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RETEACH Workbook
Grade 5
HS
PH
SP
Visit The Learning Site!www.harcourtschool.com
Title_CR_NLG5.indd 1Title_CR_NLG5.indd 1 6/18/07 5:19:59 PM6/18/07 5:19:59 PM
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ISBN 13: 978-0-15-356801-5 ISBN 10: 0-15-356801-1
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NL_SEGr5.indd template1NL_SEGr5.indd template1 6/12/07 4:08:16 PM6/12/07 4:08:16 PM
UNIT 1: USE WHOLE NUMBERS
Chapter 1: Place Value, Addition, and Subtraction1.1 Place Value Through Millions ........... RW11.2 Understand Billions ........................... RW21.3 Compare and Order Whole
Numbers ............................................. RW31.4 Round Whole Numbers .................... RW41.5 Estimate Sums and
Differences ........................................ RW51.6 Add and Subtract Whole
Numbers ............................................. RW61.7 Problem Solving Workshop
Strategy: Work Backward ................. RW7
Chapter 2: Multiply Whole Numbers2.1 Mental Math: Patterns in
Multiples ............................................ RW82.2 Estimate Products .............................. RW92.3 Multiply by 1-Digit Numbers .......... RW102.4 Multiply by Multi-Digit Numbers ........................................... RW112.5 Problem Solving Workshop
Strategy: Find a Pattern .................. RW122.6 Choose a Method ............................ RW13
Chapter 3: Divide by 1- and 2-Digit Divisors3.1 Estimate with 1-Digit
Divisors ............................................. RW143.2 Divide by 1-Digit Divisors................ RW153.3 Problem Solving Workshop
Skill: Interpret the Remainder ........ RW163.4 Zeros in Division .............................. RW173.5 Algebra: Patterns in
Division ............................................ RW183.6 Estimate with 2-Digit
Divisors ............................................. RW193.7 Divide by 2-Digit Divisors................ RW203.8 Correcting Quotients ...................... RW213.9 Practice Division .............................. RW223.10 Problem Solving Workshop
Skill: Relevant or Irrelevant Information .................... RW23
Chapter 4: Expressions and Equations4.1 Write Expressions ............................ RW244.2 Evaluate Expressions ....................... RW254.3 Properties ......................................... RW264.4 Mental Math: Use the Properties ......................................... RW274.5 Write Equations ............................... RW284.6 Solve Equations ............................... RW294.7 Functions .......................................... RW304.8 Inequalities ...................................... RW314.9 Problem Solving Workshop
Strategy: Predict and Test ............... RW32
UNIT 2: USE DECIMALS
Chapter 5: Understand Decimals5.1 Decimal Place Value ........................ RW335.2 Equivalent Decimals ........................ RW345.3 Compare and Order Decimals ........ RW355.4 Problem Solving Workshop
Skill: Draw Conclusions ................... RW36
Chapter 6: Add and Subtract Decimals6.1 Round Decimals ............................... RW376.2 Add and Subtract Decimals ............ RW386.3 Estimate Sums and Decimals ......... RW396.4 Choose a Method ............................ RW406.5 Problem Solving Workshop Skill:
Estimate or Find Exact Answer ....... RW41
Chapter 7: Multiply Decimals7.1 Model Multiplication by
a Whole Number ............................. RW427.2 Algebra: Patterns in Decimal
Factors and Products ....................... RW437.3 Record Multiplication by
a Whole Number ............................. RW447.4 Model Multiplication by
a Decimal ......................................... RW457.5 Estimate Products ............................ RW467.6 Practice Decimal Multiplication ..... RW477.7 Problem Solving Workshop
Skill: Multistep Problems ................ RW48
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Chapter 8: Divide Decimals by Whole Numbers8.1 Decimal Division .............................. RW498.2 Estimate Quotients.......................... RW508.3 Divide Decimals by Whole
Numbers ........................................... RW518.4 Problem Solving Workshop Skill:
Evaluate Answers for Reasonableness ............................... RW52
UNIT 3: DATA AND GRAPHING
Chapter 9: Data and Statistics9.1 Collect and Organize Data ............. RW539.2 Mean, Median, and Mode .............. RW549.3 Compare Data ................................. RW559.4 Analyze Graphs ............................... RW569.5 Problem Solving Workshop
Strategy: Draw a Diagram .............. RW57
Chapter 10: Make Graphs10.1 Make Bar Graphs and
Pictographs ...................................... RW5810.2 Make Histograms ................... .........RW5910.3 Algebra: Graph Ordered Pairs ........ RW6010.4 Make Line Graphs ........................... RW6110.5 Make Circle Graphs ......................... RW6210.6 Problem Solving Workshop
Strategy: Make a Graph.................. RW6310.7 Choose the Appropriate Graph...... RW64
UNIT 4: NUMBER THEORY AND FRACTION CONCEPTS
Chapter 11: Number Theory11.1 Multiples and the Least
Common Multiple ........................... RW6511.2 Divisibility ........................................ RW6611.3 Factors and Greatest
Common Factor ............................... RW6711.4 Prime and Composite Numbers...... RW6811.5 Problem Solving Workshop
Strategy: Make an Organized List ................................. RW69
11.6 Introduction to Exponents ............. RW7011.7 Exponents and Square Numbers .... RW7111.8 Prime Factorization ......................... RW72
Chapter 12: Fraction Concepts12.1 Understand Fractions ...................... RW7312.2 Equivalent Fractions ........................ RW7412.3 Simplest Form .................................. RW7512.4 Understand Mixed Numbers .......... RW7612.5 Compare and Order Fractions
and Mixed Numbers ........................ RW7712.6 Problem Solving Workshop
Strategy: Make a Model ................. RW7812.7 Relate Fractions and Decimals........ RW79
UNIT 5: FRACTION OPERATIONS
Chapter 13: Add and Subtract Fractions13.1 Add and Subtract Like Fractions .... RW8013.2 Model Addition of Unlike
Fractions ........................................... RW8113.3 Model Subtraction of Unlike
Fractions ........................................... RW8213.4 Estimate Sums and Differences ...... RW8313.5 Use Common Denominators .......... RW8413.6 Problem Solving Workshop
Strategy: Compare Strategies ......... RW8513.7 Choose a Method ............................ RW86
Chapter 14: Add and Subtract Mixed Numbers14.1 Model Addition of Mixed Numbers ........................................... RW8714.2 Model Subtraction of Mixed Numbers ........................................... RW8814.3 Record Addition and Subtraction .. RW8914.4 Subtraction with Renaming ........... RW9014.5 Practice Addition and Subtraction . RW9114.6 Problem Solving Workshop
Strategy: Use Logical Reasoning .... RW92
Chapter 15: Multiply and Divide Fractions15.1 Model Multiplication of Fractions . RW9315.2 Record Multiplication of Fractions RW9415.3 Multiply Fractions and Whole
Numbers ........................................... RW9515.4 Multiply with Mixed Numbers ....... RW9615.5 Model Fraction Division .................. RW9715.6 Divide Whole Numbers
by Fractions...................................... RW9815.7 Divide Fractions ............................... RW9915.8 Problem Solving Workshop
Skill: Choose the Operation.......... RW100
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UNIT 6: RATIO, PERCENT, AND PROBABILITY
Chapter 16: Ratios and Percent16.1 Understand and Express Ratios .... RW10116.2 Algebra: Equivalent Ratios and
Proportions ........................................RW10216.3 Ratios and Rates ............................ RW10316.4 Understand Maps and Scales........ RW10416.5 Problem Solving Workshop
Strategy: Make a Table ................. RW10516.6 Understand Percent ...................... RW10616.7 Fractions, Decimals, and Percents RW10716.8 Find Percent of a Number..............RW108
Chapter 17: Probability17.1 Outcomes and Probability ............ RW10917.2 Probability Experiments ................ RW11017.3 Probability and Predictions .......... RW11117.4 Problem Solving Workshop
Strategy: Make an Organized List ............................... RW112
17.5 Tree Diagrams ................................ RW11317.6 Combinations and
Arrangements ................................ RW114
UNIT 7: GEOMETRY AND ALGEBRA
Chapter 18: Geometric Figures18.1 Points, Lines, and Angles .............. RW11518.2 Measure and Draw Angles ........... RW11618.3 Polygons ......................................... RW11718.4 Problem Solving Workshop
Skill: Identify Relationships .......... RW11818.5 Circles ............................................. RW11918.6 Congruent and Similar Figures..... RW12018.7 Symmetry ....................................... RW121
Chapter 19: Plane and Solid Figures19.1 Classify Triangles ........................... RW12219.2 Classify Quadrilaterals................... RW12319.3 Draw Plane Figures ....................... RW12419.4 Solid Figures .................................. RW12519.5 Problem Solving Workshop
Strategy: Compare Strategies ....... RW12619.6 Nets for Solid Figures .................... RW12719.7 Draw Solid Figures from
Different Views ............................. RW128
Chapter 20: Patterns20.1 Transformations ............................ RW12920.2 Tessellations ................................... RW13020.3 Create a Geometric Pattern.......... RW13120.4 Numeric Patterns ........................... RW13220.5 Problem Solving
Workshop Strategy: Find a Pattern ................................ RW133
Chapter 21: Integers and the Coordinate Plane21.1 Algebra: Graph Relationships ...... RW13421.2 Algebra: Equations and
Functions ........................................ RW13521.3 Problem Solving
Workshop Strategy: Write an Equation ......................... RW136
21.4 Understand Integers ..................... RW13721.5 Compare and Order Integers ....... RW13821.6 Algebra: Graph Integers on
the Coordinate Plane .................... RW139
UNIT 8: MEASUREMENT
Chapter 22: Customary and Metric Measurements22.1 Customary Length ......................... RW14022.2 Metric Length ................................ RW14122.3 Change Linear Units ...................... RW14222.4 Customary Capacity
and Weight .................................... RW14322.5 Metric Capacity and Mass ............. RW14422.6 Problem Solving Workshop
Skill: Estimate or Actual Measurement ................................ RW145
22.7 Elapsed Time .................................. RW14622.8 Temperature .................................. RW147
Chapter 23: Perimeter23.1 Estimate and Measure Perimeter ....................................... RW14823.2 Find Perimeter ............................... RW14923.3 Algebra: Perimeter
Formulas ........................................ RW15023.4 Problem Solving Workshop Skill:
Make Generalizations ................... RW15123.5 Circumference ............................... RW152
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Chapter 24: Area and Volume24.1 Estimate Area ............................... RW15324.2 Algebra: Area of Squares
and Rectangles ............................. RW15424.3 Algebra: Relate Perimeter
and Area ....................................... RW15524.4 Algebra: Area of Triangles .......... RW15624.5 Algebra: Area of
Parallelograms ............................. RW15724.6 Problem Solving Workshop
Strategy: Solve a Simpler Problem ........................................ RW158
24.7 Surface Area ................................. RW15924.8 Algebra: Estimate and
Find Volume ................................. RW16024.9 Relate Perimeter, Area,
and Volume .................................. RW16124.10 Problem Solving Workshop
Strategy: Compare Strategies ..... RW162
G5-NL TOC_TEnew.indd 4G5-NL TOC_TEnew.indd 4 6/29/07 11:19:04 AM6/29/07 11:19:04 AM
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Grade5.indd 2Grade5.indd 2 6/18/07 5:51:08 PM6/18/07 5:51:08 PM
10
3,88
3
1 7
1,00
0
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
5
Estim
ate
Sum
s and
Diff
eren
ces
You
can
est
imat
e t
o fin
d an
ans
wer
tha
t is
clo
se t
o th
e ex
act
answ
er.
Use
co
mp
atib
le n
umb
ers
to e
stim
ate.
Com
patib
le n
umbe
rs a
re e
asy
to c
ompu
te m
enta
lly.
Esti
mat
e u
sing
co
mp
atib
le n
umb
ers
.10
3,88
3 1
71,
852
Esti
mat
e b
y ro
und
ing
.
1.
2.
10.
11.
12.
13.
14.
29
4,32
2
1 1
63,5
82
__
1
92
5,46
1
2 1
73,5
09
__
2
2
47,7
38
2
21,5
59
__
Thin
k: 1
04 1
72
is e
asy
to a
dd m
enta
lly, s
o10
4,00
0 1
72,
000
are
good
com
patib
lenu
mbe
rs t
o us
e fo
r an
est
imat
e.
72
,543
1 2
9,58
3
__
529,
617
2 2
37,1
50
__
$
63,8
95
1 3
7,22
8
__
745,
556
2 1
32.8
81
__
773,
645
1 1
35,7
10
__
223
,873
1 7
8,90
5
__
$
8,42
3
2 1
,825
__
56,1
08
1 4
2,33
6
__
95,2
23
1 1
03,2
29
__
554,
903
2 1
25,3
18
__
10
4,00
0
1 7
2,00
0
176,
000
8.Esti
mat
e b
y us
ing
co
mp
atib
le n
umb
ers
.
3.4.
5.6.
7.
1
9.26
3,77
6
1 6
13,8
86
MX
EN
L08A
WK
5X_R
T_C
H01
_L5.
indd
1
6/12
/07
10:
41:0
0 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
6
Add
and
Subt
ract
Who
le N
umbe
rsY
ou c
an u
se a
pla
ce-v
alue
cha
rt t
o he
lp y
ou a
dd o
r su
btra
ct.
Whe
n yo
u ad
d, y
ou fi
nd t
he s
um o
f tw
o or
mor
e nu
mbe
rs. W
hen
you
subt
ract
, you
fin
d th
e di
ffere
nce
of t
wo
num
bers
.
Inve
rse
op
era
tio
ns a
re o
pera
tions
tha
t un
do e
ach
othe
r. Th
e in
vers
e re
latio
nshi
p al
low
s yo
u to
che
ck a
dditi
on b
y us
ing
subt
ract
ion
and
to c
heck
sub
trac
tion
by u
sing
ad
ditio
n. millions
hundredthousands
tenthousands
thousands
hundreds
tens
ones
1 71 8
90
1 39
+3
25
15
5
11
14
19
4
Esti
mat
e. T
hen
find
the
sum
of
789,
039
1 3
25,1
55Es
timat
e: 8
00,0
00 1
300
,000
5 1
,100
,000
.
Star
t ad
ding
from
rig
ht t
o le
ft. R
egro
up a
s ne
eded
.
Firs
t ad
d th
e on
es. R
egro
up 1
4 on
es a
s 1
tens
and
4 o
nes.
Then
add
the
n te
ns.
Then
ad
the
hund
reds
.
Add
the
tho
usan
ds. R
egro
up 1
4 th
ousa
nds
as 1
ten
tho
usan
ds a
nd 4
tho
usan
ds.
Esti
mat
e. T
hen
find
eac
h su
m o
r d
iffe
renc
e.
1.
2.
3.
4.
5.
6.
7.
8.
9.
18,2
75 +
5,2
25 +
3,0
93
10.
2,70
5,24
3 –
1,19
2,01
3
11
. 5
00,6
01–
74,5
81
5,
382
1 8
,723
__
33,6
17
1 2
9,21
8
__
129,
336
1 6
47,2
73
__
306,
657
2 1
82,3
22
__
4,
678,
128
2 2
,119
,625
___
319,
007
2 2
27,2
42
__
603,
438
1 6
17,6
34
__
Then
add
the
ten
tho
usan
ds. R
egro
up 1
1 te
n th
ousa
nds
as 1
hun
dred
th
ousa
nds
and
1 te
n th
ousa
nd. L
ast
add
the
hund
red
thou
sand
s.
Reg
roup
11
hund
red
thou
sand
s as
1 m
illio
ns a
nd 1
hun
dred
tho
usan
ds.
1,11
4,18
4 is
clo
se t
o th
e es
timat
e of
1,0
00,0
00, s
o th
e an
swer
is r
easo
nabl
e.
48
7,01
8
1 2
41,3
93
__
259,
562
MX
EN
L08A
WK
5X_R
T_C
H01
_L6.
indd
A
6/12
/07
10:
48:3
7 A
M
RW5-RW6
Grade5.indd 3Grade5.indd 3 6/18/07 5:51:32 PM6/18/07 5:51:32 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
7
Prob
lem
Sol
ving
Wor
ksho
p S
trat
egy:
W
ork
Back
war
dO
n S
atur
day
s, S
aman
tha
has
stre
tchi
ng c
lass
fo
r 45
min
ute
s an
d b
alle
t fo
r an
ho
ur a
nd a
hal
f. A
fte
r a
30-m
inut
e b
reak
, she
has
jazz
cla
ss f
or
1 ho
ur, w
hich
is o
ver
at 1
:45
P.M
. At
wha
t ti
me
do
es
she
be
gin
?
Re
ad
to
Un
de
rsta
nd
1.
Wha
t do
es t
he p
robl
em a
sk y
ou t
o fin
d?
2.Pla
n
3.So
lve So
lve
the
prob
lem
. Use
the
spa
ce b
elow
to
show
how
you
wor
ked
back
war
d to
sol
ve t
he p
robl
em.
4.W
rite
your
ans
wer
in a
com
plet
e se
nten
ce.
Ch
eck
5.Is
the
re a
noth
er s
trat
egy
you
coul
d us
e to
sol
ve t
he p
robl
em?
Wo
rk b
ackw
ard
to
so
lve
.
The
scho
ol p
ep c
lub
spen
t $3
26 o
n su
pplie
s to
dec
orat
e th
e sc
hool
. The
y al
so s
pent
$85
to
prin
t ba
nner
s to
put
ab
ove
the
blea
cher
s. T
hey
now
hav
e $1
83 le
ft in
the
ir bu
dget
. How
muc
h m
oney
did
the
y st
art
with
in t
heir
budg
et?
From
200
6 to
200
7, t
he n
umbe
r of
st
uden
ts s
igne
d up
for
sum
mer
spo
rts
incr
ease
d by
635
. Fro
m 2
007
to 2
008
the
num
ber
incr
ease
d by
224
. In
2008
, 1,7
83
stud
ents
wer
e si
gned
up
for
sum
mer
sp
orts
. How
man
y st
uden
ts w
ere
sign
ed
up in
200
6?
7.6.
Wha
t st
rate
gy c
an y
ou u
se t
o so
lve
the
prob
lem
?
MX
EN
L08A
WK
5X_R
T_C
H01
_L7.
indd
A
6/12
/07
10:
51:1
0 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
8
Men
tal M
ath:
Pat
tern
s in
Mul
tiple
s
Find
the
pro
duc
t.80
3 1
00
You
can
use
bas
ic m
ultip
licat
ion
fact
s an
d pa
tter
ns t
o fin
d th
e pr
oduc
t w
hen
you
mul
tiply
by
a m
ultip
le o
f 10.
8 3
1
5
8 3
1 o
nes
5
8 o
nes
5
8
ba
sic
fact
8 3
10
5
8 3
1 t
ens
5
8 t
ens
5
80
ba
sic
fact
tim
es 1
0
8 3
100
5
8 3
1 h
undr
eds
5 8
0 hu
ndre
ds
5 8
00
ba
sic
fact
tim
es 1
00
80 3
100
5
80 3
1 h
undr
eds 5
80
hund
reds
5
8,0
00
Find
the
mis
sing
num
be
rs.
Find
the
pro
duc
t.
9.
2 3
80
10.
3 3
600
11.
60 3
50
12.
20 3
700
13.
9 3
4,0
00
14.
10 3
60
15.
50 3
5,0
0016
. 10
0 3
10
17.
7,00
0 3
40
18.
9,00
0 3
20
19.
80 3
6,0
0020
. 5 3
200
21.
800 3
80
22.
900 3
100
23.
300 3
500
24.
7 3
6,0
0025
. 8
3 3
0026
. 2
0 3
5,0
0027
. 80
0 3
400
28.
70 3
100
1.
5 3
3 5
5 3
30 5
5
3 3
00 5
2. 9
3 4
5
9
3 4
0 5
9
3 4
00 5
3. 6
3 2
5
6
3 2
0 5
6
3 2
00 5
4. 4
3 5
5
4
3 5
0 5
4
3 5
00 5
5. 2
3 9
5
2
3 9
0 5
2
3 9
00 5
6. 7
3 1
5
7
3 1
0 5
7
3 1
00 5
MX
EN
L08A
WK
5X_R
T_C
H02
_L01
.indd
1
6/12
/07
11:
11:3
5 A
M
RW7-RW8
Grade5.indd 4Grade5.indd 4 7/19/07 5:55:16 PM7/19/07 5:55:16 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
9
Estim
ate
Prod
ucts
You
can
use
rou
ndin
g nu
mbe
rs a
nd u
se m
ultip
licat
ion
fact
s to
es
timat
e pr
oduc
ts.
Esti
mat
e t
he p
rod
uct.
32
3 7
23
STE
P 1:
Rou
nd b
oth
fact
ors
to t
he g
reat
est
plac
e.
3
2 is
clo
ser
to 3
0 th
an 4
0.
32 r
ound
s to
30.
7
23 is
clo
ser
to 7
00 t
han
800.
72
3 ro
unds
to
700.
STE
P 2:
Use
bas
ic m
ultip
licat
ion
fact
s an
d pa
tter
ns t
o fin
d th
e pr
oduc
t of
the
rou
nded
fact
ors.
3
0 3
70
0 5
21,
000
Rem
embe
r:
If th
e di
git
to t
he r
ight
of
the
grea
test
pla
ce is
0–
4,
roun
d do
wn.
If
the
digi
t is
5–
9, r
ound
up.
Ro
und
eac
h fa
cto
r an
d e
stim
ate
the
pro
duc
t.
Esti
mat
e t
he p
rod
uct.
1. 5
2 3
31
2. 7
31 3
47
3. 6
3 3
43
4. 5
12 3
49
35
35
35
35
5.
456
3 7
66.
79
3 6
17.
53
3 1
,299
8. 2
6 3
725
9. 7
1 3
$9.
58
10.
44 3
260
11.
489
3 7
0612
. 3,4
85 3
59
13.
45 3
914
14.
38 3
4,1
18
32
3 7
23
MX
EN
L08A
WK
5X_R
T_C
H02
_L02
.indd
1
6/12
/07
11:
29:0
2 A
M
HT
OH
TO
HT
O
5
2 56 4 4
2 5
2 5 2
6 4 4
2 5 2
2 5 2
6 4 4
TT
T
2
Mul
tiply
by
1-Dig
it N
umbe
rsY
ou c
an u
se a
pla
ce-v
alue
cha
rt t
o he
lp y
ou m
ultip
ly b
y 1
-dig
it nu
mbe
rs.1
Esti
mat
e. T
hen
find
the
pro
duc
t55
6 3
4
Esti
mat
e:
600
3 9
5 2
,400
Ste
p 1
: M
ultip
ly t
he o
nes.
Ste
p 2
: M
ultip
ly t
he t
ens.
Ste
p 3
: M
ultip
ly t
he
hund
reds
.
4 3
6 o
nes
5 2
4 on
es24
one
s 5
2 t
en
4 on
es4
3 5
ten
s 5
20
tens
20 t
ens
1 2
ten
5 2
2 te
ns4
3 5
hun
dred
s 5
20
hund
reds
20 h
undr
eds
1 2
hun
dred
5
22 h
undr
eds
Rec
ord
the
4 in
the
on
es p
lace
. W
rite
the
2 ab
ove
the
tens
pla
ce.
Rec
ord
the
2 in
th
e te
ns p
lace
. W
rite
the
2 ab
ove
the
hund
reds
pl
ace.
Rec
ord
the
2 in
the
hu
ndre
ds p
lace
and
th
e ot
her
2 in
the
th
ousa
nds
plac
e.
Esti
mat
e. T
hen
find
the
pro
duc
t.
1.
42
3
3
2.
65
3
4
3.
6
04
3
5
4.
53
2
3
6
6.
46
4
3
77.
764
3
88.
1
,208
3
9
9.
3,0
45
3
8
10.
4,
365
3
6
5.
7
4 5
3
3
33
3
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
10
MX
EN
L08A
WK
5X_R
T_C
H02
_L03
.indd
1
6/12
/07
11:
31:0
6 A
M
RW9-RW10
Grade5.indd 5Grade5.indd 5 7/19/07 5:55:35 PM7/19/07 5:55:35 PM
Ste
p3:
A
dd t
he p
artia
lpr
oduc
ts.
TtTh
HT
OTt
ThH
TO
TtTh
HT
O
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
11
Mul
tiply
by
Mul
ti-Di
git N
umbe
rsY
ou c
an u
se a
pla
ce-v
alue
cha
rt a
nd r
egro
upin
g to
hel
p yo
u m
ultip
ly b
y 2-
digi
t nu
mbe
rs.
6883
67Es
tim
ate
:70
0 3
705
49,0
00
Ste
p1:
M
ultip
ly t
he o
nes.
Ste
p2:
Mul
tiply
the
ten
s.
Esti
mat
e. T
hen
find
the
pro
duc
t.
1.
253
3
17
2.
43
9
3
563.
$ 324
3
4
5
4.
576
3
43
5.
80
5
3
6 2
6.
28
7
3
387.
3
92
3 8
18.
4
66
3
2 9
9.
50
7
3 5
4
10.
1
89
3
86
11.
237 3
16 5
12.
407 3
28 5
13.
683 3
53 5
Thin
k:46
,096
is c
lose
to
the
est
imat
e of
49
,000
. So
the
answ
er
is r
easo
nabl
e.
Thin
k:603
6885
41,2
80
Thin
k:7 3
688
5 4
,816
6
5
6
8
8
3
6
7
4
8
16
4
1
2
8
0
6
8
8
3
6
7
4
8
1
6
4
6
0
9
6
4
1
2
8
0
6
8
8
3
6
7
4
8
1
6
6
5
5
4
6
5
5
4
Esti
mat
e. T
hen
find
the
pro
duc
t.
MX
EN
L08
AW
K5X
_RT
_CH
02_L
04.in
dd1
6/29
/07
3:09
:20
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
12
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Find
a P
atte
rnA
larg
e a
rt m
use
um is
pla
nnin
g a
sho
w o
f fa
mo
us o
il p
aint
ing
s.Th
e s
how
will
last
fo
r 4
day
s. If
38,
888
pe
op
le a
tte
nd t
he s
how
eac
h d
ay, h
ow
man
y p
eo
ple
will
att
end
the
sho
w in
all?
Re
ad
to
Un
de
rsta
nd
1.
Writ
e th
e qu
estio
n as
a fi
ll-in
-the
-bla
nk s
ente
nce.
Pla
n
2.
How
can
find
ing
a pa
tter
n he
lp y
ou s
olve
the
pro
blem
?
So
lve
3.
Solv
e th
e pr
oble
m. D
escr
ibe
the
stra
tegy
you
use
d.
5.
Writ
e yo
ur a
nsw
er in
a c
ompl
ete
sent
ence
.
Ch
eck
6.
How
can
you
che
ck y
our
answ
er?
Find
a p
atte
rn t
o s
olv
e.
7.
If 11
3 2
3 5
253
, 11 3
33 5
363
,11
3 3
5 5
385
, 11 3
42 5
462
, and
11 3
63 5
693
, wha
t is
the
mid
dle
digi
t in
the
pro
duct
of 1
1 an
d 81
?
8.
Wha
t is
the
sum
of t
he fi
rst
8 od
d nu
mbe
rs?
MX
EN
L08A
WK
5X_R
T_C
H02
_L05
.indd
1
6/12
/07
11:
33:2
5 A
M
RW11-RW12
Grade5.indd 6Grade5.indd 6 7/19/07 5:55:52 PM7/19/07 5:55:52 PM
This
is a
goo
d pr
oble
m
to s
olve
usi
ng a
ca
lcul
ator
bec
ause
it
requ
ires
two
calc
ulat
ions
and
use
s gr
eate
r nu
mbe
rs.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
13
Choo
se a
Met
hod
You
can
use
men
tal m
ath,
pap
er a
nd p
enci
l, or
a c
alcu
lato
r to
find
a p
rodu
ct.
You
can
use
me
ntal
mat
h to
sol
ve p
robl
ems
with
num
bers
th
at a
re e
asy
to c
ompu
te.
This
is a
goo
d pr
oble
m fo
r m
enta
l mat
h be
caus
e yo
u ca
n us
e th
e D
istr
ibut
ive
Prop
erty
to
com
pute
men
tally
.
You
can
use
pap
er a
nd p
enci
l to
solv
e pr
oble
ms
whe
n m
enta
l mat
h is
too
diff
icul
t.
This
is a
goo
d pr
oble
m t
o so
lve
usin
g pa
per
and
penc
il si
nce
a 3-
digi
t nu
mbe
r is
mul
tiplie
d by
a 2
-dig
it nu
mbe
r.
You
can
use
a c
alcu
lato
r to
sol
ve p
robl
em w
ith g
reat
er
num
bers
or
prob
lem
s w
ith m
ore
than
one
ste
p.
Mul
tiply
. 3 3
75 3
216
Find
the
pro
duc
t. C
hoo
se m
ent
al m
ath,
pap
er
and
pe
ncil,
or
a ca
lcul
ato
r.
30
5
3 2
4
453
3 6
27 5
,000
3
30
285
3 1
23 1
433
9
450
3
18 4
233
12
384
3 2
58 9
,000
3
30
702
3
8
13.
40 3
400
3 1
0 5
12.
54 3
37 3
19 5
11.
7 3
2,3
94 5
Mul
tiply
. 42
3 6
042
3 6
0 5
(40 1
2) 3
60
5
(40 3
60)
1 (2
3 6
0)
5
2,4
00 1
120
5
2,5
20
1.2.
3.4.
5.
6.7.
8.9.
10.
27
42
9
3
8
__
3,4
32
MX
EN
L08A
WK
5X_R
T_C
H02
_L06
.indd
1
6/12
/07
11:
36:3
1 A
M
Ret
each
Nam
e
RW
14©
Har
cour
t • G
rade
5
Estim
ate
with
1-Di
git D
iviso
rsC
om
pat
ible
num
be
rs a
re n
umbe
rs t
hat
are
easy
to
wor
k w
ith m
enta
lly. I
n di
visi
on, o
ne c
ompa
tible
num
ber
divi
des
even
ly in
to t
he o
ther
. Thi
nk o
f th
e m
ultip
les
of a
num
ber
to h
elp
you
find
com
patib
le n
umbe
rs.
Esti
mat
e t
he q
uoti
ent
.
Ste
p 1
: Th
ink
of t
he m
ultip
les
of 6
:
Fin
d m
ultip
les
that
are
clo
se t
o th
e fir
st 2
dig
its o
f the
div
iden
d.
18
and
24
are
both
clo
se t
o 21
. You
can
use
eith
er n
umbe
r, or
both
num
bers
to
estim
ate
the
quot
ient
.
Ste
p 2
: Es
timat
e us
ing
com
patib
le n
umbe
rs.
216
4 6
180 4
6 5
30
So, 1
80 4
6 is
abo
ut 3
0
216
4 6
240
4 6
5 4
0
So, 2
16 4
6 is
abo
ut 4
0
6
1
2
1
8
2
4
3
0
3
6
4
2
4
8
5
4
Esti
mat
e t
he q
uoti
ent
.
1.
2.
3.
4.
5
.
6.
7.
265
4 4
5 8
. 3
44 4
8 5
9.
372
4 5
5
10.
499
4 7
511
. 3
45 4
6 5
12.
189
4 8
5
6 q w
21
6
3 q w
25
2
6 q w
54
6
4 q w
25
4
5 q w
31
4
2 q w
15
7
8 q w
28
9
MX
EN
L08A
WK
5X_R
T_C
H03
_L1.
indd
1
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/07
11:
39:3
8 A
M
RW13-RW14
Grade5.indd 7Grade5.indd 7 7/19/07 5:56:10 PM7/19/07 5:56:10 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
15
Divi
de b
y 1-D
igit
Divi
sors
You
can
use
est
imat
ion
to h
elp
you
plac
e th
e fir
st d
igit
in t
he q
uotie
nt.
Then
, you
can
follo
w s
teps
to
divi
de.
Nam
e t
he p
osi
tio
n o
f th
e f
irst
dig
it o
f th
e q
uoti
ent
. The
n fin
d t
he f
irst
dig
it.
Div
ide.
Ste
p 1
: U
se e
stim
atio
n to
pla
ce t
he fi
rst
digi
t.S
tep
2:
Div
ide
the
10 h
undr
eds.
1,02
7 4
4D
ivid
e.
Mul
tiply
. 4 3
2 5
8
80
0 4
4 5
200
Subt
ract
. 10 2
8 5
2
So, t
he fi
rst
digi
t, 2,
is in
the
hun
dred
s pl
ace.
Com
pare
. 2 ,
4
Ste
p 3
: B
ring
dow
n th
e 2
tens
. The
n di
vide
the
22
tens
.
Ste
p 4
: B
ring
dow
n th
e 7
ones
. The
n di
vide
the
27
ones
.
Div
ide.
D
ivid
e.
Mul
tiply
. 4 3
5 5
20
Mul
tiply
. 4 3
6 5
24
Subt
ract
. 22 2
20 5
2Su
btra
ct. 2
7 2
24 5
3
Com
pare
. 2 ,
4C
ompa
re. 3
, 4
Writ
e th
e re
mai
nder
.
Nam
e t
he p
osi
tio
n o
f th
e f
irst
dig
it o
f th
e q
uoti
ent
. Th
en
find
the
fir
st d
igit
.
1.
2.
3.
4.
Div
ide
.4 q w
1,
027
4 q w
1,
027
2
8
222
2
0 27254 q
w
1,02
7
2
8
22
2
20 27 24 3256
2
4 q w
1,
027
8
22
2
4 q w
22
4 q
w
27
4 q w
10
8 q w
13
6
7 q w
29
7
5 q w
8,
126
7 q
w
4,97
3
3 q w
74
1
7 q w
45
6
4 q w
8,
659
6 q
w
5,23
8
4 q w
9,
449
MX
EN
L08A
WK
5X_R
T_C
H03
_L2.
indd
1
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/07
11:
44:0
3 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill:
Inte
rpre
t the
Rem
aind
erA
to
tal o
f 75
fift
h-g
rad
e s
tud
ent
s ar
e g
oin
g o
n a
field
tri
p t
o a
lo
cal p
ark.
The
sch
oo
l is
pro
vid
ing
min
ivan
s to
tak
e t
he
stud
ent
s to
the
par
k. If
eac
h m
iniv
an h
old
s 9
stud
ent
s, h
ow
m
any
min
ivan
s ar
e n
ee
de
d?
1.
Writ
e th
e qu
estio
n as
a fi
ll-in
-the
-bla
nk s
ente
nce.
2.
Wha
t is
the
quo
tient
? W
hat
is t
he r
emai
nder
, if t
here
is o
ne?
3.
How
wou
ld y
ou in
terp
ret
the
rem
aind
er?
Circ
le y
our
answ
er t
hen
expl
ain.
A
I
will
add
1 t
o th
e qu
otie
nt
B
I
will
use
the
quo
tient
and
writ
e th
e re
mai
nder
as
a fr
actio
n
C
I
will
onl
y us
e th
e re
mai
nder
D
I
will
onl
y us
e th
e qu
otie
nt
4.
How
man
y m
iniv
ans
are
need
ed fo
r th
e fie
ld t
rip t
o th
e pa
rk?
5.
How
can
you
che
ck t
o se
e if
your
ans
wer
is r
easo
nabl
e?
6.
Har
ry g
oes
on a
can
oe t
rip w
ith h
is s
cout
tr
oop.
The
y w
ill c
anoe
a t
otal
of 7
5 m
iles,
an
d w
ant
to t
rave
l 8 m
iles
each
day
. How
m
any
days
will
the
y ne
ed t
o tr
avel
the
en
tire
dist
ance
?
7.
Han
nah
and
her
fam
ily w
ant
to h
ike
8 m
iles
per
day
alon
g a
125-
mile
long
tr
ail.
How
man
y da
ys w
ill J
essi
e an
d he
r fa
mily
hik
e ex
actly
8 m
iles?
So
lve
the
fo
llow
ing
pro
ble
ms
and
the
n te
ll ho
w y
ou
wo
uld
inte
rpre
t th
e r
em
aind
er.
RW
16
9 q w
75
MX
EN
L08A
WK
5X_R
T_C
H03
_L3.
indd
1
6/12
/07
11:
45:0
7 A
M
RW15-RW16
Grade5.indd 8Grade5.indd 8 6/18/07 5:52:29 PM6/18/07 5:52:29 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
17
Zero
s in
Divi
sion
Whe
n yo
u ha
ve z
eros
in d
ivis
ion,
you
tre
at t
hem
the
sam
e w
ay y
ou w
ould
tre
at a
ny o
ther
di
git
whe
n di
vidi
ng.
Div
ide
.
Ste
p 1
: Use
est
imat
ion
to p
lace
the
firs
t di
git.
714
4 7
700
4 7
= 1
00
So,
the
firs
t di
git
is in
the
hun
dred
s pl
ace.
Ste
p 2
:
Div
ide
the
hund
reds
.
Ste
p 3
:
Brin
g do
wn
the
1 te
n.
Div
ide
the
tens
.
Ste
p 4
:
Div
ide
the
ones
.
1
< 7
, so
ther
e ar
e no
t en
ough
ten
s to
di
vide
. Writ
e 0
in
the
tens
pla
ce o
f th
e qu
otie
nt.
Mul
tiply
: 7 3
2 =
14
Subt
ract
: 14
– 1
4 =
0
Com
pare
: 0 <
7
71
0
27
____
_
01
1M
ultip
ly: 7
3 1
= 7
Subt
ract
: 7 –
7 =
0
Com
pare
: 0 <
7
02
So, 7
14 4
7 5
102
.
Div
ide
.
1.2.
3.4.
5.
6.7.
8.9.
10.
11.
605
4 3
12.
3,29
1 4
313
.84
6 4
714
.5,
420
4 5
27 0
14
214 0
20
____
_
0
7 q w
71
4
8 q w
87
2
3 q w
62
4
3 q w
92
7
5 q w
5,
412
4 q w
8,
126
3 q w
6,
024
8 q w
33
0
7 q w
84
3
2 q w
80
7
7 q w
71
4
7 q w
71
4
7 q w
1,
423
7 q w
71
4
MX
EN
L08A
WK
5X_R
T_C
H03
_L4.
indd
1
6/12
/07
11:
47:5
3 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
18
Alge
bra:
Pat
tern
s in
Divi
sion
You
can
use
bas
ic fa
cts
and
patt
erns
to
find
quot
ient
s.
Use
bas
ic f
act
pat
tern
s to
fin
d t
he q
uoti
ent
. 36
,000
4 6
0.
Ste
p 1
: Fin
d th
e ba
sic
fact
.
36,
000
4 6
0
The
bas
ic fa
ct is
36
4 6
= 6
.
Ste
p 2
: The
re a
re z
eros
in t
he d
ivid
end
and
the
div
isor
.
C
ance
l out
one
zer
o ea
ch.
36,
000
4 6
0
Ste
p 3
: Div
ide.
U
se t
he b
asic
fact
.
36,
000
4 6
0= 6
00
Use
bas
ic f
acts
and
pat
tern
s to
fin
d t
he q
uoti
ent
.
1.
40 4
2 2
. 16
0 4
8 3
. $2
70 4
90
4.
420
4 6
5. 5
00 4
50
6.
120
4 4
0 7
. 48
0 4
6 8
. 560
4 7
0
9.
210
4 3
10.
$300
4 1
011
. 63
0 4
90
12.
540
4 6
0
13.
6,30
0 4
714
. 6,
000
4 2
15.
3,00
0 4
30
16.
$4,5
00 4
50
17.
8,00
0 4
10
18.
1,40
0 4
719
. $2,
400
4 3
020
. 5,
600
4 8
MX
EN
L08A
WK
5X_R
T_C
H03
_L5.
indd
1
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/07
11:
50:0
4 A
M
RW17-RW18
Grade5.indd 9Grade5.indd 9 6/18/07 5:53:21 PM6/18/07 5:53:21 PM
Esti
mat
e t
he q
uoti
ent
.
Ste
p 1
: Roun
d t
he d
ivis
or
to t
he n
eare
st t
en.
49
roun
ds
to 5
0.
Ste
p 2
: Fin
d t
wo n
umber
s cl
ose
to t
he d
ivid
end t
hat
are
com
pat
ible
w
ith t
he r
oun
ded
div
isor.
427
is b
etw
een
400
and 5
00
Ste
p 3
: Div
ide
the
com
pat
ible
num
ber
s to
find
the
est
imat
es.
400
4 5
0 5
8
or
500
4 5
0 5
9
So, 8
and
9 a
re r
easo
nable
est
imat
es o
f the
quo
tient
.
Estim
ate
with
2-D
igit
Divi
sors
Com
pat
ible
num
ber
s ar
e nu
mber
s th
at a
re e
asy
to w
ork
with
men
tally
. In
div
isio
n, o
ne c
om
pat
ible
num
ber
div
ides
eve
nly
into
the
oth
er. T
hink
of t
he
mul
tiple
s of a
num
ber
to h
elp y
ou
find c
om
pat
ible
num
ber
s.
Wri
te t
wo
se
ts o
f co
mp
atib
le n
umb
ers
fo
r e
ach.
The
n g
ive
tw
op
oss
ible
est
imat
es.
1.
2.
3.
6. 4
09 4
63
7.
478
4 1
9 8
. 7,
145
Esti
mat
e t
he q
uoti
ent
.
11.
12.
13.
427
4 4
9
42
7 4
49
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
19
49 q w
42
7
42 q w
15
7
73 q
26
8
54 q w
34
3
81 q w
73
8
34 q w
29
3
12 q w
62
2
MX
EN
L08A
WK
5X_R
T_C
H03
_L6.
indd
1
6/12
/07
11:
51:4
4 A
M
Thin
k:
Mul
tiply
: 53
3 4
5 2
12
Subtr
act:
249
2 2
12 5
37
Com
par
e: 3
7 ,
53
Write
the
rem
aind
er t
o t
he r
ight
of t
he w
hole
nu
mber
par
t of t
he q
uotie
nt.
1.2.
3.
4.5.
6.
Divi
de b
y 2-
Digi
t Div
isors
You
can
use
estim
atio
n to
hel
p y
ou
pla
ce t
he fi
rst
dig
it in
the
quo
tient
. The
n, y
ou
can
follo
w
step
s to
div
ide.
Div
ide
.
Div
ide
. Che
ck y
our
ans
we
r.
Thin
k:
Mul
tiply
: 53
3 4
5 2
12
Subtr
act:
236
2 2
12 5
24
Com
par
e: 2
4 ,
53
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
20
Ste
p 2
: Div
ide
236.
50 q w
2,
000
40
So, t
he fi
rst
dig
it is
in t
he t
ens
pla
ce.
Ste
p 3
: Bring
dow
n th
e 9
one
s.
Then
div
ide
the
249
one
s.
So, 2
,369
4 5
3 5
44
r37.
Ste
p 1
: Use
est
imat
ion
to p
lace
the
firs
t dig
it.R
emem
ber
to u
se c
om
pat
ible
num
ber
s to
est
imat
e.
53 q w
2,
369
52 q w
61
2
63 q w
91
7
24 q w
60
8
43 q w
6,
413
27
q w
4,68
4
89 q w
1,
597
4
2 2
12 24
53 q w
2,
369
44 r
37
2 2
1 2 249
2 2
12 37
53 q w
2,
369
MX
EN
L08A
WK
5X_R
T_C
H03
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indd
1
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/07
11:
52:1
1 A
M
RW19-RW20
Grade5.indd 10Grade5.indd 10 6/18/07 5:53:31 PM6/18/07 5:53:31 PM
Corr
ectin
g Q
uotie
nts
Estim
ates
can
hel
p yo
u id
entif
y th
e fir
st d
igit
in t
he q
uotie
nt, b
ut s
omet
imes
you
will
nee
d to
co
rrec
t th
e qu
otie
nt.
Wri
te to
o h
igh,
too
low
, or
just
rig
ht f
or
the
est
imat
e b
elo
w.
Firs
t, us
e co
mpa
tible
num
bers
for
the
divi
dend
and
div
isor
.
16
2
0
416
400
Then
div
ide
usin
g yo
ur c
ompa
tible
div
iden
d an
d di
viso
r.
Nex
t, us
e th
e es
timat
ed q
uotie
nt t
o ch
eck
the
degr
ee o
f acc
urac
y.
96 >
16
96 is
muc
h gr
eate
r th
an 1
6.
So, f
or
t
he e
stim
ated
quo
tient
, 20,
is t
oo lo
w.
Div
ide
.
1.2.
3.4.
5.
6.7.
8.9.
10.
Wri
te lo
w, h
igh,
or
just
rig
ht f
or
eac
h e
stim
ate
.
206
8010
60
20
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
21
16 q w
41
6
20
2 3
2 962
0
9620
2 4
0 00
20 q w
40
0
16 q w
41
6
16 q w
41
6
58 q w
1,
325
37
q w
241
29
q w
2,27
6
82 q w
91
0
63 q w
3,
784
43 q w
9,
189
71
q w
488
37
q w
4,81
9
24 q w
21
7
16 q w
84
5
MX
EN
L08A
WK
5X_R
T_C
H03
_L8.
indd
1
6/12
/07
11:
57:0
9 A
M
Prac
tice
Divi
sion
Whe
n yo
u di
vide
, it
help
s to
rem
embe
r th
at d
ivis
ion
is a
n op
erat
ion
that
tel
ls t
he n
umbe
r of
eq
ual g
roup
s, o
r th
e nu
mbe
r in
eac
h eq
ual g
roup
.
Div
ide
Ste
p 1
: Use
est
imat
ion
to p
lace
the
firs
t di
git.
353
4 4
350
4 4
= 8
So,
the
firs
t di
git
is in
the
ten
s pl
ace.
Ste
p 2
: Div
ide
the
tens
.
M
ultip
ly: 8
3
4
= 3
2
Subt
ract
: 35
– 3
2 =
3
Com
pare
: 3 <
4
Ste
p 3
: Div
ide
the
ones
.
Mul
tiply
: 8 3
4 =
32
Subt
ract
: 33
– 3
2 =
1
Com
pare
: 1 <
4
Writ
e 1
as t
he r
emai
nder
.
8
So, 3
53 4
4 5
88
r1
1.
2
.
3
.
4.
5.
6.
219
4 7
7.
935
4 4
8.
6,1
21 4
5 9
. 9
,217
4 7
10.
8,
032
4 4
. M
ulti
ply
to
che
ck y
our
ans
we
r.
232
____
3 88
r1
Div
ide
. Mul
tip
ly t
o c
heck
yo
ur a
nsw
er.
2 3
2 332
32 1
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
22
4 q w
35
3
6 q w
11
5
9 q w
32
6
7 q w
2,
198
9 q
w
3,50
4
6 q w
3,
167
4 q w
35
3
4 q w
35
3
MX
EN
L08A
WK
5X_R
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indd
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/07
11:
59:2
3 A
M
RW21-RW22
Grade5.indd 11Grade5.indd 11 6/18/07 5:53:40 PM6/18/07 5:53:40 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
23
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill: R
elev
ant o
r Ir
rele
vant
Info
rmat
ion
A t
otal
of 4
8 fif
th g
rade
rs a
nd 4
tea
cher
s w
ent
on a
fiel
d tr
ip t
o th
e m
useu
m. T
he
tota
l cos
t fo
r th
e st
uden
ts’ t
icke
ts w
as $
576.
The
tot
al c
ost
for
the
teac
hers
’ tic
kets
w
as $
60. W
hat
was
the
pric
e fo
r ea
ch s
tude
nt t
icke
t?
1.
How
wou
ld y
ou w
rite
the
prob
lem
as
a fil
l in
the
bla
nk s
ente
nce?
2. W
hat
wou
ld y
ou c
ross
out
in t
he p
robl
em b
elow
, if y
ou w
ere
told
to
cros
s ou
t irr
elev
ant
info
rmat
ion?
Wha
t w
ould
you
circ
le, i
f you
wer
e to
ld t
o ci
rcle
rel
evan
t in
form
atio
n?
A
tot
al o
f 48
fifth
gra
ders
and
4 t
each
ers
wen
t on
a fi
eld
trip
to
the
. The
tot
al c
ost
for
the
stud
ents
’ tic
kets
w
as $
576.
The
tot
al c
ost
for
the
teac
hers
’ tic
kets
was
$60
. W
hat
was
the
pric
e fo
r ea
ch s
tude
nt t
icke
t?
3. H
ow c
ould
you
use
the
rel
evan
t in
form
atio
n in
the
spa
ce o
n th
e rig
ht t
o w
rite
an e
quat
ion
to s
olve
?
Tell
whi
ch in
form
atio
n is
re
leva
nt a
nd ir
rele
vant
to
so
lve
the
pro
ble
m. T
hen
solv
e.
5.
Shon
da e
arns
$35
per
wee
k he
lpin
g he
r
el
derly
nei
ghbo
r w
ith c
hore
s. H
er g
oal i
s to
ea
rn $
1,30
0.
She
save
s $2
5 pe
r w
eek
of
he
r ea
rnin
gs. S
he s
pend
s $1
0 at
the
sh
oppi
ng m
all w
ith h
er fr
iend
s. H
er
brot
her,
Ger
mai
ne, i
s sa
ving
$15
per
wee
k to
buy
an
MP3
pla
yer.
How
man
y w
eeks
m
ust
Shon
da s
ave
$25
to r
each
her
goa
l?
6.
Ove
r a
45-d
ay p
erio
d, M
r. G
reen
e dr
ove
1,
890
mile
s. H
e pa
id a
n av
erag
e of
$2
.57
for
gas
and
his
car
got
15 m
iles
to t
he g
allo
n. O
n av
erag
e, h
ow m
any
mile
s di
d M
r. G
reen
e dr
ive
each
day
?
4.
Wha
t w
as t
he p
rice
for
each
stu
dent
tic
ket?
MX
EN
L08A
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5X_R
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M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
24
Writ
e Ex
pres
sions
An
exp
ress
ion
has
num
bers
, ope
ratio
n si
gns,
and
som
etim
es v
aria
bles
. A
n ex
pres
sion
doe
s no
t ha
ve a
n eq
ual s
ign.
Firs
t, “7
plu
s”
7 1
Then
, “th
e qu
otie
nt o
f 72
divi
ded
by 9
”
72 4
9
So, 7
plu
s th
e qu
otie
nt o
f 72
divi
ded
by 9
7
1 (
72 4
9)
repr
esen
ts a
sum
.
An
alg
eb
raic
exp
ress
ion
is a
n ex
pres
sion
with
at
leas
t on
e va
riabl
e.
A v
aria
ble
is a
lett
er o
r sy
mbo
l tha
t st
ands
for
one
or m
ore
num
bers
.
Wri
te a
n al
ge
bra
ic e
xpre
ssio
n. T
ell
wha
t th
e v
aria
ble
re
pre
sent
s.
Dee
had
som
e bo
oks.
She
was
giv
en 9
mor
e bo
oks.
Firs
t, “s
ome
book
s”
va
riabl
e n
Then
, “gi
ven
9 m
ore
book
s”
1 9
So, “
Dee
had
som
e bo
oks.
Sh
e w
as g
iven
9 m
ore
book
s.”
can
be r
epre
sent
ed a
s
Use
clu
e w
ords
to
help
you
writ
e ex
pres
sion
s. F
or
exam
ple:
mor
e,
sum
, add
ed, a
nd
plus
indi
cate
ad
ditio
n.
Thin
k: It
is a
goo
d id
ea t
o us
e a
varia
ble
that
hel
ps
you
rem
embe
r w
hat
it re
pres
ents
. In
thi
s ca
se,
n =
boo
ks.
A n
ume
rica
l exp
ress
ion
has
only
num
ber
and
oper
atio
n si
gns.
Wri
te a
num
eri
cal e
xpre
ssio
n. T
ell
wha
t th
e e
xpre
ssio
n re
pre
sent
s.7
plus
the
quo
tient
of 7
2 di
vide
d by
9
n 1
9
let
n 5
the
num
ber
of b
ooks
Dee
had
.
Wri
te a
num
eri
cal e
xpre
ssio
n. T
ell
wha
t th
e e
xpre
ssio
n re
pre
sent
s.
1. H
ank
ran
14 m
iles
one
wee
k an
d 17
mile
s th
e ne
xt w
eek.
2. 5
2 m
ore
than
24
3. 6
plu
s th
e qu
otie
nt o
f 56
and
7
Wri
te a
n al
ge
bra
ic e
xpre
ssio
n. T
ell
wha
t th
e v
aria
ble
re
pre
sent
s.
4. K
elly
has
4 p
airs
of j
eans
. H
e gr
ew o
ut o
f 1 p
air
and
boug
ht s
ome
mor
e je
ans.
5. A
dam
hik
ed fo
r a
whi
le.
He
then
hik
ed fo
r 20
m
ore
min
utes
.
6. Q
uinn
has
thr
ee t
imes
m
ore
mar
kers
tha
n cr
ayon
s.
MX
EN
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5X_R
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/07
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09:4
3 P
M
RW23-RW24
Grade5.indd 12Grade5.indd 12 6/18/07 5:53:50 PM6/18/07 5:53:50 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
25
Eval
uate
Exp
ress
ions
To e
valu
ate
an
expr
essi
on, o
r to
find
the
val
ue o
f an
expr
essi
on, y
ou h
ave
to p
erfo
rm e
ach
oper
atio
n se
para
tely
to
det
erm
ine
the
answ
er.
Eval
uate
the
exp
ress
ion.
(6 4
3)
3 4
1 8
Rem
embe
r to
follo
w t
he o
rde
r o
f o
pe
rati
ons
.
• Pe
rfor
m t
he o
pera
tion
in p
aren
thes
is fi
rst:
(6 4
3) 3
4 1
8•
Mul
tiply
or
divi
de, f
rom
left
to r
ight
:
•
Add
or
subt
ract
, fro
m le
ft to
rig
ht:
So, (
6 4
3)
3 4
1 8
5 1
6.
Eval
uate
eac
h e
xpre
ssio
n.
1.
15 1
6 1
3 2
. 33
2 9
1 1
4 3
. (7
5 1
5)
2 8
4.
(16
3 2
) 2
10
5.
6 3
(63
4 7
) 6
. (8
1 1
11)
2 8
7.
(20
2 1
3) 3
5 8
. (2
1 1
42)
2 6
9.
3 3
(54
4 9
)10
. 38
2 (
18 4
6)
11.
64 2
(24
4 3
)12
. (1
3 2
7)
3 4
Eval
uate
the
alg
eb
raic
exp
ress
ion
for
the
giv
en
valu
e o
f th
e v
aria
ble
.
13.
2 1
(15
4 n
) if
n 5
314
. (1
6 3
w)
2 5
if
w 5
415
. (r
1 9
) 1
6
if r
5 7
16.
(96
4 1
2) 5
x
if
x 5
5
17.
(8 1
4)
3 n
if
n 5
218
. (2
2 1
k)
3 5
if
k 5
319
. z
1 (
8 3
1)
if z
5 1
8520
. m
__
6 1
14
if m
5 3
0
2
3 4
1 8
8
1 8
5 1
6
MX
EN
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e
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arco
urt •
Gra
de 5
Ret
each
RW
26
Prop
ertie
sEx
pres
sion
s us
ing
addi
tion
and
mul
tiplic
atio
n fo
llow
cer
tain
pro
pert
ies.
Ide
ntit
y Pr
op
ert
y
The
sum
of z
ero
and
any
num
ber
equa
ls
that
num
ber.
The
prod
uct
of o
ne a
nd a
ny
num
ber
equa
ls t
hat
num
ber.
5
1 0
5 5
3
3 1
5 6
Ass
oci
ativ
e P
rop
ert
y
The
way
add
ends
are
gro
uped
on
fact
ors
are
grou
ped
does
not
cha
nge
the
sum
or
prod
uct.
(9
1 3
) 1
2 5
9 1
(3
1 2
)
(7
3 4
) 3
5 5
7 3
(4
3 5
)
Co
mm
utat
ive
Pro
pe
rty
If th
e or
der
of t
he a
dden
ds o
r th
e fa
ctor
s is
cha
nged
, the
sum
or
prod
uct
stay
s th
e sa
me.
8 3
2 5
2 3
8
4 3
11
5 1
1 3
4
Ze
ro P
rop
ert
y o
f M
ulti
plic
atio
n
The
prod
uct
of a
ny n
umbe
r an
d ze
ro is
zer
o.
15 3
0 5
0
Find
the
val
ue o
f n.
Ide
ntify
the
pro
pe
rty
use
d.
81 3
n 5
81
Not
ice
that
the
pro
duct
and
the
fact
or o
ther
tha
n n
are
the
sam
e.
So, t
his
follo
ws
the
Iden
tity
Prop
erty
of M
ultip
licat
ion,
n 5
1.
Find
the
val
ue o
f n.
Ide
ntify
the
pro
pe
rty
use
d.
1.
27 3
(n
3 8
) 5
(27
3 9
) 3
82.
61
1 3
3 5
33
1 n
3.
6 3
n 5
85
3 6
4.
12 3
(6
3 8
) 5
(12
3 n
) 3
85.
(4
1 n
) 1
3 5
4 1
(7
1 3
)6.
n
3 1
20 5
0
MX
EN
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Grade5.indd 13Grade5.indd 13 6/18/07 5:54:01 PM6/18/07 5:54:01 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
27
Men
tal M
ath:
Use
the
Prop
ertie
sY
ou c
an u
se p
rope
rtie
s an
d m
enta
l mat
h to
hel
p yo
u so
lve
prob
lem
s.
The
Dis
trib
utiv
e P
rop
ert
y st
ates
tha
t yo
u ca
n br
eak
apar
t a
fact
or t
o m
ultip
ly.
You
can
use
add
ition
or
subt
ract
ion
to b
reak
apa
rt a
fact
or.
Use
pro
pe
rtie
s an
d m
ent
al m
ath
to f
ind
the
val
ue.
5 3
29
Use
the
Dis
trib
utiv
e Pr
oper
ty a
nd m
enta
l mat
h to
wor
k ou
t 5 3
29.
5 3
29
5 5
3 (
20 1
9)
5 (
5 3
20)
1 (
5 3
9)
5 1
00 1
45
5 1
45
So, 5
3 2
9 5
145
.
Use
pro
pe
rtie
s an
d m
ent
al m
ath
to f
ind
the
val
ue.
1.
27 1
26
1 3
3 2
. 4
3 2
7 3
. (2
4 1
19)
1 1
6
4.
7 3
5 3
3
5. 2
1 1
39
1 3
8 6
. 6
3 4
5
7.
3 3
9 3
2 8
. 3
3 3
60
9.
62 1
28
1 1
7
10.
7 3
3 3
9
11.
37 3
512
. 43
1 (
47 1
46)
13.
73 3
314
. 68
1 8
1 1
42
15.
12 3
4 3
5
MX
EN
L08A
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M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
28
Writ
e Eq
uatio
nsA
n e
qua
tio
n is
a n
umbe
r se
nten
ce t
hat
show
s th
at t
wo
quan
titie
s ar
e eq
ual.
Like
an
expr
essi
on, a
n eq
uatio
n ha
s nu
mbe
rs, o
pera
tion
sign
s,
and
som
etim
es v
aria
bles
. An
equa
tion
is d
iffer
ent
from
an
expr
essi
on
beca
use
an e
quat
ion
does
hav
e an
equ
al s
ign.
Wri
te a
n e
qua
tio
n. T
ell
wha
t th
e v
aria
ble
re
pre
sent
s.
Tia
has
5 fe
wer
key
s th
an O
mar
. If T
ia h
as 7
key
s, h
ow m
any
does
Om
ar h
ave?
Ste
p 1
Writ
e a
repr
esen
tativ
e se
nten
ce.
A n
umbe
r de
crea
sed
by
5 is
7.
Ste
p 2
Cho
ose
a va
riabl
e.
Let
k st
and
for
the
num
ber
of k
eys
Om
ar h
as.
Ste
p 3
Writ
e an
equ
atio
n.
k 2
5 =
7
So, t
he e
quat
ion
is
k
2 5
5 7
Let
k 5
num
ber
of k
eys
Om
ar h
as.
Wri
te a
n e
qua
tio
n fo
r e
ach.
Te
ll w
hat
the
var
iab
le r
ep
rese
nts.
1.
Bet
h’s
mot
her
mad
e 24
blu
eber
ry
muf
fins.
Bet
h’s
fam
ily a
te s
ome.
Now
th
ere
are
16 m
uffin
s le
ft. H
ow m
any
muf
fins
did
Bet
h’s
fam
ily e
at?
2. J
erro
d sa
ved
$85
to b
uy a
new
jack
et.
Afte
r he
bou
ght
the
jack
et h
e ha
d $1
8 le
ft. H
ow m
uch
did
the
jack
et c
ost?
3. R
yan
stud
ied
45 m
inut
es fo
r hi
s sc
ienc
e te
st. H
e st
udie
d 3
times
long
er fo
r hi
s sc
ienc
e te
st t
han
for
his
spel
ling
test
. H
ow lo
ng d
id R
yan
stud
y fo
r hi
s sp
ellin
g te
st?
4. M
rs. G
reen
e pl
ante
d 18
tom
ato
seed
lings
and
som
e cu
cum
ber
seed
lings
. She
pla
nted
31
seed
lings
al
toge
ther
. How
man
y cu
cum
ber
seed
lings
did
Mrs
. Gre
ene
plan
t?
MX
EN
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RW27-RW28
Grade5.indd 14Grade5.indd 14 6/18/07 5:54:13 PM6/18/07 5:54:13 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
29
Solv
e Eq
uatio
nsTo
sol
ve a
n eq
uatio
n, y
ou fi
nd a
val
ue fo
r th
e va
riabl
e th
at m
akes
the
equ
atio
n tr
ue.
That
val
ue is
the
so
luti
on.
Use
me
ntal
mat
h to
so
lve
the
fo
llow
ing
eq
uati
on.
Che
ck y
our
so
luti
on.
15 4
r 5
5
Thin
k: 1
5 di
vide
d by
wha
t nu
mbe
r eq
uals
5?
Test
som
e po
ssib
ilitie
s by
rep
laci
ng r
with
you
r pr
edic
tions
.
Try
5
1
5 4
5 5
3
No
Try
3
1
5 4
3 5
5
Yes
Che
ck: U
se a
n in
vers
e op
erat
ion
to c
heck
you
r w
ork.
3
3 5
5 1
5
Y
es
So, f
or 1
5 4
r 5
5, r
5 3
.
Whi
ch o
f th
e n
umb
ers
3, 6
, or
12 is
the
so
luti
on
of
the
eq
uati
on?
1.
t 2 5
5 1
2.
12 4
e 5
4
3.
5 3
u 5
60
4.
6 1
p 5
12
5.
f 1 1
7 5
20
6. 7
3 v
5 4
2
7.
48 4
k 5
8
8.
41 2
g 5
29
Use
me
ntal
mat
h to
so
lve
eac
h e
qua
tio
n. C
heck
yo
ur s
olu
tio
n.
9.
45 5
5 3
t10
. 32
5 p
1 9
11.
35 2
n 5
14
12.
77 4
y 5
11
13.
24 5
6 3
z14
. 18
5 a
1 2
15.
x 2
11 5
23
16.
18 4
v 5
6
MX
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Gra
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Ret
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RW
30
Func
tions
A fu
nctio
n is
a r
elat
ions
hip
betw
een
two
quan
titie
s. O
ne q
uant
ity
depe
nds
upon
the
oth
er. Y
ou c
an s
how
a fu
nctio
n us
ing
a fu
nctio
n ta
ble.
Wri
te a
n e
qua
tio
n to
re
pre
sent
eac
h fu
ncti
on.
The
n co
mp
lete
the
tab
le.
Inpu
ts30
3642
48
Outpu
ts5
68
Look
at
the
patt
ern.
Thin
k of
a r
ule.
Writ
e it
as a
n eq
uatio
n.
Rul
e: D
ivid
e by
6Eq
uatio
n: x
4 6
5 y
Rep
lace
: 42 4
6 5
y7 5
y
So, w
hen
x 5
42
in t
he fu
nctio
n ta
ble,
y 5
7.
Co
mp
lete
eac
h fu
ncti
on
tab
le.
1.
y 5
4x
2.
n 5
5m
1 4
3.
k 5
7j 1
14.
b 5
6a 2
8
x0
12
34
y0
m0
35
810
n
29
j1
35
79
k
64
a2
46
810
b16
MX
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RW29-RW30
Grade5.indd 15Grade5.indd 15 7/19/07 6:04:07 PM7/19/07 6:04:07 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
31
Ineq
ualit
ies
An
ine
qua
lity
is a
num
ber
sent
ence
tha
t sh
ows
that
tw
o am
ount
s ar
e no
t eq
ual.
Whi
ch o
f th
e n
umb
ers
8, 9
, and
10
are
so
luti
ons
of
the
ine
qua
lity,
x –
5 <
6?
Rep
lace
x w
ith e
ach
of t
he n
umbe
rs t
hat
are
poss
ible
sol
utio
ns.
So, 8
, 9, a
nd 10
are
solu
tions
to
the
ineq
uatit
y x
2 5
< 6.
Whi
ch o
f th
e n
umb
ers
8, 9
, and
10
are
so
luti
ons
of
eac
h in
eq
ualit
y? 1
. x
. 8
2.
x ,
10
3.
x ,
12
2 3
4
. x
. 6
1 1
5.
x 2
4 ,
6 6
. x
1 9
, 18
7.
x 1
5 .
13
8.
x .
15
2 6
Whi
ch o
f th
e n
umb
ers
16,
17,
and
18
are
so
luti
ons
of
eac
h in
eq
ualit
y?
9.
x .
16
10.
x ,
11
1 7
11.
x .
24
2 8
12
. x
1 1
, 18
13.
x 2
14
. 3
14.
x 1
5 .
11
15.
x 1
3 ,
22
16.
x 2
9 ,
8
8 2
5 <
6or
3 <
6 is
tru
e
9 2
5 <
6or
4 <
6 is
tru
e
10 2
5 <
6
or5
< 6
is t
rue
MX
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de 5
Ret
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RW
32
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy: P
redi
ct a
nd T
est
The
cap
acit
y o
f a
larg
e a
qua
rium
is 1
2 g
allo
ns m
ore
tha
n th
e
cap
acit
y o
f a
smal
l aq
uari
um. T
he a
qua
rium
s ho
ld a
to
tal o
f 52
gal
lons
of
wat
er.
Wha
t is
the
cap
acit
y o
f e
ach
aqua
rium
?
1.Re
ad
to
Un
de
rsta
nd
Wha
t ar
e yo
u as
ked t
o fi
nd?
3.So
lve W
hat
mig
ht 2
of y
our
pre
dic
tions
be?
Tes
t th
ese
in t
he e
qua
tions
bel
ow
.
4. Ch
eck
5.H
ow
can
you
use
the
equa
tion
c +
(c +
12)
= 52
to c
heck
your
ans
wer
?
2.Pla
n Wha
t do y
ou
know
about
the
cap
aciti
es o
f the
tw
o a
qua
rium
s?
Wha
t is
the
cap
acity
of e
ach
aqua
rium
?
6.
Cat
fish
cost
$3
each
at
the
pet
sto
re
and b
etta
s co
st $5
each
. And
y sp
ent
$19
at t
he s
tore
. How
man
y of e
ach
type
of f
ish
did
And
y buy
?
7.
Cha
rlott
e ha
s a
tota
l of 1
6 go
ldfis
h an
d
tetr
as in
her
aqua
rium
. The
pro
duc
t of
the
num
ber
s of e
ach
type
of f
ish
is 63.
If
Cha
rlott
e ha
s m
ore
gold
fish
than
tet
ras,
ho
w m
any
of e
ach
type
does
she
hav
e?
Pre
dic
t an
d t
est
to
so
lve
.
Larg
e A
qua
rium
Smal
l Aqua
rium
1(
2 12)
552
1(
2 12)
552
MX
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Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
33
Deci
mal
Pla
ce V
alue
Yo
u ca
n us
e m
od
els
to
wri
te d
eci
mal
s.
Writ
e th
e de
cim
al s
how
n by
the
sha
ded
part
of t
he m
odel
bel
ow.
Ther
e ar
e 10
0 sq
uare
s in
a h
undr
edth
s m
odel
.In
thi
s m
odel
8 s
quar
es a
re s
hade
d.U
se a
pla
ce v
alue
cha
rt t
o he
lp w
rite
the
deci
mal
.
One
sTe
nths
Hun
dre
dth
sTh
ous
and
ths
22
22
So, t
he d
ecim
al r
epre
sent
ed b
y th
e m
odel
is 0
.08.
Wri
te t
he d
eci
mal
sho
wn
by
the
sha
de
d p
art
of
eac
h m
od
el.
1.
2.
3.
4.
5.
6.
Find
the
val
ue o
f th
e u
nde
rlin
ed
dig
it.
7.
3.0
1 8
. 3.
019.
9.
814
10. 5
4.23
611
. 54.
236
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urt •
Gra
de 5
Ret
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RW
34
Equi
vale
nt D
ecim
als
Equi
vale
nt d
eci
mal
s ar
e di
ffere
nt n
ame
for
the
sam
e nu
mbe
r or
am
ount
. You
can
use
de
cim
al s
quar
es t
o fin
d eq
uiva
lent
dec
imal
s.
Wri
te e
quiv
ilant
or
not e
quiv
ilant
to
de
scri
be
the
pai
r at
de
cim
als
be
low
.
0.42
and
0.4
20.
Now
div
ide
each
of t
he 1
00 p
arts
into
10
equa
l par
ts. T
he m
odel
at
the
right
sho
ws
wha
t ea
ch s
mal
l squ
are
wou
ld lo
ok li
ke.
The
mod
el s
how
s 1,
000
equa
l par
ts. E
ach
part
rep
rese
nts
1 th
ousa
ndth
, or
0.00
1 of
th
e m
odel
.
Ther
e ar
e 42
0 sh
aded
par
ts.
This
mod
el s
how
s 42
0 th
ousa
ndth
s, o
r 0.
420.
1.
0.2
3 an
d 0.
230
2.
0.5
1 an
d 0.
500
3.
0.6
80 a
nd 0
.68
4.
4.8
7 an
d 4.
870
5.
9.8
7 an
d 9.
786.
1
.11
and
1.11
1
7.
0.8
300.
803
0.83
8.
0.9
30.
093
0.93
0
9.
1.0
071.
070
1.07
Wri
te t
he t
wo
de
cim
als
that
are
eq
uiva
lent
.
Wri
te e
quiv
alen
t or
not e
quiv
alen
t to
de
scri
be
eac
h p
air
of
de
cim
als.
Each
mod
el s
how
s th
e sa
me
amou
nt s
hade
d.
So, 0
.42
and
0.42
0 ar
e eq
uiva
lent
.
The
mod
el s
how
s 10
0 eq
ual p
arts
. Ea
ch p
art
is 1
hun
dred
th o
f the
mod
el.
42 o
f the
par
ts a
re s
hade
d.
This
mod
el s
how
s 42
hun
dred
ths,
or
0.4
2.
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Nam
e
© H
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urt •
Gra
de 5
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RW
35
Com
pare
and
Ord
er D
ecim
als
A p
lace
-val
ue c
hart
can
hel
p y
ou
com
par
e dec
imal
s. Y
ou
may
nee
d t
o a
dd z
eros
so y
ou
can
com
par
e th
e sa
me
num
ber
of d
igits
in e
ach
dec
imal
. Com
par
e th
e dig
its fr
om
left
to r
ight
.
Co
mp
are
0.2
8 an
d 0
.208
. Wri
te <
, >, o
r =
fo
r th
e
.
Use
a p
lace
val
ue c
hart
.
One
sTe
nths
Hun
dre
dth
sTh
ous
and
ths
02
80
Com
par
e th
e dig
its in
the
one
s pla
ce.
0
= 0
Com
par
e th
e dig
its in
the
ten
ths
pla
ce.
2 =
2
Com
par
e th
e dig
its in
the
hun
dre
dth
s pla
ce.
8 >
0
Sinc
e 8
hund
redth
s is
gre
ater
tha
n 0
hund
redth
s, 0
.28
is g
reat
er t
han
0.20
8.
So, 0
.28
> 0
.208
.
Co
mp
are
. Wri
te <
, >, o
r =
fo
r e
ach
.
1.
9.39
9.
9 2
. 0.
308
0.30
3.
7.24
5 7.
254
4.
8.16
8.
106
5.
0.69
0.
89 6
. 1.
83
1.83
3
7.
0.60
3 0.
6 8
. 0.
78
7.8
9.
4.71
4.
071
Ord
er
fro
m le
ast
to g
reat
est
.
10.
0.6
14, 0
.641
, 0.6
411
. 1
.576
, 1.7
65, 1
.567
12.
3.97
1, 3
.9, 4
, 3.9
01
13.
3.0
8, 3
.801
, 3.8
14.
0.1
59, 0
.154
, 0.1
4
15.
7, 6
.99,
6.0
99, 7
.001
MX
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RW
36
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill: D
raw
Con
clus
ions
Mr.
Hal
l dis
cove
rs t
hat
5 o
f hi
s st
ude
nts
spe
nt le
ss t
han
2 w
ee
ks o
n th
eir
pla
nt e
xpe
rim
ent
, 16
stud
ent
s sp
ent
2
to 4
we
eks
, and
3 s
tud
ent
s sp
ent
4 t
o 6
we
eks
. W
hat
conc
lusi
on
can
Mr.
Hal
l dra
w a
bo
ut t
he n
umb
er
of
we
eks
st
ude
nts
spe
nt o
n th
eir
pla
nt e
xpe
rim
ent
?
1.
Wha
t do
es t
he p
robl
em a
sk y
ou t
o fin
d?
Ana
lyze
Co
nclu
sio
n
How
muc
h tim
e di
d th
e gr
eate
st n
umbe
r of
stu
dent
s sp
ent
on t
he e
xper
imen
t?
How
muc
h tim
e di
d th
e le
ast
num
ber
of s
tude
nts
spen
d on
the
exp
erim
ent?
4.
Wha
t co
nclu
sion
can
Mr.
Hal
l dra
w a
bout
the
num
ber
of w
eeks
stu
dent
s sp
ent
on t
heir
plan
t ex
perim
ents
?
So
lve
by
dra
win
g a
co
nclu
sio
n.
5.
Julia
has
thr
ee p
lant
s. O
ne p
lant
rec
eive
s 1
tabl
espo
on o
f pla
nt fo
od e
ach
mon
th a
nd
grow
s to
a h
eigh
t of
7 c
m. A
sec
ond
plan
t re
ceiv
es 3
tab
lesp
oons
of p
lant
food
eac
h m
onth
and
gro
ws
to a
hei
ght
of 8
.5 c
m. A
thi
rd p
lant
is n
ot g
iven
any
pla
nt fo
od a
nd
grow
s to
a h
eigh
t of
5.7
5 cm
. Wha
t co
nclu
sion
can
you
dra
w a
bout
how
the
am
ount
of
food
giv
en a
ffect
s pl
ant
grow
th?
3.
Com
plet
e th
e ta
ble.
2.
How
wou
ld y
ou c
ompl
ete
the
stat
emen
ts b
elow
usi
ng t
he d
etai
ls fr
om t
he p
robl
em?
stu
dent
s sp
ent
less
tha
n 2
wee
ks.
stu
dent
s sp
ent
2 to
4 w
eeks
.
stu
dent
s sp
ent
4 to
6 w
eeks
.
MX
EN
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PM
RW35-RW36
Grade5.indd 18Grade5.indd 18 6/18/07 5:58:35 PM6/18/07 5:58:35 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
37
Roun
d De
cim
als
You
can
use
the
sam
e ru
les
you
lear
ned
for
roun
ding
w
hole
num
bers
to
roun
d de
cim
als.
Ro
und
0.1
34 t
o t
he p
lace
of
the
und
erl
ine
d d
igit
.
hun
dred
ths
plac
e
Ste
p 1
: Com
pare
the
dig
itto
the
rig
ht o
f the
unde
rline
d di
git
to 5
usi
ng t
he
roun
ding
rul
es.
Ste
p 2
: Rew
rite
all d
igits
to
the
right
of t
he
unde
rline
d di
git
as z
eros
. A
n eq
uiva
lent
dec
imal
can
be
writ
ten
by le
avin
g of
f the
tr
ailin
g ze
ros.
RO
UN
DIN
G R
ULE
S:
• If
the
digi
t to
the
rig
ht
is le
ss t
han
5, t
he
unde
rline
d di
git
stay
s th
e sa
me.
• If
the
digi
t to
the
rig
ht
is g
reat
er t
han
or e
qual
to
5, t
he u
nder
lined
di
git
incr
ease
s by
1.
Ro
und
eac
h nu
mb
er
to t
he p
lace
of
the
und
erl
ine
d d
igit
.
1.
7.32
5
2
. 9.
028
3.
108.
108
4.
26.1
99
5. 1
2.63
6. 1
1.32
3
7. 4
.289
8. 7
.547
9. 0
.964
10.
20.5
95
11
. 6.
8912
. 32
.514
13.
12.3
5 to
12.
414
. 0.
428
to 0
.43
15.
9.46
2 to
9.4
6
Co
mp
are
: 4 <
5
4 is
less
tha
n 5,
so
the
digi
t st
ays
the
sam
e.
Re
wri
te: 0
.130
or
0.13
Nam
e t
he p
lace
to
whi
ch e
ach
num
be
r w
as r
oun
de
d.
MX
EN
L08A
WK
5X_R
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6:43
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
38
Add
and
Subt
ract
Dec
imal
sY
ou c
an u
se li
ned
note
book
pap
er t
o ad
d an
d su
btra
ct d
ecim
als.
Sub
trac
t.
Find
the
sum
or
diff
ere
nce
.
1.
2.
3.
$13.
04
$0.9
5
$13 $1 $12
2
$ 1
3. 0
4
$ 0
. 9 5
2
$ 1
2. 0
9
12 1
9 14
So, $
13.0
4 2
$0.
95 5
$12
.09
is c
lose
to
the
estim
ate,
$12
. So
the
ans
wer
is r
easo
nabl
e.
1.
5 8
4. 5
31
$1
8. 5
2
$3
. 7 3
1
0.
4 5
0. 7
1
0. 1
2 3
7. 8
5
0. 3
2 1
1
4
6. 8
1
2.
9
0. 6
3
2
$2
1. 4
0
$1.
33
2
8.
$13
.04
2 $
0.95
__
4.
9.10
.
3.5 1
2.8
9 1
0.4
$15
2 $
1.27
22.
1 8
6. 3
9
11.
21.
05 2
2.6
512
. $
7.0
0 2
$1.
05
7.6.
5.
MX
EN
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5X_R
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indd
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0:37
PM
RW37-RW38
Grade5.indd 19Grade5.indd 19 6/18/07 5:58:44 PM6/18/07 5:58:44 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
39
Estim
ate
Sum
s and
Diff
eren
ces
You
can
est
imat
e to
find
an
answ
er t
hat
is c
lose
to
the
exac
t an
swer
. You
ca
n us
e th
e ro
undi
ng r
ules
to
help
you
est
imat
e su
ms
or d
iffer
ence
s.
Esti
mat
e t
he s
um b
y ro
und
ing
.
Firs
t, de
cide
wha
t pl
ace
to r
ound
to.
Sin
ce t
wo
of t
he
adde
nds
have
dig
its t
o th
e hu
ndre
dths
, ten
ths
wou
ld b
e a
good
pl
ace
to r
ound
to.
Rou
nd a
nd a
dd t
o es
timat
e.
Esti
mat
e t
he d
iffe
renc
e b
y ro
und
ing
.
So, t
he d
iffer
ence
is a
bout
0.5
.
Rem
embe
r th
e ro
undi
ng r
ules
:
If th
e di
git
to t
he r
ight
of t
he p
lace
you
are
ro
undi
ng t
o is
:•
less
tha
n 5,
rou
nd d
own.
• gr
eate
r th
an o
r eq
ual t
o 5,
rou
nd u
p.
1.24
7
1
.2
0.82
0
.8
3.4
+
3
.4
5.4
1
0.78
0
.8
0.30
5
0
.3
0.5
22
0.78
0.30
52
51.2
34
28.4
2
$29.
38
$42.
751
7.6
2.15
2
0.44
3
0.20
71
2.08
3
0.56
10.
41
6.17
3.5
2
1.73
1.4
1
3.28
0.86
2
15.2
7
41.8
1
$23.
07
$ 7
.83
2
3.17
1.24
70.
823.
41
Esti
mat
e b
y ro
und
ing
.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. 0
.427
+ 0
.711
12. 6
1.05
– 1
8.63
13. 4
0.51
2 +
30.
399
MX
EN
L08A
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PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
40
Choo
se a
Met
hod
Ther
e is
mor
e th
an o
ne w
ay t
o fin
d th
e su
ms
and
diffe
renc
es o
f who
le
num
bers
and
dec
imal
s. Y
ou c
an u
se m
enta
l mat
h, a
cal
cula
tor,
or p
aper
an
d pe
ncil.
1
4.21
9
1.79
31
15.8
81
Cho
ose
a m
eth
od
. Fin
d t
he s
um o
r d
iffe
renc
e.
1.
2.
3.
4.
5.
6.
7.
8.
10
1 2.
25
__
0.36
4
1 1.
558
__
71
.4
1 11
.5
__
3.3
1 5.
6
_
1.
2
2 0
.8
_
0.4
$ 1
5.79
1 3
2.81
__
48.6
0
713
1
$
Cho
ose
a m
eth
od
. Fin
d t
he d
iffe
renc
e.
Thes
e ar
e ve
ry la
rge
num
bers
,so
use
a c
alcu
lato
r.
So,
• U
se p
aper
and
a p
enci
l for
la
rger
num
bers
.
•
Use
a c
alcu
lato
r fo
r di
fficu
lt nu
mbe
rs o
r ve
ry la
rge
num
bers
.
38
.44
5
38.4
4525
.86
12.5
852
88
81
34
45
55
52
2_
..
.
11
1
11
11
73.9
4.
37
1
127.
35
928.
527
90.
4 88
.76
38.
445
2 2
5.86
__
• U
se m
enta
l mat
h fo
r pr
oble
ms
with
fe
wer
dig
its o
r ro
unde
d nu
mbe
rs.
10
MX
EN
L08A
WK
5X_R
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indd
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6/12
/07
2:4
5:39
PM
RW39-RW40
Grade5.indd 20Grade5.indd 20 6/18/07 5:58:56 PM6/18/07 5:58:56 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
41
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill:
Estim
ate
or F
ind
Exac
t Ans
wer
In a
bas
eb
all-
thro
win
g c
ont
est
, a s
core
of
50 m
ete
rs o
r m
ore
is
nee
de
d t
o a
dva
nce
to
the
fin
al r
oun
d. J
enn
a’s
first
tw
o t
hro
ws
we
re
16.6
4 m
ete
rs a
nd 1
5.33
me
ters
. Ho
w lo
ng d
oe
s he
r la
st t
hro
w n
ee
d
to b
e f
or
her
to a
dva
nce
to
the
fin
al r
oun
d?
1.
W
hat
are
you
aske
d yo
u to
find
?
2.W
ill y
ou e
stim
ate
or fi
nd a
n ex
act
answ
er t
o so
lve
this
pro
blem
? Ex
plai
n yo
ur c
hoic
e.
3. 4.
Show
how
you
sol
ve t
he p
robl
em in
the
spa
ce b
elow
.
Tell
whe
the
r yo
u ne
ed
an
est
imat
e o
r an
exa
ct a
nsw
er.
Th
en
solv
e t
he p
rob
lem
.
A w
aitr
ess
char
ged
Han
nah
$6.7
5 fo
rlu
nch.
Han
nah
wan
ts t
o tip
the
wai
tres
s ab
out
20%
of $
6.75
. How
muc
h sh
ould
she
leav
e?
Lanc
e is
pur
chas
ing
4 ne
w t
ires
for
his
car.
Each
tire
cos
ts $
110.
60. W
hat
isth
e to
tal c
ost?
6.5.
How
long
doe
s he
r la
st t
hrow
nee
d to
be
for
her
to a
dvan
ce t
o th
e fin
al r
ound
?
MX
EN
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e
© H
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urt •
Gra
de 5
Ret
each
RW
42
Mod
el M
ultip
licat
ion
by a
Who
le N
umbe
rY
ou c
an u
se t
he h
undr
edth
s m
odel
s, t
o sh
ow m
ultip
licat
ion
of
deci
mal
s an
d w
hole
num
bers
.
Find
the
pro
duc
t.2 3
0.9
6
Find
the
pro
duc
t.
1. 9
3 0
.23
2. 7
3 0
.25
3. 4
3 0
.71
4. 8
3 0
.05
5. 2
3 0
.84
6. 6
3 0
.52
7. 4
3 0
.12
8. 0
.09 3
69.
3 3
0.3
2
Ste
p1
Shad
e0.
96 t
wo
times
usi
ng a
diff
eren
t co
lor
each
tim
e.
Ste
p2
Cou
nt h
ow m
any
squa
res
are
shad
ed.
Ther
e ar
e 19
2 hu
ndre
dths
squ
ares
or
1 w
hole
and
92
hund
redt
hs. P
lace
the
de
cim
al a
fter
the
who
le n
umbe
r.
So,2
3 0
.96 5
1.9
2
MX
EN
L08
AW
K5X
_RT
_CH
07_L
1.in
dd1
6/21
/07
12:2
2:15
PM
RW41-RW42
Grade5.indd 21Grade5.indd 21 7/19/07 6:04:26 PM7/19/07 6:04:26 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
43
Alge
bra:
Pat
tern
s in
Deci
mal
Fac
tors
and
Prod
ucts
Whe
n m
ultip
lyin
g a
deci
mal
by
10, 1
00, 1
,000
or
10,0
00, f
irst
coun
t th
e nu
mbe
r of
zer
os in
the
who
le n
umbe
r. Th
en m
ove
the
deci
mal
poi
nt o
ne p
lace
to
the
right
in y
our
answ
er fo
r ev
ery
zero
tha
t yo
u co
unte
d.
Use
the
pat
tern
s to
fin
d t
he p
rod
uct.
Num
be
r o
f Z
ero
s in
W
hole
Num
be
rN
umb
er
of
Plac
es
to
Mo
ve D
eci
mal
Po
int
Ans
we
r
0.00
5 3
10
00.
005
0.00
5 3
10
11
0.0
5
0.00
5 3
100
22
0.5
0.00
5 3
1,0
003
3
5
Use
pat
tern
s to
fin
d t
he p
rod
uct.
1. 0
.185
3 1
5
0.18
5 3
10
5
0.18
5 3
100
5
0.18
5 3
1,0
00 5
2. 6
.37
3 1
5
6.37
3 1
0 5
6.37
3 1
00 5
6.37
3 1
,000
5
3. 9
.999
3 1
5
9.99
9 3
10
5
9.99
9 3
100
5
9.99
9 3
1,0
00 5
4. 0
.008
3 1
5
0.00
8 3
10
5
0.00
8 3
100
5
0.00
8 3
1,0
00 5
5. $
3.75
3 1
5
$3.7
5 3
10
5
$3.7
5 3
100
5
$3.7
5 3
1,0
00 5
6. 8
9.36
3 1
5
89.3
6 3
10
5
89.3
6 3
100
5
89.3
6 3
1,0
00 5
7. 1
.313
3 1
0 5
1.31
3 3
100
5
1.31
3 3
1,0
00 5
1.31
3 3
10,
000
5
8. $
0.25
3 1
0 5
$0.2
5 3
100
5
$0.2
5 3
1,0
00 5
$0.2
5 3
10,
000
5
9. 0
.6 3
10
5
0.6
3 1
00 5
0.6
3 1
,000
5
0.6
3 1
0,00
0 5
10. 2
.002
3 1
0 5
2.00
2 3
100
5
2.00
2 3
1,0
00 5
2.00
2 3
10,
000
5
11. 3
.94
3 1
0 5
3.9
4 3
100
5
3.9
4 3
1,0
00 5
3.9
4 3
10,
000
5
12. 0
.05
3 1
0 5
0.05
3 1
00 5
0.05
3 1
,000
5
0.05
3 1
0,00
0 5
MX
EN
L08A
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/07
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2:33
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
44
Reco
rd M
ultip
licat
ion
by a
Who
le N
umbe
rW
hen
give
n a
horizo
ntal
mul
tiplic
atio
n pro
ble
m, r
ewrite
the
pro
ble
m v
ertic
ally
. Mul
tiply
and
put
the
dec
imal
poin
t in
the
ans
wer
the
sam
e nu
mber
of p
lace
s to
the
rig
ht a
s th
e dec
imal
poin
t in
the
pro
ble
m.
Find
and
re
cord
the
pro
duc
t.
Star
t at
the
rig
ht e
nd o
f your
ans
wer
an
d c
oun
t to
the
left
the
sam
e nu
mber
of p
lace
s as
are
in t
he
dec
imal
fact
or.
Find
and
re
cord
the
pro
duc
t.
1. 0
.933
3 6
2. 5
.27
3 1
8 3
. 32.
96 3
44
4. 3
79.4
3 5
5. 0
.007
3 2
6
6. 6
.31
3 5
3 7
. 8.4
92 3
10
8. 0
.688
3 2
9. 1
21.3
3 5
10. 9
.57
3 3
4
11. 0
.04
3 8
612
. 33.
33 3
72
13. 2
90.6
3 6
14. 1
.917
3 4
115
. 5.5
85 3
28
16. 2
5.68
3 3
217
. 159
.46
3
718
. 621
.3
3 4
419
. 736
.07
3
220
. 800
.9 3
51
21. 2
37.8
9
3 2
22. 7
7.42
3 3
923
. 3.0
43 3
70
24. 0
.333
3 5
425
. 3
9.83
3
66
4.8
8
3 2
6
_
29
28
1 9
760
__
126.
88
1 1
5 4
4.88
3 2
6
Write
the
pro
ble
mve
rtic
ally
and
mul
tiply
like
who
le n
umber
s.
Ther
e ar
e tw
o d
igits
to t
he
righ
t of t
he d
ecim
al p
oin
t.
MX
EN
L08A
WK
5X_R
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/07
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8:08
PM
RW43-RW44
Grade5.indd 22Grade5.indd 22 6/18/07 5:59:21 PM6/18/07 5:59:21 PM
Firs
t, us
e a
colo
r pe
ncil
to s
hade
16
colu
mns
. Nex
t, us
e a
diffe
rent
col
or
penc
il to
sha
de 5
row
s. T
hen,
cou
nt t
he
squa
res
in t
he a
rea
in w
hich
the
col
or
shad
ing
over
laps
.
Mak
e a
mo
de
l to
fin
d t
he p
rod
uct
of
1.6
3 0
.5
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
45
Mod
el M
ultip
licat
ion
by a
Dec
imal
Whe
n yo
u us
e m
odel
s to
mul
tiply
dec
imal
s, r
emem
ber
that
eac
h sq
uare
in t
he
hund
redt
hs g
rid r
epre
sent
s 0.
01. S
o, 6
0 sq
uare
s re
pres
ent 6
0. 0
.1 r
epre
sent
ed in
the
hund
redt
hs m
odel
is t
he e
ntire
firs
t co
lum
n or
10
squa
res.
One
full
grid
rep
rese
nts
the
who
le n
umbe
r, on
e.
So, 1
.6 3
0.5
5 0
.80
Use
the
mo
de
l to
fin
d t
he p
rod
uct.
1.2.
3.
4.5.
6.
7.8.
9.
0.6 3
0.8
1.6 3
0.8
1.4 3
0.1
0.5 3
0.5
0.4 3
0.2
0.7 3
0.8
0.8 3
0.3
1.1 3
0.7
0.3 3
0.9
MX
EN
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7:12
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
46
Estim
ate
Prod
ucts
Whe
n es
timat
ing
to fi
nd t
he p
rodu
ct o
f tw
o de
cim
als,
use
a r
ange
to
dete
rmin
e if
your
ans
wer
is r
easo
nabl
e.
Esti
mat
e t
he p
rod
uct.
2.15
3 3
.92
Ro
und
Ge
t th
e r
ang
eR
ound
bot
h nu
mbe
rs d
own.
You
r or
gina
l es
timat
ed a
nsw
er
shou
ld fa
ll be
twee
nth
e ot
her
two
estim
ates
in t
hera
nge.
1.
4.25
3 7
.82
2.
$26.
83 3
11
3.
3.3 3
9.4
4.
6.71
3 4
.22
5.
29.3
3 0
.31
6.
$8.5
4 3
9
7.
$0.3
9 3
291
8.
6.6 3
8.2
9.
7.6 3
9.2
17
10.
487.
66 3
2.1
211
. 30
0.59
3 0
.312
. 0.
409 3
1.4
7
2.
15
3 3
.92
__
2
3 4
_
8
Doe
s 8
fall
betw
een
6 an
d 12
? Y
es, s
o th
is e
stim
ate
is r
easo
nabl
e.
2.
15
3 3
.92
__
2
3 3
_
6
Rou
nd b
oth
num
bers
up.
2.
15
3 3
.92
__
3
3 4
_
12
Esti
mat
e t
he p
rod
uct.
MX
EN
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RW45-RW46
Grade5.indd 23Grade5.indd 23 7/19/07 6:04:46 PM7/19/07 6:04:46 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
47
Prac
tice
Deci
mal
Mul
tiplic
atio
nW
hen
you
mul
itply
dec
imal
s, fi
rst
mul
itipl
y th
e fa
ctor
s as
who
le n
umbe
rs.
Use
est
imat
ion
to fi
gure
out
whe
re t
o pl
ace
the
deci
mal
poi
nt in
you
ran
swer
. You
can
do
this
by
estim
atin
g yo
ur fa
ctor
s. M
ultip
ly y
our
estim
ated
fact
ors
and
plac
e th
e de
cim
al p
oint
in t
he a
nsw
er. Y
our
estim
ate
shou
ld s
how
tha
t yo
ur a
ctua
l ans
wer
is r
easo
nabl
e.
Esti
mat
e. T
hen
find
the
pro
duc
t.
Mul
tiply
as
who
le n
umbe
rs.
Esti
mat
e. T
hen
find
the
pro
duc
t. 1
.
72
3 1
.3 2
.
0
.9
3 0
.4 3
.
0
.5
3 0
.7 4
.
3
.6
3 0
.8 5
.
4
.1
3 5
.3
6.
7.9
3
6.6
7.
12.
4
3 8
.5 8
.
3
.43
3
9.3
9.
60.2
3
2.6
10.
$5.
94
3 0
.07
11. 1
94.6
3 0
.2
12. 0
.381
3 1
413
. $4.
50 3
9.5
14. $
6.95
3 5
.315
. 0.0
8 3
216
. 1.1
1 3
1.1
41
3 5
6
__
2,29
6
You
kno
w y
our
actu
al a
nsw
er s
houl
d be
clo
se t
o 0.
24.
So, t
he a
ctua
l ans
wer
is 0
.229
6.
Now
est
imat
e th
e fa
ctor
s.
0.41
3
0
.56
.4
3
.6
5
.24
0.
41
3 0
.56
__
MX
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e
© H
arco
urt •
Gra
de 5
Ret
each
RW
48
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill: M
ultis
tep
Prob
lem
sA
fis
hing
bo
at c
aug
ht a
to
tal o
f 4,
012
lob
ste
rs in
one
seas
on,
se
lling
the
lob
ste
rs t
o f
ish
mar
kets
fo
r $
0.05
eac
h.H
ow
muc
h m
one
y d
id t
he b
oat
ear
n fo
r th
e s
eas
on?
1. W
hat
are
you
aske
d t
o fi
nd?
2. W
hat
step
or
step
s ar
e ne
eded
to s
olv
e pro
ble
ms?
3. M
ultip
ly.
De
scri
be
the
ste
ps
req
uire
d t
o s
olv
e. T
hen
solv
e t
he p
rob
lem
.5.
FA
ST
FAC
T in
194
0, t
he a
vera
ge
Am
eric
an a
te 9
2.4
lb o
f red
mea
t th
at
year
. The
y at
e 1.
21 t
imes
tha
t am
oun
t in
20
03. I
n 19
40, A
mer
ican
s at
e an
ave
rage
of 1
2.3
lb o
f poul
try
for
the
year
, and
5.7
9 tim
es t
hat
in 2
003.
How
man
y poun
ds
of
poul
try
and r
ed m
eat
did
the
ave
rage
A
mer
ican
eat
per
yea
r?
6. T
he U
.S. p
opul
atio
n in
199
0 w
as 2
48.7
mill
ion
peo
ple
. The
200
0 popul
atio
n w
as
1.13
tim
es t
hat.
The
2010
popul
atio
n is
pro
ject
ed t
o b
e 1.
1 tim
es the
200
0 popul
atio
n. H
ow
man
y peo
ple
, to t
hene
ares
t m
illio
n, a
re p
roje
cted
to li
ve in
the
Uni
ted S
tate
s in
201
0?
4. H
ow
muc
h m
one
y did
the
boat
ear
n fo
r th
e se
ason?
4,
012
3 $
0.05
__
MX
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RW47-RW48
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
49
Just
as
you
did
with
who
le n
umber
s, y
ou
can
use
dec
imal
m
odel
s to
div
ide
dec
imal
s in
to g
roup
s.
Use
de
cim
al m
od
els
to
mo
de
l the
quo
tie
nt.
Re
cord
yo
ur a
nsw
er.
0.28
4 4
Show
0.2
8 us
ing
a dec
imal
model
.N
ow
div
ide
the
squa
res
into
four
gro
ups
of t
he
sam
e si
ze.
Ther
e ar
e fo
ur g
roup
s of
7 sq
uare
s. S
ince
we
are
usin
g hu
ndre
dth
s m
odel
s,
the
answ
er m
ust
be
in
hund
redth
s.
So, 0
.28 4
4 5
0.0
7.
Use
de
cim
al m
od
els
or
pla
y m
one
y to
mo
de
l the
quo
tie
nt.
Re
cord
yo
ur a
nsw
er.
1.
0.3
6 4
6
2.
1.9
5 4
3
3.
$9.
75 4
5
4.
6.0
3 4
9
5.
$7.
32 4
6
6.
7.6
3 4
7
7.
0.9
4 4
2
8.
8.5
6 4
4
9.
$9.
18 4
9
11.
$9.
99 4
3
12.
4.8
8 4
8
13.
5.5
3 4
7
Deci
mal
s Div
ision
MX
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
50
Whe
n di
vidi
ng a
dec
imal
by
a tw
o-di
git
num
ber,
use
com
patib
le n
umbe
rs
for
the
divi
dend
and
div
isor
. Com
patib
le n
umbe
rs a
re
num
bers
tha
t ar
e ea
sy t
o di
vide
.
Esti
mat
e t
he q
uoti
ent
.
3.40
9 4
83
Estim
ate
Quo
tient
s
Thin
k of
som
e co
mpa
tible
num
bers
ne
ar 3
.409
and
83:
•
3.2
and
80
are
com
patib
le.
3.2 4
80
= 0
.04
•
3.6
and
60
are
com
patib
le.
3.6 4
60
= 0
.03
Whi
ch c
ompa
tible
num
bers
are
clo
ser
to t
he
orig
inal
pro
blem
?
•
3.4
and
80
are
clos
er t
o 3.
409
and
83.
Esti
mat
e t
he q
uoti
ent
.
1.
23
.7 4
9
2.
46
.8 4
5
3.
94
.2 4
6
4.
5.
3 4
8
5.
6.
4 4
9
6.
0.
312 4
5
7.
65
.8 4
22
8.
91.7
4 1
8
9.
45
.43 4
36
So, a
n ap
prop
riate
est
imat
e is
0.0
4.
Find
tw
o e
stim
ate
s fo
r th
e q
uoti
ent
.
10.
47.8
4 5
911
. 8.
91 4
27
12.
7.42
4 3
5
MX
EN
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RW49-RW50
Grade5.indd 25Grade5.indd 25 7/19/07 6:03:49 PM7/19/07 6:03:49 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
51
1.
2.
3.
4.
5.
6.
7.
8.
Find
the
quo
tie
nt.
9.
10.
11.
12.
13.
14.
15.
16.
Find
the
quo
tie
nt.
Divi
de D
ecim
als b
y W
hole
Num
bers
Whe
n div
idin
g dec
imal
s, p
laci
ng t
he d
ecim
al p
oin
t in
your
an
swer
is a
n im
port
ant
step
to fi
ndin
g th
e co
rrec
t va
lue.
Co
py
the
quo
tie
nt a
nd c
orr
ect
ly p
lace
the
de
cim
al p
oin
t.
So, 3
.15 4
5 5
0.6
3
4716
4.5
3
5 9
78.8
4
876
61.
71
028
539
$93.
99
$
241
84
38.6
4
046
534
.15
58$5
3.36
$
092
64
4.03
20
063
70.
056
00
08
285
.12
17$9
9.28
3878
.66
8245
9.2
77$2
4.6
4 4
1.74
611.
159
Step
1.
On
the
quo
tient
, put
the
dec
imal
poin
t ab
ove
the
dec
imal
poin
t of t
he d
ivid
end.
Step
2.
Plac
e a
zero
in t
he o
nes
pla
ce a
bov
e th
e
div
iden
d if
the
div
isio
n is
gre
atest
the
n th
e
div
den
d.
0
.
Step
3.
Div
ide.
5 3
.15
5 3
.15
5 3
.15
3
0
5 3
.15
.
1
5
0
.63
MX
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5:11
PM
Mid
dle
vil
le A
uto
Race
Driv
erAv
erag
eSp
eed
per
Lap
Num
ber
of L
aps
Fini
shed
Budd
y
Dani
elle
Rico
Susi
e
Bren
t
98.6
5m
ph
91.4
3m
ph
99.1
0m
ph
73.2
5m
ph
89.4
7m
ph
21 43 45 50 50
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
52
1. W
hat
are
you
aske
d t
o fi
nd?
2.
How
can
you
estim
ate
the
thre
e dec
imal
par
ties
of f
ruit,
ro
undin
g th
em to w
hole
num
ber
s?
3.
Whi
ch a
nsw
er, B
rad’s
or
Britt
any’
s, is
clo
ser
to y
our
est
imat
e?
4.
Who
se a
nsw
er is
rea
sona
ble
? Ex
pla
in.
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill: E
valu
ate
Answ
ers
for R
easo
nabl
enes
sB
ritn
ey
bo
ught
0.9
7 ki
log
ram
of
app
les,
1.0
5 ki
log
ram
s o
f b
anan
as, a
nd 0
.57
kilo
gra
m o
f o
rang
es.
Bri
tne
y sa
ys s
he
bo
ught
25.
9 ki
log
ram
s o
f fr
uit.
Bra
d s
ays
that
Bri
tne
y b
oug
ht
2.59
kilo
gra
ms
of
frui
t. W
hose
ans
we
r is
re
aso
nab
le?
US
E D
ATA
Fo
r 5-
6, u
se t
he t
able
.
5.
Bud
dy
says
he
was
fast
er t
han
Bre
nt b
y an
aver
age
spee
d o
f 19.
18 m
ph.
Bre
nt s
ays
Bud
dy
was
fast
er b
y 9.
18 m
ph.
Who
se
answ
er is
rea
sona
ble
?
6.
For
the
first
thr
ee la
ps
of t
he r
ace,
Sus
ie s
ays
her
aver
age
spee
d w
as fa
ster
tha
n D
anie
lle’s
tota
l ave
rage
spee
d. S
usie
dro
ve 9
6.4
mph,
88.5
mph,
and
83.
9 m
ph.
Is S
usie
’s a
nsw
er
re
asona
ble
? Ex
pla
in.
0.
97
1.
05
+ 0
.57
MX
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RW51-RW52
Grade5.indd 26Grade5.indd 26 6/18/07 6:00:06 PM6/18/07 6:00:06 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
53
Colle
ct a
nd O
rgan
ize
Data
A s
urv
ey
is a
way
to g
athe
r in
form
atio
n ab
out
a g
roup
.W
hen
you
are
gath
erin
g in
form
atio
n ab
out
a g
roup
,•
the
who
le g
roup
is c
alle
d t
he p
op
ula
tio
n,
• th
e peo
ple
sur
veye
d a
re c
alle
d t
he s
amp
le.
1.A p
layg
round
mak
er
wan
ts t
o f
ind
out
if c
hild
ren in
gra
de
s 4–
6 lik
e t
he
ir n
ew
p
layg
round
eq
uip
me
nt.
Te
ll w
he
the
r e
ach s
amp
le r
ep
rese
nts
the
po
pula
tio
n. I
f it
d
oe
s no
t, e
xpla
in.
A f
ruit
juic
e c
om
pan
y w
ants
to
surv
ey
child
ren a
ge
s 10
–14
. Te
ll w
he
the
r th
e s
amp
le b
elo
w r
ep
rese
nts
the
po
pula
tio
n.
If it
do
es
no
t, e
xpla
in.
A r
adio
sta
tio
n w
ants
to
fin
d o
ut
the
fav
ori
te t
ype
of
musi
c o
f p
eo
ple
that
live
in
Los
Ang
ele
s, C
alifo
rnia
. Te
ll w
he
the
r e
ach s
amp
le r
ep
rese
nts
the
po
pula
tio
n. I
f it
d
oe
s no
t, e
xpla
in.
2.3.
a ra
ndom
sam
ple
of 4
00
boys
in g
rades
4–6
a ra
ndom
sam
ple
of
child
ren
in g
rades
4–6
a ra
ndom
sam
ple
of
400
teac
hers
4.a
rand
om
sam
ple
of 2
00
peo
ple
who
live
in
Cal
iforn
ia.
a ra
ndom
sam
ple
of 2
00
adul
ts w
ho li
ve in
Los
Ang
eles
.
5.6.
a ra
ndom
sam
ple
of 2
00
peo
ple
who
live
in L
os
Ang
eles
.
A s
ample
mus
t fa
irly
rep
rese
nt t
he p
opul
atio
n.
In a
ran
do
m s
amp
le, e
very
one
in t
he p
opul
atio
n ha
s an
equa
l cha
nce
of b
eing
sur
veye
d.
A r
andom
sam
ple
of 1
00 c
hild
ren
at o
ne s
chool.
Does
the
ran
dom
sam
ple
fairly
rep
rese
nt t
he p
opul
atio
n?
• T
he s
urve
y does
invo
lve
child
ren
ages
10–
14.
• B
ut, i
t is
a s
urve
y of o
nly
one
sch
ool.
So, t
he s
urve
y does
not
repre
sent
the
popul
atio
n bec
ause
the
sa
mple
incl
udes
chi
ldre
n at
onl
y one
sch
ool.
MX
EN
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AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
54
Mea
n, M
edia
n, a
nd M
ode
Find
the
me
an, m
ed
ian,
and
mo
de
fo
r th
e d
ata
set:
8,
8, 1
4, 5
, 38,
5
1.Find
the
me
an, m
ed
ian,
and
mo
de
fo
r e
ach
set
of
dat
a.
The
me
an is
the
ave
rage
of
a se
t of d
ata.
Ste
p 1
: Add t
o fi
nd t
he s
um.
5, 5
, 8, 8
, 14,
38
5 5
78
Ste
p 2
: Div
ide
the
sum
by
the
num
ber
of a
dden
ds.
78 4
6 5
13
The
mea
n is
13.
The
me
dia
n is
the
mid
dle
nu
mber
whe
n a
set
of d
ata
is a
rran
ged in
ord
er.
Ste
p 1
: Ord
er t
he d
ata
from
le
ast
to g
reat
est.
Ste
p 2
: Fin
d t
he m
iddle
nu
mber
. Si
nce
ther
e is
an
even
num
ber
of d
ata,
the
m
edia
n is
bet
wee
n th
e tw
o
mid
dle
num
ber
s.
(8 1
8)
4 2
5 8
The
med
ian
is 8
.
8 1
8 1
14
1 5
1 3
8 1
The
mo
de
is t
he n
umber
th
at o
ccur
s m
ost
ofte
n in
a
set
of d
ata.
Ste
p 1
: Ord
er t
he d
ata
from
le
ast
to g
reat
est.
Ste
p 2
: Fin
d t
he n
umber
th
at o
ccur
s m
ost
ofte
n.
Ther
e ar
e tw
o m
odes
5
and 8
.
Som
etim
es t
here
is o
nly
one
mode
or
no m
ode.
3.
2. 4.
5.6.
7.8.
9.10
.
11.
Use
the
giv
en
me
an t
o f
ind
the
mis
sing
num
be
r in
eac
h d
ata
set.
21, 1
5, 1
7, 2
1, 1
664
1, 8
74, 6
14, 7
55
7.8,
9.4
, 10.
6, 7
.8, 7
.8, 9
.416
4, 2
15, 1
74, 1
74, 1
93
$25,
$36
, $28
, $27
21, 2
4, 2
2, 2
4, 2
1, 3
1, 2
5
350,
378
, 350
, 252
, 275
43, 5
3, 6
3, 5
3
873,
954
, 896
, 941
1.3,
1.5
5, 2
.75,
1.3
, 2.6
12.
13.
14.
15.
16.
$6
4, $
48, $
40,
;
mea
n: $
51
75, 6
2,
;
mea
n: 7
3
21
,
, 4
5, 3
2; m
ean:
33
9.
8, 1
0.2,
11.
6,
,
9.7
; mea
n: 1
0.3
10
.5, 1
2.5,
, 1
0, 1
3; m
ean:
11
$1
2, $
8, $
17,
,
$7;
mea
n: $
10.6
0
MX
EN
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RW53-RW54
Grade5.indd 27Grade5.indd 27 6/18/07 6:00:15 PM6/18/07 6:00:15 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
55
Com
pare
Dat
aY
ou c
an c
ompa
re d
ata
sets
usi
ng t
he m
ean,
ran
ge, a
nd m
edia
n.
Co
mp
are
the
me
an, m
ed
ian,
and
ran
ge
of
the
dat
a se
ts.
1.Co
mp
are
the
me
an, m
ed
ian,
and
ran
ge
of
the
dat
a se
ts.
A: P
age
s S
tud
ent
s R
ead
4310
6865
3112
7924
5252
6869
Mea
n:
573
4 1
2 5
47.
75M
ean:
54
3 4
12
5 4
5.25
The
mea
n fo
r da
ta s
et A
is g
reat
er t
han
the
mea
n fo
r da
ta s
et B
.
Ran
ge:
79 2
10
5 6
9R
ange
: 72
2 1
2 5
60
The
rang
e fo
r da
ta s
et A
is g
reat
er t
han
the
rang
e fo
r da
ta s
et B
.
Med
ian:
(5
2 1
52)
4 2
5 5
2M
edia
n:
(52
1 5
2) 4
2 5
52
The
med
ian
for
data
set
B is
the
sam
e as
the
med
ian
for
data
set
A.
2.
B: P
age
s S
tud
ent
s R
ead
3253
6812
5237
1572
6052
2268
A: W
eig
hts
of
Bo
xes
1113
1517
16
1215
209
13
B: W
eig
hts
of
Bo
xes
2213
1516
18
1118
2114
15
A: S
ong
s S
tud
ent
s H
ear
d
35
21
09
31
28
34
76
24
B: S
ong
s S
tud
ent
s H
ear
d
57
20
9
105
79
6
MX
EN
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AM
Soccer
Baseball
Basketball
HockeySp
ort
Team
Win
s
Key
:
=
8 w
ins
Fantasy
Mystery
Biography
Poetry
Typ
es
of
Bo
oks
in M
r. L
i’s C
lass
Key
:
=
fo
ur
bo
oks
Dail
y T
em
pera
ture
s
Day
Mon
.Fr
i.Th
urs.
Wed
.Tu
es.
020406080100
2731
14
30
2325
26
3340 35 30 25 20 15 10 5 0
Duke
Wom
enM
en
Tenn
esse
eCo
nnec
ticut
Mic
higa
n S
tate
Number of Wins
Scho
ol
NC
AA
Bask
etb
all
Win
s 2004–2
005
Pet
Sto
re P
op
ula
tio
n B
irds,
20
Dog
s, 1
0
Cats
, 10
Rept
iles,
5
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
56
Anal
yze
Grap
hsG
raphs
can
hel
p y
ou
dra
w c
onc
lusi
ons
, ans
wer
que
stio
ns, a
ndm
ake
pre
dic
tions
about
dat
a.
Whi
ch t
eam
had
the
most
win
s?D
uke:
27
1 3
1 5
58
Conn
ectic
ut: 2
3 1
25
5 4
8 Te
nnes
see:
14
1 3
0 5
44
Mic
higa
n: 2
6 1
33
5 5
9
So, M
ichi
gan
had t
he m
ost
win
s.
A b
ar g
rap
h u
ses
bar
s to
dis
pla
y co
unta
ble
dat
a. A
bar
gra
ph
is u
sefu
l w
hen
com
par
ing
dat
a by
group
s.
A p
icto
gra
ph d
ispla
ys c
oun
table
dat
a us
ing
pic
ture
s an
d s
ymbols
. Pic
togr
aphs
hav
e a
key
to s
how
how
man
y ea
ch p
ictu
re o
r sy
mbol s
tand
s fo
r.
How
man
y fa
ntas
y books
are
in M
r. L
i’s
clas
s?Th
e ke
y sh
ow
s ea
ch s
ymbol s
tand
s fo
r 4
books
. A h
alf s
ymbol s
tand
s fo
r 2
books
.(5
3 4
) 1
2 5
22
fant
asy
books
A c
ircl
e g
rap
h s
how
s ho
w p
arts
of d
ata
are
rela
ted t
o e
ach
oth
er a
nd t
o t
he w
hole
.A
line
gra
ph s
how
s ho
w d
ata
chan
ges
ove
r a
per
iod o
f tim
e.
Wha
t ki
nds
of a
nim
als
does
the
pet
sto
re
have
the
sam
e nu
mber
of?
cats
and
dogs
How
can
you
des
crib
e th
e tr
end
in
tem
per
atur
e fr
om
Wed
nesd
ay t
o F
riday
?
The
tem
par
ure
incr
ease
s st
eadily
.
US
E D
ATA
Fo
r 1–
4, u
se t
he
pic
tog
rap
h a
t th
e r
ight.
Twen
ty-e
ight
win
s w
oul
d b
e sh
ow
n as
. W
hich
sport
had
thi
s nu
mber
of
win
s?
1. 2.H
ow
man
y w
ins
did
bas
ebal
l hav
e?3.
How
man
y m
ore
win
s did
socc
er h
ave
than
bas
ketb
all h
ad?
4.W
hat
if fo
otb
all h
ad 2
0 w
ins.
How
woul
d t
his
amoun
t be
show
n on
the
pic
togr
aph?
MX
EN
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RW55-RW56
Grade5.indd 28Grade5.indd 28 6/18/07 6:00:28 PM6/18/07 6:00:28 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
5 7
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Draw
a D
iagr
amFi
ve s
tud
ents
wro
te a
re
po
rt a
bo
ut
U.S
. Pre
sid
ents
, 8 w
rote
a r
ep
ort
ab
out
U.S
. fir
st la
die
s, a
nd
3 w
rote
a r
ep
ort
ab
out
bo
th U
.S.
Pre
sid
ents
and
fir
st la
die
s. H
ow
man
y st
ud
ents
wro
te r
ep
ort
s?
S
olv
e
R
ea
d t
o U
nd
ersta
nd
1.H
ow
woul
d y
ou
write
the
que
stio
n as
a fi
ll-in
-the
bla
nk s
ente
nce?
2.
P
lan
How
can
dra
win
g a
dia
gram
hel
p y
ou
solv
e th
e pro
ble
m?
3.So
lve
the
pro
ble
m u
sing
the
Ven
n dia
gram
.
How
woul
d y
ou
write
your
ans
wer
as
a co
mple
te s
ente
nce.
4. Ch
eck
5.W
hat
is o
ne w
ay y
ou
coul
d c
heck
your
solu
tion.
Dra
w a
Ve
nn d
iag
ram
to
so
lve
.
Dur
ing
a one
-hour
res
earc
h per
iod, 8
peo
ple
use
d t
he In
tern
et a
nd 4
peo
ple
us
ed a
n at
las.
Tw
o o
f the
se p
eople
us
ed t
he In
tern
et a
nd a
n at
las.
How
m
any
peo
ple
use
d t
he In
tern
et o
r an
at
las
dur
ing
the
rese
arch
per
iod?
6.
23
5
U.S
. Pre
sid
ents
Firs
t La
die
s
MX
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AM
Nam
e
Wee
k 1
Biki
ng R
ecor
d
Mar
cus
Tina
Emm
aPh
ilip
Distance (in miles)
56 48 40 32 24 16 8 0
Nam
e
Bik
ing
Reco
rd
Distance (in miles)
56 48 40 32 24 16 8 0
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
58
Mak
e Ba
r Gra
phs a
nd P
icto
grap
hsPi
ctogr
aphs
use
pic
ture
s or
sym
bols
to
dis
pla
y or
org
aniz
e dat
a. B
ar g
raphs
dis
pla
y dat
a us
ing
vert
ical
or
horizo
ntal
bar
s.
Use
the
dat
a at
the
rig
ht t
o c
om
ple
teth
e bar
gra
ph.
A d
oub
le-b
ar g
rap
h is
use
d t
oco
mpar
e si
mila
r ki
nds
of d
ata.
Des
crib
e ho
w t
o m
ake
a doub
le-b
ar g
raph
of
the
dat
a in
the
tab
le a
nd t
his
dat
a: W
eek
2:
Mar
cus,
56
mile
s; T
ina,
44
mile
s; E
mm
a, 3
2 m
iles;
Phi
llip, 4
8 m
iles.
• A
dd a
key
for
Wee
k 1
and W
eek
2.•
Use
one
colo
r fo
r W
eek
1 an
d a
diff
eren
t co
lor
for
Wee
k 2.
• A
dd t
he s
econd
set
of d
ata
usin
g th
e ke
y fo
r W
eek
2.
• D
raw
a b
ar t
o s
how
40
mile
s fo
r Ti
na•
Dra
w a
bar
to s
how
mile
s fo
r Ph
illip
Wee
k 1
Bik
ing
Rec
ord
Nam
eD
ista
nce
(in m
iles)
Mac
us48
Tin
a40
Em
ma
28P
hilli
p52
Tent
s
Stov
es
Boot
s
Back
pack
s
May C
am
pin
g S
ale
s
1
9
1
3
1
1
6
1. M
ake
a bar
gra
ph
for
the
dat
a se
t.2.
Mak
e a
doub
le-b
ar g
raph
for
the
dat
a se
t
from
exe
rcis
e 1
and t
his
dat
a: J
une
Sale
s:
tent
s, 2
0; s
tove
s, 9
; boots
, 11;
bac
kpac
ks, 1
5.
MX
EN
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Grade5.indd 29Grade5.indd 29 7/19/07 5:56:28 PM7/19/07 5:56:28 PM
8 7 6 5 4 3 2 1 01-
34-
67-
910
-12
Ages
(in
year
s)
Ag
es o
f St
ud
ents
Tak
ing
Sw
imm
ing
Les
son
s
Swimming Lessons
Hei
gh
t (i
n In
ches
) o
f St
ud
ents
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
59
Ag
es
of
Ch
ild
ren
Ta
kin
g S
wim
min
g L
ess
on
s
Mak
e Hi
stog
ram
sA
his
togr
am s
how
s th
e nu
mbe
r of
tim
es, o
r fr
eque
ncy,
an
eve
nt o
r da
ta it
em o
ccur
s.
US
E D
ATA
Fo
r 1–
2, u
se t
he t
able
.
1. U
se 1
0 in
ches
for
each
inte
rval
. Li
st t
he in
terv
als.
•
To
mak
e a
hist
ogra
m, y
ou n
eed
to
find
a re
ason
able
inte
rval
.
•
The
dat
a in
the
tab
le s
how
s ag
es
from
2 t
o 12
.
•
A r
easo
nabl
e in
terv
al w
ould
be
3 ye
ars,
sta
rtin
g at
1.
•
Mak
e a
freq
uenc
y ta
ble
to d
eter
min
e th
e nu
mbe
r of
tim
es e
ach
age
appe
ars
in t
he d
ata.
•
Mak
e th
e hi
stog
ram
by
first
cho
osin
g an
app
ropr
iate
sca
le fo
r th
e ve
rtic
al
axis
. Lab
el t
he a
xis.
•
Use
the
freq
uenc
y ta
ble
to fi
nd h
ow
long
to
mak
e ea
ch b
ar.
•
Dra
w t
he b
ars
touc
hing
, but
not
ov
erla
ppin
g.
•
Mak
e ea
ch b
ar t
he s
ame
wid
th.
•
Lab
el e
ach
bar
with
the
freq
uenc
y ta
ble
inte
rval
and
the
axi
s.
•
So,
8 c
hild
ren
ages
4-6
tak
e sw
imm
ing
less
ons
the
mos
t.Freq
uenc
y Ta
ble
Inte
rval
1-3
4-6
7-9
10-1
2Fr
eque
ncy
38
36
Hei
ght
(in in
ches
) of S
tude
nts
5248
4741
5460
4246
3957
4449
4647
6143
6349
5652
5160
5442
62
Mak
e a
his
tog
ram
of
the
dat
a.
Ho
w m
any
child
ren
age
s 4-
6 ta
ke
swim
min
g le
sso
ns?
9 3 6 6
6
6
3811 5
7 10 11
4 4 12 12
2 11 4
2. M
ake
a hi
stog
ram
of t
he d
ata.
MX
EN
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PM
02
3
12345678910
45
67
89
101
02
3
1234567891011
45
67
89
1011
1
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
60
Alge
bra:
Gra
ph O
rder
ed P
airs
A c
oord
inat
e gr
id is
like
a s
heet
of g
raph
pap
er w
ith t
wo
bord
ers,
the
x-ax
is a
nd y
-axi
s. T
he x
-axi
s is
the
hor
izon
tal l
ine
at t
he b
otto
mof
the
grid
. The
y-a
xis
is t
he v
ertic
al li
ne o
n th
e le
ft si
de o
f the
grid
.A
n or
dere
d pa
ir is
the
loca
tion
of a
poi
nt o
n th
e gr
id.
Gra
ph
and
lab
el (
10, 4
) o
n th
e c
oo
rdin
ate
gri
d.
An
orde
red
pair
cont
ains
tw
o nu
mbe
rs,
x an
d y.
To
grap
h an
ord
ered
pai
r:
• St
art
at 0
.
• S
ince
x c
omes
bef
ore
y in
the
alph
abet
, rem
embe
r to
mov
e ac
ross
the
x-ax
is fi
rst.
•
Mov
e ac
ross
the
x-a
xis
to t
he n
umbe
r 10
.
• T
hen,
go
stra
ight
up
to t
he li
nela
bele
d 4
on t
he y
axi
s.
• D
raw
a d
ot a
t th
e in
ters
ectio
n of
the
two
lines
, and
labe
l the
ord
ered
pai
r(1
0, 4
).
The
x-co
ord
inat
e is
the
1st
num
ber
in t
he o
rder
ed p
air.
The
y-co
ord
inat
eis
the
2nd
num
ber
in a
n or
dere
d pa
ir.
(10,
4)
(x
, y)
Gra
ph
and
lab
el t
he f
ollo
win
g p
oin
tso
n a
coo
rdin
ate
gri
d.
1.
A (1
, 6)
2.
B (1
, 9)
3.
C (3
, 7)
4.
D (5
, 5)
5.
E (9
, 3)
6.
F (6
, 2)
Use
the
co
ord
inat
e g
rid
.W
rite
an
ord
ere
d p
air
for
eac
h p
oin
t.
7.
G 8
. H
9.
J10
. K
11.
L12
. M
(10
, 4)
H
M
K
GJ
L
MX
EN
L08A
WK
5X_R
T_C
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.indd
1
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/07
1:1
9:20
PM
RW59-RW60
Grade5.indd 30Grade5.indd 30 7/19/07 5:56:44 PM7/19/07 5:56:44 PM
Tupe
lo, M
S Av
erag
e H
ighs
15
43
2
Sacr
am
en
to,
CA
Avera
ge L
ow
sJi
m’s
Ch
eck
-Up
Heig
hts
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
R61
Mak
e Li
ne G
raph
sA
line
gra
ph s
how
s ho
w a
set
of d
ata
chan
ges
over
tim
e.
Mak
e a
line
gra
ph
of
the
dat
a in
the
tab
le.
•
To
mak
e a
line
grap
h, y
ou n
eed
to
dete
rmin
e th
e gr
eate
st n
umbe
r on
th
e ve
rtic
al s
cale
.
• T
he g
reat
est
tem
pera
ture
in t
he t
able
is
708
.
• T
he s
cale
sho
uld
go fr
om 08
to 808
us
ing
inte
rval
s of
208
.
• W
rite
the
mon
ths
alon
g th
e ho
rizon
tal
axis
usi
ng e
ven
spac
ing.
• U
se t
he d
ata
to fi
nd w
here
to
grap
h ea
ch p
oint
.
• C
onne
ct t
he p
oint
s to
sho
w t
he
aver
age
tem
pera
ture
ris
ing
from
m
onth
1 t
o m
onth
5.
Mak
e a
line
gra
ph
for
eac
h th
e d
ata
set.
1.
2
.
Aver
age
Mon
thly
Tem
pera
ture
in T
upel
o, M
S
Mon
th1
23
45
Tem
pera
ture
(in
o F)40
4454
6270
Jim
’s H
eigh
t (in
ches
)Ag
e3
45
67
Heig
ht39
4143
4648
San
Dieg
o, C
A Av
erag
e Lo
ws
Mon
thM
ar.
April
May
June
July
Tem
pera
ture
(o F).
5353
6062
66
MX
EN
L08A
WK
5X_R
T_C
H10
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/07
1:2
1:12
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
62
Fish
Cat
Rabbit
Dog
Dog
Dog
Dog
Cat
Fish
Fish
E-mail Gam
es
E-m
ail
E-m
ail
ResearchResearch
Res
earc
h
Rese
arch
Rese
archResearch
pian
opi
ano
drum
s
viol
in
guita
r
Mak
e Ci
rcle
Gra
phs
A c
ircle
gra
ph s
how
s ho
w p
arts
of t
he d
ata
are
rela
ted
to t
he w
hole
and
to
each
oth
er.
Use
the
dat
a to
mak
e a
cir
cle
gra
ph.
•To
mak
e a
circ
le g
raph
, you
nee
d to
det
erm
ine
the
tota
l num
ber
of p
arts
of t
he d
ata.
•
Ther
e ar
e 10
stu
dent
s w
ho t
ake
mus
ic le
sson
s.
•U
se t
his
num
ber
to d
ivid
e th
e ci
rcle
into
equ
al
sect
ions
.
•Th
e ci
rcle
sho
uld
be d
ivid
ed in
to fi
ve e
qual
se
ctio
ns.
•Sh
ade
and
labe
l tw
o of
the
sec
tions
pia
no.
•
Shad
e an
d la
bel o
ne s
ectio
n ea
ch a
s gu
itar,
viol
in, a
nd d
rum
s.
Use
the
dat
a to
mak
e a
cir
cle
gra
ph.
1.
2.
Mus
ic L
esso
nsIn
stru
men
tSt
uden
tsPi
ano
4Gu
itar
2Vi
olin
2Dr
ums
2
Tom
’s W
eekl
y Co
mpu
ter U
sage
Activ
ityHo
urs
E-m
ail
3 G
ames
1 R
esea
rch
6
Favo
rite
Pets
in C
J’s
Clas
s
Pet
Num
ber o
f Vot
es F
ish
3 D
ogs
4 C
ats
2 R
abbi
ts1
MX
EN
L08
AW
K5X
_RT
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10_L
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dd1
7/11
/07
4:42
:48
PM
RW61-RW62
Grade5.indd 31Grade5.indd 31 7/19/07 5:57:02 PM7/19/07 5:57:02 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
63
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Mak
e a
Grap
hH
amp
ton
Hig
h’s
bas
ketb
all t
eam
had
the
fo
llow
ing
sco
res
this
se
aso
n 63
, 67,
73,
55,
61,
53,
60,
63,
52,
61,
and
64.
Did
th
e b
aske
tbal
l te
am s
core
leas
t o
fte
n in
the
50s
, 60s
, or
70s?
Stem
Leaves
Re
ad
to
Un
de
rsta
nd
1.
Wha
t ar
e yo
u as
ked
to fi
nd?
2.Pla
n How
can
you
org
aniz
e th
e nu
mbe
rs t
o be
tter
ans
wer
the
que
stio
n?
3.So
lve H
ow w
ould
you
fill
out
the
stem
–an
d–le
af p
lot
for
this
pr
oble
m in
the
spa
ce a
t th
e rig
ht?
4.D
id t
he b
aske
tbal
l tea
m s
core
leas
t of
ten
in t
he 5
0’s,
60’
s or
70’
s?
Ch
eck
5.W
hat
othe
r gr
aph
mig
ht y
ou h
ave
used
to
solv
e th
is p
robl
em?
Mak
e a
gra
ph
to s
olv
e.
Th
e nu
mbe
r of
peo
ple
eatin
g lu
nch
at a
dine
r fo
r th
e pa
st 1
2 da
ys w
as 4
1, 1
7,
28
, 12,
37,
44,
32,
26,
18,
24,
36,
and
25. S
houl
d th
e di
ner
be m
ore
prep
ared
to s
erve
cus
tom
er n
umbe
rs in
the
10’s
, 20’
s, 3
0’s
or 4
0’s
rang
e?
6.
MX
EN
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5X_R
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Stem
Leav
es
3 45
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
64
Choo
se th
e Ap
prop
riate
Gra
phB
ar a
nd d
oubl
e-ba
r gr
aphs
, lin
e gr
aphs
, lin
e pl
ots
and
stem
-and
-leaf
plo
ts o
rgan
ize
num
eric
al d
ata.
Cho
ose
the
be
st t
ype
of
gra
ph
or
plo
t fo
r th
e d
ata.
Exp
lain
yo
ur c
hoic
e.
Cho
ose
the
be
st t
ype
of
gra
ph
or
plo
t fo
r th
e d
ata.
Exp
lain
yo
ur c
hoic
e.
1.
The
popu
latio
ns o
f mal
es a
nd fe
mal
es in
6
maj
or c
ities
.2.
Th
e da
ily g
row
th o
f a s
unflo
wer
in
inch
es.
3.
The
scor
es o
f a b
aske
tbal
l tea
m fo
r on
e se
ason
.4.
Th
e te
mpe
ratu
re e
very
hou
r fo
r 24
ho
urs.
A c
ircle
gra
ph w
ill b
est
disp
lay
how
Jen
spe
nds
the
hour
s in
one
da
y be
caus
e it
will
sho
w e
ach
hour
spe
nt in
rel
atio
n to
the
who
le.
Cir
cle
Gra
ph
A c
ircle
gra
ph s
how
s ho
w
part
s of
the
dat
a ar
e re
late
d to
the
who
le a
nd
to e
ach
othe
r.
Ste
m-a
nd-l
eaf
Plo
t
A s
tem
-and
-leaf
plo
t is
us
ed t
o or
gani
ze d
ata
by p
lace
val
ue.
Line
Plo
t
A li
ne p
lot
is u
sed
to
reco
rd d
ata
as it
is
colle
cted
.
Line
Gra
ph
A li
ne g
raph
sho
ws
chan
ge o
ver
time.
Bar
Gra
ph
A b
ar g
raph
is u
sed
to c
ompa
re
data
by
cate
gory
.
Ho
w J
en
spe
nds
the
ho
urs
of
the
day
.
1
2
3
4
Do
uble
-bar
Gra
ph
A d
oubl
e-ba
r gr
aph
is
used
to
com
pare
tw
o se
ts o
f dat
a.
MX
EN
L08A
WK
5X_R
T_C
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.indd
1
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RW63 RW64
Grade5.indd 32Grade5.indd 32 7/19/07 5:57:20 PM7/19/07 5:57:20 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
65
Mul
tiple
s and
the
Leas
t Com
mon
Mul
tiple
A m
ulti
ple
of a
num
ber
is t
he p
rodu
ct o
f tha
t nu
mbe
r an
d an
y ot
her
num
ber.
To fi
nd t
he fi
rst
eigh
t m
ultip
les,
mul
tiply
9 b
y 1,
2, 3
, 4, 5
, 6, 7
, 8, 9
, and
10.
9
3 1
= 9
9 3
2 =
18
9
3 3
= 2
7
9 3
4 =
36
9 3
5 =
45
9
3 6
= 5
4
9 3
7 =
63
9
3 8
= 7
2
9 3
9 =
81
9 3
10
= 9
0
So, t
he fi
rst
10 m
ultip
les
of 9
are
9, 1
8, 2
7, 3
6, 4
5, 5
4, 6
3, 7
2, 8
1, a
nd 9
0.
Whe
n a
num
ber
is a
mul
tiple
of t
wo
or m
ore
num
bers
it is
a c
om
mo
n m
ulti
ple
.
The
leas
t co
mm
on
mul
tip
le, o
r (L
CM
), of
tw
o or
mor
e nu
mbe
rs is
the
leas
t nu
mbe
r th
at is
a m
ultip
le o
f all
of t
he n
umbe
rs.
Wri
te t
he le
ast
com
mo
n m
ulti
ple
fo
r: 4
, 8, a
nd 1
2.
M
ultip
les
of 4
: 4
8 12
16
20 2
4 28
32
...
M
ultip
les
of 8
: 8
16
24 3
2 40
48
56 6
4 ..
.
M
ultip
les
of 1
2: 1
2 24
36
48
60 7
2 84
96
...
List
the
fir
st e
ight
mul
tip
les
of
eac
h nu
mb
er.
1.
62.
10
3. 6
and
74.
4,
5, a
nd 1
0
So, 2
4 is
the
leas
t co
mm
on m
ultip
le.
List
the
fir
st 1
0 m
ulti
ple
s o
f 9.
Wri
te t
he le
ast
com
mo
n m
ulti
ple
fo
r e
ach
set
of
num
be
rs.
MX
EN
L08A
WK
5X_R
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H11
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8:00
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
66
Divi
sibili
tyA
num
ber
is d
ivis
ible
by
anot
her
num
ber
if th
e qu
otie
nt is
a w
hole
num
ber
and
the
rem
aind
er is
0.
Test
eac
h nu
mb
er
to d
ete
rmin
e w
heth
er
it is
div
isib
le b
y 2,
3, 5
, 6, 9
, or
10.
1.
489
2. 3
64
24 4
4 =
6
2
4 is
div
isib
le b
y 4.
24 4
5 =
4 R
424
is n
ot d
ivis
ible
by
5.
Test
the
fo
llow
ing
num
be
r to
se
e if
it is
div
isib
le b
y 2,
3, 5
, 6, 9
, or
10.
5,67
5
A n
umbe
r is
div
isib
le b
y 2
if th
e la
st
num
ber
is a
n ev
en n
umbe
r.5,
675
is n
ot d
ivis
ible
by
2.
sum
not
di
visi
ble
by3
A n
umbe
r is
div
isib
le b
y 3
if th
e su
m
of it
s di
gits
is d
ivis
ible
by
3.5,
675
is n
ot d
ivis
ible
by
3.
sum
not
di
visi
ble
by9
not
a 0
3. 6
,420
4. 8
,703
od
d nu
mbe
5161
7155
23
5, 6
7 5
5161
7155
23
A n
umbe
r is
div
isib
le b
y 9
if th
e su
m
of it
s di
gits
is d
ivis
ible
by
9.
5,
675
is n
ot d
ivis
ible
by
9.
A n
umbe
r is
div
isib
le b
y 6
if is
div
isib
le b
y 2
AN
D b
y 3.
5,67
5 is
not
div
isib
le b
y 3,
so
it is
not
div
isib
le b
y 6.
A n
umbe
r is
div
isib
le b
y 10
if t
he la
st d
igit
is 0
.5,
675
is n
ot d
ivis
ible
by
10.
So,5
,675
is d
ivis
ible
by
5. It
is n
ot d
ivis
ible
by
2,3,
6,9,
or
10.
5,67
5 is
div
isib
le b
y 5.
A n
umbe
r is
div
isib
le b
y 5
if th
e la
st
digi
t is
0 o
r 5.
5, 6
, 7, 5
5, 6
7 5
5
MX
EN
L08
AW
K5X
_RT
_CH
11_L
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dd1
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/07
4:47
:59
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RW65-RW66
Grade5.indd 33Grade5.indd 33 7/19/07 5:57:40 PM7/19/07 5:57:40 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
67
Fact
ors a
nd G
reat
est C
omm
on F
acto
rA
fact
or is
a n
umbe
r m
ultip
lied
by a
noth
er n
umbe
r to
find
a p
rodu
ct.
List
the
fac
tors
of
21.
So, t
he fa
ctor
s of
21
are
1,3,
7, a
nd 2
1.
A c
om
mo
n fa
cto
r is
a fa
ctor
tha
t tw
o or
mor
e nu
mbe
rs s
hare
.
The
gre
ate
st c
om
mo
n fa
cto
r, o
r G
CF,
is t
he g
reat
est
fact
or t
hat
two
or m
ore
num
bers
hav
e in
com
mon
.
Wri
te t
he g
reat
est
co
mm
on
fact
or
for
16 a
nd 4
0.
List
the
fac
tors
of
eac
h nu
mb
er.
1. 1
4
2. 2
7
3. 1
6 4
. 32
Wri
te t
he c
om
mo
n fa
cto
rs f
or
eac
h se
t o
f nu
mb
ers
.
5. 2
1, 3
0 6
. 35
, 28
7.
18,
24
8.
45, 1
5
Wri
te t
he g
reat
est
co
mm
on
fact
or
for
eac
h se
t o
f nu
mb
ers
.
9. 2
0, 3
510
. 18
, 42
11.
30, 3
512
. 49
, 35
1 3
21
= 2
1
3 3
7 =
21
Use
a V
enn
diag
ram
to
find
the
com
mon
fact
ors.
Iden
tify
the
grea
test
com
mon
fact
or in
the
ove
rlap
sect
ion,
whi
ch in
thi
s ca
se is
8.
So, t
he g
reat
est
com
mon
fact
or o
f 16
and
40 is
8.Fa
ctor
s of
40
Fact
ors
of 1
6
16
8 4
12
5 1040
MX
EN
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e
© H
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urt •
Gra
de 5
Ret
each
RW
68
Prim
e an
d Co
mpo
site
Num
bers
A p
rim
e n
umb
er
has
exac
tly t
wo
fact
ors,
1 a
nd it
self.
A c
om
po
site
num
be
r is
a n
umbe
r th
at h
as m
ore
than
tw
o fa
ctor
s.
The
num
ber
1 is
nei
ther
prim
e no
r co
mpo
site
.
You
can
mak
e ar
rays
to
find
if a
num
ber
is p
rime
or c
ompo
site
. An
arra
y is
an
arra
ngem
ent
of o
bjec
ts
in r
ows
and
colu
mns
. A n
umbe
r w
ith e
xact
ly t
wo
arra
ys is
prim
e. A
num
ber
with
mor
e th
an t
wo
arra
ysis
com
posi
te.
Wri
te p
rim
e o
r co
mp
osi
te 3
2.Y
ou
may
use
co
une
rs o
r d
raw
arr
ays.
32
Dra
w a
rray
s to
rep
rese
nt 3
2.•
Set
an a
rray
of 1
row
of 3
2.
Labe
l it.
“1 3
32”
•
Set
an a
rray
with
8 r
ows
of 4
.
Labe
l it
“8 3
4”.
• Se
t an
arr
ay o
f 32
row
s at
1.
La
bel i
t. “3
2 3
1”
• Se
t an
arr
ay w
ith 4
row
s of
8. w
ith L
abel
it. “
4 3
8”
Sinc
e 32
can
alre
ady
be r
epre
sent
ed b
y m
ore
than
2
arra
ys t
here
is n
o re
ason
to
keep
goi
ng.
So, t
he n
umbe
r 32
is c
ompo
site
.
Wri
te p
rim
e o
r co
mp
osi
te. Y
ou
may
use
co
unte
rs o
r d
raw
arr
ays.
1. 1
52.
19
3. 1
1
4. 3
85.
45
6. 2
4
7. 6
78.
44
9. 3
18 3
41 3
32
4 3
8
32 3
1
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e
© H
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urt •
Gra
de 5
Ret
each
RW
69
Prob
lem
Sol
ving
Wor
ksho
pSt
rate
gy: M
ake
an O
rgan
ized
List
Toni
and
He
cto
r ar
e w
eav
ing
po
tho
lde
rs u
sing
the
co
lors
re
d, w
hite
, gre
en,
and
blu
e. I
f th
ey
use
onl
y tw
o c
olo
rs f
or
eac
h p
oth
old
er,
ho
w m
any
way
s ca
n th
ey
com
bin
e t
wo
co
lors
?
Re
ad
to
Un
de
rsta
nd
2.Pla
n How
can
an
orga
nize
d lis
t he
lp y
ou s
olve
the
pro
blem
?
3.So
lve How
can
you
com
plet
e th
e or
gani
zed
list
belo
w t
o he
lp s
olve
th
e pr
oble
m?
Do
not
list
any
com
bina
tion
mor
e th
an o
nce.
Fo
r ex
ampl
e, r
ed a
nd w
hite
is t
he s
ame
as w
hite
and
red
.
4.H
ow m
any
way
s ca
n To
ni a
nd H
ecto
r co
mbi
ne t
wo
colo
rs?
Ch
eck
5.Lo
ok b
ack
at t
he p
robl
em. D
oes
the
answ
er m
ake
sens
e? E
xpla
in.
Mak
e a
n o
rgan
ize
d li
st t
o s
olv
e.
A
nna
can
have
a s
andw
ich
on
whi
te, p
umpe
rnic
kel,
or w
heat
bre
ad.
S
he c
an h
ave
it gr
illed
, bak
ed, o
r co
ld.
H
ow m
any
sand
wic
h ch
oice
s do
es
A
nna
have
?
6. 1.
H
ow w
ould
you
writ
e th
e qu
estio
n as
a fi
ll-in
-the
bla
nk s
ente
nce?
Col
or C
ombi
natio
ns
red
and
whi
te; r
ed a
nd
; red
and
whi
te a
nd
; whi
te a
nd
; g
reen
and
.
MX
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de 5
Ret
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RW
70
Intr
oduc
tion
to E
xpon
ents
The
exp
one
nt is
a n
umbe
r th
at t
ells
how
man
y tim
es a
noth
er
num
ber,
the
bas
e, i
s us
ed a
s a
fact
or.
Wri
te in
exp
one
nt f
orm
. Th
en
find
the
val
ue.
10 3
10 3
10
Wha
t is
the
bas
e? 1
0
How
man
y tim
es is
the
bas
e be
ing
used
as
a fa
ctor
?
3 tim
es
So, 1
0 3
10 3
10
writ
ten
in e
xpon
ent
form
is
103 .
Wha
t is
the
val
ue o
f 10 3
10?
10
0
Wha
t is
the
val
ue o
f 100
3 1
0?
1,00
0
So, t
he v
alue
of 1
03 5 1
,000
.
Wri
te in
exp
one
nt f
orm
. The
fin
d t
he v
alue
.
1. 1
0 3
10 3
10 3
10 3
10
2. 1
0 3
10
3. 1
0 3
10 3
10 3
10 3
10 3
10 3
10
Find
the
val
ue.
5. 1
04
6. 1
09
7. 1
06
8. 1
08
4. 1
0 3
10 3
10 3
10 3
10 3
10 3
10
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e
© H
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Gra
de 5
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RW
71
Expo
nent
s and
Squ
are
Num
bers
You
alre
ady
know
tha
t an
exp
onen
t is
a n
umbe
r th
at t
ells
how
man
y tim
es
the
base
is u
sed
as a
fact
or.
In t
he la
st le
sson
, the
bas
e w
as a
lway
s 10
. The
bas
e do
es n
ot a
lway
s ha
ve t
o be
10,
tho
ugh.
Wri
te 3
3 3
3 3
3 3
3 3
3 3
in e
xpo
nent
fo
rm.
3 is
the
rep
eate
d fa
ctor
, so
3 is
the
bas
e.
The
base
is r
epea
ted
6 tim
es, s
o 6
is t
he e
xpon
ent.
3 3
3 3
3 3
3 3
3 3
3=
36
36
A b
ase
with
an
expo
nent
can
be
writ
ten
in w
ords
.
Wri
te 3
6 in
wo
rds.
The
expo
nent
6 m
eans
“th
e si
xth
pow
er.”
36 in
wor
ds is
“th
e si
xth
pow
er o
f thr
ee.”
A p
erfe
ct s
quar
e or
a s
quar
e nu
mbe
r is
the
pro
duct
of a
num
ber
and
itsel
f. A
squ
are
num
ber
can
be r
epre
sent
ed
with
the
exp
onen
t 2.
Ther
e ar
e tw
o w
ays
to w
rite
the
wor
d fo
rm fo
r an
exp
onen
t of
2 o
r 3.
Wri
te 4
3 4
and
6 3
6 3
6 in
exp
one
nt f
orm
and
in w
ord
s.
Exp
one
nt f
orm
: 42
Wo
rds:
th
e se
cond
pow
er o
r fo
ur
or
four
squ
ared
Exp
one
nt f
orm
: 63
Wo
rds:
th
e th
ird p
ower
of s
ix
or
six
cub
ed
Wri
te in
exp
one
nt f
orm
and
the
n w
rite
in w
ord
s.
1.
5 3
5 3
5 3
5 3
5 3
5 3
52.
8 3
8 3
83.
2 3
2 3
2 3
2 3
2
Find
the
val
ue.
4.
27 5
. 12
3 6
. 19
2
expo
nent
base
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RW
72
Prim
e Fa
ctor
izat
ion
Prim
e f
acto
riza
tio
n is
a w
ay t
o sh
ow a
com
posi
te n
umbe
r as
the
pro
duct
of p
rime
fact
ors.
A f
acto
r tr
ee
is a
dia
gram
tha
t sh
ows
the
prim
e fa
ctor
izat
ion
of a
com
posi
te n
umbe
r.
Find
the
pri
me
fac
tori
zati
on
of
12. Y
ou
may
use
a f
acto
r tr
ee
.
So, t
he p
rime
fact
oriz
atio
n of
12
is 2
3 2
3 3
.
You
can
use
exp
onen
ts in
prim
e fa
ctor
izat
ion
for
fact
ors
that
ap
pear
tw
o or
mor
e tim
es.
Re
wri
te 2
3 2
3 3
usi
ng e
xpo
nent
s.
2 3
2 3
3 =
22 3
3
Find
the
pri
me
fac
tori
zati
on.
Yo
u m
ay u
se a
fac
tor
tre
e.
1. 4
22.
20
3. 9
0
Re
wri
te t
he p
rim
e f
acto
riza
tio
n us
ing
exp
one
nts.
4. 2
3 3
3 5
3 5
5.
3 3
5 3
3 3
56.
2 3
5 3
3 3
5 3
2 3
5
Find
the
num
be
r fo
r e
ach
pri
me
fac
tori
zati
on.
7. 5
2 3 7
8. 2
3 3
3 5
3 2
3 7
3 3
9. 2
3 3 3
3 1
12
1
2
4
3
3
2
3
2 3
3
Writ
e th
e nu
mbe
r as
a p
rodu
ct o
f tw
o fa
ctor
s.
Writ
e ea
ch c
ompo
site
num
ber
as a
pro
duct
of t
wo
fact
ors.
Writ
e th
e nu
mbe
r be
ing
fact
ored
at
the
top.
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01
Nam
e
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Gra
de 5
Ret
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RW
73
Unde
rsta
nd F
ract
ions
A fr
actio
n de
scrib
es t
he n
umbe
r of
par
ts o
f a w
hole
or
a gr
oup.
4 __
5
1 __
7
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.
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.
6.7.
8.
Wri
te a
fra
ctio
n to
nam
e t
he p
oin
t o
n th
e n
umb
er
line
.
9.
10.
11.
12.
13.
14.
Shad
ed P
art:
1. H
ow m
any
part
s m
ake
up t
he g
roup
? 4
2. H
ow m
any
part
s ar
e sh
aded
? 1
3.
Wha
t is
the
frac
tion?
1 __
4
Uns
hade
d Pa
rt:
1. P
arts
in t
he g
roup
?
4
2. U
nsha
ded
part
s?
3
3. W
hat
is t
he fr
actio
n?
3 __ 4
B0
1
E0
1
C
01
A
01
D
01
F
4 __ 5
Num
erat
or
Den
omin
ator
Frac
tions
suc
h as
, whi
ch h
ave
1 as
the
num
erat
or, a
re c
alle
d un
it fr
actio
ns.
For
exam
ple,
show
s th
at 5
par
ts m
ake
a w
hole
, and
4 p
arts
are
bei
ng u
sed.
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de 5
Ret
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RW
74
Equi
vale
nt F
ract
ions
Equi
vale
nt f
ract
ions
are
frac
tions
tha
t na
me
the
sam
e nu
mbe
r or
am
ount
.
For
exam
ple,
a
nd
a
re e
quiv
alen
t fr
actio
ns b
ecau
se t
hey
are
equa
l in
valu
e. 3 __ 6
1 __ 2
Wri
te a
n e
qui
vale
nt f
ract
ion
for
.
Wri
te a
n e
qui
vale
nt f
ract
ion.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
2 __ 5
To w
rite
an e
quiv
alen
t fr
actio
n fo
r
:
1. C
hoos
e a
num
ber
to m
ultip
ly w
ith.
Le
t’s u
se 2
.2.
Mul
tiply
the
top
(num
erat
or) b
y 2.
2 3
2 =
43.
Mul
tiply
the
bot
tom
(den
omin
ator
) by
2.
5 3
2 =
10
2 __ 5
1 __
2 1 __ 6
2 __
5 2 __ 7
3 __ 4
2 __ 3
4 __
6 3 __
9 2 __ 4
5 __ 7
3 __ 8
4 __ 7
3 __
6 6 __ 8
1 ___
12
4 ___
10
5 __ 9
6 ___
12
7 ___
10
4 __ 5
2 __ 8
8 ___
10
3 ___
11
2 ___
12
6 __ 6
So, a
n eq
uiva
lent
frac
tion
for
2 __ 5
is 4
ove
r 10
, or
4 ___
10 .
Equi
vale
nt F
ract
ion
Rul
e:
“Wha
t yo
u do
to
the
top
(num
erat
or) y
ou m
ust
do t
o do
th
e bo
ttom
(den
omin
ator
).”
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e
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Gra
de 5
Ret
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RW
75
Sim
ples
t For
mA
frac
tion
is in
sim
ples
t fo
rm w
hen
the
num
erat
or a
nd
deno
min
ator
hav
e 1
as t
heir
only
com
mon
fact
or.
Wri
te 10
__
_ 14
in s
imp
lest
fo
rm.
Ano
ther
way
to
find
the
grea
test
co
mm
on fa
ctor
(GC
F) o
f the
nu
mer
ator
and
den
omin
ator
is t
o lis
t th
e fa
ctor
s of
the
less
er
num
ber
and
elim
inat
e th
e on
es
that
are
n’t
com
mon
.
Nam
e t
he G
CF
of
the
num
era
tor
and
de
nom
inat
or.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Wri
te e
ach
frac
tio
n in
sim
ple
st f
orm
.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
4 __ 8
6 __ 9
3 ___
12
8 ___
16
6 ___
18
20
___
90
21
___
35
30
___
45
4 __ 7
5 ___
10
2 __ 4
3 __ 9
5 ___
18
4 ___
12
16
___
24
8 __ 9
25
___
30
18
___
27
28
___
49
13
___
13
The
GC
F of
the
num
erat
or a
nd d
enom
inat
or fo
r 10
__
_ 14
is
2.
• D
ivde
the
num
erat
or a
nd d
enom
inat
or b
y th
e G
CF
2.
10 4
2
____
___
14 4
2 5
5 __ 7
So, 10
__
_ 14
in s
impl
est
for
is 5 __ 7 .
1. L
ist
the
fact
ors
of 1
0 an
d 14
.
10: 1
, 2, 5
, 10
14
: 1, 2
, 7, 1
4
2. W
hat
is t
he g
reat
est
com
mon
fa
ctor
? 2
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RW
76
Step
1.
Mul
tiply
4 2 __
5
5 3
4 5
20.
Step
2.
Add
4
2 __
5
2 1
20
5 2
2
Step
3.
Rem
embe
r to
kee
p
the
sam
e de
nom
inat
or, 5
.
Unde
rsta
nd M
ixed
Num
bers
A m
ixe
d n
umb
er
is m
ade
up o
f a w
hole
num
ber
and
a fr
actio
n.
A fr
actio
n gr
eate
r th
an 1
is s
omet
imes
cal
led
an im
pro
per
frac
tion.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
To c
hang
e
t
o a
mix
ed
num
ber,
divi
de t
he
num
erat
or o
ver
the
deno
min
ator
.
7 __ 4
11
___ 2
12
___ 5
1 5 __ 6
2 1 __ 3
19
___ 6
3 3 __
8 5
2 __ 5
29
___ 9
18
___ 7
17
___
10
6 4 __
5 4
7 ___
12
23
___ 3
7 1 __ 8
23
___
10
23
___
10
Wri
te t
he f
ract
ion
as a
mix
ed
num
be
r.
Wri
te e
ach
mix
ed
num
be
r as
a f
ract
ion.
Wri
te e
ach
frac
tio
n as
a m
ixe
d n
umb
er.
Wri
te 4
2 __
5 as
a fr
acti
on.
To c
hang
e 4
2 __ 5 to
afr
actio
n, m
ultip
ly t
he
deno
min
ator
by
the
who
le
num
ber,
then
add
the
nu
mer
ator
to
the
sum
. Th
e de
nom
inat
or s
tays
the
sam
e fo
r th
e an
swer
.
Step
1.
Div
ide
.
Step
2.
See
whe
re t
he fo
llow
ing
go:
w
hole
num
ber
re
mai
nder
di
viso
r
10 � �
23
2 3 __
_ 10
So, t
he fr
actio
n 23
__
_ 10
writ
ten
as a
mix
ed n
umbe
r is
2 3 __
_ 10
.
10 � �
23
2 r
3
220 3
So, t
he m
ixed
num
ber
4 2 __ 5 w
ritte
n as
a fr
actio
n is
22
___ 5 .
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Grade5.indd 38Grade5.indd 38 7/19/07 5:59:08 PM7/19/07 5:59:08 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
77
Com
pare
and
Ord
er F
ract
ions
and
Mix
ed N
umbe
rsFr
actio
ns a
nd m
ixed
num
bers
can
be
com
pare
d by
usi
ng
frac
tion
bars
.
Co
mp
are
. Wri
te <
, >, o
r =
fo
r e
ach
.
1.
2
. 3
.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Use
frac
tion
bars
to
com
pare
.Th
e tw
o ro
ws
are
of t
he s
ame
leng
th.
Co
mp
are
. Wri
te <
, >, o
r =
fo
r 1
3 __ 41
9 ___
12 .
So, 1
3 __ 4
1 9 __
_ 12
. 5
1 1 __
4 1 __
4 1 __
4
Firs
t, lis
t th
e m
ultip
les
of d
enom
inat
ors
M
ultip
les
of 2
1: 2
1, 4
2, 6
3
M
ultip
les
of 7
: 7, 1
4, 2
1
1 1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
1 __
_ 12
Co
mp
are
. Wri
te ,
, ., o
r 5
fo
r 5
7 ___
21
5 3 __
7
5 7 __
_ 21
5
7 ___
21
5 7 __
_ 21
5
7 ___
21
5 3 __ 7
3 3
3
____
__
7 3
3
5
9 ___
21
5 9 __
_ 21
5 7 __
_ 21
, 5
3 __ 7 b
ecau
se 5
7 ___
21 ,
5 9 __
_ 21
.
Then
, com
pare
.
Nex
t, re
nam
e th
e fr
actio
ns u
sing
the
com
mon
den
omin
ator
.
Or
you
can
look
for
the
com
mon
den
omin
ator
s.
5 __ 7
3 __ 5
15
___
24
7 ___
12
2 2 __
3 2
5 __ 6
4 __ 5
10
___
11
4 __ 9
5 __ 9
7 ___
12
5 ___
12
1 __ 2
5 __
6 2 __ 8
1 __ 4
7 3 __ 4
7 9 __
_ 12
5
7 ___
18
5 4 __ 9
4 4 __
7 4
6 ___
11
5 __ 6
13
___
16
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M
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
78
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy: M
ake
a M
odel
Am
be
r, M
arcu
s, P
aul,
and
She
lly li
ne u
p t
o m
ake
the
ir
jum
ps.
She
lly is
no
t fir
st. A
mb
er
has
at le
ast
two
pe
op
le
ahe
ad o
f he
r. P
aul i
s th
ird
. Giv
e t
he o
rde
r o
f th
e f
our
.
Re
ad
to
Un
de
rsta
nd
2.Pla
n Wha
t st
rate
gy c
an y
ou u
se t
o so
lve
the
prob
lem
?
3.So
lve H
ow c
an y
ou u
se t
he s
trat
egy
to s
olve
the
pro
blem
? Fi
ll ou
t th
e m
odel
.
4.W
hat
is t
he o
rder
of t
he fo
ur p
eopl
e?
Ch
eck
5.W
hat
othe
r st
rate
gy c
ould
you
use
to
solv
e th
e pr
oble
m?
Mak
e a
mo
de
l to
so
lve
.
Ber
nice
thr
ew t
he s
hotp
ut
10 5 _ 9 fe
et. T
erry
thr
ew t
he s
hotp
ut10
4 _ 7 ft
, an
d C
arla
thr
ew t
he
shot
put
10 2 _ 5 ft
. Who
thr
ew t
he
shot
put
the
long
est
dist
ance
? W
ho t
hrew
for
the
shor
test
dis
tanc
e?
6.1.
W
hat
are
you
aske
d to
find
?
Pau
l
1
2 3
4
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RW77-RW78
Grade5.indd 39Grade5.indd 39 7/19/07 5:59:23 PM7/19/07 5:59:23 PM
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e
© H
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Gra
de 5
Ret
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RW
79
Rela
te F
ract
ions
and
Dec
imal
sY
ou c
an r
epre
sent
a fr
actio
n as
a d
ecim
al a
nd v
ice
vers
a. F
or
exam
ple,
0.5
is t
he d
ecim
al e
quiv
alen
t to
. 1 __
2
Wri
te t
he d
eci
mal
0.2
55 a
s a
frac
tio
n o
r m
ixe
d n
umb
er
in s
imp
lest
fo
rm.
To w
rite
a de
cim
al a
s a
frac
tion,
rem
ove
the
deci
mal
po
int
and
plac
e th
e nu
mbe
r ov
er t
he a
ppro
pria
te p
ower
of
10.
Ste
p 1
.
0.25
5 ha
s th
ree
digi
ts t
o th
e rig
ht o
f the
de
cim
al p
oint
, so
the
deno
min
ator
will
hav
e th
ree
zero
s.
Ste
p 2
.
S
impl
ify.
Wri
te e
ach
de
cim
al a
s a
frac
tio
n o
r m
ixe
d n
umb
er
in s
imp
lest
fo
rm.
1.
0.8
2.
0.32
3.
0.78
4.
0.45
5.
0.36
5
6.
0.00
4 7
. 1.
7 8
. 3.
92 9
. 5.
5810
. 4.
35
Wri
te e
ach
frac
tio
n o
r m
ixe
d n
umb
er
as a
de
cim
al.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
1 __ 4
1 ___
20
3 __ 8
22
___
25
7 __ 8
437
____
_ 10
00
3 9 __
__
100
1 66
7 __
___
1000
9
7 ___
35
5 3 __ 4
0.25
5 5
25
5 __
____
1,00
0
So, 0
.255
writ
ten
as a
frac
tion
is 51
__
__
200 .
51
____
200
255
551,
000
4 45
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M
Add
and
Subt
ract
Lik
e Fr
actio
nsY
ou c
an u
se a
num
ber
line
to a
dd fr
actio
ns.
Find
the
sum
, wri
te it
in s
imp
lest
fo
rm.
Ste
p 1
The
deno
min
ator
of e
ach
frac
tion
is 1
0. D
ivid
e a
num
ber
line
into
10
equa
l par
ts.
Ste
p 2
The
first
add
end
is 7 __
_ 10
. St
art
at 0
and
jum
p 7.
Th
is b
rings
you
to
7 ___
10 .
Ste
p 3
The
seco
nd a
dden
d is
. St
art
at 7 __
_ 10
and
jum
p 1
mor
e te
nth.
You
land
at
8 ___
10 , s
o 7 __
_ 10
1 1 __
_ 10
5 8
___
10 .
Writ
ten
in s
impl
est
form
, 8 __
_ 10
5 4 __ 5 .
You
can
als
o us
e a
num
ber
line
to s
ubtr
act
frac
tions
.
Find
the
diff
ere
nce
, wri
te it
in s
imp
lest
fo
rm.
Ste
p 1
The
deno
min
ator
s ar
e 3.
D
ivid
e a
num
ber
line
into
3 eq
ual p
arts
.
Ste
p 2
The
first
frac
tion
is
.
St
art
at 0
. Jum
p 2
to 2 __
3 .
Ste
p 3
The
subt
ract
ed fr
actio
n is
. Ju
mp
back
1 to
1 __
3 .
You
land
at
1 th
ird
, so
.
is w
ritte
n in
sim
ples
t fo
rm.
Find
the
sum
or
diff
ere
nce
. Wri
te it
in s
imp
lest
fo
rm.
1.2.
3. 1 __
6 1 5 __ 6
3 ___
10 1
5 ___
10
4 __
5 2 3 __ 5
7 ___
10 1
1 ___
10 2 __ 3
1 ___
10
2 __ 3 2
1 __ 3
1 __ 3
2 __ 3 2
1 __
3 =
1 __
3 1 __
3
Nam
e
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de 5
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RW
80
MX
EN
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0:01
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RW79-RW80
Grade5.indd 40Grade5.indd 40 7/19/07 5:59:40 PM7/19/07 5:59:40 PM
Mod
el A
dditi
on o
f Unl
ike
Frac
tions
Yo
u ca
n us
e f
ract
ion
bar
s to
he
lp y
ou
add
fra
ctio
ns w
ith
unlik
e d
eno
min
ato
rs.
Trad
e f
ract
ions
bar
s o
f fr
acti
ons
wit
h un
like
de
nom
inat
ors
fo
r e
qui
vale
nt b
ars
of
frac
tio
ns w
ith
like
de
nom
inat
ors
.
Wri
te t
he s
um u
sing
fra
ctio
n b
ars.
Wri
te it
in s
imp
lest
fo
rm.
So,
.
Find
eac
h su
m u
sing
fra
ctio
n b
ars.
Wri
te it
in s
imp
lest
fo
rm.
1.2.
3. 2
__ 3 1
1 __ 4
3 __
8 1 3 __ 4
5 __ 6 1
7 ___
12
1 __ 4 1
2 ___
12
1 __ 4 1
2 ___
12 5
3 ___
12 1
2 ___
12
3 ___
12 1
2 ___
12 5
5 ___
12
1 __ 4 1
2 ___
12 5
5 ___
12
1 121 4
1 121 12
1 121 12
1 12
1 12
Ste
p 1
U
se a
1 __
4 bar
and
tw
o
1 __
_ 12
bar
s to
mod
el fr
actio
ns
w
ith u
nlik
e de
nom
inat
ors.
Ste
p 2
Tr
ade
each
1 __ 4 b
ar fo
r
th
ree
1 ___
12 b
ars.
Ste
p 3
A
dd t
he fr
actio
ns w
ith
like
deno
min
ator
s.
Nam
e
© H
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urt •
Gra
de 5
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RW
81
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EN
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0:55
PM
1 2
1 101 10
1 101 10
1 10
1 101 10
1 101 10
1 10
Mod
el S
ubtr
actio
n of
Unl
ike
Frac
tions
You
can
use
frac
tion
bars
to
help
you
sub
trac
t fr
actio
ns w
ith u
nlik
e de
nom
inat
ors.
Tra
ce fr
actio
n ba
rs o
f fra
ctio
ns w
ith u
nlik
e de
nom
inat
ors
for
equi
vale
nt b
ars
of fr
actio
ns w
ith li
ke d
enom
inat
ors.
Sub
trac
t.
Ste
p 1
U
se a
1 __ 2 fr
actio
n ba
r to
mod
el t
he fi
rst
frac
tion.
Ste
p 2
Tr
ade
the
1 __ 2 b
ar fo
r
fiv
e 1 __
_ 10
bar
s.
Ste
p 3
Su
btra
ct t
he fr
actio
ns w
ith
like
deno
min
ator
s.
So,
Use
fra
ctio
n b
ars
to f
ind
the
diff
ere
nce
. Wri
te it
in s
imp
lest
fo
rm.
1.2.
3. 2 __ 3 2
1 __ 4
7 __ 8 2
1 __
2 5 __ 6 2
1 __ 3
1 __
2 2 1 __
_ 10
5 5 __
_ 10
2 1 __
_ 10
1 __ 2 2
1 ___
10 5
4 ___
10 .
Writ
ten
in s
impl
est
form
, 4 __
_ 10
5 2 __
5 .
5 ___
10 2
1 ___
10 5
4 ___
10
1 __ 2 2
1 ___
10
Nam
e
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de 5
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RW
82
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1:13
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RW81-RW82
Grade5.indd 41Grade5.indd 41 7/19/07 5:59:59 PM7/19/07 5:59:59 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
83
Estim
ate
Sum
s and
Diff
eren
ces
You
can
rou
nd fr
actio
ns t
o 0
, to
1 __ 2 ,
or t
o 1
estim
ate
sum
s an
d di
ffere
nces
.
Estim
ate
the
sum
.
Esti
mat
e e
ach
sum
or
diff
ere
nce
.
1.2.
3.
4.5.
6.
4 __ 6 1
1 __ 9
Ste
p 1
Fi
nd 4 __ 6 o
n th
e nu
mbe
r lin
e.
Is
it c
lose
st t
o 0
, 1 __ 2 ,o
r 1?
Th
e fr
actio
n 4 __ 6 is
clo
sest
to
1 __ 2 .
Ste
p 2
Fi
nd 1 __ 9 o
n th
e nu
mbe
r lin
e.
Is it
clo
sest
to
0, 1
__ 2 ,o
r 1?
Th
e fr
actio
n 1 __ 9 is
clo
sest
to
0.
Ste
p 3
To
est
imat
e th
e su
m
4 __ 6 1
1 __ 9 ,
ad
d th
e tw
o ro
unde
d nu
mbe
rs.
So,
4 __ 6 1
1 __ 9 is
abo
ut
1 __ 2 .
1 __ 2 1
0 5
1 __ 2
4 __ 6
1
1 __
8 2 __ 6
1
7 __ 8
4 __ 6
1
3 __ 8
5 __ 6
2
3 __ 8
7 __ 8
2
5 __
6 1 __
6
1
7 __ 8
MX
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e
© H
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urt •
Gra
de 5
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each
RW
84
Use
Com
mon
Den
omin
ator
sTo
add
or
subt
ract
unl
ike
frac
tions
, you
nee
d to
ren
ame
them
as
like
frac
tions
. You
can
do
this
by
mak
ing
a lis
t of
equ
ival
ent
frac
tions
. Whe
n yo
u fin
d tw
o fr
actio
ns w
ith t
he s
ame
deno
min
ator
, the
y ar
e lik
e fr
actio
ns.
Exam
ple
1: A
dd.
5 ___
12 1
1 __ 3
Ste
p 1
W
rite
equi
vale
nt fr
actio
ns fo
r 5 __
_ 12
.
5 ___
12 , 10
__
_ 24
, 15
___
36 , 20
__
_ 48
Ste
p 2
W
rite
equi
vale
nt fr
actio
ns fo
r 1 __ 3 .
1 __
3 , 2 __
6 , 3 __ 9 ,
4 ___
12
Ste
p 3
R
ewrit
e th
e pr
oble
m u
sing
the
equ
ival
ent
frac
tions
.
Th
en a
dd.
Exam
ple
2: S
ubtr
act
9 ___
10 2
1 __
2
Ste
p 1
W
rite
equi
vale
nt fr
actio
ns fo
r 9 __
_ 10
.
9 ___
10 , 18
__
_ 20
, 27
___
30 , 36
__
_ 40
Ste
p 2
W
rite
equi
vale
nt fr
actio
ns fo
r 1 __ 2 .
1 __
2 , 2 __
4 , 3 __ 6 , 4
__ 8 ,
5 ___
10
Ste
p 3
R
ewrit
e th
e pr
oble
m u
sing
the
equ
ival
ent
frac
tions
.
Th
en s
ubtr
act.
5 ___
12 1
1 __ 3 b
ecom
es 5 __
_ 12
1 4 __
_ 12
5 9 __
_ 12
. W
ritt
en i
n s
imp
lest
for
m,
9 ___
12 5
3 __ 4 .
9 ___
10 2
1 __ 2 b
ecom
es 9 __
_ 10
2 5 __
_ 10
5 4 __
_ 10
. W
ritte
n in
sim
ples
t fo
rm,
4 ___
10 5
2 __
5 .
Stop
whe
n yo
u fin
d2
frac
tions
with
lik
e de
nom
inat
ors.
Stop
whe
n yo
u fin
d2
frac
tions
with
lik
e de
nom
inat
ors.
Find
the
sum
or
diff
ere
nce
. Wri
te t
he a
nsw
er
in s
imp
lest
fo
rm.
1.2.
3.4.
5.6.
7.8.
3 __
5 1 1 __ 3
7 __
8 2 1 __
4
1 __
2 1 2 __ 5
3 __ 4 2
2 __ 3
1 __
4 1 1 __ 6
9 ___
10 2
4 __ 5
1 __ 5 1
3 __
4
8 __ 9 2
5 __
6
MX
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RW83-RW84
Grade5.indd 42Grade5.indd 42 7/19/07 6:00:15 PM7/19/07 6:00:15 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
85
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Com
pare
Str
ateg
ies
Dav
id b
ough
t so
me
supp
lies
for
the
scie
nce
proj
ect.
He
spen
t $3
.99
for
post
er b
oard
, $1.
24
for
a gl
ue s
tick,
and
$4.
55 fo
r co
lor
penc
ils. I
f Dav
id h
ad $
1.57
whe
n he
left
the
stor
e, h
ow
muc
h m
oney
did
he
have
bef
ore
his
purc
hase
s?
Re
ad
to
Un
de
rsta
nd
2. Pla
nWha
t in
form
atio
n w
ill y
ou u
se?
3. So
lve
4.Y
ou c
an w
rite
an e
quat
ion.
Let
n 5
the
am
ount
of m
oney
Dav
id h
ad b
efor
e hi
s pu
rcha
se. S
olve
n 5
3.9
9 1
1.2
4 1
4.5
5 1
1.5
7.
Ch
eck
5.H
ow m
uch
mon
ey d
id D
avid
hav
e be
fore
his
pur
chas
es?
Co
mp
are
str
ate
gie
s to
so
lve
.
Jeff
need
s to
be
at s
choo
l at
8:00
in
the
mor
ning
. He
spen
ds 1
5 m
inut
es
gett
ing
dres
sed,
15
min
utes
eat
ing
brea
kfas
t, 5
min
utes
bru
shin
g hi
s te
eth,
and
20
min
utes
rid
ing
to s
choo
l. W
hat
time
shou
ld h
e ge
t ou
t of
bed
?
Cen
tral
Sch
ool h
as 5
th-,
6th-
, 7th
-, an
d 8t
h-gr
ader
s. If
3 __ 8 a
re 6
th-g
rade
rs, 1
__ 4 a
re
7th-
grad
ers,
and
1 __ 6 a
re 8
th-g
rade
rs,
then
wha
t fr
actio
n ar
e 5t
h-gr
ader
s?
8.7.1.
W
hat
are
you
aske
d to
find
?
Wha
t ar
e tw
o di
ffere
nt s
trat
egie
s yo
u co
uld
use
to s
olve
thi
s pr
oble
m?
6.H
ow c
an y
ou c
heck
you
r an
swer
?
MX
EN
L08A
WK
5X_R
T_C
H13
_L06
.indd
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/07
1:3
2:47
PM
Choo
se a
Met
hod
You
can
use
thi
s ch
art
to d
ecid
e w
heth
er t
o us
e m
enta
l mat
h,
pape
r an
d pe
ncil,
or
a ca
lcul
ator
to
add
or s
ubtr
act
frac
tions
.
Cho
ose
a m
eth
od
. Fin
d t
he s
um. W
rite
it in
sim
ple
st f
orm
.
1 __ 5 1
7 ___
10
7 ___
12 2
5 ___
12
3 __
4 2 3 __ 8
men
tal
mat
h
pape
r an
d pe
ncil
calc
ulat
or
Deno
min
ator
s ar
e al
ike.
X
One
deno
min
ator
is a
mul
tiple
of
the
othe
r.
X
Deno
min
ator
s ar
e no
t alik
e an
d ar
e no
t mul
tiple
s of
one
an
othe
r.
X
The
frac
tions
do
not
have
like
den
omin
ator
s: 1 __
5 1 7 __
_ 10
5
1
0
Is o
ne d
enom
inat
or a
mul
tiple
of t
he o
ther
? Y
es
5 is
a m
ultip
le o
f 10.
The
char
t st
ates
tha
t if
one
deno
min
ator
is a
mul
tiple
of t
he o
ther
, us
e pa
per
and
penc
il.
Rew
rite
usin
g eq
uiva
lent
frac
tions
: 1 __
5 1 7 __
_ 10
bec
omes
2 ___
10 1
7 ___
10 5
9 ___
10 .
So, 1
__ 5 1
7 ___
10 5
9 ___
10 .
Cho
ose
a m
eth
od
. Fin
d t
he s
um o
r d
iffe
renc
e.
Wri
te it
in s
imp
lest
fo
rm.
1.2.
3.5 __
9 1 2 __ 6
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
86
MX
EN
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H13
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.indd
1
6/15
/07
1:3
4:37
PM
RW85-RW86
Grade5.indd 43Grade5.indd 43 7/19/07 6:00:31 PM7/19/07 6:00:31 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
87
Mod
el A
dditi
on o
f Mix
ed N
umbe
rsFr
actio
n ba
rs c
an h
elp
you
add
mix
ed n
umbe
rs.
Ad
d. W
rite
the
ans
we
r in
sim
ple
st f
orm
.
STE
P 1
Dra
w a
pic
ture
to
show
the
add
ends
.
1. 5
1 __
10 1
2 3 __
10
2. 2
3 _ 8 1
1 1 _ 4
3. 3
1 _ 2 1
4 1 _ 5
4.
4
2 _ 3
1
2 3 _ 4
5.
5
5 _ 8
1
4 1 _ 2
6.
2
5 _ 6
1
3 1 _ 6
1 1 __ 4 1
1 1 __ 3
1 1 __ 4
1 1 __ 3
1 1 __ 4
1 1 __ 3
STE
P 3
Cou
nt li
ke b
ars
to fi
nd t
he s
um.
Two
1 ba
rs a
nd s
even
1 ___
12 b
ars
is 2
7 ___
12 .
Use
fra
ctio
n b
ars
to f
ind
the
sum
. Wri
te it
in s
imp
lest
fo
rm.
STE
P 2
Dra
w li
ke fr
actio
n ba
rs t
o m
odel
1 1 __
4 and
1 1 __ 3 .
So, 1
1 __
4 1 1
1 __ 3 5
2 7 __
_ 12
.
1
__
12
1
__
12
1
__
12
1
__
12
1
__
12
1
__
12
1
__
12
1 11 1
1 _
4
1 _
3
MX
EN
L08A
WK
5X_R
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9:3
1:29
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
88
Mod
el S
ubtr
actio
n of
Mix
ed N
umbe
rsFr
actio
n ba
rs c
an h
elp
you
subt
ract
mix
ed n
umbe
rs.
Dra
w a
pic
ture
to
fin
d t
he d
iffe
renc
e.
Wri
te t
he a
nsw
er
in s
imp
lest
s fo
rm.
STE
P 1
Dra
w a
pic
ture
to
show
.
STE
P 2
Ren
ame
the
bars
with
1 __
4 bar
s.
2
1 _ 2 5
2 4 _ 8 .
2
4 _ 8 2
1 3 _ 8 5
1 1 _ 8 .
So, 2
1 _ 2 2
1 3 _ 8 5
1 1 _ 8 .
Use
fra
ctio
n b
ars,
or
dra
w a
pic
ture
to
fin
d t
he d
iffe
renc
e.
Wri
te it
in s
imp
lest
fo
rm.
2 1 _ 22
1 3 _ 8
2 1 _ 2
2 1 _ 2
2 4 _ 8
2. 3
4 _ 5 2
2 1 __
10
1. 6
1 _ 3 2
4 1 _ 6
3. 6
3 _ 4 2
4 1 _ 8
2 1 _ 2
4. 4
2 _ 5 2
1 3 __
10
5. 5
5 _ 6 2
2 1 _ 4
6. 2
2 _ 3 2
1 5 __
12
STE
P 3
Subt
ract
1 3 __ 8 . C
ross
out
bar
s to
sub
trac
t 1 3
__ 8 .
C
ount
the
bar
s th
at a
re le
ft.
MX
EN
L08A
WK
5X_R
T_C
H14
_L02
.indd
1
6/13
/07
9:4
1:07
AM
RW87-RW88
Grade5.indd 44Grade5.indd 44 6/18/07 6:03:40 PM6/18/07 6:03:40 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
89
Reco
rd A
dditi
on a
nd S
ubtr
actio
nW
hen
you
add
or s
ubtr
act
mix
ed n
umbe
rs, y
ou m
ay n
eed
tore
nam
e th
e fr
actio
ns a
s fr
actio
ns w
ith a
co
mm
on
deno
min
ato
r.
Find
the
sum
. Wri
te t
he a
nsw
er
in s
imp
lest
fo
rm.
.
Ste
p 1
Mod
el
and
.
Ste
p 2
The
LC
D fo
r
a
nd
is
tw
elfth
s, s
o re
nam
e
as
a
nd
as
.
Ste
p 3
Add
the
frac
tions
.
5
9 __
12 1
1 4 __
12 5
13
__
12 .
Ste
p 4
Add
the
who
le n
umbe
rs.
5 9 __
12 1
1 4 __
12 5
6
Add
the
sum
s: 13
__
12 1
6 5
6 13
__
12.
Writ
e th
e an
swer
in s
impl
est
form
7 1 __
12 .
So, 5
3 _ 41
1 1 _ 3 5
7 1 __
12 .
Find
the
sum
or
diff
ere
nce
. Wri
te it
in s
imp
lest
fo
rm.
5 3 _ 41
1 1 _ 3
3 _ 4
1 _ 3
1. 2
2 _ 9 1
4 1 _ 6
3. 1
1 7 _ 8 2
9 5 _ 6
5 3 _ 4
1 1 _ 3
5 3 _ 4
5
9 __
12
1
1 _ 3
1
4 __
12
4. 1
8 3 _ 5 2
14
1 _ 2
2. 1
0 5 _ 6 1
5 3 _ 4
MX
EN
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6/13
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9:4
6:00
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
90
Subt
ract
ion
with
Ren
amin
gW
hen
you
subt
ract
mix
ed n
umbe
rs y
ou m
ay n
eed
to r
enam
e th
e w
hole
num
bers
.
Use
fra
ctio
n b
ars
to f
ind
the
diff
ere
nce
. Wri
te it
in s
imp
lest
fo
rm.
2 1 _ 22
1 3 _ 4
Mod
el 2
1 _ 2 u
sing
tw
o w
hole
bar
s an
d on
e 1 _ 2 ba
r.
Thin
k of
the
LC
D fo
r 1 _ 2 an
d 3 _ 4. R
enam
e 1 _ 2 as
2 _ 4.
Ren
ame
one
who
le b
ar a
s 4 _ 4.
Use
fra
ctio
n b
ars
to f
ind
the
diff
ere
nce
. Wri
te it
in s
imp
lest
fo
rm.
2. 5
3 __
10 2
3 7 __
10 5
1. 3
1 _ 8 2
1 5 _ 5 5
3.
4 1 _ 3 2
1 5 _ 6 5
4. 6
1 _ 3 2
2 3 _ 4 5
5. 5
1 _ 6 2
2 5 _ 6 5
6.
4 1 _ 5 2
1 3 _ 5 5
Subt
ract
, 2
1 _ 22
1 3 _ 4 or
1 6 _ 42
1 3 _ 4 5
3 _ 4.
So,
2 1 _ 22
1 3 _ 4 5
3 _ 4.
MX
EN
L08A
WK
5X_R
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6/13
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9:4
8:11
AM
RW89-RW90
Grade5.indd 45Grade5.indd 45 7/19/07 6:00:50 PM7/19/07 6:00:50 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
91
Prac
tice
Addi
tion
and
Subt
ract
ion
Estim
ate.
The
n w
rite
the
sum
or
diffe
renc
e in
sim
ples
t fo
rm.
Ad
d.
Esti
mat
e.
Find
the
sum
or
diff
ere
nce
. W
rite
it in
sim
ple
st f
orm
.
1.2.
3.4.
STE
P 1
STE
P 2
STE
P 3
STE
P 4
Find
the
LC
D.
Sub
trac
t.
Esti
mat
e.
Esti
mat
e.
3
1 9
_
12
Writ
e eq
uiva
lent
fr
actio
ns u
sing
the
LC
D.
Add
the
who
le
num
bers
. A
dd
the
frac
tions
.
Writ
e th
e an
swer
in
sim
ples
t fo
rm.
3: 3
, 6, 9
, 12,
15
...5:
5, 1
0, 1
5, 2
0 ...
15 2
7 5
8
STE
P 1
STE
P 2
STE
P 3
Find
the
LC
D.
5: 5
,10,
15, .
.., 4
59:
9,1
8,27
, ...,
45
Writ
e eq
uiva
lent
fr
actio
ns u
sing
the
LC
D.
15 1 __ 5 5
15
139
____
539 5
15
9 __
_ 45
Ren
ame
so y
ou c
an s
ubtr
act.
Subt
ract
th
e w
hole
num
bers
. Sub
trac
t th
e fr
actio
ns.
Writ
e th
e an
swer
in s
impl
est
form
.
15 9 __
_ 45
5
14 54
__
_ 45
2 7
15
___
45
__
7 39
__
_ 45
5 7
394
3 __
____
454
3 5 7
13
___
15
4
2 __ 9
1 8
11
___
15
__
12 3 __ 8
2 8
5 __
6
_
2 2 __ 7
1 9
1 __
2
_
15 1 __ 3
2 7
7 __
_ 12
__
7 3 __ 9 5
7 3 3
5 ____
935 5
7 15
__
_ 45
2 2
__ 3
1 8
3 __ 5
2 3
53
3 5
2 2 __ 3 5
2 3
3 3
5 3
38 3
__ 5 5
8
5 8
9 ___
15
2 10
__
_ 15
1 8
9 ___
15
__
10 19
__
_ 15
10
19
___
15 5
10
1 15
__
_ 15
1 4 __
_ 15
5 1
1 4 __
_ 15
So,
2 2 __
3
1 8
3 __
5
_
11 14
__
_ 15
5 2
10
___
15
15 1 __ 5 2
7 3 __ 9 5
8
MX
EN
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6/13
/07
9:5
4:14
AM
Player
DistanceClues
Lenn
yno
t the
sho
rtes
t hit;
7 1 __
2 ft fa
rthe
r tha
n R
aul;
3 1 __
8 ft le
ss th
an S
tanl
ey
Rau
lLe
nny
and
Son
ya h
it th
e ba
ll fa
rthe
r.
Sta
nley
not t
he s
hort
est h
it
Son
yano
t the
sho
rtes
t hit;
1 7 __
8 ft fa
rthe
r tha
n R
aul
7.6.N
ame
© H
arco
urt •
Gra
de 5
Ret
each
RW
92
Ch
ec
k
5.
How
did
the
str
ateg
y he
lp y
ou s
olve
the
pro
blem
?
Prob
lem
Sol
ving
Wor
ksho
pSt
rate
gy: U
se L
ogic
al R
easo
ning
Lenn
y, R
aul,
Sta
nle
y, a
nd S
ony
a w
rote
do
wn
the
ir lo
nge
st h
its
dur
ing
bat
ting
pra
ctic
e. T
he
sho
rte
st o
f th
e 4
hit
s w
as 1
00 3 __ 4 f
ee
t. L
enn
y hi
t th
e b
all 7
1
__ 2 f
ee
t fa
rthe
r th
an R
aul,
but
3 1
__ 8
fee
t le
ss t
han
Sta
nle
y. S
ony
a hi
t th
e b
all 1
7
__ 8 f
ee
t fa
rthe
r th
an R
aul d
id. O
f S
tanl
ey,
Rau
l, S
ony
a, a
nd L
enn
y, w
ho h
it t
he b
all t
he f
arth
est
, and
wha
t d
ista
nce
was
the
bal
l hit
?
Re
ad
to
Un
de
rsta
nd
1.
How
co
uld
you
wri
te t
he q
uest
ion
as a
fill-
in-t
he b
lank
sen
tenc
e.
Pla
n
2.
How
can
you
org
aniz
e th
e da
ta t
o so
lve
the
prob
lem
?
Am
y, B
rad,
Cla
renc
e, a
nd D
eidr
e w
rote
do
wn
the
tota
l dis
tanc
es t
hey
ran
last
w
eek.
The
fart
hest
dis
tanc
e ra
n w
as
19 1 __
8 mile
s.
Dei
dre
ran
8 5 __
8 mile
s fa
rthe
r th
an A
my
and
3 3 __
8 mile
s fa
rthe
r th
an
Bra
d.
Cla
renc
e ra
n 6
17
___
20 m
iles
less
tha
n B
rad.
W
ho r
an t
he s
hort
est
dist
ance
, an
d w
hat
was
tha
t di
stan
ce?
In O
ctob
er, 2
00 s
tude
nts
in F
iona
’s
grad
e w
atch
bas
ebal
l and
foot
ball.
If
132
stud
ents
wat
ch fo
otba
ll an
d 11
9 w
atch
bas
ebal
l, ho
w m
any
stud
ents
w
atch
bot
h ba
seba
ll an
d fo
otba
ll?
4.
Who
hit
the
ball
the
fart
hest
, and
wha
t di
stan
ce w
as t
he b
all h
it?
Use
log
ical
re
aso
ning
to
so
lve
.
So
lve
3.
C
ompl
ete
the
tabl
e.
Rem
embe
r th
at t
he s
hort
est
hit
was
100
3 __
4 feet
.
MX
EN
L08A
WK
5X_R
T_C
H14
_L06
.indd
1
6/15
/07
1:3
5:42
PM
RW91-RW92
Grade5.indd 46Grade5.indd 46 7/19/07 6:01:08 PM7/19/07 6:01:08 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
93
Mod
el M
ultip
licat
ion
of F
ract
ions
You
can
use
grid
pap
er t
o m
odel
mul
tiplic
atio
n of
frac
tions
. Y
ou w
ill u
se t
he g
rid t
o m
ake
and
shad
e a
rect
angl
e.
Find
the
pro
duc
t:
Use
the
den
omin
ator
of t
he s
econ
d fa
ctor
, 2, t
o m
ake
colu
mns
.Sh
ade
the
left
colu
mn
yello
w.
Use
the
den
omin
ator
of t
he fi
rst
fact
or, 2
, to
mak
e ro
ws.
Shad
e 1 __ 2 o
f the
yel
low
col
umn
blue
.
• T
he a
rea
of t
he r
ecta
ngle
is 2
3 2
, or
4 sq
uare
uni
ts.
• T
he a
rea
of t
he b
lue
shad
ed p
art
is 1
3 1
, or
1 sq
uare
uni
ts.
• T
he fr
actio
n of
the
par
t th
at is
sha
ded
blue
is 1
__ 4.
So,
1 __ 4
1 __ 2
1 __ 2
35
.
1.2.
3.4.
Find
the
pro
duc
t.
5.
6.7.
8.
9.10
.11
.12
.
1 __ 5 3
2 __ 3 5
3 __ 4 3
2 __ 5 5
5 __
8 3 4 __ 5 5
3 __
5 5 1 __
3 3
2 __ 3 3
5 __ 6 5
1 __ 5 3
3 __ 4 5
3 __
4 3 2 __ 3 5
7 __
8 3 2 __
3 5
1 __ 2
1 __ 2
3
Y BY
Wri
te t
he p
rod
uct
eac
h m
od
el r
ep
rese
nts.
MX
EN
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1
6/13
/07
11:
33:0
0 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
94
Reco
rd M
ultip
licat
ion
of F
ract
ions
You
can
use
an
area
mod
el t
o re
cord
mul
tiplic
atio
n of
frac
tions
.
Find
the
pro
duc
t. W
rite
the
ans
we
r in
sim
ple
st f
orm
.
Use
the
den
omin
ator
of t
he s
econ
dfa
ctor
, 4, t
o m
ake
colu
mns
.
Then
use
the
den
omin
ator
of
the
first
fact
or, 3
, to
mak
ero
ws.
Sha
de 2 __
3 of t
he
shad
ed c
olum
ns.
• Th
e ar
ea o
f the
rec
tang
le t
hat
is s
hade
d tw
ice
is 6
squ
are
units
.
• Th
e re
ctan
gle
has
an a
rea
of 1
2 sq
uare
uni
ts.
Use
the
tw
o ar
eas
to w
rite
a fr
actio
n.
area
of r
ecta
ngle
sha
ded
twic
e
__
____
____
____
____
____
____
__
area
of b
ig r
ecta
ngle
5
6 ___
12
So,
.
So, i
n si
mpl
est
form
, 2 __
3 3 3 __
4
Find
the
pro
duc
t. W
rite
in s
imp
lest
fo
rm.
1.
2.
3.
4.
5.
6.
7.
8.
9.
2 __
3 3 3 __
4 5 6 __
_ 12
5 6 __
_ 12
5 6
4 6
______
12
4 6
5 1 __
2
2 __ 3 3
3 __
4
Shad
e 3 __ 4 o
f the
col
umns
.
3 __ 5 3
5 __ 6 5
4 __ 9 3
3 __
8 5 1 __
2
3 6
__ 7 5
1 __ 3 3
2 __ 5 5
9 ___
10 3
5 __
_ 12
5 1 __
8 3 4
__ 9 5
2 __ 3 3
9 __
_ 10
5 5 __ 8 3
2 __
3 5 2 __
3 3 4
__ 5 5
MX
EN
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H15
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/07
10:
01:0
7 A
M
RW93-RW94
Grade5.indd 47Grade5.indd 47 6/18/07 6:04:09 PM6/18/07 6:04:09 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
95
Mul
tiply
Fra
ctio
ns a
nd W
hole
Num
bers
You
can
use
a m
odel
to
mul
tiply
a fr
actio
n an
d a
who
le n
umbe
r.
Find
the
pro
duc
t.
Ste
p 1
Dra
w 1
0 re
ctan
gles
.
Ste
p 2
The
den
omin
ator
of t
he fr
actio
n 4 __
5 is
5.
Thi
s m
eans
the
re a
re 5
equ
al p
arts
. So
d
ivid
e th
e re
ctan
gles
into
5 e
qual
par
ts.
Ste
p 3
The
num
erat
or o
f the
frac
tion
4 __ 5 i
s 4.
So s
hade
4 p
arts
of e
ach
rect
angl
e.
Ste
p 4
Cou
nt t
he s
hade
d pa
rts.
T
here
are
40
shad
ed p
arts
.
So, 1
0 3
4 ___
5 5
40
___ 5 5
8
1. 2
.
3.
4. 5
. 6
.
7. 8
. 9.
Find
the
pro
duc
t.
2 3
1 __
4 5 2 __ 3 3
5 5
9 3
1 __
5 5
1 __ 3 3
6 5
1 __ 4 3
10
52
3 5 __
6 5
5 3
3 __
4 5 2 __ 5 3
4 5
1 __ 2 3
3 5
10 3
4 __ 5
MX
EN
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10:
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5 A
M
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e
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urt •
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de 5
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RW
96
2 2 __ 3 5
(3 3
2)
1 2
__
____
____
_ 3
5
8 __ 3
Mul
tiply
with
Mix
ed N
umbe
rsY
ou c
an u
se p
aper
and
pen
cil t
o m
ultip
ly m
ixed
num
bers
.
Find
the
pro
duc
t.
Ste
p 1
Writ
e ea
ch m
ixed
num
ber
as a
frac
tion.
Firs
t m
ultip
ly t
he d
enom
inat
or b
y th
e w
hole
num
ber,
then
add
the
num
erat
or.
Ste
p 2
Mul
tiply
the
frac
tions
.
Ste
p 3
Writ
e th
e pr
oduc
t in
sim
ples
t fo
rm
So, 1
1 __ 2 3
2 2 __ 3 5
4.
1.2.
3.
4.5.
6.
7.8.
9.
2 1 __ 2 3
3 1 __
5 3 __
8 3 2
2 __ 3
2 5 __ 8 3
1 1 __ 2
1 4 __ 5 3
2 1 __
5 1
3 __ 4 3
1 2 __
5 2
2 __ 3 3
2 2 __ 3
1 1 __ 3 3
1 __ 2
3 5 __ 6 3
1 1 __
3 1
5 __
8 3 3
1 __ 3
Find
the
pro
duc
t.
1 1 __ 2 5
(2 3
1)
1 1
__
____
____
_ 2
5 3 __ 2
3 __ 2 3
8 __ 3 5
24
___ 6
24
___ 6 5
4
1 1 __
2 3 2
2 __
3
MX
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10:
07:4
2 A
M
RW95-RW96
Grade5.indd 48Grade5.indd 48 6/18/07 6:04:19 PM6/18/07 6:04:19 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
97
Mod
el F
ract
ion
Divi
sion
You
can
use
pic
ture
s to
mod
el d
ivis
ion
of fr
actio
ns.
Div
ide
.
Ste
p 1
Dra
w o
ne w
hole
rec
tang
le a
nd s
hade
one
fifth
of i
t.
Ste
p 2
Div
ide
the
part
s in
to t
ent
hs.
Ste
p 3
Cou
nt t
he n
umbe
r of
sha
ded
tent
hs.
Ther
e ar
e 2
tent
hs in
1 __ 5 .
So, 1
__ 5 4
1 ___
10 5
2.
Use
fra
ctio
n b
ars
to f
ind
the
quo
tie
nt.
1.2.
3.4.
5.6.
7.8.
1 __ 2 4
1 __ 4
1 __ 4 4
1 ___
12
1 __ 2 4
1 __
2 1 __ 2 4
1 ___
10
1 4
1 __ 5
3 4
1 __
3 2 4
1 __
5 3 4
1 __ 4
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 ___
10
1 __ 5 4
1 ___
10
MX
EN
L08A
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5X_R
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H15
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.indd
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/07
10:
08:2
8 A
M
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
98
Divi
de W
hole
Num
bers
by
Frac
tions
Find
the
quo
tie
nt. W
rite
it in
sim
ple
st f
orm
.
Ste
p 1
Rew
rite
4 a
s a
frac
tion.
Ste
p 2
Use
the
rec
ipro
cal o
f the
div
isor
to
writ
e a
mul
tiplic
atio
n pr
oble
m.
Ste
p 3
Mul
tiply
Ste
p 4
Rew
rite
prod
uct
as a
mix
ed fr
actio
n, if
nee
ded.
So,
4 4
7 __
8 5 4
4 __ 7 .
1.2.
3.4.
5.6.
7.8.
1 4
3 __ 4
3 4
3 __ 5
6 4
2 __ 3
4 4
1 __
2
4 4
2 __ 5
4 4
2 __ 3
3 4
6 __ 7
1 4
5 __
8
Find
the
quo
tie
nt. W
rite
it in
sim
ple
st f
orm
.
4 __ 1 4
7 __ 8
4 __ 1 3
8 __ 7
4 __ 1 3
8 __ 7 5
32
___ 7
Thin
k: W
rite
4 as
4 __ 1 .
Thin
k: T
he r
ecip
roca
l
of 7 __
8 is 8 __ 7 .
32
___ 7 5
4 4 __ 7
4 4
7 __ 8
MX
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L08A
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5X_R
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H15
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.indd
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/07
10:
10:2
7 A
M
RW97-RW98
Grade5.indd 49Grade5.indd 49 6/18/07 6:04:27 PM6/18/07 6:04:27 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
99
Ste
p 1
Lo
cate
the
div
isor
.
Ste
p 2
W
rite
the
reci
proc
al o
f the
di
viso
r.
Ste
p 3
Writ
e th
e m
ultip
licat
ion
prob
lem
.
The
divi
sor
is a
lway
s th
e se
cond
num
ber.
To w
rite
the
reci
proc
al, s
witc
h th
e nu
mer
ator
and
de
nom
inat
or o
f the
div
isor
.
Rep
lace
the
div
isor
with
its
reci
proc
al a
nd c
hang
e 4
to
3. T
he m
ixed
num
ber
3 1 __
3
writ
ten
as a
frac
tion
is 1
0
__
3
So, 3
1 __ 3 4
3 __ 8 5
80
___ 9 5
8 8 __ 9
.
We
can
use
a m
odel
to
writ
e a
divi
sion
sen
tenc
e.
Writ
e a
divi
sion
sen
tenc
e fo
r
Wri
te a
div
isio
n se
nte
nce
fo
r e
ach
mo
de
l.
1.2.
Div
ide
. Wri
te t
he a
nsw
er
in s
imp
lest
fo
rm.
3.4.
5. 6.
Divi
de F
ract
ions
We
can
rew
rite
all d
ivis
ion
prob
lem
s as
mul
tiplic
atio
n pr
oble
ms.
Div
ide
3 1 __ 3 4
3 __ 8 .
Wri
te t
he a
nsw
er
in s
imp
lest
fo
rm.
Writ
e th
e di
visi
on p
robl
em a
s a
mul
tiplic
atio
n pr
oble
m.
2 1 __ 8 4
3 __ 4
5 __ 9 4
5 __ 6
4 3 __
_ 10
4 1 __ 5
3 2 __ 7 4
5 ___
14
14 1 __ 4 5
4
3 1 __
3 4 3 __ 8
3 1 __
3 4 3 __ 8
3 1 __ 3 4
3 __ 8
8 __ 3
10
___ 3 3
8 __
3 5 80
__
_ 9
MX
EN
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H15
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/07
10:
11:2
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M
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e
© H
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urt •
Gra
de 5
Ret
each
RW
100
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill: C
hoos
e th
e O
pera
tion
Tany
a ha
s 10
fis
h in
he
r aq
uari
um. O
f th
ese
fis
h, 2 __ 5 a
re g
old
fish.
H
ow
man
y fis
h ar
e g
old
fish?
2.W
hat
oper
atio
n ca
n yo
u us
e to
sol
ve t
he p
robl
em?
Expl
ain
your
cho
ice.
3.W
hat
equa
tion
can
you
writ
e to
sol
ve?
4.H
ow m
any
fish
are
gold
fish?
5.H
ow c
an y
ou c
heck
to
see
if yo
ur a
nsw
er is
rea
sona
ble?
Tell
whi
ch o
pe
rati
on
you
wo
uld
use
to
so
lve
the
pro
ble
m.
The
n so
lve
.
Tw
elve
stu
dent
s in
Dan
ita’s
co
okin
g cl
ass
play
a m
usic
al
inst
rum
ent.
This
is 3 __ 4 o
f the
ent
ire
cl
ass.
How
man
y st
uden
ts a
re in
D
anita
’s c
ooki
ng c
lass
?
Lanc
e sp
ends
1 __
30 o
f eac
h ye
ar
stud
ying
on
wee
kend
s an
d 12
__
_ 60
of
eac
h ye
ar s
tudy
ing
on w
eekd
ays.
Wha
t fr
actio
n of
the
yea
r do
es
Lanc
e st
udy?
7.6.1.
Wha
t ar
e yo
u as
ked
to fi
nd?
MX
EN
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3 A
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RW99-RW100
Grade5.indd 50Grade5.indd 50 6/18/07 6:04:37 PM6/18/07 6:04:37 PM
5.
tria
ngle
s to
hea
rts
6.
circ
les
to a
ll sh
apes
Unde
rsta
nd a
nd E
xpre
ss R
atio
sA
rat
io is
the
com
paris
on o
f tw
o qu
antit
ies.
Rat
ios
can
com
pare
par
t to
par
t, pa
rt t
o w
hole
, or
who
le to
part
.
Wri
te t
he r
atio
in t
hre
e w
ays.
The
n na
me
the
typ
e o
f ra
tio
.
Red
str
ipes
:blu
e st
ripes
.
Cou
nt t
he s
trip
es o
f eac
h co
lor.
5 re
d st
ripes
3 bl
ue s
trip
es
The
ratio
can
be
writ
ten
as:
5
to
3
5
:3
5 __ 3
The
type
of r
atio
is: p
art
to p
art.
Wha
t is
the
rat
io o
f b
lue
str
ipe
s to
all
stri
pe
s?
3 st
ripes
8
str
ipes
in a
ll
The
ratio
can
be
writ
ten:
3 t
o 8,
3:8
, or
3 __ 8
The
type
of r
atio
is: p
art
to w
hole
.
Wri
te e
ach
rati
o in
thr
ee
way
s. T
hen
nam
e t
he t
ype
of
rati
o.
1.
circ
les
to t
riang
les
2.
all s
hape
s to
tria
ngle
s
4.
hear
ts t
o al
l sha
pes
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
101
Red
Red
Red
Red
Blu
e
Red
Blu
e
Blu
e
3.
circ
les
to h
eart
s
MX
EN
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de 5
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RW
102
Alge
bra:
Equ
ival
ent R
atio
s and
Pro
port
ions
Equi
vale
nt r
atio
s m
ake
the
sam
e co
mpa
rison
. You
can
use
pic
ture
s to
mod
el
equi
vale
nt r
atio
s.
Wri
te t
o e
qui
vale
nt r
atio
s fo
r 4
to 6
. Use
mul
tip
licat
ion
and
div
isio
n.
Writ
e th
e ra
tio a
s a
frac
tion
with
the
firs
t nu
mbe
r be
ing
the
num
erat
or a
nd the
sec
ond
num
ber
bein
g th
e de
nom
inat
or.
4
to
6 be
com
es 4 __ 6
Use
mul
tiplic
atio
n to
writ
e an
equ
ival
ent
frac
tion.
Mul
tiply
the
num
erat
or a
nd t
he
deno
min
ator
by
the
sam
e nu
mbe
r.
4 __ 6 =
(43
2)
____
__
(63
2) =
8 ___
12 =
8 to
12
Use
div
isio
n to
writ
e an
equ
ival
ent
frac
tion.
D
ivid
e th
e nu
mer
ator
and
the
den
omin
ator
by
the
sam
e nu
mbe
r.
4 __ 6 =
(44
2)
_____
(64
2) =
2 __
3 =
2 t
o 3
So, t
wo
equi
vale
nt r
atio
s fo
r 4
to 6
are
8 t
o 12
an
d 2
:3.
A p
rop
ort
ion
is a
n eq
uatio
n th
at s
how
s th
at 2
rat
ios
are
equa
l. 2 __
3 5 4 __
6 and
8 ___
12 5
2 __ 3 a
re p
ropo
rtio
ns.
Wri
te t
wo
eq
uiva
lent
rat
ios
for
eac
h ra
tio
. U
se m
ulti
plic
atio
n an
d d
ivis
ion.
1.
15
___
20
2.
12 t
o 16
3.
5:25
4.
9 to
18
Tell
whe
the
r th
e r
atio
s fo
rm a
pro
po
rtio
n. W
rite
ye
s o
r no
.
5.
5 __ 8 a
nd 1
0 __
_ 13
6
. 4 _ 7 a
nd 12
__
_ 21
7
. 35
___
40 a
nd 7 _ 8
8.
28
___
42 a
nd 5 __ 6
MX
EN
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/07
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1 A
M
RW101-RW102
Grade5.indd 51Grade5.indd 51 7/19/07 6:05:05 PM7/19/07 6:05:05 PM
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
103
Ratio
s and
Rat
esA
rat
e is
a r
atio
tha
t co
mpa
res
two
quan
titie
s ha
ving
diff
eren
t un
its o
f mea
sure
.
A u
nit
rate
is a
rat
e th
at w
hen
writ
ten
as a
frac
tion
has
a 1
as it
s de
nom
inat
or.
Wri
te t
he r
atio
in f
ract
ion
form
. The
n fin
d t
he u
nit
rate
. 77
0 m
iles
in 1
4 ho
urs
Writ
e th
e ra
te a
s fr
actio
n
R
ate
Uni
t R
ate
The
unit
rate
, 55,
is t
he n
umbe
r of
mile
s tr
avel
ed p
er h
our.
Wri
te e
ach
rati
o in
fra
ctio
n fo
rm.
1.
23
1 m
iles
in 3
hou
rs 2
. 65
mile
s in
5 h
ours
3.
$2.7
0 fo
r 6
frui
t ba
rs
Wri
te e
ach
rati
o in
fra
ctio
n fo
rm. T
hen
find
the
uni
t ra
te.
4.
72 d
og to
ys in
12
boxe
s
5.
$42
for
6 ho
urs
of w
ork
6.
27 p
lant
s in
3 r
ows
7.
24 p
aint
ings
in 1
2 da
ys 8
. 62
4 m
iles
in 1
3 ho
urs
9.
$1.1
4 fo
r 6
ears
of c
orn
The
deno
min
ator
is 1
4.
Div
ide
the
num
erat
or a
nd
deno
min
ator
by
14.
770
____
14
770 4
14
____
____
_ 14
4 1
4 5 55
__
_ 1 5 5
5
770
____
14
mile
s __
____
hour
s
MX
EN
L08A
WK
5X_R
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H16
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/07
10:
38:3
7 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
104
*
*
1 in
ch 5
200
mile
s1
inch
0
Unde
rsta
nd M
aps a
nd S
cale
sA
scal
e d
raw
ing
is a
red
uced
or
enla
rged
dr
awin
g w
hose
sha
pe is
the
sam
e as
an
actu
al o
bjec
t an
d w
hose
siz
e is
de
term
ined
by
the
scal
e.
Am
ap s
cale
is t
he r
atio
tha
t co
mpa
res
the
|di
stan
ce o
n a
map
to
the
actu
al d
ista
nce.
The
map
dis
tanc
e is
giv
en.
Fin
d t
he
actu
al d
ista
nce
of
3.6
in. T
he s
cale
is
1 in
. 5 2
00 m
i.
Writ
e ra
tios
for
the
map
sca
le a
nd t
he a
ctua
l dis
tanc
e.1:
200
dist
ance
on
map
3sc
ale
actu
al d
ista
nce
13
200
200
3.6
320
072
0
So, t
he a
ctua
l dis
tanc
e is
720
mile
s.
Co
mp
lete
the
rat
io t
able
.
The
map
dis
tanc
e is
giv
en.
Fin
d t
he a
ctua
l dis
tanc
e.
The
sca
le is
1 in
. = 3
00 m
i.
2.6.
5 in
. 3.
1.2
5 in
. 4.
3 i
n.
5. 4
.5 i
n.
Map
Dis
tanc
e (c
m)
12
3A
ctua
l Dis
tanc
e (k
m)
728
1.
Nor
thgl
en
Geo
rget
own
CO
LOR
AD
O
MX
EN
L08
AW
K5X
_RT
_CH
16_L
4.in
ddA
7/11
/07
4:50
:11
PM
RW103-RW104
Grade5.indd 52Grade5.indd 52 7/11/07 5:32:39 PM7/11/07 5:32:39 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
105
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy: M
ake
a Ta
ble
Tani
a w
eig
hs 8
0 p
oun
ds
on
Eart
h an
d 7
7 p
oun
ds
on
Ve
nus.
Tan
ia’s
fat
her
we
ighs
200
p
oun
ds.
Ho
w m
uch
wo
uld
Tan
ia’s
fat
her
we
igh
on
Ve
nus?
Re
ad
to
Un
de
rsta
nd
1. H
ow c
ould
you
writ
e th
e qu
estio
n as
a fi
ll-in
-the
-bla
nk s
ente
nce?
Pla
n
2. H
ow c
an m
akin
g a
tabl
e he
lp y
ou a
nsw
er t
he q
uest
ion?
So
lve
3. M
ake
and
com
plet
e th
e ta
ble
to s
olve
the
pro
blem
.
1 D
ivid
e. 2
00 4
80 5
2
.5
Wei
ght
on
Ear
th80
200
Wei
ght
on
Ven
us77
2 M
ultip
ly.
2.5
3 7
7 5
192
.5
4. H
ow c
ould
you
writ
e yo
ur a
nsw
er in
a c
ompl
ete
sent
ence
.
Ch
eck
5. H
ow c
an y
ou c
heck
to
see
if yo
ur a
nsw
er is
rea
sona
ble?
Exp
lain
.
6. I
n ae
robi
cs c
lass
, the
stu
dent
s ar
e gr
oupe
d by
age
: 21
–30,
31–
40, 4
1–50
, and
51+
. Th
e ag
es o
f the
stu
dent
s ar
e 25
, 68,
33
, 61,
46,
59,
46,
29,
35,
28,
21,
51
, 47,
and
79.
Mak
e a
tabl
e to
sh
ow h
ow m
any
stud
ents
are
in
each
age
gro
up.
MX
EN
L08A
WK
5X_R
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H16
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indd
A
6/13
/07
10:
55:2
1 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
106
Unde
rsta
nd P
erce
ntPe
rce
nt is
the
frac
tion
part
of a
num
ber
to 1
00. F
ract
ions
and
de
cim
als
can
be w
ritte
n as
per
cent
s. T
he s
ymbo
l for
per
cent
is %
. Th
is s
ymbo
l mea
ns “
per
hund
red.
”
Wri
te a
fra
ctio
n an
d a
pe
rce
nt t
o r
ep
rese
nt
the
sha
de
d p
art.
Th
e gr
id is
div
ided
into
100
squ
ares
.
60
squ
ares
are
sha
ded.
Writ
e th
e fr
actio
n as
par
ts:
shad
ed
____
___
who
le
5 60
__
__
100 .
Fr
actio
n: 60
__
__
100
Pe
rcen
t: 60
%
D
ecim
al:
60
____
100
Wri
te a
de
cim
al a
nd a
pe
rce
nt t
o r
ep
rese
nt
the
sha
de
d p
art.
54
of t
he h
undr
ed s
quar
es a
re s
hade
d.
D
ivid
e. 5
4 4
100 5
0.5
4
Fr
actio
n: 54
__
__
100
Pe
rcen
t: 54
%
D
ecim
al: 0
.54.
Wri
te a
fra
ctio
n an
d a
pe
rce
nt t
o r
ep
rese
nt t
he s
had
ed
par
t.
1.
2.
3.
Wri
te a
de
cim
al a
nd a
pe
rce
nt t
o r
ep
rese
nt t
he s
had
ed
par
t.
4.
5.
6
.
MX
EN
L08A
WK
5X_R
T_C
H16
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/07
10:
56:0
1 A
M
RW105-RW106
Grade5.indd 53Grade5.indd 53 6/18/07 6:05:09 PM6/18/07 6:05:09 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
107
Frac
tions
, Dec
imal
s, an
d Pe
rcen
tsPe
rcen
ts c
an b
e w
ritte
n as
dec
imal
s or
as
frac
tions
and
vic
e ve
rsa.
Wri
te 9
8% a
s a
de
cim
al a
nd a
s a
frac
tio
n in
sim
ple
st f
orm
.
•
98%
mea
ns n
inet
y-ei
ght
hund
redt
hs, o
r 0.
98.
•
98%
als
o m
eans
98
___
100
To w
rite
a fr
actio
n in
sim
ples
t fo
rm, d
ivid
e th
e nu
mer
ator
and
the
den
omin
ator
by
the
sam
e nu
mbe
r. K
eep
doin
g th
is u
ntil
1 is
the
onl
y co
mm
on fa
ctor
.
98
__
__
100 5
98 4
2
____
____
100 4
2 5
49
___
50
So, 9
8% a
s a
deci
mal
5 0
.98
98%
as
a fr
acti
on 5
49
___
50
Wri
te e
ach
pe
rce
nt a
s a
de
cim
al a
nd a
s a
frac
tio
n in
sim
ple
st f
orm
.
1.
80%
2.
35%
3.
40%
4.
32%
5.
76%
6.
51%
Wri
te e
ach
frac
tio
n o
r d
eci
mal
as
a p
erc
ent
.
7.
3 __ 5
8.
0.45
9.
9 ___
10
10.
8 ___
20
11.
0.10
12.
0.62
MX
EN
L08A
WK
5X_R
T_C
H16
_L7.
indd
A
6/13
/07
10:
56:3
2 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
108
Find
Per
cent
of a
Num
ber
You
can
use
tw
o-co
lor
coun
ters
to
mod
el p
erce
nt o
f a n
umbe
r.
1.Wha
t is
90%
of
10?
Sh
ow 1
0 co
unte
rs. E
ach
coun
ter
repr
esen
ts 1
0%.
10%
20%
30%
40%
50%
60%
70%
80%
90
%10
0%
Ea
ch c
ount
er a
lso
repr
esen
ts 1
, sin
ce 1
0 3
1 5
10.
12
34
56
78
910
Fl
ip o
ver
90%
of t
he c
ount
ers.
10%
20%
C
ount
by
1s to
fi nd
90%
of 1
0.
So, 9
0% o
f 10
is 9
.
Wha
t is
25%
of
24?
Show
24
coun
ters
.
Sinc
e 25
% 5
1 _ 4 ,
sepa
rate
the
cou
nter
s in
to
4 eq
ual g
roup
s.
12
Cha
nge
the
colo
r of
the
cou
nter
s in
1 o
f the
4 g
roup
s.C
ount
the
cou
nter
s w
hose
col
or y
ou c
hang
ed.
So, 2
5% o
f 24
is 6
.
Find
the
pe
rce
nt o
f e
ach
num
be
r.2.
3.4.
5.6.
7.8.
40%
of 8
090
% o
f 20
50%
of 1
275
% o
f 32
40%
of 3
550
% o
f 26
25%
of 2
075
% o
f 28
MX
EN
L08A
WK
5X_R
T_C
H16
_L8.
indd
1
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/07
10:
57:4
2 A
M
RW107-RW108
Grade5.indd 54Grade5.indd 54 6/18/07 6:05:19 PM6/18/07 6:05:19 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
109
Out
com
es a
nd P
roba
bilit
yA
n e
vent
is t
he s
et o
f one
or
mor
e ou
tcom
es in
a p
roba
bilit
yex
perim
ent.
The
sam
ple
sp
ace
of a
n ev
ent
is t
he s
et o
f all
poss
ible
out
com
es. I
f the
sam
ple
spac
e is
div
ided
into
equ
alpa
rts,
eac
h pa
rt is
eq
ually
like
ly t
o be
sel
ecte
d. T
he
the
ore
tica
l pro
bab
ility
can
be
expr
esse
d in
wor
ds, a
s a
ratio
, or
as a
frac
tion:
P(
even
t) 5
nu
mbe
r of
favo
rabl
e ou
tcom
es
____
____
____
____
____
____
____
___
nu
mbe
r of
equ
ally
like
ly o
utco
mes
.
The
prob
abili
ty o
f an
even
t ca
n ra
nge
from
impo
ssib
le t
oce
rtai
n, o
r fr
om 0
to
1.
Use
the
sp
inne
r to
wri
te t
he p
rob
abili
ty o
f sp
inni
ng a
co
lor
oth
er
than
blu
e.
• F
irst,
find
the
num
ber
of e
qual
ly li
kely
out
com
es
6
equa
l sec
tions
.
• T
hen,
pla
ce t
he n
umbe
r of
equ
ally
like
ly o
utco
mes
in t
he p
roba
bilit
y fo
rmul
a:
P(
colo
r ot
her
than
blu
e) 5
num
ber
of fa
vora
ble
outc
omes
____
____
____
____
____
____
____
6
•
Nex
t, co
unt
the
num
ber
of fa
vora
ble
outc
omes
4 se
ctio
ns
•
Plac
e th
e nu
mbe
r of
favo
rabl
e ou
tcom
es in
the
num
erat
or o
f the
pro
babi
lity
form
ula:
P(
colo
r ot
her
than
blu
e) 5
4 __ 6
Use
the
sp
inne
r to
fin
d t
he p
rob
abili
ty o
f sp
inni
ng e
ach
eve
nt.
1.
red
2.
blue
3.
a co
lor
othe
r th
an r
ed
Y
Y
GR
B
B
1
2
3
4
MX
EN
L08A
WK
5X_R
T_C
H17
_L1.
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1
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/07
1:3
7:43
PM
Num
ber o
f Pul
ls
412
9To
tal
Red
Blue
Green
Ma
rble
Exp
erim
ent
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
110
Prob
abili
ty E
xper
imen
tsPr
ob
abili
ty is
the
like
lihoo
d th
at a
n ev
ent
will
hap
pen.
A p
red
icti
on
is a
rea
sona
ble
gues
s ab
out
the
poss
ible
out
com
e, o
r re
sult,
of a
pro
babi
lity
expe
rimen
t.
An
exp
eri
me
ntal
pro
bab
ility
of a
n ev
ent
is t
he r
atio
of t
he n
umbe
r of
tim
es a
n ev
ent
occu
rs t
o th
e to
tal
num
ber
of t
imes
the
act
ivity
is p
erfo
rmed
.
A b
ox
cont
ains
6 b
lack
mar
ble
s an
d 3
whi
te m
arb
les.
Tob
y sh
ake
s th
e b
ox
and
pul
ls a
mar
ble
wit
hout
lo
oki
ng. H
e r
eco
rds
the
co
lor
mar
ble
on
a ta
llyta
ble
, re
pla
ces
the
mar
ble
, and
re
pe
ats
the
exp
eri
me
nt.
Pred
ict
the
num
ber
of t
imes
out
of 5
0 pu
llsth
at D
ylan
wou
ld g
et a
mar
ble
that
is e
ither
red
or b
lue.
Cou
nt t
he t
otal
num
ber
of p
ulls
/tal
ly m
arks
.
25 Add
the
tim
es a
red
mar
ble
was
pul
led
to h
ow m
any
times
a b
lue
mar
ble
was
pul
led.
12
1 4
5 1
6
Mul
tiply
by
2 si
nce
25 g
oes
into
50
2 tim
es.
16 3
2 5
32
So, y
ou c
an p
redi
ct t
hat
a re
d or
blu
e m
arbl
e w
ill b
e pu
lled
32 t
imes
out
of 5
0.
For
1–3,
use
the
tab
le.
1. S
arah
pul
led
a til
e fr
om a
bag
, rec
orde
d its
col
or,
and
put
the
tile
back
in t
he b
ag 2
0 tim
es. P
redi
ct
how
man
y tim
es o
ut o
f 40
pulls
tha
t Sa
rah
wou
ldpu
ll a
red
tile
from
the
bag
.
2. W
hat i
s th
e ex
perim
enta
l pro
babi
lity
of S
arah
pul
ling
a bl
ue ti
le?
a re
d til
e? a
gre
en ti
le?
3. B
ased
on
the
expe
rimen
tal p
roba
bilit
ies,
wou
ld y
ou
pred
ict
that
Sar
ah w
ould
pul
l a b
lue
tile
or g
reen
tile
mor
e of
ten
if sh
e pu
lled
tiles
from
the
bag
60
mor
e tim
es?
Expl
ain.
Color
Red
7 3 10
Blu
e
Gre
en
Tally
Total
Tile
Exp
erim
ent
MX
EN
L08A
WK
5X_R
T_C
H17
_L2.
indd
1
6/15
/07
1:3
8:17
PM
RW109 RW110
Grade5.indd 55Grade5.indd 55 6/18/07 6:05:30 PM6/18/07 6:05:30 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
111
Prob
abili
ty a
nd P
redi
ctio
nsEx
perim
enta
l pro
babi
lity
is a
num
ber
take
n fr
om a
n ac
tual
situ
atio
n. T
o fin
de
xpe
rim
ent
al p
rob
abili
ty o
f an
even
t, w
rite
the
ratio
of t
he n
umbe
r of
tim
esth
e ev
ent
occu
rs t
o to
tot
al n
umbe
r of
tria
ls.
Ex
perim
enta
l pro
babi
lity 5
num
ber
of t
imes
an
even
t oc
curs
____
____
____
____
____
____
____
__
tota
l num
ber
of t
rials
.
Exp
ress
the
exp
eri
me
ntal
pro
bab
ility
of
6 he
ads
in 1
4 co
into
sse
s as
a f
ract
ion
in s
imp
lest
fo
rm. T
hen
pre
dic
t th
e
out
com
e o
f 35
mo
re t
oss
es.
Use
the
sp
inne
r to
wri
te t
he p
rob
abili
ty o
f sp
inni
ng a
co
lor
oth
er
than
blu
e.
•
Firs
t, fin
d th
e nu
mbe
r of
tria
ls
14
coi
n to
sses
•
The
n, p
lace
the
num
ber
of t
rials
as
a de
nom
inat
or in
the
pro
babi
lity
form
ula:
P(he
ads)
5 nu
mbe
r of
tim
es a
n ev
ent
occu
rs
____
____
____
____
____
____
____
__
14
•
Nex
t, co
unt
the
num
ber
of t
imes
an
even
t oc
curs
4 se
ctio
ns
•
Pla
ce t
he n
umbe
r of
favo
rabl
e ou
tcom
es in
the
num
erat
or o
f the
pro
babi
lity
form
ula
and
sim
plify
:
P(co
lor
othe
r th
an b
lue)
5 6 __
_ 14
5 3 __
7
•
Fin
ally
, to
pred
ict
the
outc
ome
of fu
ture
tria
ls, m
ultip
ly t
he p
roba
bilit
y by
the
nu
mbe
r of
futu
re t
rials
.
P(co
lor
othe
r th
an b
lue)
5 3 __ 7 3
35
___ 1 5
15
So, 6
hea
ds in
14
coin
tos
ses
is 3 __ 7 in
sim
ples
t fo
rm.
You
can
exp
ect
15 h
eads
in 3
5 co
in t
osse
s.
Exp
ress
the
exp
eri
me
ntal
pro
bab
ility
of
as a
fra
ctio
n in
sim
ple
st f
orm
. The
n p
red
ict
the
out
com
es
of
futu
re t
rial
s.
1.
12 b
lue
mar
bles
in 2
0 pu
lls; 3
0 m
ore
pulls
2.
8 ev
ens
in 1
2 nu
mbe
r cu
be t
osse
s; 4
5 m
ore
toss
es
3.
4 pu
rple
in 2
4 sp
ins;
60
mor
e sp
ins
MX
EN
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PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
112
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Mak
e an
Org
aniz
ed L
istN
ia, S
am, a
nd T
oni
are
wai
ting
in li
ne t
o b
e s
eat
ed
at
Kay
e’s
Re
stau
rant
. Ho
w m
any
diff
ere
nt w
ays
can
the
y lin
e u
p?
Re
ad
to
Un
de
rta
nd
1.
Wha
t m
ath
deta
ils c
an y
ou id
entif
y?
2.
Wha
t ar
e yo
u as
ked
to fi
nd?
Pla
n
3.
Wha
t st
rate
gy c
an y
ou u
se t
o so
lve
the
prob
lem
?
So
lve
4.
How
can
you
use
the
str
ateg
y to
sol
ve t
he p
robl
em?
5.
Wha
t is
the
ans
wer
to
the
ques
tion?
Ch
eck
6.
How
do
you
know
you
r an
swer
is c
orre
ct?
Mak
e a
list
to
so
lve
.
7.
How
man
y di
ffere
nt b
reak
fast
s ca
nJo
shua
mak
e if
he c
hoos
e ex
actly
tw
odi
ffere
nt it
ems
from
this
list
eac
h m
orni
ng?
cer
eal,
frui
t, yo
gurt
, egg
8.
Supp
ose
Josh
ua’s
bro
ther
Ste
ven
choo
ses
3 ite
ms
each
mor
ning
inst
ead
of
two.
How
man
y di
ffere
nt b
reak
fast
s ca
n he
mak
e fr
om J
oshu
a’s
list.
MX
EN
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RW111-RW112
Grade5.indd 56Grade5.indd 56 6/18/07 6:05:42 PM6/18/07 6:05:42 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
113
Tree
Dia
gram
sTo
find
all
poss
ible
out
com
es o
f an
even
t, yo
u ca
n us
e ei
ther
of
the
follo
win
g:
•
Tr
ee
dia
gra
m. A
tre
e di
agra
m li
sts
all p
ossi
ble
choi
ces
or o
utco
mes
.
•
Fu
ndam
ent
al C
oun
ting
Pri
ncip
le. I
n Fu
ndam
enta
l Cou
ntin
g Pr
inci
ple,
the
to
tal p
ossi
ble
outc
omes
of t
wo
even
ts c
an b
e st
ated
as
a m
ultip
licat
ion
form
ula.
One
eve
nt h
as m
poss
ible
out
com
esA
noth
er e
vent
has
n
poss
ible
out
com
esTo
tal o
utco
mes
of
both
eve
nts
occu
rrin
g to
geth
er
35
Dra
w a
tre
e d
iag
ram
or
use
the
Fun
dam
ent
al C
oun
ting
Pri
ncip
leto
fin
d t
he t
ota
l num
be
r o
f o
utco
me
s fo
r us
ing
tw
o s
pin
ners
, bo
thw
ith
3 e
qua
l se
ctio
ns c
olo
red
re
d, b
lue
, and
ye
llow
.
Dra
w a
tre
e di
agra
m.
Firs
t, M
ake
a co
lum
n fo
r ea
ch e
vent
. Nex
t, lis
t al
l pos
sibl
e O
utco
mes
for
the 1s
t ev
ent
and
the
seco
nd e
vent
.
Spin
ner 1
Spin
ner 2
Out
com
es
red
red
red,
red
blue
red,
blu
eye
llow
red,
yel
low
blue
red
blue
, red
blue
blue
, blu
eye
llow
blue
, yel
low
yello
wre
dye
llow
, red
blue
yello
w, b
lue
yello
wye
llow
, yel
low
So, t
here
are
9 p
ossi
ble
outc
omes
.
Dra
w a
tre
e d
iag
ram
or
use
the
Fun
dam
ent
al C
oun
ting
Pri
ncip
leto
fin
d t
he t
ota
l num
be
r o
f o
utco
me
s.
1. c
hoos
ing
outfi
ts w
ith a
gr
een,
blu
e, o
r re
d sh
irt,
an
d w
hite
or
tan
shor
ts
2. t
ossi
ng a
coi
n an
d to
ssin
g a
num
ber
cube
la
bele
d 1
to 6
.
3. t
ossi
ng a
num
ber
cube
la
bele
d 1
to 6
and
sp
inni
ng a
spi
nner
la
bele
d 1
to 4
.
MX
EN
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PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
114
Com
bina
tions
and
Arr
ange
men
tsA
co
mb
inat
ion
is a
sel
ectio
n of
diff
eren
t ite
ms
in w
hich
the
orde
r is
not
impo
rtan
t. A
n ex
ampl
e of
a c
ombi
natio
n is
a t
able
plac
e se
ttin
g th
at c
onsi
sts
of a
cup
, sau
cer,
plat
e, a
nd b
owl.
List
ing
them
in t
hat
orde
r or
as
bow
l, pl
ate,
cup
, and
sau
cer
does
not
cha
nge
the
com
bina
tion
of d
ishe
s.
An
arra
nge
me
nt is
a g
roup
of i
tem
s th
at a
re o
rder
ed. A
n ex
ampl
e of
an
arra
ngem
ent
is a
ssig
ned
seat
s in
a m
ath
clas
s.O
nly
one
pers
on is
ass
igne
d an
y pa
rtic
ular
sea
t.
Mak
e a
list
or
dra
w a
tre
e d
iag
ram
to
fin
d t
he t
ota
l num
be
r o
f p
oss
ibili
tie
s.
pizz
as w
ith c
hoic
es o
f thi
ck o
r th
in c
rust
and
eith
er c
hees
e or
pe
pper
oni t
oppi
ng
Ste
p 1
Dec
ide
if it
is a
com
bina
tion
or a
n ar
rang
emen
t.
Th
e pi
zza
is t
he s
ame
if yo
u or
der
the
crus
t ei
ther
befo
re o
r af
ter
the
topp
ing.
So,
thi
s is
a c
ombi
natio
n.
O
rder
doe
s no
t m
atte
r.
Ste
p 2
Lis
t al
l pos
sibl
e ou
tcom
es e
ach
even
t. A
tab
le c
an h
elp
yo
u. M
ake
sure
eac
h to
ppin
g is
list
ed w
ith e
ach
type
of c
rust
.
Ste
p 3
Cou
nt t
he n
umbe
r of
piz
za c
hoic
es.
So, t
here
are
four
pos
sibl
e co
mbi
natio
ns.
Cru
stTo
pp
ing
thin
chee
se
thin
pepp
eron
i
thic
kch
eese
thic
kpe
pper
oni
Mak
e a
list
or
dra
w a
tre
e d
iag
ram
to
fin
d t
he t
ota
l po
ssib
le c
hoic
es.
1.
way
s to
arr
ange
the
le
tter
s T,
E, A
.2.
w
ays
to a
rran
ge le
tter
s C
,A
, T s
o th
ey fo
rm a
rea
l w
ord
3.
way
s th
at A
my,
Gre
g, L
isa
and
Mar
tin c
an s
tand
in a
tic
ket
line
MX
EN
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RW113-RW114
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C
A
DB
H
EG
F
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
115
Poin
ts, L
ines
, and
Ang
les
In g
eom
etry
, obj
ects
hav
e sp
ecia
l nam
es.
A p
oin
t is
an
exac
t lo
catio
n.
A p
lane
is a
n en
dles
s fla
t su
rfac
e.
A li
ne is
a s
trai
ght
path
in a
pla
ne. I
t ha
s no
end
poin
ts a
nd c
an b
e na
med
by
any
two
poin
ts o
n th
e lin
e.
A li
ne s
eg
me
nt is
par
t of
a li
ne b
etw
een
two
endp
oint
s an
d al
l of t
hepo
ints
bet
wee
n th
em. U
se b
oth
endp
oint
s to
nam
e a
line
segm
ent.
In
ters
ect
ing
line
s ar
e lin
es t
hat
cros
s at
exa
ctly
one
poi
nt.
Pe
rpe
ndic
ular
line
s ar
e tw
o lin
es t
hat
inte
rsec
t to
form
rig
ht a
ngle
s.
Pa
ralle
l lin
es
are
lines
in a
pla
ne t
hat
are
alw
ays
the
sam
e di
stan
ce a
part
.
Use
the
fig
ure
. Nam
e a
n e
xam
ple
of
inte
rse
ctin
g li
nes.
Dra
w t
he f
ollo
win
g o
n th
e g
rid
. Lab
el y
our
dra
win
gs.
1.pa
ralle
l lin
es
So, ‹ _
› A
D a
nd ‹ _
› B
C a
re in
ters
ectin
g lin
es.
‹ _ ›
AD
and
‹ _ ›
BC
cro
ss a
t ex
actly
one
poi
nt.
Look
for
lines
tha
t cr
oss
at e
xact
ly o
ne p
oint
.
Rig
ht a
ngle
5 90°
Acu
te a
ngle
, 90°
Ob
tuse
ang
le .
90°
Str
aig
ht a
ngle
5 180
°
2.pe
rpen
dicu
lar
lines
3.in
ters
ectin
g lin
es
4.ac
ute
angl
e
5.rig
ht a
ngle
6.ob
tuse
ang
le
MX
EN
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5X_R
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59:0
6 A
M
T
U
W
Y
ZX
T
U
W
Y
ZX
T
U
WY
Z
X1
24
5
mm
cm1
23
45
67
89
10
0180
10170
20160
30150
40140
501306012
07011
08010
010
080
110
70
120
60
13050
140
40150
30 160
20 17010
1800
90
JK
L
M NX
T
U
W
Y
ZX
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
116
Mea
sure
and
Dra
w A
ngle
sY
ou c
an u
se b
ench
mar
ks t
o es
timat
e an
gle
mea
sure
s.
0°45
°90
°13
5°18
0°
Esti
mat
e t
he m
eas
ure
of /UXY.
Thin
k: /
UX
Y a
ppea
rs t
o be
gre
ater
tha
n 45
˚./
UX
Y a
ppea
rs t
o be
less
tha
n 13
5˚.
Com
pare/
UX
Y t
o th
e be
nchm
ark 90
˚.
The
mea
sure
of /
UX
Y is
abo
ut 90˚
.
Use
a p
rotr
acto
r to
me
asur
e /
UX
Y.A
pro
trac
tor
is a
too
l for
mea
surin
g an
gles
.
Ste
p1
Thin
k: P
oint
X is
the
ver
tex.
Ray
s _
›X
Y a
nd _
›X
U fr
om t
he s
ides
of a
n an
gle.
Esti
mat
e t
he m
eas
ure
of
eac
h an
gle
. The
n us
e a
pro
trac
tor
to f
ind
the
me
asur
e.
1./
LXN
2./
KX
N3./
MX
N4./
KX
L
Ste
p2
Plac
e th
e ce
nter
poi
nt o
f the
pr
otra
ctor
on
the
vert
ex, p
oint
X
, so
that
_›
UX
pas
ses
thro
ugh 0˚
.
Ste
p3
Mea
sure/
UX
Y. S
o, /
UX
Y is
105
°.
MX
EN
L08
AW
K5X
_RT
_CH
18_L
02.in
dd1
7/11
/07
5:39
:19
PM
RW115-RW116
Grade5.indd 58Grade5.indd 58 7/19/07 6:01:28 PM7/19/07 6:01:28 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
117
Poly
gons
A p
oly
go
n is
a c
lose
d pl
ane
figur
e fo
rmed
by
thre
e or
mor
e lin
e se
gmen
ts. P
olyg
ons
are
nam
ed b
y th
e nu
mbe
r of
sid
es a
nd a
ngle
s. A
re
gul
ar p
oly
go
n ha
s si
des
that
are
all
the
sam
e le
ngth
and
ang
les
that
are
all
the
sam
e m
easu
re.
Re
gul
ar P
oly
go
nsTr
iang
leQ
uadr
ilate
ral
Pent
agon
Hex
agon
Oct
agon
Dec
agon
3 si
des
and
angl
es4
side
s an
d an
gles
5 si
des
and
angl
es6
side
s an
d an
gles
8 si
des
and
angl
es10
sid
es a
nd
angl
es
A p
olyg
on t
hat
has
side
s an
d an
gles
tha
t ar
e no
t th
e sa
me
mea
sure
is n
ot r
egul
ar, b
ut it
is
still
nam
ed b
y th
e nu
mbe
r of
sid
es a
nd a
ngle
s.
Poly
go
ns t
hat
Are
No
t R
eg
ular
Tria
ngle
Qua
drila
tera
lPe
ntag
onH
exag
onO
ctag
onD
ecag
on
3 si
des
and
angl
es4
side
s an
d an
gles
5 si
des
and
angl
es6
side
s an
d an
gles
8 si
des
and
angl
es10
sid
es a
nd
angl
es
Nam
e t
he p
oly
go
n b
elo
w a
nd t
ell
if it
is r
eg
ular
or
not
reg
ular
.
T
he p
olyg
on h
as 1
0 si
des.
It
is a
dec
agon
.
Mea
sure
the
sid
es t
o se
e if
they
are
eac
h th
e sa
me
leng
th.
T
he s
ides
are
not
eac
h th
e sa
me
leng
th.
So, t
he fi
gure
is a
pen
tago
n th
at is
not
reg
ular
.
Nam
e e
ach
po
lyg
on
and
te
ll if
it is
re
gul
ar o
r no
t re
gul
ar.
1.2.
3.4.
MX
EN
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M
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e
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urt •
Gra
de 5
Ret
each
RW
118
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill:
Iden
tify
Rela
tions
hips
Co
nne
ct t
he v
ert
ice
s w
ithi
n a
squa
re, a
nd a
re
gul
ar p
ent
ago
n.A
sq
uare
and
a r
eg
ular
pe
ntag
on
are
sho
wn
at t
he r
ight
. C
oun
t th
e li
nes
wit
hin
eac
h fig
ure
. Ho
w m
any
line
s w
oul
d y
ou
dra
w w
ithi
n a
reg
ular
he
xag
on?
1. W
hat
are
you
aske
d to
do?
2. C
onne
ct t
he v
ertic
es o
f the
hex
agon
at
the
right
.H
ow m
any
lines
did
you
dra
w?
3. U
sing
the
rel
atio
nshi
ps a
lread
y de
mon
stra
ted
by t
he t
able
, how
wou
ld y
ou fi
ll ou
t th
e m
issi
ng p
arts
of t
he t
able
?
Num
be
r o
f S
ide
s4
56
Num
be
r o
f Li
nes
Co
nne
ctin
g V
ert
ice
s2
5
6. U
sing
the
info
rmat
ion
from
the
top
of t
he
page
, pre
dict
how
man
y lin
es w
ill b
e dr
awn
for
7-, 8
-, an
d 9-
side
d fig
ures
if
you
conn
ect
thei
r ve
rtic
es.
7. T
he d
ista
nce
arou
nd a
reg
ular
hex
agon
w
ith 1
-inch
sid
es is
6 in
ches
. The
dis
tanc
e ar
ound
a r
egul
ar h
exag
on w
ith 2
-inch
si
des
is 1
2 in
ches
, and
the
dis
tanc
e ar
ound
a r
egul
ar h
exag
on w
ith 3
-inch
si
des
is 1
8 in
ches
. Wha
t is
the
dis
tanc
e ar
ound
a r
egul
ar h
exag
on w
ith 6
-inch
si
des?
+3
Ide
ntify
the
re
lati
ons
hip
. The
n so
lve
.
MX
EN
L08A
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M
RW117-RW118
Grade5.indd 59Grade5.indd 59 6/18/07 6:06:12 PM6/18/07 6:06:12 PM
V13
.5 c
m
V
So, t
he d
iam
eter
is t
wic
e th
is le
ngth
or 7
cm.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
119
Circ
les
A c
ircl
e is
rou
nd. A
ll of
its
poin
ts a
re t
he s
ame
dist
ance
from
its
cent
er. A
circ
le is
nam
ed b
y its
cen
ter
poin
t.
Look
at
circ
le B
.
A r
adiu
s of
a c
ircle
con
nect
s th
e ce
nter
of t
he c
ircle
with
any
poin
t on
the
circ
le. B
L is
a r
adiu
s. B
M a
nd B
P ar
e al
so r
adii.
A d
iam
ete
r of
a c
ircle
pas
ses
thro
ugh
the
cent
er a
nd h
as it
endp
oint
s on
the
circ
le. M
P is
a d
iam
eter
.
• Th
e le
ngth
of a
dia
met
er is
alw
ays
twic
e th
e le
ngth
of a
rad
ius.
A c
hord
has
its
endp
oint
s on
the
circ
le. N
O a
nd M
P ar
e ch
ords
.
Co
mp
lete
. The
n us
e a
co
mp
ass
to d
raw
the
cir
cle
. Lab
el t
he m
eas
ure
me
nts.
R
adiu
s: 3.5
cm
D
iam
eter
:
Y
ou k
now
tha
t th
e ra
dius
is 3.5
cm
.
Use
a c
ompa
ss s
et a
t 3.5
cm t
o dr
aw t
his
circ
le a
roun
d a
cent
er p
oint
.
Lab
el y
our
cent
er p
oint
, poi
nt V
.
NB
L
P
M
O
Use
dat
a fo
r 1–
4, u
se t
he c
ircl
e a
t th
e r
ight
.1.
Nam
e th
e ci
rcle
.2.
Nam
e a
diam
eter
.
3. N
ame
a ch
ord.
4. N
ame
a ra
dius
.
Co
mp
lete
5–6
. The
n us
e a
co
mp
ass
to d
raw
eac
h ci
rcle
.D
raw
and
lab
el t
he m
eas
ure
me
nts.
5. r
adiu
s 5
1.5
cm
diam
eter
5
6. r
adiu
s 5
di
amet
er 5
2.5
cm
X F
E
H
Cir
cle
B
MX
EN
L08A
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/07
11:
16:0
6 A
M
S
T
R
WV
U
E GH
F
J L
M
K
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
120
Cong
ruen
t and
Sim
ilar F
igur
esFi
gure
s ca
n be
con
grue
nt, s
imila
r, bo
th, o
r ne
ither
.C
ong
rue
nt f
igur
es
have
the
sam
e si
ze a
nd t
hesa
me
shap
e. S
imila
r fig
ure
s ha
ve t
he s
ame
shap
e,bu
t m
ay o
r m
ay n
ot b
e th
e sa
me
size
. Co
rre
spo
ndin
gan
gle
s an
d co
rre
spo
ndin
g s
ide
s ar
e in
the
sam
ere
late
d po
sitio
n in
diff
eren
t an
gles
.
Tell
whe
the
r th
e t
wo
fig
ure
s at
the
rig
ht a
pp
ear
to
be
cong
ruen
t,simila
r, o
r ne
ither
.
The
figu
re o
n th
e le
ft ap
pear
s to
hav
e gr
eate
r he
ight
tha
n th
e fig
ure
on t
he r
ight
.
Thi
nk, d
o th
e fig
ures
app
ear
to b
e th
e sa
me
shap
e?
•
N
o
Thi
nk, d
o th
e fig
ures
app
ear
to b
e th
e sa
me
size
?
•
N
o
So, t
he t
wo
figur
es a
re n
eith
er.
Ide
ntify
the
ang
le t
hat
corr
esp
ond
s to
/T
Fi
gure
s R
ST a
nd U
VW
are
con
grue
nt.
/
R c
orre
spon
ds t
o /
U
/S
corr
espo
nds
to /
VSo
, /T
corr
espo
nds
to /
W.
Wri
te w
heth
er
the
tw
o f
igur
es
app
ear
to
be
cong
ruen
t, simila
r, o
r ne
ither
.1.
2.3.
4.
Ide
ntify
the
co
rre
spo
ndin
g s
ide
or
ang
le.
5. E
F6.
GH
7. /
G8.
/M
MX
EN
L08A
WK
5X_R
T_C
H18
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.indd
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/07
11:
16:3
0 A
M
RW119-RW120
Grade5.indd 60Grade5.indd 60 6/18/07 6:06:23 PM6/18/07 6:06:23 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
121
Sym
met
ryIf
you
fold
the
pic
ture
of t
he li
ght
bulb
in h
alf a
long
the
das
hed
line,
th
e tw
o pa
rts
will
mat
ch e
xact
ly.
The
light
bul
b ha
s lin
e sy
mm
etry
.
Tell
whe
the
r th
e p
arts
on
eac
h si
de
of
the
line
mat
ch.
Is t
he li
ne a
line
of
sym
me
try?
Use
a p
iece
of p
aper
to
trac
e th
e fig
ure
and
line.
Cut
out
the
figu
re a
nd li
ne o
f you
pie
ce o
f pap
er.
Fold
the
cut
-out
figu
re a
long
the
das
hed
line.
So, t
he li
ne o
n th
is fi
gure
is a
line
of s
ymm
etry
.
Tell
whe
the
r th
e p
arts
on
eac
h si
de
of
the
line
mat
ch. I
s th
elin
e a
line
of
sym
me
try?
1.2.
3.4.
5.6.
7.8.
Dra
w li
nes
of
sym
me
try.
Te
ll w
heth
er
eac
h fig
ure
has
rota
tio
nal s
ymm
etr
y. W
rite
yes
or no
.9.
10.
11.
12.
A fi
gure
has
rot
atio
nal s
ymm
etry
if
it ca
n be
rot
ated
less
tha
n 36
0˚ar
ound
a c
ente
r po
int
and
still
m
atch
the
org
inal
figu
re.
MX
EN
L08A
WK
5X_R
T_C
H18
_L07
.indd
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/07
11:
30:4
8 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
122
8 cm
7 cm
2 cm
9 cm
14 c
m
15 c
m
5 in
.
5 in
.
5 in
.
10 c
m
4 cm
10 c
m
Clas
sify
Tria
ngle
sY
ou c
an c
lass
ify t
riang
les
eith
er a
ccor
ding
to
thei
r si
des
or a
ccor
ding
to
thei
r an
gles
.
Use
a r
uler
to
com
pare
sid
e le
ngth
s.U
se t
he c
orne
r of
a p
aper
to
clas
sify
ang
les.
eq
uila
tera
l tri
ang
le: A
ll si
des
are
the
sam
e le
ngth
.ac
ute
tri
ang
le: A
ll th
ree
angl
es
are
acut
e.
iso
sce
les
tria
ngle
: Tw
o si
des
are
the
sam
e le
ngth
.o
btu
se t
rian
gle
: One
ang
le is
obt
use
and
the
othe
r tw
o an
gles
are
acu
te.
scal
ene
tri
ang
le: A
ll si
des
are
of d
iffer
ent
leng
ths.
rig
ht t
rian
gle
: One
ang
le is
rig
ht
and
the
othe
r tw
o an
gles
are
acu
te.
Cla
ssify
the
tri
ang
le. W
rite
iso
sce
les,
sca
lene
, or
eq
uila
tera
l.
It
has
no e
qual
sid
e le
ngth
s.
It
is a
sca
lene
tria
ngle
.
Cla
ssify
the
tri
ang
le. W
rite
acu
te, r
ight
, or
ob
tuse
.
It
has
thre
e an
gles
, eac
h le
ss t
han
90
°.
It
is a
n ac
ute
tria
ngle
.
Cla
ssify
eac
h tr
iang
le. W
rite
isosceles
, sca
lene
, or eq
uilateral.
1.
2.
3.
Cla
ssify
eac
h tr
iang
le. W
rite
acu
te, right
, or obtus
e.
4.
5.
6.
MX
EN
L08
AW
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11
:38:
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M
RW121-RW122
Grade5.indd 61Grade5.indd 61 6/18/07 6:06:34 PM6/18/07 6:06:34 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
123
Clas
sify
Qua
drila
tera
lsY
ou c
an u
se t
his
char
t to
cla
ssify
qua
drila
tera
ls.
Cla
ssify
eac
h fig
ure
in a
s m
any
way
s as
po
ssib
le.
Wri
te q
uadr
ilate
ral,
para
llelo
gram
, squ
are,
rec
tang
le, r
hom
bus,
or
trap
ezo
id.
1.
2.
De
scri
be
eac
h q
uad
rila
tera
l usi
ng p
aral
lel,
perp
endi
cula
r, an
d c
ong
ruen
t.
3.
4.
qua
dri
late
ral
4 si
des
par
alle
log
ram
quad
rilat
eral
oppo
site
sid
es a
re p
aral
lel
oppo
site
sid
es c
ongr
uent
trap
ezo
idqu
adril
ater
alon
e pa
ir of
par
alle
l sid
es
rect
ang
lepa
ralle
logr
am4
right
ang
les
rho
mb
uspa
ralle
logr
am4
cong
ruen
t si
des
squa
rerh
ombu
sre
ctan
gle
Cla
ssify
the
fig
ure
in a
s m
any
way
s as
po
ssib
le.
Wri
te q
uadr
ilate
rals
, par
alle
logr
am, s
quar
e, r
ecta
ngle
, rho
mbu
s, o
r tr
apez
oid
.
It ha
s 4
side
s so
it is
a q
uadr
ilate
ral.
Its o
ppos
ite s
ides
are
par
alle
l and
con
grue
nt s
o it
is a
pa
ralle
lgra
m.
It ha
s 4
right
ang
les
with
opp
osite
sid
es t
hat
are
para
llel a
nd
cong
ruen
t. H
owev
er, n
ot a
ll of
its
side
s ar
e co
ngru
ent,
so it
is
a r
ecta
ngle
.
MXE
NL08
AW
K5X_
RT_C
H19_
L02.
indd
1
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/07
9:4
4:51
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
124
1 in
.
1 in
.1
in.
1 in
.
1 in
.
1 in
.
1 in
.
1 in
.1
in.
1 in
.
1 in
.
1 in
.
Draw
Pla
ne F
igur
esY
ou c
an d
raw
a p
lane
figu
re b
ased
on
a de
scrip
tion.
Dra
w t
he f
igur
e o
n a
coo
rdin
ate
pla
ne. C
lass
ify t
he f
igur
e.
A q
uadr
ilate
ral w
ith 4
rig
ht a
ngle
s; 4
con
grue
nt s
ides
eac
h m
easu
ring 1
inch
.
Ste
p1
Dra
w:
Ste
p2
Use
a p
rotr
acto
r to
mak
e su
re t
he a
ngle
form
ed
by t
he t
wo
lines
you
dre
w is
a r
ight
ang
le.
Ste
p3
Dra
w t
wo
mor
e 1
in. l
ines
to
conn
ect
your
figu
re.
Ste
p4
Use
a p
rotr
acto
r to
mak
e su
re t
he o
ther
3an
gles
are
rig
ht a
ngle
s.
Use
a p
rotr
acto
r an
d a
rul
er
to d
raw
eac
h fig
ure
. Cla
ssify
eac
h fig
ure
.
1. tr
iang
le:2
ang
les
that
mea
sure
30°
; one
si
de b
etw
een
the
angl
es t
hat
mea
sure
s 4
cent
imet
ers
2.qu
adril
ater
al: f
our 3-
cent
imet
er s
ides
; 1pa
ir of
ang
les
mea
surin
g 60
°, t
he o
ther
pa
ir m
easu
ring 12
0°
It ha
s 4
cong
ruen
t si
des
and 4
right
ang
les,
so
the
figur
e is
a s
quar
e.
MX
EN
L08
AW
K5X
_RT
_CH
19_L
03.in
dd1
7/11
/07
4:37
:23
PM
RW123-RW124
Grade5.indd 62Grade5.indd 62 7/19/07 6:01:49 PM7/19/07 6:01:49 PM
Solid
Fig
ures
You
can
iden
tify
a p
oly
hed
ron
by t
he s
hape
of i
ts fa
ces.
They
are
sol
id fi
gure
s w
ith fa
ces
that
are
pol
ygon
s.
A p
yram
id is
a p
olyh
edro
n w
ith o
nly
one
poly
gon
bas
e.
A p
rism
has
tw
o co
ngru
ent
and
para
llel
poly
gons
as
bas
es.
tr
iang
ular
p
yram
idA
ll fa
ces
are
tria
ngle
s.tr
iang
ular
p
rism
The
two
base
s ar
e tr
iang
les.
rect
ang
ular
p
yram
idTh
e on
e ba
se
is a
rec
tang
le.
rect
ang
ular
p
rism
All
face
s ar
e re
ctan
gles
.
squa
re
pyr
amid
The
one
base
is
a s
quar
e.cu
be
All
face
s ar
e sq
uare
s.
pe
ntag
ona
l p
yram
idTh
e on
e ba
se
is a
pen
tago
n.p
ent
ago
nal
pri
smTh
e tw
o ba
ses
are
pent
agon
s.
hexa
go
nal
pyr
amid
The
one
base
is
a h
exag
on.
hexa
go
n p
rism
The
two
base
s ar
e he
xago
n.
A s
olid
figu
re w
ith c
urve
s is
no
t a
po
lyhe
dro
n.co
neTh
e on
e ba
se
is a
circ
le.
cylin
de
rTh
e tw
o ba
ses
are
circ
les.
sphe
resh
ape
of a
ba
ll an
d ha
s no
bas
e
Cla
ssify
the
so
lid f
igur
e. W
rite
pris
m, p
yram
id, c
one
, cyl
inde
r, o
r sp
here
.
Th
e fig
ure
is a
pol
ygon
.
Th
e fig
ure
has
only
one
bas
e.
So, t
he fi
gure
is a
pyr
amid
.
Cla
ssify
eac
h so
lid f
igur
e. W
rite
pris
m, p
yram
id, c
one
, cyl
inde
r, o
r sp
here
.
1.
2.
3.
4.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
125
MXE
NL08
AW
K5X_
RT_C
H19_
L04.
indd
1
6/13
/07
10:
44:3
1 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
126
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy: C
ompa
re
Stra
tegi
esM
acke
nzie
use
d 1
2 st
raw
s as
ed
ge
s to
bui
ld a
pyr
amid
. Ho
w m
any
sid
es
did
the
bas
e
of
her
pyr
amid
hav
e?
Ho
w m
any
vert
ice
s d
id h
er
pyr
amid
hav
e?
Re
ad
to
Un
de
rsta
nd
1. W
rite
the
ques
iton
as a
fill-
in-t
he-b
lank
sen
tenc
e.
Pla
n
So
lve
by
com
par
ing
str
ate
gie
s.
6. A
pyr
amid
has
9 fa
ces.
how
man
y ed
ges
do
es it
hav
e? H
ow m
any
vert
ices
doe
s it
ha
ve?
7. M
acke
nzie
use
d 15
str
aws
to m
ake
a
pris
m. H
ow m
any
face
s do
es t
he p
rism
have
? H
ow m
any
vert
ices
doe
s th
e
pris
m h
ave?
So
lve
2. W
hat
are
two
stra
tegi
es y
ou c
an u
se t
o so
lve
the
prob
lem
?
3. So
lve
the
prob
lem
. Des
crib
e th
e st
rate
gy u
sed.
4. W
rite
your
ans
wer
in a
com
plet
e se
nten
ce.
Ch
eck
5. H
ow c
an y
ou c
heck
you
r an
swer
?
MX
EN
L08
AW
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11
:42:
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M
RW125-RW126
Grade5.indd 63Grade5.indd 63 7/19/07 6:02:07 PM7/19/07 6:02:07 PM
a.b. d.
c.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
127
Net
s for
Sol
id F
igur
esY
ou c
an id
entif
y so
lid fi
gure
s by
the
ir ne
ts. A
ne
t ca
n be
cut
out
, fol
ded,
and
tap
ed t
oget
her
to fo
rm a
pol
yhed
ron.
Mat
ch t
he s
olid
fig
ure
wit
h it
s ne
t.
Nam
e t
he s
olid
fig
ure
fo
r th
e n
et.
Th
e fig
ure
is a
tria
ngul
ar p
rism
.
It
has 5
face
s; 3
rec
tang
ular
and
2 t
riang
ular
.
Lo
ok fo
r th
e ne
t w
ith 5
face
s.
N
ets
a an
d b
each
hav
e 5
face
s.
N
et b
has
3 r
ecta
ngul
ar a
nd 2
tria
ngul
ar fa
ces.
So, t
he s
olid
figu
re m
atch
es n
et b
.
Mat
ch e
ach
solid
fig
ure
wit
h it
s ne
t.
1.2.
3.4.
a.b
.c.
d.
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M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
128
1.2.
3.4.
Draw
Sol
id F
igur
es fr
om D
iffer
ent V
iew
sY
ou c
an u
se t
his
char
t to
hel
p yo
u id
entif
y so
lids
from
diff
eren
t vi
ews.
Ide
ntify
the
so
lid f
igur
e t
hat
has
the
giv
en
vie
ws.
Th
e fr
ont
and
side
vie
ws
are
tria
ngle
s,
so
the
figu
re is
a p
yram
id o
r co
ne.
Th
e to
p vi
ew is
a s
quar
e.
A
ll in
side
seg
men
ts m
eet
at a
poi
nt.
So t
he fi
gure
is a
squ
are
pyra
mid
.
top
vie
w
f
ront
vie
w
s
ide
vie
w
pris
m o
r cy
linde
rsh
ape
of b
ase
r
ecta
ngle
rect
angl
e
pyra
mid
or
cone
t
riang
le
tr
iang
le
sphe
reci
rcle
c
ircle
circ
le
top
fro
nt
s
ide
top
fro
nt
s
ide
Ide
ntify
the
so
lid f
igur
e t
hat
has
the
giv
en
vie
ws.
to
p
fro
nt
si
de
top
fro
nt
sid
e
top
fro
nt
sid
e
shap
e of
bas
e w
itha
poin
t in
the
cen
ter
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
129
Tran
sformations
A t
rans
form
atio
n is
a c
hang
e in
pos
ition
of a
figu
re t
hat
does
not
chan
ge it
s si
ze o
r sh
ape.
A t
rans
lati
on
is a
tra
nsfo
rmat
ion
that
slid
es t
he fi
gure
alo
ng a
stra
ight
line
. An
exam
ple
is a
sle
d sl
idin
g do
wnh
ill.
A r
efle
ctio
n is
a t
rans
form
atio
n th
at fl
ips
the
figur
e ov
er a
line
.A
n ex
ampl
e is
tur
ning
a p
age
in a
boo
k.
A r
ota
tio
n is
a t
urn.
An
exam
ple
is t
he w
ay t
he h
ands
on
anan
alog
clo
ck m
ove.
Nam
e t
he t
rans
form
atio
n. W
rite
tran
slation,
reflection,
or
rotatio
n.
So, t
he t
rans
form
atio
n is
a r
efle
ctio
n.
Nam
e e
ach
tran
sfo
rmat
ion.
Wri
te tran
slation,
reflection,
or rotatio
n.
1.
2.3.
4.5.
6.
A
sk h
ow t
he fi
gure
cha
nges
pos
ition
.
• C
an y
ou p
ush
it al
ong
a st
raig
ht li
ne?
N
o, s
o it
is n
ot a
tra
nsla
tion.
• C
an y
ou t
urn
it ab
out
a po
int?
N
o, s
o it
is n
ot a
rot
atio
n.
• C
an y
ou p
ick
it up
and
tur
n it
over
?
Yes
, so
it is
a r
efle
ctio
n.
MX
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42 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
130
Tessellatio
nsA
te
sse
llati
on
is a
des
ign
that
use
s cl
osed
figu
res.
In a
te
ssel
latio
n, o
ne o
r m
ore
figur
es a
re p
ut t
oget
her
to c
over
a
surf
ace
with
out
any
gaps
or
over
laps
.
Pre
dic
t w
heth
er
the
fig
ure
or
figur
es
will
or
will
no
t te
sse
llate
. W
rite
yes
or no
.
1.
3.
4.
5.
6.
Exam
ple
– T
ess
ella
tio
n
Pred
ict
whe
ther
the
figu
re w
ill t
esse
late
.Te
st y
our
pred
ictio
n.
So, y
es,
the
figur
e w
ill t
esse
llate
.If
you
try
to c
over
a s
urfa
ce w
ith c
lose
d fig
ures
, an
you
have
gap
s or
ove
rlaps
, the
n yo
u ha
ve n
ot
mad
e a
tess
ella
tion.
2.
Exam
ple
– N
ot
a Te
sse
llati
on
The
pair
of fi
gure
s a
nd
do n
ot
tess
ella
te.
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1.
2.
3.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
131
Crea
te a
Geo
met
ric P
atte
rnY
ou c
an u
se r
efle
ctio
ns, r
otat
ions
, or
tran
slat
ions
to
crea
te a
pa
tter
n.
Tran
slat
ion
A t
rans
latio
n is
a s
lide
that
oc
curs
with
out
turn
ing
or
pick
ing
the
figur
e up
.
Re
flect
ion
Ro
tati
on
A r
efle
ctio
n is
a fl
ip o
ver
a lin
e. It
mus
t be
pic
ked
up fo
r th
is t
rans
form
atio
n.
A r
otat
ion
is a
tur
n ab
out
a po
int.
De
scri
be
ho
w t
he p
atte
rn m
ight
hav
e b
ee
n cr
eat
ed
.
It
is n
ot a
tra
nsla
tion
beca
use
the
figur
e tu
rns
intw
o di
ffere
nt w
ays.
Th
is fi
gure
mus
t be
pic
ked
up a
nd fl
ippe
d to
form
thi
s pa
tter
n.
Th
is p
atte
rn m
ight
hav
e be
en c
reat
ed b
y re
flect
ing
the
figur
e ov
er a
ver
tical
line
.
De
scri
be
ho
w t
he p
atte
rn m
ight
hav
e b
ee
n cr
eat
ed
.
Th
e tr
iang
les
are
all f
acin
g th
e sa
me
way
. Eac
h tr
iang
le
can
slid
e to
the
nex
t po
sitio
n w
ithou
t be
ing
pick
ed u
p.
Th
is p
atte
rn m
ight
hav
e be
en c
reat
ed b
y tr
ansl
atin
g th
e fig
ure
to t
he r
ight
.
Tell
how
eac
h p
atte
rn m
ight
hav
e b
ee
n cr
eat
ed
.
MXE
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M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
132
Num
eric
Pat
tern
sY
ou c
an fi
nd a
rul
e fo
r a
num
ber
patt
ern
and
use
it to
ext
end
the
patt
ern.
Ide
ntify
the
rul
e f
or
the
pat
tern
.
567,
189
, 63,
21,
...
Ste
p 1
See
if y
ou c
an e
limin
ate
two
oper
atio
ns
• D
oes
the
patt
ern
incr
ease
?
N
o. S
o el
imin
ate
addi
tion
and
mul
tiplic
atio
n.
•
Doe
s th
e pa
tter
n de
crea
se?
Yes
. So
the
rule
is e
ither
sub
trac
tion
or d
ivis
ion.
Ste
p 2
Tes
t su
btra
ctio
n, o
ne o
f the
rem
aini
ng o
pera
tions
.Th
ink
of a
num
ber
to s
ubtr
act
from
567
to
get
189.
56
7 2
378
5 1
89
189 2
378
fi 6
3
• A
re t
he d
iffer
ence
s th
e sa
me
as t
he p
atte
rn in
the
pro
blem
?
N
o. S
o th
e ru
le s
ubtr
act
378
does
n’t
wor
k.
Th
en, t
est
divi
sion
, the
oth
er r
emai
ning
ope
ratio
n.
56
7 4
189
5 3
18
9 4
63 5
3
63 4
21 5
3
•
Are
the
quo
tient
s th
e sa
me?
Y
es, s
o th
e ru
le is
div
isio
n.
Ste
p 3
Writ
e th
e ru
le.
So t
he r
ule
is d
ivid
e by
3.
Ide
ntify
the
rul
e o
f e
ach
pat
tern
.
1. 1
5, 4
5, 1
35, 4
05, .
..2.
313
, 307
, 301
, 295
, ...
3. 2
40, 1
20, 6
0, 3
0, ..
.
4. 1
01, 1
20, 1
39, 1
58, .
..5.
388
8, 6
48, 1
08, 1
8, ..
.6.
612
, 598
, 584
, 570
, ...
7. 7
, 35,
63,
91,
...
8. 8
28, 8
18, 8
08, 7
98, .
..9.
12,
48,
192
, 768
, ...
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Grade5.indd 66Grade5.indd 66 7/19/07 6:02:56 PM7/19/07 6:02:56 PM
Prob
lem
Sol
ving
Wor
ksho
pSt
rate
gy:
Find
a P
atte
rn
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
133
Suri
is p
aint
ing
a de
sign
tha
t is
a p
atte
rn o
f squ
ares
. Sh
e pa
ints
the
squ
ares
bla
ck, w
hite
, bla
ck, g
ray,
bla
ck,
whi
te, b
lack
, gra
y. W
hat
colo
r co
uld
the
next
tw
o sq
uare
s be
?
Re
ad
to
Un
de
rsta
nd
1.
Writ
e th
e qu
estio
n as
a fi
ll-in
-the
bla
nk s
ente
nce.
Pla
n2.
W
hat
kind
of p
atte
rn d
oes
Suri
use,
a r
epea
ting
patt
ern
or
a se
quen
ce?
So
lve
3.
Col
or t
he fi
rst 10
squ
ares
to
show
Sur
i’s p
atte
rn s
o fa
r.
Then
dra
w a
line
und
er t
he p
art
that
rep
eats
.
?
4.
Ans
wer
the
que
stio
n.W
hat
colo
rs w
ill t
he n
ext
two
squa
res
be?
Ch
eck
5.
How
did
find
ing
the
patt
ern
help
you
sol
ve t
he p
robl
em?
Loo
k fo
r a
pat
tern
to
so
lve
.6.
N
ate
uses
a d
esig
n m
ade
of fi
gure
s to
de
cora
te a
cer
amic
bow
l he
mad
e in
po
tter
y cl
ass.
His
des
ign
so fa
r lo
oks
like
this
:
Wha
t ar
e th
e ne
xt t
wo
figur
es in
his
pa
tter
n?
7.
Am
elia
use
s pa
ralle
logr
ams
to d
ecor
ate
a pi
ctur
e fr
ame.
Her
des
ign
so fa
r is
sho
wn
belo
w.
Wha
t w
ill t
he n
ext
thre
e fig
ures
in A
mel
ia’s
de
sign
be?
?
MX
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57 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
134
Num
ber
of p
enta
gons
, x1
23
4
Num
be
r o
f si
de
s, y
510
1520
Nu
mb
er o
f Pen
tag
on
s
Number of Sides
(1,5
)(2,1
0)(3,1
5)(4,2
0)
46
810
4681012 1416182022
2
2 0
y
x
Num
be
r o
f fa
ces,
x6
1218
Num
be
r o
f cu
be
s, y
12
3
N
umb
er
of
Tria
ngle
s, x
12
34
N
umb
er
of
Ve
rtic
es,
y3
69
12
46
810
1214
1618
46
2
2 0x
y
46
8
8
10
1012 46
2
2 0 x
y
Alge
bra:
Gra
ph R
elat
ions
hips
You
can
gra
ph o
rder
ed p
airs
to
show
rel
atio
nshi
ps b
etw
een
two
amou
nts.
For
exa
mpl
e, y
ou c
an s
how
the
rel
atio
nshi
p be
twee
n a
num
ber
of p
enta
gons
and
the
num
ber
of s
ides
.
Wri
te t
he o
rde
red
pai
rs a
nd t
hen
gra
ph
the
m.
A t
able
can
sho
w t
he r
elat
ions
hip.
1.Wri
te t
he o
rde
red
pai
rs.
The
n g
rap
h th
em
.
Each
col
umn
can
be w
ritte
n as
an
orde
red
pair
(x,y
):1
pent
agon
, 5 s
ides
: (1,5)
2 pe
ntag
ons,
10
side
s: (2
,10)
3 pe
ntag
ons,
15
side
s: (3
,15)
4 pe
ntag
ons,
20
side
s: (4
,20)
You
can
gra
ph t
he o
rder
ed p
airs
for
the
rela
tions
hip
show
n in
the
tab
le.
For
this
rel
atio
nshi
p, t
he y
-coo
rdin
ate
is a
lway
s5
times
the
x-c
oord
inat
e.
2.
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:02
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RW133-RW134
Grade5.indd 67Grade5.indd 67 7/11/07 5:33:43 PM7/11/07 5:33:43 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
135
inp
ut, x
34
5
out
put
, y
12
3+ 4
+ 6+ 8
+ 4+ 6+ 8
+ 2
+ 2 0
y
x
Find
the
rul
e t
o c
om
ple
te t
he f
unct
ion
tab
le.
The
n w
rite
an
eq
uati
on.
inp
ut, x
12
34
5
out
put
, y5
1015
25
inp
ut, x
01
23
4
out
put
, y1
410
13
You
can
writ
e an
equ
atio
n an
d m
ake
a fu
nctio
n ta
ble
to s
how
the
rel
atio
nshi
p be
twee
n tw
o am
ount
s.
Find
a r
ule
and
co
mp
lete
the
fun
ctio
nta
ble
. The
n w
rite
an
eq
uati
on.
•
Find
a r
ule.
Eac
h va
lue
of y
is 4
times
the
val
ue o
f x.
So, t
he r
ule
is m
ultip
ly 4
.
•
Com
plet
e th
e fu
nctio
n ta
ble.
Use
the
rul
e to
find
the
val
ue o
f yw
hen
x 5
0.
0 3
4 5
0
So, t
he v
alue
of y
whe
n x
5 0
is 0
.
•
Use
the
rul
e to
writ
e an
equ
atio
n.
Rul
e: m
ultip
ly b
y 4.
Equa
tion:
4x
x0
12
34
y4
812
16
x0
12
34
y0
48
1216
1 3
4 5
42
3 4
5 8
3 3
4 5
12
4 3
4 5
16
Use
the
func
tion
tabl
e to
writ
e th
e or
dere
d pa
irs.
(0,0
), (
1,4)
, (2,
8), (
3,12
), (
4,16
)
Gra
ph t
he o
rder
ed p
airs
. +
4 +
6 +
8
+8 +12
+16
+18
+2
+4 0
y
x
Alge
bra:
Equ
atio
ns a
nd F
unct
ions
1.C
om
ple
te t
he f
unct
ion
tab
le. T
hen
gra
ph
the
ord
ere
d p
air.
Rul
e: S
ubtr
act
3.Eq
uatio
n: y
5 x
2
2.3.
MXE
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:01
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
136
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Writ
e an
Equ
atio
nN
ancy
has
tra
vele
d 3
9 m
iles
on
a tr
ain
to v
isit
he
rg
rand
mo
the
r. T
he t
ota
l dis
tanc
e o
f th
e t
rip
is 8
4 m
iles.
Ho
w m
any
mo
re m
iles
will
Nan
cy t
rave
l to
re
ach
her
gra
ndm
oth
er’
s ho
use
.
Re
ad
to
Un
de
rsta
nd
1.
Wha
t in
form
atio
n is
giv
en?
Pla
n
2.
Wha
t st
rate
gy c
an y
ou u
se t
o so
lve
the
prob
lem
?
So
lve
3.
Writ
e an
equ
atio
n. L
et n
rep
rese
nt t
he n
umbe
r of
mile
s N
ancy
stil
l mus
t tr
avel
.
4. S
olve
the
equ
atio
n fo
r n.
5. H
ow m
any
mor
e m
iles
does
Nan
cy n
eed
to t
rave
l?
Ch
eck
5.
Look
bac
k at
the
pro
blem
. Exp
lain
why
you
r an
swer
mak
esse
nse
for
the
prob
lem
.
Wri
te a
n e
qua
tio
n to
so
lve
.
84 5
6.
Ric
hard
spe
nds
$150
eac
h m
onth
on
lunc
hes.
How
muc
h do
es h
e sp
end
for
lunc
hes
in o
ne y
ear?
7.
Mar
y sa
ves
$20
each
wee
k fr
om h
erpa
yche
ck. S
he h
as $
360
in h
er s
avin
gs
acco
unt.
How
muc
h w
ill s
he h
ave
in t
he
acou
nt 1
2 w
eeks
from
now
?
MXE
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:17
AM
RW135-RW136
Grade5.indd 68Grade5.indd 68 7/19/07 6:05:23 PM7/19/07 6:05:23 PM
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
137
Unde
rsta
nd In
tege
rsIn
teg
ers
are
the
set
of w
hole
num
bers
and
the
ir op
posi
tes.
Inte
gers
can
be
grap
hed
on a
num
ber
line.
Posi
tive
inte
ge
rs a
re a
lway
s gr
eate
r th
an 0
.N
umbe
rs w
ith n
o si
gn in
fron
t of
the
m a
re
posi
tive
inte
gers
.N
eg
ativ
e in
teg
ers
are
alw
ays
less
tha
n 0.
The
abso
lute
val
ue o
f an
inte
ger
is it
s di
stan
cefr
om 0
on
a nu
mbe
r lin
e.
Ide
ntify
the
inte
ge
rs g
rap
hed
on
the
num
be
r lin
e.
Wri
te a
n in
teg
er
to r
ep
rese
nt e
ach
situ
atio
n.
2.
a ga
in o
f 12
yard
s3.
7
degr
ees
belo
w fr
eezi
ng4.
8
floor
s up
1.
• T
he p
oint
tha
t is
2 u
nits
left
of 0
rep
rese
nts 22.
•Th
e po
int
that
is o
ne p
lace
to
the
right
of 0
rep
rese
nts
1.
•
The
poi
nt t
hat
is 5
uni
ts r
ight
of 0
rep
rese
nts
5.
So, t
he in
tege
rs a
re 22,
1, a
nd 5
.
Wri
te a
n in
teg
er
to r
ep
rese
nt 7
9 d
eg
ree
s b
elo
w 0
.
Not
ice
how
the
wor
d “b
elow
” is
in
the
situ
atio
n.
Use
the
cha
rt a
t th
e rig
ht t
o se
e w
hich
ca
tego
ry “
belo
w”
falls
und
er.
“B
elow
” is
a w
ord
used
with
neg
ativ
e in
tege
rs.
So, t
he in
terg
er t
hat
repr
esen
ts 7
9 de
gree
s be
low
0 is
279
.
Wri
te a
n in
teg
er
that
de
scri
be
s a
pro
fit o
f $
80
0.
No
prof
it or
loss
is a
t $0
, and
pro
fit is
a p
ositi
ve w
ord.
So, t
he in
tege
r fo
r a
prof
it of
$80
0 is
180
0, o
r ju
st 8
00.
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
nega
tive
inte
gers
posi
tive
inte
gers
Som
e w
ords
use
d w
ithne
gativ
e in
tege
rs
loss
de
crea
sebe
hind
back
war
dbe
low
dow
nun
der
with
draw
alto
the
left
Som
e w
ords
use
d w
ithpo
sitiv
e in
tege
rs
gain
or
prof
itin
crea
seah
ead
forw
ard
abov
e up ov
erde
posi
tto
the
rig
ht
Iden
tify
the
inte
gers
gra
phed
on
the
num
ber
line.
MXE
NL08
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:55
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
138
Two
inte
gers
are
equ
al o
nly
whe
n th
ey h
ave
the
sam
e si
gn a
nd s
ame
num
ber.
Com
pare
and
Ord
er In
tege
rsY
ou c
an u
se a
num
ber
line
to c
ompa
re a
nd o
rder
inte
gers
by
gra
phin
g th
e in
tege
rs o
n a
num
ber
line.
Co
mp
are
24
and
28.
Use
,, .
, or
5.
Sinc
e 2
4 is
rig
ht o
f 28,
24.2
8
Co
mp
are
14
and
28.
Use
,, .
, or
5.
Sinc
e 14
is t
o th
e rig
ht o
f 28,
14.
28
Co
mp
are
14
and
19.
Use
,, .
, or
5.
Sinc
e 14
is t
o th
e le
ft of
19,
14,
19
Co
mp
are
24
and
19.
Use
,, .
, or
5.
Sinc
e 24
is t
o th
e le
ft of
19,
24,
19
Co
mp
are
. Wri
te ,
, ., o
r 5
.
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
+4
-4
+6
-6
+8
-8
+10
-10
+2
- 20
1. 17
2
13.
14
0
7. 11
2
18.
15
2
6
2. 29
2
8
4. 26
1
6
9. 23
2
3
6. 22
1
65.
22
2
10
MXE
NL08
AW
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dd
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1:38
:57
AM
RW137-RW138
Grade5.indd 69Grade5.indd 69 6/18/07 6:07:57 PM6/18/07 6:07:57 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
139
axis
-axi
s- 2
- 3- 1
- 1- 4
- 5
- 2 - 3 - 4
- 5
+ 4+ 5
+ 4
+ 5
+ 2+ 3
+ 2+ 3 + 1
+ 10 y-
x
E
F
A
B
C
U
E
H
Q
Z
D
N
S
axis
-axi
s- 2
- 4
- 2 - 4
+ 4
+ 4
+ 2
+ 2 0 y-
x
Alge
bra:
Gra
ph In
tege
rs o
n th
e Co
ordi
nate
Pla
ne
1. 7.For
2–
9, i
de
ntify
the
ord
ere
d p
air
for
eac
h p
oin
t.
2. 4.
3. 5. 6.
8.9.
SN
DZ U
E QH
Just
as
a co
ordi
nate
grid
is fo
rmed
by
two
perp
endi
cula
r ra
ys,
a co
ord
inat
e p
lane
is fo
rmed
by
two
inte
rsec
ting
and
num
ber
perp
endi
cula
r lin
es.
The
horiz
onta
l lin
e is
the
x-a
xis.
The
ver
tical
line
is t
he y
-axi
s.
The
ori
gin
is t
he p
lace
whe
re t
he x
- an
d y-
axis
inte
rsec
t.
The
x-co
ordi
nate
s to
the
rig
ht o
f the
y-a
xis
are
posi
tive
inte
gers
. Th
e x-
coor
dina
tes
to t
he le
ft of
the
y-a
xis
are
nega
tive
inte
gers
.
The
y-co
ordi
nate
s ab
ove
the
x-ax
is a
re p
ositi
ve in
tege
rs.
The
y-co
ordi
nate
s be
low
the
x-a
xis
are
nega
tive
inte
gers
.
Wha
t o
rde
red
pai
r na
me
s th
e lo
cati
on
of
po
int
P o
n th
e c
oo
rdin
ate
pla
ne?
•
Po
int
C is
1 u
nit
left
of t
he
y-ax
is, s
o th
e x-
coor
dina
te is
21.
•
Po
int
C is
3 u
nits
abo
ve t
he
x-ax
is, s
o th
e y-
coor
dina
te is
3.
So, t
he o
rder
ed p
air
(21,
3) n
ames
the
loca
tion
of p
oint
C.
Wha
t o
rde
red
pai
r na
me
s th
e lo
cati
on
of
po
int
F?
•
Po
int
F is
4 u
nits
rig
ht o
f the
y-a
xis,
so
the
x-co
ordi
nate
is 4
.
•
Po
int
F is
4 u
nits
bel
ow
the
x-a
xis,
so
the
y-co
ordi
nate
is 2
4.
So, t
he o
rder
ed p
air
(4, 2
4) n
ames
the
loca
tion
of p
oint
F.
To g
raph
(23,
12)
, you
wou
ld s
tart
at
(0,)
and
the
go
to
23
and
t
o 1
2
MXE
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:21
AM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
140
Cust
omar
y Le
ngth
Whe
n yo
u m
easu
re w
ith p
reci
sio
n, y
ou u
se t
he s
mal
lest
uni
t po
ssib
le.
Mea
surin
g to
the
nea
rest
1 _ 2 in
ch is
mor
e pr
ecis
e th
an m
easu
ring
to t
he n
eare
st in
ch.
Mea
surin
g to
the
nea
rest
1 _ 4 in
ch is
eve
n m
ore
prec
ise.
Esti
mat
e t
he le
ngth
of
the
cra
yon
in in
che
s. T
hen
me
asur
e t
he le
ngth
to
the
ne
are
st 1 _ 8
inch
.
So, t
o th
e ne
ares
t 1 __
8 inch
, the
cra
yon
is 2
5 __ 8 in
ches
long
.
1.
near
est
inch
2
. ne
ares
t 1 __ 2 in
ch.
3. n
eare
st 1 __ 8 in
ch.
Tell
whi
ch m
eas
ure
me
nt is
mo
re p
reci
se.
4.
5 7
__ 8 in
ches
or
6 in
ches
5. 3
feet
or
34 1 __ 2 in
ches
6.
7 in
ches
or
7 1 __ 4 in
ches
Me
asur
e t
he le
ngth
to
the
ne
are
st 1 __ 8 in
ch.
7.8.
Estim
ate:
Abo
ut 3
inch
es lo
ng
Th
e cr
ayon
is a
lmos
t ex
actly
on
the
2 5 __ 8 in
. mar
k.
Now
mea
sure
the
cra
yon
to t
he n
eare
st 1 __
8 inch
usi
ng a
rul
er.
Esti
mat
e t
he le
ngth
of
the
cat
erp
illar
in in
che
s. T
hen
me
asur
e t
he le
ngth
to
the
ne
are
st in
ch.
MXE
NL08
AW
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dd
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13/0
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1:57
:52
AM
RW139-RW140
Grade5.indd 70Grade5.indd 70 6/18/07 6:08:07 PM6/18/07 6:08:07 PM
cm1
23
45
6
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
141
Met
ric L
engt
hSo
me
com
mon
met
ric u
nits
of l
engt
h ar
e:
So, t
o th
e ne
ares
t ce
ntim
eter
, the
pap
ercl
ip is
4 c
m lo
ng.
1.
2.
3.
4.
5.
S
TEP
1
A
lign
the
left
side
of t
he
pape
rclip
with
the
zer
o m
ark
on t
he r
uler
. Thi
s is
al
read
y sh
own
in t
he p
ictu
re.
STE
P 2
Not
ice
whe
re t
he p
aper
clip
en
ds o
n th
e ru
ler.
The
pape
rclip
is e
xact
ly
4 ce
ntim
eter
s lo
ng.
cen
tim
ete
r (c
m)
mill
ime
ter
(mm
)m
ete
r (m
)
1 m
5 1
00 c
m
1 m
5 1
,000
cm
kilo
me
ter
(km
)
1 km
5 1
,000
m
1 m
m1
cm
You
can
use
a c
entim
eter
rul
er t
o m
easu
re le
ngth
.
Ho
w lo
ng is
the
pap
erc
lip t
o t
he n
ear
est
ce
ntim
ete
r?
The
se 2
citi
es a
re in
diff
eren
t st
ates
, so
ther
e is
a c
onsi
dera
ble
amou
nt o
f dis
tanc
e be
twee
n th
em.
Wri
te t
he a
pp
rop
riat
e m
etr
ic u
nit
fo
r m
eas
uri
ng t
he d
ista
nce
fro
m B
ost
on
to D
alla
s.
So, t
he m
ost
appr
opria
te u
nit
for
mea
surin
g th
e di
stan
ce fr
om B
osto
n to
Dal
las
is t
he k
ilom
eter
.
Esti
mat
e t
he le
ngth
of
the
str
ing
in c
ent
ime
ters
. The
n m
eas
ure
the
leng
th
to t
he n
eare
st c
entim
eter
.to
the
nea
rest
mill
imet
er.
wid
th o
f a p
aper
clip
wire
di
stan
ce fr
om A
ustin
to
Phoe
nix
Wri
te t
he a
pp
rop
riat
e m
etr
ic u
nit
fo
r m
eas
uri
ng e
ach
.
hei
ght
of a
sky
scra
per
6.w
idth
of t
he N
orth
Am
eric
an C
ontin
ent
1 cm
5 1
0 m
m
MXE
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dd
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13/0
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2:01
:10
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
142
Chan
ge L
inea
r Uni
tsM
ultip
ly t
o ch
ange
from
larg
er t
o sm
alle
r lin
ear
units
.
Div
ide
to c
hang
e fr
om s
mal
ler
to la
rger
line
ar u
nits
Find
the
sum
or
diff
ere
nce
.
Cha
nge
the
uni
t.
1.33
yd
5
ft
2. 5
00 c
m 5
m
3.
13 c
m 5
mm
4.5.
6.
Ad
d.6
ft
7 in
. 14
ft 8
in.
Add
feet
. Add
inch
es.
Subt
ract
.R
enam
e, b
ecau
se 1
5 in
. > 1
ft.
15 in
. 51
ft 3
in.
10 ft
15
in. =
10
ft 1
1 ft
3 in
. 511
ft 3
in.
So,6
ft 7
in. 1
4 ft
8 in
. 511
ft 3
in.
Sub
trac
t.9
yd 1
ft
22
yd 2
ft
Ther
e ar
e no
t en
ough
feet
to
subt
ract
2
feet
from
1 fo
ot, s
o re
nam
e.6
ft
7 in
.1
4 ft
8
in.
10 ft
1
5 in
.8
yd
4 ft
.2
2 y
d
2 ft
.
6 yd
2
ft.
3 ft
5
in.
1 5
ft
4 in
.5
yd
2 ft
.1
8 y
d
2 ft
.
10 ft
4 in
.2
3 ft
1
1 in
.
So,9
yd
1 ft
22
yd 2
ft 5
6 yd
2 ft
9 yd
1 ft
58
yd 4
ft
Cus
tom
ary
Uni
tso
f Le
ngth
12in
.51
ft3
ft 5
1 yd
5,28
0 ft
51
mi
1,76
0 yd
51
mi
Me
tric
Uni
tso
f Le
ngth
10 m
m 5
110
0 cm
51
m1,
000
m 5
1 km
STE
P1
Dec
ide:
Mul
tiply
or
Div
ide
cmm
smal
ler
larg
er
STE
P2
Thin
k:
100
m 5
1 m
,so
800
m5
c
m
STE
P3
Div
ide.
800
4 1
00 5
8
So,8
00 c
entim
eter
s 5
8 m
.
Cha
nge
the
uni
t: 8
00 c
m 5
m
STE
P1
Dec
ide:
Mul
tiply
or
Div
ide
feet
inch
esla
rger
smal
ler
STE
P2
Thin
k:
1 ft
512
in.,
so3
ft5
in
STE
P3
Mul
tiply
.3
3 1
2 5
36
So,3
feet
536
inch
es.
Cha
nge
the
uni
t: 3
fee
t 5
inch
es
MX
EN
L08
AW
K5X
_RT
_CH
22_L
3.in
dd1
7/11
/07
4:38
:55
PM
RW141-RW142
Grade5.indd 71Grade5.indd 71 7/11/07 5:34:28 PM7/11/07 5:34:28 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
143
Cust
omar
y Ca
paci
ty a
nd W
eigh
tM
ultip
ly t
o ch
ange
from
larg
er t
o sm
alle
r un
its.
Div
ide
to c
hang
e fr
om s
mal
ler
to la
rger
uni
ts.
Cha
nge
the
uni
t: 8
c 5
q
t.
STE
P 1
Dec
ide:
Mul
tiply
or
Div
ide
cups
qua
rts
so d
ivid
e to
cha
nge
units
smal
ler
la
rger
STE
P 2
Thin
k:
4 c
5 1
qt,
so 8
c 5
oz
STE
P 3
Div
ide.
8 c
4 4
5 2
qt
Cus
tom
ary
Uni
ts
of
Cap
acit
y
8 fl
oz =
1 c
2 c
= 1
pt
2 pt
= 1
qt
4 c
= 1
qt
4 qt
= 1
gal
So, t
here
are
2 q
t in
8 c
.
Cha
nge
the
uni
t: 2
1 __ 2 lb
5
oz.
STE
P 1
Dec
ide:
Mul
tiply
or
Div
ide
poun
ds
oun
ces
larg
er
s
mal
ler
STE
P 2
Thin
k:
1 lb
5 1
6 oz
, so
2 1 __ 2 lb
5
oz
STE
P 3
Mul
tiply
.
2 1 __
2 3 1
6 5
So, 2
1 __ 2 p
ound
s is
equ
al t
o 40
oun
ces.
Cha
nge
the
uni
t.
1.
14 p
t =
qt
2. 1
2,00
0 lb
=
T3.
7
c =
fl
oz
4.
7.5
lb =
oz
5.
9 ga
l =
qt6.
16
c =
qt
Cus
tom
ary
Uni
ts
of
We
ight
16 o
z 5
1 lb
2,00
0 lb
5 1
T
16
___ 1 5
40
8
7.
20 q
t 5
gal
8.
3
T 5
lb
9.
16 fl
oz
5
pt
5 __ 2 3
MXE
NL08
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13/0
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2:16
:47
PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
144
Cust
omar
y Ca
paci
ty a
nd M
ass
Mul
tiply
to
chan
ge fr
om la
rger
to
smal
ler
units
.
Div
ide
to c
hang
e fr
om s
mal
ler
to la
rger
uni
ts.
Cha
nge
the
uni
t: 9
,000
g 5
k
g
STE
P 1
Dec
ide:
Mul
tiply
or
Div
ide
gram
s
kilo
gram
ssm
alle
r
la
rger
STE
P 2
Thin
k:
9,00
0 g
5 1
kg,
so
9,0
00 g
5
kg
STE
P 3
Div
ide.
Me
tric
Uni
ts o
f C
apac
ity
1,00
0 m
L =
1 L
250
mL
= 1
met
ric c
up
4 m
etric
cup
s =
1 L
1,00
0 lit
ers
= 1
kL
So, t
here
are
9 k
ilogr
ams
in 9
,000
gra
ms.
Cha
nge
the
uni
t: 1
0 g
5
mg
.
STE
P 1
Dec
ide:
Mul
tiply
or
Div
ide
gram
s
m
illig
ram
sla
rger
smal
ler
STE
P 2
Thin
k:
1 g
= 1
,000
mg,
so
10 g
=
mg
STE
P 3
Mul
tiply
.
10 3
1,0
00
510
,000
mg
So, t
here
are
10,
000
mill
igra
ms
in 1
0 gr
ams.
Cha
nge
the
uni
t.
1.
6,00
0 L
5
kL
2.
350
mg
5
g
3.
1,7
50 m
L 5
m
etric
cu
4.
10,0
00 m
L 5
L
5.
4 k
g 5
g
6.
4 L
5
met
ric c
ups
Me
tric
Uni
ts o
f M
ass
1,00
0 m
g =
1 g
1,00
0 g
= 1
kg
7.
20 m
etric
cup
s 5
L
8.
9
g 5
m
g9.
2,
500
ml 5
kL
5 9
kg
4,00
0 __
____
1,0
00
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:07:
46 P
M
RW143-RW144
Grade5.indd 72Grade5.indd 72 6/18/07 6:08:24 PM6/18/07 6:08:24 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
145
Prob
lem
Sol
ving
Wor
ksho
pSk
ill:
Estim
ate
or A
ctua
l Mea
sure
men
tJe
nnife
r w
ent
jog
gin
g la
st w
ee
k. S
he jo
gg
ed
2.6
5 ki
lom
ete
rs o
n Tu
esd
ay a
nd t
wic
e a
s fa
r o
n W
ed
nesd
ay.
Ho
w f
ar d
id J
enn
ifer
jog
last
we
ek?
1.
How
cou
ld y
ou w
rite
the
ques
tion
as a
fill-
in-t
he b
lank
sen
tenc
e.
2.
Can
you
est
imat
e, o
r do
you
nee
d to
find
an
actu
al
mea
sure
men
t? W
hy?
3. W
hat
oper
atio
ns w
ill y
ou u
se t
o so
lve
the
prob
lem
?
4.
Wha
t eq
uatio
ns c
an y
ou u
se t
o so
lve
the
prob
lem
?
5.
How
far
did
Jenn
ifer
jog
last
wee
k?
6.
How
can
you
che
ck y
our
answ
er?
Tell
whe
the
r yo
u ne
ed
an
est
imat
e o
r an
act
ual m
eas
ure
me
nt.
The
n so
lve
.
7.
Col
in w
ants
to
mak
e pi
llow
s fr
om a
32
9-cm
-long
pie
ce o
f cor
duro
y.
He
need
s to
use
146
-cm
-long
pie
ces
for
two
pillo
ws
and
an 83-
cm-lo
ng
piec
e fo
r th
e th
ird. D
oes
Col
in h
ave
enou
gh c
ordu
roy?
8.
Julie
nee
ds t
o bu
y fa
bric
to
mak
e co
stum
es fo
r th
e sc
hool
pla
y. S
he
does
n’t
wan
t an
y le
ft-ov
er fa
bric
. She
ne
eds 1.2
met
ers
for
thre
e co
stum
es
and 1.9
met
ers
for
two
cost
umes
. How
m
uch
fabr
ic s
houl
d Ju
lie b
uy?
MX
EN
L08
AW
K5X
_RT
_CH
22_L
6.in
dd
16/
6/07
11
:16:
38 A
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
146
Elap
sed
Tim
eY
ou c
an u
se a
clo
ck t
o fin
d el
apse
d tim
e in
hou
rs a
nd m
inut
es.
Wri
te t
he e
lap
sed
tim
e.
Ste
p 1
Cou
nt t
he d
ays.
M
ay 5
– M
ay 3
5 5
2 3
5 2
day
s
Ste
p 2
Cou
nt t
he h
ours
and
min
utes
.
Wri
te t
he t
ime
fo
r e
ach.
June
SM
TW
RF
S
12
3
45
67
89
10
1112
1314
1516
17
1819
2021
2223
24
2526
2728
3930
891011
12
76
5432
1
891011
12
76
5432
1
891011
12
76
5432
1
Cou
nt h
ours
unt
il yo
u re
ach 2
P.M
.C
ount
the
num
ber
on
min
utes
in 5
min
ute
inte
rval
s.5
min
ute
inte
rval
: en
ding
on
the 6
So, t
he e
laps
ed t
ime
is 2
day
s 1
hr, a
nd 30
min
.
1.
On
June
2, L
arry
pla
nned
a w
hite
wat
er r
aftin
g tr
ip fo
r Ju
ne 30.
How
man
y w
eeks
aw
ay is
Lar
ry’s
trip
?
Jan.
SM
TW
RF
S
12
34
56
7
89
1011
1213
14
1516
1718
1920
21
2223
2425
2627
28
2930
31
Feb.
SM
TW
RF
S
12
34
56
78
910
11
1213
1415
1617
18
1920
2122
2324
25
2627
28
March
SM
TW
RF
S
12
34
56
78
910
11
1213
1415
1617
18
1920
2122
2324
25
2627
2829
3031
You
can
use
a c
alen
dar
to fi
nd e
laps
ed t
ime
in w
eeks
and
day
s.
Wha
t is
ela
pse
d t
ime
fro
m J
anua
ry 1
2 to
Mar
ch 5
?
Ste
p 1
Cou
nt w
eeks
.
J
anua
ry 12
to M
arch
5 is
7 w
eeks
.
Ste
p 2
Cou
nt d
ays.
M
arch
2 t
o M
arch
5 is
3 d
ays.
So, t
he e
laps
ed t
ime
from
Jan
uary
12
to M
arch
5 is
7 w
eeks
and
3 d
ays.
Sta
rt: M
ay 3
, 1 P
.M.
End
: May
5, 2
:30
P.M
.
2.
Star
t: 4:35
P.M
.
Ela
psed
tim
e: 2
hr 5
min
End
:
3.
Star
t:
Ela
psed
tim
e: 3
hr 29
min
End
: 9:52
P.M
.
4.
Star
t: Ju
ne 21,
3:20
P.M
.
Ela
psed
tim
e:
End
: Jun
e 25
, 5:30
P.M
.
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:18:
11 P
M
RW145-RW146
Grade5.indd 73Grade5.indd 73 6/18/07 6:08:35 PM6/18/07 6:08:35 PM
3249212
– 5 – 15
– 25
– 35
– 45152535455565758595105 5
°C°F
95ºC
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
147
Tem
pera
ture
Deg
rees
Fah
renh
eit
(°F)
are
the
cus
tom
ary
units
for
mea
surin
g te
mpe
ratu
re. D
egre
es C
els
ius
(°C
) are
the
met
ric u
nits
for
mea
surin
g te
mpe
ratu
re.
The
ther
mom
eter
at
the
right
mea
sure
s de
gree
s Fa
hren
heit
and
degr
ees
Cel
sius
.
Tem
pera
ture
s be
low
zer
o ar
e w
ritte
n as
neg
ativ
e nu
mbe
rs. F
or
exam
ple,
9 d
egre
es b
elow
zer
o is
writ
ten
-9 º
C o
r -9
ºF.
Use
the
the
rmom
eter
to
calc
ulat
e ch
ange
s in
tem
pera
ture
.
Find
102
ºF
to 9
5 ºF
•
Thin
k: 1
02 º
F is
hig
her
than
95
ºF
•
Cha
nge:
Dec
reas
e
•
Equa
tion:
Sub
trac
tion
102 2
95 5
7
So, t
he c
hang
e in
tem
pera
ture
from
102
ºF
to 9
5 ºF
is 7
ºF.
Find
the
cha
nge
in t
em
pe
ratu
re f
rom
-2
5 º
C t
o 1
0 º
C.
•
Thin
k: -
25 º
C is
low
er t
han
10 º
C
•
Cha
nge:
Incr
ease
•
Equa
tion:
25 1
10 5
35
So, t
he c
hang
e in
tem
pera
ture
from
-25
ºC
to
10 º
C is
35
ºC.
Use
the
the
rmo
me
ter
to s
olv
e.
1.
The
tem
pera
ture
is 4
9 ºF
. Sev
en h
ours
late
r, th
e te
mpe
ratu
re is
39
ºF.
By
how
man
y de
gree
s Fa
hren
heit
did
the
tem
pera
ture
cha
nge?
Use
the
the
rmo
me
ter
to f
ind
the
cha
nge
in t
em
pe
ratu
re.
2.
7 ºC
to
61 º
C3.
-35
ºC
to
50 º
C
4.
75 º
C t
o 39
ºC
5.
52 º
C t
o -4
4 ºC
6.
111
ºF t
o 77
ºF
7.
-20
ºF t
o 45
ºF
8.
8 ºF
to
103
ºF9.
69
ºF
to -
13 º
F
MXE
NL08
AW
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:21:
26 P
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
148
Estim
ate
and
Mea
sure
Per
imet
erTh
e p
eri
me
ter
of a
figu
re is
the
dis
tanc
e ar
ound
the
figu
re. Y
ou c
an u
se a
pi
ece
of s
trin
g to
est
imat
e th
e pe
rimet
er o
f a fi
gure
.
Find
the
pe
rim
ete
r o
f th
e p
oly
go
n in
ce
ntim
ete
rs.
Ste
p 1
Lay
a p
iece
of s
trin
g a
roun
d th
e fig
ure.
Ste
p 2
Cut
the
str
ing
whe
re it
mee
ts it
self.
Ste
p 3
Lay
the
str
ing
in a
str
aigh
t lin
e an
d
mea
sure
its
leng
th w
ith a
cen
timet
er r
uler
.
Find
the
pe
rim
ete
r o
f e
ach
po
lyg
on
in c
ent
ime
ters
.
1.2.
3.4.
centimeters
12
34
56
78
910
1112
1314
15
The
strin
g is
abo
ut 1
3 ce
ntim
eter
s lo
ng, s
o th
e pe
rimet
er o
fth
e po
lygo
n is
abo
ut 1
3 ce
ntim
eter
s.
MXE
NL08
AW
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/07
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3:49
PM
RW147-RW148
Grade5.indd 74Grade5.indd 74 7/19/07 6:03:13 PM7/19/07 6:03:13 PM
5 in
.
7 in
.
4 ft
6 ft
3 ft
4 cm
7 ft
7 ft
3 in
.
7 cm
2 cm6
in.
4 in
.4
in.
5 ft
3 m
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
149
Find
Per
imet
erFi
nd t
he p
eri
me
ter
of
eac
h p
oly
go
n.
4 ft
8 ft
Sinc
e th
e si
des
of a
reg
ular
pol
ygon
ar
e eq
ual,
you
can
use
mul
tiplic
atio
nto
find
the
per
imet
er o
f a r
egul
ar p
olyg
on.
Each
of t
he 3
sid
es a
re 6
in. l
ong.
M
ultip
ly. 6
3 3
5 1
8
Sinc
e op
posi
te s
ides
of a
par
alle
logr
am a
re
equa
l, yo
u ca
n us
e ad
ditio
n to
find
the
pe
rimet
er o
f a p
aral
lelo
gram
.
8 1
4 1
8 1
4 5
24
So, t
he p
erim
eter
of t
he p
olyg
on is
18
in.
So, t
he p
erim
eter
of t
he p
olyg
on is
24
ft.
Find
the
pe
rim
ete
r o
f e
ach
po
lyg
on.
1.
2.
3.
4.
5.
6.
MXE
NL08
AW
K5X_
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/07
1:2
8:39
PM
6 ft
8 ft
6 ft
4.5
m3.
6 cm
5.7
cm
3.6
cm
5.7
cm
3 in
.
6 in
.
2 yd
3 yd
8 yd
5 yd
10 in
.
10 in
.
3 in
.3
in.
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
150
1.
2.
3.
4.
5.
6.
Alge
bra:
Per
imet
er F
orm
ulas
Sinc
e th
e si
des
of a
reg
ular
pol
ygon
are
equa
l, yo
u ca
n us
e a
form
ula
to fi
nd t
hepe
rimet
er o
f a r
egul
ar p
olyg
on.
Find
the
pe
rim
ete
r o
f e
ach
po
lyg
on
by
usin
g a
fo
rmul
a.
Perim
eter
(P) 5
(num
ber
of s
ides
) 3 S
P
5 5
3 8
P
5 4
0
So, t
he p
erim
eter
of t
he p
enta
gon
is 4
0 cm
.
Perim
eter
(P) 5
(num
ber
of s
ides
) 3 S
P
5 4
3 9
P
5 3
6
So, t
he p
erim
eter
of t
he s
quar
e is
36
in.
Find
the
pe
rim
ete
r o
f e
ach
po
lyg
on
by
usin
g a
fo
rmul
a.
9 in
.
8 cm
MXE
NL08
AW
K5X_
RT_C
H23_
L03.
indd
1
6/13
/07
1:3
5:08
PM
RW149-RW150
Grade5.indd 75Grade5.indd 75 7/19/07 6:03:31 PM7/19/07 6:03:31 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
152
Circ
umfe
renc
eTh
e di
stan
ce a
roun
d, o
r ci
rcum
fere
nce
, of a
circ
le is
p (a
bout
3.14)
mul
itplie
d by
the
di
stan
ce a
cros
s th
e ci
rcle
.
The
dist
ance
acr
oss
a ci
rcle
is c
alle
d th
e di
amet
er.
Hal
f the
dis
tanc
e ac
ross
a c
ircle
is c
alle
d th
e ra
dius
.
If yo
u kn
ow t
he d
iam
eter
, d, o
f the
circ
le, y
ou c
an u
se t
he fo
rmul
a C
= p
3 d
to
find
the
circ
umfe
renc
e, C
.
To t
he n
ear
est
hun
dre
dth
, fin
d t
he c
ircu
mfe
renc
e o
f a
circ
le t
hat
has
a ra
diu
s o
f 1
in.
The
radi
us is
hal
f of t
he d
iam
eter
so
mul
tiply
the
mea
sure
men
tat
the
rad
ius
by 2
.
1
3 2
5 2
in.
Use
the
form
ula
now
tha
t yo
u ha
ve t
he le
ngth
of t
he d
iam
eter
C
= p
3 d
C
= 3.14 3
2
C
= 6.28
So, t
he c
ircum
fere
nce
of t
he c
ircle
is a
bout
6.28
in.
To t
he n
ear
est
te
nth,
fin
d t
he c
ircu
mfe
renc
e o
f a
circ
le t
hat
has
1.
a di
amet
er o
f 4 m
2.
a ra
dius
of 1
0 in
.3.
a
diam
eter
of 1
0 in
.
4.
a ra
dius
of 9
ft5.
a
diam
eter
of 2
0 cm
6.
a ra
dius
of 2
yd
1in
.
MX
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10
:45:
40 A
M
Mak
e g
ene
riza
tio
ns t
o s
olv
e.
6.
The
lid o
f a d
esig
ner
box
is in
the
sh
ape
of a
reg
ular
hex
agon
. The
pe
rimet
er is
84
inch
es. W
hat
is t
he
leng
th o
f eac
h si
de o
f the
lid?
7.
Jen
has 96
feet
of f
enci
ng t
o m
ake
a re
ctan
gula
r sh
aped
pen
in h
er
back
yard
. The
wid
th o
f the
pen
is
15 fe
et. W
hat
is t
he le
ngth
of t
he p
en?
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
151
Prob
lem
Sol
ving
Wor
ksho
p Sk
ill:
Mak
e Ge
nera
lizat
ions
Two
tis
sue
bo
xes
are
co
ngru
ent
cub
es.
If
the
pe
rim
ete
r o
f th
e b
ase
of
one
ti
ssue
bo
x is
16
in.,
wha
t is
the
leng
th o
f o
ne s
ide
of
the
bas
e o
f th
e o
the
r ti
ssue
bo
x?
Re
ad t
o U
nde
rsta
nd
1.
Wha
t ar
e yo
u as
ked
to fi
nd?
Plan
2.
How
doe
s kn
owin
g th
at t
he t
wo
tissu
e bo
xes
are
cong
ruen
t he
lp y
ou?
So
lve
3.
Whi
ch fo
rmul
a w
ould
you
use
to
solv
e th
e pr
oble
m?
4.
Wha
t is
the
leng
th o
f one
sid
e of
the
bas
e of
the
oth
er t
issu
e bo
x?
Che
ck
5.
How
can
you
che
ck y
our
answ
er?
MX
EN
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6/07
10
:45:
15 A
M
RW151-RW152
Grade5.indd 76Grade5.indd 76 6/18/07 6:09:02 PM6/18/07 6:09:02 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
154
Alge
bra:
Are
a of
Squ
ares
and
Rec
tang
les
You
can
find
the
are
a of
a s
quar
e or
rec
tang
le b
y us
ing
form
ulas
. Fi
nd t
he a
rea
of
the
fig
ure
at
the
rig
ht.
Are
a o
f a
squa
re.
O
ne p
art
of t
he fi
gure
is a
12.4
km
-by-
12.4
km
squ
are.
To
find
the
are
a of
the
squ
are,
use
the
form
ula
A
5 S
2 .
A
5 1
2.42 o
r A
5 1
53.7
6
Are
a o
f a
rect
ang
le.
Th
e ot
her
part
of t
he fi
gure
is a
6 k
m-b
y-9.
6 km
rec
tang
le.
To
find
the
are
a of
the
rect
angl
e, u
se t
he fo
rmul
a A
5 1
3 w
.
A
5 9
.6 3
6 o
r A
5 5
7.6
Are
a o
f a
po
lyg
on.
To
find
the
tot
al a
rea,
add
the
are
a of
the
squ
are
to t
he a
rea
of t
he r
ecta
ngle
.
15
3.76
1 5
7.6
5 2
11.3
6
So, t
he a
rea
of t
he fi
gure
is 2
11.3
6 km
2 .
1.2.
3.
5 in
.
5 in
.
5 in
.
12 in
.
10 y
d
4 yd
8 cm
8 cm
Find
the
are
a o
f e
ach
figur
e.
12.4
km
9.6
km
12.4
km
6 km
MXE
NL08
AW
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RT_C
H24_
L2.in
dd
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13/0
7 1
:39:
00 P
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
153
Estim
ate
Area
The
are
a of
a fi
gure
is t
he n
umbe
r of
sq
uare
uni
ts n
eede
d to
cov
er it
.
You
can
use
cen
timet
er g
rid p
aper
to
estim
ate
the
area
of a
figu
re.
Esti
mat
e t
he a
rea
of
the
sha
de
d f
igur
e.
Each
sq
uare
on
the
gri
d is
1 in
.2
Ste
p 1
Cou
nt t
he n
umbe
r of
par
tial s
quar
es.
Skip
squ
ares
with
onl
y a
tiny
corn
er.
Tota
l num
ber
of p
artia
l sq
uare
s: 8
Ste
p 2
Div
ide
the
tota
l by
2.8
4 2
5 4
Ste
p 3
Cou
nt t
he n
umbe
r of
full
squa
res,
in
clud
ing
thos
e m
issi
ng o
nly
a tin
y co
rner
.
Tota
l num
ber
of fu
ll sq
uare
s: 1
8
Ste
p 4
Add
the
val
ues
from
Ste
ps 2
and
3.
4 1
18
5 2
2
1.2.
3.
4.5.
6.
So, t
he e
stim
ated
are
a of
the
sha
ded
figur
e is
abo
ut 2
2 in
.2
Esti
mat
e t
he a
rea
of
the
sha
de
d f
igur
e. E
ach
squa
re o
n th
e g
rid
is 1
cm
2.
MXE
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:38:
22 P
M
RW153-RW154
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
155
Alge
bra:
Rel
ate
Perim
eter
and
Are
aY
ou c
an u
se g
rid p
aper
to
rela
te p
erim
eter
and
are
a.
For
the
pe
rim
ete
r 44
yar
ds,
fin
d t
he le
ngth
and
wid
th o
f th
e r
ect
ang
le
wit
h th
e g
reat
est
are
a. U
se w
hole
num
be
rs o
nly.
Ste
p 1
D
ivid
e th
e pe
rimet
er b
y 2.
The
leng
th a
nd
w
idth
of a
rec
tang
le w
hose
per
imet
er is
44
have
a s
um o
f 22.
Ste
p 2
M
ake
as m
any
diffe
rent
rec
tang
les
as p
ossi
ble.
Find
the
are
a of
eac
h.
C
ompl
ete
a ta
ble
show
ing
poss
ibili
ties.
Stop
whe
n yo
u re
aliz
e yo
u ha
ve fo
und
the
gr
eate
st a
rea.
So ,
the
rect
angl
e w
ith t
he g
reat
est
area
for
a re
ctan
gle
with
a p
erim
eter
of 4
4 ya
rds
is
11 y
d 3
11
yd.
For
a g
ive
n p
eri
me
ter,
the
sq
uare
has
the
gre
ate
st a
rea.
1.
20
in.
2.
38 k
m3.
32
cm
4.
26 in
.5.
8
yd
You
can
use
fact
ors
of
a gi
ven
area
to
find
the
leng
th a
nd w
idth
of
rect
angl
es.
Leng
thW
idth
Peri
me
ter
Are
a1
2144
21
220
4440
319
4457
418
4472
517
4485
616
4496
715
4410
5
814
4411
2
913
4411
7
1012
4412
0
1111
4412
1
1210
4412
0
139
4411
7
For
the
giv
en
pe
rim
ete
r, f
ind
the
leng
th a
nd w
idth
of
the
re
ctan
gle
wit
h th
e g
reat
est
are
a. U
se w
hole
num
be
rs o
nly.
MX
EN
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16/
6/07
1:
45:3
8 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
156
Alge
bra:
Are
a of
Tria
ngle
sY
ou c
an u
se g
rid p
aper
to
find
a re
latio
nshi
p be
twee
n th
e ar
eas
of
tria
ngle
s an
d re
ctan
gles
.
So t
he a
rea
is 1
7.5
m2 .
Find
the
are
a o
f e
ach
tria
ngle
.
A 5
1 __ 2 b
h
A 5
1 __ 2 3
7 3
5
A 5
17.
5
1.2.
3.5.
6 m
7 m
5 m
10.5
in.
8 in
.
7.6
m
5 1
__ 8 ft
7 ft
Ste
p 1
D
raw
a r
ecta
ngle
on
grid
pap
er.
Mak
e ea
ch s
ide
grea
ter
than
10
units
.
Fi
nd a
nd r
ecor
d th
e ar
ea o
f the
rec
tang
le.
Ste
p 2
C
ut o
ut the
rec
tang
le.
Ste
p 3
Pl
ace
a do
t on
one
sid
e of
the
rec
tang
le.
Use
a r
uler
to
draw
a li
ne fr
om t
he d
ot t
o ea
ch o
ppos
ite c
orne
r.
Y
ou h
ave
mad
e a
tria
ngle
.
Ste
p 4
Fo
ld t
he r
ecta
ngle
alo
ng t
wo
side
s of
the
tria
ngle
.
•
Doe
s it
cove
r th
e tr
iang
le w
ithou
t ov
erla
ppin
g?
• W
hat
does
tha
t sa
y ab
out
the
area
of t
he t
riang
le?
Rep
eat
step
s 1
thro
ugh
4 w
ith t
he s
ame-
size
rec
tang
le,
but
plac
e th
e do
t in
Ste
p 3
on a
diff
eren
t si
de.
How
doe
s th
e ar
ea o
f the
tria
ngle
rel
ate
to t
he a
rea
of t
he r
ecta
ngle
?
If b
is t
he b
ase
and
h is
the
he
ight
, yo
u ca
n us
e t
he f
orm
ula
A 5
1 __ 2 b
h to
fin
d t
he a
rea,
A, o
f a
tria
ngle
.
Find
the
are
a o
f th
e t
rian
gle
.
MXE
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7 1
:43:
06 P
M
RW155-RW156
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Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
157
Alge
bra:
Are
a of
Par
alle
logr
ams
You
can
use
grid
pap
er a
nd t
he b
ase
and
heig
ht o
f a
para
llelo
gram
to
find
its a
rea.
Wri
te t
he b
ase
and
he
ight
of
the
par
alle
log
ram
.Th
en
find
its
are
a in
sq
uare
uni
ts.
Ste
p 1
Dra
w a
par
alle
logr
am o
n gr
id p
aper
and
cut
it o
ut.
Ste
p 2
Dra
w a
line
seg
men
t at
the
bot
tom
row
to
form
a
right
tria
ngle
.
Ste
p 3
Cut
out
the
tria
ngle
on
the
bott
om a
nd m
ove
it to
th
e to
p of
the
par
alle
logr
am t
o fo
rm a
rec
tang
le.
Ste
p 4
Cou
nt t
he g
rid s
quar
es t
o fin
d th
e ar
ea o
f the
par
alle
logr
am.
The
base
is 3
uni
ts a
nd t
he h
eigh
t is
5 u
nits
.
The
form
ula
for
the
area
of a
par
alle
logr
am is
bas
e 3
hei
ght
So, t
he a
rea
of t
he p
aral
lelo
gram
is:
A
5 b
h
A
5 3
3 5
A
5 1
5 un
its2
1.2.
3.
5
3
Find
the
are
a o
f e
ach
par
alle
log
ram
.
bh
MX
EN
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47:0
8 PM
Nam
e
© H
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urt •
Gra
de 5
Ret
each
RW
158
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Solv
e a
Sim
pler
Pro
blem
Fran
’s s
un c
atch
er
has
6 ro
ws
of
5 sq
uare
s. E
ach
squa
re
has
3 ro
ws
of
3 sm
all s
qua
res.
Ho
w m
any
smal
l sq
uare
s ar
e
in F
ran’
s su
n ca
tche
r?
Re
ad
to
Un
de
rsta
nd
1.
Wha
t ar
e yo
u as
ked
to fi
nd?
Pla
n
2.
How
can
pic
ture
s he
lp y
ou b
reak
the
pro
blem
into
sim
pler
par
ts?
So
lve
3.
The
first
row
of t
he s
un c
atch
er is
sho
wn.
H
ow m
any
smal
l squ
ares
are
in t
his
1st
row
?
4.
Mul
tiply
you
r an
swer
from
que
stio
n 3
by 6
. How
man
y sm
all s
quar
es a
re in
Fra
n’s
sun
catc
her?
Ch
eck
5.
Wha
t is
ano
ther
way
to
solv
e th
is p
robl
em?
Use
the
so
lve
a si
mpl
er p
robl
em s
trat
eg
y to
so
lve
.
6.
Find
the
are
a of
the
figu
re.
7.
If th
e ar
ea o
f a r
ecta
ngle
is 3
6 cm
lo
ng a
nd o
ne o
f its
sid
es is
8 c
m w
hile
its
per
imet
er is
25
cm, t
hen
wha
t is
th
e le
ngth
of i
ts o
ther
3 s
ides
co
mbi
ned?
5m
8m
2m
7m
MX
EN
L08
AW
K5X
_RT
_CH
24_L
6.in
dd1
7/20
/07
11:4
4:26
AM
RW157-RW158
Grade5.indd 79Grade5.indd 79 7/20/07 11:49:04 AM7/20/07 11:49:04 AM
E
BD
CA
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
159
Surf
ace
Area
You
can
find
the
sur
face
are
a, th
e to
tal a
rea
of t
he s
urfa
ce o
f a s
olid
figu
re,
by a
ddin
g th
e ar
ea o
f eac
h fa
ce.
Ano
ther
way
to
find
the
surf
ace
are
a is
to
use
a ne
t.
Use
the
ne
t to
fin
d t
he s
urfa
ce a
rea
of
the
fig
ure
in s
qua
re u
nits
.
Find
the
are
a of
the
squ
are
A
5 s
3 s
A
5 4
3 4
A
5 1
6Fi
nd t
he a
rea
of o
ne t
riang
le
A 5
1 _ 2 3
b 3
h
A 5
1 _ 2 3
4 3
4
A 5
8A
dd t
he a
reas
of t
he s
quar
e an
d th
e 4
tria
ngle
s;
16 1
8 1
8 1
8 1
8
So, t
he s
urfa
ce a
rea
of t
he s
quar
e py
ram
id is
48
squa
re u
nits
or
48 u
nits
2.
Use
the
ne
t to
fin
d t
he s
urfa
ce a
rea
of
eac
h p
rism
in
sq
uare
uni
ts.
1.
2.
MXE
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:44:
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M
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e
© H
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urt •
Gra
de 5
Ret
each
RW
160
Alge
bra:
Est
imat
e an
d Fi
nd V
olum
eJu
st a
s yo
u us
ed s
quar
es t
o fin
d th
e ar
ea o
f a r
ecta
ngle
, yo
u ca
n us
e cu
bes
to fi
nd t
he v
olu
me
of a
rec
tang
ular
pris
m.
Are
a is
mea
sure
d in
squ
ares
, or
in s
quar
e un
its.
Vol
ume
is m
easu
red
in c
ubes
, or
in c
ubic
uni
ts.
Each
face
of a
squ
are
is o
ne s
quar
e un
it.
Find
the
vo
lum
e o
f th
e r
ect
ang
ular
pri
sm.
So, t
he v
olum
e of
the
pris
m is
27
cubi
c fe
et, o
r 27
ft3 .
Find
the
vo
lum
e o
f e
ach
rect
ang
ular
pri
sm in
cub
ic c
ent
ime
ters
.
1.2.
3.
3 ft
3 ft
3 ft
5 in
5 in
4 in
10 c
m8
cm
8 cm
2.5
m4.
8 m
3.1
m
Th
e vo
lum
e w
ill t
ell y
ou h
ow m
uch
sp
ace
can
fi t in
side
the
box
, the
cube
, or
any
othe
r fi g
ure.
The
form
ula
for
volu
me
is
Vol
ume 5
leng
th 3
wid
th 3
hei
ght
V
5 3
ft 3
3ft 3
3ft
V
5 2
7ft3
MXE
NL08
AW
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RT_C
H24_
L8.in
dd
16/
13/0
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:48:
05 P
M
RW159-RW160
Grade5.indd 80Grade5.indd 80 6/18/07 6:11:10 PM6/18/07 6:11:10 PM
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
161
Rela
te P
erim
eter
, Are
a, a
nd V
olum
ePe
rimet
er is
the
dis
tanc
e ar
ound
a fl
at fi
gure
.
Are
a is
the
spa
ce o
n a
flat
surf
ace
of a
flat
figu
re.
Vol
ume
is t
he a
mou
nt o
f spa
ce a
sol
id fi
gure
tak
es u
p.
Wri
te t
he u
nits
yo
u co
uld
use
fo
r m
eas
urin
g t
he a
rea
of
this
tri
ang
le
Are
a
A
5 1
3 w
5
8 in
3 4
in 5
32
in2
in
3
in
5
in2
So, u
nit
for
area
5 in
2
U
se u
nit
squa
re o
r cm
2 , ft2 ,
yd2 ,
mi2 ,
m2 ,
or k
m2 .
Peri
me
ter
Per
imet
er 5
sid
e 1
sid
e 1
sid
e1 s
ide 1
sid
e 1
sid
e
5 5
m 1
5 m
1 5
m 1
5 m
1 5
m 1
5 m
5 3
0 m
So, u
nit
for
perim
eter
5 m
U
se t
he u
nit
or c
m.,
ft, y
d, m
i, in
, or
k.
Vo
lum
e
V
olum
e 5
l 3
w 3
h
5 3
ft 3
3 ft
3 7
ft 5
63
ft3
ft
3
ft 3
ft 5
ft3
So, t
he u
nit
for
volu
me 5
ft3
U
se c
ubic
uni
ts o
r cm
3, y
d3, i
n3, m
i3, m
3, o
r km
3.
1. a
rea
of t
his
para
llelo
gram
2. v
olum
e of
thi
s tr
iang
ular
pris
m3.
pe
rimet
er o
f thi
s re
gula
r pe
ntag
on
7 cm
4 ft
5 cm
3 cm
8 in8 in
.
4 in
.
5 m
Wri
te t
he u
nits
yo
u w
oul
d u
se f
or
me
asur
ing
eac
h.
7 ft
3 ft
3 ft
MXE
NL08
AW
K5X_
RT_C
H24_
L9.in
dd
16/
13/0
7 1
:53:
06 P
M
Nam
e
© H
arco
urt •
Gra
de 5
Ret
each
RW
162
Prob
lem
Sol
ving
Wor
ksho
p St
rate
gy:
Com
pare
Str
ateg
ies
Tash
i is
rep
airi
ng t
iles
on
a ki
tche
n flo
or.
The
flo
or
is in
the
sha
pe
of
a re
ctan
gle
. The
are
a o
f th
e f
loo
r is
156
ft2 .
The
leng
th is
13
fee
t. W
hat
is t
he w
idth
of
the
flo
or?
Re
ad
to
Un
de
rsta
nd
1.
Wha
t is
the
sha
pe a
nd s
ize
of t
he k
itche
n flo
or?
Pla
n
2.
Nam
e tw
o di
ffere
nt s
trat
egie
s yo
u ca
n us
e to
sol
ve t
his
prob
lem
.
So
lve
3.
One
way
to
solv
e is
to
use
Mak
e a
Mod
el. T
he le
ngth
is 1
3 fe
et, s
o us
e gr
id p
aper
to
draw
a
13 3
1 r
ecta
ngle
. The
n se
e if
it ha
s 15
6 sq
uare
s. If
it d
oes
not,
draw
a 1
3 3
2 a
nd t
hen
a 13
3 3
rec
tang
le. C
ontin
ue u
ntil
you
have
156
squ
ares
insi
de t
he r
ecta
ngle
.
Ano
ther
way
to
solv
e th
e pr
oble
m is
to
Writ
e an
Equ
atio
n an
d so
lve
it.Si
nce
A 5
l 3
w, 1
56 5
13 3
w.
Whi
ch is
the
bet
ter
met
hod?
Exp
lain
.
Ch
eck
4.
How
can
you
che
ck t
hat
the
mis
sing
leng
th is
12
feet
?
Mak
e a
mo
de
l or
wri
te a
n e
qua
tio
n to
so
lve
.
5. T
hree
str
aigh
t sl
ices
are
mad
e ac
ross
api
zza.
Wha
t is
the
mos
t nu
mbe
r of
pie
ces
poss
ible
? [H
int:
Piec
es d
o no
t ha
ve t
o be
equa
l in
size
or
shap
e.]
6. T
he v
olum
e of
an
elev
ator
is 1
200
ft3 .Its
floo
r is
a 1
0 ft-
by-1
0 ft
squa
re.
Wha
t is
its
heig
ht?
MX
EN
L08
AW
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_RT
_CH
24_L
10.in
dd
16/
6/07
1:
50:5
2 PM
RW161-RW162
Grade5.indd 81Grade5.indd 81 6/18/07 6:11:35 PM6/18/07 6:11:35 PM
