

247,

384–

388

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 1

Estim

ate

leng

th w

ith a

nd w

ithou

t to

ols;

mea

sure

leng

th w

ith t

ools

to

the

near

est

1

_

8 i

nch

and

mill

imet

er;

estim

ate

the

mea

sure

of

angl

es w

ith a

nd w

ithou

t to

ols;

use

too

ls t

o dr

aw a

ngle

s w

ith g

iven

mea

sure

s.

Data

Ana

lysi

sS

tude

nts

appl

y th

eir

unde

rsta

ndin

g of

who

le n

umbe

rs, f

ract

ions

, and

de

cim

als

as t

hey

cons

truc

t an

d an

alyz

e do

uble

-bar

and

line

gra

phs

and

use

orde

red

pair

s on

coo

rdin

ate

grid

s.

337–

342, 4

05–

410, 5

68–

572,

704–

709, 7

10–

715, 7

16–

721,

803–

808, 8

09–

813, 8

14–

819,

820–

824, 9

46–

951

Da

ta a

nd

Ch

an

ce G

oa

l 1

Col

lect

and

org

aniz

e da

ta o

r us

e gi

ven

data

to

crea

te b

ar, l

ine,

and

cir

cle

grap

hs

with

rea

sona

ble

title

s, k

eys,

and

inte

rval

s.

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 4

Use

ord

ered

pai

rs o

f nu

mbe

rs t

o na

me,

loca

te, a

nd p

lot

poin

ts in

all

four

qu

adra

nts

of a

coo

rdin

ate

grid

.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

5Ev

eryd

ay M

athe

mat

ics

Grad

e 5

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

5 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

nsB

uild

ing

on t

heir

wor

k in

gra

de 4

, stu

dent

s ex

tend

the

ir u

nder

stan

ding

of

pla

ce v

alue

to

num

bers

thr

ough

mill

ions

and

mill

iont

hs in

var

ious

co

ntex

ts.

80–

84, 1

60–

164, 5

47–

551,

552–

556, 9

77–

979

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs a

nd d

ecim

als;

iden

tify

plac

es in

suc

h nu

mbe

rs

and

the

valu

es o

f th

e di

gits

in t

hose

pla

ces;

use

exp

ande

d no

tatio

n to

rep

rese

nt

who

le n

umbe

rs a

nd d

ecim

als.

They

app

ly w

hat

they

kno

w a

bout

mul

tiplic

atio

n of

who

le n

umbe

rs t

o la

rger

num

bers

. Stu

dent

s al

so e

xplo

re c

onte

xts

that

the

y ca

n de

scri

be

with

neg

ativ

e nu

mbe

rs (

e.g.

, situ

atio

ns o

f ow

ing

mon

ey o

r m

easu

ring

el

evat

ions

abo

ve a

nd b

elow

sea

leve

l).

115–

119, 1

20–

125, 1

26–

131,

80–

84, 5

42–

546, 5

47–

551,

552–

556, 5

73–

577

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs a

nd d

ecim

als;

iden

tify

plac

es in

suc

h nu

mbe

rs

and

the

valu

es o

f th

e di

gits

in t

hose

pla

ces;

use

exp

ande

d no

tatio

n to

rep

rese

nt

who

le n

umbe

rs a

nd d

ecim

als.

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

Use

num

eric

al e

xpre

ssio

ns in

volv

ing

one

or m

ore

of t

he b

asic

fou

r ar

ithm

etic

op

erat

ions

, gro

upin

g sy

mbo

ls, a

nd e

xpon

ents

to

give

equ

ival

ent

nam

es f

or

who

le n

umbe

rs;

conv

ert

betw

een

base

-10, e

xpon

entia

l, an

d re

peat

ed-f

acto

r no

tatio

ns.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Dem

onst

rate

aut

omat

icity

with

mul

tiplic

atio

n fa

cts

and

prof

icie

ncy

with

div

isio

n fa

cts

and

exte

nsio

ns.



NCT

M C

urri

culu

m F

ocal

Poi

nts

and

Ever

yday

Mat

hem

atic

s, G

rade

6

Rep

rinte

d w

ith p

erm

issio

n f

rom

Curr

iculu

m F

ocal P

oin

ts f

or

Pre

kin

derg

art

en t

hro

ug

h G

rad

e 8

Math

em

atics:

A Q

uest

for

Co

here

nce,

co

pyrig

ht

2006 b

y t

he N

atio

nal C

ouncil

of

Teachers

of

Math

em

atics.

All

rig

hts

reserv

ed

. P

lease v

isit w

ww

.nctm

.org

/fo

calp

oin

ts f

or

mo

re info

rmatio

n.

NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 6

Ever

yday

Mat

hem

atic

s Gr

ade

6 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade

Leve

l Goa

ls

Num

ber

and

Oper

atio

ns: D

eve

lop

ing

an

un

de

rsta

nd

ing

of

an

d f

lue

nc

y w

ith

mu

ltip

lic

ati

on

an

d d

ivis

ion

of

fra

cti

on

s a

nd

de

cim

als

Stu

dent

s us

e th

e m

eani

ngs

of f

ract

ions

, mul

tiplic

atio

n an

d di

visi

on,

and

the

inve

rse

rela

tions

hip

betw

een

mul

tiplic

atio

n an

d di

visi

on t

o m

ake

sens

e of

pro

cedu

res

for

mul

tiply

ing

and

divi

ding

fra

ctio

ns a

nd

expl

ain

why

the

y w

ork.

283–

288, 2

89–

293, 5

30–

536,

537–

541

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f fr

actio

ns a

nd m

ixed

nu

mbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

They

use

the

rel

atio

nshi

p be

twee

n de

cim

als

and

frac

tions

, as

wel

l as

the

rel

atio

nshi

p be

twee

n fin

ite d

ecim

als

and

who

le n

umbe

rs (

i.e.,

a fin

ite d

ecim

al m

ultip

lied

by a

n ap

prop

riat

e po

wer

of

10 is

a w

hole

nu

mbe

r), t

o un

ders

tand

and

exp

lain

the

pro

cedu

res

for

mul

tiply

ing

and

divi

ding

dec

imal

s.

123–

126, 1

29–

134, 1

41–

146,

294–

299

Nu

mb

er

an

d N

um

era

tion

Goa

l 1

Rea

d an

d w

rite

who

le n

umbe

rs a

nd d

ecim

als;

iden

tify

plac

es in

suc

h nu

mbe

rs

and

the

valu

es o

f th

e di

gits

in t

hose

pla

ces;

use

exp

ande

d no

tatio

n, n

umbe

r-an

d-w

ord

nota

tion,

exp

onen

tial n

otat

ion,

and

sci

entif

ic n

otat

ion

to r

epre

sent

w

hole

num

bers

and

dec

imal

s.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f w

hole

num

bers

, dec

imal

s,

and

sign

ed n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 5

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber,

deci

mal

, fra

ctio

n, a

nd m

ixed

nu

mbe

r ad

ditio

n, s

ubtr

actio

n, m

ultip

licat

ion,

and

div

isio

n pr

oble

ms;

exp

lain

how

th

e es

timat

es w

ere

obta

ined

.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 6

Ever

yday

Mat

hem

atic

s Gr

ade

6 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade

Leve

l Goa

ls

Stu

dent

s us

e co

mm

on p

roce

dure

s to

mul

tiply

and

div

ide

frac

tions

and

de

cim

als

effic

ient

ly a

nd a

ccur

atel

y.123–

126, 1

29–

134, 2

83–

288,

289–

293, 5

30–

533, 5

37–

541,

See

als

o w

ww

.eve

ryd

aym

ath

emati

cs.c

om

for

Algo

rithm

s Ha

ndbo

ok p

p. 5

0–

61

and

72–

81.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f w

hole

num

bers

, dec

imal

s,

and

sign

ed n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f fr

actio

ns a

nd m

ixed

nu

mbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

They

mul

tiply

and

div

ide

frac

tions

and

dec

imal

s to

sol

ve p

robl

ems,

in

clud

ing

mul

tiste

p pr

oble

ms

and

prob

lem

s in

volv

ing

mea

sure

men

t.289–

293, 6

45–

650, 9

14–

918,

919–

926, 9

27–

930, 9

31–

934,

935–

937, 9

38–

943

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 2

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f w

hole

num

bers

, dec

imal

s,

and

sign

ed n

umbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 4

Use

men

tal a

rith

met

ic, p

aper

-and

-pen

cil a

lgor

ithm

s, a

nd c

alcu

lato

rs t

o so

lve

prob

lem

s in

volv

ing

the

mul

tiplic

atio

n an

d di

visi

on o

f fr

actio

ns a

nd m

ixed

nu

mbe

rs;

desc

ribe

the

str

ateg

ies

used

and

exp

lain

how

the

y w

ork.

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 5

Mak

e re

ason

able

est

imat

es f

or w

hole

num

ber,

deci

mal

, fra

ctio

n, a

nd m

ixed

nu

mbe

r ad

ditio

n, s

ubtr

actio

n, m

ultip

licat

ion,

and

div

isio

n pr

oble

ms;

exp

lain

how

th

e es

timat

es w

ere

obta

ined

.

Num

ber

and

Oper

atio

ns: C

on

ne

cti

ng

ra

tio

an

d r

ate

to

mu

ltip

lic

ati

on

an

d d

ivis

ion

Stu

dent

s us

e si

mpl

e re

ason

ing

abou

t m

ultip

licat

ion

and

divi

sion

to

solv

e ra

tio a

nd r

ate

prob

lem

s (e

.g.,

“If

5 it

ems

cost

$3.7

5 a

nd a

ll ite

ms

are

the

sam

e pr

ice,

the

n I c

an f

ind

the

cost

of

12 it

ems

by f

irst

di

vidi

ng $

3.7

5 b

y 5 t

o fin

d ou

t ho

w m

uch

one

item

cos

ts a

nd t

hen

mul

tiply

ing

the

cost

of

a si

ngle

item

by

12”)

.

202–

207, 6

90–

695, 6

96–

702,

710–

715, 7

22–

728, 7

29–

732,

741–

747

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Use

rat

ios

and

scal

ing

to m

odel

siz

e ch

ange

s an

d to

sol

ve s

ize-

chan

ge

prob

lem

s; r

epre

sent

rat

ios

as f

ract

ions

, per

cent

s, a

nd d

ecim

als,

and

usi

ng

a co

lon;

mod

el a

nd s

olve

pro

blem

s in

volv

ing

part

-to-

who

le a

nd p

art-

to-p

art

ratio

s; m

odel

rat

e an

d ra

tio n

umbe

r st

orie

s w

ith p

ropo

rtio

ns;

use

and

expl

ain

cros

s m

ultip

licat

ion

and

othe

r st

rate

gies

to

solv

e pr

opor

tions

.

By

view

ing

equi

vale

nt r

atio

s an

d ra

tes

as d

eriv

ing

from

, and

ex

tend

ing,

pai

rs o

f ro

ws

(or

colu

mns

) in

the

mul

tiplic

atio

n ta

ble,

an

d by

ana

lyzi

ng s

impl

e dr

awin

gs t

hat

indi

cate

the

rel

ativ

e si

zes

of

quan

titie

s, s

tude

nts

exte

nd w

hole

num

ber

mul

tiplic

atio

n an

d di

visi

on

to r

atio

s an

d ra

tes.

Thu

s, t

hey

expa

nd t

he r

eper

toir

e of

pro

blem

s th

at

they

can

sol

ve b

y us

ing

mul

tiplic

atio

n an

d di

visi

on, a

nd t

hey

build

on

thei

r un

ders

tand

ing

of f

ract

ions

to

unde

rsta

nd r

atio

s.

202–

207, 6

90–

695, 6

96–

702,

703–

709, 7

22–

728, 7

29–

732,

755–

760

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Use

rat

ios

and

scal

ing

to m

odel

siz

e ch

ange

s an

d to

sol

ve s

ize-

chan

ge

prob

lem

s; r

epre

sent

rat

ios

as f

ract

ions

, per

cent

s, a

nd d

ecim

als,

and

usi

ng

a co

lon;

mod

el a

nd s

olve

pro

blem

s in

volv

ing

part

-to-

who

le a

nd p

art-

to-p

art

ratio

s; m

odel

rat

e an

d ra

tio n

umbe

r st

orie

s w

ith p

ropo

rtio

ns;

use

and

expl

ain

cros

s m

ultip

licat

ion

and

othe

r st

rate

gies

to

solv

e pr

opor

tions

.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 6

Ever

yday

Mat

hem

atic

s Gr

ade

6 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade

Leve

l Goa

ls

Stu

dent

s so

lve

a w

ide

vari

ety

of p

robl

ems

invo

lvin

g ra

tios

and

rate

s.202–

207, 6

90–

695, 6

96–

702,

703–

709, 7

10–

715, 7

22–

728,

729–

732, 7

35–

738, 7

41–

747,

748–

754, 7

55–

760, 7

61–

765,

859–

863

Op

era

tion

s a

nd

Com

pu

tati

on

Goa

l 6

Use

rat

ios

and

scal

ing

to m

odel

siz

e ch

ange

s an

d to

sol

ve s

ize-

chan

ge

prob

lem

s; r

epre

sent

rat

ios

as f

ract

ions

, per

cent

s, a

nd d

ecim

als,

and

usi

ng

a co

lon;

mod

el a

nd s

olve

pro

blem

s in

volv

ing

part

-to-

who

le a

nd p

art-

to-p

art

ratio

s; m

odel

rat

e an

d ra

tio n

umbe

r st

orie

s w

ith p

ropo

rtio

ns;

use

and

expl

ain

cros

s m

ultip

licat

ion

and

othe

r st

rate

gies

to

solv

e pr

opor

tions

.

Alge

bra:

Wri

tin

g,

inte

rpre

tin

g,

an

d u

sin

g m

ath

em

ati

ca

l e

xp

ressio

ns a

nd

eq

ua

tio

ns

Stu

dent

s w

rite

mat

hem

atic

al e

xpre

ssio

ns a

nd e

quat

ions

tha

t co

rres

pond

to

give

n si

tuat

ions

, the

y ev

alua

te e

xpre

ssio

ns, a

nd t

hey

use

expr

essi

ons

and

form

ulas

to

solv

e pr

oble

ms.

180–

182, 1

85–

190, 1

91–

195,

196–

201, 2

02–

205, 2

08–

213,

703–

707, 7

22–

726, 7

29–

732,

798–

803, 8

16–

821, 8

46–

851,

852–

858

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns

and

use

them

to

solv

e pr

oble

ms;

rep

rese

nt p

atte

rns

and

rule

s us

ing

alge

brai

c no

tatio

n; r

epre

sent

fun

ctio

ns u

sing

wor

ds, a

lgeb

raic

not

atio

n, t

able

s, a

nd

grap

hs;

tran

slat

e fr

om o

ne r

epre

sent

atio

n to

ano

ther

and

use

rep

rese

ntat

ions

to

solv

e pr

oble

ms

invo

lvin

g fu

nctio

ns.

They

und

erst

and

that

var

iabl

es r

epre

sent

num

bers

who

se e

xact

va

lues

are

not

yet

spe

cifie

d, a

nd t

hey

use

vari

able

s ap

prop

riat

ely.

180–

182, 1

85–

190, 1

91–

195,

196–

201, 2

02–

205, 2

08–

213,

571–

576, 7

03–

707, 7

22–

726,

729–

732, 7

98–

803, 8

16–

821,

846–

851, 8

52–

858

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns

and

use

them

to

solv

e pr

oble

ms;

rep

rese

nt p

atte

rns

and

rule

s us

ing

alge

brai

c no

tatio

n; r

epre

sent

fun

ctio

ns u

sing

wor

ds, a

lgeb

raic

not

atio

n, t

able

s, a

nd

grap

hs;

tran

slat

e fr

om o

ne r

epre

sent

atio

n to

ano

ther

and

use

rep

rese

ntat

ions

to

solv

e pr

oble

ms

invo

lvin

g fu

nctio

ns.

Stu

dent

s un

ders

tand

tha

t ex

pres

sion

s in

diff

eren

t fo

rms

can

be

equi

vale

nt, a

nd t

hey

can

rew

rite

an

expr

essi

on t

o re

pres

ent

a qu

antit

y in

a d

iffer

ent

way

(e.

g., t

o m

ake

it m

ore

com

pact

or

to f

eatu

re

diff

eren

t in

form

atio

n).

191–

195, 7

92–

797, 7

98–

803,

804–

809, 8

10–

815, 8

46–

851

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Det

erm

ine

whe

ther

equ

aliti

es a

nd in

equa

litie

s ar

e tr

ue o

r fa

lse;

sol

ve o

pen

num

ber

sent

ence

s an

d ex

plai

n th

e so

lutio

ns;

use

a pa

n-ba

lanc

e m

odel

to

solv

e lin

ear

equa

tions

in o

ne o

r tw

o un

know

ns;

use

tria

l-an

d-er

ror

and

equi

vale

nt

equa

tion

stra

tegi

es t

o so

lve

linea

r eq

uatio

ns in

one

unk

now

n.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 3

Des

crib

e an

d ap

ply

the

conv

entio

nal o

rder

of

oper

atio

ns.

Stu

dent

s kn

ow t

hat

the

solu

tions

of

an e

quat

ion

are

the

valu

es o

f th

e va

riab

les

that

mak

e th

e eq

uatio

n tr

ue.

571–

576, 5

77–

583, 5

84–

589,

590–

595, 7

03–

709, 7

22–

726,

729–

732, 8

10–

815, 8

16–

821,

840–

845, 8

46–

851, 8

52–

858

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Det

erm

ine

whe

ther

equ

aliti

es a

nd in

equa

litie

s ar

e tr

ue o

r fa

lse;

sol

ve o

pen

num

ber

sent

ence

s an

d ex

plai

n th

e so

lutio

ns;

use

a pa

n-ba

lanc

e m

odel

to

solv

e lin

ear

equa

tions

in o

ne o

r tw

o un

know

ns;

use

tria

l-an

d-er

ror

and

equi

vale

nt

equa

tion

stra

tegi

es t

o so

lve

linea

r eq

uatio

ns in

one

unk

now

n.



NCT

M C

urri

culu

m F

ocal

Poi

nts

for

Grad

e 6

Ever

yday

Mat

hem

atic

s Gr

ade

6 Te

ache

r’s

Less

on G

uide

Pag

es

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade

Leve

l Goa

ls

They

sol

ve s

impl

e on

e-st

ep e

quat

ions

by

usin

g nu

mbe

r se

nse,

pr

oper

ties

of o

pera

tions

, and

the

idea

of

mai

ntai

ning

equ

ality

on

both

si

des

of a

n eq

uatio

n.

571–

576, 5

77–

583, 5

84–

589,

590–

595, 7

03–

709, 7

22–

726,

729–

732, 8

10–

815, 8

16–

821,

840–

845, 8

46–

851, 8

52–

858

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 2

Det

erm

ine

whe

ther

equ

aliti

es a

nd in

equa

litie

s ar

e tr

ue o

r fa

lse;

sol

ve o

pen

num

ber

sent

ence

s an

d ex

plai

n th

e so

lutio

ns;

use

a pa

n-ba

lanc

e m

odel

to

solv

e lin

ear

equa

tions

in o

ne o

r tw

o un

know

ns;

use

tria

l-an

d-er

ror

and

equi

vale

nt

equa

tion

stra

tegi

es t

o so

lve

linea

r eq

uatio

ns in

one

unk

now

n.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 3

Des

crib

e an

d ap

ply

the

conv

entio

nal o

rder

of

oper

atio

ns.

They

con

stru

ct a

nd a

naly

ze t

able

s (e

.g.,

to s

how

qua

ntiti

es t

hat

are

in e

quiv

alen

t ra

tios)

, and

the

y us

e eq

uatio

ns t

o de

scri

be s

impl

e re

latio

nshi

ps (

such

as

3x

= y

) sh

own

in a

tab

le.

202–

207, 2

08–

213, 2

30–

235,

690–

695, 6

96–

700

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns

and

use

them

to

solv

e pr

oble

ms;

rep

rese

nt p

atte

rns

and

rule

s us

ing

alge

brai

c no

tatio

n; r

epre

sent

fun

ctio

ns u

sing

wor

ds, a

lgeb

raic

not

atio

n, t

able

s, a

nd

grap

hs;

tran

slat

e fr

om o

ne r

epre

sent

atio

n to

ano

ther

and

use

rep

rese

ntat

ions

to

solv

e pr

oble

ms

invo

lvin

g fu

nctio

ns.

NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

6Ev

eryd

ay M

athe

mat

ics

Grad

e 6

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade-

Leve

l Goa

ls

Num

ber

and

Oper

atio

nsS

tude

nts’

wor

k in

div

idin

g fr

actio

ns s

how

s th

em t

hat

they

can

ex

pres

s th

e re

sult

of d

ivid

ing

two

who

le n

umbe

rs a

s a

frac

tion

(vie

wed

as

part

s of

a w

hole

).

202–

207, 3

00–

306, 5

37–

541

Nu

mb

er

an

d N

um

era

tion

Goa

l 3

Use

GC

Fs, L

CM

s, a

nd d

ivis

ibili

ty r

ules

to

man

ipul

ate

frac

tions

.

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Find

equ

ival

ent

frac

tions

and

fra

ctio

ns in

sim

ples

t fo

rm b

y ap

plyi

ng

mul

tiplic

atio

n an

d di

visi

on r

ules

and

con

cept

s fr

om n

umbe

r th

eory

; co

nver

t be

twee

n fr

actio

ns, m

ixed

num

bers

, dec

imal

s, a

nd p

erce

nts.

Stu

dent

s th

en e

xten

d th

eir

wor

k in

gra

de 5

with

div

isio

n of

who

le

num

bers

to

give

mix

ed n

umbe

r an

d de

cim

al s

olut

ions

to

divi

sion

pr

oble

ms

with

who

le n

umbe

rs.

135–

139, 1

41–

144, 3

00–

306

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Find

equ

ival

ent

frac

tions

and

fra

ctio

ns in

sim

ples

t fo

rm b

y ap

plyi

ng

mul

tiplic

atio

n an

d di

visi

on r

ules

and

con

cept

s fr

om n

umbe

r th

eory

; co

nver

t be

twee

n fr

actio

ns, m

ixed

num

bers

, dec

imal

s, a

nd p

erce

nts.

They

rec

ogni

ze t

hat

ratio

tab

les

not

only

der

ive

from

row

s in

the

m

ultip

licat

ion

tabl

e bu

t al

so c

onne

ct w

ith e

quiv

alen

t fr

actio

ns.

202–

207, 2

30–

235, 6

90–

695,

696–

702

Nu

mb

er

an

d N

um

era

tion

Goa

l 5

Find

equ

ival

ent

frac

tions

and

fra

ctio

ns in

sim

ples

t fo

rm b

y ap

plyi

ng

mul

tiplic

atio

n an

d di

visi

on r

ules

and

con

cept

s fr

om n

umbe

r th

eory

; co

nver

t be

twee

n fr

actio

ns, m

ixed

num

bers

, dec

imal

s, a

nd p

erce

nts.



NCT

M C

onne

ctio

ns t

o th

e Fo

cal

Poin

ts f

or G

rade

6Ev

eryd

ay M

athe

mat

ics

Grad

e 6

Teac

her’

s Le

sson

Gui

de P

ages

Ever

yday

Mat

hem

atic

s Gr

ade

6 Gr

ade-

Leve

l Goa

ls

Stu

dent

s di

stin

guis

h m

ultip

licat

ive

com

pari

sons

fro

m a

dditi

ve

com

pari

sons

.191–

195, 7

92–

797, 8

28–

833

Nu

mb

er

an

d N

um

era

tion

Goa

l 2

Sol

ve p

robl

ems

invo

lvin

g pe

rcen

ts a

nd d

isco

unts

; ex

plai

n st

rate

gies

use

d;

iden

tify

the

unit

who

le in

situ

atio

ns in

volv

ing

frac

tions

, dec

imal

s, a

nd p

erce

nts.

Alge

bra

Stu

dent

s us

e th

e co

mm

utat

ive,

ass

ocia

tive,

and

dis

trib

utiv

e pr

oper

ties

to s

how

tha

t tw

o ex

pres

sion

s ar

e eq

uiva

lent

.786–

791, 7

92–

797, 7

98–

801,

804–

809, 8

10–

815

Nu

mb

er

an

d N

um

era

tion

Goa

l 4

App

ly t

he o

rder

of

oper

atio

ns t

o nu

mer

ical

exp

ress

ions

to

give

equ

ival

ent

nam

es

for

ratio

nal n

umbe

rs.

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

prop

ertie

s of

ari

thm

etic

and

mul

tiplic

ativ

e an

d ad

ditiv

e in

vers

es.

They

als

o ill

ustr

ate

prop

ertie

s of

ope

ratio

ns b

y sh

owin

g th

at t

wo

expr

essi

ons

are

equi

vale

nt in

a g

iven

con

text

(e.

g., d

eter

min

ing

the

area

in t

wo

diff

eren

t w

ays

for

a re

ctan

gle

who

se d

imen

sion

s ar

e x

+ 3

by

5).

786–

791, 7

92–

797, 8

04–

809

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 4

Des

crib

e an

d ap

ply

prop

ertie

s of

ari

thm

etic

and

mul

tiplic

ativ

e an

d ad

ditiv

e in

vers

es.

Seq

uenc

es, i

nclu

ding

tho

se t

hat

aris

e in

the

con

text

of

findi

ng

poss

ible

rul

es f

or p

atte

rns

of f

igur

es o

r st

acks

of

obje

cts,

pro

vide

op

port

uniti

es f

or s

tude

nts

to d

evel

op f

orm

ulas

.

180–

184, 1

91–

195, 1

96–

201,

846–

851

Pa

ttern

s, F

un

cti

on

s, a

nd

Alg

eb

ra G

oa

l 1

Exte

nd, d

escr

ibe,

and

cre

ate

num

eric

pat

tern

s; d

escr

ibe

rule

s fo

r pa

tter

ns

and

use

them

to

solv

e pr

oble

ms;

rep

rese

nt p

atte

rns

and

rule

s us

ing

alge

brai

c no

tatio

n; r

epre

sent

fun

ctio

ns u

sing

wor

ds, a

lgeb

raic

not

atio

n, t

able

s, a

nd

grap

hs;

tran

slat

e fr

om o

ne r

epre

sent

atio

n to

ano

ther

and

use

rep

rese

ntat

ions

to

solv

e pr

oble

ms

invo

lvin

g fu

nctio

ns.

Mea

sure

men

t an

d Ge

omet

ryP

robl

ems

that

invo

lve

area

s an

d vo

lum

es, c

allin

g on

stu

dent

s to

fin

d ar

eas

or v

olum

es f

rom

leng

ths

or t

o fin

d le

ngth

s fr

om v

olum

es o

r ar

eas

and

leng

ths,

are

esp

ecia

lly a

ppro

pria

te. T

hese

pro

blem

s ex

tend

th

e st

uden

ts’

wor

k in

gra

de 5

on

area

and

vol

ume

and

prov

ide

a co

ntex

t fo

r ap

plyi

ng n

ew w

ork

with

equ

atio

ns.

65–

70, 1

96–

201, 8

28–

833,

834–

839, 8

46–

851

Mea

sure

men

t a

nd

Refe

ren

ce F

ram

es

Goa

l 2

Cho

ose

and

use

appr

opri

ate

form

ulas

to

calc

ulat

e th

e ci

rcum

fere

nce

of c

ircl

es

and

to s

olve

are

a, p

erim

eter

, and

vol

ume

prob

lem

s.


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