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RETEACH Workbook

Grade 4

HS

PH

SP

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Unit 1: UNDERSTAND WHOLE NUMBERS AND OPERATIONS

Chapter 1: Understand Place Value1.1 Place Value Through

Hundred Thousands . . . . . . . . . . . . RW11.2 Model Millions . . . . . . . . . . . . . . . . . . RW21.3 Place Value Through

Millions . . . . . . . . . . . . . . . . . . . . . . . . . . . RW31.4 Compare Whole Numbers. . . . . RW41.5 Order Whole Numbers. . . . . . . . . RW51.6 Problem Solving Workshop

Strategy: Use Logical Reasoning . . . . . . . . . . . . . . . . . . . . . . . . RW6

Chapter 2: Add and Subtract Whole Numbers

2.1 Algebra: Relate Addition and Subtraction . . . . . . . . . . . . . . . . . RW7

2.2 Round Whole Numbers Through Millions . . . . . . . . . . . . . . . RW8

2.3 Mental Math: Estimate Sums and Differences . . . . . . . . . . RW9

2.4 Mental Math Strategies . . . . . . . . . . . . . . . . . . . . . . RW10

2.5 Problem Solving Workshop Skill: Estimate or Exact Answer . . . . . . . . . . . . . . . . . . RW11

2.6 Add and Subtract 3-Digit and4-Digit Numbers. . . . . . . . . . . . . . . RW12

2.7 Choose a Method. . . . . . . . . . . . . RW13

Chapter 3: Algebra: Use Addition and Subtraction

3.1 Addition Properties . . . . . . . . . . RW143.2 Write and Evaluate

Expressions . . . . . . . . . . . . . . . . . . . . . RW153.3 Addition and Subtraction

Equations . . . . . . . . . . . . . . . . . . . . . RW163.4 Problem Solving Workshop

Strategy: Work Backward . . . RW173.5 Patterns: Find a Rule . . . . . . . . RW18

Unit 2: MULTIPLICATION AND DIVISION FACTS

Chapter 4: Multiplication and Division Facts4.1 Algebra: Relate

Operations . . . . . . . . . . . . . . . . . . . . . RW194.2 Algebra: Relate Multiplication

and Division . . . . . . . . . . . . . . . . . . . RW204.3 Multiply and Divide

Facts Through 5 . . . . . . . . . . . . . . . RW214.4 Multiply and Divide

Facts Through 10 . . . . . . . . . . . . . RW224.5 Multiplication Table

Through 12 . . . . . . . . . . . . . . . . . . . . RW234.6 Patterns on the

Multiplication Table . . . . . . . . . . RW244.7 Problem Solving Workshop

Skill: Choose the Operation . . . . . . . . . . . . . . . . . . . . . . RW25

4.8 Algebra: Find Missing Factors . . . . . . . . . . . . . . . . . . . . . . . . . . RW26

Chapter 5: Algebra: Use Multiplication and Division Facts

5.1 Multiplication Properties . . . . . . . . . . . . . . . . . . . . . . RW27

5.2 Multiplication and Division Expressions . . . . . . . . . . . . . . . . . . . . . RW28

5.3 Order of Operations . . . . . . . . . RW295.4 Multiplication and Division

Equations . . . . . . . . . . . . . . . . . . . . . . RW305.5 Problem Solving Workshop

Strategy: Predict and Test . . . RW315.6 Explore Inequalities . . . . . . . . . . RW325.7 Patterns: Find a Rule . . . . . . . . . RW33

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Unit 3: TIME, TEMPERATURE, AND DATA

Chapter 6: Time and Temperature 6.1 Telling Time . . . . . . . . . . . . . . . . . . . . RW346.2 Elapsed Time . . . . . . . . . . . . . . . . . . RW356.3 Elapsed Time on

a Calendar . . . . . . . . . . . . . . . . . . . . . RW366.4 Algebra: Change Units

of Time . . . . . . . . . . . . . . . . . . . . . . . . . RW376.5 Problem Solving Workshop

Skill: Sequence Informaiton . . . . . . . . . . . . . . . . . . . . RW38

6.6 Temperature . . . . . . . . . . . . . . . . . . . RW396.7 Explore Negative Numbers . . RW40

Chapter 7: Collect and Organize Data7.1 Collect and Organize Data . . RW417.2 Venn Diagrams . . . . . . . . . . . . . . . . RW427.3 Find Mean, Median,

and Mode . . . . . . . . . . . . . . . . . . . . . . RW437.4 Line Plots . . . . . . . . . . . . . . . . . . . . . . . RW447.5 Choose a Reasonable

Scale and Interval . . . . . . . . . . . . . RW457.6 Problem Solving Workshop

Skill: Make Generalizations . . . . . . . . . . . . . . . RW46

Chapter 8: Interpret and Graph Data8.1 Bar Graphs . . . . . . . . . . . . . . . . . . . . . RW478.2 Make Bar and Double-Bar

Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . RW488.3 Circle Graphs . . . . . . . . . . . . . . . . . . RW498.4 Algebra: Use a Coordinate

Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RW508.5 Line Graphs . . . . . . . . . . . . . . . . . . . . RW518.6 Make Line Graphs . . . . . . . . . . . . RW528.7 Problem Solving Workshop

Strategy: Make a Graph . . . . . RW538.8 Choose an

Appropriate Graph . . . . . . . . . . . RW548.9 Algebra: Graph

Relationships . . . . . . . . . . . . . . . . . . RW558.10 Problem Solving Workshop

Skill: Identify Relationships . . . RW56

Unit 4: MULTIPLY BY 1-DIGIT AND 2-DIGIT NUMBERS

Chapter 9: Multiply by 1-Digit Numbers 9.1 Mental Math: Multiplication

Patterns . . . . . . . . . . . . . . . . . . . . . . . . RW579.2 Mental Math: Estimate

Products . . . . . . . . . . . . . . . . . . . . . . . . RW589.3 Model 2-Digit by 1-Digit

Multiplication . . . . . . . . . . . . . . . . . RW599.4 Record 2-Digit by 1-Digit

Multiplication . . . . . . . . . . . . . . . . . RW609.5 Multiply 3-Digit and 4-Digit

Numbers and Money . . . . . . . . . RW619.6 Multiply with Zeros . . . . . . . . . . RW629.7 Problem Solving Workshop

Skill: Evaluate Reasonableness . . . . . . . . . . . . . . . RW63

Chapter 10: Multiply by 2-Digit Numbers10.1 Mental Math: Multiplication

Patterns . . . . . . . . . . . . . . . . . . . . . . . . RW6410.2 Multiply by Tens . . . . . . . . . . . . . . RW6510.3 Mental Math: Estimate

Products . . . . . . . . . . . . . . . . . . . . . . . . RW6610.4 Problem Solving Workshop

Strategy: Solve a Simpler Problem . . . . . . . . . . . . . . RW67

10.5 Model 2-Digit by 2-Digit Multiplication . . . . . . . . . . . . . . . . . RW68

10.6 Record 2-Digit by 2-Digit Multiplication . . . . . . . . . . . . . . . . . RW69

10.7 Multiply 2-Digit and 3-Digit Numbers and Money . . . . . . . . . RW70

10.8 Choose a Method . . . . . . . . . . . . . RW7110.9 Problem Solving Workshop

Skill: Multistep Problems . . . . RW72


Unit 5: DIVIDE BY 1-DIGIT AND 2-DIGIT DIVISORS

Chapter 11: Understand Division11.1 Divide with Remainders . . . . . RW7311.2 Model 2-Digit by 1-Digit

Division . . . . . . . . . . . . . . . . . . . . . . . . . RW7411.3 Record 2-Digit by 1-Digit

Division . . . . . . . . . . . . . . . . . . . . . . . . . RW7511.4 Problem Solving Workshop

Strategy: Draw a Diagram . . . RW7611.5 Mental Math:

Division Patterns . . . . . . . . . . . . . . RW7711.6 Mental Math:

Estimate Quotients . . . . . . . . . . . RW7811.7 Place the First Digit . . . . . . . . . . RW79

Chapter 12: Practice Division12.1 Problem Solving Workshop

Skill: Interpret the Remainder . . . . . . . . . . . . . . . . . . . . . RW80

12.2 Divide 3-Digit Numbers and Money . . . . . . . . . . . . . . . . . . . . RW81

12.3 Zeros in Division . . . . . . . . . . . . . . RW8212.4 Choose a Method . . . . . . . . . . . . . RW83

Chapter 13: Divide by 2-Digit Divisors13.1 Estimate Quotients . . . . . . . . . . . RW8413.2 Model Division by 2-Digit

Divisors . . . . . . . . . . . . . . . . . . . . . . . . . RW8513.3 Record Division . . . . . . . . . . . . . . . RW8613.4 Adjusting Quotients . . . . . . . . . . RW8713.5 Problem Solving Workshop

Skill: Too Much/Too LittleInformation . . . . . . . . . . . . . . . . . . . . RW88

Chapter 14: Number Theory and Patterns14.1 Factors and Multiples . . . . . . . . RW8914.2 Divisibility Rules . . . . . . . . . . . . . . . RW9014.3 Prime and Composite

Numbers . . . . . . . . . . . . . . . . . . . . . . . RW9114.4 Number Patterns . . . . . . . . . . . . . . RW9214.5 Problem Solving Workshop

Strategy: Find a Pattern . . . . . RW93

Unit 6: FRACTIONS AND DECIMALS

Chapter 15: Understand Fractions and Mixed Numbers

15.1 Read and Write Fractions . . . RW9415.2 Model Equivalent

Fractions . . . . . . . . . . . . . . . . . . . . . . . RW9515.3 Compare and Order

Fractions . . . . . . . . . . . . . . . . . . . . . . . RW9615.4 Read and Write Mixed

Numbers . . . . . . . . . . . . . . . . . . . . . . . RW9715.5 Compare and Order

Mixed Numbers . . . . . . . . . . . . . . . RW9815.6 Problem Solving Workshop

Skill: Sequence Information . . . . . . . . . . . . . . . . . . . . RW99

Chapter 16: Add and Subtract Fractions and Mixed Numbers

16.1 Model Addition . . . . . . . . . . . . . . RW10016.2 Model Subtraction . . . . . . . . . . RW10116.3 Record Addition

and Subtraction . . . . . . . . . . . . . . RW10216.4 Problem Solving Workshop

Strategy: Write an Equation . . . . . . . . . . . . . . . . . . . . . . RW103

16.5 Add and Subtract Mixed Numbers . . . . . . . . . . . . . . RW104

16.6 Model Addition and Subtraction of Unlike Fractions . . . . . . . . . . . . . . . . . . . . . . RW105

Chapter 17: Understand Decimals and Place Value

17.1 Relate Fractions and Decimals . . . . . . . . . . . . . . . . . RW106

17.2 Decimals to Thousandths . . . RW10717.3 Equivalent Decimals . . . . . . . . . RW10817.4 Relate Mixed Numbers

and Decimals . . . . . . . . . . . . . . . . . RW10917.5 Compare and

Order Decimals . . . . . . . . . . . . . . . RW11017.6 Problem Solving Workshop

Skill: Draw Conclusions . . . . . RW111


Chapter 18: Add and Subtract Decimals and Money

18.1 Round Decimals . . . . . . . . . . . . . . RW11218.2 Estimate Decimal Sums

and Differences . . . . . . . . . . . . . . RW11318.3 Model Addition . . . . . . . . . . . . . . RW11418.4 Model Subtraction . . . . . . . . . . RW11518.5 Record Addition

and Subtraction . . . . . . . . . . . . . . RW11618.6 Make Change . . . . . . . . . . . . . . . . RW11718.7 Problem Solving Workshop

Strategy: Compare Strategies . . . . . . . . . . . . . . . . . . . . . RW118

Unit 7: GEOMETRY

Chapter 19: Lines, Rays, Angles, and Plane Figures

19.1 Points, Lines, and Rays . . . . . . RW11919.2 Measure and

Classify Angles . . . . . . . . . . . . . . . RW12019.3 Line Relationships . . . . . . . . . . . RW12119.4 Polygons . . . . . . . . . . . . . . . . . . . . . . RW12219.5 Classify Triangles . . . . . . . . . . . . . RW12319.6 Classify Quadrilaterals . . . . . . RW12419.7 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . RW12519.8 Problem Solving

Workshop Strategy: Compare Strategies . . . . . . . . . RW126

Chapter 20: Motion Geometry20.1 Congruent and

Similar Figures . . . . . . . . . . . . . . . RW12720.2 Turns and Symmetry . . . . . . . .RW12820.3 Transformations . . . . . . . . . . . . . . RW12920.4 Problem Solving Workshop

Strategy: Act It Out . . . . . . . . . RW13020.5 Tessellations . . . . . . . . . . . . . . . . . . RW13120.6 Geometric Patterns . . . . . . . . . . RW132

Chapter 21: Solid Figures21.1 Faces, Edges, and Vertices . . RW13321.2 Draw Figures . . . . . . . . . . . . . . . . . RW13421.3 Patterns for Solid Figures . . RW13521.4 Different Views of

Solid Figures . . . . . . . . . . . . . . . . . . RW13621.5 Problem Solving Workshop

Strategy: Make a Model . . . . RW13721.6 Combine and

Divide Figures . . . . . . . . . . . . . . . . . RW138

Unit 8: MEASUREMENT AND PROBABILITY

Chapter 22: Customary and Metric Measurement22.1 Measure Fractional Parts . . . RW13922.2 Algebra: Change

Customary Linear Units . . . . . RW14022.3 Weight . . . . . . . . . . . . . . . . . . . . . . . . RW14122.4 Customary Capacity . . . . . . . . . RW14222.5 Problem Solving

Workshop Strategy: Compare Strategies . . . . . . . . . RW143

22.6 Metric Length . . . . . . . . . . . . . . . . RW14422.7 Algebra: Change Metric

Linear Units . . . . . . . . . . . . . . . . . . . RW14522.8 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . RW14622.9 Metric Capacity . . . . . . . . . . . . . . RW14722.10 Problem Solving Workshop

Strategy: Make a Table . . . . . RW148

Chapter 23: Perimeter, Area, and Volume23.1 Estimate and Measure

Perimeter . . . . . . . . . . . . . . . . . . . . . RW14923.2 Algebra: Find Perimeter . . . . RW15023.3 Area of Plane Figures . . . . . . . RW15123.4 Algebra: Find Area . . . . . . . . . . RW15223.5 Problem Solving Workshop

Skill: Use a Formula . . . . . . . . . RW15323.6 Relate Perimeter

and Area . . . . . . . . . . . . . . . . . . . . . . RW15423.7 Estimate and Find Volume

of Prisms . . . . . . . . . . . . . . . . . . . . . . RW155

Chapter 24: Probability24.1 List All Possible

Outcomes . . . . . . . . . . . . . . . . . . . . . RW15624.2 Problem Solving Workshop

Strategy: Make an Organized List . . . . . . . . . . . . . . . . RW157

24.3 Predict Outcomes of Experiments . . . . . . . . . . . . . . . . . . RW158

24.4 Probability as a Fraction . . . . RW15924.5 Experimental Probability . . . RW16024.6 Combinations and

Arrangements . . . . . . . . . . . . . . . RW161


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ed

3.

428,

391,

032

4. W

rite

the

valu

e of

the

dig

it 6.

5.

Writ

e th

e di

git in

the

ten

mill

ions

pl

ace.

PER

IOD

Hun

dred

s T

ens

One

s

0

5 8

Hun

dred

s T

ens

One

s

0

0 9,

Hun

dred

s T

ens

O

nes

3

6 5,

MIL

LIO

NS

THO

US

AN

DS

ON

ES

MX

EN

L08

AW

K4X

_RT

_CH

01_L

3.in

ddA

6/14

/07

11:5

2:16

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

4

1. 4

,092

,332

9

93,7

932.

$2,

092,

092

$2

,102

,002

3. 3

37,4

93

377

,943

4. $

5,00

0,00

0

$500

,000

5. 7

,192

,322

1

,797

,322

6. 4

,002

,384

4

,020

,000

7. 8

,344

,475

8

,344

,475

8. $

5,49

2,00

0

$492

,000

9. 4

,928

,388

4

,923

,488

10.

7,08

4,12

2

7,18

7,08

4

11.

1,20

3,43

7

1,20

3,44

512

. $5

28,8

07,4

14

$5,6

99,0

01

Co

mp

are

. Wri

te <

, >

, or

= f

or

eac

h

.

Com

pare

Who

le N

umbe

rsU

sele

ss t

han

( < ),

gre

ate

r th

an (

> ),

or

eq

ual t

o (

= )

to c

ompa

re n

umbe

rs.

Co

mp

are

. Wri

te <

, >

, or

= .

$631

,328

or

$640

,009

• W

rite

one

num

ber

unde

r th

e ot

her.

L

ine

up li

ke p

lace

s. $631

,328

$6

40,0

09

• C

ompa

re the

dig

its, b

egin

ning

with

the

gre

ates

t pl

ace

valu

e po

sitio

n.

$631

,328

$6

40,0

09

• C

ircle

the

firs

t di

gits

tha

t ar

e di

ffere

nt.

$6

31,3

28

$6

40,

009

• Si

nce

4 te

n th

ousa

nds

> 3

ten

tho

usan

ds, t

hen

$640

,009

> $

631,

328.

MX

EN

L08

AW

K4X

_RT

_CH

01_L

4.in

ddA

6/14

/07

11:5

4:30

AM

RW3-RW4


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

5

Ord

er W

hole

Num

bers

Whe

n yo

u pu

t w

hole

num

bers

in o

rder

, you

find

whi

ch n

umbe

rs h

ave

the

low

est an

d hi

ghes

t va

lues

.

Yo

u ca

n co

mp

are

eac

h nu

mb

er’

s d

igit

s to

ord

er

the

m f

rom

leas

t to

gre

ate

st.

Wri

te t

he n

umb

ers

in o

rde

r fr

om

leas

t to

gre

ate

st.

$55,

997;

$57

,000

; $56

,038

• A

lign

the

num

bers

und

er e

ach

othe

r.

$55

,997

$

57,0

00

$56

,038

• C

ompa

re the

dig

its b

egin

ning

with

the

gre

ates

t pl

ace

valu

e po

sitio

n.

• C

ircle

the

firs

t gr

oup

of d

igits

tha

t ar

e di

ffere

nt.

$55,

997

$

57,0

00

$56

,038

Sinc

e5

< 6

, and

6 <

7, t

he n

umbe

rs in

ord

er fr

om le

ast to

gre

ates

t ar

e:$5

5,99

7; $

56,0

38; $

57,0

00.

Wri

te t

he n

umb

ers

in o

rde

r fr

om

leas

t to

gre

ate

st.

1.78

7,92

5; 1

,056

,000

; 789

,100

2.94

,299

; 82,

332;

100

,554

3.

$1,3

54,2

99; $

1,94

2,33

2; $

1,30

0,55

4

4.7,

234,

000;

6,3

11,0

94; 2

,102

,444

$57,

000

$55,

997

• St

art w

ith the

dig

it in

the

ten

tho

usan

ds p

lace

.•

The

digi

t 5

is the

sam

e in

all

3 nu

mbe

rs•

The

next

dig

it is

in the

tho

usan

ds p

lace

.$5

6,03

8

• C

ompa

re the

dig

its 5

,7, a

nd 6

.•

Seve

n is

gre

ater

tha

n 5

and

6, s

o 57

,000

is the

gre

ates

t nu

mbe

r.•

Five

is le

ss tha

n 6,

so

55,9

97 is

less

tha

n 56

,038

.

MX

EN

L08

AW

K4X

_RT

_CH

01_L

5.in

dd1

6/14

/07

11:5

6:45

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

6

Shar

i’s s

occe

r te

am w

on e

very

gam

e of

th

e se

ason

. Her

tea

m p

laye

d ev

ery

Satu

rday

and

Sun

day

for

thre

e w

eeks

ea

ch m

onth

. Sup

pose

the

y pl

ayed

thi

s sc

hedu

le fo

r th

ree

mon

ths.

How

man

y ga

mes

did

Sha

ri’s

socc

er tea

m w

in?

Mr.

Lee’

s cl

ass

sold

140

muf

fins

and

235

cook

ies

at the

bak

e sa

le. T

he

num

ber

of c

upca

kes

sold

was

bet

wee

n th

e nu

mbe

r of

muf

fins

and

cook

ies

sold

. Wha

t is

the

gre

ates

t nu

mbe

r of

cu

pcak

es the

cla

ss c

ould

hav

e so

ld?

6.7.

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Use

Logi

cal R

easo

ning

Wha

t if

a b

aske

tbal

l te

am h

as a

sco

re t

hat

is a

2-d

igit

num

be

r.

The

sum

of

the

dig

its

is 8

. The

diff

ere

nce

be

twe

en

the

dig

its

is 2

. Th

e t

ens

dig

it is

less

tha

n th

e o

nes

dig

it.

Wha

t is

the

sco

re?

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

Pla

n

2.

How

can

usi

ng lo

gica

l rea

soni

ng h

elp

you

solv

e th

e pr

oble

m?

So

lve

3.

Wha

t is

the

sco

re?

Des

crib

e ho

w y

ou u

sed

logi

cal r

easo

ning

.

4.

Writ

e yo

ur a

nsw

er in

a c

ompl

ete

sent

ence

.

Ch

eck

5.

Loo

k at

the

pro

blem

. D

oes

the

answ

er m

ake

sens

e fo

r th

e pr

oble

m?

Expl

ain.

So

lve

by

usin

g lo

gic

al r

eas

oni

ng.

MX

EN

L08

AW

K4X

_RT

_CH

01_L

6.in

dd

A6/

6/07

1:

55:0

3 PM

RW5-RW6


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

7

6 1

5

13

1.Alge

bra:

Rel

ate

Addi

tion

and

Subt

ract

ion

You

can

sol

ve a

dditi

on a

nd s

ubtr

actio

n se

nten

ces

by u

sing

re

late

d fa

cts.

For

exa

mpl

e, 7

+ 8

= 1

5 an

d 8

+ 7

= 1

5 ar

e re

late

d ad

ditio

n fa

cts;

15

– 8

= 7

and

15

– 7

= 8

are

rel

ated

su

btra

ctio

n fa

cts.

Wri

te a

re

late

d f

act.

Use

it t

o c

om

ple

te t

he n

umb

er

sent

enc

e.

2

9 5

6

Use

an

inve

rse

oper

atio

n.

• A

n in

vers

e o

pe

rati

on

is a

n op

posi

te o

pera

tion.

Add

ition

is the

inve

rse,

or

oppo

site

, ope

ratio

n of

sub

trac

tion.

Sub

trac

tion

is the

opp

osite

of a

dditi

on.

2 9

5 6

U

se in

vers

e op

erat

ions

to

writ

e th

e pr

oble

m a

s an

add

ition

sen

tenc

e.

9 1

6 5

15

U

se a

fac

t fa

mily

to

find

the

mis

sing

num

ber.

9 1

6 5

15

6

1 9

5 1

5

15

2 6

5 9

1

5 2

9 5

6

Wri

te a

re

late

d f

act.

Use

it t

o c

om

ple

te t

he n

umb

er

sent

enc

e.

9 2

5

73

1

5 1

13.

2.16

2

5 7

4.

5.10

2

5 4

6. 7

.8.

12

1

5 2

01

5 2

5

89

2

5 3

5 1

5

21

12.

11.

1 8

5 1

6 1

12

5 1

5 2

4 5

13

10.

9.So, 1

5 2

9 5

6.

MX

EN

L08

AW

K4X

_RT

_CH

02_L

1.in

dd

Sec1

:16/

14/0

7 1

1:58

:51

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

8

Ro

und

3,476

,321

to

the

pla

ce o

f th

e

und

erl

ine

d d

igit

.

• Id

entif

y th

e un

derli

ned

digi

t.

j Th

e un

derli

ned

digi

t is

a 4

in the

hund

red

thou

sand

s pl

ace.

• Lo

ok a

t th

e nu

mbe

r to

the

rig

ht o

f the

unde

rline

d di

git.

j If

that

num

ber

is 0

-4, t

he

un

derli

ned

digi

t st

ays

the

sam

e.

j If

that

num

ber

is 5

-9, t

he

un

derli

ned

digi

t in

crea

ses

by

1.

j Th

e nu

mbe

r to

the

rig

ht o

f the

unde

rline

d di

git is

a 7

,

so the

und

erlin

ed d

igit,

4, w

ill

in

crea

se b

y on

e; 4

+ 1

= 5

.

• C

hang

e al

l the

dig

its to

the

right

of t

he

hu

ndre

d th

ousa

nds

plac

e to

zer

os.

So, 3

,476

,321

rou

nded

to

the

near

est

hund

red

thou

sand

is 3

,500

,000

.

Ro

und

8,851

,342

to

the

ne

are

st

tho

usan

d.

• Id

entif

y th

e di

git in

the

tho

usan

ds p

lace

.

j

The

1 to

the

left

of the

com

ma

is

in

the

tho

usan

ds p

lace

.

• Lo

ok a

t th

e nu

mbe

r to

the

rig

ht o

f the

the

digi

t in

the

tho

usan

ds p

lace

.

j

If th

at n

umbe

r is

0-4

, the

1 s

tays

the

sam

e.

j

If th

at n

umbe

r is

5-9

, th

e 1

incr

ease

s

by 1

.

j

The

num

ber

to the

rig

ht o

f the

thou

sand

s pl

ace

is a

3, s

o th

e

one

in the

tho

usan

ds p

lace

will

stay

the

sam

e.

• C

hang

e al

l the

dig

its to

the

right

of t

he

th

ousa

nds

plac

e to

zer

os.

So, 8

,851

,342

rou

nded

to

the

near

est

thou

sand

is 8

,851

,000

.

Whe

n yo

u ro

und

a n

umbe

r, yo

u re

plac

e it

with

a n

umbe

r th

at is

eas

ier

to w

ork

with

bu

t no

t as

exa

ct. Y

ou c

an r

ound

num

bers

to

diffe

rent

pla

ce v

alue

s.

Ro

und

eac

h nu

mb

er

to t

he p

lace

val

ue o

f th

e u

nde

rlin

ed

dig

it.

1.3

,45

22.

18

03.

$7

2,4

71

4.5

,72

3,0

00

Ro

und

eac

h nu

mb

er

to t

he n

ear

est

te

n, t

hous

and

, hu

ndre

d t

hous

and

, and

mill

ion.

6.1

,33

3,4

56

5.2

,47

2,9

12

Roun

d W

hole

Num

bers

Thr

ough

Mill

ions

MX

EN

L08

AW

K4X

_RT

_CH

02_L

2.in

dd

A6/

6/07

5:

26:4

9 PM

RW7-RW8


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

9

Men

tal M

ath:

Est

imat

e Su

ms a

nd D

iffer

ence

sW

hen

you

estim

ate,

you

use

num

bers

tha

t ar

e no

t as

exa

ct a

s th

e or

igin

al n

umbe

rs b

ut a

re e

asie

r to

use

. To

find

a di

ffere

nce

that

is

clos

e to

the

exa

ct n

umbe

r, su

m o

f you

can

est

imat

e.

Use

ro

und

ing

to

est

imat

e.

Esti

mat

e t

he s

um.

Rou

nd e

ach

num

ber

to the

nea

rest

th

ousa

nds

plac

e.

Add

the

est

imat

ed n

umbe

rs.

So, t

he e

stim

ated

sum

is 1

5,00

0.

So, t

he e

stim

ated

diff

eren

ce is

50,

000.

Use

ro

und

ing

to

est

imat

e.

1.

2.

3.

4.

5.

Use

fro

nt-e

nd e

stim

atio

n to

est

imat

e.

6.

7.

8.

9.

10.

78

,402

1 9

1,11

3

__

54,8

23

2 3

6,91

1

__

640

1 4

80

__

7,98

0

2 1

,341

__

32

,116

1 7

1,93

0

__

17,4

50

1 8

1,94

2

__

Use

fro

nt-e

nd e

stim

atio

n to

est

imat

e.

9,

362

1 5

,781

__

9,00

0

1 6

,000

__

15,0

00

Subt

ract

the

fron

t di

gits

.

Cha

nge

all t

he o

ther

dig

its to

zero

s.

73

,206

2 2

1,35

8__

70

,000

2 2

0,00

0

__

5

0,00

0

7,

000,

115

1

4,0

00,2

12

___

22

,001

2

4,66

3

__

6,78

9

2

834

__

1,64

2

2

926

__

MX

EN

L08

AW

K4X

_RT

_CH

02_L

3.in

dd

16/

14/0

7 1

2:02

:41

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

10

1.

714

+ 2

24

2.

322

+ 1

38 3

. 83

– 6

1

4.

150

+ 3

50 5

. 29

3 –

168

10.

187

+ 2

50

6.

428

– 3

64 7

. 69

+ 8

1 8

. 65

4 –

270

9.

36 –

22

Men

tal M

ath

Stra

tegi

esTh

ere

are

stra

tegi

es tha

t ca

n he

lp y

ou a

dd o

r su

btra

ct m

enta

lly. T

he B

reak

Ap

art

Str

ate

gy,

the

Fri

end

ly N

umb

er

Str

ate

gy,

and

the

Sw

app

ing

Str

ate

gy

are

thre

e of

the

str

ateg

ies

you

can

use.

Bre

ak A

par

t S

trat

eg

y

86

7 2

42

5

• B

reak

apa

rt the

plac

e va

lues

.

• Su

btra

ct e

ach

pl

ace

valu

e

sepa

rate

ly.

800

2 4

00 5

400

60 2

20

5 4

07

2 5

5 2

• A

dd th

e di

ffere

nces

.

400

1 4

0 1

2 5

442

So, 8

67 2

425

5 4

42.

You

can

als

o us

e th

is

met

hod

for

addi

tion

and

subt

ract

ion.

Frie

nd

ly N

um

be

r S

tra

teg

y

86

7 2

42

5

• Fr

iend

ly n

umbe

rs e

nd

with

a z

ero.

• A

dd to

mak

e on

e of

you

r nu

mbe

rs e

nd w

ith z

ero.

86

7 1

3 5

870

870

is a

frie

ndly

num

ber.

• A

djus

t th

e ot

her

num

ber

by

ad

ding

the

sam

e am

ount

.

42

5 1

3 5

428

• Su

btra

ct the

new

num

bers

.

87

0 2

428

5 4

42

So, 8

87 2

425

5 4

42.

You

can

onl

y us

e th

is m

etho

d fo

r su

btra

ctio

n.

Sw

ap

pin

g S

tra

teg

y

14

5 1

21

3

• Sw

ap d

igits

with

the

sam

e pl

ace

valu

e.

• A

dd to

mak

e on

e of

yo

ur n

umbe

rs e

nd w

ith

zero

.

14

5 1

5 5

150

• A

djus

t th

e ot

her

num

ber

by s

ubtr

actin

g

the

sam

e am

ount

.

213

2 5

5 2

08

• A

dd the

new

num

bers

.

150

1 2

08 5

358

So, 1

45 1

213

5 3

58.

You

can

onl

y us

e th

is

met

hod

for

addi

tion.

Ad

d o

r S

ubtr

act

me

ntal

ly. T

ell

the

str

ate

gy

you

use

d.

Ad

d o

r su

btr

act

me

ntal

ly. T

ell

the

str

ate

gy

you

use

d.

MX

EN

L08

AW

K4X

_RT

_CH

02_L

4.in

dd

A6/

14/0

7 1

2:05

:20

PM

RW9-RW10


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

11

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Estim

ate

or E

xact

Ans

wer

To b

e an

airl

ine

pilo

t, yo

u m

ust fly

a tot

al o

f at le

ast 1

,500

hou

rs.

Dan

flew

827

hou

rs la

st y

ear

and

582

hour

s th

is y

ear.

How

man

y m

ore

hour

s m

ust D

an fl

y to

be

an a

irlin

e pi

lot?

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

Shou

ld y

ou u

se a

n es

timat

e or

an

exac

t an

swer

?

3. S

olve

the

pro

blem

in the

spa

ce b

elow

. Wha

t op

erat

ions

did

you

use

?

4.

How

man

y m

ore

hour

s m

ust D

an fl

y to

be

an a

irlin

e pi

lot?

Writ

e yo

ur a

nsw

er in

a c

ompl

ete

sent

ence

.

5.

How

can

you

tel

l if y

our

answ

er is

cor

rect

?

Op

era

tio

ns U

sed

:S

olv

e:

Exp

lain

whe

the

r to

est

imat

e o

r fin

d a

n e

xact

ans

we

r.

The

n so

lve

the

pro

ble

m.

Ja

ne too

k a

mat

h te

st. S

he a

nsw

ered

13

4 ite

ms

corr

ectly

in the

firs

t se

ctio

n an

d 11

3 ite

ms

corr

ectly

in the

sec

ond

sect

ion.

How

man

y ite

ms

did

Jane

an

swer

cor

rect

ly in

all?

La

nce

spen

t 18

0 da

ys in

Mr.

Lee’

s cl

ass

and

176

days

in M

rs. M

ac’s

cl

ass.

Abo

ut h

ow m

any

days

did

La

nce

spen

d in

Mr.

Lee’

s an

d M

rs. M

ac’s

cla

sses

in a

ll?

6.7.

MX

EN

L08

AW

K4X

_RT

_CH

02_L

5.in

dd

A6/

14/0

7 1

2:07

:23

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

12

Add

and

Subt

ract

3-Di

git a

nd 4

-Dig

it N

umbe

rsY

ou c

an u

se p

lace

val

ue to

add

and

subt

ract

larg

e nu

mbe

rs.

An

estim

ate

will

hel

p yo

u kn

ow if

you

r an

swer

is r

easo

nabl

e.

Esti

mat

e. T

hen

find

the

sum

.72

6

1 2

,643

__

R

ound

the

num

bers

to

estim

ate:

1,0

00 1

3,0

00 5

4,0

00.

So

, 7261

2,64

3 is

abo

ut 4

,000

.

Add

the

tho

usan

ds.

Add

the

hun

dred

sR

egro

up 1

3 hu

ndre

ds.

Add

the

ten

s.A

dd the

one

s.

Sinc

e3,

369

is c

lose

to

the

estim

ate

of 4

,000

, the

ans

wer

is r

easo

nabl

e.

Subt

ract

.R

egro

up10

ten

s as

9 te

ns, 1

0 on

es.

Reg

roup

10 h

undr

eds

as9

hund

reds

, 10

ten

s.

Reg

roup

6 th

ousa

nds

as5

thou

sand

s,

10 h

undr

eds.

Esti

mat

e. T

hen

find

the

diff

ere

nce

.

6,

000

2 4

,275

__

R

ound

the

num

bers

to

estim

ate:

6,0

00 2

4,0

00 5

2,0

00.

So

, 6,0

002

4,27

5 is

abo

ut 2

,000

.

Sinc

e1,

725

is c

lose

to

the

estim

ate

of 2

,000

, the

ans

wer

is r

easo

nabl

e.

Esti

mat

e. T

hen

find

the

sum

or

diff

ere

nce

.

1

41

5

1 3

42

__

2,31

4

1 3

,986

__

613

1 4

,967

__

1.

2,

527

2 1

,761

__

5,00

0

2 4

,291

__

2.3.

4.5.

6.7.

8.9.

10.

72

6

1 2

,643

__

9 1

72

6

1 2

,643

__

6

9

726

1 2

,643

__

36

9

726

1 2

,643

__

3,36

9

6,0

00

4,2

75

22

22

510

510

510

510

1010

99

1,7

25

6,0

00

4,2

75

6,0

00

4,2

75

6,0

00

4,2

75

5,

796

1

123

__

7,

000

2

251

__

1,37

6

2

149

__

1,

427

1

427

__

1010

10

4,

931

2 3

,187

__

99

9

MX

EN

L08

AW

K4X

_RT

_CH

02_L

6.in

ddA

6/28

/07

11:2

1:49

AM

RW11 RW12


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

13

Choo

se a

Met

hod

Cho

ose

the

met

hod

that

wor

ks b

est fo

r so

lvin

g an

add

ition

or

a s

ubtr

actio

n pr

oble

m. T

hink

abo

ut the

que

stio

n be

ing

aske

d,

and

then

use

one

of t

he fo

llow

ing

thre

e m

etho

ds.

Pap

er

and

Pe

ncil

Pape

r an

d pe

ncil

can

be u

sed

with

gre

ater

num

bers

an

d w

hen

the

answ

er h

as to

be e

xact

.

Cal

cula

tor

A c

alcu

lato

r ca

n be

use

d w

ith g

reat

er n

umbe

rs tha

t ne

ed

to b

e re

grou

ped

and

whe

n th

e an

swer

has

to

be e

xact

.

Me

ntal

Mat

h

Men

tal m

ath

can

be u

sed

whe

n us

ing

roun

ded

or s

mal

ler

num

bers

.

Find

the

diff

ere

nce

. 94

5,32

2 2

461

,070

Sinc

e nu

mbe

rs a

regr

eate

r an

d yo

u ne

ed a

nex

act an

swer

, use

pe

ncil

and

pap

er.

Wor

k fr

om r

ight

to

left.

R

egro

up w

hen

need

ed.

C

ontin

ue s

ubtr

actin

g.

Find

the

sum

or

diff

ere

nce

. Wri

te t

he m

eth

od

use

d.

1.

378,

452

2 1

93,5

11

__

2.

1,

420,

300

1 1

,678

,500

___

3.

62

0,00

0

2 4

19,5

00

__

4.

3,

721,

682

1 6

,161

,248

___

5.

7,57

3,15

0

1 1

,200

,000

___

6.

62

1,89

9

1 2

13,6

55

__

7.

6,

280,

000

1 1

,300

,000

___

8.

70

1,00

0

2 1

07,0

00

__

94

5,32

2

246

1,07

0

__

94

5,32

2

246

1,07

0

__

252

94

5,32

2

246

1,07

0

__

484,

252

122

122

148

So, 9

45,3

22 2

461

,070

5 4

84,2

52.

MX

EN

L08

AW

K4X

_RT

_CH

02_L

7.in

dd

A6/

6/07

5:

32:2

2 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

14

Addi

tion

Prop

ertie

sC

om

mut

ativ

e P

rop

ert

y

The

Com

mut

ativ

e Pr

oper

ty s

tate

s th

at y

ou c

an a

dd n

umbe

rs in

any

ord

er

and

the

sum

will

be

the

sam

e. F

or e

xam

ple,

2 1

3 5

5 a

nd 3

1 2

5 5

.Th

e or

der

in w

hich

you

add

num

bers

doe

s no

t ch

ange

the

sum

.

Find

the

mis

sing

num

be

r.47

1 j

5 5

6 1

47

So, j

5 5

6.

Ass

oci

ativ

e P

rop

ert

y

The

Ass

ocia

tive

Prop

erty

sta

tes

that

the

way

add

ends

are

gro

uped

doe

s no

t ch

ange

th

e su

m. F

or e

xam

ple,

(4

1 2

) 1

5 a

nd (

5 1

4)

1 2

bot

h eq

ual 1

1.Th

e w

ay y

ou g

roup

num

bers

doe

s no

t ch

ange

the

sum

.

Find

the

mis

sing

num

be

r.j

1 (

31 1

18)

5 (

24 1

31)

1 1

8

So, j

5 2

4.

Ide

ntit

y Pr

op

ert

y

The

Iden

tity

Prop

erty

sta

tes

that

whe

n yo

u ad

d ze

ro to

any

num

ber,

the

sum

is tha

t nu

mbe

r. Fo

r ex

ampl

e, 4

5 1

0 5

45.

Find

the

mis

sing

num

be

r.62

1 j

5 6

2

Add

ing

a ze

ro to

62 d

oes

not ch

ange

62.

So, j

5

0.

Find

the

mis

sing

num

be

r. N

ame

the

pro

pe

rty

you

use

d.

1.

39 1

j 5

39

2.

(7 1

j )

1 8

5 7

1 (2

1 8

)

3.

j 1

40

5 4

0 1

22

4.

3 1

(14

1 6

) 5

(3

1 j

) 1

6

56 1

47

uses

the

sam

e nu

mbe

rsas

47

1 j

, but

in a

diff

eren

t or

der.

47 1

56

5 5

6 1

47

(24

1 3

1) 1

18

uses

the

sam

e nu

mbe

rs a

s j

1 (

31 1

18)

, exc

ept

grou

ped

toge

ther

diff

eren

tly.

24 1

(31

1 1

8) 5

(24

1 3

1) 1

18

62 1

0 5

62

MX

EN

L08

AW

K4X

_RT

_CH

03_L

1.in

dd

A6/

14/0

7 1

2:13

:23

PM

RW13-RW14


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

15

Writ

e an

d Ev

alua

te E

xpre

ssio

nsA

n e

xpre

ssio

n is

par

t of

a n

umbe

r se

nten

ce tha

t ha

s nu

mbe

rsan

d si

gns

such

as

“1,”

“2

,” a

nd “

3,”

but

it d

oes

not ha

ve a

n “5

” si

gn. Y

ou c

an fi

nd the

val

ue o

f an

expr

essi

on b

y ad

ding

,su

btra

ctin

g, m

ultip

lyin

g, a

nd/o

r di

vidi

ng. W

hen

ther

e ar

epa

rent

hese

s in

an

expr

essi

on, y

ou n

eed

to d

o th

e op

erat

ion

inpa

rent

hese

s fir

st.

Find

the

val

ue o

f th

e e

xpre

ssio

n.

13 2

n if

n 5

6

• R

epla

ce n

with

6 in

the

equ

atio

n.

13 2

6

• S

olve

the

exp

ress

ion.

13

2 6

5 7

So, t

he v

alue

of t

he e

xpre

ssio

n is

7.

Wri

te a

n e

xpre

ssio

n w

ith

a va

riab

le. T

ell

wha

t th

e v

aria

ble

re

pre

sent

s.

Alis

ha fo

und

som

e sh

ells

on

the

beac

h. S

he g

ave

4 of

the

m to

Lanc

e.

• C

hoos

e a

varia

ble

to r

epre

sent

the

unk

now

n Le

t s r

epre

sent

the

she

lls

num

ber

in the

pro

blem

.

Alis

ha fo

und.

• A

lisha

gav

e aw

ay 4

she

lls, s

o 4

wer

e su

btra

cted

from

her

tot

al.

• W

rite

the

expr

essi

on.

s 2

4

So, t

he e

xpre

ssio

n is

s 2

4, a

nd the

var

iabl

e s

repr

esen

ts the

she

lls A

lisha

foun

d.

Find

the

val

ue o

f e

ach

exp

ress

ion.

1.

12 1

j if

j 5

82.

5

1 (

7 1

7)

3.

14 2

(5

1 3

)

4.

j 1

6 i

f j

5 4

5.

(11

1 2

) 2

46.

15

1 (

3 1

7)

Wri

te a

n e

xpre

ssio

n w

ith

a va

riab

le. T

ell

wha

t th

e v

aria

ble

re

pre

sent

s.

7. K

arl h

as s

ome

book

s. H

e gi

ves

3 to

Joe

.8.

M

ary

has

5 pe

ns. S

he fi

nds

som

e m

ore.

MX

EN

L08

AW

K4X

_RT

_CH

03_L

2.in

dd

A6/

14/0

7 1

2:16

:14

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

16

Addi

tion

and

Subt

ract

ion

Equa

tions

An

eq

uati

on

is a

num

ber

sent

ence

sta

ting

that

tw

o am

ount

s ar

e eq

ual.

Wri

te a

n e

qua

tio

n. U

se 1

, 2 o

r 5

fo

r e

ach

.

So

lve

the

eq

uati

on.

13 2

n 5

4

• U

se m

enta

l mat

h to

find

whi

ch n

umbe

r n

repr

esen

ts.

1

3 2

n 5

4Th

ink:

“13

min

us w

hat nu

mbe

r eq

uals

4?”

• R

epla

ce n

with

9 in

the

equ

atio

n. 1

3 2

9 5

4

• T

he a

nsw

er is

cor

rect

.

4

5 4

So, n

5 9

.

Wri

te a

n e

qua

tio

n fo

r e

ach.

Use

1, 2

or

5 f

or

eac

h .

So

lve

the

eq

uati

on.

3. 8

1 j

5 2

04.

n 2

9 5

85.

4 1

k 5

12

6. y

1 5

5 1

37.

6 1

j 5

12

8. 1

1 2

j 5

8

Emil

has

18 s

tam

ps. H

e us

es s

ome

stam

ps, s

, and

now

has

12

stam

ps le

ft.

18

s

12

• S

ince

the

var

iabl

e, s

, rep

rese

nts

the

num

ber

of s

tam

ps E

mil

used

,yo

u w

ill s

ubtr

act.

• S

ince

12

is the

num

ber

of s

tam

psEm

il ha

s le

ft, y

ou w

ill u

se the

equa

l sig

n.

So, t

he e

quat

ion

is 1

8 2

s 5

12.

1

8 2

s

12

1

8 2

s

5 1

2

30

b 1

216

s

5 1

0

1. Ti

m h

as 1

6 so

cks.

Afte

r he

lose

sso

me

of h

is s

ocks

, s, h

e ha

s 10

soc

ksle

ft.

2. Th

ere

are

30 b

owls

on

the

shel

f.Th

ere

are

som

e w

hite

bow

ls, b

, and

12

gre

en b

owls

.

MX

EN

L08

AW

K4X

_RT

_CH

03_L

3.in

dd

A6/

14/0

7 1

2:17

:47

PM

RW15-RW16


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

17

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Wor

k Ba

ckw

ard

Man

y vo

lunt

ee

r te

ams

mus

t p

atro

l and

cle

an t

he li

on

pre

serv

e.

Twe

lve

te

ams

leav

e t

he p

rese

rve

on

pat

rol.

Se

ven

team

s ar

rive

to

cle

an. T

here

are

23

team

s at

the

pre

serv

e n

ow

. Ho

w m

any

volu

nte

er

team

s w

ere

the

re o

rig

inal

ly?

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2. H

ow c

an y

ou u

se the

wor

k ba

ckgr

ound

str

ateg

y to

sol

ve thi

s pr

oble

m?

So

lve

3. S

olve

the

pro

blem

. Sho

w y

our

wor

k be

low

.

4. H

ow m

any

volu

ntee

r te

ams

wer

e th

ere

orig

inal

ly?

Ch

eck

5. Is

the

re a

noth

er s

trat

egy

you

coul

d us

e to

sol

ve the

pro

blem

?

Wo

rk b

ackw

ard

s to

so

lve

.

6. P

ete

got ho

me

at 4

:30

P.M

. It to

ok h

im

20 m

inut

es to

wal

k fr

om s

choo

l to

the

libra

ry. H

e w

as a

t th

e lib

rary

for

1 ho

ur.

It to

ok P

ete

5 m

inut

es to

wal

k fr

om the

lib

rary

at ho

me.

At w

hat tim

e di

d Pe

te

leav

e sc

hool

?

7. T

here

wer

e so

me

cars

in a

par

king

lot in

the

mor

ning

. Lat

er, 1

5 m

ore

cars

arriv

ed the

n 9

cars

left.

The

re a

re n

ow

60 c

ars

in the

par

king

lot.

How

man

y ca

rs w

ere

orig

inal

ly in

the

par

king

lot?

MXE

NL08

AW

K4X_

RT_C

H03_

L4.in

dd

A6/

6/07

12

:22:

11 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

18

Patt

erns

: Fin

d a

Rule

The

inpu

t an

d ou

tput

num

bers

in a

tab

le a

re r

elat

ed b

y a

rule

. Fi

nd the

rel

atio

nshi

p be

twee

n th

e tw

o nu

mbe

rs.

Find

a r

ule

. Wri

te y

our

rul

e a

s an

eq

uati

on.

Use

the

eq

uati

on

to e

xte

nd y

our

pat

tern

.

• F

ind

a re

latio

nshi

p be

twee

n ea

ch in

put nu

mbe

r an

d ou

tput

num

ber.

1

2 1

8 5

20

25 1

8 5

33

31

1 8

5 3

9

• T

he r

ule

is to

add

8 to

eac

h in

put nu

mbe

r to

get

the

out

put nu

mbe

r.W

rite

the

rule

.

a

1 8

5 b

• U

se the

rul

e to

com

plet

e th

e ta

ble.

4

3 1

8 5

51

59 1

8 5

67

• T

he r

ule

is a

dd 8

to

a, o

r a 1

8 5

b.

So, u

se 5

1 an

d 67

to

exte

nd y

our

patter

n. Fi

nd a

rul

e. U

se y

our

rul

e t

o f

ind

the

ne

xt t

wo

ord

ere

d p

airs

.

1.

2.

3.

4.

Inpu

ta

1225

3143

59

Outp

utb

2033

39j

j

Inpu

ta

1225

3143

59

Outp

utb

2033

3951

67

Inpu

tc

6258

47j

j

Outp

utd

5753

4231

24

Inpu

te

3146

5973

90

Outp

utf

4257

70j

j

Inpu

tg

1824

57j

j

Outp

uth

915

4862

84

Inpu

tj

1417

2234

56

Outp

utk

47

12j

j

MX

EN

L08

AW

K4X

_RT

_CH

03_L

5.in

dd

A6/

14/0

7 1

2:20

:28

PM

RW17-RW18


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

19

Alge

bra:

Rel

ate

Ope

ratio

nsW

hen

you

add

or m

ulti

ply

, you

join

equ

al-s

ize

grou

ps.

Whe

n yo

u su

btra

ct o

r d

ivid

e, y

ou s

epar

ate

into

equ

al-s

ize

grou

ps o

r fin

d ou

t ho

w m

any

are

in e

ach

grou

p.

Wri

te t

he r

ela

ted

mul

tip

licat

ion

sent

enc

e.

Dra

w a

pic

ture

tha

t sh

ow

s th

e s

ent

enc

e.

3 1

3 1

3 1

3 1

3 5

15

Star

t w

ith 0

on

the

num

ber

line.

Ski

p co

unt by

3’s

five

tim

es. S

top

at 1

5.

5 gr

oups

of 3

is the

sam

e as

5 3

3.

So, t

he r

elat

ed m

ultip

licat

ion

sent

ence

is 5

3 3

5 1

5.

Wri

te t

he r

ela

ted

div

isio

n se

nte

nce

.D

raw

a p

ictu

re t

hat

sho

ws

the

se

nte

nce

.

9 2

3 2

3 2

3 5

0

Star

t w

ith 9

on

the

num

ber

line.

Sub

trac

t 3

thre

e tim

es. S

top

at 0

.

A g

roup

of 9

div

ided

into

equ

al g

roup

s of

3 is

the

sam

e as

9 4

3.

So, t

he r

elat

ed d

ivis

ion

sent

ence

is 9

4 3

5 3

.

1.

2 1

2 1

2 1

2 1

2 5

10

2. 1

5 2

3 2

3 2

3 2

3 2

3 5

0

3. 5

1 5

1 5

1 5

5 2

04.

18

2 6

2 6

2 6

5 0

Wri

te t

he r

ela

ted

mul

tip

licat

ion

or

div

isio

n se

nte

nce

. D

raw

a p

ictu

re t

hat

sho

ws

the

se

nte

nce

.

MX

EN

L08

AW

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_RT

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1.in

dd

16/

6/07

12

:17:

09 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

20

Alge

bra:

Rel

ate

Mul

tiplic

atio

n an

d Di

visio

nM

ultip

licat

ion

and

divi

sion

are

opp

osite

ope

ratio

ns, o

r in

vers

e o

pe

rati

ons

. A f

act

fam

ilyis

a s

et o

f mul

tiplic

atio

n an

d di

visi

on s

ente

nces

tha

t us

e th

e sa

me

num

bers

.

Wri

te t

he f

act

fam

ily f

or

the

se

t o

f nu

mb

ers

.2,

3, 6

2 3

3 5

6

2 ro

ws

of 3

tria

ngle

s ea

ch.

So, 2

3 3

5 6

.

6 4

3 5

2

D

ivid

e 6

tria

ngle

s in

to 3

equ

al g

roup

s.

T

here

are

2 tria

ngle

s in

eac

h gr

oup.

S

o, 6

4 3

5 2

.

3 3

2 5

6

3 ro

ws

of 2

tria

ngle

s ea

ch.

So, 3

3 2

5 6

.

6 4

2 5

3

D

ivid

e 6

tria

ngle

s in

to 2

equ

al g

roup

s.

T

here

are

3 tria

ngle

s in

eac

h gr

oup.

S

o, 6

4 2

5 3

.

So, t

he fa

ct fa

mily

for

2, 3

, 6 is

2 3

3 5

6, 3

3 2

5 6

, 6 4

3 5

2, 6

4 2

5 3

.

Wri

te t

he f

act

fam

ily f

or

the

se

t o

f nu

mb

ers

.

1.

2, 9

, 18

2.

3, 5

, 15

3.

1, 8

, 84.

2,

8, 1

6

5.

5, 8

, 40

6.

3, 8

, 24

MX

EN

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dd

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12

:18:

33 P

M

RW19-RW20


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

21

Mul

tiply

and

Div

ide

Fact

s Thr

ough

5Y

ou c

an u

se a

mul

tiplic

atio

n ta

ble

to h

elp

you

find

pro

duc

ts a

nd q

uoti

ent

s.

Find

the

pro

duc

t.

3 3

3

• Th

e fir

st fa

ctor

is 3

. Fi

nd the

row

labe

led

3.

• Th

e se

cond

fact

or is

3.

Find

the

col

umn

labe

led

3.

• Tr

ace

with

you

r fin

gers

acr

oss

row

3 a

nd d

own

colu

mn

3.

• W

here

row

3 a

nd c

olum

n 3

mee

t is

the

pr

oduc

t, 9.

• So

, 3 3

3 5

9.

Find

the

quo

tie

nt.

15 4

5

• Th

e di

viso

r is

5. F

ind

the

row

labe

led

5.

• Th

e di

vide

nd is

15.

Fol

low

acr

oss

row

5

until

you

find

the

num

ber

15.

• Fr

om 1

5, fo

llow

you

r fin

ger

up to

the

top

of the

col

umn.

• Th

e nu

mbe

r yo

u fin

d is

the

quo

tient

, 3.

• So

, 15

4 5

5 3

.

Find

the

pro

duc

t o

r q

uoti

ent

.

1. 8

4 2

2. 5

3 3

3. 4

3 2

4. 2

5 4

5 5

. 24

4 3

6. 4

3 6

7. 2

0 4

58.

3 3

69.

24

4 4

10.

7 3

4

MX

EN

L08

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3.in

dd

A6/

14/0

7 1

2:22

:25

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

22

Mul

tiply

and

Div

ide

Fact

sTh

roug

h 10

Find

the

pro

duc

t o

r q

uoti

ent

. Sho

w t

he s

trat

eg

y yo

u us

ed

.50

4 1

0

• So

lve

usin

g in

vers

e o

pe

rati

ons

, so

use

mul

tiplic

atio

n fa

cts

you

know

.Th

ink:

10

3 5

5 5

0

• So

, 50

4 1

0 5

5.

3 3

9

• D

raw

a r

ecta

ngul

ar a

rray

tha

t is

3 u

nits

wid

e an

d 9

units

long

.

• C

ount

the

squ

ares

in the

arr

ay.

• Th

ere

are

a to

tal o

f 27

squa

res.

• So

, 3 3

9 5

27.

Find

the

pro

duc

t o

r q

uoti

ent

. Sho

w t

he s

trat

eg

y yo

u us

ed

.

1.

5 3

4 5

3.

30 4

5 5

5.

7 3

8 5

7.

24 4

6 5

9.

5 3

9 5

11.

6 3

3 5

13.

44 4

4 5

15.

32 4

8 5

2.

12 3

2 5

4.

81 4

9 5

6.

7 3

7 5

8.

6 3

9 5

10.

9 3

7 5

12.

28 4

4 5

14.

4 3

0 5

16.

64 4

8 5

MX

EN

L08

AW

K4X

_RT

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04_L

4.in

dd

A6/

14/0

7 1

2:24

:00

PM

RW21-RW22


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

23

Mul

tiplic

atio

n Ta

ble

Thro

ugh

12Y

ou c

an u

se a

mul

tiplic

atio

n ta

ble

to m

ultip

ly a

nd d

ivid

e nu

mbe

rs thr

ough

12.

Use

the

mul

tip

licat

ion

tab

le t

o

find

the

pro

duc

t o

f 7 3

11.

•

The

first

fact

or is

7.

Find

the

row

labe

led

7.

•

The

seco

nd fa

ctor

is 1

1.

Fi

nd the

col

umn

labe

led

11.

•

Trac

e yo

ur fi

nger

acr

oss

row

7

an

d do

wn

colu

mn

11.

•

Whe

re r

ow 7

and

col

umn

11

m

eet is

the

pro

duct

, 77.

•

So, 7

3 1

1 5

77.

Use

the

rul

e t

o f

ind

the

mis

sing

num

be

rs in

the

tab

le a

t th

e r

ight

.

Find

the

pro

duc

t o

r q

uoti

ent

. Sho

w t

he s

trat

eg

y yo

u us

ed

.

1. 2

3 1

22.

12 4

33.

4 3

11

4. 6

6 4

6

ALG

EBR

A U

se t

he r

ule

.

5.

Mul

tip

ly t

he in

put

by

12.

6

.

Div

ide

the

inp

ut b

y 11

.

Inpu

tO

utpu

t

36 726

108

914

412

Inpu

tO

utpu

t

4

60

7

108

Inpu

tO

utpu

t

33 77

8 10

•

Find

the

row

for

the

divi

sor,

12.

•

Tr

ace

your

fing

er a

cros

s ro

w 1

2

to

find

the

div

iden

d, 3

6.

•

Si

nce

36 is

in the

col

umn

for

3,

3 is

the

out

put.

•

So

, 36 4

12 5

3.

colu

mn

row

Div

ide

the

inp

ut b

y 12

.

MX

EN

L08

AW

K4X

_RT

_CH

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5.in

dd

A6/

14/0

7 1

2:26

:11

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

24

Patt

erns

on

the

Mul

tiplic

atio

n Ta

ble

Whe

n a

who

le n

umbe

r is

mul

tiplie

d by

ano

ther

who

le n

umbe

r, th

e pr

oduc

t is

cal

led

a m

ulti

ple

. For

exa

mpl

e, 5

3 3

5 1

5, s

o 15

is a

mul

tiple

of 5

and

3.

Whe

n a

num

ber

is m

ultip

lied

by it

self,

the

pro

duct

is c

alle

d a

squa

re n

umb

er.

Fo

r ex

ampl

e, 4

3 4

5 1

6, s

o 16

is a

squ

are

num

ber.

Whi

ch m

ulti

ple

s ha

ve o

nly

eve

n nu

mb

ers

?

Look

at th

e ro

ws

in the

mul

tiplic

atio

n ta

ble.

Look

for

a ro

w w

here

all

the

mul

tiple

s ar

e ev

en

num

bers

.

All

the

mul

tiple

s fo

r th

e nu

mbe

rs 2

, 4, 6

, 8, 1

0,

and

12 a

re e

ven.

Use

the

mul

tip

licat

ion

tab

le t

o f

ind

the

sq

uare

num

be

r fo

r 11

3 1

1.

Find

the

col

umn

and

row

for

the

num

ber

11.

Use

you

r fin

ger

to m

ove

acro

ss r

ow 1

1 an

d do

wn

colu

mn

11.

You

r fin

gers

mee

t at

121

.

11 3

11

= 1

21, s

o 12

1 is

the

squ

are

num

ber.

Find

the

sq

uare

num

be

r.

1.

5 3

5

2.

8 3

8

3.

2 3

2 4

. 4 3

4

5.

7 3

7

6.

10 3

10

7.

9 3

9

8.

12 3

12

Use

the

mul

tip

licat

ion

tab

le.

9.

Do

any

num

bers

hav

e on

ly

odd-

num

bere

d m

ultip

les?

10.

Wha

t pa

tter

n do

you

see

in the

mul

tiple

s o

f 5?

colu

mn

row row

colu

mn

MX

EN

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2:27

:46

PM

RW23-RW24


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

25

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Choo

se th

e O

pera

tion

The

hiki

ng trip

is o

n Fr

iday

. On

Thur

sday

, the

hig

h te

mpe

ratu

re

was

678

F. R

usse

ll ho

pes

that

it w

ill b

e 8º

war

mer

on

Frid

ay.

Wha

t te

mpe

ratu

re is

Rus

sell

hopi

ng fo

r?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. W

ill y

ou s

epar

ate

a gr

oup

into

sm

alle

r gr

oups

to

solv

e? W

hy o

r w

hy n

ot?

3. W

hich

ope

ratio

n w

ill y

ou u

se to

answ

er the

que

stio

n?

4. W

hat w

as the

hig

h te

mpe

ratu

re o

n Th

ursd

ay?

How

man

y de

gree

s w

ill y

ou

add

to T

hurs

day’

s te

mpe

ratu

re to

get Fr

iday

’s tem

pera

ture

?

5. W

hat te

mpe

ratu

re is

Rus

sell

hopi

ng fo

r on

Frid

ay?

Writ

e a

num

ber

sent

ence

to

sol

ve.

6. H

ow c

an y

ou c

heck

you

r an

swer

?

Tell

whi

ch o

pe

rati

on

you

wo

uld

use

to

so

lve

the

pro

ble

m.

The

n so

lve

the

pro

ble

m.

7. M

aria

has

24

appl

es. S

he w

ants

to

give

an

equa

l num

ber

of a

pple

s to

ea

ch o

f her

4 fr

iend

s. H

ow m

any

appl

es s

houl

d M

aria

giv

e ea

chfr

iend

?

8. G

ina

has

5 bo

xes.

She

pla

ces

6 bo

oks

in e

ach

box.

How

man

ybo

oks

does

Gin

a ha

ve in

all?

MX

EN

L08

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7.in

dd

A6/

6/07

12

:21:

58 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

26

Alge

bra:

Fin

d M

issin

g Fa

ctor

sY

ou c

an d

raw

mod

els

and

use

fact

fam

ilies

to

find

mis

sing

fact

ors.

Find

the

mis

sing

fac

tor.

10 3

g 5

70

Use

a m

odel

to

find

the

valu

e of

g.

Use

the

pro

duct

, 70,

and

the

fact

or, 1

0,to

dra

w 7

0 co

unte

rs in

row

s of

10.

Cou

nt the

num

ber

of r

ows

to fi

nd the

mis

sing

fact

or.

Ther

e ar

e 7

row

s of

10,

so

the

mis

sing

fact

or, g

, is

7.

So, 1

0 3

7 5

70,

or

g 5

7.

Find

the

mis

sing

fac

tor.

3

11 5

132

Use

a r

elat

ed d

ivis

ion

sent

ence

to

find

the

mis

sing

fact

or,

.

Thin

k of

the

fact

fam

ily a

nd w

rite

the

rela

ted

divi

sion

sen

tenc

e. 1

32 4

11 5

Solv

e fo

r .

1

32 4

11 5

12,

or

5 1

2

The

mis

sing

fact

or is

12.

So, 1

2 3

11 5

132

, or

5 1

2.

Find

the

mis

sing

fac

tor.

1.

7 3

n 5

28

2.

5 3

p 5

30

3.

m 3

5 5

45

4.

6 3

n 5

48

5.

8 3

g 5

72

6.

w 3

11 5

55

7.

4 3

j 5

32

8.

9 3

v 5

27

9.

4 3

g 5

12

10.

9 3

v 5

81

11.

3 3

n 5

36

12.

10 3

z 5

80

MX

EN

L08

AW

K4X

_RT

_CH

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8.in

dd

A6/

14/0

7 1

2:29

:57

PM

RW25-RW26


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

27

Mul

tiplic

atio

n Pr

oper

ties

Use

pro

pe

rtie

s an

d m

ent

al m

ath

to f

ind

the

pro

duc

t.

Ass

oci

ativ

e P

rop

ert

y

The

Ass

ocia

tive

Prop

erty

sta

tes

that

the

w

ay fa

ctor

s ar

e gr

oupe

d do

es n

ot c

hang

e th

e pr

oduc

t.

8 3

3 3

3

• (8

3 3

) 3

3 u

ses

the

sam

e nu

mbe

rs a

s 8

3 (3

3 3

), ex

cept

the

y ar

e gr

oupe

d to

geth

er d

iffer

ently

.

• C

hoos

e w

hich

set

of f

acto

rs y

ou c

an

easi

ly u

se m

enta

l mat

h to

sol

ve.

8

3 (

3 3

3)

or

(

8 3

3)

3 3

8 3

9 5

72

2

4 3

3 5

72

So, 8

3 3

3 3

5 7

2.

Dis

trib

utiv

e P

rop

ert

y

The

Dis

trib

utiv

e Pr

oper

ty s

tate

s th

at y

ou

can

brea

k ap

art on

e of

the

fact

ors

and

mul

tiply

its

part

s by

the

oth

er fa

ctor

.

7 3

11

• B

reak

11

apar

t so

you

can

eas

ily u

se

men

tal m

ath.

11 5

10

1 1

• M

ultip

ly b

oth

part

s by

7, a

nd the

n ad

d th

e pr

oduc

ts.

7

3 1

0 5

70

7 3

1 5

770

1 7

5 7

7

So, 7

3 1

1 5

77.

Co

mm

utat

ive

Pro

pe

rty

The

Com

mut

ativ

e Pr

oper

ty s

tate

s yo

u ca

n m

ultip

ly n

umbe

rs in

any

ord

er a

nd the

pro

duct

w

ill b

e th

e sa

me.

The

ord

er in

whi

ch y

ou m

ultip

ly n

umbe

rs d

oes

not ch

ange

the

pro

duct

. Fo

r ex

ampl

e, 1

0 3

3 a

nd 3

3 1

0 bo

th h

ave

a pr

oduc

t of

30.

Ze

ro P

rop

ert

y

The

Zer

o Pr

oper

ty s

tate

s th

at the

pro

duct

of

any

num

ber

and

zero

is z

ero.

For

ex

ampl

e, 1

2 3

0 5

0, 5

6 3

0 5

0, a

nd

864

3 0

5 0

.

Ide

ntit

y Pr

op

ert

y

The

Iden

tify

Prop

erty

sta

tes

that

the

pr

oduc

t of

any

num

ber

and

1 is

tha

t nu

mbe

r. Fo

r ex

ampl

e, 9

3 1

5 9

, 56

3 1

5 5

6, a

nd 8

64 3

1 5

864

.

Use

the

pro

pe

rtie

s an

d m

ent

al m

ath

to f

ind

the

pro

duc

t.

1.

6 3

2 3

4 2

. 8

3 1

2 3

. 6

3 0

3 4

4.

2 3

13

3 1

5.

5 3

14

6

. 5

3 5

3 3

7.

4 3

6 3

3 8

. 2

3 0

3 3

9

. 8

3 1

3 5

MX

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:17:

04 P

M

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e

© H

arco

urt •

Gra

de 4

Ret

each

RW

28

Mul

tiplic

atio

n an

d Di

visio

n Ex

pres

sions

Som

e m

ultip

licat

ion

and

divi

sion

exp

ress

ions

hav

e va

riabl

es tha

tre

pres

ent nu

mbe

rs. A

var

iabl

e is

a le

tter

or

sym

bol t

hat st

ands

for

any

num

ber.

Exam

ples

of m

ultip

licat

ion

and

divi

sion

exp

ress

ions

are

9 3

k a

nd h

4 3

.

Wri

te a

n e

xpre

ssio

n th

at m

atch

es

the

wo

rds.

A h

and

ful o

f ke

ys, k

, div

ide

d e

qua

lly a

nd p

ut o

n 4

key

chai

ns.

Writ

e a

varia

ble

to r

epre

sent

the

unk

now

n nu

mbe

r in

the

pro

blem

.

The

keys

are

bei

ng s

epar

ated

into

four

gro

ups,

so the

ope

ratio

n is

div

isio

n.

So, t

he e

xpre

ssio

n is

k 4

4.

Let k

repr

esen

t th

e ke

ys.

k 4

4

Find

the

val

ue o

f th

e e

xpre

ssio

n.

s 4

3 if

s 5

27

The

varia

ble

s re

pres

ents

27.

Rep

lace

s w

ith 2

7.

Solv

e us

ing

divi

sion

.

So, t

he v

alue

of s

4 3

is 9

, if s

5 2

7.

s 4

327

4 3

27 4

3 5

9

Wri

te a

n e

xpre

ssio

n th

at m

atch

es

the

wo

rds.

1. 7

times

mor

e bo

oks

than

bef

ore

2. th

e pr

ice

of s

ome

pens

, at $1

eac

h

3. 40

pla

tes

divi

ded

equa

lly a

nd p

ut o

n so

me

tabl

es4.

so

me

shoe

s di

vide

d eq

ually

and

pu

t in

to 8

box

es

Find

the

val

ue o

f th

e e

xpre

ssio

n.

5. b

3 5

if b

5 4

6. 6

0 4

k if

k 5

10

7.

8 3

w if

w 5

6

8. m

4 4

if m

5 3

69.

s 3

9 if

s 5

510

. h

4 7

if h

5 4

9

MX

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Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

29

Ord

er o

f Ope

ratio

nsTo

sol

ve p

robl

ems

that

hav

e m

ore

than

one

typ

e of

ope

ratio

n, y

oune

ed to

know

the

ord

er

of

op

era

tio

ns. T

his

is the

set

of r

ules

that

tel

ls w

hich

ope

ratio

n to

do

first

. Firs

t, so

lve

any

oper

atio

ns in

par

enth

eses

. N

ext,

mul

tiply

and

div

ide

from

left

to r

ight

. The

n, a

dd a

nd s

ubtr

act fr

om le

ft to

rig

ht.

Follo

w t

he o

rde

r o

f o

pe

rati

ons

to

fin

d t

he v

alue

of

eac

h e

xpre

ssio

n.

3 3

(6 2

2) 1

4 4

2

Perf

orm

the

cal

cula

tions

insi

de the

par

enth

eses

firs

t.

Thin

k: 6

2 2

5 4

Mul

tiply

and

div

ide

from

left

to r

ight

.

Thin

k: 3

3 4

5 1

2 an

d

4

4 2

5 2

.

Add

and

sub

trac

t fr

om le

ft to

rig

ht.

Thin

k: 1

2 1

2 5

14

So, 3

3 (

6 2

2) 1

4 4

2 5

14

Follo

w t

he o

rde

r o

f o

pe

rati

ons

to

fin

d t

he v

alue

of

the

exp

ress

ion.

1.

(1

2 1

8) 4

(4 2

2)

4.

14

2 (

5 1

2)

7.

7 3

6 1

6

10.

9 1

(6 3

2) 4

2

2.

(8

3 3

) 2

6

5.

(2

3 4

) 1

(5 3

1)

8.

(8

3 4

) 4

(3 1

1)

11.

10

4 2

3 5

3.

4 3

(3 3

2)

6.

5 1

10 4

2

9.

2 3

(12

4 4

) 2

3

12.

(8

3 5

) 2

8 4

4

3 3

(6 2

2) 1

4 4

2

3 3

4

1 4

4 2

3 3

4 1

4 4

2

12

1

2

12

1

2

14

MX

EN

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2:

30:5

2 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

30

Mul

tiplic

atio

n an

d Di

visio

n Eq

uatio

nsW

rite

an

eq

uati

on.

Cho

ose

a v

aria

ble

fo

r th

e u

nkno

wn.

Te

ll w

hat

the

var

iab

le r

ep

rese

nts.

The

to

tal n

umb

er

of

ring

s d

ivid

ed

eq

ually

am

ong

4 f

rie

nds

is 2

rin

gs

for

eac

h fr

iend

.

• D

ecid

e on

a v

aria

ble

to r

epre

sent

the

unkn

own

amou

nt.

• C

hoos

e an

ope

ratio

n to

sol

veth

e pr

oble

m.

• Th

e rin

gs a

re d

ivid

ed in

to e

qual

grou

ps o

f 2 r

ings

.

The

varia

ble,

r, w

ill r

epre

sent

the

tota

l num

ber

of r

ings

.

The

rings

are

bei

ng s

epar

ated

into

four

equ

al g

roup

s, s

o th

e op

erat

ion

is d

ivis

ion.

r 4

4 5

2

So, t

he e

quat

ion

is r

4 4

5 2

, and

the

var

iabl

e, r

, rep

rese

nts

the

tota

l num

ber

of r

ings

.

So

lve

the

eq

uati

on.

a 3

7 5

63

• U

se m

enta

l mat

h to

find

whi

chnu

mbe

r is

rep

rese

nted

by

a.

• Y

ou k

now

tha

t 9 3

7 5

63.

• R

epla

ce a

with

9.

So, t

he v

alue

of a

is 9

.

Wha

t nu

mbe

r tim

es 7

equ

als

63?

a 5

9

9 3

7 5

63

6

3 5

63

Wri

te a

n e

qua

tio

n fo

r e

ach.

Cho

ose

a v

aria

ble

fo

r th

e u

nkno

wn.

Te

ll w

hat

the

va

riab

le r

ep

rese

nts.

1. 5

bow

ls w

ith a

num

ber

of g

rape

s in

ea

ch is

30

grap

es.

2. $4

0 di

vide

d eq

ually

am

ong

a nu

mbe

r of

frie

nds

is $

10 e

ach.

So

lve

the

eq

uati

on.

3. b

3 7

5 3

54.

60 4

k 5

10

5. 9

3 j 5

45

6. 3

6 4

m 5

67.

s 3

3 5

12

8. h

4 6

5 7

MX

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:21:

29 P

M

RW29-RW30


Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Pred

ict a

nd T

est

Mar

c lik

es

to s

olv

e w

ord

scr

amb

les

and

maz

es.

Ye

ste

rday

he

so

lve

d

10 w

ord

scr

amb

les

and

maz

es

in a

ll. H

e s

olv

ed

2 m

ore

wo

rd s

cram

ble

s th

an m

aze

s. H

ow

man

y w

ord

scr

amb

les

did

Mar

c so

lve

ye

ste

rday

?

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2.

How

can

you

use

the

pre

dict

and

tes

t st

rate

gy to

help

you

sol

ve the

pro

blem

?

So

lve

3. U

se the

tab

le to

help

you

sol

ve the

pro

blem

.

Pre

dic

tT

est

Diff

eren

ce

Tot

al

Do

es i

t C

he

ck

?

6

scra

mbl

es

4 m

azes

5

scra

mbl

es

5 m

azes

Num

ber

of w

ord

scra

mbl

es M

arc

solv

ed:

Ch

eck

4. D

oes

the

answ

er m

ake

sens

e fo

r th

e pr

oble

m?

Expl

ain.

Pre

dic

t an

d t

est

to

so

lve

.

5.

The

prod

uct of

tw

o nu

mbe

rs is

42.

Th

eir

sum

is 1

3. W

hat ar

e th

e nu

mbe

rs?

6.

Sue

has

thre

e tim

es a

s m

uch

mon

ey

as D

an. T

oget

her

they

hav

e $4

0. H

ow

muc

h m

oney

doe

s ea

ch p

erso

n ha

ve?

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

31

MX

EN

L08

AW

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dd

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7 1

:22:

47 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

32

Expl

ore

Ineq

ualit

ies

Re

me

mb

er:

$ m

eans

“gr

eate

r th

an o

r eq

ual t

o,”

and

# m

eans

“le

ss tha

n or

equ

al to.

”

m 2

2 $

3

• Su

bstit

ute

each

num

ber

for

m.

• Su

bstit

ute

2 fo

r m

. 2 2

2 $

3.

0

$ 3

, so

2 do

es n

ot w

ork.

• Su

bstit

ute

3 fo

r m

. 3 2

2 $

3.

1

$ 3

, so

3 do

es n

ot w

ork.

• Su

bstit

ute

5 fo

r m

. 5 2

2 $

3.

3

$ 3

, so

5 w

orks

.

• Su

bstit

ute

6 fo

r m

. 6 2

2 $

3.

4

$ 3

, so

6 w

orks

.

So, 5

and

6 m

ake

the

ineq

ualit

y tr

ue.

a ,

7

Try

each

of t

he n

umbe

rs to

see

whi

ch o

nes

mak

e th

e in

equa

lity

true

.

Try 2.

2 ,

7.

Sinc

e 2

, 7

, 2 w

orks

.

Try 3.

3 ,

7.

Sinc

e 3

, 7

, 3 w

orks

.

Try 5.

5 ,

7.

Sinc

e 5

, 7

, 5 w

orks

.

Try 6.

6 ,

7.

Sinc

e 6

, 7

, 6 w

orks

.

So, 2

, 3, 5

, and

6 m

ake

the

ineq

ualit

y tr

ue.

Whi

ch o

f th

e n

umb

ers

2, 5

, 8, a

nd 9

mak

e t

he in

eq

ualit

y tr

ue?

1.

b #

7

2.

n 3

2 $

14

3.

a ,

12

Whi

ch o

f th

e n

umb

ers

4, 6

, 8, a

nd 10

mak

e t

he in

eq

ualit

y tr

ue?

4.

5 $

k 5

. h

3 2

$ 1

3 6.

10

# 4

1 w

Whi

ch o

f th

e n

umb

ers

1, 3

, 7, a

nd 9

mak

e t

he in

eq

ualit

y tr

ue?

Whi

ch o

f th

e n

umb

ers

2, 3

, 5, a

nd 6

mak

e t

he in

eq

ualit

y tr

ue?

7. 7

# y

8.

r 3

3 #

25

9.

3 #

h

MX

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:24:

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Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

33

Patt

erns

: Fin

d a

Rule

Whe

n yo

u ar

e as

ked

to to

find

a ru

le, l

ook

for

a pa

tter

n in

the

nu

mbe

rs. S

ee h

ow d

iffer

ent pa

irs o

f num

bers

are

rel

ated

.

Find

a r

ule

. Wri

te y

our

rul

e a

s an

eq

uati

on.

U

se y

our

rul

e t

o f

ind

the

mis

sing

num

be

rs.

Inpu

t, b

9070

6050

3020

10

Outp

ut, c

97

6

• Lo

ok fo

r a

patter

n in

the

num

bers

to

find

a ru

le.

Stud

y ho

w the

inpu

t nu

mbe

rs r

elat

e to

the

out

put nu

mbe

rs.

• Th

e fir

st p

air

is 9

0 an

d 9.

90

4 1

0 5

9.

The

seco

nd p

air

is 7

0 an

d 7.

70

4 1

0 5

7.

The

rule

is “

divi

de b

y 10

.”

• W

rite

your

rul

e as

an

equa

tion.

You

are

div

idin

g b

by 1

0 to

get

c, s

o th

e eq

uatio

n is

b 4

10

5 c

.

• Fi

nd the

mis

sing

num

bers

in the

tab

le u

sing

the

equ

atio

n.50

4 1

0 5

5; 3

0 4

10

5 3

; 20

4 1

0 5

2; 1

0 4

10

5 1

So, t

he r

ule

is “

divi

de b

y 10

,” the

equ

atio

n fo

r th

e ru

le is

b 4

10

5 c

, and

the

mis

sing

nu

mbe

rs in

the

tab

le a

re 5

, 3, 2

, and

1.

Find

a r

ule

. Wri

te y

our

rul

e a

s an

eq

uati

on.

Use

yo

ur r

ule

to

fin

d

the

mis

sing

num

be

rs.

1.2.

3.

4.

5.

6.

Inpu

t, a

36

79

10

Outp

ut, b

12

2428

Inpu

t, p

24

68

9

Outp

ut, m

12

2436

Inpu

t, r

23

56

8

Outp

ut, s

18

2745

Inpu

t, r

69

1821

33

Outp

ut, s

2

36

Inpu

t, e

5030

2510

Outp

ut, f

10

65

1

Inpu

t, w

8070

40

Outp

ut, z

40

3520

105

MX

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:25:

41 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

34

Telli

ng T

ime

On

a di

gita

l clo

ck, t

he c

olo

n (:)

sep

arat

es the

hou

rs fr

om the

m

inut

es. T

he n

umbe

r to

the

left

of the

col

on tel

ls the

hou

r. Th

e nu

mbe

r to

the

rig

ht o

f the

col

on tel

ls the

min

utes

.

Wri

te t

he t

ime

as

sho

wn

on

a d

igit

al c

lock

.

47 m

inut

es

afte

r tw

elv

e

• D

eter

min

e th

e ho

ur.

• S

ince

it is

47

min

utes

afte

r tw

elve

, the

ho

ur is

stil

l 12:

00.

• D

eter

min

e th

e m

inut

es.

• 4

7 m

inut

es h

ave

gone

by,

so

the

min

ute

is 0

:47.

• S

o, the

tim

e is

12:

47.

20 m

inut

es

be

fore

fo

ur

• D

eter

min

e th

e ho

ur.

• It

is 2

0 m

inut

es b

efor

e fo

ur, s

o it

is n

ot

four

yet

. The

hou

r is

3:0

0.

• D

eter

min

e th

e m

inut

e.

• T

here

are

60

min

utes

in a

n ho

ur. I

n 20

m

inut

es, i

t w

ill b

e 4:

00.

• S

ubtr

act:

60 m

inut

es 2

20

min

utes

5

40 m

inut

es, o

r 40

min

utes

hav

e pa

ssed

. The

min

ute

is 0

:40.

• S

o, the

tim

e is

3:4

0.

Wri

te t

he t

ime

as

sho

wn

on

a d

igit

al c

lock

.

1.

5

min

utes

afte

r se

ven

2

. 50

min

utes

afte

r th

ree

3.

22 m

inut

es b

efor

e fiv

e

4.

46 m

inut

es a

fter

nine

5. 5

min

utes

afte

r tw

o 6

. 20

min

utes

afte

r ei

ght

7.

11 m

inut

es a

fter

twel

ve

8.

30 m

inut

es b

efor

e th

ree

9.

25 m

inut

es a

fter

four

10.

5 m

inut

es b

efor

e el

even

11.

10

min

utes

bef

ore

one

12.

48

min

utes

afte

r se

ven

MX

EN

L08

AW

K4X

_RT

_CH

06_L

1.in

dd

A6/

14/0

7 1

:27:

25 P

M

RW33-RW34


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

35

Elap

sed

Tim

eEl

apse

d t

ime

tel

ls h

ow m

uch

time

has

pass

ed fr

om the

beg

inni

ng to

the

end

of a

n ac

tivity

. The

end

tim

e is

the

tim

e w

hen

an a

ctiv

ity e

nds.

Find

the

ela

pse

d t

ime

.

star

t: 12:00

A.M

. e

nd: 3:07

P.M

.

• C

ount

the

hou

rs. B

e su

re to

chec

k th

e A

.M. a

nd P

.M.

• B

egin

at 12

:00

A.M

. (st

art tim

e) a

nd

coun

t to

12:

00 P

.M. T

hat is

12

hour

s.

12:0

0 P.

M. t

o 3:

00 P

.M. i

s 3

hour

s.12

1 3

5 1

5 ho

urs.

• C

ount

the

min

utes

.

• 3

:00

P.M

. to

3:07

P.M

. is

7 m

inut

es.

• A

dd the

min

utes

to

the

hour

s.

• 1

5 ho

urs 1

7 m

inut

es 5

15

hour

s 7

min

utes

.

• S

o, the

ela

psed

tim

e is

15

hour

s 7

min

utes

, or

15 h

r 7

min

.

Use

a c

lock

to

fin

d t

he e

nd t

ime

.

star

t: 3:07

A.M

.

ela

pse

d t

ime

: 8 h

r 55

min

• A

dd the

ela

psed

tim

e to

the

sta

rt tim

e.

8

hr 5

5 m

in

1 3

hr

7

min

___

11

hr 6

2 m

in

• R

egro

up m

inut

es to

hour

s.

• Th

ere

are

60 m

inut

es in

1 h

our.

62 m

in 5

1 h

r 2

min

• 11

hr 1

1 h

r 1

2 m

in 5

12

hr 2

min

.

• So

, the

end

tim

e is

12:

02 P

.M.

Find

the

ela

pse

d t

ime

. 1

. st

art:

1:05

P.M

.

end:

2:0

5 P.

M.

2.

star

t: 9:

14 P

.M.

end:

11:

28 P

.M.

3.

star

t: 2:

10 A

.M.

end:

2:3

5 A

.M.

4.

star

t: 6:

45 P

.M.

end:

8:2

0 P.

M.

5.

star

t: 7:

00 P

.M.

end:

12:

10 A

.M.

6.

star

t: 6:

50 A

.M.

end:

4:5

0 P.

M.

7.

star

t: 10

:30

P.M

. en

d: 3

:20

A.M

.

8.

star

t: 1:

10 A

.M.

end:

7:3

5 P.

M.

Find

the

end

tim

e.

9.

star

t: 3:

07 A

.M.

elap

sed

time:

7 hr

5 m

in

10.

sta

rt: 6

:29

P.M

.el

apse

d tim

e:1

hr 1

2 m

in

11.

star

t: 9:

47 A

.M.

elap

sed

time:

7 hr

30

min

12.

star

t: 5:

30 A

.M.

elap

sed

time:

48 m

in

MX

EN

L08

AW

K4X

_RT

_CH

06_L

2.in

dd

A6/

14/0

7 1

:29:

45 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

36

Elap

sed

Tim

e on

a C

alen

dar

Janu

ary

Sun

Mon

Tue

Wed

Thu

Fri

Sat

12

34

5

67

89

1011

12

1314

1516

1718

19

2021

2223

2425

26

2728

2930

31

Febr

uary

Sun

Mon

Tue

Wed

Thu

Fri

Sat

12

34

56

78

9

1011

1213

1415

16

1718

1920

2122

23

2425

2627

2829

Use

the

cal

end

ars

abo

ve.

1.

Terr

y go

es o

n va

catio

n fr

om

Janu

ary

6 to

Jan

uary

21.

For

ho

w m

any

days

is T

erry

on

vaca

tion?

2. M

arty

’s c

lass

stu

dies

Aus

tral

ia fo

r3

wee

ks. T

he c

lass

beg

ins

the

unit

of s

tudy

on

Janu

ary

14. W

hen

does

the

unit

of s

tudy

end

?

3. Th

ere

is s

now

on

the

grou

nd in

Mar

sha’

s ya

rd fr

om J

anua

ry 1

3th

to F

ebru

ary

9th.

Fo

r ho

w m

any

days

was

the

re s

now

on

the

grou

nd?

4. Si

mon

e ta

kes

a 5-

wee

k cl

ass

at the

lo

cal c

olle

ge. H

er c

lass

end

s on

Feb

ruar

y 23

. Whe

n do

es S

imon

e’s

clas

s be

gin?

Li w

ent

to

the

de

ntis

t o

n Ja

nuar

y 15

. If

tod

ay

is F

eb

ruar

y 29

, ho

w m

any

we

eks

and

day

s is

it

sin

ce s

he s

aw t

he d

ent

ist?

From

Jan

uary

15

to F

ebru

ary

15 is

one

mon

th.

Then

cou

nt the

day

s fr

om F

riday

, Feb

ruar

y 15

to

Frid

ay, F

ebru

ary

29.

That

is 1

4 da

ys.

So, i

t ha

s be

en 6

wee

ks a

nd 3

day

s si

nce

Li s

aw the

den

tist.

Fran

k fin

ishe

d b

uild

ing

a m

od

el s

hip

on

Feb

ruar

y 26

. If

he w

ork

ed

on

it f

or 8

we

eks

, w

hen

did

he

sta

rt t

he s

hip

?

Cou

nt the

wee

ks b

ackw

ards

from

Tue

sday

, Fe

brua

ry 2

6 to

Tue

sday

, Jan

uary

1.

Eigh

t w

eeks

bac

kwar

ds fr

om F

ebru

ary

26 is

Ja

nuar

y 1.

So, J

anua

ry 1

is w

hen

Fran

k st

arte

d th

e sh

ip.

MX

EN

L08

AW

K4X

_RT

_CH

06_L

3.in

dd

A6/

14/0

7 1

:30:

43 P

M

RW35-RW36


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

37

Alge

bra:

Cha

nge

Units

of T

ime

Whe

n yo

u ch

ange

tim

e fr

om lo

nger

uni

ts (s

uch

as y

ears

) to

shor

ter

units

(suc

h as

day

s), y

ou m

ultip

ly. W

hen

you

chan

ge tim

e fr

om s

hort

er

units

(suc

h as

min

utes

) to

long

er u

nits

(suc

h as

hou

rs),

you

divi

de.

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

. 1

. 5

wee

ks 5

j d

ays

2.

24 m

onth

s 5

j y

ears

3.

3 ho

urs

5 j

min

utes

4

. 16

8 ho

urs

5 j

day

s

5.

3 ye

ars

5 j

wee

ks

6.

5 ho

urs

5 j

min

utes

7.

2 ye

ars

5 j

day

s 8

. 1,

460

days

5 j

yea

rs

9.

9 ho

urs

5 j

min

utes

10

. 1,

440

min

utes

5 j

hou

rs

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

3 ye

ars 5

j d

ays

• Y

ears

are

long

er tha

n da

ys, s

o m

ultip

ly.

• 1

year

5 3

65 d

ays.

Mul

tiply

the

num

ber

of y

ears

by

365.

• 3

3 3

65 5

1,0

95.

• So

, you

mul

tiply

to

show

3 y

ears

5 1

,095

mon

ths.

120

hour

s 5

j d

ays

• H

ours

are

sho

rter

tha

n da

ys, s

o di

vide

.

• 1

day

5 2

4 ho

urs.

Div

ide

the

num

ber

of h

ours

by

24.

• 12

0 4

24

5 5

• So

, you

div

ide

to s

how

120

hou

rs 5

5 d

ays.

Units

of T

ime

1 m

inut

e =

60 s

econ

ds

1 ho

ur =

60

min

utes

1 da

y =

24 h

ours

1 w

eek

= 7

days

1 ye

ar =

12

mon

ths

1 ye

ar =

365

day

s

1 ye

ar =

abo

ut 5

2 w

eeks

1 le

ap y

ear =

366

day

s

MXE

NL08

AW

K4X_

RT_C

H06_

L4.in

dd

A6/

6/07

12

:18:

35 P

M

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Sequ

ence

Info

rmat

ion

Jo’s

cla

ss a

rriv

ed

at

the

sci

enc

e m

use

um a

t 10

:15

A.M

. to

st

udy

the

inve

ntio

n o

f th

e a

irp

lane

. The

stu

de

nts

have

3

hour

s 30

min

ute

s in

all

at t

he m

use

um. T

he c

lass

wan

ted

40

min

ute

s fo

r lu

nch

and

25

min

ute

s in

the

gift

sho

p a

fte

r lu

nch.

Wha

t w

as t

he la

test

tim

e t

hey

coul

d s

top

fo

r lu

nch?

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

At w

hat tim

e w

ill the

stu

dent

s le

ave

the

mus

eum

? Ex

plai

n.

3.

Wha

t is

the

late

st tim

e th

e st

uden

ts c

an s

top

for

lunc

h? W

rite

your

ans

wer

in a

co

mpl

ete

sent

ence

.

4.

How

can

you

che

ck y

our

answ

er to

this

pro

blem

?

So

lve

the

pro

ble

m.

5. Ji

m’s

fath

er w

as b

orn

in 1

920.

Jim

’s

mot

her

was

bor

n in

192

4. J

im’s

mot

her

was

30

year

s ol

d w

hen

Jim

was

bor

n.

Jim

’s fa

ther

was

32

year

s ol

d w

hen

Jim

’s s

iste

r Pa

t w

as b

orn.

Whi

ch c

hild

isol

der:

Jim

or

Pat?

6. Ti

na m

ows

the

law

n an

d cl

eans

out

th

e ga

rage

tod

ay. S

he s

tart

s at

9:0

0 A

.M.

and

it ta

kes

a to

tal o

f 3 h

ours

. Firs

t, sh

e m

ows

the

law

n. T

hen,

she

tak

es a

15

min

ute

brea

k. N

ext,

she

clea

ns the

ga

rage

, whi

ch tak

es h

er 1

hou

r 30

min

utes

. At w

hat tim

e do

es T

ina

take

he

r br

eak?

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

38

MX

EN

L08

AW

K4X

_RT

_CH

06_L

5.in

dd

A6/

14/0

7 1

:31:

47 P

M

RW37-RW38


10 3030 10

DC

20020

25 15 5

°C

5 2515

°F

1001020304050607080 2025

30

5515253545556575 15

G

DJK

E H F

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

39

Tem

pera

ture

Som

e th

erm

omet

ers

show

tem

pera

ture

in d

eg

ree

s Fa

hre

nhe

it (

8F ).

O

ther

s sh

ow tem

pera

ture

in d

eg

ree

s C

els

ius

( 8C

). So

me

ther

mom

eter

s us

e bo

th F

ahre

nhei

t an

d C

elsi

us.

Use

the

the

rmo

me

ter

to f

ind

the

te

mp

era

ture

sho

wn

by

eac

h le

tte

r.

C • Th

e th

erm

omet

er u

ses

degr

ees

Cel

sius

.

• Th

e le

tter

C is

at th

e lin

e fo

r 25

8 C

elsi

us.

So, t

he tem

pera

ture

at th

e le

tter

C is

258

C.

D • Th

e th

erm

omet

er u

ses

degr

ees

Cel

sius

.

• Th

e le

tter

D is

bel

ow the

num

ber

0, s

o it

is

a ne

gativ

e te

mpe

ratu

re.

• D

is a

t th

e lin

e fo

r –20

8 C

elsi

us.

So, t

he tem

pera

ture

at th

e le

tter

D is

–20

8C.

Use

the

the

rmo

me

ter

be

low

to

fin

d t

he t

em

pe

ratu

re s

how

n b

y e

ach

lett

er.

1. C

2. D

3. E

4. F

5. G

6. H

7. I

8. J

9. K

MX

EN

L08

AW

K4X

_RT

_CH

06_L

6.in

dd

A6/

14/0

7 1

:33:

59 P

M

20

10

0

ED

10

20

0

AH

KE

QF

JP

GL

BR

ND

CM

– 10– 20

– 30+ 10

+ 20+ 30

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

40

Expl

ore

Neg

ativ

e N

umbe

rsN

eg

ativ

e n

umb

ers

are

num

bers

less

tha

n ze

ro, s

uch

as –

3 or

–16

. N

egat

ive

num

bers

hav

e a

smal

l min

us (–

) sig

n ne

xt to

them

. Po

siti

ve n

umb

ers

are

num

bers

gre

ater

tha

n ze

ro, s

uch

as 5

and

12.

Z

ero

is n

eith

er a

pos

itive

nor

a n

egat

ive

num

ber.

Nam

e t

he n

umb

ers

re

pre

sent

ed

by

po

ints

D a

nd E

on

the

num

be

r lin

e.

• F

irst,

find

the

letter

on

the

num

ber

line.

Th

en, c

ount

from

0 to

see

whi

ch n

umbe

r is

at th

e po

int fo

r ea

ch le

tter

.

• P

oint

D is

loca

ted

at 1

2. T

here

is a

plu

s si

gn b

efor

e th

e nu

mbe

r. W

hen

ther

e is

a

plus

sig

n, the

num

ber

is p

ositi

ve. P

oint

D is

at

112

, or

posi

tive

12.

• P

oint

E is

loca

ted

at –

16. T

he m

inus

sig

n (–) i

n fr

ont of

the

num

ber

show

s th

at the

nu

mbe

r is

neg

ativ

e. P

oint

E is

at

–16

, or

nega

tive

16.

• S

o, p

oint

D is

at

112

and

poi

nt E

is a

t –16

.

Nam

e t

he n

umb

er

rep

rese

nte

d b

y e

ach

lett

er

on

the

num

be

r lin

e b

elo

w.

1.

A

2.

B 3

. C

4.

D

5.

E

6.

F 7

. G

8.

H

9.

J10

. K

11.

L12

. M

13.

N14

. P

15.

Q16

. R

MXE

NL08

AW

K4X_

RT_C

H06_

L7.in

dd

A6/

12/0

7 1

2:53

:00

PM

RW39-RW40


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

41

Colle

ct a

nd O

rgan

ize

Data

You

can

use

a fr

eque

ncy

tabl

e to

org

aniz

e da

ta.

Use

the

Fie

ld D

ay P

arti

cip

ants

fre

que

ncy

tab

le.

If 2

six

th g

rad

ers

de

cid

e n

ot

to p

arti

cip

ate

and

5

fifth

gra

de

rs s

ign

up, h

ow

man

y to

tal s

tud

ent

s w

ill p

arti

cip

ate

fro

m t

he f

ifth

and

six

th g

rad

es?

• A

djus

t th

e nu

mbe

r of

six

th g

rade

rs.

Two

deci

de n

ot to

part

icip

ate,

so

subt

ract

2.

50 2

2 5

48.

• A

djus

t th

e nu

mbe

r of

fifth

gra

ders

. Fi

ve s

ign

up, s

o ad

d fiv

e.

55 1

5 5

60.

• A

dd the

num

ber

of fi

fth a

nd s

ixth

gra

ders

.

48

1 6

0 5

108

So, 1

08 fi

fth a

nd s

ixth

gra

ders

will

par

ticip

ate.

For

1–

2, u

se t

he F

ield

Day

Par

tici

pan

ts f

req

uenc

y ta

ble

ab

ove

.

1.

How

man

y to

tal s

tude

nts

will

par

ticip

ate

from

the

sec

ond

grad

e an

d th

ird g

rade

?

2.

How

man

y m

ore

stud

ents

will

pa

rtic

ipat

e fr

om s

ixth

gra

de tha

n fr

om K

inde

rgar

ten?

For

3–

5, u

se t

he F

avo

rite

Dra

win

g T

oo

l fre

que

ncy

tab

le b

elo

w.

Tell

whe

the

r e

ach

stat

em

ent

is t

rue

or

fals

e. E

xpla

in.

3.

Mor

e st

uden

ts c

hose

mar

kers

tha

n co

lor

penc

ils.

Fav

ori

te D

raw

ing

To

ol

Draw

ing

Tool

Freq

uenc

y

Colo

r pen

cil

3

Cra

yon

8

Mar

ker

7

4.

Mor

e st

uden

ts c

hose

col

or p

enci

ls

than

cra

yons

. 5.

A

tot

al o

f 20

stud

ents

res

pond

ed

to the

sur

vey

abou

t fa

vorit

e dr

awin

g to

ols.

Grad

e(N

umbe

r of S

tude

nts)

K

45

1

42

2

54

3

58

4

41

5

55

6

50

Fiel

d Da

y Pa

rtici

pant

s Fr

eque

ncy

MX

EN

L08

AW

K4X

_RT

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07_L

1.in

dd

A6/

14/0

7 1

:42:

25 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

42

Venn

Dia

gram

sY

ou c

an u

se a

Ve

nn d

iag

ram

to

sort

and

des

crib

e da

ta. R

emem

ber

the

sect

ion

whe

re the

circ

les

over

lap

show

s da

ta the

circ

les

have

in c

omm

on.

Wha

t la

be

l co

uld

yo

u us

e f

or

Se

ctio

n B

?

• Se

ctio

n A

con

tain

s m

ultip

les

of 2

.

• Se

ctio

n C

con

tain

s m

ultip

les

of 3

.

• So

, the

labe

l for

Sec

tion

B s

houl

d be

m

ultip

les

of 2

and

3.

Why

are

the

num

be

rs 6

and

12

sort

ed

in

Se

ctio

n B

of

the

dia

gra

m?

• Th

e nu

mbe

r 6

is a

mul

tiple

of b

oth

2 an

d 3.

• Th

e nu

mbe

r 12

is a

mul

tiple

of b

oth

2 an

d 3.

So, t

he n

umbe

rs 6

and

12

are

in S

ectio

n B

be

caus

e bo

th a

re m

ultip

les

of 2

and

3.

A

B

C

For 1–

2, u

se t

he V

enn

dia

gra

m a

t th

e r

ight

.

1.

Wha

t la

bel s

houl

d yo

u us

e fo

r

se

ctio

n B

?

A

B

C

2.

Wha

t ar

e tw

o nu

mbe

rs tha

t co

uld

go in

Sec

tion

B?

For 3–

4, u

se t

he V

enn

dia

gra

m a

t th

e r

ight

.

MU

LTIP

LES

OF

2M

ULT

IPLE

O

FM

ULT

IPLE

S

OF

NU

MBE

RS

LESS

TH

AN

60

NU

MBE

RS

GRE

ATE

R TH

AN

50

2

4

8 3

915

6 12

14

10

EVEN

N

UM

BERS

MU

LTIP

LES

OF

5

A

B

C3.

Wha

t ar

e tw

o nu

mbe

rs tha

t co

uld

go in

Sec

tion

B?

4. C

an the

num

ber

15 g

o in

Se

ctio

n B

? Ex

plai

n.

MX

EN

L08

AW

K4X

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07_L

2.in

dd

A6/

14/0

7 1

:44:

30 P

M

RW41-RW42


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

43

Find

Mea

n, M

edia

n, a

nd M

ode

The

me

an is

the

ave

rage

num

ber

in a

set

of d

ata.

The

me

dia

n is

the

m

iddl

e nu

mbe

r in

a s

et o

f dat

a. T

he m

od

e is

the

num

ber

that

occ

urs

the

mos

t of

ten

in a

set

of d

ata.

Rain

fall

Mon

thAp

rM

ayJu

nJu

lAu

gSe

p

Inch

es8

75

94

3

Find

the

me

dia

n.

• L

ist th

e nu

mbe

rs in

ord

er fr

om

leas

t to

gre

ates

t. 3,

4, 5

, 7, 8

, 9

• F

ind

the

num

ber

in the

mid

dle

of

the

list.

• B

oth

5 an

d 7

are

the

mid

dle

num

bers

. So,

add

the

tw

o m

iddl

e nu

mbe

rs a

nd d

ivid

e by

2.

• 5

1 7

5 1

2; 1

2 4

2 5

6.

S

o, the

med

ian

is 6

.

Find

the

mo

de

.

• A

ll of

the

num

bers

app

ear

once

.

Find

the

me

an.

• A

dd the

num

bers

in the

cha

rt.

• 8

1 7

1 5

1 9

1 4

1 3

5 3

6

• C

ount

how

man

y nu

mbe

rs a

re in

the

set

.

• T

here

are

6 n

umbe

rs.

• D

ivid

e th

e su

m b

y 6.

• 3

6 4

6 5

6

So

, the

mea

n is

6.

Find

the

me

an, m

ed

ian,

and

mo

de

.

1.G

am

es

Pla

yed

Mon

thJa

nFe

bM

arAp

r

Gam

es7

57

9

2.B

oo

ks

Read

Mon

thJa

nFe

bM

arAp

r

Boo

ks10

54

5

3. 4

.Sci

en

ce C

lub

Age

78

910

11

Freq

uenc

y5

14

64

Thu

nd

ers

torm

s

Mon

thM

ayFe

bM

arAp

r

Gam

es8

36

9

mea

nm

edia

nm

ode

mea

nm

edia

nm

ode

mea

nm

edia

nm

ode

mea

nm

edia

nm

ode

So, t

here

is n

o m

ode.

MX

EN

L08

AW

K4X

_RT

_CH

07_L

3.in

dd

A6/

7/07

11

:44:

48 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

44

Sp

ort

s D

ata

Sur

vey

Que

stio

n: H

ow m

any

hour

s do

you

sp

end

play

ing

spor

ts e

ach

wee

k?

Re

spo

nse

s: 1

, 4, 4

, 5, 2

, 6, 6

, 4, 1

, 0,

5, 4

, 2, 4

, 1

Line

Plo

tsY

ou c

an u

se a

line

plo

t to

gra

ph d

ata.

The

ran

ge

is the

diff

eren

ce

betw

een

the

low

est an

d hi

ghes

t va

lues

in the

dat

a. A

clu

mp

is

whe

re d

ata

is c

once

ntra

ted

in o

ne p

lace

. A h

ole

is w

here

the

re is

no

data

.

Use

the

To

uris

t Ti

me

Sur

vey

dat

a to

ans

we

r th

e q

uest

ions

be

low

.

Wha

t is

the

ran

ge

of

the

dat

a?

• O

n th

e lin

e pl

ot, f

ind

the

grea

test

nu

mbe

r of

hou

rs s

pent

vis

iting

.

• 5

is the

gre

ates

t nu

mbe

r.

• F

ind

the

leas

t nu

mbe

r of

hou

rs

on the

line

plo

t.

• 0

is the

leas

t nu

mbe

r.

• F

ind

the

diffe

renc

e be

twee

n th

e gr

eate

st a

nd the

leas

t nu

mbe

rs.

• 5

2 0

5 5

.

S

o, the

ran

ge is

5.

Ho

w m

any

hour

s d

id m

ost

to

uris

ts s

pe

nd?

Ho

w c

an y

ou

tell?

• L

ook

at the

line

plo

t to

see

whi

ch h

our

has

the

mos

t re

spon

ses.

• T

he li

ne p

lot ha

s th

e m

ost X

s at

1.

So, m

ost pe

ople

vis

ited

for

1 ho

ur b

ecau

se the

line

plo

t ha

s th

e m

ost X

s at

1.

For 1–

4, u

se t

he S

po

rts

Sur

vey

Dat

a at

the

rig

ht.

1.

Mak

e a

tally

tab

le to

show

the

dat

a.

2.

Mak

e a

line

plot

to

show

the

dat

a.

Tour

ist

Tim

e S

urve

yQ

uest

ion:

How

man

y ho

urs

did

you

spen

d vi

sitin

g K

enne

dy

Spac

e C

ente

r V

isito

r C

ompl

ex?

Re

spo

nse

s: 2

, 3, 4

, 1,

2, 5

, 1, 3

, 2, 1

, 2, 1

, 3,

3, 2

, 1, 0

, 3, 4

, 5, 1

3.

Wha

t is

the

ran

ge o

f the

dat

a?

4.

For

how

man

y ho

urs

do m

ost

peop

le p

lay

spor

ts?

How

can

you

tel

l?

MX

EN

L08

AW

K4X

_RT

_CH

07_L

4.in

dd

A6/

14/0

7 1

:45:

41 P

M

RW43-RW44


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

45

Choo

se a

Rea

sona

ble

Scal

e an

d In

terv

alTh

e in

terv

al o

f a g

raph

is the

diff

eren

ce b

etw

een

one

num

ber

and

the

next

on

the

scal

e of

a g

raph

. Gra

phs

use

diffe

rent

inte

rval

s ba

sed

on the

dat

a be

ing

grap

hed.

Cho

ose

5, 10,

25,

or 10

0 as

the

mo

st r

eas

ona

ble

in

terv

al f

or

eac

h se

t o

f d

ata.

200,

350

, 100

, 250

, 500

The

num

bers

are

all

in the

hun

dred

s. A

n in

terv

al o

f 5 o

r 10

wou

ld b

e to

o sm

all t

o fit

all

the

data

on

the

grap

h. A

n in

terv

al o

f 100

wou

ld s

how

the

dat

a cl

early

on

the

grap

h.

So, a

n in

terv

al o

f 100

wou

ld b

e m

ost re

ason

able

.

25, 79,

50,

45,

90

The

num

bers

are

all

less

tha

n 10

0, s

o an

inte

rval

of 1

00 w

ould

be

too

big.

An

inte

rval

of

5 w

ould

be

too

smal

l to

fit a

ll th

e da

ta. A

n in

terv

al o

f 10

wou

ld s

how

the

dat

a cl

early

on

the

gra

ph.

So, a

n in

terv

al o

f 10

wou

ld b

e m

ost re

ason

able

.

Cho

ose

5, 10,

or 10

0 as

the

mo

st r

eas

ona

ble

in

terv

al f

or

eac

h se

t o

f d

ata.

Exp

lain

yo

ur c

hoic

e.

1.

180,

195

, 183

, 210

, 200

2.

8, 1

3, 1

2, 6

, 9

3.

19, 1

3, 1

1, 7

, 5 4

. 35

, 65,

72,

21,

83

5.

90, 1

30, 2

00, 1

50, 3

10 6

. 56

0, 6

20, 7

20, 8

90, 3

60

7.

8, 1

, 3, 7

, 4, 6

8.

93, 6

6, 2

5, 7

5, 7

1

9.

52, 6

0, 3

4, 8

8, 6

9 10

. 75

7, 2

83, 8

91, 3

62, 5

49

MX

EN

L08

AW

K4X

_RT

_CH

07_L

5.in

dd

A6/

7/07

11

:47:

56 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

46

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Mak

e Ge

nera

lizat

ions

Use

the

tab

le a

t th

e r

ight

. Mak

e a

ge

nera

lizat

ion.

Miy

u is

25

year

s o

ld. S

he d

oe

s ae

rob

ics

for

a ha

lf ho

ur a

nd t

hen

me

asur

es

her

hear

t ra

te. I

f it

is

180

be

ats

pe

r m

inut

e, s

houl

d M

iyu

slo

w h

er

exe

rcis

ing

, or

can

she

co

ntin

ue a

t th

e s

ame

p

ace

? Ex

pla

in y

our

ans

we

r.

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

Look

at th

e in

form

atio

n in

the

gra

ph. M

ake

a ge

nera

lizat

ion.

3.

Find

a r

elat

ions

hip

betw

een

Miy

u’s

hear

t ra

te a

nd the

info

rmat

ion

in the

cha

rt.

Wha

t co

nclu

sion

can

you

dra

w a

bout

Miy

u’s

hear

t ra

te?

4.

Shou

ld M

iyu

slow

her

exe

rcis

ing

or c

an s

he c

ontin

ue a

t th

e sa

me

pace

? Ex

plai

n.

5.

How

can

you

che

ck y

our

answ

er?

For 6–

7, u

se t

he t

able

ab

ove

. Mak

e a

ge

nera

lizat

ion.

The

n so

lve

the

pro

ble

m.

6. W

hat if

the

grap

h sh

owed

a m

axim

um

targ

et h

eart

rat

e of

145

? Fo

r w

hat ag

e w

ould

tha

t ra

te b

e?

7. Si

d is

45

year

s ol

d. W

hile

ex

erci

sing

, his

hea

rt b

eats

190

tim

es

per

min

ute.

Sho

uld

Sid

slow

his

ex

erci

sing

?

MX

EN

L08

AW

K4X

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6.in

dd

A6/

13/0

7 2

:18:

04 P

M

RW45-RW46


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

47

Bar G

raph

sY

ou c

an u

se b

ar g

rap

hs to

com

pare

dat

a. In

ver

tical

bar

gra

phs,

th

e ba

rs r

un fr

om the

bot

tom

of t

he g

raph

to

the

top

of the

gra

ph.

In h

oriz

onta

l bar

gra

phs,

the

bar

s ru

n fr

om le

ft to

rig

ht.

Use

the

Cam

p C

hoic

es

bar

gra

ph.

Whi

ch c

amp

was

cho

sen

by

the

fe

we

st s

tud

ent

s?

The

shor

test

bar

is the

one

for

dram

a. S

o, d

ram

a ca

mp

was

cho

sen

by the

few

est nu

mbe

r of

stu

dent

s.

Ho

w m

any

stud

ent

s ch

ose

sp

ace

cam

p?

The

bar

for

spac

e ca

mp

goes

up

to the

line

for

20.

So, 2

0 st

uden

ts c

hose

spa

ce c

amp.

25 20 15 10 5 0

Cam

p

Cam

p C

ho

ices

Number of Students

Space

Drama

Sports

Adventur

e

Outdoor

25 20 15 10 5 0

Frui

t

Appl

eOr

ange

Grap

esPe

ach

Favo

rite

Fru

it

Number of Students

20 15 10 5 0

Spor

t

Foot

ball

Hock

eySo

ccer

Favo

rite

Sp

ort

Number of Students

For 3–

4, u

se t

he F

avo

rite

Sp

ort

bar

gra

ph.

For 1–

2, u

se t

he F

avo

rite

Fru

it b

ar g

rap

h.

1.

Whi

ch fr

uit do

mos

t st

uden

ts p

refe

r?

2.

How

man

y st

uden

ts p

refe

r or

ange

s?

3.

Whi

ch tw

o sp

orts

re

ceiv

ed the

sam

e nu

mbe

r of

vot

es?

4.

How

man

y vo

tes

did

hock

ey r

ecei

ve in

all?

MX

EN

L08

AW

K4X

_RT

_CH

08_L

1.in

dd

A6/

7/07

1:

16:3

4 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

48

Mak

e Ba

r and

Dou

ble-

Bar G

raph

sY

ou c

an u

se d

oub

le-b

ar g

rap

hs to

com

pare

si

mila

r ki

nds

of d

ata.

Dou

ble-

bar

grap

hs h

ave

two

bars

for

each

cat

egor

y.

Use

the

dat

a in

the

tab

le a

t th

e r

ight

to

mak

etw

o b

ar g

rap

hs. T

hen

mak

e a

do

uble

-bar

gra

ph.

• M

ake

a ba

r gr

aph

for

the

boys

’ dat

a.

• D

ecid

e on

a s

cale

for

the

bar

grap

h.

• Ti

tle a

nd la

bels

will

be

the

sam

e as

in the

tab

le.

• U

se the

num

bers

sho

wn

in the

tab

le to

draw

a

b

ar fo

r ea

ch tim

e of

day

.

• U

se the

sam

e st

eps

to m

ake

a ba

r

grap

h fo

r th

e gi

rls’ f

avor

ite tim

e of

day

.

• U

se the

sam

e sc

ale

and

inte

rval

.

• M

ake

a do

uble

-bar

gra

ph fo

r bo

th s

ets

if da

ta.

• U

se the

num

bers

sho

wn

in the

tab

le. D

raw

a

b

ar fo

r ea

ch tim

e of

day

for

the

girls

and

the

boy

s.

• U

se o

ne c

olor

for

the

boys

and

ano

ther

for

the

girls

.

• In

sert

a k

ey to

show

wha

t ea

ch c

olor

rep

rese

nts.

Use

the

dat

a in

the

tab

le a

t th

e r

ight

to

mak

e

two

bar

gra

phs

. The

n m

ake

a d

oub

le-b

ar g

rap

h.

Use

the

sp

ace

pro

vid

ed

be

low

.

35 30 25 20 15 10 5 0Morning

Afternoo

nEven

ing

Number of Students

Tim

e of

Day

Favo

rite

Tim

e of

Day

(Boy

s)

Favo

rite

Tim

e o

f D

ay

Tim

eBo

ysGi

rls

Mor

ning

68

Afte

rnoo

n20

17

Eve

ning

3331

Vic

tori

es

T

eam

Ap

rilM

ay J

une

J

ays

83

9

Robi

ns7

128

35 30 25 20 15 10 5 0Morning

Afternoo

nEven

ing

Number of Students

Tim

e of

Day

Favo

rite

Tim

e of

Day

(Girl

s)

Girl

sKe

y:Bo

ys35 30 25 20 15 10 5 0

Morning

Afternoo

nEven

ing

Number of Students

Tim

e of

Day

Favo

rite

Tim

e of

Day

MX

EN

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dd

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17:0

5 PM

RW47-RW48


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

49

Circ

le G

raph

sY

ou c

an u

se c

ircl

e g

rap

hs to

show

dat

a in

diff

eren

t ca

tego

ries.

Ea

ch c

olor

in the

gra

ph s

tand

s fo

r a

cate

gory

. Eac

h ca

tego

ry is

di

vide

d in

to s

ectio

ns to

show

the

dat

a.

Use

the

Fav

ori

te B

reak

fast

gra

ph.

Whi

ch t

ype

of

bre

akfa

st r

ece

ive

d t

he le

ast

num

be

r o

f vo

tes?

• T

he le

ast nu

mbe

r of

vot

es is

rep

rese

nted

by

the

sm

alle

st s

ectio

n.•

In the

gra

ph, t

he s

mal

lest

sec

tion

is la

bele

d eg

gs.

• S

o, the

typ

e of

bre

akfa

st w

ith the

leas

t nu

mbe

r of

vo

tes

is e

ggs.

Whi

ch t

ype

of

bre

akfa

st r

ece

ive

d t

he g

reat

est

nu

mb

er

of

vote

s?•

The

gre

ates

t nu

mbe

r of

vot

es is

rep

rese

nted

by

the

la

rges

t se

ctio

n.•

In the

gra

ph, t

he la

rges

t se

ctio

n is

labe

led

cere

al.

• So

, the

typ

e of

bre

akfa

st w

ith the

gre

ates

t nu

mbe

r of

vo

tes

is c

erea

l.

Favo

rite

Su

bje

ct

Favo

rite

An

imal

Favo

rite

Bre

akfa

st Eggs

Cere

al

Waf

fl es

Scie

nce

6M

ath

2 Read

ing

3

Wri

ting

8

Hor

se 1

5

Cat

4D

og 1

1

Mon

key

8

1.

Whi

ch s

ubje

ct r

ecei

ved

the

leas

t nu

mbe

r of

vot

es?

2.

Whi

ch s

ubje

ct r

ecei

ved

alm

ost ha

lf

of the

vot

es?

3.

How

man

y pe

ople

sai

d th

at d

ogs

wer

e th

eir

favo

rite

anim

al?

4.

How

man

y vo

tes

did

hors

es r

ecei

ve?

For 3–

4, u

se t

he F

avo

rite

Ani

mal

gra

ph.

For 1–

2, u

se t

he F

avo

rite

Sub

ject

gra

ph.

MX

EN

L08

AW

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_RT

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3.in

dd

A6/

7/07

1:

21:2

1 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

50

Alge

bra:

Use

a C

oord

inat

e Gr

idY

ou c

an p

lot po

ints

on

a co

ordi

nate

grid

usi

ng o

rde

red

pai

rs. A

n or

dere

d pa

ir ha

s tw

o nu

mbe

rs in

par

enth

eses

, lik

e th

is, (

3,6)

. Th

e fir

st n

umbe

r te

lls h

ow fa

r to

mov

e ho

rizon

tally

acr

oss

the

grid

. Th

e se

cond

num

bers

tel

ls h

ow fa

r to

mov

e ve

rtic

ally

up

the

grid

.

Wri

te t

he p

oin

t fo

r e

ach

ord

ere

d p

air

on

the

co

ord

inat

e g

rid

.

( 2,4

)

• St

art at

0. T

he fi

rst nu

mbe

r in

the

ord

ered

pai

r is

2,

so

mov

e ac

ross

tw

o un

its to

the

right

.

• Th

e se

cond

num

ber

is 4

, so

mov

e up

four

uni

ts. Y

ou e

nd a

t po

int A

.

• So

, the

poi

nt fo

r ( 2

,4) i

s A

.

( 3,8

)

• St

art at

0. M

ove

3 un

its to

the

right

and

8 u

nits

up.

• So

, the

poi

nt fo

r ( 3

,8) i

s G

.

Use

the

co

ord

inat

e g

rid

be

low

. Wri

te t

he p

oin

t fo

r e

ach

ord

ere

d p

air.

1. (1

,5)

2. (0

,4)

3. (7

,2)

4. (9

,9)

5. (2

,0)

6. (9

,1)

7. (6

,5)

8. (0

,3)

678910 5 4 3 2 1 02

31

45

78

910

6x-

axis

y-axis

678910 5 4 3 2 1 02

31

45

78

910

6x-

axis

y-axis

A

HG

D

F

C

E

N

PD

I HJ

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L

A

BK

R

MF

MX

EN

L08

AW

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_RT

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08_L

4.in

dd

A6/

7/07

1:

21:4

8 PM

RW49-RW50


Nam

eN

ame

© H

arco

urt •

Gra

de 4

Ret

each

RW

51

Line

Gra

phs

Line

gra

phs

sho

w h

ow d

ata

chan

ges

over

tim

e. T

rend

s re

fer

to

part

s of

the

line

gra

ph w

here

dat

a st

ays

the

sam

e, in

crea

ses,

or

decr

ease

s ov

er a

per

iod

of tim

e.

Wha

t w

as t

he h

eig

ht o

f th

e w

ate

r in

the

gla

ss o

n Th

ursd

ay?

• Fi

nd the

line

labe

led

for

Thur

sday

on

the

bottom

of t

he

grap

h. F

ollo

w the

line

up

to the

po

int.

See

whi

ch h

oriz

onta

l lin

e th

e po

int to

uche

s.

• Fo

llow

the

poi

nt a

ccro

ss to

the

left.

See

whe

re the

line

to

uche

s th

e sc

ale.

• Th

e lin

e to

uche

s th

e sc

ale

at

6 cm

, so

the

heig

ht o

f the

w

ater

in the

gla

ss o

n Th

ursd

ay w

as 6

cm

.

Be

twe

en

whi

ch t

wo

day

s d

id t

he w

ate

r le

vel s

tay

the

sam

e?

• W

hen

a lin

e be

twee

n tw

o po

ints

doe

s no

t go

up

or

dow

n, it

mea

ns tha

t th

e da

ta s

tays

the

sam

e. T

he

line

betw

een

the

poin

ts fo

r Tu

esda

y (T

ue) a

nd

Wed

nesd

ay (W

ed) s

tays

leve

l. So

, the

wat

er le

vel

stay

ed the

sam

e be

twee

n Tu

esda

y an

d W

edne

sday

.

For 1–

2, u

se t

he P

lant

Gro

wth

gra

ph.

For 3–

4, u

se t

he G

ame

s W

on

gra

ph.

1.

How

tal

l was

the

pla

nt in

Dec

embe

r?

3.

Dur

ing

whi

ch 3

mon

ths

did

the

team

w

in the

sam

e nu

mbe

r of

gam

es?

2.

Dur

ing

whi

ch tw

o m

onth

s di

d th

e pl

ant’s

hei

ght in

crea

se the

mos

t?4.

H

ow m

any

gam

es d

id the

tea

m w

in in

A

pril?

Height (in Centimeters)

Hei

ght o

f Wat

er in

a G

lass

Out

door

s

Day

Mon

Thu

Fri

Wed

Tue

12 10 8 6 4 2 0

Height (in centimeters)

Pla

nt

Gro

wth

Day

Oct

Jan

Feb

Mar

Apr

Dec

Nov

14 12 10 8 6 4 2 0

Number of Games

Gam

es

Wo

n

Day

Oct

Jan

Feb

Mar

Apr

Dec

Nov

10 8 6 4 2 0

MX

EN

L08

AW

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5.in

dd

A6/

7/07

1:

22:2

5 PM

Height (in Inches)

Plan

t Gro

wth

End

of W

eek

14

32

20 15 10 5 0

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

52

Mak

e Li

ne G

raph

sU

se t

he d

ata

in t

he t

able

to

mak

e a

line

gra

ph.

• W

rite

a tit

le fo

r th

e gr

aph:

Pla

nt G

row

th.

• C

hoos

e a

scal

e an

d in

terv

al. S

ince

the

sm

alle

st n

umbe

r in

the

tab

le is

6 a

nd the

gr

eate

st n

umbe

r is

15,

you

can

use

a s

cale

of 5

. Writ

e th

e la

bel a

nd s

cale

num

bers

al

ong

the

left

side

of t

he g

raph

.

• W

rite

the

labe

ls fo

r th

e w

eeks

alo

ng the

bot

tom

of t

he g

raph

.

• Pl

ot the

poi

nts.

The

n dr

aw li

ne s

egm

ents

to

conn

ect th

e po

ints

from

left

to r

ight

.

For 1–

2, u

se t

he d

ata

to m

ake

line

gra

phs

in t

he b

oxe

s b

elo

w.

1.2.

Hale

y’s

Aft

ern

oo

n B

ikeri

de

Tim

e (in

m

inut

es)

3060

9012

0

Dist

ance

(in

mile

s)9

1621

27

Pla

nt

Gro

wth

End

of W

eek

12

34

Heig

ht (i

n in

ches

)6

913

15

Bia

nca

’s W

riti

ng

Pro

gre

ssTo

tal P

ages

13

911

Tim

e (in

m

inut

es)

1530

4560

MX

EN

L08

AW

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6.in

dd

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7/07

1:

22:5

2 PM

RW51-RW52


10 8 6 4 2 0La

keOc

ean

Natio

nal

Park

Amus

emen

tPa

rk

Number of Students

Plac

e

Pla

ces

We’d

Lik

e t

o V

isit

Girls

Boys

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

53

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Mak

e a

Grap

hM

anue

l’s c

lass

vo

ted

fo

r th

eir

fav

ori

te p

lace

s to

vi

sit.

Mak

e a

do

uble

-bar

gra

ph

to s

how

the

dat

a.

The

n fin

d w

hich

pla

ce h

ad t

he g

reat

est

diff

ere

nce

in

vo

tes

be

twe

en

gir

ls a

nd b

oys

.

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

Pla

n

2.

How

can

mak

ing

a gr

aph

help

you

sol

ve thi

s pr

oble

m?

So

lve

3.

C

ompl

ete

the

grap

h be

low

. Whi

ch p

lace

had

the

gre

ates

t di

ffere

nce

betw

een

boys

an

d gi

rls.

Ch

eck

4.

How

can

you

che

ck to

see

that

you

r an

swer

is c

orre

ct?

Use

the

Win

ter

Sno

wfa

ll ta

ble

to

mak

e a

gra

ph.

5.

Pla

ces

Man

uel’s

Cla

ss

Wo

uld

Lik

e t

o V

isit

Plac

eGi

rlsBo

ys L

ake

21

Oce

an4

2 N

atio

nal P

ark

27

Am

usem

ent P

ark

68

Win

ter

Sn

ow

fall

Mon

thIn

ches

of S

now

Dec

25

Jan

30

Feb

15

MX

EN

L08

AW

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_RT

_CH

08_L

7.in

dd

A6/

13/0

7 1

:56:

10 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

54

Choo

se a

n Ap

prop

riate

Gra

phY

ou c

an u

se a

var

iety

of g

raph

s to

pre

sent

dat

a.Th

e gr

aph

you

use

depe

nds

on the

typ

e of

dat

a yo

u ar

e sh

owin

g.

bar

gra

ph

or

do

uble

-bar

gra

phs

Use

to

show

and

com

pare

dat

a ab

out

diffe

rent

cat

egor

ies

or g

roup

s. E

xam

ple:

cars

so

ld in

Jun

e, J

uly,

and

Aug

ust.

circ

le g

rap

hsU

se to

com

pare

par

ts o

f a g

roup

to

aw

hole

gro

up. E

xam

ple:

the

favo

rite

subj

ects

of a

gro

up o

f stu

dent

s.

line

gra

phs

Use

to

show

how

dat

ach

ange

ove

r tim

e.Ex

ampl

e: g

row

th o

f a tr

eeo

ver

one

yea

r.

line

plo

tsU

se to

show

the

freq

uenc

y of

the

dat

a al

ong

a nu

mbe

r lin

e. E

xam

ple:

num

ber

of s

tude

nts

who

ha

ve 1

, 2, o

r 3

sibl

ings

.

pic

tog

rap

hsU

se to

show

and

com

pare

da

ta a

bout

diff

eren

t ca

tego

ries

or g

roup

s.

Exam

ple:

num

ber

of s

hirt

s,

sho

es, a

nd p

ants

in

a cl

ose

t.

Cho

ose

and

exp

lain

the

be

st t

ype

of

gra

ph

or

plo

t fo

r th

e d

ata.

the

num

be

r o

f in

che

s o

f ra

in o

ver 5

day

s

• T

his

data

focu

ses

on h

ow the

am

ount

of r

ain

chan

ged

or s

taye

d th

e sa

me

over

the

5 d

ays.

• A

line

gra

ph w

ould

be

the

best

typ

e of

gra

ph fo

r th

is d

ata

beca

use

a lin

e gr

aph

show

s ho

w d

ata

chan

ges

over

tim

e.

Cho

ose

and

exp

lain

the

be

st t

ype

of

gra

ph

or

plo

t fo

r th

e d

ata.

1. th

e ty

pes

of s

hoes

tha

t w

ere

mos

tof

ten

purc

hase

d in

Apr

il2.

ho

w a

fam

ily s

pent

$10

0

3. ho

w m

uch

a vi

ne g

rew

eac

h ye

ar4.

ho

w m

any

hom

e ru

ns a

tea

m h

it at

ea

ch o

f 5 g

ames

5. th

e nu

mbe

r of

stu

dent

s pl

ayin

g di

ffere

nt s

port

s at

rec

ess

6. th

e te

mpe

ratu

re in

a s

mal

l tow

n ov

er

3 m

onth

s

MX

EN

L08

AW

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8.in

dd

A6/

7/07

1:

23:4

8 PM

RW53-RW54


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

55

Alge

bra:

Gra

ph R

elat

ions

hips

A f

unct

ion

is a

n eq

uatio

n th

at tel

ls y

ou the

val

ue o

f a n

umbe

r, y,

in

ter

ms

of a

noth

er n

umbe

r, x.

For

exa

mpl

e: y

5 x

1 9

. A

func

tio

n ta

ble

con

nect

s an

inpu

t va

lue,

x, t

o an

out

put va

lue,

y.

Find

a r

ule

. Wri

te y

our

rul

e a

s an

eq

uati

on.

Use

the

eq

uati

on

tofin

d t

he m

issi

ng n

umb

ers

.

Yard

s, x

12

3

45

6j

Feet

, y3

j9

12

15

j 2

1

• L

ook

for

a pa

tter

n in

the

tab

le to

find

a ru

le.

Try

to fi

gure

out

how

the

inpu

t nu

mbe

rs r

elat

e to

the

out

put nu

mbe

rs.

• T

he v

alue

of y

is a

lway

s 3

times

mor

e th

an the

val

ue o

f x.

So, t

he r

ule

is to

mul

tiply

x b

y 3.

• W

rite

the

rule

as

an e

quat

ion:

y 5

3x.

• U

se the

equ

atio

n to

find

the

mis

sing

num

bers

in the

tab

le.

2

3 3

5 6

6 3

3 5

18

7

3 3

5 2

1

So

, the

rul

e is

y 5

3x,

and

the

mis

sing

num

bers

in the

tab

le a

re 6

, 18,

and

7.

Find

a r

ule

. Wri

te y

our

rul

e a

s an

eq

uati

on.

U

se t

he e

qua

tio

n to

ext

end

the

pat

tern

.

1. 2. 3.

Inpu

t, x

100

j80

7060

j40

Outp

ut, y

50

45j

35j

2520

Penn

ies,

x 1

020

j 4

0j

60

j

Dim

es, y

1

j3

4

5

67

Inpu

t, x

1j

3

4

j6

7

Outp

ut, y

714

2128

35j

j

MX

EN

L08

AW

K4X

_RT

_CH

08_L

9.in

dd

A6/

7/07

1:

24:2

4 PM

1216 1418 10 8 6 4 2 02

31

45

Leng

th o

f Si

de

Perimeter

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

56

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Iden

tify

Rela

tions

hips

Ro

bin

is f

enc

ing

in a

sq

uare

gar

de

n th

at is

2 y

ard

s o

n

eac

h si

de

. Jo

sé is

fe

ncin

g in

a s

qua

re g

ard

en

that

is

4 ya

rds

on

eac

h si

de

. Ho

w m

uch

mo

re f

enc

ing

do

es

José

ne

ed

tha

n R

ob

in?

Use

the

gra

ph

to t

he r

ight

.

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

How

is the

leng

th o

f eac

h si

de r

elat

ed to

the

perim

eter

?

3.

How

muc

h fe

ncin

g do

es J

osé

need

? H

ow m

uch

fenc

ing

does

Rob

in n

eed?

4.

How

muc

h m

ore

fenc

ing

does

Jos

é ne

ed tha

n R

obin

?

5.

How

can

you

che

ck y

our

answ

er?

Use

the

Pip

es

tab

le t

o s

olv

e.

6.

How

man

y po

unds

wou

ld a

n 18

-foo

t

le

ngth

of p

ipe

wei

gh?

7.

Dan

nee

ds 2

3 fe

et o

f pip

e to

fix

his

drai

n. H

e bu

ys tw

o 10

-foo

t se

ctio

ns

and

one

3-fo

ot s

ectio

n. W

hat w

ill b

e th

e to

tal w

eigh

t of

the

pip

es?

Pip

es

Leng

th, f

t6

1012

Wei

ght,

lb18

3036

x-ax

is

y-axis

MX

EN

L08

AW

K4X

_RT

_CH

08_L

10.in

dd

A6/

7/07

1:

27:2

9 PM

RW55-RW56


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

57

Men

tal M

ath:

Mul

tiplic

atio

n Pa

tter

nsY

ou c

an u

se b

asic

fact

s an

d m

enta

l mat

h to

mul

tiply

who

le

num

bers

by

mul

tiple

s of

10,

100

, 1,0

00.

Use

me

ntal

mat

h to

co

mp

lete

the

pat

tern

.

Use

me

ntal

mat

h to

co

mp

lete

the

pat

tern

.

1.

7 3

5 5

35

7

3 5

0 5

7

3 5

00 5

2.

3 3

4 5

12

3

3 4

0 5

3

3 4

00 5

3.

9 3

8 5

72

9

3 8

0 5

9

3 8

00 5

4.

8 3

6 5

48

8

3

5

480

8

3 6

00 5

5.

2 3

7 5

14

2

3

5 1

40

2

3 7

00 5

6.

4 3

4 5

16

4

3

516

0

4

3 4

00 5

7.

6 3

3 5

6

3 3

0 5

6

3

5

1,8

00

6

3 3

,000

5

8.

7 3

7 5

7

3 7

0 5

7

3

5

4,9

00

7

3 7

,000

5

9.

4 3

6 5

4

3 6

0 5

4

3

5

2,4

00

4

3 6

,000

5

Use

me

ntal

mat

h to

co

mp

lete

the

pat

tern

.

• 4

3 5

is a

bas

ic m

ath

fact

. 4

3 5

5 2

0

• U

se the

bas

ic fa

ct to

find

4 3

50.

Sinc

e 50

has

one

zer

o, a

dd o

ne z

ero

to the

pro

duct

. 4

3 50

5 2

00

• U

se the

bas

ic fa

ct to

find

4 3

500

.Si

nce

500

has

two

zero

s, a

dd tw

o ze

ros

to the

pro

duct

.

4 3

500

5 2

,000

• 9

3 6

is a

bas

ic m

ath

fact

. 9

3 6

5 5

4

• U

se the

bas

ic fa

ct to

find

9 3

60.

Sinc

e 60

has

one

zer

o, a

dd o

ne z

ero

to the

pro

duct

. 9

3 60

5 5

40

• U

se the

bas

ic fa

ct to

find

9 3

600

.Si

nce

600

has

two

zero

s, a

dd tw

o ze

ros

to the

pro

duct

.

9 3

600

5 5

,400

MX

EN

L08

AW

K4X

_RT

_CH

09_L

1.in

dd

A6/

13/0

7 1

:29:

38 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

58

Men

tal M

ath:

Est

imat

e Pr

oduc

tsY

ou c

an u

se r

ound

ing

and

com

patib

le n

umbe

rs to

estim

ate

prod

ucts

.

Ro

und

the

gre

ate

r fa

cto

r. T

hen

use

me

ntal

mat

h to

est

imat

e t

he p

rod

uct.

6 3

325

• R

ound

325

to

the

near

est hu

ndre

d.

325

roun

ds to

300

• U

se m

enta

l mat

h.

6 3

3 5

18

6 3

30

5 1

80

6 3

300

5 1

,800

So, 6

3 3

25 is

abo

ut 1

,800

.

Esti

mat

e t

he p

rod

uct.

Wri

te t

he m

eth

od

.

7 3

$7.

59

• U

se a

com

patib

le n

umbe

r th

at is

eas

y to

com

pute

men

tally

. 7

3 $

7.59

7 3

$8

• U

se m

enta

l mat

h.

7

3 8

5 5

6

7 3

80

5 5

60

7 3

800

5 5

,600

• A

dd a

dec

imal

and

dol

lar

sign

. 7

3 $

8.00

5 $

56.0

0

So, 7

3 $

7.59

is a

bout

$56

.00.

Ro

und

the

gre

ate

r fa

cto

r. T

hen

use

me

ntal

mat

h to

est

imat

e t

he p

rod

uct.

1. 6

3 3

162.

5 3

$2.

893.

4 3

$3.

20

Esti

mat

e t

he p

rod

uct.

Wri

te t

he m

eth

od

.

4. 3

3 5

085.

7 3

22

6. 8

3 3

,061

MX

EN

L08

AW

K4X

_RT

_CH

09_L

2.in

dd

A6/

13/0

7 1

:31:

00 P

M

RW57-RW58


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

59

Mod

el 2

-Dig

it by

1-Di

git

Mul

tiplic

atio

nY

ou c

an u

se a

rray

s to

mul

tiply

2-d

igit

num

bers

by

1-di

git nu

mbe

rs.

Use

gri

d p

ape

r to

mo

de

l the

pro

duc

t.R

eco

rd y

our

ans

we

r.

9 3

14

• D

raw

a 9

by

14 a

rray

.

• M

ake

two

smal

ler

arra

ys. U

se p

rodu

cts

you

know

.

• Fi

nd the

pro

duct

of t

he tw

o sm

alle

r ar

rays

. 9

3 1

0 5

90

9 3

4 5

36

• Fi

nd the

sum

of t

he p

rodu

cts.

9

0 1

36

5 1

26

So

, 9 3

14

5 1

26.

Use

gri

d p

ape

r to

mo

de

l the

pro

duc

t.R

eco

rd y

our

ans

we

r.

1. 3

3 1

32.

6

3 1

6

3. 5

3 1

74.

4

3 1

4

14

9

410

9

9 3

109

3 4

MX

EN

L08

AW

K4X

_RT

_CH

09_L

3.in

dd

A6/

14/0

7 2

:05:

11 P

M

Nam

eN

ame

© H

arco

urt •

Gra

de 4

Ret

each

RW

60

Reco

rd 2

-Dig

it by

1-Di

git

Mul

tiplic

atio

nU

se p

artia

l pro

duct

s an

d pl

ace

valu

e w

ith r

egro

upin

g to

mul

tiply

.

Use

par

tial

pro

duc

ts.

Mul

tiply

. 81

3 5

.Es

timat

e th

e pr

oduc

t. 8

1 ro

unds

to

80; 8

0 3

5 5

400

.

Mul

tiply

the

1 o

ne, o

r 1,

by

5.

8

1

3

5

_

o

r

1

3

5

_

5

Mul

tiply

the

8 ten

s, o

r 80

, by

5.

81

3 5

_

o

r

80

3 5

_

400

Add

the

par

tial p

rodu

cts.

5

1 4

00 5

405

So, 8

1 3

5 5

405

. Sin

ce 4

05 is

clo

se to

the

estim

ate

of 4

00, i

t is

rea

sona

ble.

Use

pla

ce v

alue

and

re

gro

upin

g.

Mul

tiply

. 57

3 4

.Es

timat

e th

e pr

oduc

t. 5

7 ro

unds

to

60; 6

0 3

4 5

240

.

Mul

tiply

the

7 o

nes

by 4

.

57

3 4

_

8

4

3 7

one

s 5

28

ones

Reg

roup

the

2 ten

s.

Mul

tiply

the

5 ten

s by

4

3 5

tens

5 2

0 te

nsA

dd the

reg

roup

ed ten

s. 20

ten

s 1

2 ten

s 5

22

tens

So, 5

7 3

4 5

228

. Sin

ce 2

28 is

clo

se to

the

estim

ate

of 2

40, i

t is

rea

sona

ble.

Esti

mat

e. T

hen

reco

rd t

he p

rod

uct.

1.

63

3 3

_

2.

8

3

3 5

_

3.

3

6

3 8

_

4.

6

7

3 6

_

5.

59

3 3

_

6.

7

2

3 4

_

7.

1

8

3 4

_

8.

4

2

3 7

_

22

57

3 4

_

228

2

MX

EN

L08

AW

K4X

_RT

_CH

09_L

4.in

dd

A6/

13/0

7 1

:42:

31 P

M

RW59-RW60


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

61

Mul

tiply

3-Di

git a

nd 4

-Dig

it N

umbe

rs a

nd M

oney

Whe

n yo

u m

ultip

ly 3

-dig

it an

d 4-

digi

t nu

mbe

rs, y

ou m

ay n

eed

to r

egro

up.

Whe

n yo

u m

ultip

ly m

oney

, ign

ore

the

dolla

r si

gn a

nd the

dec

imal

,an

d ad

d th

em to

your

fina

l pro

duct

.

Esti

mat

e. T

hen

find

the

pro

duc

t.

Estim

ate

the

prod

uct.

$

13.9

4 ro

unds

to

$10;

$10

3 7

5 $

70

Mul

tiply

the

4 o

nes

by 7

.R

egro

up the

2 ten

s. 4

3 7

5 2

8

Mul

tiply

the

2 ten

s by

7.

Add

the

reg

roup

ed ten

s. 2

3 7

5 1

4R

egro

up 1

hun

dred

. 1

4 1

2 5

16

Mul

tiply

the

3 h

undr

eds

by 7

.A

dd the

reg

roup

ed h

undr

eds.

3

3 7

5 2

1R

egro

up 2

tho

usan

ds.

2

1 1

1 5

22

Mul

tiply

the

1 tho

usan

d by

7.

Add

the

reg

roup

ed tho

usan

ds.

1

3 7

5 7

7

1 2

5 9

Add

a d

olla

r si

gn a

nd a

dec

imal

poi

nt $

92.6

8to

you

r an

swer

.

So, $

13.2

4 3

7 5

$92

.68.

Sinc

e $9

2.68

is c

lose

to

the

estim

ate

of $

70, i

t is

rea

sona

ble.

Esti

mat

e. T

hen

find

the

pro

duc

t.

1.

$3,

184

2.

82

83.

$26.

374.

6

,916

$1

3.24

3

7

$1

3.24

3

7

8

$1

3.24

3

7

68

$1

3.24

3

7

2

68

$1

3.24

3

7

92

682 21

21

21

22

3

23

2

3

2

3

7

MX

EN

L08

AW

K4X

_RT

_CH

09_L

5.in

dd

A6/

6/07

10

:53:

00 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

62

Mul

tiply

with

Zer

osEs

tim

ate

. The

n fin

d t

he p

rod

uct.

Estim

ate

the

prod

uct.

$

10.2

0 ro

unds

to

$10;

$10

3 7

5 $

70

Mul

tply

the

0 o

ne, o

r 0,

by

7.

Plac

e th

e 0

in the

one

s pl

ace.

Mul

tiply

the

2 te

ns, o

r 20

, by

7.

Reg

roup

1 h

undr

ed.

Mul

tiply

the

0 h

undr

eds,

or

0, b

y 7.

Add

the

reg

roup

ed h

undr

ed to

the

prod

uct:

0 1

1 5

1.

Mul

tiply

the

1 tho

usan

d, o

r 1,

000,

by

7.

Add

the

dec

imal

and

the

dol

lar

sign

to

the

final

pro

duct

.

So, $

10.2

0 3

7 5

$71

.40.

Sin

ce $

71.4

0 is

clo

se to

the

estim

ate

of $

70, i

t is

rea

sona

ble.

Esti

mat

e. T

hen

find

the

pro

duc

t.

1.2.

3.4.

0

3 7

0

2

3 7

14

0

3 7

0

1

3 7

7

10

20

10

20

10

20

10

20 40 140

$71.

40

1 1 1

4,

046

0

$1

0.20

3

7

7140

3

7

3

7

3

7

3

7

3

2

7,

040

3

3

$1

3.09

3

7

5,00

83

2

MX

EN

L08

AW

K4X

_RT

_CH

09_L

6.in

dd

A6/

13/0

7 1

:48:

03 P

M

RW61-RW62


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

63

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Eval

uate

Rea

sona

blen

ess

The

Fra

nklin

s ha

ve a

bre

akfa

st b

udg

et.

Eac

h w

ee

k, t

hey

spe

nd $5

for

cere

al, $7

for

bac

on,

$4

for

bre

ad, $2

for

eg

gs,

an

d $3

for

juic

e. H

ow

muc

h w

ill t

hey

spe

nd o

n b

reak

fast

in

4 w

ee

ks?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. W

hich

ope

ratio

ns w

ill y

ou u

se to

solv

e th

is p

robl

em?

3. H

ow m

uch

will

the

y sp

end

on b

reak

fast

in 4

wee

ks?

Estim

ate.

The

n so

lve.

4. D

oes

your

ans

wer

see

m r

easo

nabl

e? W

hy o

r w

hy n

ot?

Each

mon

th M

ario

spe

nds

$5 o

n sp

orts

ca

rds,

$12

on

snac

ks, a

nd $

9 on

sch

ool

supp

lies.

If M

ario

spe

nds

the

sam

e am

ount

eac

h m

onth

, how

muc

h do

es

Mar

io s

pend

all

toge

ther

in 3

mon

ths?

Am

y’s

scho

ol is

mad

e up

of

524

stud

ents

. The

prin

cipa

l gav

e ea

ch s

tude

nt 3

pen

cils

. Am

y sa

ys

1,04

8 pe

ncils

wer

e ha

nded

out

. Is

Am

y’s

answ

er r

easo

nabl

e? W

hy o

r w

hy n

ot?

5.6.

So

lve

the

pro

ble

m.

MX

EN

L08

AW

K4X

_RT

_CH

09_L

7.in

dd

16/

13/0

7 1

:49:

40 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

64

Men

tal M

ath:

Mul

tiplic

atio

n Pa

tter

nsY

ou c

an u

se b

asic

mat

h fa

cts

and

patter

ns to

help

you

mul

tiply

w

hole

num

bers

by

mul

tiple

s of

10,

100

, and

1,0

00.

Use

pat

tern

s an

d m

ent

al m

ath

to f

ind

the

pro

duc

t.

12 3

6,0

00

• 12

3 6

is a

bas

ic m

ath

fact

.

12 3

6 5

72

• U

se the

bas

ic m

ath

fact

to

find

12 3

6,0

00.

• Si

nce

6,00

0 ha

s th

ree

zero

s, a

dd 1

2 3

6,000

5 7

2,00

0

thre

e ze

ros

to the

pro

duct

.

So, 1

2 3

6,0

00 5

72,

000.

11 3

1,0

00

• 11

3 1

is a

bas

ic m

ath

fact

. 1

1 3

1 5

11

• U

se the

bas

ic m

ath

fact

to

find

11 3

1,0

00.

• Si

nce

1,00

0 ha

s th

ree

zero

s, a

dd 1

1 3

1,000

5 1

1,00

0

thre

e ze

ros

to the

pro

duct

.

So, 1

1 3

1,0

00 5

11,

000.

Use

pat

tern

s an

d m

ent

al m

ath

to f

ind

the

pro

duc

t.

2. 3

5 3

100

3. 6

7 3

100

4. 4

8 3

1,0

00

6.

31 3

1,0

00

7.

49 3

10

8.

50 3

10

1

0. 1

0 3

300

1

1. 4

2 3

1,0

00

12.

90 3

900

14.

93 3

100

1

5. 6

0 3

100

1

6. 1

7 3

1,0

00

1. 2

2 3

10

5. 4

0 3

100

9. 8

0 3

20

13.

19 3

10

MX

EN

L08

AW

K4X

_RT

_CH

10_L

1.in

dd

16/

14/0

7 9

:26:

58 A

M

RW63-RW64


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

65

Mul

tiply

by

Tens

The

mul

tiple

s of

10

are

10, 2

0, 3

0, 4

0, a

nd s

o on

.

Whe

n on

e of

the

fact

ors

in a

mul

tiplic

atio

n pr

oble

m is

am

ultip

le o

f 10,

you

can

find

the

pro

duct

by

solv

ing

asi

mpl

er p

robl

em a

nd w

ritin

g a

zero

.

Cho

ose

a m

eth

od

. The

n fin

d t

he p

rod

uct.

78 3

60

• So

lve

the

sim

pler

pro

blem

78 3

6.

• Th

ere

is o

ne z

ero

in 6

0.W

rite

one

zero

in the

pro

duct

.

• Si

nce

78 3

6 5

468

, the

n 78

3 6

0 5

4,6

80

So, 7

8 3

60 5

4,6

80.

Cho

ose

a m

eth

od

. The

n fin

d t

he p

rod

uct.

1. 9

0 3

18

2. 4

6 3

50

3. 5

0 3

32

4. 2

2 3

40

5. 6

0 3

28

6. 3

6 3

30

7. 1

2 3

20

8. 4

7 3

40

9. 6

6 3

60

78 3

6 5

468

4,68

0

MX

EN

L08

AW

K4X

_RT

_CH

10_L

2.in

dd

A6/

14/0

7 9

:25:

41 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

66

Men

tal M

ath:

Est

imat

e Pr

oduc

tsY

ou c

an u

se c

ompa

tible

num

bers

and

rou

ndin

g to

est

imat

e pr

oduc

ts.

Use

ro

und

ing

and

me

ntal

mat

h to

est

imat

e t

he p

rod

uct.

$23 3

62

• R

ound

bot

h nu

mbe

rs to

the

near

est te

n.

$23

roun

ds to

$20.

62 r

ound

s to

60.

• R

ewrit

e th

e pr

oble

m u

sing

the

rou

nded

num

bers

. $2

0 3

60

• U

se m

enta

l mat

h.

$2 3

6 5

$12

$20 3

6 5

$12

0

$2

0 3

60 5

$1,

200

So, $

23 3

62

is a

bout

$1,

200.

Use

co

mp

atib

le n

umb

ers

and

me

ntal

mat

h to

est

imat

e p

rod

ucts

.

41 3

178

41

3 1

78

• U

se c

ompa

tible

num

bers

tha

t ar

e ea

sy to

co

mpu

te m

enta

lly.

40 3

200

4 3

2 5

8•

Use

men

tal m

ath.

4 3

20 5

80

4 3

200

5 8

00

40

3 2

00 5

8,0

00So

, 41 3

178

is a

bout

8,0

00.

Esti

mat

e t

he p

rod

uct.

Cho

ose

the

me

tho

d.

1.

78 3

21

2.

$ 46

3 5

9

3.

81 3

33

4.

67 3

102

5.

$42 3

88

6.

51 3

36

7.

73 3

73

8.

$44 3

99

9.

92 3

19

10.

26 3

37

11.

193 3

18

12.

58 3

59

MX

EN

L08

AW

K4X

_RT

_CH

10_L

3.in

dd

A6/

6/07

5:

20:1

7 PM

RW65-RW66


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

67

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Solv

e a

Sim

pler

Pro

blem

Whe

n M

agg

ie g

oe

s o

n b

ird

ing

wat

chin

g t

rip

s, s

he s

ee

s at

le

ast 45

bir

ds.

She

has

tak

en 12

bir

d w

atch

ing

tri

ps

a ye

ar o

ver

the

pas

t 16

ye

ars.

Ho

w m

any

bir

ds

has

Mag

gie

se

en?

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

Pla

n

2.

How

can

you

use

the

sol

ve a

sim

pler

pro

blem

str

ateg

y to

sol

ve thi

s pr

oble

m?

So

lve

3.

Solv

e a

sim

pler

pro

blem

. Sho

w y

our

wor

k be

low

.12

3 1

6 5

192

192

3 4

5 5

192

3 (

1

)

5

(19

2 3

)

1 (

192

3

) 1

(19

2 3

)

5

1

1

5

Ch

eck

4.

How

did

thi

s st

rate

gy h

elp

you

solv

e th

e pr

oble

m?

So

lve

a s

imp

ler

pro

ble

m.

5.

New

cast

le’s

pop

ulat

ion

is

385,

296

peop

le. D

unn’

s po

pula

tion

is 1

94,8

25 p

eopl

e. H

ow m

any

mor

e pe

ople

live

in N

ewca

stle

tha

n D

unn?

6.

Cin

dy s

hoot

s 50

free

thr

ows

a da

y,

7 da

ys a

wee

k. H

ow m

any

free

th

row

s do

es s

he s

hoot

in 5

wee

ks?

MX

EN

L08

AW

K4X

_RT

_CH

10_L

4.in

dd

A6/

6/07

5:

21:0

2 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

68

19

18

9

810

10

Mod

el 2

-Dig

it by

2-D

igit

Mul

tiplic

atio

nY

ou c

an u

se a

rray

s to

mul

tiply

2-d

igit

num

bers

by

2-di

git

num

bers

.

Use

the

mo

de

l and

par

tial

pro

duc

ts t

o s

olv

e.

19 3

18

The

arra

y is

19

units

wid

e an

d 18

uni

ts lo

ng.

Div

ide

the

arra

y in

to 4

sm

alle

r ar

rays

.U

se p

rodu

cts

you

know

.

Find

the

pro

duct

s of

sm

alle

r ar

rays

. 10

3 1

0 5

100

10

3 8

5 8

0

10 3

9 5

90

9

3 8

5 7

2

Find

the

sum

of t

he p

rodu

cts.

10

0 1

80

1 9

0 1

72

5 3

42

So, 1

9 3

18

5 3

42.

Use

the

mo

de

l and

par

tial

pro

duc

ts t

o s

olv

e.

1. 2

1 3

25

2. 1

6 3

14

3. 2

4 3

15

MX

EN

L08

AW

K4X

_RT

_CH

10_L

5.in

dd

A6/

14/0

7 9

:28:

58 A

M

RW67-RW68


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

69

Reco

rd 2

-Dig

it by

2-D

igit

Mul

tiplic

atio

nEs

tim

ate

. The

n fin

d t

he p

rod

uct.

Estim

ate

the

prod

uct.

8

9 ro

unds

to

90; 4

7 ro

unds

to

50 9

0 3

50

5 4

,500

Mul

tiply

the

9 o

nes

by the

7 o

nes.

Reg

roup

the

6 ten

s.7

ones

3 9

one

s 5

63

ones

Mul

tiply

the

8 ten

s by

the

7 o

nes.

Add

the

reg

roup

ed ten

s.7

ones

3 8

ten

s 5

56

tens

56 ten

s 1

6 ten

s 5

62

tens

Mul

tiply

the

9 o

nes

by the

4 ten

s, o

r 40

.R

egro

up the

3 te

ns.

9 on

es 3

40

5 3

60

Mul

tiply

the

8 ten

s by

the

4 ten

s, o

r 40

.A

dd the

reg

roup

ed ten

s.8

tens

3 4

ten

s 5

32

tens

32 ten

s 1

3 ten

s 5

35

tens

Add

the

par

tial p

rodu

cts.

So, 8

9 3

47

5 4

,183

. Sin

ce 4

,183

is c

lose

to

the

estim

ate

of 4

,500

, it is

rea

sona

ble.

Esti

mat

e. T

hen

cho

ose

eit

her

me

tho

d t

o f

ind

the

pro

duc

t.

1.2.

3.4.

893

47 3 89

3 47

623 89

3 4

762

3 893

47 89

3 4

7

60 623

3560 623

1 3

560

41836 6 3 3

89

3 4

7

_

76

3 3

1

_

24

3 3

5

_

14

3 2

8

_

64

3 5

6

_

MX

EN

L08

AW

K4X

_RT

_CH

10_L

6.in

dd

A6/

6/07

5:

21:5

0 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

70

Mul

tiply

2-D

igit

and

3-Di

git N

umbe

rs a

nd M

oney

Whe

n yo

u m

ultip

ly m

oney

, it is

the

sam

e as

mul

tiply

ing

who

le n

umbe

rs.

Just

add

the

dec

imal

( . )

and

dol

lar

sign

( $

) to

your

ans

wer

.

Esti

mat

e. T

hen

find

the

pro

duc

t.

Estim

ate

the

prod

uct.

$

3.99

rou

nds

to $

4; 3

0 st

ays

at 3

0.$4

3 3

0 5

$12

0

Mul

tiply

399

by

0 on

es.

3

99 3

0 o

nes

5 0

00

Mul

tiply

399

by

3 te

ns, o

r 30

.A

dd the

reg

oupe

d te

ns.

3

99 3

30

5 1

1,97

0

Add

the

par

tial p

rodu

cts.

Add

the

dec

imal

. The

re w

ill a

lway

s be

2

digi

ts to

the

right

of t

he d

ecim

al in

mon

ey.

$1

19.7

0A

dd a

dol

lar

sign

to

your

ans

wer

.

So, $

3.99

3 3

0 5

$11

9.70

.Si

nce

$119

.70

is c

lose

to

the

estim

ate

of $

120,

it is

rea

sona

ble.

Esti

mat

e. T

hen

find

the

pro

duc

t.

1.2.

3.4.

5.6.

7.8.

$3

.99

3

30

$3

.99

3

30

$3

.99

3

30

$3

.99

3

30

000

000

1197

0

000

1 1

1970

1197

0

22

12

6

3

21

_

$2

.45

3

16

__

24

1

3

33

_

40

6

3

24_

$4

.19

3

42

__

$6

.20

3

44

__

38

6

3

37

_

18

7

3

29_

MX

EN

L08

AW

K4X

_RT

_CH

10_L

7.in

dd

A6/

14/0

7 9

:30:

59 A

M

RW69-RW70


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

71

Choo

se a

Met

hod

The

num

bers

you

mul

tiply

hel

p yo

u de

cide

whe

ther

to

use

men

tal m

ath,

pap

er a

nd p

enci

l, or

a c

alcu

lato

r.

Thes

e ar

e ea

sy n

umbe

rs w

ith w

hich

to

wor

k. U

se m

enta

l mat

h to

sol

ve.

Estim

ate:

800

3 3

0 5

24,

000

Thin

k: 8

3 3

5 2

4

8

0 3

3 5

240

8

00 3

3 5

2,4

00

8

00 3

30 5

24,

000

So, 8

00 3

30 5

24,

000.

Sinc

e 24

,000

is c

lose

to

the

estim

ate

of

24,0

00, i

t is

rea

sona

ble.

Thes

e nu

mbe

rs w

ould

be

hard

er to

mul

tiply

usi

ng m

enta

l mat

h. U

se p

aper

an

d pe

ncil

to s

olve

.

Estim

ate:

300

3 2

0 5

6,0

00

Thin

k:

So, 3

25 3

24 5

7,8

00.

Sinc

e 7,

800

is c

lose

to

the

estim

ate

of

6,00

0, it

is r

easo

nabl

e.

Esti

mat

e. T

hen

find

the

pro

duc

t. W

rite

the

me

tho

d y

ou

use

d.

1.

2.

3.

4.

5.

6.

7.

8.

1300

6500

7800

49

3 6

2

_

50

3 2

0

_

60

3 4

0

_

45

3 2

5

_

93

3 7

9

_

$2

4

3 6

4

_

36

1

3 8

3

_

799

3

45

_

32

5

3 2

4

_

32

5

3 2

4

_

800

3 3

0

_

MX

EN

L08

AW

K4X

_RT

_CH

10_L

8.in

dd

A6/

14/0

7 9

:34:

55 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

72

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Mul

tiste

p Pr

oble

ms

The

Sp

ace

Ne

ed

le, b

uilt

in 196

2 fo

r th

e S

eat

tle

Wo

rld

’s F

air,

is

605

fee

t ta

ll. T

he P

etr

ona

s To

we

rs, a

n o

ffic

e b

uild

ing

in

Mal

aysi

a, h

as 88

sto

rie

s. E

ach

sto

ry is

ab

out

14

fee

t hi

gh.

A

bo

ut h

ow

muc

h ta

ller

is t

he P

etr

ona

s To

we

rs t

han

the

Sp

ace

Ne

ed

le?

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

Can

you

com

pare

the

hei

ghts

of t

he tw

o st

ruct

ures

usi

ng ju

st o

ne s

tep?

W

hy o

r w

hy n

ot?

3.

Whi

ch o

pera

tion,

or

oper

atio

ns, d

o yo

u ne

ed to

use

to s

olve

the

pro

blem

?

4.

Abo

ut h

ow m

uch

talle

r is

the

Pet

rona

s To

wer

tha

n th

e Sp

ace

Nee

dle?

5.

How

can

you

che

ck y

our

answ

er?

So

lve

the

pro

ble

m.

6. G

ina

owns

a m

usic

sto

re. O

n Sa

turd

ay,

her

stor

e so

ld 5

6 C

Ds

for

$15.

00 e

ach.

O

n Su

nday

her

sto

re s

old

32 C

Ds

for

$8.5

0 ea

ch. H

ow m

uch

mon

ey d

id

Gin

a’s

stor

e m

ake

in a

ll on

Sat

urda

y an

d Su

nday

?

7.

Dan

and

Pet

er b

oth

enjo

y hi

king

th

roug

h th

e w

oods

. Dan

hik

ed

6 m

iles

a da

y fo

r 5

days

. Pet

er

hike

d 9

mile

s a

day

for

4 da

ys.

How

man

y m

ore

mile

s di

d Pe

ter

hike

tha

n D

an?

MX

EN

L08

AW

K4X

_RT

_CH

10_L

9.in

dd

A6/

7/07

9:

21:5

1 A

M

RW71-RW72


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

73

Divi

de w

ith R

emai

nder

sW

hen

a nu

mbe

r ca

nnot

be

divi

ded

even

ly, t

here

is a

n am

ount

left

over

. Thi

s le

ftove

r am

ount

is c

alle

d th

e re

mai

nde

r.

Use

co

unte

rs t

o f

ind

the

quo

tie

nt a

nd r

em

aind

er.

24 4

5

• Y

ou a

re d

ivid

ing

the

divi

dend

, 24,

so

use

24 c

ount

ers.

• Si

nce

you

are

divi

ding

24

by 5

, dra

w 5

circ

les.

Then

div

ide

the

24 c

ount

ers

into

5 e

qual

-siz

ed g

roup

s.

• Th

ere

are

4 co

unte

rs in

eac

h ci

rcle

, so

the

quot

ient

is 4

.Th

ere

are

4 co

unte

rs le

ft ov

er, s

o th

e re

mai

nder

is 4

.

So, 2

4 4

5 5

4 r

4.

Use

co

unte

rs t

o f

ind

the

quo

tie

nt a

nd r

em

aind

er.

42 4

5

• Y

ou a

re d

ivid

ing

the

divi

dend

, 42,

so

use

42 c

ount

ers.

• Si

nce

you

are

divi

ding

42

by 5

, dra

w 5

circ

les.

Div

ide

42 c

ount

ers

into

5 e

qual

-siz

ed g

roup

s.

• Th

ere

are

8 co

unte

rs in

eac

h ci

rcle

, so

the

quot

ient

is 8

.Th

ere

are

2 co

unte

rs le

ft ov

er, s

o th

e re

mai

nder

is 2

.

So, 4

2 4

5 5

8 r

2.

Use

co

unte

rs t

o f

ind

the

quo

tie

nt a

nd r

em

aind

er.

3.

29

4 9

2. 1

4 4

3

4. 3

1 4

7

1. 1

9 4

6

5. 6

5 4

86.

44

4 5

MX

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each

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Mod

el 2

-Dig

it by

1-Di

git D

ivisi

onY

ou c

an u

se b

ase–

ten

bloc

ks to

mod

el d

ivis

ion

of a

2-di

git

num

ber,

such

as

26, b

y a

1-di

git nu

mbe

r, su

ch a

s 3.

Use

bas

e–

ten

blo

cks

to f

ind

the

quo

tie

nt a

nd r

em

aind

er.

7 q w

84

•

Sho

w 8

4 as

8 ten

s an

d 4

ones

. Th

en d

raw

7 c

ircle

s, s

ince

you

are

div

idin

g 84

by

7.

• P

lace

an

equa

l num

ber

of ten

s in

eac

h ci

rcle

. If

ther

e ar

e an

y te

ns le

ft ov

er, r

egro

up the

m a

s on

es.

Now

pla

ce a

n eq

ual n

umbe

r of

one

s in

eac

h gr

oup.

• C

ount

the

num

ber

of ten

s an

d on

es in

eac

h ci

rcle

to

find

the

quot

ient

. Th

ere

is 1

ten

and

2 o

nes

in e

ach

circ

le.

10 1

2 5

12,

so

the

quot

ient

is 1

2.

• Th

ere

are

no le

ftove

r bl

ocks

, so

ther

e is

no

rem

aind

er.

• So

, 84

4 7

5 1

2.

Use

bas

e–

ten

blo

cks

to f

ind

the

quo

tie

nt a

nd r

em

aind

er.

1.

71 4

8

2. 3

5 4

83.

19

4 6

5. 5

q w

28

6. 7

q w

35

4. 6

q w

49

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RW73-RW74


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e

© H

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Gra

de 4

Ret

each

RW

75

Reco

rd 2

-Dig

it by

1-Di

git D

ivisi

onY

ou c

an u

se lo

ng d

ivis

ion

to d

ivid

e 2-

digi

t nu

mbe

rs b

y 1-

digi

t nu

mbe

rs. S

omet

imes

the

num

bers

will

div

ide

even

ly. O

ther

tim

esth

ere

will

be

a re

mai

nder

.

Cho

ose

a m

eth

od

. The

n d

ivid

e a

nd r

eco

rd.

Use

Lon

g D

ivis

ion.

Beg

in in

the

ten

s pl

ace,

di

vide

8 b

y 5.

8 4

5 5

1W

rite

1 in

the

ten

s pl

ace.

Mul

tiply

. 1 3

5 5

5.

Subt

ract

. 8 2

5 5

3.

Com

pare

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5.

Then

brin

g do

wn

the

5 on

es a

nddi

vide

35

by 5

.

35 4

5 5

7W

rite

7 in

the

one

s pl

ace.

Mul

tiply

. 7 3

5 5

35.

Subt

ract

. 35 2

35 5

0.

Com

pare

. 0 ,

5.

So, 8

5 4

5 5

17.

Cho

ose

a m

eth

od

. The

n d

ivid

e a

nd r

eco

rd.

5 q w

85

5 q w

85

2

5 31

2 5 3517

2 3

5 0

5 q w

85

1. 7

� �

82

2. 6

� �

74

3. 3

� �

57

4. 7

� �

95

5.

9 q w

89

6 q

w

36

6.

MXE

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M

Nam

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© H

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RW

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Prob

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Sol

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Wor

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Draw

a D

iagr

amA

ri r

uns

a tr

aini

ng s

cho

ol f

or

pe

t ac

tors

. Las

t ye

ar h

e t

rain

ed

3

tim

es

as m

any

do

gs

as c

ats.

If t

he t

ota

l num

be

r o

f d

og

s an

d c

ats

he t

rain

ed

last

ye

ar is

84,

ho

w m

any

cats

did

he

tr

ain?

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2. H

ow c

an d

raw

ing

a di

agra

m h

elp

you

solv

e th

is p

robl

em?

So

lve

3. C

ompl

ete

and

labe

l the

dia

gram

bel

ow to

solv

e th

e pr

oble

m.

dogs

21

4. H

ow m

any

cats

did

Ari

trai

n?

Ch

eck

5. D

oes

the

answ

er m

ake

sens

e fo

r th

e pr

oble

m?

Expl

ain.

Dra

w a

dia

gra

m t

o s

olv

e.

6. A

cra

te h

as 5

tim

es a

s m

any

appl

es a

s or

ange

s. If

the

re a

re 1

20 p

iece

s of

frui

t in

the

cra

te, a

nd the

re a

re o

nly

appl

es

and

oran

ges

in the

cra

te, t

hen

how

m

any

oran

ges

are

ther

e?

7. J

ulio

has

6 tim

es a

s m

any

gree

n m

arbl

es a

s re

d m

arbl

es. I

f he

has

210

mar

bles

alto

geth

er, h

ow m

any

mar

bles

are

gre

en?

Tota

l pet

sis

84

pets

.

MX

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L08

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K4X

_RT

_CH

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RW75-RW76


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e

© H

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Gra

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each

RW

77

Men

tal M

ath:

Div

ision

Pat

tern

sU

se m

ent

al m

ath

to c

om

ple

te t

he p

atte

rn.

64 4

8 5

640 4

8 5

6,40

0 4

8 5

• So

lve

for

the

basi

c m

ath

fact

.

64

4 8

5 8

• In

640

4 8

, the

re is

one

zer

o ad

ded

to

the

divi

dend

. So

ther

e w

ill b

e on

e ze

ro a

dded

to

the

quot

ient

.

Use

me

ntal

mat

h to

co

mp

lete

the

pat

tern

.

90 4

9 5

900 4

9 5

9,00

0 4

9 5

• So

lve

for

the

basi

c m

ath

fact

.

90

4 9

5 1

0•

In

900

4 9

, the

re is

one

zer

o ad

ded

to th

e di

vide

nd. S

o th

ere

will

be

one

zero

add

ed to

th

e qu

otie

nt.

• In

9,0

00 4

9, t

here

are

2 z

eros

add

ed to

th

e di

vide

nd, s

o th

ere

will

be

2 ze

ros

ad

ded

to the

quo

tient

.So

, 90 4

9 5

10;

900

4 9

5 1

00; a

nd 9

,000

4 9

5 1

,000

.

Use

me

ntal

mat

h to

co

mp

lete

the

pat

tern

.

4.

18 4

3 5

1

80 4

3 5

1

,800

4 3

5

5.

36 4

4 5

3

60 4

4 5

3

,600

4 4

5

6.

30 4

3 5

3

00 4

3 5

3

,000

4 3

5

640 4

8 5

80

6,40

0 4

8 5

800

• In

6,4

00 4

8, t

here

are

2 z

eros

add

ed to

the

divi

dend

, so

ther

e w

ill b

e 2

zero

sad

ded

to the

quo

tient

.

So, 6

4 4

8 5

8; 6

40 4

8 5

80;

and

6,4

00 4

8 5

800

.

900 4

9 5

100

9,00

0 4

9 5

1,0

00

1.

25 4

5 5

2

50 4

5 5

2

,500

4 5

5

2.

63 4

9 5

6

30 4

9 5

6

,300

4 9

5

3.

56 4

7 5

5

60 4

7 5

5

,600

4 7

5

MX

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L08

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K4X

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5.in

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A6/

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7 1

0:35

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AM

Nam

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Gra

de 4

Ret

each

RW

78

Men

tal M

ath:

Est

imat

e Q

uotie

nts

You

can

use

com

patib

le n

umbe

rs a

nd r

ound

ing

to e

stim

ate

quot

ient

s.

Esti

mat

e t

he q

uoti

ent

.14

7 4

3

Use

com

patib

le n

umbe

rs to

estim

ate.

• Lo

ok a

t th

e nu

mbe

rs in

the

pro

blem

. 1

5 4

3Th

ink

of s

ome

basi

c m

ath

fact

s yo

u kn

ow 1

50 4

3th

at a

re s

imila

r to

the

num

bers

in the

pro

blem

. 1

50 i

s cl

ose

to 1

47

• U

se m

enta

l mat

h to

div

ide.

1

5 4

3 5

5

15

0 4

3 5

50

So, a

goo

d es

timat

e fo

r 14

7 4

3 is

50.

Esti

mat

e t

he q

uoti

ent

.

4,52

2 4

9

Use

rou

ndin

g to

est

imat

e.

• R

ound

the

div

iden

d to

the

nea

rest

tho

usan

d. 4

,522

rou

nds

to 5

,000

Rou

nd the

div

isor

to

the

near

est te

n.

9

roun

ds to

10

• R

ewrit

e th

e pr

oble

m u

sing

the

rou

nded

num

bers

.

5,00

0 4

10

• U

se m

enta

l mat

h to

div

ide.

5

4 1

5 5

5

,000

4 1

0 5

500

Esti

mat

e t

he q

uoti

ent

.

1.

409 4

4

2.

212

4 5

3.

7,8

39 4

2

4.

311 4

6

5.

2,10

3 4

4

6.

731

4 7

7.

239

4 4

8.

403 4

8

9.

119 4

3

10.

5,7

88 4

6

11.

787

4 4

12.

273

4 4

So, a

goo

d es

timat

e fo

r 4,

522 4

9 is

500

.

MX

EN

L08

AW

K4X

_RT

_CH

11_L

6.in

dd

A6/

14/0

7 1

0:37

:29

AM

RW77-RW78


Nam

e

RW

79©

Har

cour

t • G

rade

4R

etea

ch

Plac

e th

e Fi

rst D

igit

Whe

n yo

u us

e lo

ng d

ivis

ion,

it is

impo

rtan

t to

pla

ce the

firs

t di

git of

the

quot

ient

in the

cor

rect

pla

ce. Y

ou c

an u

se e

stim

atio

n an

d pl

ace

valu

e to

hel

pyo

u fin

d w

here

to

plac

e th

e fir

st d

igit.

Tell

whe

re t

o p

lace

the

fir

st d

igit

. The

n d

ivid

e.

• U

se c

ompa

tible

num

bers

to

estim

ate.

1

2 4

6 5

212

6 4

6 is

abo

ut the

sam

e as

120

4 6

. 12

0 4

6 5

20

• 12

0 4

6 5

20,

so

the

first

dig

it is

in the

ten

s pl

ace.

• N

ow p

lace

the

firs

t di

git in

the

qu

otie

nt in

the

ten

s pl

ace

and

divi

de.

So, t

he fi

rst di

git is

the

ten

s pl

ace,

and

126

4 6

5 2

1.

6 q w

12

6

2 1

2 062

6 021

8 q w

88

9

2 8 082

809

2 8 1

111

r1

Tell

whe

re t

o p

lace

the

fir

st d

igit

. The

n d

ivid

e.

889 4

8

1. 8

q w

286

2.

5 q w

74

3

3. 5

36 4

44.

647

4 9

• Se

e if

the

num

ber

in the

hun

dred

s pl

ace

of 8

89 is

larg

er tha

n 8.

• If

the

num

ber

in the

hun

dred

s pl

ace

is s

mal

ler

than

the

div

isor

,th

en y

ou b

egin

at th

e ne

xt p

lace

to

the

right

: the

ten

s pl

ace.

• If

the

num

ber

in the

hun

dred

s pl

ace

is la

rger

tha

n or

equ

al to

the

divi

sor,

then

you

can

pla

ce the

firs

t di

git in

the

hun

dred

s pl

ace.

• H

ere,

8 is

equ

al to

8, s

o th

e fir

st d

igit

will

be

in the

hun

dred

s pl

ace.

Now

div

ide.

• So

, the

firs

t di

git is

in the

hun

dred

s pl

ace,

and

889

4 8

5 1

11 r

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Tell

whe

re t

o p

lace

the

fir

st d

igit

. The

n d

ivid

e.

6 q w

12

6

MX

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K4X

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7.in

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Prob

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Wor

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p Sk

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Inte

rpre

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Rem

aind

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lve.

Writ

e a,

b, o

r c,

to

expl

ain

how

to

inte

rpre

t th

e re

mai

nder

.

Gui

de

s le

ad g

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s o

f 9

pe

op

le o

n b

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g t

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s in

the

par

k.Th

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96

pe

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le w

ho d

eci

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d t

o g

o o

n th

e t

our

s.H

ow

man

y p

eo

ple

will

be

on

a to

ur t

hat

is n

ot

full?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. D

ivid

e 96

by

9. H

ow m

any

grou

ps o

f 9 p

eopl

e w

ill the

gui

des

lead

on

biki

ng tou

rs?

3. W

hat is

the

rem

aind

er?

How

wou

ld y

ou in

terp

ret th

e re

mai

nder

?

4. H

ow m

any

peop

le a

re o

n a

tour

tha

t is

not

full?

So

lve

. Wri

te a

, b, o

r c

or

to e

xpla

in h

ow

to

inte

rpre

t th

e r

em

aind

er.

5. Ea

ch v

an h

olds

9 p

eopl

e. T

here

are

82 p

eopl

e tr

avel

ing.

How

man

y va

ns w

ill b

e co

mpl

etel

y fu

ll?

6. Ea

ch c

ar in

an

amus

emen

t pa

rkrid

e ca

n ho

ld 6

peo

ple.

The

re a

re

34 p

eopl

e w

aitin

g in

line

. How

man

y pe

ople

will

be

on a

car

tha

t is

not

full?

a. In

crea

se the

quo

tient

by

1.

b. Q

uotie

nt s

tays

the

sam

e. D

rop

the

rem

aind

er.

c. U

se the

rem

aind

er a

s th

e an

swer

.

MX

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L08

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K4X

_RT

_CH

12_L

1.in

dd

16/

13/0

7 1

2:11

:44

PM

RW79-RW80


Divi

de 3-

Digi

t Num

bers

and

Mon

eyY

ou c

an u

se m

ultip

licat

ion

to c

heck

div

isio

n pr

oble

ms.

Div

ide

and

che

ck.

924 4

8 o

r

• Es

timat

e to

see

whe

re to

plac

e th

e fir

st d

igit.

9

24 is

abo

ut 9

00Si

nce

ther

e is

1 h

undr

ed, p

lace

the

firs

t di

git in

the

8

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9hu

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lace

. 9

00 4

9 5

100

• D

ivid

e. P

lace

the

1 in

the

hun

dred

s pl

ace.

• M

ultip

ly. P

lace

the

8 b

elow

the

9 h

undr

eds.

• Su

btra

ct. B

ring

dow

n th

e 2

tens

.

Rep

eat th

e st

eps.

• D

ivid

e. P

lace

the

1 in

the

ten

s pl

ace.

8

� �

924

o

r 8

� �

12•

Mul

tiply

. Pla

ce the

8 b

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the

2 ten

s.

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btra

ct. B

ring

dow

n th

e 4

ones

.

Rep

eat th

e st

eps.

• D

ivid

e. P

lace

the

5 in

the

one

s pl

ace.

8

� �

924

o

r 8

� �

44•

Mul

tiply

. Pla

ce the

40

belo

w the

44

ones

.

• Su

btra

ct. T

he r

emai

nder

is 4

.

So, 9

24 4

8 5

115

r4.

Che

ck.

• M

ultip

ly the

quo

tient

by

the

divi

sor.

115 3

8 5

920

• A

dd the

pro

duct

and

the

rem

aind

er.

920 1

4 5

924

• Si

nce

the

sum

is the

div

iden

d, y

our

answ

er is

cor

rect

.

Div

ide

and

che

ck.

1. 6 �

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81

MXE

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K4X_

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13/0

7 1

2:17

:38

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

82

Zero

s in

Divi

sion

Div

ide

and

che

ck.

Estim

ate

to s

ee w

here

to

plac

e th

e fir

st d

igit.

• S

ince

the

re is

1 h

undr

ed, p

lace

the

first

dig

it in

the

hun

dred

s pl

ace.

• D

ivid

e. P

lace

the

1 in

the

hun

dred

s pl

ace.

1

1

• M

ultip

ly. P

lace

the

9 b

elow

the

9 h

undr

eds.

• Su

btra

ct. B

ring

dow

n th

e 2

tens

.

or

Rep

eat th

e st

eps.

• D

ivid

e. P

lace

the

0 in

the

ten

s pl

ace.

• M

ultip

ly. P

lace

the

0 b

elow

the

2 ten

s. or

• Su

btra

ct. B

ring

dow

n th

e 7

ones

.

Rep

eat th

e st

eps.

• D

ivid

e. P

lace

the

3 in

the

one

s pl

ace.

• M

ultip

ly. P

lace

the

27

belo

w the

27

ones

. o

r

• Su

btra

ct. T

here

is n

o re

mai

nder

.

So, 9

27 4

9 5

103

.

Che

ck.

• M

ultip

ly the

quo

tient

by

the

divi

sor.

103 3

9 5

927

• A

dd the

pro

duct

and

the

rem

aind

er.

927 1

0 5

927

• Si

nce

the

sum

is the

div

iden

d, y

our

answ

er is

cor

rect

.

Div

ide

and

che

ck.

1.

2.3.

29 02

29 0

29 02 20 27

22

7 0

10 103

29 02 20 27

9 � �

92

7

9 � �

92

7

9 � �

92

7

9 � �

9

20 20

9 � �

2

5 � �

52

4

7 � �

91

0

4 � �

81

5

22

7 0

9 � �

27

3

9 � �

92

7

927

is a

bout

900

9 is

abo

ut 9

900 4

9 5

100

MXE

NL08

AW

K4X_

RT_C

H12_

L3.in

dd

16/

6/07

12

:04:

27 P

M

RW81-RW82


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

83

26.0

8 4

8 5

3.2

6

Choo

se a

Met

hod

You

can

use

pap

er a

nd p

enci

l, a

calc

ulat

or, o

r m

enta

l mat

h to

div

ide.

Div

ide

. Wri

te t

he m

eth

od

use

d.

2,19

8 4

3 o

r

• U

se p

aper

and

pen

cil.

So, 2

,198

4 3

5 7

32 r

2.

o

r $2

6.08

4 8

• U

se a

cal

cula

tor

to s

olve

.

So, $

26.0

8 4

8 5

$3.

26.

o

r 4,

270

� 7

• U

se m

enta

l mat

h fo

r nu

mbe

rs tha

t ar

e ea

sy to

wor

k w

ith.

Her

e, the

div

iden

d co

ntai

ns m

ultip

les

of 7

, the

div

isor

. 4

,270

4 7

So, u

se m

enta

l mat

h to

sol

ve.

4

,200

4 7

5 6

00 7

0 4

7 5

10

• A

dd the

quo

tient

s. 6

00 1

10

5 6

10

So, 4

,270

4 7

5 6

10, o

r .

Div

ide

. Wri

te t

he m

eth

od

use

d.

1.2.

3.4.

221 09 2

9 08 26 2

732

r2

610

3 � �

2,

198

3 � �

21

98

8 � �

3,

729

5 �

�

3,07

5

8 � �

4,

888

4 �

�

1,47

2

4 � �

4,

270

7 � �

4,

270

8 � �

$2

6.08

MXE

NL08

AW

K4X_

RT_C

H12_

L4.in

dd

16/

13/0

7 1

2:19

:56

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

84

Estim

ate

Quo

tient

sY

ou c

an u

se c

ompa

tible

num

bers

or

basi

c m

ath

fact

s to

est

imat

e qu

otie

nts.

Wri

te t

he n

umb

ers

yo

u w

oul

d u

se t

o e

stim

ate

eac

h q

uoti

ent

. The

n e

stim

ate

.

Esti

mat

e.

23 � �

18

8 o

r 18

8 4

23

• Fi

nd c

ompa

tible

num

bers

.18

8 4

23

180

4 2

0

• U

se m

enta

l mat

h.18

4 2

5 9

180

4 2

0 5

9

• So

, a r

easo

nabl

e es

timat

e fo

r18

8 4

23

is 9

.

Esti

mat

e.

3,

600

4 3

7

• Fi

nd c

ompa

tiabl

e nu

mbe

rs.

3,

600

4 3

7

3,60

0 4

36

• U

se a

bas

ic fa

ct a

nd a

pat

tern

of

mul

tiple

s of

10.

36

4 3

6 5

1

360

4 36

5 1

0

3,60

0 4

36

5 1

00

• So

, a r

easo

nabl

e es

timat

e fo

r 3,

600

4 3

7 is

100

.

Wri

te t

he n

umb

ers

yo

u w

oul

d u

se t

o e

stim

ate

the

quo

tie

nt.

The

n e

stim

ate

.

1. 3

2 � �

5,

893

2.

703

4 6

83.

38 �

�

2,12

1

4. 4

,146

4 5

95.

69 �

�

4,82

5

6. 6

55 4

80

7. 7

3 � �

2,

994

8.

8,2

45 4

92

9. 5

1 � �

5,

024

MXE

NL08

AW

K4X_

RT_C

H13_

L1.in

dd

16/

6/07

12

:13:

59 P

M

RW83-RW84


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

85

Mod

el D

ivisi

on b

y 2-

Digi

t Div

isors

You

can

mod

el d

ivis

ion

by u

sing

bas

e-te

n bl

ocks

.

Use

bas

e-t

en

blo

cks

to d

ivid

e.

99 4

31

Firs

t, m

odel

the

div

iden

d, 9

9, a

s 9

tens

and

9 o

nes.

The

divi

sor

is 3

1, s

o m

ake

as m

any

grou

ps o

f 31

as y

ou c

an.

Ther

e ar

e 3

grou

ps o

f 31.

So,

3 is

the

quo

tient

.

Ther

e ar

e 6

ones

left

over

, so

6 is

the

rem

aind

er.

So, 9

9 4

31

5 3

r6

Use

bas

e-t

en

blo

cks

to d

ivid

e.

1.

65 4

21

2.

96 4

47

3.

117

4 1

94.

17

5 4

58

MXE

NL08

AW

K4X_

RT_C

H13_

L2.in

dd

16/

6/07

12

:14:

16 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

86

Reco

rd D

ivisi

onY

ou c

an u

se lo

ng d

ivis

ion

or p

artia

l quo

tient

s to

div

ide.

Div

ide

and

re

cord

. •

Estim

ate

to p

lace

the

firs

t di

git.

80

0 4

40

5 2

0 an

d 84

0 4

40

5 1

2

In b

oth

estim

ates

, the

firs

t di

git is

in the

ten

s pl

ace,

so y

ou w

ill p

lace

the

firs

t di

git in

the

ten

s pl

ace.

• R

egro

up 8

hun

dred

s 4

tens

as

84 ten

s.

Then

div

ide

the

84 ten

s by

35.

• D

ivid

e. 8

4 4

35

• M

ultip

ly. 3

5 3

2

• Su

btra

ct. 8

4 2

70

• C

ompa

re. 1

4 ,

35

• B

ring

dow

n th

e on

es. D

ivid

e 14

1 by

35.

• D

ivid

e. 1

41 4

35

• M

ultip

ly. 3

5 3

4

• Su

btra

ct. 1

41 2

140

• C

ompa

re. 1

, 3

5

Ther

e is

1 le

ft ov

er s

o th

e re

mai

nder

is 1

.

So, 8

41 4

35

5 2

4 r1

Div

ide

and

re

cord

.

1.

36 � �

77

5

2.

27 � �

36

1

3.

689

4 1

64.

97

8 4

61

2

270 14

�14

0 124

35 � �

84

1

35 � �

84

1

270 14

1

35 � �

84

1

MXE

NL08

AW

K4X_

RT_C

H13_

L3.in

dd

16/

13/0

7 1

2:06

:47

PM

RW85-RW86


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

87

Adju

stin

g Q

uotie

nts

Whe

n an

est

imat

e is

too

hig

h or

too

low

, you

mus

t ad

just

the

quo

tient

.

Ad

just

the

est

imat

ed

dig

it in

the

quo

tie

nt, i

f ne

ed

ed

. Th

en

div

ide

.

3

45 � �

21

0

• Tr

y th

e es

timat

e, 3

.

• Tr

y 4.

• So

, 210

4 4

5 5

4 r

30.

Ad

just

the

est

imat

ed

dig

it in

the

quo

tie

nt, i

f ne

ed

ed

.Th

en

div

ide

.

1.

27 � �

13

8

2.

39 � �

17

1

3.

16 � �

13

4

4.

41 � �

22

2

5.

33 � �

25

6

6.

48 � �

30

6

3

213

5 75

45 � �

21

0

75 .

45

Sinc

e th

e re

mai

nder

isla

rger

tha

n th

e di

viso

r,th

e es

timat

e is

too

low

.

3 3

45 5

135

4 3

45 5

180

4

218

0 30

45 � �

21

0

53

1

66

6

MXE

NL08

AW

K4X_

RT_C

H13_

L4.in

dd

16/

6/07

12

:14:

50 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

88

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Too

Muc

h/To

o Li

ttle

Info

rmat

ion

A r

adio

sta

tio

n p

lays

49

min

ute

s o

f m

usic

eac

h ho

ur. T

he r

est

of

the

ho

ur, c

om

me

rcia

ls a

re p

laye

d. I

n 2

hour

s, t

he s

tati

on

pla

yed

32

song

s. H

ow

man

y m

inut

es

of

com

me

rcia

ls d

oe

s th

est

atio

n p

lay

in 3

ho

urs?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. W

hat in

form

atio

n is

giv

en in

thi

s pr

oble

m?

3. W

hat in

form

atio

n do

you

nee

d to

sol

ve thi

s pr

oble

m?

4. Is

the

re a

ny e

xtra

info

rmat

ion

give

n in

thi

s pr

oble

m?

If ye

s, w

hy is

the

info

rmat

ion

not ne

eded

?

5. H

ow m

any

min

utes

of c

omm

erci

als

does

the

sta

tion

play

in 3

hou

rs?

De

cid

e if

the

pro

ble

m h

as t

oo

muc

h o

r to

o li

ttle

info

rmat

ion.

The

n, s

olv

e if

po

ssib

le. I

f th

ere

is t

oo

litt

le in

form

atio

n, id

ent

ifyth

e m

issi

ng in

form

atio

n.

6. Pe

ter

earn

s $9

per

hou

r. H

e w

orke

d on

Tu

esda

y an

d Th

ursd

ay. H

ow

muc

h m

oney

did

Pet

er e

arn?

7. Th

ere

are

25 s

tude

nts

in the

cla

ss,

12 b

oys

and

13 g

irls.

If e

ach

stud

ent

has

6 pe

ncils

, how

man

y pe

ncils

are

in

the

clas

sroo

m?

MX

EN

L08

AW

K4X

_RT

_CH

13_L

5.in

dd

A6/

13/0

7 1

2:08

:48

PM

RW87-RW88


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

89

Fact

ors a

nd M

ultip

les

A f

acto

r is

a n

umbe

r m

ultip

lied

by a

noth

er n

umbe

r to

find

apr

oduc

t. Fo

r ex

ampl

e, 3

and

5 a

re fa

ctor

s of

15.

The

num

ber

15is

a m

ulti

ple

of 5

and

3, b

ecau

se 3

3 5

5 1

5.

Use

arr

ays

to f

ind

all

the

fac

tors

of 6.

Mak

e an

arr

ay w

ith 6

squ

ares

.

This

arr

ay c

onta

ins

2 ro

ws

of 3

squ

ares

each

. So,

2 a

nd 3

are

bot

h fa

ctor

s of

6.

Mak

e a

diffe

rent

arr

ay w

ith 6

squ

ares

.

An

arra

y w

ith 1

row

of 6

con

tain

s 6

squa

res.

So,

1 a

nd 6

are

bot

h fa

ctor

s of

6.

So, t

he fa

ctor

s of

6 a

re 1

, 2, 3

, and

6.

List

the

fir

st t

we

lve

mul

tip

les

of 8.

Mul

tiply

8 b

y th

e nu

mbe

rs 1

thr

ough

12.

List

eac

h pr

oduc

t.

1 3

8 5

82 3

8 5

16

3 3

8 5

24

4 3

8 5

32

5 3

8 5

40

6 3

8 5

48

7 3

8 5

56

8 3

8 5

64

9 3

8 5

72

10 3

8 5

80

11 3

8 5

88

12 3

8 5

96

So, t

he fi

rst tw

elve

mul

tiple

s of

8 a

re:

8, 1

6, 2

4, 3

2, 4

0, 4

8,

56, 6

4, 7

2, 8

0, 8

8, 9

6.

Use

arr

ays

to f

ind

all

the

fac

tors

of

eac

h p

rod

uct.

1. 1

02.

16

3. 2

1

List

the

fir

st t

we

lve

mul

tip

les

of

eac

h nu

mb

er.

4. 3

5. 5

6. 7

7. 1

0

2

3

6

1

MX

EN

L08

AW

K4X

_RT

_CH

14_L

1.in

dd

A6/

13/0

7 1

1:59

:30

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

90

Divi

sibili

ty R

ules

You

can

use

spe

cial

rul

es to

find

out if

a nu

mbe

r is

div

isib

le b

yan

othe

r nu

mbe

r.

Num

ber

Divi

sibi

lity

Rule

Exam

ple

2

The

last

dig

it m

ust b

e ev

en.

456.

The

last

dig

it is

an

even

num

ber

(0, 2

, 4, 6

, 8).

5

The

last

dig

it m

ust b

e 0

or 5

.45

. The

last

dig

it is

a 5

.

10

The

last

dig

it m

ust b

e 0.

90. T

he la

st d

igit

is a

0.

25

The

last

dig

its m

ust b

e 00

, 25,

50,

or

75.

3,25

0. T

he la

st tw

o di

gits

are

50.

Tell

whe

the

r 48

0 is

div

isib

le b

y 2,

5, 10,

or 25

.

• 48

0 is

div

isib

le b

y 2

beca

use

the

last

dig

it is

eve

n.

• 48

0 is

div

isib

le b

y bo

th 5

and

10

beca

use

the

last

dig

it is

a 0

.

• 48

0 is

not

div

isib

le b

y 25

bec

ause

the

last

dig

its a

re n

ot 0

0, 2

5, 5

0, o

r 75

.

So, 4

80 is

div

isib

le b

y 2,

5, a

nd 1

0.

Tell

whe

the

r th

e n

umb

er

is d

ivis

ible

by 2,

5, 10,

or 25

.

1.

675

2.

100

3.

320

4.

126

5.

250

6.

254

7.

30 8

. 83

5

9.

400

10.

975

11.

104

12.

56

13.

470

14.

250

15.

248

16.

575

MXE

NL08

AW

K4X_

RT_C

H14_

L2.in

dd

A6/

6/07

12

:23:

41 P

M

RW89-RW90


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

91

Prim

e an

d Co

mpo

site

Num

bers

A p

rim

e n

umb

er

has

only

tw

o fa

ctor

s: 1

and

itse

lf. T

he n

umbe

r 3

is p

rime

beca

use

it ha

s on

ly 1

and

3 a

s its

fact

ors.

A c

om

po

site

nu

mb

er

has

mor

e th

an tw

o fa

ctor

s. T

he n

umbe

r 15

isco

mpo

site

bec

ause

its

fact

ors

are

1, 3

, 5, a

nd 1

5.

Mak

e a

rray

s to

fin

d t

he f

acto

rs. W

rite

prim

e o

r co

mpo

site

fo

r e

ach

num

be

r.

11 Mak

e al

l the

arr

ays

you

can

with

11

squ

are

tiles

.

List

the

fact

ors.

The

fact

ors

of 1

1 ar

e 1

and

11.

Sinc

e 11

onl

y ha

s 2

fact

ors,

1 a

nd 1

1, it

is p

rime.

21 Mak

e al

l the

ar

rays

you

can

with

21

squa

retil

es.

List

the

fact

ors.

The

fact

ors

of 2

1 ar

e 1,

3, 7

, and

21.

Sinc

e 21

has

mor

e th

an 2

fact

ors,

1, 3

, 7, a

nd 2

1, it

is c

om

posi

te.

Mak

e a

rray

s to

fin

d t

he f

acto

rs. W

rite

prim

e o

r co

mpo

site

fo

r e

ach

num

be

r.

1. 1

62.

19

1 3

11

11 3

1

7 3

33 3

721

3 1

1 ro

w o

f 11

tiles

11 r

ows

of 1

tile

eac

h

21 r

ows

of 1

tile

ea

ch

3 ro

ws

of

7 til

es7

row

s of

3

tiles

1 3

21

1 ro

w o

f 21

tile

s

MX

EN

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14_L

3.in

dd

A6/

13/0

7 1

2:01

:07

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

92

Num

ber P

atte

rns

To fi

nd n

umbe

r pa

tter

ns, f

ind

a ru

le. I

f a n

umbe

r pa

tter

n in

crea

ses,

try

add

ition

or

mul

tiplic

atio

n. If

a n

umbe

r pa

tter

n de

crea

ses,

try

sub

trac

tion

or d

ivis

ion.

Find

a r

ule

. The

n fin

d t

he n

ext

tw

o n

umb

ers

in y

our

pat

tern

.18

, 28,

38,

48,

j ,

j

Look

at th

e fir

st tw

o nu

mbe

rs: 1

8 an

d 28

. W

hat ru

le c

hang

es 1

8 to

28?

Sinc

e 28

is la

rger

tha

n 18

, try

add

ition

or

mul

tiplic

atio

n.

Try

“add

10”

bec

ause

18 1

10 5

28.

Test

“ad

d 10

”: 2

8 1

10 5

38.

38

1 1

0 5

48.

The

rule

wor

ks.

Find

the

nex

t tw

o nu

mbe

rs.

Thin

k: 4

8 1

10 5

58;

58 1

10 5

68.

So, t

he r

ule

is “

add

10,”

and

the

nex

t tw

o nu

mbe

rs in

the

pat

tern

are

58

and

68.

Find

a r

ule

. The

n fin

d t

he n

ext

tw

o n

umb

ers

in y

our

pat

tern

.

1.

25,

20,

15,

10,

j ,

j

2.

13,

19,

25,

31,

j ,

j

3.

30,

33,

32,

35,

34,

37,

j ,

j

4.

10,

21,

32,

43,

j ,

j

5.

2, 4

, 8, 1

6, j

, j

6. 3

, 5, 1

0, 1

2, 2

4, 2

6, j

,j

7.

100

, 80,

60,

40,

j ,

j

8. 7

, 12,

10,

15,

13,

18,

j ,

j

9.

800

, 400

, 200

, 100

, j ,

j

10.

3, 2

, 12,

11,

j ,

j

MX

EN

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_RT

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4.in

dd

A6/

13/0

7 1

2:02

:52

PM

RW91-RW92


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

93

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Find

a P

atte

rnTh

e t

able

be

low

sho

ws

the

num

be

r o

f w

ind

ow

s o

n e

ach

floo

r o

f a

skys

crap

er.

Ho

w m

any

win

do

ws

mig

ht b

e o

n th

e t

ent

h flo

or?

Flo

or1

23

4 5

6 7

8 9

Num

ber o

f Win

dow

s4

86

108

1210

1412

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

Pla

n

2.

How

wou

ld the

find

a p

atte

rn s

trat

egy

help

you

sol

ve the

pro

blem

?

So

lve

3.

Wha

t is

the

pat

tern

for

the

tabl

e? H

ow m

any

win

dow

s m

ight

be

on the

ten

th fl

oor?

Ch

eck

4.

Doe

s th

e an

swer

mak

e se

nse

for

the

prob

lem

? Ex

plai

n.

Find

a p

atte

rn t

o s

olv

e.

5. Pe

ter

star

ted

an e

xerc

ise

prog

ram

. On

the

first

day

he

did

5 pu

shup

s. O

n th

ese

cond

day

he

did

10 p

ushu

ps. O

n th

eth

ird d

ay h

e di

d 15

pus

hups

. If t

he

patter

n co

ntin

ues

how

man

y pu

shup

s w

ill P

eter

do

on the

six

th d

ay?

6. Ja

n ro

de h

er b

ike

for

10 m

inut

es the

first

day

, 12

min

utes

the

sec

ond

day,

and

14 m

inut

es the

thi

rd d

ay. I

f the

patter

n co

ntin

ues,

how

man

y m

inut

es w

ill J

an r

ide

her

bike

on

the

fifth

day

?

MX

EN

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_RT

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14_L

5.in

dd

A6/

13/0

7 1

2:03

:46

PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

94

Read

and

Writ

e Fr

actio

nsA

frac

tion

is a

num

ber

that

nam

es p

art of

a w

hole

or

part

of a

set

. You

use

frac

tions

eve

ry d

ay. Y

ou c

an e

at a

fr

actio

n of

a b

anan

a. Y

ou c

an r

ead

a fr

actio

n of

a b

ook.

Wri

te a

fra

ctio

n fo

r th

e s

had

ed

par

t.

Find

the

de

nom

inat

or,

or

bottom

num

ber.

The

deno

min

ator

is the

tot

al n

umbe

r of

eq

ual p

arts

in the

figu

re. T

he r

ecta

ngle

is

divi

ded

into

ten

equ

al p

arts

, so

10 is

the

de

nom

inat

or.

Wri

te a

fra

ctio

n fo

r th

e u

nsha

de

d p

art.

Find

the

de

nom

inat

or,

or

bottom

num

ber.

The

de

nom

inat

or

is 1

0.

Find

the

num

era

tor,

or

top

num

ber.

The

num

erat

or is

the

num

ber

of

shad

ed p

arts

. The

re a

re s

even

par

ts

shad

ed, s

o 7

is the

num

erat

or.

Find

the

num

era

tor,

or

top

num

ber.

The

num

era

tor

is the

num

ber

of

unsh

aded

par

ts. T

here

are

thr

ee

part

s un

shad

ed, s

o 3

is the

nu

mer

ator

.

Wri

te a

fra

ctio

n fo

r th

e s

had

ed

par

t. W

rite

a f

ract

ion

for

the

uns

had

ed

par

t.

1.

2.

3.

4.

5.

6.

num

erat

or

deno

min

ator

7 __

10

, or

sev

en ten

ths

of the

rec

tang

le is

sha

ded.

num

erat

or

deno

min

ator

3 __

10

, or

thr

ee ten

ths

of the

rec

tang

le is

uns

hade

d.

shad

ed

unsh

aded

shad

ed

unsh

aded

shad

ed

unsh

aded

shad

ed

unsh

aded

shad

ed

unsh

aded

shad

ed

unsh

aded

MX

EN

L08

AW

K4X

_RT

_CH

15_L

1.in

dd

16/

14/0

7 1

1:34

:36

AM

RW93-RW94


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

95

Mod

el E

quiv

alen

t Fra

ctio

nsEq

uiva

lent

fra

ctio

ns a

re tw

o or

mor

e fr

actio

ns tha

t na

me

the

sam

e am

ount

. The

re a

re s

ever

al w

ays

to m

odel

eq

uiva

lent

frac

tions

.

Wri

te t

wo

eq

uiva

lent

fra

ctio

ns.

• Li

ne u

p on

e 1 _ 4 b

ar w

ith the

bar

for

1 to

sho

w 1 _ 4

.

• Li

ne u

p 1 _ 8 b

ars

to s

how

the

sam

e am

ount

as

1 _ 4 .

• 2 _ 8 s

how

s th

e sa

me

amou

nt a

s 1 _ 4 .

• Li

ne u

p 1 __

12

bar

s to

sho

w the

sam

e am

ount

as

1 _ 4 .

•

3 __

12 s

how

s th

e sa

me

amou

nt a

s 1 _ 4 .

• So

, 2 _ 8 a

nd 3 __

12

are

equ

ival

ent to

1 _ 4 .

Wri

te t

wo

eq

uiva

lent

fra

ctio

ns.

• Th

e nu

mbe

r lin

es s

tart

at 0

and

end

at 1

.

• W

hen

the

num

ber

lines

are

line

d up

, the

fr

actio

n 2 _ 6 l

ines

up

with

the

frac

tions

1 _ 3 a

nd 4 __

12

.

• So

, 1 _ 3 a

nd 4 __

12

are

equ

ival

ent to

2 _ 6 .

Wri

te t

wo

eq

uiva

lent

fra

ctio

ns f

or

eac

h m

od

el.

1.

2

.

3.

4

.

1 60 6

2 63 6

4 65 6

6 6

1 30 3

2 33 3 12 12

0 121 12

2 123 12

4 125 12

6 127 12

8 129 12

10 1211 12

MX

EN

L08

AW

K4X

_RT

_CH

15_L

2.in

dd

A6/

14/0

7 1

1:37

:20

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

96

Com

pare

and

Ord

er F

ract

ions

You

can

com

pare

tw

o fr

actio

ns to

find

out w

hich

is la

rger

, ev

en w

hen

the

two

frac

tions

hav

e di

ffere

nt d

enom

inat

ors.

Y

ou c

an a

lso

com

pare

a g

roup

of f

ract

ions

and

put

the

m in

or

der

by s

ize.

Co

mp

are

3 __ 4 a

nd 1 __ 2 . W

rite

,, .

or

5.

Ord

er

the

fra

ctio

ns 4 __ 6 , 1

__ 3 , a

nd 9 __

_ 12

fro

m

leas

t to

gre

ate

st.

Co

mp

are

. Wri

te <

, >, o

r =

fo

r e

ach

.

Ord

er

the

fra

ctio

ns f

rom

leas

t to

gre

ate

st.

3 __ 6

4 __ 5

,,

,,

3 __ 5

4 __ 9

4 __

_ 12

5 __

_ 12

1 __ 2

3 __ 8

3 __ 4

1 __ 4

9 __

_ 10

,

,

6 __ 7

2 __ 4

4.5.

6.

1.2.

3.

Line

up

thre

e 1 _ 4 b

ars

with

the

bar

for

1 to

sho

w 3 _ 4

.

Line

up

one

1 _ 2 b

ar to

show

1 _ 2 .

Arr

ange

the

frac

tion

bars

in o

rder

from

lo

nges

t to

sho

rtes

t.

3 _ 4 is

a lo

nger

row

tha

n 1 _ 2 .

So, 3 _ 4

> 1 _ 2

.

Star

t w

ith the

frac

tion

bar

for

1.

Line

up

frac

tion

bars

for

4 _ 6 ,

1 _ 3 , a

nd 9 __

12

.

Arr

ange

the

frac

tion

bars

in o

rder

fr

om s

hort

est to

long

est.

So, t

he o

rder

from

leas

t to

gre

ates

t

is 1 _ 3

, 4 _ 6 ,

9 __

12 .

5 __ 8

1 __ 3

MX

EN

L08

AW

K4X

_RT

_CH

15_L

3.in

dd

A6/

13/0

7 1

1:10

:23

AM

RW95-RW96


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

97

Read

and

Writ

e M

ixed

Num

bers

A m

ixe

d n

umb

er

is a

num

ber

mad

e up

of a

who

le n

umbe

ran

d a

frac

tion.

Wri

te a

mix

ed

num

be

r fo

r th

e m

od

el a

t th

e r

ight

.

The

circ

le o

n th

e le

ft re

pres

ents

1 w

hole

figu

re s

hade

d.

The

circ

le o

n th

e rig

ht r

epre

sent

s 5 _ 6 s

hade

d.

So, 1

5 _ 6 fi

gure

s ar

e sh

aded

.

Re

nam

e 5

5 _ 6 a

s a

frac

tio

n.

Mod

el 5

5 _ 6 .

Ren

ame

each

1 w

hole

as

6 _ 6 .

The

tota

l num

ber

of s

ixth

s is

the

num

erat

or o

f the

frac

tion.

The

num

erat

or is

35.

So, 5

5 _ 6 r

enam

ed a

s a

frac

tion

is 35

__

6

.

Wri

te a

mix

ed

num

be

r fo

r e

ach

pic

ture

in s

tand

ard

fo

rm a

nd w

ord

fo

rm.

1.

2

.

3.

Re

nam

e t

he f

ract

ion

as a

mix

ed

num

be

r an

d e

ach

mix

ed

num

be

r as

a f

ract

ion.

Yo

u m

ay w

ish

to d

raw

a p

ictu

re.

4.

5

_

3 5

. 3

3

_

4

MX

EN

L08

AW

K4X

_RT

_CH

15_L

4.in

dd

A6/

20/0

7 1

1:33

:21

AM

1 92 9

3 94 9

5 96 9

7 98 9

2

24

68

1012

1416

171 18

1818

1818

1818

1818

1818

1818

1818

1818

35

79

1113

151

2

1 211

2

1 62 6

3 64 6

5 63

41 8

2 83 8

4 86 8

7 85 8

12

1 52 5

4 53 5

12

1 82 8

3 85 8

4 87 8

6 83

4

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

98

Com

pare

and

Ord

er M

ixed

Num

bers

You

can

use

mod

els

to c

ompa

re a

nd o

rder

mix

ed n

umbe

rs.

Co

mp

are

6 1 _ 6 a

nd 6

1 _ 2 . U

se ,

, ., o

r 5

.

• D

raw

a n

umbe

r lin

e di

vide

d in

to s

ixth

s.Lo

cate

6 1 _ 6 .

• D

raw

a n

umbe

r lin

e di

vide

d in

to h

alve

s.Lo

cate

6 1 _ 2 .

• S

ince

6 1 _ 2 is

to

the

right

of 6

1 _ 6 o

n th

e nu

mbe

r lin

e,

then

6 1 _ 6 <

6 1 _ 2 .

Ord

er

the

mix

ed

num

be

rs 1

7 _ 9 , 1

1 _ 2 , 1

12

__

18 f

rom

gre

ate

st t

o le

ast.

• D

raw

a n

umbe

r lin

e di

vide

din

to n

inth

s. L

abel

1 7 _ 9 .

• D

raw

a n

umbe

r lin

e di

vide

d in

to h

alve

s. L

abel

1 1 _ 2 .

• D

raw

a n

umbe

r lin

e di

vide

din

to e

ight

eent

hs. L

abel

1 12

__

18

.

• S

ince

1 7 _ 9 is

fart

hest

to

the

right

, it is

the

gre

ates

t nu

mbe

r.

• S

ince

1 1 _ 2 is

fart

hest

to

the

left,

it is

the

leas

t nu

mbe

r.

• S

o, the

ord

er fr

om g

reat

est to

leas

t is

1 7 _ 9 , 1

12

__

18 ,

1 1 _ 2 .

Co

mp

are

the

mix

ed

num

be

rs. U

se <

, >

, or

= f

or

eac

h .

1.

2.

Ord

er

the

mix

ed

num

be

rs f

rom

gre

ate

st t

o le

ast.

3. 2

7 __ 9 , 2

4 __ 5 , 3

1 __ 3

4. 4

1 __ 5 , 4

10

___

11 , 4

3 __ 4

5. 2

3 __ 5 , 3

3 __ 7 , 1

2 __ 8

6. 5

1 __ 5 , 6

2 __ 5 , 6

3 __ 8

1 3 __ 8

1 2 __ 5

3 3 __ 6

3 4 __ 8

MX

EN

L08

AW

K4X

_RT

_CH

15_L

5.in

dd

A6/

20/0

7 1

1:37

:01

AM

RW97-RW98


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

99

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Sequ

ence

Info

rmat

ion

Glo

ria

giv

es

her

youu

nge

st c

at 1

1 _ 2 c

ups

of

foo

d. S

he g

ive

s th

e o

lde

st c

at 7 _ 8

cup

of

foo

d a

nd h

er

seco

nd-o

lde

st c

at1

3 _ 4 c

ups

of

foo

d. D

raw

a n

umb

er

line

to

re

pre

sent

the

am

oun

ts

of

foo

d. W

hich

cat

s g

ets

the

mo

st f

oo

d?

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

2.

Wha

t ar

e th

e th

ree

amou

nts

of fo

od tha

t yo

u ar

e co

mpa

ring?

3.

Labe

l the

thr

ee n

umbe

r lin

es b

elow

for

each

am

ount

of f

ood.

4.

Ord

er the

food

am

ount

s fr

om g

reat

est to

leas

t.

5.

Whi

ch c

at g

ets

the

mos

t fo

od?

6.

How

can

you

che

ck y

our

answ

er?

7. M

ike

has

thre

e m

ovie

s an

d w

ants

to

wat

ch the

long

est on

e. T

he fi

rst m

ovie

is 2

1 _ 3 h

ours

long

, the

sec

ond

mov

ie is

1 7 _ 8 h

ours

long

, and

the

thi

rd m

ovie

is 2

2 _ 5 h

ours

long

. Ord

er the

leng

th o

f

the

mov

ies

from

long

est to

sho

rtes

t.

8. J

oe, L

ara,

and

Mik

e sh

are

a bu

cket

of

popc

orn.

Joe

eat

s 2

2 _ 3 c

ups,

Lar

a ea

ts

2 3 _ 6 c

ups,

and

Mik

e ea

ts 2

1 _ 5 c

ups.

Who

eats

the

mos

t po

pcor

n? W

ho e

ats

the

leas

t? O

rder

the

am

ount

s of

pop

corn

from

leas

t to

gre

ates

t.

20

1

20

1

20

1

MX

EN

L08

AW

K4X

_RT

_CH

15_L

6.in

dd

A6/

13/0

7 1

1:53

:32

AM

01

1 102 10

3 104 10

6 107 10

8 109 10

5 10

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

100

Mod

el A

dditi

onN

umbe

r lin

es c

an h

elp

you

add

frac

tions

. Fra

ctio

ns tha

t ha

ve the

sam

e de

nom

inat

or, o

r bo

ttom

num

ber,

are

calle

d lik

e f

ract

ions

. Whe

n fr

actio

ns h

ave

the

sam

e de

nom

inat

or, y

ou c

an a

dd the

num

erat

ors,

or

top

num

bers

. The

den

omin

ator

alw

ays

stay

s th

e sa

me.

Find

the

sum

.

1 __ 2 1

1 __ 2

•

The

deno

min

ator

tel

ls the

num

ber

of e

qual

par

ts the

num

ber

line

is d

ivid

ed in

to.

•

The

deno

min

ator

is 2

, so

the

num

ber

line

is d

ivid

ed in

to 2

equ

al p

arts

.

•

Labe

l the

num

ber

line

with

0,

1 _ 2 , a

nd 2 _ 2

.

•

Shad

e th

e pa

rt fr

om 0

to

1 _ 2 . T

o ad

d 1 _ 2 , m

ove

from

1 _ 2 to

1, o

r 2 _ 2 .

•

Sinc

e th

ere

are

2 eq

ual p

arts

in a

ll, tha

t m

eans

the

re a

re

2 ou

t of

2 e

qual

par

ts, o

r 2 _ 2 .

•

So, 1 _ 2

1 1 _ 2

5 2 _ 2

, or

1.

Find

the

sum

.

01 2

2 2

5 ___

10 1

2 ___

10

3 __ 7 1

2 __ 7

5 __ 6 1

1

__ 6

1.2.

3.

1 __ 3 1

1 __ 3

3 __ 7 1

1 __

7 4 __ 7 1

2 __ 7

Mo

de

l the

sum

. Re

cord

yo

ur a

nsw

er.

5.6.

4.

MX

EN

L08

AW

K4X

_RT

_CH

16_L

01.in

dd

16/

13/0

7 9

:03:

50 A

M

RW99-RW100


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

101

Mod

el S

ubtr

actio

nN

umbe

r lin

es c

an b

e us

ed to

subt

ract

frac

tions

.Fr

actio

ns w

ith c

omm

on d

enom

inat

ors

are

calle

d lik

efr

acti

ons

. Whe

n yo

u ha

ve li

ke fr

actio

ns, y

ou o

nly

need

to

subt

ract

the

num

erat

ors.

The

den

omin

ator

sta

ys the

sam

e.

Find

the

diff

ere

nce

.

1 __ 2 2

1 __ 2

• Th

e de

nom

inat

or is

2 s

o dr

aw a

num

ber

line

and

divi

de it

into

2 e

qual

par

ts.

• B

egin

at 1 _ 2 .

• Su

btra

ct 1 _ 2

by

coun

ting

back

1 p

art on

the

num

ber

line.

• Y

ou s

topp

ed o

n 0

so, 0 _ 2

of t

he n

umbe

r lin

e re

mai

ns.

Find

the

diff

ere

nce

.

Mo

de

l the

diff

ere

nce

. Re

cord

yo

ur a

nsw

er.

1.2.

3.

01 2

2 2

01

1 102 10

3 104 10

6 107 10

8 109 10

5 10

4.5.

6.

That

mea

ns 1 _ 2

2 1 _ 2

5 0 _ 2

, or

0.

4 __ 5 2

1 __ 5

6 ___

10 2

2 __

_ 10

9 ___

14 2

3 __

_ 14

5 __ 7 2

3 __ 7

3 ___

10 2

1 __

_ 10

2 __

6 2 1

__ 6

MX

EN

L08

AW

K4X

_RT

_CH

16_L

02.in

dd

A6/

14/0

7 1

1:24

:19

AM

Find

and

re

cord

the

sum

or

diff

ere

nce

.

So, 4

__ 6 1

3 __ 6 5

7 __ 6 , o

r 1 1

__ 6 .

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

102

Reco

rd A

dditi

on a

nd S

ubtr

actio

nW

hen

you

add

or s

ubtr

act fr

actio

n w

ith li

ke d

enom

inat

ors,

ad

d or

sub

stra

ct o

nly

the

num

erat

ors.

Find

and

re

cord

the

sum

.

4 __ 6 1

3 __ 6

3 __

5 1 4 __ 5

1 __

9 1 7 __ 9

8 ___

11 2

3 ___

11

5 __

6 1 4 __ 6

6 ___

10 2

4 ___

10

10

___

12 2

5 ___

12

2 __ 3 1

1 __

3

3 __

4 2 1 __

4

15

15

4 pa

rts

shad

ed

__

____

____

____

6

equa

l par

ts

3 pa

rts

shad

ed

__

____

____

____

6

equa

l par

ts

7 pa

rts

shad

ed

__

____

____

____

6

equa

l par

ts

4 __ 6 1

3 __ 6

4 1

3

____

_ 6

7 __ 6 , o

r 1 1

__ 6

Add

the

num

erat

ors.

Writ

e th

e de

nom

inat

or.

1.2.

3.4.

5.6.

7.8.

MX

EN

L08

AW

K4X

_RT

_CH

16_L

03.in

dd

A6/

13/0

7 9

:19:

05 A

M

RW101-RW102


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

103

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Writ

e an

Equ

atio

nJe

n ta

kes

a d

ance

cla

ss o

f hi

p-h

op

and

jazz

. The

cla

ss la

sts

9 __

10 h

our

. Hip

-ho

p la

sts

4 __

10 h

our

. Whi

ch p

art

last

s lo

nge

r?

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd?

Writ

e th

e qu

estio

n as

a fi

ll-in

-the

-bla

nk s

ente

nce.

Pla

n

2. C

hoos

e a

varia

ble.

Wha

t do

es the

var

iabl

e re

pres

ent?

So

lve

3.

W

rite

an e

quat

ion

to fi

nd the

leng

th o

f the

jazz

cla

ss. W

hich

par

t of

the

cl

ass

last

s lo

nger

?

Ch

eck

4.

H

ow c

an y

ou c

heck

you

r w

ork?

5.

Of G

eorg

e’s

book

s, 5 _ 8

are

abo

ut a

nim

als.

Th

ere

are

2 _ 8 b

ooks

in h

is a

nim

al

colle

ctio

n ab

out do

gs. W

hat fr

actio

n of

th

e bo

oks

are

abou

t ot

her

anim

als?

W

rite

an e

quat

ion

to s

olve

.

6.

Mar

ia s

pend

s 6 __

14

of h

er d

ay in

sc

hool

. She

stu

dies

mat

h fo

r 2 __

14

of

her

time

at s

choo

l. W

hat fr

actio

n of

M

aria

’s d

ay is

spe

nt s

tudy

ing

othe

r su

bjec

ts in

sch

ool?

Writ

e an

eq

uatio

n to

sol

ve.

MXE

NL08

AW

K4X_

RT_C

H16_

L04.

indd

A

6/6/

07

12:1

2:26

PM

15 5

2

Add

and

Subt

ract

Mix

ed N

umbe

rsY

ou c

an a

dd a

nd s

ubtr

act m

ixed

num

bers

with

like

den

omin

ator

s. A

dd

or s

ubtr

act th

e fr

actio

ns fi

rst.

Then

add

or

subt

ract

the

who

le n

umbe

rs.

You

can

che

ck y

our

answ

er b

y m

akin

g a

mod

el o

f the

equ

atio

n.

Mo

de

l and

re

cord

the

sum

. 3 1

__

6 1 1

3 __

6

Star

t w

ith the

frac

tions

. Add

the

nu

mer

ator

s fir

st. K

eep

the

deno

min

ator

s th

e sa

me.

Add

the

who

le n

umbe

rs.

If ne

eded

, cha

nge

any

impr

oper

fr

actio

ns in

the

sum

to

a m

ixed

num

ber.

Writ

e th

e an

swer

in s

impl

est fo

rm.

1 __

6 + 3 __ 6 =

4 __

6

3 1 __

6 1 1

3 __ 6 5

4 4 __ 6

4 4 __

6 5 4

2 __ 3

Mo

de

l and

re

cord

the

diff

ere

nce

. 1 4

__

5 2 2 __

5

Star

t w

ith the

frac

tions

. Sub

trac

t th

e nu

mer

ator

s. K

eep

the

deno

min

ator

s th

e sa

me.

Subt

ract

the

who

le n

umbe

rs.

If ne

eded

, cha

nge

any

impr

oper

fr

actio

ns in

the

diff

eren

ce to

a m

ixed

nu

mbe

r.

Writ

e th

e an

swer

in s

impl

est fo

rm.

4 __

5 2 2 __

5 5 2 __ 5

1 2

0 5

1

1 4 __

5 2 2 __

5 5

1.3.

2.4.

5.7.

6.8.

3 +

1 =

4

3 1 __

6 1 1

3 __ 6 5

4 4 __ 6 , o

r 4 2

__ 3

Mo

de

l and

re

cord

the

sum

or

diff

ere

nce

.

1 2 __ 5

1 4 __

5 2 2 __ 5 5

1 2 __ 5

1 2 __ 5

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

104

1 2 __ 5

1 1 __ 5

1

6 5 __

6

2 1 __

6 2

3 7 __ 8

2 6 __ 8

2

2 7 __

_ 10

3 5 __

_ 10

1

3 4 __

7

2 4 __

7 2

2 8 __

_ 11

2 8 __

_ 11

1

4 8 __ 9

3 5 __ 9

1

4 1 __

3

4 2 __

3 1

MX

EN

L08

AW

K4X

_RT

_CH

16_L

05.in

dd

A6/

14/0

7 1

1:27

:52

AM

RW103-RW104


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

105

Mod

el A

dditi

on a

nd S

ubtr

actio

nof

Unl

ike

Frac

tions

Unl

ike

fra

ctio

ns a

re fr

actio

ns tha

t ha

ve d

iffer

ent de

nom

inat

ors,

or

bot

tom

num

bers

. You

can

use

a m

odel

to

add

or s

ubtr

act

unlik

e fr

actio

ns.

Mo

de

l to

fin

d t

he s

um.

2 __

3 1 1 __

6 U

se p

atte

rn b

lock

s to

mod

el e

ach

frac

tion.

Mod

el 2 _ 3

.

Use

tw

o 1 _ 3 p

atte

rn b

lock

s.

Use

pat

tern

blo

cks

to fi

nd a

frac

tion

equi

vale

nt to

2 _ 3 .

2 __ 3 5

4 __ 6

Mod

el 1 _ 6

.

Use

one

1 _ 6 p

atte

rn b

lock

.

You

foun

d th

at 2 _ 3

is e

quiv

alen

t to

4 _ 6 .

Sinc

e 4 _ 6 a

nd 1 _ 6

hav

e th

e sa

me

deno

min

ator

, you

can

add

the

frac

tions

.

Add

. 4 __ 6 1

1 __ 6 5

5 __ 6

Mo

de

l to

fin

d t

he s

um o

r d

iffe

renc

e.

1.

1 __ 3 1

1 __ 4 5

2.

2 __ 4 1

3 __

8 5

3.

7

___

10 2

1 __ 5 5

4.

2 __ 5 1

3 ___

10 5

6.

1 __ 6 1

2 __ 3 5

5.

7

__ 9 2

2 __

3 5

So,

2 _ 3 1

1 _ 6 5

5 _ 6 .

MX

EN

L08

AW

K4X

_RT

_CH

16_L

06.in

dd

A6/

13/0

7 1

1:13

:16

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

106

110 100 10 0

5 10 0.5

110 100 10 0

5 10 0.5

4 10 0.4

3 10 0.3

2 10 0.2

1 10 0.1

9 10 0.9

8 10 0.8

7 10 0.7

6 10 0.6

10 100 10

5 10 0.5

01

10 100 10

5 10 0.5

01

Rela

te F

ract

ions

and

Dec

imal

sA

de

cim

al is

a n

umbe

r w

ith o

ne o

r m

ore

digi

ts to

the

right

of t

he d

ecim

al p

oint

.

Wri

te t

he f

ract

ion

and

de

cim

al s

how

n b

y th

e m

od

el.

The

num

ber

line

is d

ivid

ed in

to 1

0 eq

ual p

arts

.

Firs

t, la

bel t

he p

arts

bel

ow the

num

ber

line

usin

g te

nths

st

artin

g at

0 a

nd s

topp

ing

at 1

.

Then

, lab

el the

par

ts a

bove

the

num

ber

line

usin

g fr

actio

ns

star

ting

at 0 __

10

and

end

ing

at 10

__

10

.

Loca

te the

poi

nt o

n th

e nu

mbe

r lin

e.Th

e fr

actio

n 6 __

10

and

the

dec

imal

0.6

nam

es the

poi

nt.

So, t

he fr

actio

n an

d th

e de

cim

al s

how

n by

the

mod

el is

6

__

10 a

nd 0

.6.

Wri

te t

he f

ract

ion

as a

de

cim

al.

Writ

e a

zero

in the

one

s pl

ace,

and

then

writ

e th

e de

cim

al p

oint

.W

rite

a ze

ro in

the

ten

ths

plac

e.W

rite

a 5

in the

hun

dred

ths

plac

e.So

, 5 ___

100 w

ritte

n as

a d

ecim

al is

0.0

5.

Wri

te t

he f

ract

ion

and

de

cim

al s

how

n b

y e

ach

mo

de

l.

Wri

te e

ach

frac

tio

n as

a d

eci

mal

. Yo

u m

ay d

raw

a p

ictu

re.

3.

3 __

__

100

4.

4 __

8 5.

20

__

__

100

6.

9 ___

10

5 __

__

100

1.2.

Rea

d th

e fr

actio

n al

oud

as fi

ve h

undr

edth

s.

0.j

j

0.0j

0.05

j

MX

EN

L08

AW

K4X

_RT

_CH

17_L

1.in

dd

A6/

20/0

7 1

1:41

:12

AM

RW105-RW106


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

107

Deci

mal

s to

Thou

sand

ths

Wri

te 0.303

as

a fr

acti

on.

Rea

d th

e de

cim

al a

loud

.

Thin

k: T

he d

ecim

al is

“th

ree

hund

red

thre

e th

ousa

ndth

s.”

Writ

e th

e nu

mbe

rs to

the

right

of t

he

deci

mal

as

the

num

erat

or.

Thin

k: T

he n

umbe

r to

the

rig

ht o

f the

de

cim

al is

303

.

The

deno

min

ator

is the

pla

ce v

alue

of t

he

digi

t fa

rthe

st r

ight

.

Thin

k: T

he d

igit

fart

hest

rig

ht is

in the

th

ousa

ndth

s pl

ace

so the

den

omin

ator

is

1,0

00.

So, 0

.303

writ

ten

as a

frac

tion

is

303

____

1,00

0 .

Wri

te 0.007

as

a fr

acti

on.

Rea

d th

e de

cim

al a

loud

.

Thin

k: T

he d

ecim

al is

“se

ven

thou

sand

ths.

”

Writ

e th

e nu

mbe

r to

the

rig

ht o

f the

de

cim

al a

s th

e nu

mer

ator

.

Thin

k: T

he n

umbe

r to

the

rig

ht o

f the

de

cim

al is

7.

The

deno

min

ator

is the

val

ue o

f the

dig

it w

hich

is fa

rthe

st r

ight

.

Thin

k: T

he d

igit

fart

hest

rig

ht is

in the

th

ousa

ndth

s pl

ace

so the

den

omin

ator

is

1,00

0.

So, 0

.007

writ

ten

as a

frac

tion

is

7 __

__

1,00

0 .

Wri

te e

ach

de

cim

al a

s a

frac

tio

n.

1.

0.72

5

2. 0

.238

3.

0.09

3 4

. 0.

512

5.

0.78

4

6. 0

.056

7.

0.20

7

8. 0

.003

9.

0.69

2 1

0. 0

.555

11.

0.4

13 1

2. 0

.708

0.3

12 1

4. 0

.009

15.

0.4

75 1

6. 0

.902

13.

MX

EN

L08

AW

K4X

_RT

_CH

17_L

2.in

dd

A6/

14/0

7 1

0:53

:02

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

108

Equi

vale

nt D

ecim

als

You

can

use

mod

els

to s

how

equ

ival

ent de

cim

als.

Use

a h

und

red

ths

mo

de

l. A

re t

he t

wo

de

cim

als

eq

uiva

lent

?W

rite

5 o

r fi

.M

od

el 0.71

and

0.17.

Shad

e in

the

app

ropr

iate

num

ber

of s

quar

es in

eac

h hu

ndre

dths

mod

el.

0

.71

is 7

1 hu

ndre

dths

.Sh

ade

in 7

1 sq

uare

s. 0

.17

is 1

7 hu

ndre

dths

.Sh

ade

in 1

7 sq

uare

s.

See

if th

e m

odel

s ha

ve the

sam

e nu

mbe

r of

squ

ares

sha

ded.

0.71

has

mor

e sq

uare

s sh

aded

tha

n 0.

17.

The

deci

mal

mod

els

show

tha

t 0.

71 is

not

equ

ival

entt to

0.17

.

So, 0

.71 fi

0.1

7.

Wri

te a

n e

qui

vale

nt d

eci

mal

fo

r 0.20

. Yo

u m

ay u

se a

de

cim

al m

od

el.

Shad

e 20

squ

ares

of a

hun

dred

ths

mod

el to

show

0.2

0.

Ther

e ar

e tw

o co

lum

ns s

hade

d.

Shad

e tw

o co

lum

ns o

f a ten

ths

mod

el.

The

deci

mal

mod

els

show

tha

t 0.

2 is

equ

ival

ent to

0.2

0.

1. 0.

3 an

d 0.

302.

0.

72 a

nd 0

.27

3. 0.

60 a

nd 0

.64.

0.

05 a

nd 0

.50

5. 0

.96.

0.2

7. 0

.50

8. 0

.7

Use

a t

ent

hs m

od

el a

nd a

hun

dre

dth

s m

od

el.

Are

the

tw

od

eci

mal

s e

qui

vale

nt?

Wri

te 5

or fi

.

Wri

te a

n e

qui

vale

nt d

eci

mal

fo

r e

ach.

Yo

u m

ay u

se d

eci

mal

mo

de

ls.

So, 0

.2 5

0.2

0.

MX

EN

L08

AW

K4X

_RT

_CH

17_L

3.in

dd

A6/

14/0

7 1

0:58

:42

AM

RW107-RW108


2.010 10

0 10 1.011

11

11

11

11

15 10 1.5

4 10 1.4

3 10 1.3

2 10 1.2

1 10 1.1

9 10 1.9

8 10 1.8

7 10 1.7

6 10 1.6

2.010 10

0 10 1.011

15 10 1.5

0 10 3.0

3

4.010 103

35 10 3.5

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

109

Rela

te M

ixed

Num

bers

and

Dec

imal

sW

rite

an

eq

uiva

lent

mix

ed

num

be

r an

d a

de

cim

al f

or

the

mo

de

l.

The

num

ber

line

belo

w is

div

ided

into

10

equa

l par

ts.

Loca

te the

poi

nt o

n th

e nu

mbe

r lin

e.Th

e de

cim

al is

1.8

and

the

mix

ed n

umbe

r is

1 8 __

10

.

So, t

he m

ixed

num

ber

and

the

deci

mal

sho

wn

by the

mod

el is

1 8 __

10

and

1.8

.

Wri

te t

he d

eci

mal

fo

r th

e m

ixe

d n

umb

er.

The

n w

rite

the

wo

rd f

orm

.2

3 _ 4

•W

rite

the

frac

tion

3 _ 4 u

sing

a d

enom

inat

or o

f 100

.3 _ 4 5

3 3

25

____

_ 4

3 2

5 5

75

___

100

•W

rite

75

___

100 a

s a

deci

mal

.0.

75

•W

rite

the

who

le n

umbe

r.2.

75

So,2

3 _ 4 w

ritte

n as

a d

ecim

al is

2.7

5, o

r tw

o an

d se

vent

y-fiv

e hu

ndre

dths

.

Wri

te a

n e

qui

vale

nt m

ixe

d n

umb

er

and

a d

eci

mal

fo

r e

ach

mo

de

l. 1.

2.

3. 5

.14.

2 7 __

_ 10

0

Firs

t, la

bel t

he p

arts

bel

ow the

num

ber

line

usin

g te

nths

st

artin

g at

1.0

and

sto

ppin

g at

2.0

.Th

en, l

abel

the

par

ts a

bove

the

num

ber

line

usin

g fr

actio

ns

star

ting

at 1

0 __

10 a

nd s

topp

ing

at 1

10

__

10 .

Wri

te a

n e

qui

vale

nt m

ixe

d n

umb

er

and

a d

eci

mal

fo

r e

ach.

The

n w

rite

the

wo

rd f

orm

. Yo

u m

ay u

se a

mo

de

l.

MX

EN

L08

AW

K4X

_RT

_CH

17_L

4.in

ddA

6/28

/07

11:2

5:55

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

110

2.00

2.10

$2.0

9

2.20

2.30

2.40

2.50

$2.5

0$2

.55

$2.9

0

2.60

2.70

2.80

2.90

3.00

Com

pare

and

Ord

er D

ecim

als

Use

the

num

be

r lin

e t

o o

rde

r th

e d

eci

mal

s fr

om

leas

t to

gre

ate

st.

$2.9

0, $

2.09

, $2.

50, $

2.55

Loca

te the

dol

lar

amou

nts

on the

num

ber

line

belo

w.

The

deci

mal

s on

a n

umbe

r lin

e go

in o

rder

from

left

to r

ight

.Th

e de

cim

al tha

t is

fart

hest

to

the

left

is the

leas

t.Th

e de

cim

al tha

t is

fart

hest

to

the

right

is the

gre

ates

t.

2.09

is the

fart

hest

left,

so

it is

the

leas

t.2.

90 is

the

fart

hest

rig

ht, s

o it

is the

gre

ates

t.

So, t

he o

rder

of t

he d

ecim

als

from

leas

t to

gre

ates

t is

$2.

09,$

2.50

,$2.

55,$

2.90

.

Co

mp

are

. Wri

te <

, >, o

r 5

fo

r th

e

.

$1.64

$1.46

Writ

e on

e de

cim

al a

bove

the

oth

er.

1.64

Be

sure

to

line

up the

dec

imal

poi

nts.

1.46

Bot

h de

cim

als

star

t w

ith o

ne. C

ontin

ue c

ompa

ring.

Com

pare

the

nex

t di

git to

the

rig

ht, t

he ten

ths

plac

e.

1

.64

Sinc

e6

> 4

, tha

t m

eans

1.6

4 >

1.4

6.1.46

So,$

1.64

>

$1.4

6.

Use

the

num

be

r lin

e a

bo

ve t

o o

rde

r th

e d

eci

mal

s fr

om

leas

t to

gre

ate

st.

3. 0

.43

0.

344.

2.36

2.

095.

$0.5

7

$1.5

7

6.$1

.80

$1

.83

7.2.

27

3.25

8.1.

031.

30

2. $

2.33

, $2.

43, $

2.34

, $2.

401.

$2.

20, $

2.21

, $2.

02, $

2.10

Co

mp

are

. Wri

te <

, >, o

r 5

fo

r e

ach

.

MX

EN

L08

AW

K4X

_RT

_CH

17_L

5.in

ddA

6/28

/07

11:2

4:57

AM

RW109-RW110


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

111

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Draw

Con

clus

ions

Me

gan

live

s 2.4

mile

s fr

om

Bo

b’s

Bo

oks

tore

. Jo

sh li

ves

2.15

mile

s aw

ay, a

nd S

am li

ves 2.09

mile

s aw

ay. W

ho

live

s cl

ose

st t

o B

ob

’s B

oo

ksto

re?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. H

ow c

an y

ou u

se in

form

atio

n in

the

pro

blem

to

draw

a c

oncl

usio

n?

3. U

se the

info

rmat

ion

in the

pro

blem

to

orde

r th

e di

stan

ces

from

nea

rest

to

fart

hest

di

stan

ce fr

om B

ob’s

Boo

ksto

re.

4. W

ho li

ves

clos

est to

Bob

’s B

ooks

tore

? W

rite

your

ans

wer

in a

com

plet

e se

nten

ce.

5. H

ow c

an y

ou c

heck

you

r an

swer

?

So

lve

the

pro

ble

m.

6. D

ane

spen

t $1

2.50

on

scho

ol s

uppl

ies.

Pe

ter

spen

t $1

2.05

. Mar

shal

l spe

nt

$12.

25. W

hich

stu

dent

spe

nt the

leas

t on

sch

ool s

uppl

ies:

Dan

e, P

eter

, or

Mar

shal

l?

7. Sa

rah

leap

ed 2

.1 m

eter

s. K

elly

leap

ed

2.23

met

ers.

Lin

da le

aped

2.

19 m

eter

s. W

ho le

aped

the

fart

hest

di

stan

ce: S

arah

, Kel

ly, o

r Li

nda?

MX

EN

L08

AW

K4X

_RT

_CH

17_L

6.in

dd

A6/

6/07

2:

42:5

1 PM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

112

Roun

d De

cim

als

Whe

n yo

u ro

und

a de

cim

al to

the

near

est te

nth,

the

rou

nded

nu

mbe

r w

ill s

top

at the

ten

ths

plac

e. F

or e

xam

ple,

whe

n 4.

73 is

ro

unde

d to

the

nea

rest

ten

th, i

t be

com

es 4

.7.

Ro

und

5.55

to t

he n

ear

est

te

nth.

• Id

entif

y th

e te

nths

pla

ce.

Thin

k: T

he 5

to

the

right

of t

he d

ecim

al

poin

t is

in the

ten

ths

plac

e.

• L

ook

at the

dig

it to

the

rig

ht o

f the

te

nths

pla

ce.

If th

at n

umbe

r is

less

tha

n 5,

the

dig

it in

th

e te

nths

pla

ce s

tays

the

sam

e. If

it is

5

or m

ore,

the

dig

it is

incr

ease

d by

1.

Thin

k: T

he d

igit

to the

rig

ht o

f the

ten

ths

plac

e is

5, s

o ro

und

the

num

ber

in the

te

nths

pla

ce u

p. R

ound

5.5

5 to

5.6

.

So, 5

.55

roun

ded

to the

nea

rest

ten

th

is 5

.6.

Ro

und

$48

.92

to t

he n

ear

est

do

llar.

• W

hen

you

roun

d to

the

nea

rest

dol

lar,

you

roun

d to

the

one

s pl

ace.

Iden

tify

the

ones

pla

ce.

Thin

k: T

he 8

is in

the

one

s pl

ace.

• If

the

num

ber

to the

rig

ht o

f the

one

s pl

ace

is le

ss tha

n 5,

the

dig

it in

the

on

es p

lace

sta

ys the

sam

e. If

it is

5 o

r m

ore,

the

dig

it is

incr

ease

d by

1.

Thin

k: 9

is g

reat

er tha

n 5,

so

roun

d th

e nu

mbe

r in

the

one

s pl

ace

up. R

ound

$4

8.92

to

$49.

So, $

48.9

2 ro

unde

d to

the

nea

rest

dol

lar

is $

49.

Ro

und

eac

h nu

mb

er

to t

he n

ear

est

te

nth

and

eac

h m

one

yam

oun

t to

the

ne

are

st d

olla

r.

1.

1.

92

2.

$21.

10

3.

56.1

6 4

. $2

4.60

5.

6.07

6

. 28

.34

7.

6.37

8

. 55

.92

9.

$81.

99

10.

8.35

11

. 48

.48

12.

8.77

13.

39.9

4 14

. $6

.03

15.

18.2

6 16

. 77

.98

MX

EN

L08

AW

K4X

_RT

_CH

18_L

1.in

dd

A6/

6/07

2:

12:0

6 PM

RW111-RW112


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

113

Estim

ate

Deci

mal

Sum

s and

Diff

eren

ces

You

can

use

rou

ndin

g to

est

imat

e de

cim

al s

ums

and

diffe

renc

es.

Esti

mat

e t

he d

iffe

renc

e.

Rou

nd e

ach

num

ber

to the

nea

rest

dol

lar.

Estim

ate

by s

ubtr

actin

g th

e ro

unde

d nu

mbe

rs.

So, a

n es

timat

e fo

r $9

1.24

2 $

39.8

7 is

$51

.

Esti

mat

e t

he s

um.

Rou

nd e

ach

num

ber

to the

nea

rest

who

le n

umbe

r.Es

timat

e by

add

ing

the

roun

ded

num

bers

.

So, a

n es

timat

e fo

r 4.

0 1

8.9

1 1

.4 is

14.

Esti

mat

e t

he s

um o

r d

iffe

renc

e.

3.

9

1 1

.8

_

$52.

35

2 4

0.22

__

5.0

2

4.9

6

1 1

.78

__

$3

1.99

1 1

2.12

__

81.2

5

2 2

9.50

__

23.9

2

2 1

1.03

__

3

.95

7.1

1

1 2

.18

__

58.7

5

2 4

5.11

__

$1.8

7

1 1

.09

__

6.

67

1 2

.25

__

8.6

7

5.8

9

1 6

.92

__

$72.

99

2 3

2.87

__

$9

1.24

2 $

39.8

7

__

$91.

24

2 $

39.8

7

__

$91

2 $

40

__

$9

1

2 $

40

__

$51

4

.0

8.9

1 1.

4

_

4

9

1

1

_

4

9

1 1

_

14

1. 4. 7. 10.

2. 5. 8. 11.

3. 6. 9. 12.

4

.0

8.9

1 1.

4

_

MX

EN

L08

AW

K4X

_RT

_CH

18_L

2.in

dd

A6/

14/0

7 1

0:16

:38

AM

Mod

el A

dditi

onY

ou c

an u

se d

ecim

al m

odel

s to

hel

p yo

u ad

d de

cim

als.

Use

a m

od

el t

o f

ind

the

sum

.

Ste

p 1

: Sha

de s

quar

es to

repr

esen

t 1.

6.

Thin

k : S

hade

all

the

squa

res

in o

ne m

odel

and

6 s

quar

es in

ano

ther

hu

ndre

dths

mod

el.

Ste

p 2

: Sha

de in

squ

ares

to

repr

esen

t 1.

0.

Thin

k: A

dd o

ne w

hole

mod

el to

the

mod

el o

f 1.6

.

Ste

p 3

: Cou

nt a

ll th

e sq

uare

s.

Thin

k: T

here

are

2 w

hole

mod

els

shad

ed a

nd 6

squ

ares

of a

hu

ndre

dths

mod

el, o

r 2.

6.

So, 1

.6 1

1.0

5 2

.6.

Use

mo

de

ls t

o f

ind

the

sum

. 1

. 2.

1 1

0.5

9 2

. 1.

4 1

0.2

2

3.

1.

27

1 1

.15

__

4

.

0.81

1 0

.43

__

1.

6

1 1

.0

_

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

114

MX

EN

L08

AW

K4X

_RT

_CH

18_L

3.in

dd

A6/

14/0

7 1

0:17

:53

AM

RW113-RW114


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

115

Mod

el S

ubtr

actio

nY

ou c

an u

se d

ecim

al m

odel

s to

hel

p yo

u su

btra

ct d

ecim

als.

Use

a m

od

el t

o f

ind

the

diff

ere

nce

.

Ste

p 1

: Sha

de 1

1 co

lum

ns o

n de

cim

al m

odel

s to

sho

w 1

.1.

Shad

e al

l the

col

umns

on

one

mod

el a

nd 1

col

umn

on a

noth

er m

odel

.

Ste

p 2

: Dra

w X

s on

4 o

f the

sha

ded

colu

mns

to

show

0.4

.

Ste

p 3

: Cou

nt the

sha

ded

colu

mns

tha

t do

not

hav

e X

s on

the

m.

Ther

e ar

e 7

colu

mns

tha

t do

not

hav

e X

s on

the

m.

So, 1

.1 2

0.4

5 0

.7.

Use

mo

de

ls t

o f

ind

the

diff

ere

nce

.

1.

1.4

2 0

.61

2.

1.6

2 1

.08

3.

0.

84

2 0

.17

__

4

.

1.39

2 1

.14

__

1.

1

2 0

.4

_

MX

EN

L08

AW

K4X

_RT

_CH

18_L

4.in

dd

A6/

14/0

7 1

0:19

:01

AM

Reco

rd A

dditi

on a

nd S

ubtr

actio

nEq

uiva

lent

de

cim

als

can

help

add

or

subt

ract

num

bers

tha

t do

not ha

ve the

sam

e nu

mbe

r of

dig

its a

fter

the

deci

mal

poi

nt.

Esti

mat

e. T

hen

reco

rd t

he s

um.

Estim

ate

the

sum

by

roun

ding

to

the

near

est do

llar.

Add

the

est

imat

es.

Line

up

the

deci

mal

poi

nts

and

add

to fi

nd the

sum

.

Sinc

e$4

3.29

is c

lose

to

the

estim

ate

of $

43, i

t is

rea

sona

ble.

So, $

23.4

41

$19.

855

$43.

29.

Esti

mat

e. T

hen

reco

rd t

he d

iffe

renc

e.

Estim

ate

the

diffe

renc

e by

rou

ndin

g to

the

near

est w

hole

num

ber.

Line

up

the

deci

mal

poi

nts.

Pla

ce z

eros

to

the

right

of t

he d

ecim

al p

oint

so

each

num

ber

has

the

sam

e nu

mbe

r of

dig

its a

fter

the

deci

mal

poi

nt.

Subt

ract

as

you

wou

ld w

ith w

hole

num

bers

.Pl

ace

the

deci

mal

poi

nt in

the

diff

eren

ce.

Sinc

e57

.12

is c

lose

to

the

estim

ate

of 5

7, it

is r

easo

nabl

e.So

, 67.

12

9.98

557

.12.

Esti

mat

e. T

hen

reco

rd t

he d

iffe

renc

e.

1.

7.

5

2 2

.3

_

2.

$4.5

5

1 1

.97

__

3.

24.

3

2 1

8.18

__

4.

71.7

1 2

.34

__

5.

8.

2

1 3

.5

_

6.

$45.

36

2 1

9.55

__

7.

61.7

1 1

2.84

__

8.

52.9

9

1 6

5.20

__

$23.

44

1 $

19.8

5

__

$23

1 $

20

__

11

$23.

44

1 $

19.8

5

__

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

116

67.1

2 9

.98

__

67.1

2 9

.98

__

67

2 1

0

_

57

67.1

0

2 9

.98

__

67

.10

2 9

.98

__

57.

12

5

1610

10

$23.

44

1 $

19.8

5

__

$43

$43.

29

MX

EN

L08

AW

K4X

_RT

_CH

18_L

5.in

ddA

6/21

/07

12:3

9:53

PM

RW115-RW116


Mak

e Ch

ange

Mak

e c

hang

e. L

ist

the

bill

s an

d c

oin

s.

Co

unt

On

Cos

t: $6

.57

Paid

with

:

Ste

p 1

: Cou

nt o

n fr

om the

cos

t us

ing

coin

s an

d bi

lls.

Thin

k: T

he it

em c

ost $

6.57

. Cou

nt o

n:

$6.5

8 $

6.59

$6

.60

$6.

70 $6

.75

$7.

00

$8

.00

$

9.00

$1

0.00

Ste

p 2

: Cou

nt the

coi

ns a

nd b

ills.

Thin

k: 3

pen

nies

, 1 d

ime,

1 n

icke

l, 1

quar

ter,

3

$1 b

ills 5

$3.

43.

So, t

he c

hang

e is

$3.

43.

Sub

trac

t

Cos

t: $3

.34

Paid

with

:

Ste

p 1

: Sub

trac

t th

e co

st fr

om the

am

ount

pai

d.

Thin

k:

Ste

p 2

: Cou

nt o

n to

mak

e th

e ch

ange

, $1.

66. S

tart

with

the

bill

s.

Thin

k: $

1.00

, $1.

25, $

1.50

, $1.

60,

$1.6

5, $

1.66

.

So, t

he c

hang

e is

1 $

1 bi

ll,

2 qu

arte

rs, 1

dim

e, 1

nic

kel,

and

1 pe

nny.

Mak

e c

hang

e. L

ist

the

bill

s an

d c

oin

s.

1. C

ost:

$5.9

5

P

aid

with

:

2. C

ost:

$2.2

8

P

aid

with

:

3. C

ost:

$13.

39

P

aid

with

:

4. C

ost:

$4.6

3

P

aid

with

:

$5.0

0

2 $

3.34

__

$1.6

6

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

117

MX

EN

L08

AW

K4X

_RT

_CH

18_L

6.in

dd

A6/

14/0

7 1

0:26

:12

AM

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Com

pare

Str

ateg

ies

The

ve

ndin

g m

achi

ne h

as 10

dri

nks

left

. So

me

are

juic

e a

nd s

om

e a

re

wat

er.

The

re a

re m

ore

can

s o

f ju

ice

left

tha

n b

ott

les

of

wat

er.

Wha

t p

oss

ible

co

mb

inat

ions

of

wat

er

and

juic

e c

oul

d b

e le

ft in

the

mac

hine

?

5. J

en h

as a

red

shi

rt, a

blu

e sh

irt, a

nd a

gr

een

shirt

. She

has

bla

ck p

ants

and

bl

ue p

ants

. How

man

y di

ffere

nt

shirt

-pan

t co

mbi

natio

ns c

an J

en m

ake?

Te

ll w

hich

str

ateg

y yo

u us

ed.

6. T

he m

ail c

omes

at 1:

00 P

.M. o

n M

onda

y, 2

:00

P.M

. on

Tues

day,

1:

00 P

.M. o

n W

edne

sday

, and

2:

00 P

.M. o

n Th

ursd

ay. W

hat tim

e w

ill

the

mai

l com

e on

Frid

ay?

Tell

whi

ch

stra

tegy

you

use

d.

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

118

Cho

ose

a s

trat

eg

y to

so

lve

.

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

So

lve

3. S

how

how

you

use

you

r st

rate

gy to

solv

e th

e pr

oble

m. W

hat po

ssib

le c

ombi

natio

ns

of w

ater

and

juic

e co

uld

be le

ft in

the

mac

hine

?

Pla

n

2. W

hat st

rate

gy c

ould

you

use

to

solv

e th

is p

robl

em?

Ch

eck

4. H

ow c

an y

ou c

heck

to

see

if yo

ur a

nsw

er is

rea

sona

ble?

MX

EN

L08

AW

K4X

_RT

_CH

18_L

7.in

dd

A6/

14/0

7 1

0:28

:06

AM

RW117-RW118


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

119

Poin

ts, L

ines

, and

Ray

sTe

rmDe

finiti

onEx

ampl

e

poin

t n

ames

an

exac

t loc

atio

n in

spac

e

line

segm

ent

par

t of a

line

; has

two

end

poin

ts

line

stra

ight

pat

h of

poi

nts

that

cont

inue

s w

ithou

t end

in b

oth

dire

ctio

ns

ray

par

t of a

line

that

has

one

endp

oint

and

con

tinue

s

with

out e

nd in

one

dire

ctio

n

plan

e f

lat s

urfa

ce th

at c

ontin

ues

with

out e

nd in

all

dire

ctio

ns

Nam

e a

ge

om

etr

ic t

erm

th

at r

ep

rese

nts

eac

h o

bje

ct.

flag

po

le

• A fl

agpo

le d

oes

not

co

ntin

ue w

ithou

t en

d in

both

dire

ctio

ns. T

he top

s

and

bottom

of t

he p

ole

are

like

endp

oint

s.

• So,

a fl

agpo

le c

an b

e re

pres

ente

d by

a li

ne

segm

ent.

lase

r b

eam

• A la

ser

beam

com

es o

ut

of

a b

ox o

r po

inte

r an

d

sh

ines

with

out en

d in

one

dire

cito

n.

• So,

a la

ser

beam

can

be

repr

esen

ted

by a

ray

.

Nam

e a

ge

om

etr

ic t

erm

tha

t re

pre

sent

s th

e o

bje

ct.

1.

h

ighw

ay 2

. tip

of a

mar

ker

3.

scr

ewdr

iver

4.

fo

otba

ll fie

ld 5

. g

rain

of s

alt

6.

tw

o w

ay a

rrow

7.

m

eado

w 8

. p

iece

of r

ope

9.

flas

hlig

ht b

eam

10.

ext

ensi

on c

ord

11.

farm

fiel

d12

. r

ailro

ad tra

ck

A

DE

BC

FG

KM

L

MX

EN

L08

AW

K4X

_RT

_CH

19_L

1.in

dd

A6/

14/0

7 9

:40:

41 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

120

Mea

sure

and

Cla

ssify

Ang

les

An

ang

le is

form

ed w

hen

two

rays

or

two

line

segm

ents

sha

re the

sam

e en

dpoi

nt, o

r ve

rte

x. A

n ac

ute

ang

le m

easu

res

grea

ter

than

0º

and

less

tha

n 90

º. A

n o

btu

se a

ngle

mea

sure

s gr

eate

r th

an 9

0º a

nd le

ss tha

n 18

0º. A

str

aig

ht a

ngle

fo

rms

a st

raig

ht li

ne a

nd m

easu

res

180º

. A r

ight

ang

le

mea

sure

s 90

º an

d fo

rms

a sq

uare

cor

ner.

• Whe

n yo

u m

easu

re a

n an

gle

usin

g a

prot

ract

or, r

emem

ber

to p

lace

the

cen

ter

poin

t of

the

pro

trac

tor

on the

ver

tex

of

the

angl

e.

• /FG

H m

easu

res

60º.

So, /

FGH

is a

n ac

ute

angl

e.

Use

a p

rotr

acto

r to

me

asur

e t

he a

ngle

. The

n w

rite

acu

te, right

, straigh

t, o

r obtus

e.

1.

2

.

3.

4.

5

.

6.

Use

a p

rotr

acto

r to

me

asur

e t

he a

ngle

. The

n w

rite

acu

te, right

, straigh

t, o

r obtus

e.

• /LM

N m

easu

res

90º.

An

angl

e th

at

mea

sure

s 90

º is

a r

ight

ang

le.

So, /

LMN

is a

rig

ht a

ngle

.

F

G

HM

N

L

AB

CL

M

N

K

L

J

D

EF

X

Y

Z

SR

Q

MX

EN

L08A

WK

4X_R

T_C

H19

_L2.

indd

A

6/13

/07

6:5

3:08

AM

RW119-RW120


AB

C

Dac

ute

obtu

se

M N

LK

right

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

121

Line

Rel

atio

nshi

psPa

irs o

f lin

es m

ay b

e in

ters

ect

ing

, par

alle

l, or

pe

rpe

ndic

ular

.

Nam

e a

ny li

ne r

ela

tio

nshi

ps

you

see

in

the

fig

ure

be

low

. Wri

te in

ters

ectin

g,

para

llel,

or

perp

endi

cula

r.

• The

line

s di

vidi

ng the

bla

ck a

nd w

hite

st

ripes

are

par

alle

l lin

es b

ecau

se the

lin

es n

ever

inte

rsec

t.

• The

line

s th

at fo

rm the

top

of t

he k

ite

are

inte

rsec

ting

lines

bec

ause

the

y cr

oss

at o

ne p

oint

and

form

an

angl

e.

• The

line

s w

here

the

kite

pol

es

inte

rsec

t th

e st

ripes

are

per

pend

icul

ar

lines

bec

ause

the

y in

ters

ect an

d fo

rm

right

ang

les.

Nam

e a

ny li

ne r

ela

tio

nshi

ps

you

see

in e

ach

figur

e.

Wri

te in

ters

ectin

g, p

aral

lel,

or

perp

endi

cula

r.

1.

2

.

3.

4

.

Type

of L

ine

Defi n

ition

Exam

ple

inte

rsec

ting

lines

that

cro

ss

each

oth

er a

t one

po

int a

nd fo

rm

four

ang

les

para

llel

lines

that

nev

er

inte

rsec

t and

are

al

way

s th

e sa

me

dist

ance

apa

rt

perp

endi

cula

rlin

es th

at in

ter-

sect

and

form

fo

ur ri

ght a

ngle

s

So, t

he fi

gure

con

tain

s pa

ralle

l, in

ters

ectin

g, a

nd p

erpe

ndic

ular

line

s.

MX

EN

L08

AW

K4X

_RT

_CH

19_L

3.in

dd

A6/

14/0

7 9

:43:

59 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

122

Poly

gons

Poly

go

ns a

re c

lose

d pl

ane

figur

es tha

t ar

e fo

rmed

by

3 or

mor

e st

raig

ht

side

s th

at a

re li

ne s

egm

ents

. A r

eg

ular

po

lyg

on’

s si

des

are

all t

he s

ame

leng

th, a

nd it

s an

gles

are

all

the

sam

e m

easu

re.

Nam

e t

he p

oly

go

n. T

ell

if it

ap

pe

ars

to b

e r

egul

ar o

r no

t reg

ular

.

• The

pol

ygon

has

3 s

ides

, so

it is

a tria

ngle

.

• The

sid

es o

f the

tria

ngle

are

not

all

the

sam

e le

ngth

, so

the

tria

ngle

is n

ot r

egul

ar.

• The

pol

ygon

has

8 s

ides

, so

it is

an

octa

gon.

• The

ang

les

of the

oct

agon

are

all

the

sam

e m

easu

re, a

nd the

sid

es a

re a

ll th

e sa

me

leng

th, s

o th

e oc

tago

n is

reg

ular

.

Nam

e t

he p

oly

go

n. T

ell

if it

ap

pe

ars

to b

e r

egul

ar o

r no

t reg

ular

.

1.

2.

3.

4.

5.

6.

Poly

gon

Side

s an

d An

gles

Exam

ple

trian

gle

3 si

des

3 an

gles

3 ve

rtice

s

quad

rilat

eral

4 si

des

4 an

gles

4 ve

rtice

s

pent

agon

5 si

des

5 an

gles

5 ve

rtice

s

hexa

gon

6 si

des

6 an

gles

6 ve

rtice

s

octa

gon

8 si

des

8 an

gles

8 ve

rtice

s

deca

gon

10 s

ides

10 a

ngle

s10

ver

tices

MX

EN

L08

AW

K4X

_RT

_CH

19_L

4.in

dd

A6/

14/0

7 9

:45:

18 A

M

RW121-RW122


2 cm

2 cm

2 cm

3 cm

3 cm

2 cm

2 cm

3 cm

4 cm

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

123

Clas

sify

Tria

ngle

sC

lass

ify e

ach

tria

ngle

. Wri

te

isosc

eles

, sca

lene

, or eq

uilateral.

The

n w

rite

right

, acu

te, o

r obtus

e.

• Eac

h si

de is

a d

iffer

ent le

ngth

, so

the

tria

ngle

is s

cale

ne.

• The

re is

one

obt

use

angl

e, s

o th

e tr

iang

le is

obt

use.

• All

thre

e si

des

are

equa

l, so

the

tr

iang

le is

equ

ilate

ral.

• The

re a

re 3

acu

te a

ngle

s, s

o th

e tr

iang

le is

acu

te.

Cla

ssify

eac

h tr

iang

le. W

rite

isosc

eles

, sca

lene

, or eq

uilateral.

The

n w

rite

right

, acu

te, o

r obtus

e.

1.

2

.

3.

4.

5

.

6.

Tria

ngle

Desc

riptio

nEx

ampl

e

equ

ilate

ral

3

equ

al s

ides

iso

scel

es

two

equa

l sid

es

sca

lene

n

o eq

ual s

ides

rig

ht

1 ri

ght a

ngle

acu

te

3 a

cute

ang

les

obt

use

1

obt

use

angl

e

9 ft

6 ft

7 ft

9 yd

9 yd

9 yd

12 c

m12

cm

4 cm

7 ft

5 ft

6 ft

8 in

.8

in.

8 in

.

12 m

3 m

8 m

6 yd

6 yd

9 yd

8 m

10 m

9 m

MX

EN

L08A

WK

4X_R

T_C

H19

_L5.

indd

A

6/13

/07

7:2

2:27

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

124

Clas

sify

Qua

drila

tera

lsA

qua

dri

late

ral i

s a

poly

gon

with

4 s

ides

and

4 a

ngle

s.

Cla

ssify

eac

h fig

ure

in a

s m

any

of

the

fo

llow

ing

way

s as

po

ssib

le. W

rite

qu

adrilateral

, parallelogram

, rho

mbu

s,

rectan

gle,

squ

are,

or trap

ezoid

.

• Th

e fig

ure

has

4 si

des,

so

it is

a

quad

rilat

eral

. It ha

s 2

pairs

of p

aral

lel s

ides

, so

it is

a p

aral

lelo

gram

. All

4 si

des

are

equa

l, so

it is

a r

hom

bus.

Dra

w a

n e

xam

ple

of

a q

uad

rila

tera

lth

at h

as n

o p

aral

lel s

ide

s.

• Th

is fi

gure

has

4 s

ides

and

non

e of

the

si

des

are

para

llel.

Cla

ssify

eac

h fig

ure

in a

s m

any

of

the

fo

llow

ing

way

s as

po

ssib

le. W

rite

qu

adrilateral

, parallelogram

, rho

mbu

s, rec

tang

le, s

quare,

or trap

ezoid

.

1.

2.

3.

Dra

w a

n e

xam

ple

of

eac

h q

uad

rila

tera

l.4.

It

has

2 p

airs

of p

aral

lel

side

s. 5

. It

has

2 e

qual

sid

es.

6.

It h

as o

nly

1 pa

ir of

pa

ralle

l sid

es.

Quad

rilat

eral

Desc

riptio

n

para

llelo

gram

- 2 p

airs

of p

aral

lel s

ides

- opp

osite

sid

es a

re e

qual

squ

are

- 2 p

airs

of p

aral

lel s

ides

- 4 e

qual

sid

es- 4

righ

t ang

les

rec

tang

le- 2

pai

rs o

f par

alle

l sid

es- o

ppos

ite s

ides

are

equ

al- 4

righ

t ang

les

rho

mbu

s- 2

pai

rs o

f par

alle

l sid

es- 4

equ

al s

ides

tra

pezo

id- e

xact

ly 1

pai

r of p

aral

lel

side

s

MX

EN

L08A

WK

4X_R

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A

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/07

7:2

9:24

AM

RW123-RW124


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

125

Circ

les

A c

ircl

e is

a c

lose

d fig

ure

mad

e up

of p

oint

s th

at a

re the

sam

e di

stan

ce fr

om the

cen

ter.

Circ

les

have

man

y pa

rts.

Co

nstr

uct

circ

le J

wit

h a

2-ce

ntim

ete

r ra

diu

s. L

abe

l eac

h o

f th

e f

ollo

win

g.

a. c

hord

: EF

b. d

iam

ete

r: B

C

• R

adiu

s JC

is 2

cm

.

• C

ircle

J h

as c

hord

EF

on it

. Cho

rd

EF is

a li

ne s

egm

ent w

hose

tw

o en

dpoi

nts

are

on the

circ

le.

• D

iam

eter

BC

pas

ses

thro

ugh

the

cent

er o

f the

circ

le.

Co

nstr

uct

circ

le P

wit

h a 3-

cent

ime

ter

rad

ius.

Lab

el e

ach

of

the

fo

llow

ing

. C

ons

truc

t ci

rcle

L w

ith

a 1-

inch

rad

ius.

La

be

l eac

h o

f th

e f

ollo

win

g.

1.

chor

d: M

K

2.

radi

us: P

B

3.

diam

eter

: AD

4.

chor

d: X

Y

5.

radi

us: L

A

6.

diam

eter

: CN

Part

of

Circ

leDe

scrip

tion

Exam

ple

chor

d

a lin

e se

gmen

t w

hose

two

endp

oint

s ar

e on

th

e ci

rcle

diam

eter

radi

us

B

A

chor

d: AB

diam

eter

: CD

radi

us: PK

CD

P PK

P

J

BE

F

C

a lin

e se

gmen

t with

on

e en

dpoi

nt a

t th

e ce

nter

and

on

e en

dpoi

nt o

n th

e ci

rcle

a ch

ord

that

pa

sses

thro

ugh

the

cent

er o

f the

ci

rcle

MX

EN

L08

AW

K4X

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7.in

dd

A6/

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:57:

02 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

126

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Com

pare

Str

ateg

ies

Ge

org

e w

ants

to

de

sig

n a

swim

min

g p

oo

l tha

t ha

s at

leas

t tw

o

sets

of

par

alle

l sid

es

and

at

leas

t tw

o s

ets

of

eq

ual s

ide

s.

Whi

ch o

f th

e s

wim

min

g p

oo

l dia

gra

ms

at t

he r

ight

wo

uld

loo

k lik

e G

eo

rge

’s d

esi

gn?

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2. W

hat st

rate

gy w

ould

you

use

to

solv

e th

is p

robl

em?

So

lve

3. C

ompl

ete

the

Ven

n di

agra

m a

nd tab

le b

elow

to

solv

e th

e pr

oble

m.

Use

Lo

gic

al R

eas

oni

ng

Mak

e a

Tab

le

4. W

hich

of t

he s

wim

min

g po

ol d

iagr

ams

wou

ld lo

ok li

ke G

eorg

e’s

desi

gn?

So

lve

.6.

P

ete’

s sa

ndbo

x ha

s at

leas

t 1

right

an

gle.

Whi

ch fi

gure

from

abo

ve lo

oks

like

the

sand

box?

7. A

pie

ce o

f fab

ric h

as a

t le

ast 1

set

of p

aral

lel s

ides

and

at le

ast 2

obtu

se a

ngle

s. W

hich

figu

res

from

ab

ove

mat

ch thi

s de

scrip

tion?

Fig

ure

AB

CD

E

2 s

ets

of p

aral

lel s

ides

yes

no

2 s

ets

of e

qual

sid

esye

sno

AC

B

2 se

tspa

ralle

lsi

des

2 se

tseq

ual

side

s

A

ED

Ch

eck

5. W

hich

str

ateg

y w

as m

ore

help

ful:

use

logi

cal r

easo

ning

or

mak

e a

tabl

e? W

hy?

MX

EN

L08

AW

K4X

_RT

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dd

A6/

7/07

3:

53:5

8 PM

RW125-RW126


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

127

Cong

ruen

t and

Sim

ilar F

igur

esC

ong

rue

nt f

igur

es

have

the

exa

ct s

ame

size

and

sha

pe.

Sim

ilar

figur

es

have

the

sam

e sh

ape

but di

ffere

nt s

izes

.

• Th

e fig

ures

bot

h ha

ve the

exa

ct s

ame

shap

e. T

his

mea

ns the

y ar

e si

mila

r.

• Th

e fig

ures

bot

h ha

ve the

exa

ct s

ame

size

. Sin

ce th

ey a

re a

lso

the

exac

t sam

e sh

ape,

the

figu

res

are

cong

ruen

t.

• So

, the

figu

res

are

both

co

ngru

ent a

nd

sim

ilar.

• Th

e fig

ures

are

the

sam

e sh

ape,

so

they

ar

e si

mila

r.

• B

oth

of the

figu

res

are

the

sam

e si

ze.

Bec

ause

the

y al

so h

ave

the

sam

e sh

ape,

th

ey a

re c

ong

ruen

t.

• So

, the

figu

res

are

both

co

ngru

ent a

nd

sim

ilar.

Tell

whe

the

r th

e t

wo

fig

ure

s ar

e c

ong

ruen

t and

sim

ilar,

sim

ilar,

or

neith

er.

1.2.

3.

4.5.

6.

7.8.

9.

Tell

whe

the

r th

e t

wo

fig

ure

s ar

e c

ong

ruen

t and

sim

ilar,

sim

ilar,

or

neith

er.

MX

EN

L08

AW

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dd

A6/

14/0

7 9

:59:

19 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

128

Turn

s and

Sym

met

ryIf

you

draw

a li

ne to

divi

de a

figu

re in

hal

f, an

d th

e tw

o ha

lves

are

id

entic

al to

one

anot

her,

then

the

figu

re h

as li

ne s

ymm

etr

y. If

a

figur

e ca

n be

rot

ated

aro

und

a ce

nter

poi

nt a

nd s

till l

ook

the

sam

e in

at le

ast tw

o po

sitio

ns, i

t ha

s ro

tati

ona

l sym

me

try.

• In

figur

e A

, the

bug

is e

xact

ly the

sam

e on

eac

h si

de. S

o, the

bug

has

line

sy

mm

etry

.

• In

figur

e B

, the

bug

has

bee

n ro

tate

d. It

lo

oks

diffe

rent

from

figu

re A

. So,

the

bug

do

es n

ot h

ave

rota

tiona

l sym

met

ry.

• So,

the

bug

has

line

sym

met

ry.

• In

figur

e A

, the

pen

ny d

oes

not lo

ok

exac

tly the

sam

e on

bot

h si

des

of the

lin

e. S

o, the

pen

ny d

oes

not ha

ve li

ne

sym

met

ry.

• In

figur

e B

, the

pen

ny h

as b

een

rota

ted.

It

now

look

s di

ffere

nt fr

om fi

gure

A. S

o, the

pe

nny

does

not

hav

e ro

tatio

nal

sym

met

ry.

• So,

the

ans

wer

is n

eith

er.

Tell

whe

the

r th

e f

igur

e a

pp

ear

s to

hav

e li

ne s

ymm

etry

, ro

tatio

nal s

ymm

etry

, bo

th,

or

neith

er.

1.2.

3.

4.5.

6.

7.8.

9.

Tell

whe

the

r th

e f

igur

e a

pp

ear

s to

hav

e li

ne s

ymm

etry

, ro

tatio

nal s

ymm

etry

, bo

th, o

r ne

ither

.

AB

AB

MX

EN

L08

AW

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2.in

dd

A6/

14/0

7 1

0:00

:23

AM

RW127-RW128


refle

ctio

n

tran

slat

ion

rota

tio

n

refle

ctio

ntr

ansl

atio

nro

tati

on

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

129

Tran

sfor

mat

ions

A t

rans

form

atio

n is

the

mov

emen

t of

a fi

gure

. The

re a

re thr

ee

kind

s of

tra

nsfo

rmat

ions

. A t

rans

lati

on

mov

es a

figu

re b

y sl

idin

g it

in a

str

aigh

t lin

e. A

ro

tati

on

mov

es a

figu

re b

y tu

rnin

g it

arou

nd a

si

ngle

poi

nt. A

re

flect

ion

mov

es a

figu

re b

y fli

ppin

g it

over

a

line. Te

ll ho

w t

he f

igur

e w

as m

ove

d.

Wri

te tran

slation,

rotatio

n, o

r refle

ction.

• If

the

figur

e w

as a

ref

lect

ion,

or

flipp

ed o

ver,

it w

ould

look

like

tw

o ar

row

s po

intin

g in

opp

osite

di

rect

ions

.

• If

the

figur

e w

as a

tra

nsla

tion,

the

ar

row

s w

ould

be

in d

iffer

ent

plac

es b

ut b

oth

poin

ting

in the

sa

me

dire

ctio

n.

• Th

e fig

ure

has

been

tur

ned,

or

rota

ted.

So,

the

figu

re is

a r

otat

ion.

Dra

w f

igur

es

to s

how

a t

rans

lati

on,

a r

ota

tio

n,

and

a r

efle

ctio

n o

f th

e f

igur

e.

Tell

how

eac

h fig

ure

was

mo

ved

. Wri

te tran

slation,

rotatio

n, o

r refle

ction.

1.2.

3.

Dra

w f

igur

es

to s

how

a t

rans

lati

on,

a r

ota

tio

n, a

nd a

re

flect

ion

of

the

fig

ure

.

4.

MX

EN

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dd

A6/

14/0

7 1

0:02

:21

AM

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

130

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy: A

ct it

Out

Bud

dy

use

d p

atte

rn b

lock

s to

mak

e t

he t

wo

rab

bit

s sh

ow

n at

the

rig

ht. A

re t

he r

abb

its

cong

rue

nt?

Exp

lain

ho

w y

ou

kno

w.

Re

ad

to

Un

de

rsta

nd

1.

Wha

t ar

e yo

u as

ked

to fi

nd o

ut in

thi

s pr

oble

m?

Pla

n

2.

How

wou

ld y

ou u

se the

act

it o

ut s

trat

egy

to s

olve

the

pro

blem

?

So

lve

3.

Solv

e th

e pr

oble

m. D

escr

ibe

the

stra

tegy

you

use

d.

4.

Are

the

rab

bits

con

grue

nt?

Ch

eck

5.

Wha

t ot

her

stra

tegy

cou

ld y

ou u

se to

solv

e th

e pr

oble

m?

Act

it o

ut t

o s

olv

e.

6.

Tim

mad

e th

e su

n be

low

usi

ng p

atte

rn

bloc

ks. D

oes

Tim

’s s

un h

ave

line

sym

met

ry?

7.

Mar

a m

ade

the

leaf

bel

ow u

sing

patter

n bl

ocks

. Doe

s M

ara’

s le

af

ha

ve li

ne s

ymm

etry

?

MX

EN

L08

AW

K4X

_RT

_CH

20_L

4.in

dd

A6/

14/0

7 1

0:04

:01

AM

RW129-RW130


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

131

Tess

elat

ions

A t

ess

ella

tio

n is

a r

epea

ting

patter

n of

clo

sed

figur

es tha

t co

vers

a

surf

ace.

The

re a

re n

o op

en s

pace

s or

gap

s in

a tes

sella

tion,

an

d no

ne o

f the

figu

res

in the

tes

sella

tion

ever

ove

rlap.

If a

figu

re

can

be u

sed

to m

ake

a te

ssel

latio

n, the

figu

re is

sai

d to

“t

esse

llate

.”

• Th

e po

inte

d pa

rt o

f the

figu

re fi

ts

perf

ectly

into

the

inde

nted

par

t. B

ecau

se

of thi

s, the

re a

re n

o ga

ps, a

nd the

figu

re

does

not

ove

rlap.

• So

yes

, the

figu

re w

ill tes

sella

te.

• W

hen

you

use

the

figur

e ab

ove

to try

to

mak

e a

tess

ella

tion,

the

re a

re g

aps

and

the

surf

ace

is n

ot c

over

ed.

• So

no

, the

figu

re w

ill n

ot tes

sella

te.

Tell

whe

the

r th

e f

igur

e w

ill t

ess

ella

te. W

rite

yes

or no

.

1.2.

3.

4.5.

6.

7.8.

9.

Tell

whe

the

r th

e f

igur

e w

ill t

ess

ella

te. W

rite

yes

or no

.

MX

EN

L08

AW

K4X

_RT

_CH

20_L

5.in

dd

A6/

7/07

9:

19:0

6 A

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

132

Geom

etric

Pat

tern

sG

eo

me

tric

pat

tern

s ar

e pa

tter

ns b

ased

on

colo

r, si

ze, s

hape

, pos

ition

, an

d nu

mbe

r of

figu

res.

A g

eom

etric

pat

tern

rep

eats

ove

r an

d ov

er a

gain

.

• The

pat

tern

use

s la

rge

and

smal

l circ

les.

Th

e la

rge

and

smal

l circ

les

alte

rnat

e: 1

la

rge,

1 s

mal

l, 1

larg

e, 1

sm

all,

and

so o

n.

• The

dot

on

each

pai

r of

circ

les

alte

rnat

es

betw

een

bein

g on

the

top

of e

ach

pair

of

circ

les

and

bein

g on

the

right

sid

e of

eac

h pa

ir of

circ

les.

• So,

the

nex

t tw

o fig

ures

wou

ld b

e:

• The

pat

tern

use

s la

rge

tria

ngle

s an

d sm

all

tria

ngle

s.

• Eac

h la

rge

tria

ngle

is fo

llow

ed b

y 3

smal

l tr

iang

les.

The

n th

e pa

tter

n re

peat

s ag

ain:

on

e la

rge

tria

ngle

, thr

ee s

mal

l tria

ngle

s,

and

so o

n.

• So,

the

nex

t tw

o fig

ures

wou

ld b

e:

Find

a p

oss

ible

pat

tern

. The

n d

raw

the

ne

xt t

wo

fig

ure

s in

yo

ur p

atte

rn

1.2.

3.4.

5.6.

Find

a p

oss

ible

pat

tern

. The

n d

raw

the

ne

xt t

wo

fig

ure

s in

yo

ur p

atte

rn.

MX

EN

L08A

WK

4X_R

T_C

H20

_L6.

indd

A

6/13

/07

8:4

5:25

AM

RW131-RW132


Face

s, Ed

ges,

and

Vert

ices

• T

he f

ace

is a

pol

ygon

tha

t is

a fl

at s

urfa

ce.

• T

he e

dg

e is

whe

re tw

o fa

ces

mee

t.

• A

ve

rte

x is

whe

re 3

or

mor

e ed

ges

mee

t.

• T

he b

ase

is the

flat

sur

face

on

whi

ch a

sol

id fi

gure

res

ts.

A c

ube

has

12 e

dges

.

A r

ect

ang

ular

pri

sm h

as 1

2 ed

ges.

A t

rian

gul

ar p

yram

id

has

6 ed

ges.

A s

qua

re p

yram

idha

s 8

edge

s.

A r

ect

ang

ular

p

yram

id h

as 8

edg

es.

Nam

e a

so

lid f

igur

e t

hat

has 8

ed

ge

s.

A s

quar

e py

ram

id a

nd a

rec

tang

ular

pyr

amid

bot

h ha

ve 8

edg

es.

Nam

e a

so

lid f

igur

e t

hat

is d

esc

rib

ed

.

1. 5

face

s2.

4

face

s3.

al

l squ

are

face

s

4. 8

edge

s5.

4

vert

ices

6. al

l rec

tang

ular

face

s

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

133

face

ed

ge

vert

ex

bas

e

MX

EN

L08

AW

K4X

_RT

_CH

21_L

1.in

dd

A6/

13/0

7 4

:01:

35 P

M

You

can

use

squ

are

dot pa

per

to d

raw

thr

ee-d

imen

sion

al fi

gure

s.

Wha

t so

lid f

igur

e d

oe

s th

e n

umb

er

cub

eat

the

rig

ht lo

ok

like

?

The

num

ber

cube

has

6 fa

ces

that

are

squ

are.

It h

as 1

2 ed

ges

and

8 ve

rtic

es. A

sol

id fi

gure

with

6 s

quar

e fa

ces,

12

edge

s an

d 8

vert

ices

is

a c

ube

or a

rec

tang

ular

pris

m.

Use

do

t p

ape

r to

dra

w t

he n

umb

er

cub

e.

Rem

embe

r: a

ll si

des

of a

cub

e ar

e th

e sa

me

leng

th.

Dra

w a

squ

are

Dra

w s

lant

ed li

ne s

egm

ents

fr

om the

ver

tices

.C

onne

ct the

end

poi

nts

and

draw

hid

den

lines

.

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

134

Draw

Fig

ures

2. W

hat so

lid fi

gure

doe

s

the

shoe

box

look

like

?

1

32

1. W

hat so

lid fi

gure

doe

s

the

soup

can

look

like

?

For1–

2, u

se d

ot

pap

er

to d

raw

eac

h fig

ure

.

MX

EN

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ddA

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/07

12:3

8:44

PM

RW133-RW134


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

135

Patt

erns

for S

olid

Fig

ures

A n

et

is a

pat

tern

tha

t ca

n be

fold

ed to

mak

e a

thre

e-di

men

sion

al fi

gure

.

Dra

w a

ne

t th

at c

an b

e c

ut t

o m

ake

a m

od

el o

f e

ach

solid

fig

ure

.

A

squ

are

pyra

mid

has

5 fa

ces.

The

base

of a

squ

are

pyra

mid

is a

squ

are.

Dra

w the

bas

e.

The

othe

r fa

ces

of a

squ

are

pyra

mid

are

tria

ngle

s.D

raw

the

oth

er 4

face

s of

the

sol

id fi

gure

.

A

cub

e ha

s 6

face

s.

The

base

of a

cub

e is

a s

quar

e.D

raw

the

bas

e.

Dra

w the

oth

er 5

face

s.

Dra

w a

ne

t th

at c

an b

e c

ut t

o m

ake

a m

od

el o

f e

ach

solid

fig

ure

.

1.

2.

3.

MX

EN

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:33:

00 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

136

top

view

Diff

eren

t Vie

ws o

f Sol

id F

igur

esY

ou c

an id

entif

y so

lid fi

gure

s by

the

way

the

y lo

ok fr

om d

iffer

ent vi

ews.

For

exa

mpl

e,

you

can

look

at th

e ob

ject

from

the

top

, fro

m the

fron

t, an

d fr

om the

sid

es.

Nam

e t

he s

olid

fig

ure

tha

t ha

s th

e f

ollo

win

g v

iew

s.

• Th

e to

p vi

ew s

how

s th

at the

figu

re h

as a

rec

tang

ular

bas

e.

Th

e cr

osse

d lin

es s

how

tha

t th

e si

des

com

e to

geth

er to

form

a p

oint

.

• Th

e fr

ont an

d si

de v

iew

s sh

ow tha

t th

e fa

ces

are

tria

ngle

s.

• A

sol

id fi

gure

with

a r

ecta

ngul

ar b

ase

and

tria

ngul

ar fa

ces

is a

rec

tang

ular

pyr

amid

.

So

, the

figu

re is

a r

ecta

ngul

ar p

yram

id.

Nam

e t

he s

olid

fig

ure

tha

t ha

s th

e f

ollo

win

g v

iew

s.

1.

2. 3.

top

view

fron

t vi

ewsi

de v

iew

top

view

fron

t vi

ewsi

de v

iew

top

view

fron

t vi

ewsi

de v

iew

fron

t vi

ewsi

de v

iew

MX

EN

L08

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4.in

dd

A6/

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7 4

:34:

47 P

M

RW135 RW136


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

137

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Mak

e a

Mod

elA

licia

use

d t

he f

ew

est

po

ssib

le c

ube

s to

mak

e a

bui

ldin

g w

hose

vi

ew

s ar

e s

how

n b

elo

w. H

ow

man

y cu

be

s d

id A

licia

use

?

top

view

fron

t vi

ewsi

de v

iew

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

So

lve

3. U

se c

entim

eter

cub

es to

build

the

top

vie

w. T

hen

add

mor

e cu

bes

to c

reat

e th

efr

ont vi

ew a

nd the

sid

e vi

ew. H

ow m

any

cube

s di

d A

licia

use

?

Ch

eck

4. H

ow c

an y

ou c

heck

you

r an

swer

to

the

prob

lem

?

Mak

e a

mo

de

l to

so

lve

.

5. J

ames

use

s ce

ntim

eter

cub

es to

mak

ea

rect

angu

lar

pris

m. T

he p

rism

is7

cube

s w

ide,

2 c

ubes

hig

h an

d 2

cube

s lo

ng. H

ow m

any

cube

s do

es J

ames

us

e?

6. P

am u

ses

bloc

ks to

mak

e a

squa

re.

Each

sid

e of

the

squ

are

is 4

blo

cks

long

and

one

blo

ck h

igh.

The

n Pa

m

pulls

out

4 b

lock

s in

the

cen

ter

of the

sq

uare

. How

man

y bl

ocks

are

left?

Pla

n

2. H

ow w

ould

you

use

the

mak

e a

mod

el s

trat

egy

to h

elp

solv

e th

e pr

oble

m?

MX

EN

L08

AW

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5.in

dd

A6/

13/0

7 4

:38:

28 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

138

Com

bine

and

Div

ide

Figu

res

You

can

mak

e ne

w fi

gure

s by

div

idin

g a

figur

e. Y

ou c

an a

lso

mak

e a

new

fig

ure

by c

ombi

ning

tw

o fig

ures

tog

ethe

r.

Dra

w a

pic

ture

to

sho

w t

he n

ew

fig

ure

.

Div

ide

a re

ctan

gle

to m

ake

two

tria

ngle

s.

Ste

p 1

: Dra

w a

rec

tang

le.

Ste

p 2

: Dra

w a

line

tha

t di

vide

s th

e

rect

angl

e in

hal

f.

Thin

k: D

raw

the

line

from

one

ver

tex

to

an

othe

r so

tha

t th

e re

ctan

gle

is

di

vide

d in

to tw

o tr

iang

les,

rat

her

th

an tw

o sq

uare

s.

Dra

w a

pic

ture

to

sho

w t

he n

ew

fig

ure

.

Com

bine

tw

o re

ctan

gles

to

mak

e a

squa

re.

Ste

p 1

: Dra

w a

rec

tang

le w

ith the

wid

th

tw

ice

as lo

ng a

s th

e le

ngth

.

Ste

p 2

: Dra

w a

sec

ond

rect

angl

e th

at

to

uche

s th

e fir

st o

ne. I

t sh

ould

be e

xact

ly the

sam

e si

ze a

s th

e

first

one

.

Thin

k: D

raw

the

rec

tang

les

side

by

side

,

so tha

t th

eir

long

er s

ides

are

touc

hing

. Thi

s fo

rms

a sq

uare

.

Dra

w a

pic

ture

to

sho

w t

he n

ew

fig

ure

.

1.

Com

bine

tw

o sq

uare

s to

mak

e a

rect

angl

e. 2

. D

ivid

e a

rect

angl

e to

mak

e a

tria

ngle

an

d a

trap

ezoi

d.

3.

Com

bine

tw

o tr

apez

oids

to

mak

e a

pa

ralle

logr

am.

4.

Div

ide

a sq

uare

to

mak

e 4

tria

ngle

s.

MX

EN

L08

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:49:

47 P

M

RW137 RW138


12

3in

ches

Mea

sure

Fra

ctio

nal P

arts

The

units

tha

t ar

e us

ually

use

d to

mea

sure

leng

th a

re in

ch (i

n.),

foot

(ft),

yar

d (y

d), a

nd m

ile (m

i).

Esti

mat

e t

o t

he n

ear

est

inch

. The

n m

eas

ure

to

the

ne

are

st 1

_ 2 i

nch.

• Est

imat

e th

e le

ngth

of t

he fo

wer

bud

to

the

near

est in

ch.

• The

leng

th o

f the

flow

er b

ud is

bet

wee

n 2

and

3 in

ches

.

• To

the

near

est in

ch, t

he fl

ower

bud

is 3

inch

es lo

ng.

• Mea

sure

the

flow

er b

ud to

the

near

est ha

lf in

ch.

• Th

e le

ngth

of t

he fl

ower

bud

is b

etw

een

2 1 _ 2 in

ches

and

3 in

ches

.

• The

leng

th o

f the

flow

er b

ud is

clo

ser

to 2

1 _ 2 in

ches

. Es

tim

ate

to

the

ne

are

st in

ch. T

hen

me

asur

e t

o t

he n

ear

est

1 _ 2 i

nch.

1.

2.

3.4.

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

139

So, t

he le

ngth

of t

he fl

ower

bud

to

the

near

est ha

lf in

ch is

2 1 _ 2 in

ches

.

MX

EN

L08

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1.in

dd

A6/

14/0

7 2

:17:

49 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

140

Alge

bra:

Cha

nge

Cust

omar

y Li

near

Uni

tsY

ou c

an u

se m

ultip

licat

ion

and

divi

sion

to

chan

ge fr

om o

ne u

nit

of le

ngth

to

anot

her.

You

mul

tip

ly in

ord

er to

chan

ge fr

om a

la

rger

uni

t to

a s

mal

ler

unit,

suc

h as

feet

to

inch

es. Y

ou d

ivid

e in

or

der

to c

hang

e fr

om a

sm

alle

r un

it to

a la

rger

uni

t, su

ch a

s fe

et

to y

ards

.

C

usto

mar

y U

nits

of

Leng

th

1

foo

t (f

t) 5

12

inch

es

(in.)

1

yard

(yd

) 5 3

fe

et

(ft)

, o

r 36

inch

es

(in.)

1

mile

(mi) 5

5,280

fe

et

(ft)

, o

r 1,76

0 ya

rds

(yd

)

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

1.

120

ft 5

j y

d 2

. 3

ft 5

j in

. 3

. 27

yd

5 j

ft

4.

48 in

. 5 j

ft 5

. 8

ft 5

j in

. 6

. 60

ft 5

j y

d

7.

13 ft

5 j

in.

8.

3,52

0 yd

5 j

mi

9.

2 m

i 5 j

ft

10.

75 y

d 5

j ft

11.

15,8

40 ft

5 j

mi

12.

312

in. 5

j ft

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

48 ft

5 j

yd

• Th

ink

abou

t ho

w the

tw

o un

its a

re r

elat

ed.

• 3

feet

5 1

yar

d.

• Th

ink:

sho

uld

I mul

tiply

or

divi

de?

• Fe

et a

re s

mal

ler

than

yar

ds, s

o di

vide

.

4

8

4 3

5

16

4

8 fe

et fe

et in

1 y

ard

to

tal y

ards

So, y

ou d

ivid

e to

sho

w 4

8 ft

5 1

6 yd

.

MX

EN

L08

AW

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2.in

dd

A6/

13/0

7 3

:23:

30 P

M

RW139-RW140


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

141

Wei

ght

The

cust

omar

y un

its fo

r m

easu

ring

wei

ght ar

e o

unce

s (o

z),

po

und

s (lb

), an

d to

ns (T

). Th

ere

are

16 o

unce

s in

one

pou

nd.

Ther

e ar

e 2,

000

poun

ds in

one

ton

. You

can

use

mul

tiplic

atio

nan

d di

visi

on to

chan

ge fr

om o

ne u

nit to

ano

ther

.

C

usto

mar

y U

nits

of

We

ight

1

po

und

(lb

) 5 1

6 o

unce

s (o

z)

1

ton

(T) 5

2,0

00 p

oun

ds

(lbs)

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

1.

2 T

5 j

lb 2

. 64

lb 5

j o

z 3

. 8,

000

lb 5

j T

4.

32 o

z 5

j lb

5.

3 lb

5 j

oz

6.

4 T

5 j

lb

7.

10,0

00 lb

5 j

T 8

. 64

oz

5 j

lb 9

. 2,

000

lb 5

j T

10.

144

oz 5

j lb

11.

7 T

5 j

lb12

. 11

lb 5

j o

z

13.

160

oz 5

j lb

14.

16,0

00 lb

5 j

T15

. 13

lb 5

j o

z

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.7

lb 5

j o

z

• Te

ll ho

w the

tw

o un

its a

re r

elat

ed.

• 1

poun

d 5

16

ounc

es.

• Th

ink:

sho

uld

I mul

tiply

or

divi

de?

• Po

unds

are

larg

er tha

n ou

nces

, so

mul

tiply

.

7

3

1

6

5 11

2

7

lb o

z in

1 lb

to

tal o

z

So, y

ou m

ultip

ly to

show

7 lb

5 1

12 o

z.

MX

EN

L08

AW

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3.in

dd

A6/

13/0

7 3

:24:

50 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

142

Cust

omar

y Ca

paci

tyC

apac

ity

tells

how

muc

h liq

uid

a co

ntai

ner

can

hold

. The

cus

tom

ary

units

for

mea

surin

g ca

paci

ty a

re t

eas

po

on

(tsp

), ta

ble

spo

on

(tb

sp),

cup

(c),

pin

t (p

t), q

uart

(qt)

, gal

lon

(g),

and

fluid

oun

ce (f

l oz)

.

1 cu

p =

8 fl

uid

ounc

es

1 ga

llon

= 4

quar

ts1

pint

= 2

cup

s 1

quar

t =

2 pi

nts

1 ta

bles

poon

= 3

teas

poon

s

Co

mp

lete

the

tab

le. C

hang

e t

he u

nits

.

• D

eter

min

e ho

w m

any

pint

s ar

e in

1 g

allo

n.

• 1

gal

lon

5 8

pin

ts

• M

ultip

ly o

r di

vide

by

8.

5

g 3

8 5

40

pt 5

6 pt

4 8

5

7 g

8

3 8

pt

5 6

4 pt

So, 5

g 5

40

pt, 7

g 5

56

pt, a

nd 8

g 5

64

pt.

Co

mp

lete

the

tab

le. C

hang

e t

he u

nits

.

• D

eter

min

e ho

w m

any

cups

ar

e in

1 q

uart

.

• 4

cup

s 5

1 q

uart

• M

ultip

ly o

r di

vide

by

4.

2

8 c

4 4

5 7

qt

4

0 c

4 4

5 1

0 qt

1

6 qt

3 4

5 6

4 c

So, 2

8 c

5 7

qt,

40 c

5 1

0 qt

, and

64

c 5

16

qt.

Co

mp

lete

eac

h ta

ble

. Cha

nge

the

uni

ts.

1.2.

3.4.

Cups

, c28

40j

Quar

ts, qt

jj

16

Gallo

ns, g

5j

8

Pint

s, pt

j56

j

Gallo

ns, g

57

8

Pint

s, pt

4056

64

Cups

, c28

4064

Quar

ts, qt

710

16

Gallo

ns, g

68

Quar

ts, qt

8

Teas

poon

s, t

615

Tabl

espo

ons,

s7

Cups

, c12

18

Pint

s, pt

7

Cups

, c64

96

Gallo

ns, g

1

MX

EN

L08

AW

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22_L

4.in

dd

A6/

13/0

7 3

:34:

32 P

M

RW141-RW142


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

143

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Com

pare

Str

ateg

ies

An

ang

ler

is a

pe

rso

n w

ho f

ishe

s fo

r fu

n. M

ia is

an

ang

ler.

M

ia c

aug

ht a

fis

h th

at w

eig

hed

6 p

oun

ds.

Ho

w m

any

oun

ces

did

the

fis

h w

eig

h?

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2. W

hich

str

ateg

y w

ould

you

use

to

solv

e th

is p

robl

em: m

ake

a ta

ble

or d

raw

a

diag

ram

? W

hy?

So

lve

3. H

ow w

ill y

ou s

olve

the

pro

blem

usi

ng the

str

ateg

y yo

u ch

ose?

Sho

w y

our

wor

k.

4. H

ow m

any

ounc

es d

id M

ia’s

fish

wei

gh?

Ch

eck

5. H

ow c

an y

ou c

heck

to

see

if yo

ur a

nsw

er is

cor

rect

?

Cho

ose

a s

trat

eg

y to

so

lve

.

6. M

ary’

s ju

mp

rope

is 7

feet

long

. How

m

any

inch

es lo

ng is

the

rop

e?

7. A

pol

e is

5 fe

et lo

ng. A

bat

on is

36

inch

es lo

ng. H

ow m

uch

long

er

is the

pol

e th

an the

bat

on?

MX

EN

L08

AW

K4X

_RT

_CH

22_L

5.in

dd

A6/

7/07

12

:59:

38 P

M

scho

olho

me

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

144

Met

ric L

engt

hSo

me

met

ric u

nits

of l

engt

h ar

e m

illim

ete

r (m

m),

cent

ime

ter

(cm

),d

eci

me

ter

(dm

), m

ete

r (m

), an

d ki

lom

ete

r (k

m).

A c

oin

is

abou

t 1

mill

imet

er thi

ck. Y

our

inde

x fin

ger

is a

bout

1 c

entim

eter

wid

e. A

CD

is a

little

mor

e th

an 1

dec

imet

er w

ide.

A b

aseb

all b

atis

abo

ut 1

met

er lo

ng. A

kilo

met

er is

abo

ut 4

city

blo

cks

long

.

Cho

ose

the

mo

st r

eas

ona

ble

uni

t o

f m

eas

ure

. Wri

te m

m, c

m, d

m, m

, or

km.

• T

he m

ap s

how

s a

long

dis

tanc

e be

twee

n sc

hool

and

hom

e, s

o km

wou

ld b

e th

e m

ost re

ason

able

uni

t of

mea

sure

.

• T

he p

ictu

re s

how

s a

shor

t di

stan

ce

acro

ss a

coi

n, s

o m

m o

r cm

wou

ld b

e th

e m

ost re

ason

able

uni

ts o

f mea

sure

.

Cho

ose

the

mo

st r

eas

ona

ble

uni

t o

f m

eas

ure

. Wri

te m

m, c

m, d

m, m

, or

km.

1.2.

3.4.

5.6.

MX

EN

L08

AW

K4X

_RT

_CH

22_L

6.in

dd

A7/

2/07

8:

24:2

5 A

M

RW143-RW144


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

145

Alge

bra:

Cha

nge

Met

ric L

inea

r Uni

tsTo

cha

nge

met

ric li

near

uni

ts, m

ultip

ly o

r di

vide

by

10

, 10

0, o

r 1

,00

0. Y

ou m

ulti

ply

in o

rder

to

chan

ge la

rger

uni

ts to

smal

ler

units

, suc

h as

met

ers

to c

entim

eter

s. Y

ou d

ivid

e in

ord

er to

chan

ge s

mal

ler

units

to

larg

er u

nits

, suc

h as

met

ers

to k

ilom

eter

s.

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

. 1

. 40

cm

5 j

mm

2.

500

mm

5 j

dm

3.

6 km

5 j

m

4.

5,00

0 cm

5 j

m 5

. 4

m 5

j d

m 6

. 20

0 m

m 5

j c

m

7.

600

cm 5

j d

m 8

. 8,

000

m 5

j k

m 9

. 30

dm

5 j

m

10.

7 dm

5 j

mm

11.

13 m

5 j

cm

12.

25 d

m 5

j c

m

Me

tric

Uni

ts o

f Le

ngth

1 ce

ntim

ete

r (c

m) 5

10

mill

ime

ters

(mm

)

1 d

eci

me

ter

(dm

) 5 10

cent

ime

ters

(cm

)

1 m

ete

r (m

) 5 1,000

mill

ime

ters

(mm

)

1 ki

lom

ete

r (k

m) 5

1,000

me

ters

(m)

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

566

mm

5 j

dm

• Thi

nk a

bout

how

the

tw

o un

its a

re r

elat

ed.

• 1 d

ecim

eter

5 1

00 m

illim

eter

s.

• Thi

nk: s

houl

d I m

ultip

ly o

r di

vide

?

• Mill

imet

ers

are

smal

ler

than

dec

imet

ers,

so

div

ide.

56

6

4 10

0

5 5

.66

mill

imet

ers

m

m in

1 d

m to

tal

d

ecim

eter

s

So, y

ou d

ivid

e to

sho

w 5

66 m

m 5

5.6

6 dm

.

MX

EN

L08

AW

K4X

_RT

_CH

22_L

7.in

dd

A6/

13/0

7 3

:38:

34 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

146

Mas

sM

ass

is h

ow m

uch

mat

ter

an o

bjec

t co

ntai

ns. T

wo

met

ric u

nits

of

mas

s ar

e th

e g

ram

(g) a

nd k

ilog

ram

(kg

). A

dol

lar

bill

has

a m

ass

of a

bout

1 g

ram

. A tex

tboo

k ha

s a

mas

s of

abo

ut 1

kilo

gram

. One

kilo

gram

is e

qual

to

1,00

0 gr

ams.

You

div

ide

in

orde

r to

cha

nge

gram

s to

kilo

gram

s. Y

ou m

ulti

ply

in o

rder

to

chan

ge k

ilogr

ams

to g

ram

s.

3 kg

5 j

g

• Thi

nk a

bout

how

the

tw

o un

its a

re

rela

ted.

• 1 k

ilogr

am 5

1,0

00 g

ram

s.

• Thi

nk: s

houl

d I m

ultip

ly o

r di

vide

?

• Kilo

gram

s ar

e la

rger

tha

n gr

ams,

so

mul

tiply

.

3

3 1,

000

5 3

,000

kilo

gram

s g

in 1

kg

to

tal

gr

ams

So, y

ou m

ultip

ly to

show

3 k

g 5

3,0

00 g

.

j k

g 5

10,00

0 g

• Thi

nk a

bout

how

the

tw

o un

its a

re r

elat

ed.

• 1 k

ilogr

am 5

1,0

00 g

ram

s.

• Thi

nk: s

houl

d I m

ultip

ly o

r di

vide

?

• Gra

ms

are

smal

ler

than

kilo

gram

s,

so d

ivid

e.

1

0,00

0

4 1

,000

5

10

gr

ams

g

in 1

kg

to

tal

ki

logr

ams

So, y

ou d

ivid

e to

sho

w 1

0,00

0 g

5 1

0 kg

.

1.

5,00

0 g

5 j

kg

2.

7

kg 5

j g

3.

j g

5 2

kg

4.

8 kg

5 j

g

5. 9,

000

g 5

j k

g 6

. 3,

000

g 5

j k

g

7.

11,0

00 g

5 j

kg

8.

22

,000

g 5

j k

g 9

. 13

kg

5 j

g

10.

j k

g 5

15,

000

g 1

1. 6

7,00

0 g

5 j

kg

12.

32 k

g 5

j g

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

MX

EN

L08

AW

K4X

_RT

_CH

22_L

8.in

dd

A6/

14/0

7 2

:28:

46 P

M

RW145-RW146


Met

ric C

apac

ityC

apac

ity

is the

am

ount

of l

iqui

d a

cont

aine

r ca

n ho

ld. T

he m

etric

un

its fo

r ca

paci

ty in

clud

e lit

er

(L) a

nd m

illili

ter

(mL)

. A d

ropp

er

hold

s ab

out 1

mill

ilite

r. A

n av

erag

e-si

zed

wat

er b

ottle

hol

ds a

bout

1

liter

. 1 li

ter

5 1

,000

mill

ilite

rs. Y

ou c

an u

se m

ultip

licat

ion

and

divi

sion

to

chan

ge m

etric

uni

ts o

f cap

acity

.

12,000

mL 5

j L

• Thi

nk a

bout

how

the

tw

o un

its

are

rela

ted.

• 1 li

ter

5 1

,000

mill

ilite

rs.

• Thi

nk: s

houl

d I m

ultip

ly o

r di

vide

?

• Mill

ilite

rs a

re s

mal

ler

than

lite

rs, s

o di

vide

.

12,0

00 4

1

,000

5

12

mill

ilite

rs m

L in

1 L

to

tal

li

ters

So, y

ou d

ivid

e to

sho

w 1

2,00

0 m

L 5

12

L.

3 L 5

j m

L

• Thi

nk a

bout

how

the

tw

o un

its

are

rela

ted.

• 1,0

00 m

illili

ters

5 1

lite

r

• Thi

nk: s

houl

d I m

ultip

ly o

r di

vide

?

• Lite

rs a

re la

rger

tha

n m

illili

ters

, so

mul

tiply

.

3

3

1,

000

5 3,

000

liter

m

L in

1 L

to

tal

m

illili

ters

So, y

ou m

ultip

ly to

show

3 L

5 3

,000

mL.

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

. 1

. 5

L 5

j m

L 2

. 2,

000

mL

5 j

L 3

. 7

L 5

j m

L

4.

4,00

0 m

L 5

j L

5.

8 L

5 j

mL

6.

9 L

5 j

mL

7.

12 L

5 j

mL

8.

15,0

00 m

L 5

j L

9.

10,0

00 m

L 5

j L

10.

25,0

00 m

L 5

j L

11.

6 L

5 j

mL

12.

11,0

00 m

L 5

j L

Co

mp

lete

. Te

ll w

heth

er

you multip

ly o

r divide

.

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

147

MX

EN

L08

AW

K4X

_RT

_CH

22_L

9.in

dd

A6/

13/0

7 3

:45:

18 P

M

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy: M

ake

a Ta

ble

Juan

is m

akin

g k

eyc

hain

s to

se

ll at

a c

raft

fai

r. H

e p

lans

to

sell

the

m f

or $4

eac

h. H

ow

muc

h m

one

y w

ill J

uan

mak

e if

he

se

lls 14

keyc

hain

s?

Re

ad

to

Un

de

rsta

nd

1. W

rite

the

ques

tion

as a

fill-

in-t

he-b

lank

sen

tenc

e.

Pla

n

2. H

ow c

an m

akin

g a

tabl

e he

lp y

ou s

olve

the

pro

blem

?

So

lve

3. C

ompl

ete

the

tabl

e to

sol

ve the

pro

blem

.

4. H

ow m

uch

mon

ey w

ill J

uan

mak

e if

he s

ells

14

keyc

hain

s? W

rite

your

ans

wer

in a

co

mpl

ete

sent

ence

.

Ch

eck

5. H

ow c

an y

ou c

heck

to

see

if yo

ur a

nsw

er is

cor

rect

?

Mak

e a

tab

le t

o s

olv

e.

6. D

ave

prac

tices

sho

otin

g fr

ee thr

ows

2 ou

t of

eve

ry 5

day

s. H

ow m

any

days

will

Dav

e ha

ve p

ract

iced

afte

r 40

sch

ool d

ays?

7. K

elly

sta

cks

CD

s on

her

shel

ves.

Eac

h sh

elf h

olds

15

CD

s.H

ow m

any

CD

s w

ill K

elly

hav

est

acke

d if

she

fille

d 5

shel

ves?

Key

rings

Sol

d1

24

68

1012

14

Am

ount

Ear

ned

$4$8

$16

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

148

MX

EN

L08

AW

K4X

_RT

_CH

22_L

10.in

dd

A6/

13/0

7 3

:44:

44 P

M

RW147-RW148


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

149

Estim

ate

and

Mea

sure

Per

imet

erPe

rim

ete

r is

the

dis

tanc

e ar

ound

a fi

gure

. You

can

use

str

ing

and

a ru

ler

to fi

nd the

per

imet

er o

f an

obje

ct.

Use

str

ing

to

est

imat

e a

nd m

eas

ure

the

pe

rim

ete

r o

f yo

ur d

esk

.

• Ta

ke a

long

pie

ce o

f str

ing

and

wra

p it

all t

he w

ay a

roun

d yo

ur d

esk.

Mak

e su

re tha

t th

e st

ring

is tig

ht a

gain

st the

sid

es o

f you

r de

sk.

If th

e st

ring

is too

loos

e, y

our

estim

ate

and

mea

sure

men

t w

ill b

e le

ss a

ccur

ate.

• M

ark

or c

ut the

str

ing

to s

how

the

per

imet

er o

f the

des

k.

• La

y th

e st

ring

on the

gro

und

so tha

t it

is s

trai

ght.

Now

est

imat

e: a

bout

how

man

y in

ches

long

is the

str

ing?

• M

easu

re the

str

ing

with

a r

uler

. Rec

ord

the

perim

eter

in in

ches

.

• A

typ

ical

sch

ool d

esk

has

a pe

rimet

er o

f abo

ut 9

6 in

ches

.

Find

the

pe

rim

ete

r o

f th

e f

igur

e a

t th

e r

ight

.

• Th

e sh

ape

is a

rec

tang

le.

Cou

nt the

num

ber

of u

nits

on

each

sid

e of

the

rect

angl

e.

• 4

units

1 3

uni

ts 1

4 u

nits

1 3

uni

ts 5

14

units

• So

, the

figu

re h

as a

per

imet

er o

f 14

units

.

Use

str

ing

to

est

imat

e a

nd m

eas

ure

the

pe

rim

ete

r o

f e

ach

ob

ject

.1.

a

text

book

2. a

fold

er3.

a

box

of c

rayo

ns

Find

the

pe

rim

ete

r o

f e

ach

figur

e.

4.

5.

6.

7.

8.

9.

MX

EN

L08

AW

K4X

_RT

_CH

23_L

1.in

dd

A6/

13/0

7 4

:53:

04 P

M

110

m

110

m

42 m

42

m

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

150

Alge

bra:

Fin

d Pe

rimet

er

You

can

find

the

per

imet

er o

f a fi

gure

by

addi

ng the

leng

ths

of

the

side

s, o

r by

usi

ng a

form

ula.

Find

the

pe

rim

ete

r.

• A

dd the

leng

ths

of a

ll of

the

sid

es.

• Th

e fig

ure

has

side

s th

at a

re 2

0 cm

, 16

cm

, 15

cm, a

nd 1

4 cm

long

.

• 20

1 1

6 1

15

1 1

4 5

65.

So, t

he p

erim

eter

of t

he fi

gure

is

65 c

m.

• U

se the

form

ula

P 5

(2

3 l)

1 (

2 3

w).

This

m

eans

Per

imet

er 5

(2

3 le

ngth

) 1

(2

3 w

idth

).

• S

ubst

itute

the

mea

sure

men

ts fo

r th

e le

ngth

and

w

idth

into

the

form

ula.

P

5 (2

3 l)

1 (

2 3

w)

P

5 (2

3 1

10)

1 (

2 3

42)

P

5 2

20 1

84

P

5 3

04

So, t

he p

erim

eter

of t

he fi

gure

is 3

04 m

.

Find

the

pe

rim

ete

r.1.

2.3.

4.5.

6.

MX

EN

L08

AW

K4X

_RT

_CH

23_L

2.in

dd

A6/

13/0

7 4

:55:

36 P

M

RW149-RW150


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

151

Area

of P

lane

Fig

ures

Are

a re

fers

to

how

man

y sq

uare

uni

ts a

re n

eede

d to

cov

er a

su

rfac

e.

• C

ount

the

squ

ares

.

• T

here

are

24

full

squa

res.

• Th

ere

are

4 al

mos

t-fu

ll sq

uare

s.

•Th

ere

are

6 ha

lf-fu

ll or

alm

ost ha

lf-fu

ll sq

uare

s.64

25

abo

ut 3

full

squa

res

• A

dd a

ll of

the

squ

ares

you

cou

nted

.

241

41

35

31

So, t

he a

rea

of the

figu

re is

abo

ut

31 s

quar

e ce

ntim

eter

s, o

r 31

sq

cm.

• C

ount

the

squ

ares

.

• T

here

are

16

full

squa

res.

• T

here

are

4 a

lmos

t-fu

ll sq

uare

s.

•Th

ere

are

8 ha

lf-fu

ll or

alm

ost ha

lf-fu

ll sq

uare

s.84

25

abo

ut 4

full

squa

res

• A

dd a

ll of

the

squ

ares

you

cou

nted

.

161

41

45

24

So, t

he a

rea

of the

figu

re is

abo

ut

24 s

quar

e ce

ntim

eter

s, o

r 24

sq

cm.

Esti

mat

e t

he a

rea

of

eac

h fig

ure

. Eac

h un

it is

1 s

q c

m.

1.2.

3.

4.5.

6.

Esti

mat

e t

he a

rea

of

eac

h fig

ure

. Eac

h un

it is

1 s

q c

m.

MX

EN

L08

AW

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_RT

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3.in

ddA

6/21

/07

12:3

7:18

PM

9 m

i

9 m

i

6 m

16 m

8 cm

8 cm

3 m

i

11 m

i

5 yd

2 yd

4 in

.

12 in

.7

ft

7 ft

13 k

m

10 k

m

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

152

Alge

bra:

Fin

d Ar

ea

You

can

use

mul

tiplic

atio

n an

d fo

rmul

as in

ord

er to

find

the

area

of

a fi

gure

. You

mus

t kn

ow the

figu

re’s

leng

th a

nd w

idth

.

Find

the

are

a.

• Id

entif

y th

e fig

ure’

s le

ngth

. The

leng

th is

th

e di

stan

ce a

cros

s th

e fig

ure.

The

leng

th

of the

figu

re is

9 m

iles.

• Id

entif

y th

e fig

ure’

s w

idth

. The

wid

th is

th

e di

stan

ce fr

om the

top

of t

he fi

gure

to

the

bottom

of t

he fi

gure

. The

wid

th o

f the

fig

ure

is 9

mile

s.

• M

ultip

ly the

leng

th b

y th

e w

idth

.

9

3 9

5 8

1

So, t

he a

rea

is 8

1 sq

uare

mile

s.

Find

the

are

a.

• U

se the

form

ula

A 5

l 3

w. T

his

mea

ns

to m

ultip

ly the

figu

re’s

leng

th b

y its

w

idth

.

• Th

e fig

ure

is 1

6 m

long

and

6

m w

ide.

A

5 l 3

w

A

5 1

6 3

6

A

5 9

6

Find

the

are

a.1.

2.3.

4.5.

6.

So, t

he a

rea

is 9

6 sq

m.

MX

EN

L08

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4.in

dd

A6/

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:24:

34 P

M

RW151-RW152


Nam

eNam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

153

Prob

lem

Sol

ving

Wor

ksho

p Sk

ill:

Use

a Fo

rmul

aA

dia

gra

m o

f R

yan’

s sa

nd c

astl

e is

sho

wn

at t

he

rig

ht. H

e d

eci

de

s to

line

the

pe

rim

ete

r o

f hi

s ca

stle

wit

h sh

ells

. If

eac

h sh

ell

is t

wo

inch

es

long

, ho

wm

any

she

lls d

oe

s R

yan

nee

d?

1. W

hat ar

e yo

u as

ked

to fi

nd?

2. W

hat is

the

mis

sing

leng

th o

f Rya

n’s

cast

le?

2

4 in

. 2 1

0 in

. 5

3. H

ow c

an y

ou u

se the

form

ula

belo

w to

find

the

perim

eter

of t

he c

astle

?

P

5 a

1 b

1 c

1 d

1 e

1 f

P

5

1

1

1

1

1

P 5

in

ches

4. H

ow c

an y

ou u

se the

per

imet

er o

f the

san

d ca

stle

to

find

the

num

ber

of s

hells

Rya

n ne

eds?

Sol

ve the

pro

blem

.

Use

the

fo

rmul

a to

so

lve

.

5. Th

e di

agra

m b

elow

sho

ws

asw

imm

ing

pool

. The

ow

ner

need

s to

plac

e 1-

inch

tile

s ar

ound

the

poo

l’spe

rimet

er. H

ow m

any

tiles

will

the

ow

ner

need

?

6. Th

e di

agra

m b

elow

sho

ws

afie

ld d

ivid

ed in

to fo

ur s

ectio

ns.

Wha

t is

the

are

a of

Sec

tion

C?

7 in

.7

in.

20 in

.

10 in

.

24 in

.

10 in

.24 in

.

10 in

.

18 in

.

10 in

.

28 in

.

?

10 m

2 m

C4

m

6 m

MX

EN

L08

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_RT

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5.in

dd

A6/

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7 5

:28:

34 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

154

Rela

te P

erim

eter

and

Are

aTh

e p

eri

me

ter

and

are

a of

a fi

gure

are

rel

ated

. Som

etim

es w

hen

you

chan

ge the

per

imet

er, t

he a

rea

also

cha

nges

. Sim

ilarly

, you

m

ay c

hang

e th

e pe

rimet

er o

f a fi

gure

whe

n yo

u ch

ange

the

are

a.

Find

the

pe

rim

ete

r an

d a

rea

of

the

fig

ure

s at

the

rig

ht.

The

n d

raw

ano

the

r fig

ure

tha

t ha

s th

e s

ame

pe

rim

ete

r b

ut a

diff

ere

nt a

rea.

• 9

1 1

1 9

1 1

5 2

0Th

e fig

ure’

s pe

rimet

er is

20

mi.

• 9

3 1

5 9

The

figur

e’s

area

is 9

squ

are

in.

• Th

e fig

ure

you

draw

mus

t ha

ve a

per

imet

er o

f 20,

and

an a

rea

that

is n

ot e

qual

to

9.

• 10

1 1

2 1

10

1 1

2 5

44

The

figur

es p

erim

eter

is 4

4 m

m.

• 12

3 1

0 5

120

The

figur

e’s

area

is 1

20 m

m.

• So

, the

figu

re y

ou d

raw

mus

t ha

ve a

n ar

ea o

f 120

, an

d a

perim

eter

not

equ

al to

44.

1.2.

3.

Find

the

pe

rim

ete

r an

d a

rea

of

eac

h fig

ure

. The

n d

raw

ano

the

r fig

ure

th

at h

as t

he s

ame

pe

rim

ete

r b

ut a

diff

ere

nt a

rea.

9 in

.1

in.

6 m

8 m

8 ft

8 ft

4 cm

6 cm

Find

the

per

imet

er o

f eac

h fig

ure.

The

n dr

aw a

noth

er

figur

e th

at h

as the

sam

e ar

ea b

ut a

diff

eren

t pe

rimet

er.

10 m

m

12 m

m

MX

EN

L08

AW

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6.in

dd

A7/

2/07

8:

25:0

8 A

M

RW153-RW154


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

155

Estim

ate

and

Find

Vol

ume

of P

rism

s V

olu

me

is the

am

ount

of s

pace

a s

olid

figu

re o

ccup

ies.

Vol

ume

is

mea

sure

d in

cub

ic u

nits

, suc

h as

cub

ic in

ches

, cub

ic

cent

imet

ers,

and

cub

ic fe

et. Y

ou c

an fi

nd the

vol

ume

of a

sol

id

figur

e by

cou

ntin

g or

mul

tiply

ing.

Co

unt

or

Mul

tip

ly

• Cou

nt a

ll of

the

cub

es tha

t ar

e us

ed to

mak

e th

e pr

ism

.

• The

re a

re tot

al o

f 8 c

ubes

.

• The

pris

m is

6 u

nits

long

, 4 u

nits

wid

e,

and

3 un

its h

igh.

• Mul

tiply

leng

th 3

wid

th 3

hei

ght.

6

3

4

3

3 5

72

So, t

he fi

gure

has

a v

olum

e of

72

cub

ic u

nits

.

Co

unt

or

mul

tip

ly t

o f

ind

the

vo

lum

e.

1.2.

3.

4.5.

6.

64

3

So, t

he p

rism

has

a v

olum

e of

8

cubi

c un

its.

MX

EN

L08

AW

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_RT

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7.in

dd

A6/

14/0

7 2

:51:

38 P

M

6

32

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

156

List

All

Poss

ible

Out

com

esU

se t

he t

able

.Li

st a

ll th

e p

oss

ible

out

com

es

for

the

exp

eri

me

nt.

• M

ake

an o

rgan

ized

list

.

• W

rite

the

poss

ible

out

com

es fo

r th

e co

in: h

eads

, tai

ls.

• W

rite

the

poss

ible

out

com

es fo

r th

e nu

mbe

r cu

be: 1

, 2, 3

, 4, 5

, 6.

• Pa

ir th

e po

ssib

le o

utco

mes

of t

he c

oin

with

the

pos

sibl

e ou

tcom

es o

f the

num

ber

cube

. Hea

ds, 1

; Hea

ds, 2

; Hea

ds, 3

; Hea

ds, 4; H

eads

, 5; H

eads

, 6;

Tails

, 1; T

ails

, 2; T

ails

, 3; T

ails

, 4; T

ails

, 5; T

ails

, 6.

So, t

he p

ossi

ble

outc

omes

for

toss

ing

the

coin

and

rol

ling

the

num

ber

cube

are

H

eads

, 1; H

eads

, 2; H

eads

, 3; H

eads

, 4; H

eads

, 5; H

eads

, 6;

Tails

, 1; T

ails

, 2; T

ails

, 3; T

ails

, 4 T

ails

, 5; T

ails

, 6.

For

1, u

se t

he p

ictu

res

be

low

. Lis

t al

l the

po

ssib

le o

utco

me

s o

f th

e e

xpe

rim

ent

.

For 2

– 3

, use

the

tab

le b

elo

w.

Sal

ly’s

Exp

eri

me

ntR

oll

a N

umb

er

Cub

e a

nd T

oss

a C

oin

Num

ber

Coi

nsH

eads

Tails

12

34

56

Tod

d’s

Exp

eri

me

ntS

pin

the

Po

inte

r an

d T

oss

a C

oin

Num

ber

Coi

nsH

eads

Tails

12

34

5

2. H

ow m

any

poss

ible

out

com

es a

re the

re?

3. H

ow m

any

times

did

the

out

com

e Ta

ils, 5

occ

ur?

1. tos

sing

the

cub

e la

bele

d 1

to 6

and

sp

inni

ng the

poi

nter

on

the

spin

ner

MX

EN

L08

AW

K4X

_RT

_CH

24_L

1.in

dd

A6/

13/0

7 6

:34:

39 P

M

RW155-RW156


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

157

Prob

lem

Sol

ving

Wor

ksho

p St

rate

gy:

Mak

e an

Org

aniz

ed L

istY

uri i

s p

layi

ng a

gam

e u

sing

a c

oin

and

a s

pin

ner.

H

e t

oss

es

the

co

in a

nd s

pin

s th

e p

oin

ter.

Lis

t al

l th

e p

oss

ible

out

com

es.

Re

ad

to

Un

de

rsta

nd

1. W

hat ar

e yo

u as

ked

to fi

nd?

Pla

n

2. H

ow c

an m

akin

g an

org

aniz

ed li

st h

elp

you

solv

e th

is p

robl

em?

So

lve

3. M

ake

an o

rgan

ized

list

. Sho

w y

our

wor

k be

low

.

4. H

ow m

any

poss

ible

out

com

es a

re the

re?

Ch

eck

5. D

oes

the

answ

er m

ake

sens

e fo

r th

e pr

oble

m?

Expl

ain.

Mak

e a

n o

rgan

ize

d li

st t

o s

olv

e.

6. T

om s

pins

the

poi

nter

bel

ow a

nd tos

ses

a nu

mbe

r cu

be la

bele

d 1

to 6

. Lis

t al

l th

e po

ssib

le o

utco

mes

.

7. M

ary

spin

s th

e po

inte

r be

low

and

al

so tos

ses

a co

in. H

ow m

any

poss

ible

out

com

es a

re the

re?

head

s, 1

head

s, 2

,

,

, 1

,

, 3

ta

ils, 4

MX

EN

L08

AW

K4X

_RT

_CH

24_L

2.in

dd

A6/

13/0

7 6

:39:

06 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

158

Pred

ict O

utco

mes

of E

xper

imen

tsY

ou c

an p

redi

ct the

like

lihoo

d of

som

e ev

ents

. Som

e ev

ents

are

ce

rtai

n, w

hich

mea

ns the

y w

ill a

lway

s ha

ppen

. Som

e ev

ents

are

im

po

ssib

le, w

hich

mea

ns the

y w

ill n

ever

hap

pen.

An

even

t is

like

ly

if it

has

a gr

eate

r th

an e

ven

chan

ce o

f hap

peni

ng. A

n un

like

ly e

vent

ha

s a

less

tha

n ev

en c

hanc

e of

hap

peni

ng. T

wo

even

ts c

an b

e e

qua

lly li

kely

, whi

ch m

eans

tha

t th

ey h

ave

the

sam

e ch

ance

of

happ

enin

g.

Tell

whe

the

r e

ach

eve

nt is

like

ly, u

nlikely,

certain

, or im

poss

ible.

toss

ing

a nu

mbe

r gr

eate

r th

an 1

on

a cu

be la

bele

d 1

to 6

The

num

ber

cube

has

the

num

bers

1, 2

, 3, 4

, 5, a

nd 6

. Fiv

e of

the

se n

umbe

rs a

re g

reat

er tha

n 1,

so

it is

like

ly tha

t yo

u w

ill tos

s a

2, 3

, 4, 5

, or

6.

So, t

he e

vent

is li

kely

.

spin

ning

a m

ultip

le o

f 4 o

n a

spin

ner

with

4 e

qual

par

ts la

bele

d 4,

8, 1

2, a

nd 1

6

• Eac

h of

the

num

bers

on

the

spin

ner

is a

mul

tiple

of 4

. Th

eref

ore,

the

poi

nter

will

alw

ays

land

on

a m

ultip

le o

f 4.

• Eve

nts

that

will

alw

ays

happ

en a

re c

erta

in.

So, t

he e

vent

is c

erta

in.

Tell

whe

the

r th

e e

vent

is like

ly, u

nlikely,

certain

, or im

poss

ible

.

1. p

ullin

g a

red

mar

ble

from

a b

ag tha

t co

ntai

ns 6

gre

en m

arbl

es, 4

whi

te

mar

bles

, and

2 y

ello

w m

arbl

es

2. tos

sing

a n

umbe

r gr

eate

r th

an 2

on

a nu

mbe

r cu

be la

bele

d 1

to 6

3. s

pinn

ing

an o

dd n

umbe

r on

a s

pinn

er

with

four

equ

al p

arts

labe

led

3, 5

, 7, a

nd 9

4. p

ullin

g a

whi

te m

arbl

e fr

om a

bag

that

co

ntai

ns 1

2 gr

een

mar

bles

, 2 w

hite

m

arbl

es, a

nd 1

4 ye

llow

mar

bles

5. to

ssin

g a

num

ber

less

tha

n 6

on a

nu

mbe

r cu

be la

bele

d 1

to 6

6. s

pinn

ing

a m

ultip

le o

f 6 o

n a

spin

ner

with

five

equ

al p

arts

labe

led

21, 2

4, 3

1, 3

5, a

nd 4

7

MX

EN

L08

AW

K4X

_RT

_CH

24_L

3.in

dd

A6/

14/0

7 2

:55:

59 P

M

RW157-RW158


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

159

Prob

abili

ty a

s a F

ract

ion

You

can

use

num

bers

to

expr

ess

the

mat

hem

atic

al p

rob

abili

ty o

f an

even

t ha

ppen

ing.

If a

n ev

ent w

ill n

ever

hap

pen,

its

mat

hem

atic

al

prob

abili

ty is

0. I

f the

eve

nt is

cer

tain

to

happ

en, i

ts m

athe

mat

ical

pr

obab

ility

is 1

. All

othe

r m

athe

mat

ical

pro

babi

litie

s ar

e w

ritte

n as

fr

actio

ns. T

he n

umer

ator

tel

ls the

num

ber

of fa

vora

ble

outc

omes

, an

d th

e de

nom

inat

or tel

l the

num

ber

of p

ossi

ble

outc

omes

.

Use

the

sp

inne

r. T

he s

pin

ner

has

eq

ual s

ect

ions

. Wri

te t

he

pro

bab

ility

of

not

spin

ning

gra

y as

a f

ract

ion.

For 1–

3, u

se t

he s

pin

ner

be

low

. The

sp

inne

r ha

s e

qua

l se

ctio

ns. W

rite

the

p

rob

abili

ty a

s a

frac

tio

n.

For 4–

6, u

se t

he s

pin

ner

be

low

. The

sp

inne

r ha

s e

qua

l se

ctio

ns. W

rite

the

p

rob

abili

ty a

s a

frac

tio

n.

1.

spi

nnin

g w

hite

4

. n

ot s

pinn

ing

whi

te

2.

not

spi

nnin

g gr

ay

5.

spi

nnin

g gr

ay o

r w

hite

3.

spi

nnin

g gr

ay o

r do

tted

6

. n

ot s

pinn

ing

whi

te o

r bl

ack 5 __ 6

• The

re a

re 2

whi

te s

ectio

ns, 3

bla

ck s

ectio

ns, a

nd 1

gra

y se

ctio

n. S

o th

ere

are

6 po

ssib

le o

utco

mes

.

• A fa

vora

ble

outc

ome

wou

ld b

e to

land

on

a se

ctio

n th

at is

not

gr

ay. T

here

are

5 fa

vora

ble

outc

omes

: lan

ding

on

the

2 w

hite

se

ctio

ns a

nd o

n th

e 3

blac

k se

ctio

n’s.

• Rep

lace

the

info

rmat

ion

in the

frac

tion

belo

w.

P

roba

bilit

y of

not

gra

y 5

favo

rabl

e ou

tcom

es ( 2

whi

te, 3

bla

ck)

________________________________________

poss

ible

out

com

es (1

gra

y, 2

whi

te, 3

bla

ck) 5

So, t

he p

roba

bilit

y of

not

spi

nnin

g gr

ay is

5 _ 6 .

MX

EN

L08

AW

K4X

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24_L

4.in

dd

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:42:

33 P

M

Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

160

The

exp

eri

me

ntal

pro

bab

ility

of a

n ev

ent ca

n be

foun

d by

cond

uctin

g re

peat

ed tria

ls. C

ompa

re the

num

ber

of tim

es the

eve

ntac

tual

ly o

ccur

s to

the

tot

al n

umbe

r of

tim

es y

ou r

epea

t th

e ac

tivity

.U

se t

he s

pin

ner

at t

he r

ight

and

the

tab

le b

elo

w.

The

sp

inne

r ha

s e

qua

l se

ctio

ns.

Mary

ell

en

’s R

esu

lts

Outc

omes

Whi

teGr

ayBl

ack

Spot

ted

Tally

Wha

t is

the

exp

erim

enta

l pro

babi

lity

of the

poi

nter

not

sto

ppin

g on

gra

y?•

To

find

the

exp

erim

enta

l pro

babi

lity,

look

at th

e re

sults

in M

arye

llen’

s ta

lly tab

le.

• ex

perim

enta

l pro

babi

lity:

nu

mbe

r of

tim

es t

he p

oint

er d

oes

not

stop

on

gray

__

____

____

____

____

____

____

____

____

_

tota

l num

ber

of t

imes

Mar

yelle

n sp

ins

the

poin

ter 5

32

__

40 , o

r 4 _ 5

So, t

he e

xper

imen

tal p

roba

bilit

y of

the

poi

nter

not

sto

ppin

g on

gra

y is

32

__

40 ,

or 4 _ 5

.

Wha

t is

the

mat

hem

atic

al p

roba

bilit

y of

the

poi

nter

not

sto

ppin

g on

gra

y?

•

To fi

nd t

he m

athe

mat

ical

pro

babi

lity,

look

at

the

poss

ible

out

com

eson

the

spi

nner

.

Expe

rimen

tal P

roba

bilit

y

• m

athe

mat

ical

pro

babi

lity:

So, t

he m

athe

mat

ical

pro

babi

lity

of t

he p

oint

er n

ot s

topp

ing

on g

ray

is 3 _ 4

.

Use

the

num

be

r cu

be

and

the

tab

le b

elo

w.

1. W

hat is

the

exp

erim

enta

l pro

babi

lity

of tos

sing

4?

Wha

t is

the

mat

hem

atic

al

prob

abili

ty?

2. W

hat is

the

exp

erim

enta

l pro

babi

lity

of tos

sing

an e

ven

num

ber?

Wha

t is

the

mat

hem

atic

al

prob

abili

ty?

Resu

lts

Outc

omes

12

34

56

Tally

6

32

favo

rabl

e ou

tcom

es (b

lack

, whi

te, s

pott

ed)

____

____

____

____

____

____

____

_

to

tal p

ossi

ble

outc

omes

(bla

ck, w

hite

, gre

y) 5

3 _ 4

MX

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:47:

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M

RW159-RW160


Nam

e

© H

arco

urt •

Gra

de 4

Ret

each

RW

161

Com

bina

tions

and

Arr

ange

men

tsY

ou c

an u

se t

ree

dia

gra

ms

to fi

nd c

ombi

natio

ns a

ndar

rang

emen

ts. A

co

mb

inat

ion

is a

cho

ice

in w

hich

the

ord

er d

oes

not m

atte

r. Fo

r ex

ampl

e, in

a b

lue

shirt

/bla

ck p

ants

com

bina

tion,

the

orde

r in

whi

ch y

ou li

st the

tw

o ite

ms

does

not

mat

ter.

In a

nar

rang

em

ent

, tho

ugh,

the

ord

er o

f the

item

doe

s m

atte

r.

Mak

e a

tre

e d

iag

ram

to

list

and

fin

d t

he n

umb

er

of

po

ssib

le c

om

bin

atio

ns.

Sun

dae

Cho

ice

sFl

avo

r: v

anill

a, s

traw

berr

y, c

hoco

late

Top

pin

g: c

aram

el, c

herr

y

• L

ist th

e fla

vor

choi

ces

in the

left

colu

mn.

• L

ist th

e to

ppin

g ch

oice

s in

the

cen

ter

colu

mn.

• D

raw

line

s to

con

nect

eac

h fla

vor

to

each

top

ping

.

• W

rite

each

com

bina

tion

in the

thi

rdco

lum

n.

• C

ount

the

tot

al n

umbe

r of

com

bina

tions

• T

here

are

a tot

al o

f 6 d

iffer

ent co

mbi

natio

ns.

Mak

e a

tre

e d

iag

ram

to

list

and

fin

d t

he n

umb

er

of

po

ssib

le c

om

bin

atio

ns.

Flav

orTo

ppin

gCo

mbi

natio

ns

vani

llaca

ram

el

cher

ry

vani

lla, c

aram

el

vani

lla, c

herr

y

stra

wbe

rry

cara

mel

cher

ry

stra

wbe

rry,

car

amel

stra

wbe

rry,

che

rry

choc

olat

eca

ram

el

cher

ry

choc

olat

e, c

aram

el

choc

olat

e, c

herr

y

1. S

alad

cho

ices

Lett

uce

: ice

berg

, rom

aine

Dre

ssin

g: r

anch

, gar

lic, F

renc

h

2. D

ecor

atin

g ch

oice

sR

ug C

olo

r: b

row

n, g

ray,

tan

Wal

l Co

lor:

whi

te, t

an, y

ello

w, g

reen

MX

EN

L08

AW

K4X

_RT

_CH

24_L

6.in

dd

A6/

13/0

7 6

:49:

45 P

M

RW161


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