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G ra d e 4
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CONTENTS
Unit 1: UNDERSTAND NUMBERS AND
OPERATIONS
Chapter 1: Place Value and Number Sense
1.1 Benchmark Numbers . . . . . . . . . . . . . 1
1.2 Understand Place Value . . . . . . . . . . 2
1.3 Place Value Through Hundred
Thousands . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Place Value Through Millions . . . . . 4
1.5 Problem Solving Skill: Use
a Graph . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: Compare and
Order Numbers
2.1 Compare Numbers . . . . . . . . . . . . . . 6
2.2 Order Numbers . . . . . . . . . . . . . . . . . 7
2.3 Problem Solving Strategy:
Make a Table . . . . . . . . . . . . . . . . . . . . 8
2.4 Round Numbers . . . . . . . . . . . . . . . . . 9
Chapter 3: Add and Subtract
Greater Numbers
3.1 Estimate Sums and Differences . . . 10
3.2 Use Mental Math Strategies . . . . . . 11
3.3 Add and Subtract 4-Digit
Numbers . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Subtract Across Zeros . . . . . . . . . . . 13
3.5 Add and Subtract Greater
Numbers . . . . . . . . . . . . . . . . . . . . . . 14
3.6 Problem Solving Skill: Estimate or
Find Exact Answers . . . . . . . . . . . . . . 15
Chapter 4: Algebra: Use Addition
and Subtraction
4.1 Expressions . . . . . . . . . . . . . . . . . . . . 16
4.2 Use Parentheses . . . . . . . . . . . . . . . . 17
4.3 Match Words and Expressions . . . . 18
4.4
4.5
4.6
4.7
Use Variables . . . . . . . . . . . . . . . . . . . 19
Find a Rule . . . . . . . . . . . . . . . . . . . . 20
Add Equals to Equals . . . . . . . . . . . . 21
Problem Solving Strategy:
Make a Model . . . . . . . . . . . . . . . . . 22
Unit 2: DATA, GRAPHING,
AND TIME
Chapter 5: Collect and Organize
Data
5.1 Collect and Organize Data . . . . . . . 23
5.2 Find Median and Mode . . . . . . . . . 24
5.3 Line Plot . . . . . . . . . . . . . . . . . . . . . . . 25
5.4 Stem-and-Leaf Plot . . . . . . . . . . . . . 26
5.5 Compare Graphs . . . . . . . . . . . . . . . 27
5.6 Problem Solving Strategy:
Make a Graph . . . . . . . . . . . . . . . . . . 28
Chapter 6: Analyze and
Graph Data
6.1 Double-Bar Graphs . . . . . . . . . . . . . 29
6.2 Read Line Graphs. . . . . . . . . . . . . . . 30
6.3 Make Line Graphs . . . . . . . . . . . . . . . 31
6.4 Choose an Appropriate
Graph . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.5 Problem Solving Skill: Draw
Conclusions . . . . . . . . . . . . . . . . . . . . 33
Chapter 7: Understand Time
7.1 Before and After the Hour . . . . . . . 34
7.2 A.M. and P.M. . . . . . . . . . . . . . . . . . . 35
7.3 Elapsed Time . . . . . . . . . . . . . . . . . . 36
7.4 Problem Solving Skill: Sequence
Information . . . . . . . . . . . . . . . . . . . . 37
7.5 Elapsed Time on a Calendar . . . . . 38
Unit 3: MULTIPLICATION AND
DIVISION FACTS
Chapter 8: Practice Multiplication
and Division Facts
8.1 Relate Multiplication
and Division . . . . . . . . . . . . . . . . . . . 39
8.2 Multiply and Divide Facts
Through 5 . . . . . . . . . . . . . . . . . . . . . 40
8.3 Multiply and Divide Facts
Through 10 . . . . . . . . . . . . . . . . . . . . . 41
8.4 Multiplication Table
Through 12 . . . . . . . . . . . . . . . . . . . . . 42
8.5 Multiply 3 Factors . . . . . . . . . . . . . . 43
8.6 Problem Solving Skill: Choose
the Operation . . . . . . . . . . . . . . . . . 44
Chapter 9: Algebra: Use Multiplication
and Division Facts
9.1 Expressions with Parentheses . . . . 45
9.2 Match Words and Expressions . . . 46
9.3 Multiply Equals by Equals . . . . . . . 47
9.4 Expressions with Variables . . . . . . . 48
9.5 Equations with Variables . . . . . . . . 49
9.6 Find a Rule . . . . . . . . . . . . . . . . . . . . 50
9.7 Problem Solving Strategy:
Work Backward . . . . . . . . . . . . . . . . . 51
Unit 4: MULTIPLY BY 1- AND 2-DIGIT
NUMBERS
Chapter 10: Multiply by 1-Digit Numbers
10.1 Mental Math: Multiplication
Patterns . . . . . . . . . . . . . . . . . . . . . . . 52
10.2 Estimate Products . . . . . . . . . . . . . . 53
10.3 Model Multiplication . . . . . . . . . . . 54
10.4 Multiply 3-Digit Numbers . . . . . . . 55
10.5 Multiply 4-Digit Numbers . . . . . . . 56
10.6 Problem Solving Strategy:
Write an Equation . . . . . . . . . . . . . . 57
Chapter 11: Understand Multiplication
11.1 Mental Math: Patterns
with Multiples . . . . . . . . . . . . . . . . . 58
11.2 Multiply by Multiples of 10 . . . . . . 59
11.3 Estimate Products . . . . . . . . . . . . . . 60
11.4 Model Multiplication . . . . . . . . . . . 61
11.5 Problem Solving Strategy: Solve a
Simpler Problem . . . . . . . . . . . . . . . 62
Chapter 12: Multiply by 2-Digit
Numbers
12.1 Multiply by 2-Digit Numbers . . . . . 63
12.2 More About Multiplying by
2-Digit Numbers . . . . . . . . . . . . . . . 64
12.3 Multiply Greater Numbers . . . . . . 65
12.4 Practice Multiplication . . . . . . . . . . 66
12.5 Problem Solving Skill: Multistep
Problems . . . . . . . . . . . . . . . . . . . . . . 67
Unit 5: DIVIDE BY 1-AND 2-DIGIT
DIVISORS
Chapter 13: Understand Division
13.1 Divide with Remainders . . . . . . . . . 68
13.2 Model Division . . . . . . . . . . . . . . . . . 69
13.3 Division Procedures . . . . . . . . . . . . 70
13.4 Problem Solving Strategy: Predict
and Test . . . . . . . . . . . . . . . . . . . . . . . 71
13.5 Mental Math: Division
Patterns . . . . . . . . . . . . . . . . . . . . . . . 72
13.6 Estimate Quotients . . . . . . . . . . . . . 73
Chapter 14: Divide by 1-Digit Divisors
14.1 Place the First Digit . . . . . . . . . . . . . 74
14.2 Divide 3-Digit Numbers . . . . . . . . . 75
14.3 Zeros in Division . . . . . . . . . . . . . . . 76
14.4 Divide Greater Numbers . . . . . . . . 77
14.5 Problem Solving Skill: Interpret
the Remainder . . . . . . . . . . . . . . . . . 78
14.6 Find the Mean . . . . . . . . . . . . . . . . . 79
Chapter 15: Divide by 2-Digit Divisors
15.1 Division Patterns to
Estimate . . . . . . . . . . . . . . . . . . . . . . 80
15.2 Model Division . . . . . . . . . . . . . . . . . 81
15.3 Division Procedures . . . . . . . . . . . . 82
15.4 Correcting Quotients . . . . . . . . . . . 83
15.5 Problem Solving Skill: Choose
the Operation . . . . . . . . . . . . . . . . . 84
Chapter 16: Patterns with Factors
and Multiples
16.1 Factors and Multiples . . . . . . . . . . . 85
16.2 Factor Numbers . . . . . . . . . . . . . . . . 86
16.3 Prime and Composite
Numbers . . . . . . . . . . . . . . . . . . . . . . 87
16.4 Find Prime Factors . . . . . . . . . . . . . . 88
16.5 Problem Solving Strategy:
Find a Pattern . . . . . . . . . . . . . . . . . . 89
Unit 6: FRACTIONS AND DECIMALS
Chapter 17: Understand Fractions
17.1 Read and Write Fractions . . . . . . . 90
17.2 Equivalent Fractions . . . . . . . . . . . . 91
17.3 Equivalent Fractions . . . . . . . . . . . . 92
17.4 Compare and Order
Fractions . . . . . . . . . . . . . . . . . . . . . . 93
17.5 Problem Solving Strategy:
Make a Model . . . . . . . . . . . . . . . . . 94
17.6 Mixed Numbers . . . . . . . . . . . . . . . . 95
Chapter 18: Add and Subtract Fractions
and Mixed Numbers
18.1 Add Like Fractions . . . . . . . . . . . . . . 96
18.2 Subtract Like Fractions . . . . . . . . . . 97
18.3 Add and Subtract Mixed
Numbers . . . . . . . . . . . . . . . . . . . . . . 98
18.4 Problem Solving Skill: Choose
the Operation . . . . . . . . . . . . . . . . . 99
18.5 Add Unlike Fractions . . . . . . . . . . 100
18.6 Subtract Unlike Fractions . . . . . . . 101
Chapter 19: Understand Decimals
19.1 Relate Fractions and
Decimals . . . . . . . . . . . . . . . . . . . . . 102
19.2 Decimals Greater Than 1 . . . . . . . . 103
19.3 Equivalent Decimals . . . . . . . . . . . 104
19.4 Compare and Order Decimals . . . 105
19.5 Problem Solving Strategy: Use
Logical Reasoning . . . . . . . . . . . . . 106
19.6 Relate Mixed Numbers and
Decimals . . . . . . . . . . . . . . . . . . . . . 107
Chapter 20: Add and Subtract Decimals
20.1 Round Decimals . . . . . . . . . . . . . . . 108
20.2 Estimate Sums and
Differences . . . . . . . . . . . . . . . . . . . 109
20.3 Add Decimals . . . . . . . . . . . . . . . . . 110
20.4 Subtract Decimals . . . . . . . . . . . . . . 111
20.5 Add and Subtract Decimals . . . . . 112
20.6 Problem Solving Skill: Evaluate
Reasonableness of Answers . . . . . 113
Unit 7: MEASUREMENT, ALGEBRA,
AND GRAPHING
Chapter 21: Customary Measurement
21.1 Choose the Appropriate Unit . . . 114
21.2 Measure Fractional Parts . . . . . . . . 115
21.3 Algebra: Change Linear Units . . . . 116
21.4 Capacity . . . . . . . . . . . . . . . . . . . . . . 117
21.5 Weight . . . . . . . . . . . . . . . . . . . . . . . 118
21.6 Problem Solving Strategy:
Compare Strategies . . . . . . . . . . . . 119
Chapter 22: Metric Measurement
22.1 Linear Measure . . . . . . . . . . . . . . . . 120
22.2 Algebra: Change Linear Units . . . . 121
22.3 Capacity . . . . . . . . . . . . . . . . . . . . . . 122
22.4 Mass . . . . . . . . . . . . . . . . . . . . . . . . . 123
22.5 Problem Solving Strategy: Draw
a Diagram . . . . . . . . . . . . . . . . . . . . . 124
Chapter 23: Algebra: Explore Negative
Numbers
23.1 Temperature: Fahrenheit . . . . . . . . 125
23.2 Temperature: Celsius . . . . . . . . . . . 126
23.3 Negative Numbers . . . . . . . . . . . . . 127
23.4 Problem Solving Skill: Make
Generalizations . . . . . . . . . . . . . . . 128
Chapter 27: Solid Figures and Volume
27.1 Faces, Edges, and Vertices . . . . . . 145
27.2 Patterns for Solid Figures . . . . . . . 146
27.3 Estimate and Find Volume
of Prisms . . . . . . . . . . . . . . . . . . . . . 147
27.4 Problem Solving Skill: Too Much/
Too Little Information . . . . . . . . . 148
Chapter 24: Explore the
Coordinate Grid
24.1 Use a Coordinate Grid . . . . . . . . . 129
24.2 Length on the Coordinate
Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 130
24.3 Use an Equation . . . . . . . . . . . . . . . . 131
24.4 Graph an Equation . . . . . . . . . . . . . 132
24.5 Problem Solving Skill: Identify
Relationships . . . . . . . . . . . . . . . . . . 133
Chapter 28: Measure and Classify
Plane Figures
28.1 Turns and Degrees . . . . . . . . . . . . . 149
28.2 Measure Angles . . . . . . . . . . . . . . . 150
28.3 Circles . . . . . . . . . . . . . . . . . . . . . . . . 151
28.4 Circumference . . . . . . . . . . . . . . . . . 152
28.5 Classify Triangles . . . . . . . . . . . . . . 153
28.6 Classify Quadrilaterals . . . . . . . . . 154
28.7 Problem Solving Strategy:
Draw a Diagram . . . . . . . . . . . . . . . . 155
Unit 8: GEOMETRY
Chapter 25: Plane Figures
25.1 Lines, Rays, and Angles . . . . . . . . . 134
25.2 Line Relationships . . . . . . . . . . . . . 135
25.3 Congruent Figures and
Motion . . . . . . . . . . . . . . . . . . . . . . . 136
25.4 Symmetric Figures . . . . . . . . . . . . . 137
25.5 Problem Solving Strategy:
Make a Model . . . . . . . . . . . . . . . . . 138
Chapter 26: Perimeter and Area
of Plane Figures
26.1 Perimeter of Polygons . . . . . . . . . . 139
26.2 Estimate and Find
Perimeter . . . . . . . . . . . . . . . . . . . . . 140
26.3 Estimate and Find Area . . . . . . . . . 141
26.4 Relate Area and Perimeter . . . . . . 142
26.5 Relate Formulas and Rules . . . . . . 143
26.6 Problem Solving Strategy:
Find a Pattern . . . . . . . . . . . . . . . . . 144
Unit 9: PROBABILITY
Chapter 29: Outcomes
29.1 Record Outcomes . . . . . . . . . . . . . 156
29.2 Tree Diagrams . . . . . . . . . . . . . . . . . 157
29.3 Problem Solving Strategy:
Make an Organized List . . . . . . . . 158
29.4 Predict Outcomes of
Experiments . . . . . . . . . . . . . . . . . . . 159
Chapter 30: Probability
30.1 Probability as a Fraction . . . . . . . . 160
30.2 More About Probability . . . . . . . . 161
30.3 Test for Fairness . . . . . . . . . . . . . . . 162
30.4 Problem Solving Skill: Draw
Conclusions . . . . . . . . . . . . . . . . . . . 163
LESSON 1.1
Name
Benchmark Numbers
You can use a known number, called a benchmark,
to help you estimate another number that is difficult
to count or measure.
• Is the tree about 20 or about 200 feet tall?
You know that the man is 6 feet tall.
So, 6 is the benchmark.
Look at the tree. It looks like it is a little more
than 3 times as tall as the man.
Think: 3 × 6 = 18
6 ft
So, the tree is about 20 feet tall.
Use the benchmark to decide which is the more
reasonable estimate for the unknown amount.
Circle that number.
1.
number of beans in jar
2.
pounds of potatoes in the box
50 beans
30 lb
100
3.
or
200
gallons of water in the pool
100
4.
or
1,000
paper clips in the container
© Harcourt
200 clips
400 gallons
800
or
1,200
100
or
1,000
Reteach
RW1
LESSON 1.2
Name
Understand Place Value
The value of a digit in a number
depends on its position within
the number.
Thousands Hundreds
3,
The value of the digit 7 in 3,750 is 700
because it is in the hundreds place.
Tens
Ones
5
0
7
3 × 1,000 7 × 100 5 × 10 0 × 0
To find the value of the digit 5,
first decide the place-value position of
the 5. The 5 is in the
place.
3,000
700
50
0
5 10 50, so the 5 has a value of 50.
To increase a number by 5,000, increase the digit in the
thousands place by 5.
3,750 increased by 5,000 is 8,750. Notice that the only
change was the increase by 5 in the thousands place.
Look at the number 48,792.
1. In what place-value position is
the 4?
2.
Complete to find the value of
the 8.
8
3.
What is the value of the 9?
4.
What is the value of the 2?
For 5–10, use the numbers at the top of each column.
88,203 to 81,203
In what place value was a
digit changed?
5.
6.
Was the digit increased or
decreased?
7.
8.
What is the change in the
value of the number?
9.
10.
RW2
Reteach
© Harcourt
45,893 to 85,893
LESSON 1.3
Name
Place Value Through Hundred Thousands
In the place-value system, each place is ten times as great
as the place to its right.
Complete the table.
THOUSANDS
Period
Place
Hundreds
Tens
Ones
Hundreds
1,000
Value
Think:
ONES
100
thousands
10
thousands
1
thousand
100
ones
Tens
Ones
10
1
10
ones
1
one
To write a number in word form, write the number in
each period and follow it with the name of the period. Do
not name the ones period. Remember to use commas to
separate each period of numbers.
367,412 is three hundred sixty-seven thousand, four
hundred twelve.
Write each number in word form.
1.
56,398
2.
308,481
To write a number in expanded form, write the sum of the value
of each digit. Do not include digits for which the value is 0.
© Harcourt
367,012 300,000 60,000 7,000 10 2
Write each number in expanded form.
3.
458,913
4.
290,384
Reteach
RW3
LESSON 1.4
Name
Place Value Through Millions
The period after the thousands period is the millions period.
Write the values that complete the table.
MILLIONS
Period
Place
Hundreds
Tens
Ones
Value
Think:
100
million
THOUSANDS
10
million
Hundreds
Tens
ONES
Ones
Hundreds
1,000,000
10,000
100
1
million
100
10
1
thousand thousand thousand
100
ones
Tens
Ones
10
ones
1
one
To write a number given in word form with the millions period, use a
place-value chart.
Fill in the place-value chart to find the standard form of three hundred
fifty-seven million, two hundred twelve thousand, one hundred eight.
MILLIONS
Period
THOUSANDS
Place
Hundreds
Tens
Ones
Value
_______
5
7,
Hundreds
Tens
_______ _______
ONES
Ones
Hundreds
Tens
Ones
2,
_______
0
_______
Write the value of the bold digit.
1.
32,404,202
2.
489,304,203
3.
44,203,203
4.
5,306,200
Complete.
349,305,203
three hundred forty-nine
two hundred three
6.
,three hundred five
62,617,204
-two million, six hundred
hundred four
RW4
Reteach
thousand,
,
© Harcourt
5.
LESSON 1.5
Name
Problem Solving Skill
Use a Graph
A pictograph is a graph that uses pictures to represent numbers in
order to compare information. The pictures are related to the data.
Box Office Video records their national rentals each week using a
pictograph. Each picture of a VCR represents 1,000,000 rentals.
BOX OFFICE VIDEO NATIONAL RENTALS
Week 6
Week 5
Week 4
Week 3
Week 2
Week 1
Key: Each
= 1,000,000 rentals.
The graph shows four VCR symbols during Week 1.
So, 4 1,000,000 4,000,000 video rentals were made during Week 1.
Half of a symbol represents 500,000 video rentals.
© Harcourt
So, during Week 2 there were 4,000,000 500,000 4,500,000 rentals made.
1.
What other symbol might Box
Office Video have chosen to
represent 1,000,000 videos?
2.
During which week were the
greatest number of rentals
made? the fewest?
3.
How many rentals were made
during Week 3?
4.
How many rentals were made
during Week 1?
5.
During Week Five, 7,000,000
videos were rented. Add this
information to the graph.
6.
During Week Six, 6,500,000
videos were rented. Add this
information to the graph.
Reteach
RW5
LESSON 2.1
Name
Compare Numbers
Which number is greater, 3,491 or 3,487?
Step 1
Write one number under the other.
Step 2
Compare the digits, beginning with the greatest
place-value position.
Step 3
Circle the first digits that are different.
ts ts
gi igi
i
d d
e e
m
m
sa sa
↓ ↓
3,487
3,491
Think: Since 8 tens < 9 tens,
3,487 < 3,491.
Circle the greater number in each pair of numbers. Then write < or >.
627
637
2.
627
4.
637
7.
5.
312,629
54,329
28,617
54,329
10.
28,617
327,203
327,320
327,320
6,410
6,503
6,410
RW6
6,503
Reteach
3,200
998
4,423
1,062
1,028
40,894
763,492
739,814
763,492
15.
44,988
40,890
40,894
40,890
12.
24,318
45,225
44,988
45,225
9.
1,062
1,028
25,614
24,318
25,614
6.
4,408
4,423
4,408
14.
3,285
1,456
998
1,456
11.
3.
2,400
3,200
2,400
8.
327,203
13.
3,296
314,619
312,629
314,619
3,296
3,285
739,814
5,301
5,401
5,301
5,401
© Harcourt
1.
LESSON 2.2
Name
Order Numbers
Abraham is buying a new car. He has prices from 3 dealers.
Dealer
Price
Autoworld
$25,614
Car Land
$28,327
Cars & More
$28,619
List the prices in order from greatest to least.
Step 1
Compare the digits
in each place-value
position, beginning
with the ten thousands.
Step 2
Compare the digits in
the next place-value
position.
$28,327
↓
$25,614
↓
$25,614
$28,327
$28,327
$28,619
$28,619
Think: 5 ten thousands
8 ten thousands, so
25,614 is the least
number.
↓
↓
Step 3
Compare $28,327
and $28,619.
↓
$28,619
Think: 6 hundreds 3
hundreds, so $28,619 $28,327.
So, the prices in order from greatest to least are $28,619;
$28,327; $25,614.
© Harcourt
Circle the greatest number and underline the least number.
1.
738
689
691
2.
3,670
3,760
3,761
3.
14,285
14,119
14,148
4.
252,861
251,312
253,112
5.
6,228,119
6,282,113
6,228,214
6.
6,260
62,260
26,260
7.
7,932
8,239
7,392
8.
52,441
52,414
52,144
9.
10,804
107,408
107,084
Reteach
RW7
LESSON 2.3
Name
Problem Solving Strategy
Make a Table
A table can help organize information to make it easier to
answer questions.
Students are doing the long jump. Emil jumps 157
centimeters, Rob jumps 135 centimeters, Kate jumps 135
centimeters, Hayley jumps 148 centimeters, and Kyle
jumps 105 centimeters. Find out how their jumps compare.
Make a table to show how far each student jumped.
Name
Length of Jump
Use the table to answer 1–8.
jump?
3. Which two students jumped the
same distance?
5. Look at the length of Hayley’s
jump. What digit is in the tens
place?
7. Pete jumps 152 centimeters. In
what place is Pete?
RW8
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2. Which student had the shortest
jump?
4. Who jumped farther, Rob or
Hayley?
6. Whose jump is closest to 100
centimeters?
© Harcourt
1. Which student had the longest
8. Whose jump is 3 less than 160
centimeters?
LESSON 2.4
Name
Round Numbers
Round 314,289 to the nearest ten thousand.
Follow these steps to round a number to a given
place value:
• Circle the digit to be rounded.
314,289
• Underline the digit to its right.
314,289
• If the underlined digit is 5 or more, the circled digit
increases by 1. If the underlined digit is less than 5,
the circled digit stays the same.
4 < 5, so 1
stays the
same
• Write zeros in all places to the right of the circled digit.
310,000
Use the steps above to help round each number to
the place of the circled digit.
1.
243, 873
2.
2,375,467
3.
54,239
4.
5,394,372
5.
25,658,345
6.
43,872
7.
34,456,379
8.
359,752
Round each number to the nearest ten thousand.
© Harcourt
9.
2,418,652
10.
65,528,514
11.
651,842
12.
896,328
15.
65,978,154
16.
658,548
Round each number to the nearest million.
13.
25,548,475
14.
6,514,879
Reteach
RW9
LESSON 3.1
Name
Estimate Sums and Differences
When you estimate, you find an answer that is close to the
exact answer. You can use rounding to help estimate a sum
or difference.
Round 4,782 to the nearest thousand.
Follow these rules:
4,782
• Circle the digit to be rounded.
4,782
• Underline the digit to its right.
7>5
• If the underlined digit is 5 or more, the circled
digit increases by 1. If the underlined digit is less
than 5, the circled digit stays the same.
5,000
• Write zeros in all places to the right of the
circled digit.
So, 4,782 rounded to the nearest thousand is 5,000.
Circle the digit to be rounded. Underline the digit to its right.
Round 649,385 to the given place value.
1.
nearest ten thousand 2. nearest hundred
649,385
3.
649,385
nearest hundred
thousand
649,385
Round to the greatest place value.
2,617
5.
304,921
6.
89,314
Round each number to the greatest place value. Write the
estimated sum or difference.
7.
397,216 →
− 149,348 → →
RW10 Reteach
8.
652,381 →
+ 348,927 → +
→
© Harcourt
4.
LESSON 3.2
Name
Use Mental Math Strategies
Mental math strategies help you compute without using
pencil and paper.
Use the break apart strategy to find the sums and differences.
1.
3.
48 27
2.
19 58
Add the tens.
Add the tens.
Add the ones.
Add the ones.
Add the sums.
Add the sums.
59 37
4.
98 22
Subtract the tens.
Subtract the tens.
Subtract the ones.
Subtract the ones.
Add the differences.
Add the differences.
Use the make a ten strategy to find the sums and differences.
5.
© Harcourt
7.
89 24
6.
82 48
Change one number
to a multiple of 10.
Change the number being
subtracted to a multiple of 10.
Adjust the other number.
Adjust the other number.
Add.
Subtract.
42 63
8.
92 19
Change one number
to a multiple of 10.
Change the number being
subtracted to a multiple of 10.
Adjust the other number.
Adjust the other number.
Add.
Subtract.
Reteach
RW11
LESSON 3.3
Name
Add and Subtract 4-Digit Numbers
You may need to regroup when adding or subtracting. A grid can help
you to regroup neatly. Record your regroupings in the top 2 rows.
+
1
4,
2,
7,
5
6
1
1
6
1
8
7
8
5
3
$4,
$2,
$1,
12
2
3
6
6
10
0
1
1
9
12
2
8
4
Copy each problem into the grid. Find the sum or difference. Remember to leave
the first 2 rows of boxes empty for regrouping.
1.
$2,456
$1,782
2.
+
4.
3.
$4,211
$2,732
5.
8,742
2,178
6.
$7,419
$5,297
9.
5,320
6,942
© Harcourt
8.
6,163
5,254
+
+
3,877
4,399
2,631
1,975
+
+
+
7.
2,312
1,452
+
+
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+
LESSON 3.4
Name
Subtract Across Zeros
When you add or subtract numbers
with many digits, it’s important that
you write the numbers carefully so
that digits with the same place value
line up. This also helps you write
regrouped digits in the correct places.
3
4
3
9
10
0,
2,
7,
9
10
0
9
0
9
10
0
6
3
10
0
3
7
Copy each problem into the grid. Find the difference.
Remember to leave the first 2 rows of boxes empty for regrouping.
1.
900
831
2.
5,002
2,764
3.
7,001
3,208
,
4.
,
6,000
1,955
5.
80,000
31,523
6.
10,000
4,661
,
© Harcourt
7.
40,000
20,118
8.
70,000
55,049
9.
60,000
33,604
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RW13
LESSON 3.5
Name
Add and Subtract Greater Numbers
When you’re adding or subtracting greater numbers, a
grid can help you make sure you’re lining up the digits
correctly. Record your regroupings in the top 2 rows.
6
3
3
5
1
4
3
4,
2,
1,
12
2
8
4
0
1
0
0
17
7
9
8
Copy each problem into the grid. Find the sum or difference. Remember to leave
the first two rows of boxes empty for regrouping.
1.
352,134
277,679
2.
312,852
291,785
3.
994,117
504,334
,
4.
132,741
83,296
5.
514,902
423,978
6.
788,435
662,929
,
,
225,068
102,266
8.
734,668
519,862
9.
412,396
325,052
© Harcourt
7.
,
RW14
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LESSON 3.6
Name
Problem Solving Skill
Estimate or Find Exact Answers
When solving a word problem, you must decide if you need to
estimate or give the exact answer. Read the problem carefully.
Words and phrases like about how much and estimate indicate
that an estimate will do.
Jennifer and Kyle are planning a trip to the art museum.
Admission to the museum is $6.95, lunch costs $3.80, and
a drink costs $0.85. About how much money should each
bring for the trip?
Is an exact answer required? No, the words about how
much tell you that an estimate will do.
© Harcourt
Tell whether an estimate or an exact answer is needed. Solve.
1.
About how much money should
Jennifer bring on the trip to the
art museum?
2.
Kyle brought a $20.00 bill to
the museum. How much change
should he get after buying his
admission?
3.
There will be 120 students
going on the field trip.
Permission slips have been
turned in by 87 students.
How many permission slips
still need to be turned in?
4.
Greg has 504 baseball cards
and 398 football cards. About
how many cards does Greg
have altogether?
5.
On vacation, the Clarks drove
1,225 miles to visit relatives.
Then they drove 488 miles to
the beach and 1,348 miles
home. How many miles did the
Clarks drive on their vacation?
6.
At the sports awards luncheon,
there will be 52 soccer
players, 38 baseball players,
44 basketball players, and
28 cheerleaders. About how
many people will attend?
Reteach
RW15
LESSON 4.1
Name
Expressions
An expression is a part of a number sentence that has no
equal sign. Expressions can include many different operations.
Find the value 4 (10 9).
41
Always do the problem inside
parentheses first. Subtract 9 from
10 to get 1.
5
Find the final answer.
Add or subtract the numbers in the parentheses. Write that sum or difference.
1.
5 (12 9)
2.
(12 13) 10 3. 20 (3 16)
5
10
4.
(28 12) 7
20 7
Find the value of each expression.
5.
7 (2 4)
6.
7
8.
13 (3 6)
(13 9) 5
9.
48 (2 11)
11
(55 17) 4
10.
4
12.
(65 43) 9
48 14.
(21 17) 11
13 5
11.
7.
40 (18 7)
40 13.
(15 7) 2 8
9
56 (12 9)
15.
16.
12 68 (14 13)
68 © Harcourt
56 12 (3 4 1)
28
17.
(16 132 5) 48
48
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18.
59 (15 12)
59 19.
1,210 (234 761)
1,210 LESSON 4.2
Name
Use Parentheses
An expression can have different values depending on where
you place the parentheses.
14 (3 9)
14 12
(14 3) 9
← Numbers are the same.
Parentheses in different places.
11 9
← Problem in parentheses solved.
2
20
← Different answers.
Show how placing the parentheses differently can give
two different answers.
1.
16 9 2
16 9 2
2.
12 5 3
12 5 3
Choose the expression that shows the given value. Write a or b.
3.
6
4.
13
5.
45
a
22 (10 6)
a
(25 8) 4
a
56 (20 9)
b
(22 10) 6
b
25 (8 4)
b
(56 20) 9
Find the value that gives the expression a value of 16.
6.
(8 7) 7.
(15 )5
8.
30 (9 )
Show where the parentheses should be placed to make the
expression equal to the given value.
© Harcourt
9.
15 10 1; 6
10.
40 15 6; 31
11.
20 5 6; 9
14.
954=26
Place the parentheses so that the equations are true.
12.
541=33
13.
10 6 3 = 12 − 5
Reteach
RW17
LESSON 4.3
Name
Match Words and Expressions
Marni had 12 pairs of shoes in her closet. She gave 3 pairs to her little sister
and 4 pairs to her cousin. How many pairs of shoes does she have left?
Write an expression using parentheses to match the meaning of the words.
Use parentheses to group together a problem which should be solved first.
12 (3 4)
↑
number of
pairs of shoes
Marni had
↑
total number of
pairs of shoes
she gave away
Add 3 4 to find the total number
of pairs of shoes Marni gave away.
Then subtract this number from 12.
So, Marni has 5 pairs of shoes left.
Write an expression that matches the words.
1.
Mary earned $6 washing cars.
She earned $4 more baby-sitting,
and then spent $3 on a snack.
($
$
↑
amount of
money earned
)$
↑
amount of
money spent
2.
Our class began the year with 23
students. Two students moved
away, and 3 new students enrolled.
)
↑
↑
number of students
number of
after 2 moved away new students
(
3.
There are 17 marbles in a jar. Ten
are removed and 6 are added.
4.
Kyra cares for 3 injured birds.
She adopts 4 more and then
gives away 5.
5.
Dan’s father bought 6 pears and
9 apples at the grocery store.
The family ate 4 pieces of fruit.
6.
There are 28 people on the bus.
At the Main Street stop, 4 more
get on and 6 get off.
7.
Aidan had 17 toy race cars. He
gave 3 to his friend and bought
5 more.
8.
Ricky brought a package of 12
snack bars on a bike trip. He
gave 3 to his friends and ate 2.
RW18
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© Harcourt
Write an expression for the words.
LESSON 4.4
Name
Use Variables
Use variables to hold the place of unknown numbers. Unless the directions
specify a certain letter to use as a variable, choose any letter you want.
Write an expression for the number of pens in the desk.
There were some pens in the desk.→ Use p to stand for the unknown
number of pens in the desk.
9 pens were added.
→ Add 9 to p.
p9
Write an equation to represent the situation.
Robin researched some animals
for her Science report.
→ Use a to stand for the unknown
number of animals Robin
researched.
She researched three more.
→ Add 3 to a.
Robin researched a total of
13 animals.
→ Make a 3 equal to 13.
a 3 13
Write an expression. Choose a variable for the unknown.
1. There were some ducks in the
2. Rosie counted some stars in the
pond. Three of the ducks went
sky. An hour later, there were
onto the shore.
20 more.
© Harcourt
3.
Some volleyball teams were in
the gym. Twelve teams went
outside to practice.
4.
There were some snow shovels
in the shed. Rob returned 4
shovels to the shed.
Write an equation. Choose a variable for the unknown.
5. Paula baked 36 chocolate chip
6. Cindy began the day with 24
cookies. She baked some more.
pencils. She gave some away to
Now she has 61 cookies.
her friends. Now she has 6
pencils.
Reteach
RW19
LESSON 4.5
Name
Find a Rule
The table shows inputs and outputs. Find a rule which will give you the
output value when you use the input value.
Input
Output
n
b
2
8
4
10
3
9
10
16
Think: What can you add to 2 to get 8?
Possible rule: Add 6.
Test your rule on each row of the input/output
table. If your rule holds true, you can write an
equation for the rule.
Because you add 6 to every input to get the
correct output, the equation is n 6 b.
Complete the output for the given rule.
1.
Rule: Add 5.
Input
2.
Output
Rule: Subtract 7.
Input
3.
Output
Rule: Add 9.
Input
7
12
13
9
20
10
3
9
2
8
11
6
Output
4.
RW20
Input
Output
a
5.
b
Input
x
Output
y
3
15
20
5
17
12
8
Reteach
6.
Input
r
Output
p
12
14
4
10
2
56
46
24
25
17
17
7
20
9
1
22
12
© Harcourt
Find the rule. Then write the rule as an equation.
LESSON 4.6
Name
Add Equals to Equals
An equation is like a balanced scale. Both sides of an equation have the
same value, just like both sides of a balanced scale have the same weight.
If you add two blocks to each side of the
scale, it will still be balanced.
If you add five blocks to one side, you must
add five blocks to the other side for it to
balance.
What if Walt placed six more
blocks on the left side of the
scale below? How many blocks
must he place on the other side
to keep the scale balanced?
2.
What can you do to balance the
scale below?
3.
Tricia is trying to balance the
scale below, but she has no
more blocks. How can she
balance the scale?
4.
How can you balance this scale?
© Harcourt
1.
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RW21
LESSON 4.7
Name
Problem Solving Strategy
Make a Model
Sometimes it helps to make a model of a problem. Simple
objects like counters can represent part of the problem.
Example Greta and Nigel are playing a game in which the
players earn and lose points.
After Round 3, Greta has 10 points and Nigel has 7 points. In Round 4,
Greta earns 4 points and Nigel loses 2 points. In Round 5, Greta loses 3
points and Nigel gains 5 points.
Which player has more points after Round 5?
Use counters to model the problem. Each counter represents 1 point.
The counters
represent the number
of points for each
player after 3 rounds.
Greta
Nigel
To model Round 4,
give Greta 4 more
points and take away
2 of Nigel’s points.
Greta
Nigel
To model Round 5,
take away 3 of Greta’s
points and give Nigel
5 points.
Greta
Nigel
So, Greta has 11 points and Nigel has 10 points. Greta has more points.
1.
Chris bought 2 dogs. Sparky
weighed 12 pounds and Fido
weighed 15 pounds. During the
first month, Sparky gained
2 pounds and Fido gained
1 pound. During the second
month, Sparky lost 2 pounds
and Fido gained 3 pounds. How
much does each dog weigh after
2 months?
RW22
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2.
Roger was collecting shells on
the beach to glue onto a picture
frame. He found 11 shells. His
sister gave him 7 more and he
threw 3 back into the ocean. He
gave 2 shells to his mother. How
many shells does Roger have
now?
© Harcourt
Make a model and solve.
LESSON 5.1
Name
Collect and Organize Data
Mr. Cohen used tally marks to keep a record of the number of
gerbils that he sold each week in his pet shop.
Week
Gerbils Sold
1
2
3
4
A frequency table helps you organize the data from a tally
table. The frequency column shows the total for each week.
The cumulative frequency column shows a running total of the
number sold.
Use the tally table to complete the frequency table below.
NUMBER OF GERBILS SOLD
Week
Frequency
(Gerbils Sold)
Cumulative
Frequency
1
Cumulative Frequency:
← sold in Week 1
2
← sold in Weeks 1 and 2
3
← sold in Weeks 1, 2, and 3
4
← sold in Weeks 1, 2, 3, and 4
© Harcourt
For 1–3, use the frequency table.
1.
During which week did Mr. Cohen sell the most gerbils?
2.
During which two weeks did Mr. Cohen sell the same
number of gerbils?
3.
How many gerbils did Mr. Cohen sell in all?
Reteach
RW23
LESSON 5.2
Name
Find Median and Mode
When a list of data is written in order from least to greatest,
the number in the middle is called the median. The number
that occurs most often is called the mode.
John’s scores on his math tests Monday through Friday were:
72, 88, 95, 65, and 88.
Step 1: List the scores in order from least to greatest.
65, 72, 88, 88, 95
Step 2: Cross out one from each end until only one number is left.
65, 72, 88, 88, 95
Step 3: The number in the middle is the median.
65, 72, 88, 88, 95
Step 4: The number that occurs most often is the mode.
65, 72, 88, 88, 95
In this case, 88 is both the mode and the median.
1.
In the art classes there are 4 nine-year-olds, 5 ten-year-olds, 6 elevenyear-olds, and 8 twelve-year-olds.
2.
In Hawaii the temperatures for a given week were 80, 84, 79, 90, 86,
84, and 89.
3.
The car traveled 250 miles the first day, 600 the second day, 455 the
third day, 320 the fourth day, and 800 the fifth day.
4.
The baseball team sold 80 tickets on Monday, 66 on Tuesday, 55 on
Wednesday, 68 on Thursday, and 80 on Friday.
RW24
Reteach
© Harcourt
List the data from least to greatest. Find the median and the mode.
LESSON 5.3
Name
Line Plot
A line plot is a graph that shows the
frequency of data along a number line.
You can use a line plot to find the
range. The range is the difference
between the greatest and the least
values in a set of data.
✗
✗
✗
✗
✗
✗
✗
✗
✗
0
1
✗ ✗
✗ ✗
✗ ✗
✗
2 3 4
Homerun Total
5
6
How many players hit 3 home runs?
Step 1 Locate 3 on the diagram.
Step 2 Count the number of Xs above the number 3.
Three players hit 3 home runs.
For 1–4, use the line plot.
© Harcourt
1.
players hit 1 home run.
2.
More players hit
number of home runs.
home runs than any other
3.
The same number of players hit
home runs.
4.
An outlier is a value that is seperated from the rest of the data. The
outlier in the line plot above is
.
5.
Four new players joined the team for the 2000-2001
season. Use the data in the table to make a line plot.
and
Homerun Total
Number of
Homeruns
0
1
2
3
4
5
Number of
Players
0
2
4
7
2
1
6
Reteach
RW25
LESSON 5.4
Name
Stem-and-Leaf Plot
Becky measured the height in
inches of each sunflower in her
garden. She made a list of the
heights.
20
35
12
23
27
18
inches
inches
inches
inches
inches
inches
14
12
32
24
27
33
inches
inches
inches
inches
inches
inches
17
15
27
21
33
inches
inches
inches
inches
inches
• To organize the data in a stem-and-leaf plot write
the numbers in order from least to greatest. Group
the data in rows according to the digit in the tens place.
12, 12, 14, 15, 17, 18
20, 21, 23, 24, 27, 27, 27
32, 33, 33, 35
• The stem is the tens digit. It is written only once.
The leaves are the ones digits.
Stem (Tens)
•
1
2
3
1  2 means 12.
3  5 means 35.
Leaves (Ones)
2 2 4 5 7 8
0 1 3 4 7 7 7
2 3 3 5
The shortest sunflower is 12 inches.
The tallest sunflower is 35 inches.
1.
How many sunflowers are 33 inches tall?
2.
How many sunflowers are 20 inches or taller?
3.
The mode is the number that occurs most often in a
set of data. What is the mode for the sunflower heights?
4.
The median is the middle number in an ordered
set of numbers. What is the median sunflower height?
5.
The range is the difference between the greatest
and the least values in a data set.
What is the range of the sunflower heights?
RW26
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© Harcourt
For 1–5, use the stem-and-leaf plot.
LESSON 5.5
Name
Compare Graphs
Tom and his friends earned money cutting lawns during the summer.
Lawns Cut This Summer
Number of Lawns
Tom
6
Jack
10
Barb
8
Kim
4
Lia
3
The interval of
a scale is the
difference
between any
two numbers. The
interval of this
graph is 2.
To make a bar graph with a different
interval, you would follow these steps.
Lawns Cut This Summer
Number of Lawns Cut
Name
The scale is the series of
numbers placed at fixed distances.
The scale of this graph is 0 -12.
12
10
8
6
4
2
0
Tom Jack
Barb Kim
Name
Lia
Step 1 Choose an interval for the graph. Label the graph.
Step 2 Draw the bars.
Step 3 Check to see if your graph is correct.
Use the table above to complete the bar graphs.
1.
What is the scale and the interval of the graph on the bottom left?
2.
What is the scale and the interval of the graph on the bottom right?
Lawns Cut
This Summer
Tom
Tom
Jack
Jack
Name
Name
© Harcourt
Lawns Cut This Summer
Barb
Barb
Kim
Kim
Lia
Lia
0
1
2
3 4 5 6 7 8 9 10 11 12
Number of Lawns Cut
0 2 4 6 8 10 12
Number of Lawns Cut
Reteach
RW27
LESSON 5.6
Name
Problem Solving Strategy
Make a Graph
The table below shows how many birthdays are in each month for
Mr. Robinson’s class.
Month
Number of Birthdays
January
3
February
5
March
2
April
6
May
0
June
2
July
1
August
1
September
1
October
3
November
4
December
0
Follow these hints to complete the bar
graph below:
Step 1: On the side, write: Number of
Birthdays.
Step 2: What label should you write on
the bottom?
Step 3: What is the title of the graph?
Step 4: What should the length of each
bar show?
7
6
5
4
2
1
0
RW28 Reteach
© Harcourt
3
LESSON 6.1
Name
Double-Bar Graphs
Peter made a table of the number of shells that he collected
on two beaches. Then he started a double-bar graph to
compare the data. Each bar represents the number of each
kind of shell collected.
SHELLS COLLECTED ON WEST AND DUNE BEACHES
Type of Shell
West Beach
Dune Beach
Clam
6
8
Oyster
9
1
Scallop
10
2
6
6
Periwinkle
© Harcourt
Number of Shells
12
Key
10
West Beach
8
6
4
2
0
Dune Beach
Clam
1.
What scale did Peter choose?
2.
What is the interval of the scale?
3.
Complete Peter’s graph. Remember to include
bars, a title, and labels.
For 4–5, use the graph.
4.
On which beach were more oyster shells collected?
5.
On which beach were equal numbers of clam shells
and periwinkle shells collected?
Reteach
RW29
LESSON 6.2
Name
Read Line Graphs
Mr. Lowe made this line graph to show the number of
books he sold each month for the first seven months of the
year. This line graph shows how book sales change over time.
BOOKS SOLD JANUARY THROUGH JULY
Number of Books
500
300
200
100
0
•
•
400
•
•
•
•
•
Jan
Feb Mar Apr May Jun
Month
Jul
How many books were sold in April?
Step 1 Find the line labeled Apr. Put your finger on the
line. Follow that line up to the point ( • ).
Step 2 Move your finger along the line to the left to
find the number of books sold in April.
Mr. Lowe sold 400 books during April.
For 1–5, use the graph.
Mr. Lowe sold the greatest number of books during
and
.
2.
During June, Mr. Lowe sold about
3.
The difference in the number of books sold in
February and April was about
books.
© Harcourt
1.
.
4.
Mr. Lowe sold about
more books in June than in January.
5.
There was an increase in the number of books sold from
.
RW30 Reteach
LESSON 6.3
Name
Make Line Graphs
JACOB’S SOCCER GOALS
A line graph is used to display data
which change over time.
Make a line graph for the data at
the right.
1.
The greatest number of goals is 20
and the least is
, so the
scale should go from 0 to a little
more than 20.
Year
Number of Goals
1998
6
1999
8
2000
9
2001
17
2002
20
Since the data are close together, an interval of
would be appropriate.
Fill in the missing parts of the line graph:
Title:
21
15
12
3
0
© Harcourt
1998
2.
To plot 6 goals for Jacob in 1998, move your finger or
pencil straight up from 1998. Stop when you reach the
line for 6. Make a dot for 6 goals for Jacob in 1998.
3.
Plot the rest of the data in the table on the graph the
same way.
4.
Do you notice any trend in the number of soccer goals
Jacob scored over the years?
Reteach
RW31
LESSON 6.4
Name
Choose an Appropriate Graph
A line graph shows change over
time.
Books Read Each Week
Number of Books
Kinds of Books Read
•
•
•
1
2 3
Week
y
Bi ster
og y
ra
p
Sc hy
ie
nc
An e
im
al
•
60
50
40
30
20
10
0
M
Number of Books
60
50
40
30
20
10
0
A bar graph compares data.
4
Kind of Book
Use a line graph to show whether
data increase or decrease.
Use a bar graph to see how many of
each type of book are in the data.
A line plot shows how many times
each number occurs in the data.
A stem-and-leaf plot shows data
arranged by place value.
Number of Pages in 15 Books
✗ ✗
✗ ✗ ✗ ✗
✗ ✗ ✗ ✗ ✗
✗ ✗ ✗ ✗ ✗ ✗
2
3
4
5
6 7
Stem
Leaves
2
3 4 5 5 8 9
3
1 2 8 9
4
0 1 3 4 9
Number of Books Read
Use a line plot to record how many
students read books.
Use a stem-and-leaf plot to
organize a long list of numbers
by grouping them.
Write the kind of graph or plot you would choose.
to show a decrease in absences
during the school year
2.
to organize the list of weights of
the football team members
3.
to compare the numbers of
students signing up for hockey,
football, and cheerleading
4.
to record the number of students
who have 0, 1, 2, 3, or 4 siblings
© Harcourt
1.
RW32 Reteach
LESSON 6.5
Name
Problem Solving Skill: Draw Conclusions
Bobby saved his money for a new telescope. He put his
money into a savings account at Colesville Federal Bank.
BOBBY'S SAVING ACCOUNT
220
•
200
180
Amount of Money
(in dollars)
160
•
140
120
•
100
•
80
40
•
•
60
•
•
•
•
20
0
Jan
•
•
Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
The bank gives its customers a free poster for each month in which
their balance is greater than $150.00. Since Bobby’s balance was greater
than $150.00 in July and August, he received posters in those months.
During which month was
Bobby’s bank account balance
the greatest?
2.
During which month was
Bobby’s bank account balance
the least?
3.
When did Bobby most likely
buy his telescope?
4.
When did Bobby most likely
make a $30 withdrawal to buy
a book about the planets?
5.
During which months did
Bobby most likely earn money
at a part-time job?
6.
About how much money was in
Bobby’s account in October?
© Harcourt
1.
Reteach
RW33
LESSON 7.1
Name
Before and After the Hour
Digital Clock
Analog Clock
The second hand
shows the seconds
and moves around
the clock in 1 minute.
The minute hand is the
long hand.
11 12 1
2
3
4
10
9
8
7 6 5
2:35
The hour hand is the
short hand.
Read: 5:15 and 10 seconds, or 15
minutes and 10 seconds after five
Read: 35 minutes after two
Complete to write the time shown on the clock. Include the seconds.
1.
2.
11 12 1
11 12 1
2
3
4
10
9
8
7 6 5
7 6 5
and
seconds
3.
and
seconds
4.
11 12 1
11 12 1
2
3
4
10
9
8
2
3
4
10
9
8
7 6 5
minutes and
2
3
4
10
9
8
7 6 5
seconds
minutes and
after
seconds
after
Draw the hour hand and the minute hand to show the time.
6.
11 12 1
2
3
4
10
9
8
7.
11 12 1
11 12 1
2
3
4
10
9
8
7 6 5
7 6 5
:35
3:25
2
:35
8:05
2
2
3
4
10
9
8
7 6 5
:35
9:57
2
© Harcourt
5.
Write second, minutes, hour, or day.
8.
It took Joe 45
9.
A snap of your fingers takes about 1
10.
Anne’s piano lesson was 1
RW34
Reteach
to wash and vacuum his car.
.
long.
LESSON 7.2
Name
A.M. and P.M.
A.M. means “the time between
P.M. means “the time between noon
midnight and noon.”
and midnight.”
At 8:15 A.M. Kathy goes to school.
At 8:15 P.M. Kathy goes to bed.
Write A.M. or P.M.
1.
George eats supper at 5:30
2.
Carlos goes to school from 8:40
3.
Maria has a violin lesson from 3:45
.
to 2:40
.
to 4:30
.
Write the time, using A.M. or P.M.
4.
the time the library opens
5.
the time Kitty sees the sunset
11 12 1
11 12 1
2
3
4
10
9
8
7 6 5
6.
7 6 5
the time Lu takes a spelling test
at school
7.
the time Frank gets home from
school
11 12 1
11 12 1
2
3
4
10
9
8
© Harcourt
7 6 5
the time Jack has soccer practice
9.
the time Dad cooks breakfast
11 12 1
2
3
4
10
9
8
7 6 5
2
3
4
10
9
8
7 6 5
8.
2
3
4
10
9
8
11 12 1
2
3
4
10
9
8
7 6 5
Reteach
RW35
LESSON 7.3
Name
Elapsed Time
Skip counting can help you find the elapsed time. What is
the elapsed time from 7:20 A.M. to 9:40 A.M.?
11 12 1
10
2 hr 9
8
1 hr
11 12 1
2
3
4
2
3
4
10
9
8
7 6 5
7 6 5
20
15
It is 2 hours from 7:20 A.M.
to 9:20 A.M.
10
5
It is 20 minutes from 9:20 A.M.
to 9:40 A.M.
So, it is 2 hours 20 minutes from 7:20 A.M. to 9:40 A.M.
Find the elapsed time.
from 12:30 A.M. to 12:45 A.M.
2.
from 4:25 P.M. to 5:20 P.M.
11 12 1
11 12 1
2
3
4
10
9
8
7 6 5
3.
7 6 5
from 10:15 A.M. to 11:45 A.M.
4.
from 11:15 P.M. to 2:40 A.M.
11 12 1
11 12 1
2
3
4
10
9
8
7 6 5
from 7:05 A.M. to 9:20 P.M.
6.
from 6:35 A.M. to 6:45 P.M.
11 12 1
2
3
4
10
9
8
7 6 5
RW36
Reteach
2
3
4
10
9
8
7 6 5
5.
2
3
4
10
9
8
11 12 1
2
3
4
10
9
8
7 6 5
© Harcourt
1.
LESSON 7.4
Name
Problem Solving Skill: Sequence Information
Zoo Schedule of Shows
Kevin is going to be at the zoo from
8:30 A.M. until 1:30 P.M. He will be
meeting his friends for lunch from
12:00 P.M. to 1:00 P.M.
He has decided to see the 9:00–10:30
Manatee Madness show. He put this
into his schedule.
Kevin’s Schedule
9:00–10:30
Manatee Madness
10:00–10:45
Monkey Business
11:00–12:00
Birds of the Wild
12:00–12:30
Insect Round-Up
1:00–2:30
Manatee Madness
2:00–2:45
Monkey Businesss
3:00–4:00
Birds of the Wild
4:00–4:30
Insect Round-Up
Sara’s Schedule
8:30 to 9:00
8:30 to 9:00
9:00 to 9:30
Manatee Madness
9:00 to 9:30
9:30 to 10:00
Manatee Madness
9:30 to 10:00
10:00 to 10:30
Manatee Madness
10:00 to 10:30
10:30 to 11:00
10:30 to 11:00
11:00 to 11:30
11:00 to 11:30
11:30 to 12:00
11:30 to 12:00
12:00 to 12:30
12:00 to 12:30
12:30 to 1:00
12:30 to 1:00
1:00 to 1:30
1:00 to 1:30
Put Kevin’s lunch with his
friends into the schedule.
2.
Kevin wants to see another show.
Which show can he see? Put the
show into Kevin’s schedule.
3.
Can Kevin see another show?
Explain.
4.
Sara is also going to the zoo from
8:30 A.M. to 1:30 P.M. She wants to
see three shows. Make a schedule
for Sara.
© Harcourt
1.
Reteach
RW37
LESSON 7.5
Name
Elapsed Time on a Calendar
You can use a calendar to find
elapsed time. How many days will
the Harvest Festival last?
• Look at the October calendar.
Find 9.
• Count each day of the festival.
So, the festival will last 10 days.
Wilson Farm began advertising the
Harvest Festival 5 weeks before the
first day of the festival. On what date
did Wilson Farm begin advertising?
• Begin at October 9. It is a Thursday.
• Use the calendars to count back
5 weeks, or 5 Thursdays.
So, Wilson Farm began advertising September 4.
For Problems 1–4, use the calendars.
2.
3.
4.
On how many Saturdays will the
Harvest Festival be held?
Tom bought a pumpkin on the first
Friday of the festival. How many
weeks will it be from then until
October 31?
How many days is it from September 28
to the last day of the Harvest Festival?
Wilson Farm began harvesting
pumpkins 4 weeks before the
beginning of the festival. On what
date did the farm begin harvesting
pumpkins?
RW38 Reteach
September
Sun
Mon
Tue
Wed
Thu
Fri
Sat
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
26
27
28
29
30
Wed
Thu
Fri
Sat
1
2
3
4
October
Sun
Mon
Tue
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
© Harcourt
1.
LESSON 8.1
Name
Relate Multiplication and Division
Multiplication and division are inverse operations. One
operation undoes the other. You can use models to show this.
2 × 5 = 10
10 ÷ 5 = ?
Divide 10 into 5
equal groups.
12 ÷ 3 = ?
Divide 12 into 3
equal groups.
3 × 4 = 12
How many are in
each group? 4
12 ÷ 3 = 4
How many are in
each group? 2
10 ÷ 5 = 2
Circle equal groups to show the division. Complete the division equation.
1.
3 × 6 = 18
18 ÷ 6 =
2.
7 × 2 = 14
3.
14 ÷ 2 =
4 × 5 = 20
20 ÷ 5 =
Write the fact family by writing two multiplication equations
and two division equations which are shown by these models.
4. 6. © Harcourt
5. 7.
What multiplication equation can
you use to help you find 28 ÷ 4?
8.
What multiplication equation can
you use to help you find 12 ÷ 4?
Reteach
RW39
LESSON 8.2
Name
Multiply and Divide Facts Through 5
You can use models to help you remember
fact families.
How many stars are in the array?
How many rows are there?
How many columns are there?
Since there are 3 rows of 6 stars, multiply 3 6
to get the number of stars.
3 6 18
Use the model to write the other three facts in the
fact family for 3, 6, and 18.
Find the product or quotient. You may wish to use a model.
64
2.
28
3.
28 4 4.
36 9 5.
24 8 6.
62
7.
75
8.
55
9.
15 3 10.
95
11.
54
12.
30 5 © Harcourt
1.
RW40
Reteach
LESSON 8.3
Name
Multiply and Divide Facts Through 10
The break apart strategy can help you find products of greater numbers.
What is 6 9?
The unshaded squares represent the
multiplication problem 4 6.
What is 4 6?
What multiplication problem is
represented by the light gray squares?
What multiplication problem is
represented by the darker gray squares?
9 6 (4 6) (
↓
9 6 24
So, 9 6 6) (
↓
6)
↓
.
You can check your answer by counting the squares.
© Harcourt
Use shading and the break apart strategy to solve.
1.
78
2.
88
3.
76
4.
77
Reteach
RW41
LESSON 8.4
Name
Multiplication Table Through 12
Look at the 10 Facts Chart.
0
1 10
2 10
3 10
4 10
5 10
6 10
7 10
8 10
9 10
10 10
11 10
12 10
10
10 Facts Chart
0 10 20 30 40 50 60 70 80 90 100 110 120
What is the rule for finding the product of 10 and another number?
You can use the basic facts for 10 and a model or other facts to help you
find the product of factors greater than 10.
12 11 ?
↓
10 2
What is 12 11?
Step 1
Write an addition expression
using break apart numbers.
Step 2
Use the 10 Facts Chart and
a model to find the products.
(10 11)
↓
Step 3
Find the sum of the products.
110
(2 11) ?
↓
132
22
So, 11 12 132.
Complete each step.
12 12 ?
↓
10 2
2.
(10 12) (2 12) ?
So, 12 12 RW42
Reteach
11 11 ?
↓
10 (10 11) (
.
So, 11 11 11) ?
.
© Harcourt
1.
LESSON 8.5
Name
Multiply 3 Factors
The Grouping Property may help you when you multiply.
Remember that when the grouping of factors changes, the
product remains the same. Look at the example below
which shows the Grouping Property.
(3 2) 2 gives the same answer as 3 (2 2).
Choose the way of grouping that is easier for you to multiply.
What is 5 2 3?
• Group the factors two ways.
(5 2) 3 ?
5 (2 3) ?
• Choose the way that is easier
for you.
• Solve.
Think: 5 2 10. It’s
easy to multiply tens.
10 3 30
5 6 30
© Harcourt
Circle the grouping that is easier for you. Show how to solve it.
1.
(2 2) 3 ? ,
or 2 (2 3) ?
2.
(2 5) 3 ? ,
or 2 (5 3) ?
3.
(4 2) 3 ? ,
or 4 (2 3) ?
4.
(5 3) 3 ? ,
or 5 (3 3) ?
5.
(2 2) 8 ? ,
or 2 (2 8) ?
6.
(4 3) 3 ? ,
or 4 (3 3) ?
7.
(5 2) 2 ? ,
or 5 (2 2) ?
8.
(2 6) 1 ? ,
or 2 (6 1) ?
Reteach
RW43
LESSON 8.6
Name
Problem Solving Skill
Choose the Operation
Add
Join groups
of different sizes
There are 24 students in Mr. Clark’s
class and 26 students in Ms. Young’s
class. How many students are there in
both classes?
24 26 50
Subtract
Take away or
compare groups
How many more students are in Ms.
Young’s class than in Mr. Clark’s class?
26 24 2
Multiply
Join equal-sized
groups
Find how many
in each group or how
many groups
2 more students
Mr. Clark gave 3 pencils to each of his
24 students. How many pencils in all did
Mr. Clark give his students?
24 3 72
Divide
50 students
72 pencils
Mr. Clark divided his class of 24 students
into 4 equal groups to work on a science
project. How many students were in
each group?
24 4 6
6 students
1.
Maria used 3 rolls of film on her
vacation. Each roll had 12
exposures. How many
photographs did she take?
2.
Maria bought a photograph
album for $7.00 and glue for
$4.00. What change should she
get from $20.00?
3.
Maria put 7 photographs on
each page of her album. How
many pages did she use for her
84 photographs of the Rocky
Mountains?
4.
During a 4-day hiking trip,
Maria hiked 5 km, 7 km, 9 km,
and 10 km. What was the total
length of her hike?
RW44
Reteach
© Harcourt
Name the operation or operations you use. Then solve.
LESSON 9.1
Name
Expressions with Parentheses
An expression is a part of a number sentence
that does not contain an equal sign.
If an expression contains parentheses, do the
problem in the parentheses first.
Evaluate (5 12) 20
↓
↓
60 1. Do the problem in the parentheses first.
Think: 5 12 60.
20
2. Write that answer and bring down the
rest of the problem.
40
3. Solve the new problem.
Solve the problem inside the parentheses and write the result on
the line.
1.
3 (12 9)
2.
(10 14) 4
3
3.
4
15 (2 3)
4.
(4 6) 8
15 8
Solve the problem inside the parentheses and write the result on
the line. Then find the final answer.
5.
(4 7) 8
6.
20 (15 3)
7.
20 8
(36 4) 8
8
Find the value of the expression.
© Harcourt
8.
11.
20 (14 2)
(4 5) (14 7)
9.
12.
54 (18 2)
10.
4 (7 5)
(81 9) 5
13.
50 (5 5)
Reteach
RW45
LESSON 9.2
Name
Match Words and Expressions
The words in a problem can help you decide which
operation to choose.
Words like total, and, altogether,
and plus might tell you to add.
Words like gave away, spent, and
less than might tell you to subtract.
Robin bought 2 books and 3
magazines. How many items did
she buy altogether?
Rick had 10 cookies and gave
away 4. How many does he have
left?
235
10 4 6
Words like each and per might tell
you to multiply.
Words like share, split, and divide
might tell you to divide.
Mary bought 3 sandwiches for
$4.00 each. How much did she
spend?
Ike divided 52 cards among 4
friends. How many did each friend
get?
3 $4.00 $12.00
52 4 13
Sometimes you have to use more than one operation
to write an expression. Use parentheses to show the
operation that should be done first.
50
Monique picked 50 apples.
↓
↓
She gave away 5 apples to each
original
gave away
of her 3 friends.
number
tells you to
of apples
subtract
(3 5)
↓
number of
apples
given
away
1.
Tim had 4 pages with 6 stamps
on each page and then his
mother used 5 stamps.
2.
Vicki had 85¢. Then she lost
two dimes.
a.
(4 6) 5
a.
(85 2) 10
b.
4 (6 5)
b.
85 (2 10)
RW46
Reteach
© Harcourt
Choose the expression that matches the words. Underline the
words that tell you the operations to use.
LESSON 9.3
Name
Multiply Equals by Equals
You can think of an equation as being like a balance.
Look at the balance. It models the equation
6 3 2. Since there are 6 blocks on each
side of the balance, it is level.
When you multiply both sides of an equation
by the same number, the equation stays true.
How many blocks will be on the left-hand
side if we multiply the number of blocks
on that side by 3?
How many blocks will be on the
right-hand side if we multiply the
number of blocks on that side by 3?
Since both sides of the balance have
18 blocks, the balance is still level.
632
6 3 (3 2) 3
18 18
1.
Original equation
Multiply both sides by 3.
Find the value of each side.
Multiply both sides by 5.
Find the value of each side.
2.
Original equation
Multiply both sides by 4.
© Harcourt
Find the value of each side.
3.
583
Original equation
Original equation
5
(8 3) 25 4 nickels 2 dimes
20¢ 7429
Multiply both sides by 5.
Find the value of each side.
Reteach
RW47
LESSON 9.4
Name
Expressions with Variables
A variable is a letter or symbol that stands for a number.
When the number is given, replace the variable with that number.
Find the value of 7 p if p 9.
Replace p in 7 p with 9.
7 9 63.
Variables are helpful because we don’t always have all the
information needed to write an expression.
Marguerite has 12 stuffed animals to share with her friends
at her slumber party. How many animals will each girl get?
The number of stuffed animals each girl gets depends on the
number of girls at the party. Since that information is not given,
use a variable to stand for that number.
12 g is the number of stuffed animals each girl will get.
If there are 4 girls at the party, then g 4 and 12 g becomes 12 4,
which is 3. Each girl will get 3 stuffed animals.
If there are 6 girls at the party, then g 6 and 12 g becomes 12 6,
which is 2. Each girl will get 2 stuffed animals.
Find the value of the expression.
1.
4 n if n 5
2. 12
w if w 4
45
3.
30 pancakes divided by the
number of people, p, at breakfast
4.
12 eggs times the number of
egg cartons, c
5.
a number of tennis balls, t,
divided among 8 players
6.
twice the number of football
helmets, h
RW48
Reteach
© Harcourt
Write an expression that matches the words.
LESSON 9.5
Name
Equations with Variables
An equation is a number sentence. Equations are different
from expressions because equations contain equal signs.
Turn word sentences into number sentences by looking
for important words.
The words is or are tell you where the equal sign goes.
Write an equation for the words.
4 trays with 5 glasses on each tray are the total number of glasses.
↓
↓
↓
45
t
t represents the total number of glasses on the trays.
Write an equation using a variable. Tell what the variable
represents.
1.
A number of desks divided into 5 rows is 4 desks in each row.
↓
↓
↓
d
d is the total number of desks.
2.
4 chairs at each of some tables are 48 chairs.
↓
↓
↓
t is
A number of blankets in each of 7 stacks is 49 blankets.
© Harcourt
3.
Reteach
RW49
LESSON 9.6
Name
Find a Rule
Find a rule for the input/output table.
Look at the first row of numbers in the table.
Think: What can you do to 5 to find 20?
You could add 15 or multiply by 4.
Look at the next row of the table.
Think: What can you do to 6 to find 24?
You could add 18 or multiply by 4.
input
output
a
b
5
20
6
24
2
8
10
40
3
12
Think: The rule multiply by 4 works for the first
two rows. Check the rule on the other rows.
Does 2 4 8? Yes
Does 10 4 40? Yes
Does 3 4 12? Yes; so, the rule is multiply by 4. a 4 b
The input number, multiplied by 4, equals the output number.
Write the answer.
1.
What operations on 5 give a value of 10?
a.
b.
2.
input
output
t
input
output
w
s
b
15
5
5
40
9
3
2
16
12
4
4
32
18
6
8
64
24
8
9
72
Equation:
RW50
Reteach
3.
Rule:
Equation:
Rule:
© Harcourt
Find a rule, and write the rule as an equation.
LESSON 9.7
Name
Problem Solving Strategy
Work Backward
Sometimes you can work backward to solve
a problem.
Operation
Opposite
Operation
Bruce collected some sand dollars. He gave 10
to his younger sister. His friend, Susan, gave him
7 sand dollars. He put 4 back in the water. He
had 14 sand dollars left. How many did he collect?
Add
Subtract
Subtract
Add
Divide
Multiply
Multiply
Divide
•
First, write the steps of the problem in order
from left to right. Use numbers and operations
to describe the steps.
Number of
sand dollars
collected.
?
•
→
He gave
10 to
sister.
10
Susan
gave
him 7.
→
7
He put
4 back.
→
4
Number
left
→
14
←
14
Then, reverse the order and use the opposite
operations. Find the answer.
21
←
10
←
7
←
4
So, Bruce collected 21 sand dollars.
Reverse the order and use opposite operations. Find the answer.
1.
?
→
3
←
2.
?
→
© Harcourt
←
3.
→
8
←
11
→
←
→
11
←
25
→
→
27
←
52
←
Sasha and Chuck played a number game. Sasha
chose a number and subtracted 3. Next, she added
12. Then, she subtracted 20. Last, she added 11. The
result was 30. What number did Sasha begin with?
Reteach
RW51
LESSON 10.1
Name
Mental Math: Multiplication Patterns
60 is the product of 6 and 10.
6 10 60. You can use basic multiplication facts and
patterns to find the product.
Solve 60 2 ? .
Step 1 Think of the basic fact. Multiply.
6 2 12
Step 2 Count the number of zeros in the factors.
60 has 1 zero.
Step 3 Place the number of zeros in the factors
at the end of the basic-fact product.
The answer is 120.
Write the basic multiplication fact.
1.
70 4 280
2.
2,000 2 4,000
3.
600 8 4,800
5.
77
6.
54
Complete each pattern.
7.
10.
33
30 3 70 7 50 4 300 3 700 7 500 4 3,000 3 7,000 7 5,000 4 62
8.
48
9.
39
60 2 40 8 30 9 600 2 400 8 300 9 6,000 2 4,000 8 3,000 9 3
18
6 180
6 1,800
3,000 6 RW52
Reteach
11.
75
12.
5 350
700 5 7,000 9
36
4 360
900 4 35,000
4 36,000
© Harcourt
4.
LESSON 10.2
Name
Estimate Products
When you don’t need an exact answer, you can estimate
by rounding the factors.
To round:
• Circle the digit to be rounded.
• Underline the digit to its right.
• If the underlined digit is 5 or
greater, increase the circled digit
by 1. Otherwise, the circled digit
stays the same.
Round to the greatest place value.
551 →
2
→
600 ←
2
1,200
Find the
estimated
product.
• Write zeros in all places to the
right of the circled digit.
Round the larger factor to its greatest place value. Then estimate
the product.
1.
4.
© Harcourt
7.
10.
623 →
600
2 → 2
1,200
328 → 2.
5.
435 → 3.
575 → 4 → 2 → 944 → 6.
894 → 7 → 8 → 6 → $37 → 8.
155 → 9.
$984 → 5 → 5 → 4 → 448 → 11.
637 → 5 → 9 → 12.
$275 → 7 → Reteach RW53
LESSON 10.3
Name
Model Multiplication
243
3
729
Step 1
Hundreds
Ones
Use base-ten blocks to model the number 243.
Hundreds
Step 2
Tens
Tens
Ones
Use base-ten blocks to model the problem 3 243.
Hundreds
Tens
3 groups of
4 tens, or
12 tens
3 groups
of 2 hundreds,
or 6 hundreds
Step 3
Ones
3
groups
of 3
ones,
or 9
ones
Regroup the tens.
Hundreds
Tens
Ones
© Harcourt
So, 3 243 729.
Use base-ten blocks to model the multiplication and find the product.
124
4
135
3
1. RW54
2. Reteach
123
5
3. 232
3
4. LESSON 10.4
Name
Multiply 3-Digit Numbers
Break a 3-digit number down by place value to multiply it
by another number.
Multiply. 5 361
300 60 1
5
5
5
1,500 300 5
Add the products. 1,500 300 5 1,805
So, 5 361 1,805.
1.
Multiply. 7 819
800 10 9
7
7
7
5,600 70 63
Add the products. 5,600 So, 7 819 5,733.
Use the above method to find the products.
453
6
4.
357
6
6.
218
7
8.
627
3
© Harcourt
2.
3.
216
4
5.
846
2
7.
851
4
9.
524
5
Reteach
RW55
LESSON 10.5
Name
Multiply 4-Digit Numbers
You can use place value and the expanded form of a
number to help you find the product of a one-digit number
and a four-digit number.
EXAMPLE:
Find the product 4 2,281.
Step 1 Write the expanded form of the four-digit number.
2,281 2,000 200 80 1
Step 2 Multiply each addend by the one-digit number.
Step 3 Add these numbers.
EXPAND
2,000
200
80
1
MULTIPLY by 4
8,000
800
320
4
9,124
SUM
So, 4 2,281 9,124.
Use place value and the expanded form of the four-digit number
to find the product.
3 3,418
EXPAND
3,000
400
10
8
2.
MULTIPLY by 3
5 $3,067
EXPAND
MULTIPLY by 5
SUM
SUM
3.
9 2,847
EXPAND
4.
MULTIPLY by 9
EXPAND
SUM
RW56
Reteach
7 7,209
© Harcourt
1.
MULTIPLY by 7
SUM
LESSON 10.6
Name
Problem Solving Strategy
Write an Equation
A variable represents an unknown number.
EXAMPLE:
Mary has seven lambs. Each lamb produces 15 pounds of
wool. How much wool do the lambs produce in all?
Use the equation 7 15 n to answer the problem. Use
the letter n to represent the unknown number, the total
number of pounds of wool produced by the seven lambs.
Since 7 15 105, the total pounds produced is n 105.
© Harcourt
Write an equation and solve.
1.
In a tall building there are nine flights of stairs. If each
flight of stairs has 21 steps, how many steps are there in
the building?
2.
June has 90 minutes before her
piano lesson. If she does homework for 48 minutes, how much
time will she have before her
piano lesson?
3.
How many days are in 52 weeks
and one day?
4.
In one hour, five squirrels each
store 12 acorns. How many
acorns do they store in all?
5.
Peter divides 48 index cards into
four stacks. How many cards are
in each stack?
Reteach
RW57
LESSON 11.1
Name
Mental Math: Patterns with Multiples
A multiple is the product of a given whole number and another whole
number. For example, 60 is a multiple of 6 and 10: 6 10 60.
You can use basic multiplication facts and mental math to find
the product when you multiply by a multiple of 10, 100, or 1,000.
Solve 60 20 ? .
Step 1 Think of the basic fact. Multiply.
6 2 12
Step 2 Count the number of zeros in the factors.
60 has 1 zero and
20 has 1 zero.
Step 3 Write that number of zeros to the right of
the basic-fact product.
The answer is 1,200.
Write the basic multiplication fact.
1.
70 4 280
2.
2,000 2 4,000
3.
600 8 4,800
Tell how many zeros will be in the product.
4.
300 60 5.
5,000 300 6.
40 800 8.
800 400 9.
500 800 Find the product.
30 300 9,
32
,
40
10.
900 3,000 11. 12,000
12.
9,000 30,000 13.
14.
400 3,000 15. 1,200
RW58
Reteach
,
900 60 500 1,100 © Harcourt
7.
LESSON 11.2
Name
Multiply by Multiples of 10
The multiples of 10 are 10, 20, 30, 40, and so on.
When one of the factors in a multiplication problem is a
multiple of 10, you can find the product by solving a
simpler problem and writing a zero.
Example:
15
30
Solve the problem 3 15, then write one zero.
3 15 45 so, 30 15 450.
Write a simpler problem to be solved.
1.
22
40
2.
52
60
Find the product.
3.
12
40
4.
?
5.
?
82
40
?
Simpler Problem:
Simpler Problem:
Simpler Problem:
4 12 6 29 4 82 Write 1 zero so,
Write 1 zero so,
Write 1 zero so,
40 12 56
30
60 29 .
7.
75
40
40 82 .
8.
.
26
80
© Harcourt
6.
29
60
Reteach
RW59
LESSON 11.3
Name
Estimate Products
You can round numbers and use basic facts to estimate
products.
For 2-digit numbers:
For 3-digit numbers:
If the ones digit is 0–4, the digit in
the tens place stays the same.
If the tens digit is 0–4, the digit in the
hundreds place stays the same.
If the ones digit is 5–9, the digit in
the tens place increases by 1.
If the tens digit is 5–9, the digit in the
hundreds place increases by 1.
For example: Round 41–44 to 40.
Round 45–49 to 50.
For example: Round 700–749 to 700.
Round 750–799 to 800.
50
45 →
42 → 40
2,000
749 →
700
44 → 40
28,000
1.
4.
41 →
36 → 160 →
41 → 2.
157 →
57 → 3.
125 →
25 → 5.
187 →
72 → 6.
236 →
45 → 7.
349 →
74 → 8.
456 →
56 → 9.
568 →
27 → 10.
638 →
16 → 11.
774 →
55 → 12.
836 →
43 → 13.
719 →
85 → 14.
468 →
68 → 15.
229 →
54 → RW60
Reteach
© Harcourt
Round each factor and estimate the product.
LESSON 11.4
Name
Model Multiplication
A model can be used to simplify multiplication.
This is a model for 12 26.
Multiply to find the number of
squares in each rectangle. Then
add the partial products.
6
20
10
10
2
26
12
2 20 40
10 6 60
10 20 + 200
2
312
6
20
When you add the number of squares on each rectangle in
the grid, you find that the product of 12 and 26 is 312.
Multiply and solve by using the model and writing partial products.
1.
26
14
46
6
20
10
10
4 20 10 6 4
4
6
20
10 20 © Harcourt
2.
23
17
20
73
3
10
10
7 20 10 3 7
7
20
3
10 20 Reteach
RW61
LESSON 11.5
Name
Problem Solving Strategy
Solve a Simpler Problem
Solve. 60 217 = ? , or 217
60
• You can find the product by breaking the three-digit
number apart.
• Multiply, using the simpler numbers.
First, multiply 60 times 7. 60 7 420
partial products
Second, multiply 60 times 10. 60 10 600
Third, multiply 60 times 200. 60 200 12,000
Fourth, add the partial products to find the product.
420 600 12,000 13,020
Find each product by using partial products. Show the four steps
you used.
1.
241
6
2.
237
30
3.
675
20
5.
295
70
6.
327
50
616
6 40 240
6 200 1,200
6 240 1,200
1,446
555
40
© Harcourt
4.
RW62
Reteach
LESSON 12.1
Name
Multiply by 2-Digit Numbers
Solve 12 15 by using the partial-products method.
Step 1
Step 2
Step 3
Step 4
15
12
15
12
15
12
15
12
Partial Products
1 5
1 2
25
1 0
Step 5
2 10 2 0
Add partial products together.
10 5 5 0
10 10 1 0 0
10 20 50 100 180
product → 1 8 0
The product of 12 and 15 is 180.
Multiply by using the partial-products method.
© Harcourt
1.
3 1
1 7
2.
4 6
2 8
3.
7 9
5 6
71
86
69
7 30 8 40 6 70 10 1 20 6 50 9 10 30 20 40 50 70 product →
product →
product →
4.
82
25
5.
63
47
6.
92
34
Reteach
RW63
LESSON 12.2
Name
More About Multiplying by 2-Digit Numbers
Find the product 28 136.
• Estimate by rounding. Since 30 100 is 3,000, your
answer should be close to 3,000.
• Multiply.
136
28
1,088 ← 136 8
2,720 ← 136 20
3,808
Remember:
136 8 is the same as
• Compare the product to your estimate.
Since 3,808 is close to your estimate of
3,000, it is a reasonable product.
48
86 240
8 30 800
8 100
1,088
Estimate and multiply.
1 2 6
126 → 100
45 → 50
5,000
2.
4 5
276 →
38 →
6 3 0
3.
737 →
47 →
RW64
Reteach
5,
0 4 0
5,
6 7 0
7 3 7
4 7
2 7 6
3 8
4.
545 →
24 →
5 4 5
2 4
© Harcourt
1.
LESSON 12.3
Name
Multiply Greater Numbers
When multiplying greater numbers, it is more important than ever to work
neatly. Using a grid may help to keep your regroupings from getting mixed up.
3,846
27
5
3 4
3,
8 4 6
← Use one row for the first regroupings.
2 7
2 6, 9 2 2
1
1
5
3 4
3,
8 4 6
← Use another row for the second regroupings.
2 7
2 6, 9 2 2
7 6, 9 2 0
1 0 3, 8 4 2
Multiply.
© Harcourt
1.
2.
6,
5 5 1
8 4
4,
3 7 5
4 7
Reteach
RW65
LESSON 12.4
Name
Practice Multiplication
Betty is selling her CD collection. She sold 12 CDs at $14.95
each. How much money did she receive for her CDs?
• What do you need to do? Multiply 12 $14.95.
• First, estimate by rounding. Since 12 $14.95 is close to
10 $15.00, your answer should be close to $150.00.
• Multiply.
14.95
12
2990 ← 1495 2
14950 ← 1495 10
$179.40 ← Find the sum and place the decimal point.
• Write the answer in dollars and cents.
• Compare the product with your estimate. Since $179.40 is
close to your estimate of $150.00, it is a reasonable product.
1.
3.
5.
7.
$7.98
51
→
$8.00
50
$400.00
$80.10 →
53
913
44
→
645
83
→
RW66
→
391
28
→
→
391
28
$80.10 4. 2,356 →
45
53
→
2356
45
→
913 6. 5,469 →
32
44
→
5469
32
→
645
83
$86.32 →
61
→
$86.32
61
Reteach
2.
→
$7.98
51
798
39900
$406.98
8.
© Harcourt
Estimate and multiply.
LESSON 12.5
Name
Problem Solving Skill
Multistep Problems
To solve a problem, you need to make several decisions. One way
to approach a problem is to organize the data, select the operation
needed, and then solve.
The fourth-grade class sold treats at a flea market. On Friday, they
sold 144 chocolate chip cookies and 36 crispy treats. On Saturday,
they sold 136 chocolate chip cookies and 46 crispy treats. Each
chocolate chip cookie sells for $0.25, and each crispy treat sells for
$0.50. How much money did they make?
Break the problem down into steps.
1. Find the total number of chocolate chip cookies sold.
Operation: addition
Equation: 144 136 2. Find the total number of crispy treats sold.
Operation: addition
Equation: 36 46 3. Find the amount of money earned from each type of cookie.
Operation:
Equations: 280 $0.25 82 $0.50 4. Solve the problem.
Operation: addition
© Harcourt
Equation:
Organize the information. Select the operations. Solve.
1.
Molly is the star forward on her
basketball team. In Saturday’s
game she scored eleven 2-point
baskets and four 3-point baskets.
How many points did Molly score?
2.
Each egg carton at the dairy
holds 144 eggs. There are 4
cartons on the truck and 34
cartons in the warehouse. How
many eggs are there altogether?
Reteach
RW67
LESSON 13.1
Name
Divide with Remainders
What is 23 5?
The model below shows one way to
think about this division problem.
Think: If you divide 23 into 5 equal groups, how many are in each group?
23 5 There are 5 groups of 4, with 3 left over.
4 r3
3
52
20
3
23 5 4 r3
The remainder is the amount left over when a
number cannot be divided evenly. You can
multiply and subtract to find the remainder.
Complete 1–5 to find the remainder.
6r
1.
53
2
30
4r
2.
73
1
28
9r
3.
43
9
6r
31
9
4.
6r
5.
96
2
6.
74
3
RW68
Reteach
7.
51
9
8.
32
2
9.
21
7
© Harcourt
Divide.
LESSON 13.2
Name
Model Division
Divide 92 into 4 equal groups.
Write 92 4.
Step 1
Step 2
Show 92 as 9 tens and 2 ones.
2
Begin by dividing the
2
9 tens. Place an equal 49
8
number of tens into
1
each group.
2 tens in
each group
8 tens used
1 ten left
Draw 4 circles to show 4 groups.
Step 3
Regroup the 1 ten left
into 10 ones. Now
there are 12 ones.
2
2
bring
49
8↓ ones
12 down
Step 4
Divide the 12 ones. Place an equal
number of ones into each group.
23 3 ones in
2
each group
49
8
12
12 12 ones used
0 0 ones left
So, 92 4 23.
Use base-ten blocks to model each problem. Record the numbers
as you complete each step.
24
8
2.
39
6
3.
45
2
4.
23
5
5.
56
5
© Harcourt
1.
Reteach
RW69
LESSON 13.3
Name
Division Procedures
What is 47 3? Remember these steps:
Divide Multiply Subtract
1
7
34
3
1
Compare Bring down
1
34
313
431
13
Divide the tens.
Multiply.
Subtract.
Compare.
1
34
7
3↓
17
Bring down?
15
7
34
3
17
15
2
Yes. Bring the 7 down
to make 17.
5
31
7
3 5 15
17 15 2
23
Divide the ones.
Multiply.
Subtract.
Compare.
15 r2
7
34
3
17
15
2
Bring down?
Nothing is left to bring down.
If the number left over is
less than the divisor,
record it as the remainder.
1.
56
5
RW70 Reteach
2.
35
2
3.
25
1
4.
49
1
© Harcourt
Solve. Use the D, M, S, C, B steps shown above.
LESSON 13.4
Name
Problem Solving Strategy
Predict and Test
1.
There are 87 marbles. After the marbles are divided evenly into bags,
there are 2 marbles left over. There are fewer than 10 bags. How many
bags are there? How many marbles are in each bag?
Predict:
Test: Remember,
there must be 2
marbles remaining.
3 Bags?
29
38
7
6
27
27
0
No remainder
4 Bags?
21 r3
7
48
8
7
4
3
5 Bags?
Remainder of 3
Remainder of 2
You found the answer!
17 r2
7
58
5
37
35
2
There are 5 bags with 17 marbles in each bag, and
2 marbles are left over.
2.
There are 82 cookies. When the cookies are divided evenly on plates,
there are 4 cookies left over. There are fewer than 10 plates. How
many plates are there? How many cookies are there on each plate?
HINT: Predict numbers between 0 and 9. You may need to
predict more than three times.
© Harcourt
Predict:
Plates?
Plates?
Plates?
Test: Remember,
there must be 4
cookies remaining.
Reteach
RW71
LESSON 13.5
Name
Mental Math: Division Patterns
You can find how many fives are in 2,000 by following these steps.
• Rewrite each problem as a
multiplication fact.
20 5 4
5 4 20
5 4 0 20 0
• Circle all the numbers in the
basic fact to see how many
zeros to write in the quotient.
200 5 40
2,000 5 ?
5 4 00 20 00
So, 2,000 5 400.
• Write the quotient. If there is
a zero in the basic fact, the
quotient has one fewer zero than
the dividend. Otherwise the
quotient has the same number of
zeros as the dividend. You can
estimate a quotient by using
numbers that are easy to divide.
Use a basic division fact and patterns to find each quotient.
1.
4.
45 9 2.
21 3 3.
81 9 450 9 210 3 810 9 4,500 9 2,100 3 8,100 9 36 6 5.
72 8 6.
12 4 360 6 720 8 120 4 3,600 6 7,200 8 1,200 4 Basic Fact:
RW72 Reteach
Basic Fact:
Basic Fact:
© Harcourt
Divide mentally. Write the basic division fact and the quotient.
7. 480 6 8. 1,500 3 9. 36,000 ÷ 6 LESSON 13.6
Name
Estimate Quotients
Compatible numbers are numbers that are close to the actual numbers
and can be divided evenly. They can help you estimate a quotient.
Estimate.
43
9
5
Step 1
Round the dividend to a number
that can be divided evenly by the
divisor.
Step 2
Rewrite the division problem with
the compatible numbers, and solve.
The number 395 can be rounded to
360. This number can be divided
evenly by 4. The number 395 can
also be rounded to 400. This number
can also be divided evenly by 4.
90
43
6
0
100
44
0
0
So, one estimate of the quotient is
90. A second estimate is 100.
Write two pairs of compatible numbers for each. Give two
possible estimates.
1.
74
6
0
2.
53
7
0
3.
64
4
3
5.
76
,5
2
1
6.
93
1
5
__________________
83
,0
6
2
© Harcourt
4.
Reteach
RW73
LESSON 14.1
Name
Place the First Digit
When you begin a division problem, you must decide
where to place the first digit of the quotient.
2
371
6↓
11
Example B
hundreds
tens
ones
tens
ones
Example A
Divide the tens.
Multiply.
326
Subtract.
761
Compare.
13
Bring down? Yes. Bring
down 1 to
make 11.
3 4, so ask yourself
4
3 9
2 how many 4’s are in 39.
Divide the ones.
339
2 3 r2 Multiply.
Subtract.
11 9 2
37
1
Compare.
23
6↓
Bring down? Nothing is left.
1 1
If there is a
9
remainder,
2
record it.
98
4
3
92
36↓
0
32
32
0
Divide the tens.
Multiply. Subtract.
Compare. Bring down.
Divide the ones.
Multiply. Subtract. Compare.
Nothing is left.
Write an x where the first digit in the quotient should be placed.
1.
53
4
– 3 0
4
2.
4
37
– 6 ↓
1 4
3.
1
2
9
5
1 8 ↓
1 5
4.
45
3
9
4
1 3
5.
58
3
– 5 ↓
3 3
6.
32
9
RW74
Reteach
7.
24
7
8.
7 1 9
5 0
0
0 0
0
0 0 0
0
9. 5 7 5
0 0
4 0
0
0
© Harcourt
Divide.
LESSON 14.2
Name
Divide 3-Digit Numbers
These examples show how to divide a three-digit number.
Example A
Example B
1
9
352
3↓
22
Divide the hundreds.
Multiply.
Subtract.
Compare.
Bring down?
615
9
There are not enough
hundreds to divide.
17
2
9
35
3
22
21↓
19
Divide the tens.
Multiply.
Subtract.
Compare.
Bring down?
2
9
615
12↓
39
176 r1 Divide the ones.
2
9
35
Multiply.
3
Subtract.
22
Compare.
21
Bring down? If there is a
19
remainder,
18
record it.
1
Divide the tens.
Multiply.
Subtract.
Compare.
Bring down?
Divide the ones.
26 r3 Multiply.
59
61
Subtract.
12
Compare.
39
Bring down? If there is a
36
remainder,
3
record it.
Write an x where the first digit in the quotient should be placed.
1.
53
4
9
– 3 0 ↓
4 9
2.
37
4
5
– 6 ↓
1 4
3.
21
9
8
– 1 8 ↓
1 8
4.
45
3
2
– 4 ↓
1 3
© Harcourt
Divide.
5.
53
4
9
6.
37
4
5
7.
21
9
8
8.
45
3
2
Reteach
RW75
LESSON 14.3
Name
Zeros in Division
In some division problems, you need to place a zero in the
tens place or the ones place of the quotient.
Zero in ones place
Zero in tens place
H T O
2 0 8
8 2
43
8
0 3
0
3 2
3 2
0
428
Write 2
in quotient.
43
400
Write 0
in quotient.
43
2 4 8 32
Write 8
in quotient.
48
326
Write 2
in quotient.
31
5 3 5 15
Write 5
in quotient.
32
300
Write 0
in quotient.
H T O
2 5 0 r2
5
2
37
6
1 5
1 5
0 2
0
2
37
Divide and check.
36
1
4
2.
4.
7
9 1
3
5.
44
2
8
38
4
0
3.
6.
28
1
9
56
5
2
© Harcourt
1.
RW76
Reteach
LESSON 14.4
Name
Divide Greater Numbers
To divide greater numbers, repeat the division pattern
until there is nothing left to bring down and the difference
is less than the divisor.
T
H T O
3 2, 5 6 9
0000
1.
2.
3.
4.
5.
6.
7.
8.
9.
Divide the hundreds.
Multiply and subtract.
Bring down the 6 tens.
Divide the tens.
Multiply and subtract.
Bring down the 9 ones.
Divide the ones.
Multiply and subtract.
There is nothing left to bring down.
1 3. The remainder is 1.
Write divide, multiply and subtract, or bring down to show the
next step.
8
7 2
1.
2.
5 4, 2 8 5
6 4, 3 3 7
4 0
4 2
2 8
1 3
© Harcourt
Divide.
3.
41
,8
7
3
4.
7$
4
5
.9
2
5.
23
,4
9
8
6.
6$
3
5
.4
6
7.
87
,8
9
3
8.
39
,8
2
2
Reteach
RW77
LESSON 14.5
Name
Problem Solving Skill
Interpret the Remainder
You can use a remainder in different ways. Look for word clues
in problems to help you decide what to do with the remainder.
Look at the list of clues below. Use it to help solve the problems.
a. If the problem asks how
much is left over, use the
remainder for your answer.
b. If the problem asks how
many groups are needed
to divide a number into
groups, increase the
quotient by 1.
c. If the problem asks how
many groups can be made
from a larger number,
drop the remainder.
Example: Thomas has to divide his 24
classmates into groups of 5. How many
students will be left over?
24 5 4 r4
4 students will be left over.
Example: How many tables will be
needed to seat the 24 classmates at
tables of 5?
24 5 4 r4
5 tables will be needed (4 tables will not
be enough.)
Example: How many 10-person
spelling bee teams can be made with
24 classmates?
24 10 2 r4
There will be 2 teams. (There are not
enough classmates to make a third
team of 10.)
1.
Mark is cutting ribbon to wrap a
gift. He has a spool of 45 inches of
ribbon. How many 8-inch lengths
of ribbon can he make?
2.
Scott needs 27 sheets of paper
for an art project. Paper comes in
packages of 20 sheets. How
many packages of paper should
Scott buy?
3.
Anne is making bead bracelets.
Each bracelet takes 5 beads.
Anne has 37 beads. How many
beads will she have left over?
4.
Alex is giving 6 brownies to each
teacher. She has 38 brownies.
How many teachers can she give
brownies to?
RW78 Reteach
© Harcourt
Look for the clues in these problems. Solve. Then write a, b, or c
from above to tell how you interpreted the remainder.
LESSON 14.6
Name
Find the Mean
The mean is the number found by dividing the sum
of a set of numbers by the number of addends.
To find the mean:
Step 1
Add the numbers.
Step 2
Count the number of addends.
Step 3
Divide the sum by the number of addends.
Examples:
Find the mean of 21, 14, 26, and 47.
Find the mean of 35, 47, 28, 29, and 31.
Step 1
21 14 26 47 108
Step 1 35 47 28 29 31 170
Step 2
There are 4 addends.
Step 2 There are 5 addends.
Step 3
108 4 27
Step 3 170 5 34
The mean is 27.
The mean is 34.
Find the mean of each set of numbers.
1.
5, 6, and 10
2.
Step 1 5 6 10 $1.89; $2.85; $3.24
Step 1
Step 2 There are
Step 3
addends.
3
The mean is
Step 2 There are
addends.
Step 3
.
The mean is
.
© Harcourt
Find the mean.
3.
4, 5, 5, 8, 8
4.
19, 29, 12, 48
5.
$3,112; $2,018; $4,527
6.
32, 71, 89, 64
7.
151, 109, 133
8.
$1.19, $0.78, $0.61
Reteach
RW79
LESSON 15.1
Name
Division Patterns to Estimate
In the problem 594 18 33, 594 is the dividend, 18 is the
divisor, and 33 is the quotient. To estimate the quotient,
follow Steps 1–3.
Step 1
Step 2
Step 3
Round the divisor to
the nearest tens place.
Round the dividend to
the nearest multiple of
the new divisor.
Divide, using the
rounded numbers.
18 → 20
595 → 600
600 20 30
Write the numbers you would use to estimate the quotient.
Then write the estimate.
1.
36 11 n
2.
78 13 n
3.
56 19 n
Write the numbers you would use to estimate the quotient.
Then write the basic fact that helps you find the quotient.
The first one has been done for you.
4.
446 92 n
5.
52 11 n
6.
162 79 n
8.
272 28 n
9.
235 56 n
450 90 5
45 9 5
7.
76 24 n
10.
Dividend
Divisor
40
20
20
11.
12.
4,000
RW80 Reteach
20
Quotient
13.
20
Dividend
Divisor
90
30
30
14.
15.
9,000
30
Quotient
30
© Harcourt
Complete the tables.
LESSON 15.2
Name
Model Division
You can divide by using models.
What is 35 12?
Step 1
Step 2
Step 3
Count out 35 objects.
Make as many groups
of 12 as you can.
Record your
work. Count
2 r11
the number
5
123
of groups
24
and how
11
many are
left over.
© Harcourt
Make a model to divide.
1.
154
2
2.
174
9
3.
251
0
3
4.
216
5
5.
113
5
6.
311
2
8
7.
169
3
8.
195
9
9.
248
4
Reteach
RW81
LESSON 15.3
Name
Division Procedures
What is 231 19? Follow Steps 1–4 when you divide a
3-digit number by a 2-digit number.
Step 1
Step 2
How many
groups of 19 are
in 2? Zero groups.
Write an x in the
hundreds place.
How many
Multiply.
groups of 19 are
Subtract 19 and
in 23? One group. bring down the 1.
Write a 1 in the
tens place.
x
3
1
192
Step 3
Step 4
How many groups
of 19 are in 41?
Two groups.
Write a 2 in the
ones place.
Subtract.
x1
3
1
192
19
41
x1
3
1
192
x12
3
1
192
19
41
38
3
Compare the remainder, 3, with the divisor, 19. The
remainder is less than the divisor. Write the remainder
as part of the quotient.
So, 231 19 12 r3.
Divide.
118
0
6
2.
121
8
5
3.
151
2
2
4.
161
1
3
6.
131
8
3
7.
185
8
1
8.
221
1
9
© Harcourt
1.
5.
174
1
4
RW82 Reteach
LESSON 15.4
Name
Correcting Quotients
Divide. 321 38 n
383
2
1
When you are dividing, sometimes the first digit you write
in the quotient is too high or too low.
Too High
9
383
2
1
342
Since 342 > 321, the
quotient 9 is too high.
Use a lesser number.
Too Low
Just Right
7
383
2
1
266
55
Since the remainder,
55, is greater than the
divisor, 38, the quotient
7 is too low. Use a
greater number.
8
2
1
383
304
17
Since the remainder,
17, is less than the
divisor, 38, the quotient
8 is just right.
© Harcourt
Write too high, too low, or just right for each estimate. Explain.
6
3
6
1. 181
108
28
7
0
3
2. 161
112
8
3
9
3. 171
136
3
5
1
9
4. 231
115
4
6
3
6
5. 322
192
44
7
7
0
6. 281
196
Divide.
7.
433
6
5
8.
343
3
8
9.
523
7
9
10.
684
5
4
Reteach
RW83
LESSON 15.5
Name
Problem Solving Skill: Choose the Operation
Add
Joining groups
of different sizes
Erin read a biography of Thomas Jefferson
that was 312 pages long. Next she read a
fiction book which was 289 pages long.
How many pages did she read altogether?
312 289 601
Subtract
Taking away or
comparing groups
How many pages longer was the biography
than the fiction book?
312 289 23
Multiply
Joining equal-sized
groups
Finding how many
in each group or how
many equal-sized
groups
23 pages longer
Erin thinks it takes about 3 minutes to read
one page of a book. How many minutes
should it take her to read the biography?
3 312 936
Divide
601 pages
936 minutes
Erin wants to read the fiction book in 17
sittings. How many pages should she read
at each sitting?
289 17 17
17 pages
1. There are 12 players on Coach
Walker’s soccer team, 14
players on Coach Lee’s team,
and 24 adults. How many
people are there?
2. If each of the players at the
soccer game is wearing 2 shin
guards, how many shin guards
are at the game?
3. Coach Walker has brought 3
balls to the soccer game. Before
the game, he wants equal groups
of his players to warm up with
the balls. How many players
should be in each group?
4. Each quarter of the soccer
game lasts 12 minutes. If
9 minutes have gone by,
how many minutes remain
in the quarter?
RW84
Reteach
© Harcourt
Solve. Name the operation you use. Write add, subtract,
multiply, or divide.
LESSON 16.1
Name
Factors and Multiples
Use a multiplication table to find the multiples of a given
number. A multiple is the product of a given whole
number and another whole number.
Look at the row or column for a
given number. All the numbers in
that row or column are multiples of
that number.
Example: Look at the row or the
column for 6.
Some multiples of 6 are 6, 12, 18,
24,
,
66, and
,
, 48, 54,
,
.
1
2
3
4
5
6
7
8
9 10 11 12
1
1
2
3
4
5
6
7
8
9 10 1 1 12
2
2
4
6
8 10 12 14 16 18 20 22 24
3
3
6
9 12 15 18 21 24 27 30 33 36
4
4
8 12 16 20 24 28 32 36 40 44 48
5
5 10 15 20 25 30 35 40 45 50 55 60
6
6 12 18 24 30 36 42 48 54 60 66 72
7
7 14 21 28 35 42 49 56 63 70 77 84
8
8 16 24 32 40 48 56 64 72 80 88 96
9
9 18 27 36 45 54 63 72 81 90 99 108
10
10 20 30 40 50 60 70 80 90 100 110 120
11
11 22 33 44 55 66 77 88 99 110 121 132
You can also use the table to find
factors of a given number. To find
factors of 36, search the table for 36. When you see 36,
look at the row and column numbers to find the factors.
12
Some factors of 36 are 3, 4, 6, 9, and
12 24 36 48 60 72 84 96 108 120 132 144
.
Use the multiplication table to find factors for each product.
1.
45
2.
24
3.
20
6.
5
Use the multiplication table to write 4 multiples for
each number.
4
5.
9
© Harcourt
4.
Write as many factors as you can for each product.
7.
18
8.
36
Reteach
RW85
LESSON 16.2
Name
Factor Numbers
An array is an arrangement of objects in rows and
columns. Arrays can show how factors and products are
related.
Use arrays to break down 56 into factors.
STEP 1 Outline a rectangle that has 56 squares. Use
your basic facts to help decide the length and
width of the rectangle.
What two factors have a product of 56?
STEP 2 Split the rectangle into two equal parts to find
other factors.
These arrays show
(
) 56
STEP 3 Can you split your array into different equal parts
to find more factors?
These arrays show
(
) 56
Write an equation for the arrays shown.
1.
2.
Show two ways to break apart the model.
4.
© Harcourt
3.
Write two ways to break down the number.
5. 42
RW86
6.
Reteach
36
LESSON 16.3
Name
Prime and Composite Numbers
A prime number has exactly two factors, 1 and the
number itself.
A composite number has more than two factors.
The number 1 is a special number because it is neither
prime nor composite. It has only 1 factor, 1.
Make arrays to find the factors. Write prime or composite for
each number.
8
2. 23
3.
35
4. 9
5.
24
6. 11
© Harcourt
1.
Reteach
RW87
LESSON 16.4
Name
Find Prime Factors
You can find the prime factors of a number by using
a factor tree.
Begin by writing the number.
80
2 40
Use what you know about
multiplying to write the
number as a product.
If one factor is prime,
underline it. You can not break
that factor down any further.
2 2 20
2245
22225
Break down the factors that
are not underlined.
Continue until all factors
are prime.
They should all be underlined
at the bottom of your tree.
So, 80 2 2 2 2 5.
Use factor trees to write each as a product of prime factors.
30
2. 45
3.
50
4. 64
© Harcourt
1.
RW88
Reteach
LESSON 16.5
Name
Problem Solving Strategy
Find a Pattern
Carrie has finished part of her square
quilt. How many patches of each color
will be in the final quilt?
Look for patterns to help you
make a plan for solving.
Since this is a square quilt and
there are 9 patches along the
top, you know there must
be 9 rows.
B
R
B
R
B
R
RWB
WB R
RWB
WB R
RWB
WB R
B Blue
Look closely at the quilt. There
is a pattern of 3 blues, 3 reds,
and 3 whites in each row.
Multiply each color by the
9 rows. So, there are 27 blue,
27 red, and 27 white patches.
Blue
RW
WB
RW
WB
RW
WB
R Red
Red
B
R
B
R
B
R
RW
WB
RW
WB
RW
WB
W White
White
3 9 27 3 9 27 3 9 27
Fill in the missing pieces of the pattern.
© Harcourt
1.
2.
128, 64, 32, 16,
3.
4, 6, 8, 9, 10, 12, 14, 15,
,
,
4.
Reteach
RW89
LESSON 17.1
Name
Read and Write Fractions
A fraction is a number that names a part of a whole.
Read:
two thirds
two out of three
two divided by three
Read:
one half
one out of two
one divided by two
parts shaded 2
total parts 3
parts shaded
number of equal parts
1
2
Write a fraction for each model.
1.
parts shaded
total parts
3.
parts shaded
total parts
5.
2.
parts shaded
total parts
4.
parts with stars
number of equal parts parts with cats
number of equal parts 6.
part with baseballs
number of equal parts
Write two ways to read each fraction.
3
4
8.
1
2
2
3
9. © Harcourt
7. RW90 Reteach
LESSON 17.2
Name
Equivalent Fractions
1
1
4
Fractions that name the same amount are called
1
8
1
1
6
1
8
1
1
6
1
1
6
1
1
6
equivalent fractions. 14, 28, and 146 are different
names for the same number. So, 14 28 146 , which
makes them equivalent fractions.
Find the equivalent fraction.
1.
2.
2
3 3.
1
2 6
8
4.
3
5 3
4 Color the correct number of parts to show an equivalent
fraction. Then write the equivalent fraction.
5.
6.
© Harcourt
6
9 7.
2
4 8.
1
3 1
4 Reteach
RW91
LESSON 17.3
Name
Equivalent Fractions
You can use different fractions to name the same amount.
Fractions that name the same amount are called equivalent fractions.
You can find equivalent fractions in three ways.
Use a number line.
0
1
1
2
0
2
4
1
4
3
4
1
You can see that 12 24,
so they are equivalent
fractions.
Multiply both the
numerator and the
denominator by the
same number.
Divide both the
numerator and the
denominator by the
same number.
1 13 3
3 33 9
6 6÷2 3
8 8÷2 4
The fraction 13 names
the same amount
as 39, so they are
equivalent fractions.
The fractions 68 and 34
name the same
amount, so they are
equivalent fractions.
Use the number lines to find out if the fractions are equivalent.
Write yes or no.
1
4
3
12
8
12
3
4
1. 2. 0
0
1 2
12 12
3
4
2
4
1
4
4
3 12
12
5
12
7
6 12
12
8
12
1
10 11
12
9 12
12
1
2
3
25
35
4
5
43
53
3
8
32
82
4. 1
7
14
74
6. 3. 5. Divide both the numerator and the denominator to find the
simplest form of each fraction.
7
28
77
28 7
16
24
16 8
24 8
12
16
12 4
16 4
8. 10
15
10 5
15 5
10. RW92
Reteach
7. 9. © Harcourt
Multiply both the numerator and the denominator to name an
equivalent fraction.
LESSON 17.4
Name
Compare and Order Fractions
You can compare fractions that have the
same denominators.
2
5
Compare 25 and 35.
Compare the shaded areas
in the fraction models.
2 3, so 25 35.
3
5
You can also compare fractions that have
different denominators.
Compare 23 and 12.
Compare the shaded areas
in the fraction models.
Since 23 has a larger shaded
area, 23 12.
2
3
1
2
Compare the fraction models. Write , , or in the
1.
2.
4
6
5
6
4.
© Harcourt
3.
3
4
6
8
5.
2
3
2
4
7.
1
3
2
3
3
5
2
4
3
6
4
1
0
5
8
6.
5
8
6
8
8.
3
9
.
9.
3
4
3
5
Reteach
RW93
LESSON 17.5
Name
Problem Solving Strategy
Make a Model
Beth made a snack mix for the class party. She used 12 lb
of peanuts, 23 lb of pretzels, and 14 lb of dried fruit. List
the ingredients in order from greatest to least amount.
You can make a model to help solve the problem. Use
fraction bars to model the fractions. Line up the fraction
bars so you can tell which bar has the most shaded and
which bar has the least shaded.
1
2
1
2
lb of peanuts
2
3
lb of pretzels
1
3
1
4
lb of dried fruit
1
4
1
3
The bar for pretzels has the most shaded and the bar for
dried fruit has the least shaded. So, the ingredients in order
from greatest to least are pretzels, peanuts, and dried fruit.
Make a model to solve.
The amusement park has a
colorful sidewalk. Starting with
the first square, every third
square is orange, and every fifth
square is green. Draw a model of
the first 12 squares of the sidewalk
in the space below. What fraction
of the first 12 squares are either
orange or green?
2.
Patti made punch from 12 gallon
of grape juice, 23 gallon of kiwi
juice, and 58 gallon of papaya
juice. Draw a model for each
in the space below. Then list
the ingredients in order from
greatest to least.
1
2
1
3
grape juice
1
3
1 1 1 1 1
8 8 8 8 8
RW94
Reteach
kiwi juice
papaya juice
© Harcourt
1.
LESSON 17.6
Name
Mixed Numbers
A mixed number is made up of a whole number and
a fraction.
These models represent a mixed number.
1
3
2
3
3
3
4
3
5
3
6
3
7
3
8
3
9
3
10 11
3 3
There are 3 whole figures shaded.
3 shaded.
The last figure is 2
3 figures are shaded.
So, 32
31
of the figures are shaded.
Or, you can say 1
© Harcourt
Write a fraction and a mixed number for each picture.
1.
2.
3.
4.
Rename each fraction as a mixed number. You may wish to draw a picture.
3
2
6. 13
5
7. 13
3
9. 26
6
10. 5. 8. 22
4
18
8
Reteach
RW95
LESSON 18.1
Name
Add Like Fractions
When you add like fractions, add only the numerators.
2
1
5 5 ?
+
2 parts shaded
5 parts
=
1 part shaded
5 parts
3 parts shaded
5 parts
2
1 → Add the numerators.
3
→ 21 →
5 5 → Write the denominator. → 5 → 5
Find the sum. Show how you added the numerators.
2
3
23
5
6 6 6 6
2. 1
4
8 8 4. 2
1
4 4 6. 4
5
10 10
8. 4
6
12 12
10. 3
2
55
12. 3. 5. 7. 9. 11. RW96
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3
3
7 7 2
2
5 5 1
1
3 3 4
1
99
2
1
8 8 5
6
88
© Harcourt
1. LESSON 18.2
Name
Subtract Like Fractions
When you subtract like fractions, subtract only
the numerators.
2
6
8 8 ?
X
X
6 parts shaded
8 pa rt s
2 parts shaded
8 pa rt s
4 parts shaded
8 pa rt s
1
6 2 → Subtract the numerators. → 6 2 → 4
,
or
8
8 → Write the denominator.
→ 8 →
8
2
Find the difference. Show how you subtracted the numerators.
5
3
53
2, or 1
3
6 6 6 6
2. 7
4
8 8 4. 2
1
4 4 6. 7
5
10 1
0
8. 10
6
12 12 10. 9
3
10 1
0
12. 1. 3. 5. © Harcourt
7. 9. 11. 6
3
7 7 4
2
5 5 2
1
3 3 4
1
9 9 2
1
88
9
1
12 1
2
Reteach
RW97
LESSON 18.3
Name
Add and Subtract Mixed Numbers
When adding mixed numbers, add the fractions first.
Then add the whole numbers.
43
Add. 72
6
6
Step 1 Add the
fractions.
Step 2 Add the whole
numbers.
6
72
6
43
6
72
6
43
5
6
6
115
Step 3
.
The sum is115
6
43
115
.
So, 72
6
6
6
When subtracting mixed numbers, subtract the fractions
first. Then subtract the whole numbers.
4
16
Subtract. 85
6
Step 1 Subtract the
fractions.
Step 2 Subtract the
whole numbers.
85
6
85
6
4
16
4
16
1
6
76
Step 3 The difference
1
is 76.
5
4
1
So, 86 16 76
1
Find the sum or difference.
9
52
2.
9
24
5.
5
84
6.
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5130
4.
2160
7
93
7.
72
7
8 63
52
8 RW98
3.
6
71
31
5
9.
6
83
8
13
9
45
8.
31
9
0
1
81
12
12
4
62
31
4
3
10.
8
97
11.
4 32
4 71
© Harcourt
1.
LESSON 18.4
Name
Problem Solving Skill
Choose the Operation
In deciding if you should add or subtract to solve a problem, think
about the answer. Should the answer be greater than the numbers in the
problem? If so, then consider adding. If not, then consider subtracting.
Eldrick and Erin wrote their
ages as mixed numbers. Eldrick
is 10172 years old and Erin is 9112
years old. How much older than
Erin is Eldrick?
Think: Should the answer be
more than or less than 10172?
less than
Solve:
10172
1
912
1162 or 112 years
Read the problem. Answer the Think question. Then solve.
1.
© Harcourt
2.
Devon needs 2 pieces of lumber
to build a frame for his treehouse. One should be 1212 feet
long and the other should be 312
feet long. How long are the 2
pieces of lumber together?
Think: Should the answer be
greater than or less than 1212?
Alicia is knitting a scarf that will
be 523 feet long. So far she has
knitted 313 feet. How much more
does she have to knit?
Think: Should the answer be
greater than or less than 523?
Solve:
Solve:
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RW99
LESSON 18.5
Name
Add Unlike Fractions
Yesterday, Jill walked 14 mile, and Bob walked
far did they walk in all?
To answer this, you add
1
4
1
3
mile. How
13.
The fractions 14 an 13 are unlike fractions because the
denominators are not the same.
To add unlike fractions, you can use the following steps:
Step 1
Step 2
Use fraction bars to
model 14 13.
Find the like fraction
Write an equation for
bars that are equivalent the sum shown by the
to 14 13 in length.
like bars.
Step 3
1
1
4
1
3
1
1
1
4
1
4
1
3
1
3
1 1 1 1 1 1 1
12 12 12 12 12 12 12
?
7
1
1
4 3 12
1.
1
1
6 4
2.
2
2 5 10
3.
1
5
3 6
4.
2
1
8
4
5.
1
1
5 2
6.
1
1
3 4
7.
4
1
6 4
8.
1
1
3 2
9.
3
3 5 10
10.
1
2
4 6
3
3
8 4
12.
1
2
6 3
11.
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© Harcourt
Use fraction bars to find the sum.
LESSON 18.6
Name
Subtract Unlike Fractions
Subtract. 4
3
5 10
To subtract unlike fractions, you can follow these steps.
Step 1
Step 2
Place four 15 fraction bars under a
1 whole bar. Then place three 110
fraction bars under the four 15 bars.
Compare the bars. Find like fraction
bars that fit exactly under the
difference 45 130 .
1
1
1
5
1
5
1
5
1
5
1
5
1
5
1 1 1
10 10 10
1 1 1
10 10 10
1
5
1
5
?
1 1 1 1 1
10 10 10 10 10
So,
4
5
130 5
10,
or
1
2
.
© Harcourt
Use fraction bars to find the difference. Write the answer in simplest form.
1.
3
1
4 2
2.
5
1
8 4
3.
2
1
3 2
4.
7 2
10 5
5.
6
1
8 2
6.
1
1
1
12 3
7.
2
1
3 6
8.
3
1
4 2
9.
9 1
10 2
10.
3
3
4 8
1
1
2 6
12.
1
3 2 10
11.
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RW101
LESSON 19.1
Name
Relate Fractions and Decimals
You can write a fraction or a decimal to tell what part is shaded.
Model
Fraction
Decimal
O
4 shaded parts
10 parts
0
T
.
O
25 shaded parts
100 parts
0
.
H
four tenths
4
T
•.
Read
H
twenty-five
hundredths
2
5
Complete the table.
Model
Fraction
Decimal
O
1.
shaded parts
parts
Read
T
H
T
H
T
H
T
H
T
H
.
•
O
2.
.
•
O
3.
.
O
4.
© Harcourt
•
.
•
O
5.
.
•.•
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LESSON 19.2
Name
Decimals Greater Than 1
6110
A mixed number is made up of a
whole number and a fraction.
Decimal
Number
6.1
A decimal greater than 1 is made up
of a whole number and a decimal.
$500.75
Whole
Number
Two whole blocks and three tenths
of a block are shaded.
Write: 2130, or 2.3
Write a mixed number and a decimal greater than 1 for
each model.
1.
2.
3.
4.
Write each mixed number as a decimal.
5.
3140
6.
23
9
10
0
7.
4140
8.
9
3
100
© Harcourt
Write each decimal as a mixed number.
9.
5.6
10.
7.02
11.
3.78
12.
6.09
13.
2.8
14.
6.07
15.
8.02
16.
9.4
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RW103
LESSON 19.3
Name
Equivalent Decimals
Equivalent decimals are different names for the same amount.
0.3
Ones
=
0.30
Tenths
0
.
3
0
.
3
The decimal models
show that 0.3 and 0.30
are equivalent decimals.
The same amount is
shaded in each model.
Hundredths
0
The place-value chart also
shows that 0.3 = 0.30. Both
numbers have the digit 3 in
the tenths place.
Shade an equivalent amount in each pair of models. Write the
decimal for each shaded amount.
1.
2.
3.
4.
6.
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© Harcourt
5.
LESSON 19.4
Name
Compare and Order Decimals
Write these numbers in order from greatest to least:
0.4, 0.35, 0.37, 0.05.
Step 1 Write the numbers in a place-value chart.
Step 2 Write a zero in the empty hundredths place.
Step 3 Compare the digits in each place, beginning at the left.
Ones
Tenths
Hundredths
0
.
4
0
0
.
3
5
0
.
3
7
0
.
0
5
Greatest
0.40, 0.37, 0.35, 0.05
Least
Compare. Then write the greater number.
1.
0.2
0.3
2.
2.25
2.27
3.
0.36
0.34
4.
3.5
3.9
5.
0.37
0.3
6.
0.02
0.10
7.
0.38
0.40
8.
1.04
1.2
Compare. Write , , or in the .
9.
10.
© Harcourt
0.2
0.19
11.
0.50
0.39
12.
0.2
13. 0.34 0.39
14. 2.1 2.09
0.25
0.47
15. 0.24 0.34
16. 0.04 0.4
0.41
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RW105
LESSON 19.5
Name
Problem Solving Strategy
Use Logical Reasoning
Laverne, Jody, Pearl, Leroy, and Cal are each performing
in a variety show. Laverne will perform after Leroy. Pearl
will perform third. Jody will perform before Pearl. Cal will
perform before Jody. In what order will they perform?
Use notecards to help you solve the problem. Write each person’s name
on a slip of paper. When you know a person’s position, place the slip of
paper with his or her name in that position.
rne
Lave
Jody
Pearl
Leroy
Cal
Pearl will perform third.
?
?
Pearl
?
?
Jody will perform before Pearl. Cal will perform before Jody.
Cal
Jody
Pearl
?
?
Leroy
Laverne
Laverne will perform after Leroy.
Cal
Jody
Pearl
So, the order is: Cal, Jody, Pearl, Leroy, Laverne.
1.
Vicky has 5 classes before lunch each day. She has math
before science. She has Spanish before math. She has
English right before lunch. She has history after science.
In what order does Vicky have classes in the morning?
2.
There are 6 children in the Ingraham family. Denise is
older than Steven, but younger than Eliot. Kayla is
younger than all her brothers and sisters except for Kirk.
Bernie is younger than Steven. Write the names of the
Ingraham children in order from oldest to youngest.
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© Harcourt
Use notecards and logical reasoning.
LESSON 19.6
Name
Relate Mixed Numbers and Decimals
Change 23 to a decimal.
5
Change 4.70 to a mixed number.
3
6 0.6
5 10
Read 4.70 as “4 and 70 hundredths.”
So, 23 2.6
5
7
0
4.70 4
100
Write the mixed number in simplest form.
So, 4.70 47.
10
Write an equivalent decimal for each mixed number.
1.
32 3
3.
5
10
2.
23 4
3.
43 10
Write an equivalent mixed number for each decimal.
4.
3.6 3 and
6.
4.90 tenths 3
10
3
5.
2.8 7.
3.75 Write an equivalent mixed number or decimal.
8.
63
5
9.
4.20
10.
1
2
10
2
0
8
100
12.
31
2
13.
7.75
14.
2
3
5
15.
4.6
16.
7.9
© Harcourt
11.
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RW107
LESSON 20.1
Name
Round Decimals
Round 2.53 to the nearest whole number.
Step 1
Underline the digit in the ones place.
2 .53
Step 2
Look at the digit to the right. If the digit is 5 or greater,
the digit in the rounding place is increased by 1.
→
2 .53
So, round 2 to 3.
Step 3
Rewrite with zeros in all places to the right.
3.00
Use similar steps to round 1.26 to the nearest tenth.
→
1. 2 6
Round 2 to 3.
1.30
Complete the chart.
1.
3.25
2.
14.85
3.
21.51
4.
65.28
5.
75.31
6.
80.09
7.
114.72
8.
151.38
RW108 Reteach
Round to the nearest whole number.
© Harcourt
Round to the nearest tenth.
LESSON 20.2
Name
Estimate Sums and Differences
You can estimate the sum of $25.36 and $14.84 by rounding to the nearest
dollar and then adding.
$2 5. 0 0
$2 5. 3 6 →
$1 5. 0 0
$1 4. 8 4
$4 0. 0 0
You can estimate the difference between $38.63 and $12.49 by rounding to
the nearest dollar and then subtracting.
$3 8. 6 3 → $3 9. 0 0
$1 2. 4 9
$1 2. 0 0
$2 7. 0 0
Estimate the sum or difference.
© Harcourt
1.
$3.51 →
$1.21
$4. 0 0
$1. 0 0
2.
15. 84
22. 31
3.
$9.51
$6.22
4.
85.43
19.84
5.
$8.62
$4.25
6.
$19.65
$ 8.42
7.
$78.23
$48.61
8.
69.58
10.71
9.
50.28
49.72
10.
9.54
34.19
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RW109
LESSON 20.3
Name
Add Decimals
How can you add 1.65 to 0.37?
Remember to keep digits lined up by place value and their
decimal points.
Step 1
Step 2
Identify the place value of each digit.
1.65
0.37
1 is in the ones place.
0 is in the ones place.
6 is in the tenths place.
3 is in the tenths place.
5 is in the hundredths place. 7 is in the hundredths place.
Line up each digit
Step 3 Add, starting from the right.
by its place value.
Regroup if you need to.
O
T
H
.
0 •. 3
•.
1 •6
O
T
H
.
0 •. 3
2 •. 0
7
1
5
7
11 •161 5
2
So, 1.65 0.37 equals 2.02.
Rewrite each problem in the place-value grid. Find the sum.
0.5
O
T
2.
0.9 0.6
H
O
•.
T
6. 1.4
H
•.
•.
RW110 Reteach
•.
O
7.
T
•.
O
8. 29.09 15.8
T
T H
•.
•.
O
T H
•.
•.
H
•.
21.34 40.82
H
T
•.
•.
T
•.
1.46 0.17
H
•.
•.
T
4.
•.
1.5
O
•.
O
•.
1.2
O
H
•.
•.
5. 0.6
1.67 0.58
•.
•.
T
3.
•.
•.
© Harcourt
1. 0.8
LESSON 20.4
Name
Subtract Decimals
How can you subtract 0.58 from 1.72?
Remember to keep digits lined up by place value and
their decimal points.
Step 1
Step 2
Identify the place value of each digit.
1.72
0.58
1 is in the ones place.
0 is in the ones place.
7 is in the tenths place.
5 is in the tenths place.
2 is in the hundredths place. 8 is in the hundredths place.
Line up each digit
Step 3 Subtract, starting from the
by its place value.
right. Regroup if you need to.
O
T
.
0 •. 5
•.
1 •7
H
O
2
1 •67
1
.
0 •. 5
1 •. 1
8
T
H
2
11
12
8
1
4
So, 1.72 0.58 equals 1.14.
Rewrite each problem in the place-value grid. Find the difference.
0.7 0.1
1.
O
2. 0.9
T
H
O
•.
•.
T O
T
•.
•.
0.51
O
T
4. 1.91
H
•.
18.9 5.2
T O
H
T
•.
7.
52.61 32.84
H
T O
•.
•.
•.
H
•.
•.
6.
T
•.
•.
0.85
O
•.
•.
H
•.
31.7 24.5
© Harcourt
T
3. 1.72
•.
•.
5.
0.5
T
H
•.
•.
•.
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RW111
LESSON 20.5
Name
Add and Subtract Decimals
How can you add 15.3 and 2.45?
Step 1
Write the decimals, keeping
the decimal points in a line.
T
O
.
2 •. 4
•.
1
5•3
Step 2
Write an equivalent decimal
with a zero in the hundredths
place so that both numbers
have the same number of
digits after the decimal point.
T
O
5
T H
.
2 •. 4
7 •. 7
1
5•3 0
1
Step 3
T H
5
5
Add as you would with whole numbers.
Follow the same steps when you subtract.
Rewrite each problem in the place-value grid. Add or subtract.
1.34 0.5
O
T
2.
21.67 4.2
T
H
O
•.
T
5.
O
H
•.
•.
RW112
O
•.
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•.
6.
25.2 8.71
T H
T
•.
•.
•.
T H
•.
7 1.57
•.
T
•.
1.8 0.52
O
T H
•.
•.
4.
5 24.89
•.
•.
3.
O
T H
•.
•.
•.
© Harcourt
1.
LESSON 20.6
Name
Problem Solving Skill
Evaluate Reasonableness of Answers
Carrie walks 2.5 miles to Sue’s house. Then
she and Sue walk 1.6 miles to Joan’s house.
All three girls then walk 3.1 miles to Carrie’s
house. How many miles did Carrie walk?
Complete the chart. Find the actual distance and
the estimated distance.
Distance in miles between:
Actual
Estimate
Carrie’s and Sue’s house
2.5
3.0
Sue’s and Joan’s house
1.6
2.0
Joan’s and Carrie’s house
3.1
3.0
Total
7.2
8.0
Compare the actual distance to the estimated distance.
7.2 miles is close to the estimate of 8 miles. So the answer
is reasonable. Carrie walked 7.2 miles.
For 1–4, use the information above.
Circle the more reasonable answer.
1.
Sue decided to walk home from
Carrie’s house. How far did she
walk in all?
a. 2.5 miles
© Harcourt
3.
5.
b. 7.2 miles
How much closer does Carrie
live to Sue than to Joan?
a. 0.6 mile
Joan decided to walk home
from Carrie’s house. How far
did she walk in all?
a. 6.2 miles
4.
b. 5.6 miles
Carrie walked to Sue’s house
and back home 3 times last
week. About how far did she
walk in all?
a. 18 miles
2.
How much closer does Joan live
to Sue than to Carrie?
a. 4.7 miles
6.
b. 3.1 miles
b. 1.5 miles
How much closer does Sue live
to Joan than to Carrie?
a. 0.9 miles
b. 4.1 miles
b. 9 miles
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RW113
LESSON 21.1
Name
Choose the Appropriate Unit
The customary system of measurement uses inches, feet,
yards, and miles to measure length, width, distance, and
height. You can use the following sentence to help you
remember the order of the sizes of these measurements.
In
Fourth grade
You
Measure.
Inch
Foot
Yard
Mile
Use inches to measure small items, such as a pen or your
finger. Use feet for greater measurements, such as heights.
Use yards to measure distances, such as the length of a
football field. Use miles to measure great distances, such as
the distance between cities.
Name the greater measurement.
1.
234 mi or 234 yd
4.
100 yd or 100 in.
2.
1 in. or 1 ft
3.
45 ft or 45 yd
5.
80 in. or 80 mi
6.
17 ft or 17 mi
Choose the most reasonable unit of measure. Write in., ft, yd, or mi.
The width of your hand is about 3
8.
.
9.
11.
10.
.
The length of one lap around a
running track is 220
The width of your desk is 2
12.
.
234 mi or 234 yd
notebook paper is 11
RW114
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14.
1 in. or 1 ft
.
The length of a piece of
Circle the greater measurement.
13.
.
to Boston is about 200
The length of your classroom is
25
The distance from New York City
15.
45 ft or 45 yd
.
© Harcourt
7.
LESSON 21.2
Name
Measure Fractional Parts
MATERIALS scissors, paper, inch ruler
You can make the following paper model to
help identify the marks on a ruler.
• Cut a long strip of printer paper. Label
the left edge 0 and the right edge 1.
• Fold the strip in half four times.
• Unfold the strip of paper. You should now
have 16 equal sections and 15 folds. Count
over 8 folds from 0 and draw a line on that
fold. Label this line 12.
• Starting again at 0, count over 4 folds and
12 folds. Draw lines on the folds. Label
the first one 14 and the second one 34.
1
2
0
0
1
4
1
2
1
1
3
4
1
1
Use a ruler to measure the length to the nearest 4 inch and to the nearest 8 inch.
1.
2.
3.
4.
Draw a chain of 3 large paper clips. Make each
paper clip 134 inches long.
6.
What is the length of your paper clip
chain measured to the nearest 14 inch?
© Harcourt
5.
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RW115
LESSON 21.3
Name
Algebra: Change Linear Units
To change units, think which measurement is larger. Then
think about how the two units relate.
2 yd 24 in. ft
ft
Yards are larger than feet. There
are 3 feet in every yard. To find the
number of feet in 2 yards, multiply.
Inches are smaller than feet. One
foot is 12 inches. To find the
number of feet for 24 inches, divide.
236
24 12 2
So, 2 yd 6 ft.
Larger Unit to Smaller Unit
So, 24 in. 2 ft.
Smaller Unit to Larger Unit
Number of Miles 1,760 Yards
Number of Inches 12 Feet
Number of Miles 5,280 Feet
Number of Inches 36 Yards
Number of Yards 3 Number of Yards 36 Number of Feet 12 1.
Feet
Inches
Inches
Number of Feet 3 Yards
Number of Yards 1,760 Number of Feet 5,280 Miles
Miles
List the linear units of measurement from least to greatest.
Think about how to change the unit. Write either larger unit to
smaller unit or smaller unit to larger unit. Write multiply or
divide to tell how to change the units.
9 feet to 3 yards
3.
2 miles to 10,560 feet 4. 36 inches to 1 yard
© Harcourt
2.
Change the unit.
5.
3 yd RW116
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ft 6. 72 in. ft 7. 3 mi ft 8. 300 ft yd
LESSON 21.4
Name
Capacity
To change units of capacity, think which unit is larger.
Then think about how the two units relate.
Smaller Unit
Larger Unit
2 cups 1 pint
2 pints 1 quart
4 quarts 1 gallon
When changing from larger units to
smaller units, multiply.
When changing from smaller units
to larger units, divide.
2 gal 6c
qt
248
So, 2 gallons equals 8 quarts.
pt
623
So, 6 cups equals 3 pints.
Find the equivalent measurement by drawing a picture.
12 c pt
2.
4 qt pt
3.
2 gal c
© Harcourt
1.
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RW117
LESSON 21.5
Name
Weight
Lightest Unit
Ounce (oz)
Heaviest Unit
1 Pound (lb) 16 oz
1 Ton (T) 2,000 lb
To change units of weight, think which unit is heavier.
Then think about how the two units relate.
3 lb 32 oz oz
lb
Ounces are lighter than pounds.
One pound is 16 ounces.
Multiply 3 by 16.
Pounds are heavier than ounces.
There are 16 ounces in a pound.
Divide 32 by 16.
3 16 48
32 16 2
So, 3 pounds equals 48 ounces.
So, 32 ounces equals 2 pounds.
Think about how to change the unit. Write either lighter unit to
heavier unit or heavier unit to lighter unit. Write multiply or divide
to tell how to change the units.
1.
2 tons to 4,000 pounds 2. 64 ounces to 4 pounds 3. 1 pound to 16 ounces
Circle the more reasonable measurement.
5.
300 T or 300 lb
6.
4 oz or 4 lb
© Harcourt
4.
12 oz or 12 lb
Write the equivalent measurement.
7.
3 lb RW118
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oz
8.
7 T
lb
9.
50,000 lb T
LESSON 21.6
Name
Problem Solving Strategy
Compare Strategies
You can use many different strategies to solve a problem.
Sometimes one strategy is more helpful or easier to
understand than another.
Charlotte is making a household cleaner. She mixes
together 3 gallons of water and one quart of ammonia.
She is going to pour the cleaner into some quart-sized
bottles. How many bottles can she fill?
You can use the strategies draw a picture and make a table
to solve this problem.
Draw a Picture
Make a Table
}
water
ammonia
gallons
1
2
3
quarts
4
8
12
The picture shows that the cleaner will
be made with 13 quarts of liquid.
The table shows that 3 gallons 12 quarts
So, Charlotte can fill 13 bottles.
12 quarts water 1 quart
ammonia 13 total quarts.
So, Charlotte can fill 13 bottles.
Choose a strategy to solve.
© Harcourt
1.
Sheri is making a household
cleaner. She mixes 3 pints of
bleach and 2 quarts of water.
How many one-pint squeeze
bottles can she fill?
2.
Bethany is putting a border
around the trees in her yard.
Each tree needs 12 feet of
border. She buys 20 yards of
border. How many trees can she
surround?
Reteach
RW119
LESSON 22.1
Name
Linear Measure
The centimeter, decimeter, and meter are units of length.
1 cm
The width of a large
paper clip is about 1
centimeter (cm).
1 dm
1m
The width of a
computer disk is
about 1 decimeter (dm).
The width of a door
is about 1 meter (m).
Use the references above to help you decide if the measurement
is close to a cm, dm, or m.
1.
the thickness
of a calculator
2.
the width of
a small
paperback
book
3.
the height of a counter
5.
the thickness
of a workbook
4.
6.
the length of a
large dog
the height of
a big apple
7.
width of a softball
a.
1 cm
b.
1 dm
c.
1m
8.
width of a basketball
a.
25 cm
b.
25 dm
c.
25 m
9.
height of an average ceiling
a.
3 cm
b.
3 dm
c.
3m
length of a bed
a.
200 cm
b.
200 dm
c.
200 m
10.
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© Harcourt
Choose the most reasonable unit of measure. Write a, b, or c.
LESSON 22.2
Name
Algebra: Change Linear Units
You have to multiply or divide when you change units.
Multiply when you change from a larger unit to a smaller unit.
3 meters ? dm
3 meters ? cm
Larger Unit
1 Meter
3 m 10 30 dm
3 m 100 300 cm
Smaller Unit
100 Centimeters
10 Decimeters
10
10
Divide when you change from a smaller unit to a larger unit.
300 cm ? dm
300 cm ? m
Smaller Unit
100 Centimeters
300 cm 10 30 dm
300 cm 100 3 m
10 Decimeters
Larger Unit
1 Meter
10
10
Circle the larger unit.
1.
meter or
decimeter
2.
centimeter or
decimeter
3.
meter or
centimeter
Would you multiply by 10 or by 100 to change the larger units
to the smaller units? Write 10 or 100.
4.
11 m ? cm
5.
20 dm ? cm
6.
8 m ? dm
Would you divide by 10 or by 100 to change the smaller units to
the larger units? Write 10 or 100.
7.
180 cm ? dm
8.
200 cm ? m
9.
800 dm ? m
© Harcourt
Complete the table.
Meters
10.
9
13.
17
Centimeters
900
100
11.
12.
Decimeters
1,000
170
120
1,200
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RW121
LESSON 22.3
Name
Capacity
You can measure capacity, or how much liquid a container can
hold when it is filled, by using the units milliliter (mL) or liter (L).
A box the size of a
base-ten ones unit
holds 1 milliliter
(mL).
1,000 boxes the
size of a ones unit
hold a liter (L).
A small springwater bottle
also holds
1,000 mL,
or 1 L.
Use the pictures above to help you choose the equivalent
measurement. Circle a, b, or c.
3 liters ? milliliters
a.
2.
3,000 mL
50 L
b.
c.
5,000 mL
4,000 mL
c.
400 mL
10 mL
c.
100 mL
3,000 mL
c.
30 mL
200 mL
c.
2,000 mL
5L
4 mL
b.
1,000 mL
b.
3 L ? milliliters
a.
6.
c.
1 L ? milliliters
a.
5.
300 mL
4 liters ? milliliters
a.
4.
b.
5 liters ? milliliters
a.
3.
30 mL
300 mL
b.
2 L ? milliliters
a.
20 mL
b.
© Harcourt
1.
Write the equivalent measurement.
7.
4L
mL
8.
1L
mL
9.
8L
mL
10.
2L
mL
11.
5L
mL
12.
3L
mL
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LESSON 22.4
Name
Mass
You can measure mass with the units grams (g) and
kilograms (kg).
A base-ten ones unit has a
mass of about1 gram (g).
100
100
100
100
100
100
100
100
100
100
If you put 1,000 ones units in a
bag, the bag would have a mass
of 1 kilogram (kg), or 1,000
grams.
Or you can think of a
large container of
oatmeal. It has a mass
of about 1 kg, or 1,000 g.
Circle each object that has a mass greater than 1 kg.
1.
an airplane
2.
a strand of hair
3.
a pencil
4.
a bed
5.
a full backpack
6.
a computer disk
Choose the more reasonable unit of measure. Write g or kg.
7.
a shirt
9.
a refrigerator
10.
a big bag of pet food
a baby
12.
a pair of glasses
11.
8.
a photograph
Circle the more reasonable unit of measure.
spool of thread a. 10 g
b.
10 kg
14.
car tire
a.
20 g
b.
20 kg
15.
bag of flour
a.
5g
b.
5 kg
16.
hammer
a.
2g
b.
2 kg
17.
scarf
a.
25 g
b.
25 kg
© Harcourt
13.
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RW123
LESSON 22.5
Name
Problem Solving Strategy
Draw a Diagram
Row1
Row 2
Row 3
Row 4
Megan is planting rows of watermelon
plants. The plants have to be spaced
2 meters apart. Her garden is 8 meters
by 8 meters. She will start her first row
1 meter in from the edge of the garden.
How many rows of watermelon plants
can she plant?
• To solve, use grid paper to draw
a diagram of the garden. Have the
length of a square represent 1 meter.
• Since each square represents 1 meter,
start the first row 1 space from the
edge. Draw the next row 2 meters or
2 spaces over. Continue drawing
rows until no more can fit.
=1 meter
=1 meter
So, Megan can plant 4 rows.
Draw a diagram to solve.
Megan’s watermelon plants need to be spaced 2 meters
away from every other watermelon plant. She will plant
the first plant in each row 1 meter from the edge of the
garden. How many watermelon plants can Megan fit in
her 8 meter by 8 meter garden?
2.
Tom wants to plant a watermelon garden like Megan’s.
He has a square area that is 12 meters by 12 meters.
How many watermelon plants can he plant?
3.
If Megan and Tom plant marigold plants every 1 meter
around the perimeter of their gardens starting at a
corner, how many marigold plants will they plant?
© Harcourt
1.
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LESSON 23.1
Name
Temperature: Fahrenheit
You can find the temperature by reading a thermometer.
A thermometer is like a vertical number line. The scale of a
thermometer is usually marked in intervals of 5° or 10°.
As it gets warmer, the alcohol in the thermometer expands, and
we say the temperature rises.
As it gets colder, the alcohol in the thermometer contracts, and
we say the temperature falls.
Most household thermometers measure temperatures from
about 20°F to 120°F. This thermometer reads 80°F.
3.
112°F
Use skip-counting to find the change in temperature.
20°F to 40°F
4.
5. 30°F to 75°F
6.
0°F to 40°F
© Harcourt
Fill in the thermometer to show the given temperature.
1.
60°F
2.
30°F
Estimate the temperature of each of the following.
7.
a cup of hot cocoa
8.
a hot summer day
9.
a warm bath
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RW125
LESSON 23.2
Name
Temperature: Celsius
The customary unit for measuring temperature is
degrees Fahrenheit (°F).
The metric unit for measuring temperature is
degrees Celsius (°C).
Water freezes at 0°C. Water boils at 100°C. Room
temperature is about 20°C.
The greater the temperature, the warmer it is.
So, 80°C is warmer than 72°C because 80 > 72.
Temperature can be compared
by using subtraction. So, 80°C is
8° warmer than 72°C.
80°
– 72°
8°
Compare the temperatures. Write or to show which
is warmer.
1.
90°
70°
50°
30°
10°
10°
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
80°
60°
40°
20°
0°
2.
90°
70°
50°
30°
10°
10°
°C
50°C
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
80°
60°
40°
20°
0°
3.
90°
70°
50°
30°
10°
10°
10°C
80°
60°
40°
20°
0°
°C
°C
40°C
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
30°C
80°C
70°C
Write the temperatures. Write or to show which is warmer.
Find the change in temperature.
4.
90°
70°
50°
30°
10°
10°
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
°C
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80°
60°
40°
20°
0°
5.
90°
70°
50°
30°
10°
10°
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
°C
80°
60°
40°
20°
0°
6.
90°
70°
50°
30°
10°
10°
90°
70°
50°
30°
10°
10°
80°
60°
40°
20°
0°
°C
80°
60°
40°
20°
0°
LESSON 23.3
Name
Negative Numbers
A number line can help you to compare two numbers.
The number that is to the right is the greater number.
Compare 5 and 4. Write a number sentence
using , , or .
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Since 4 is to the right of 5, 4 5 or, 5 4.
Locate each pair of numbers on the number line by drawing dots.
Then write a number sentence using or to compare them.
1.
1 and 3
2.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
3.
2 and 7
4 and 2
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
4.
1 and 3
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Draw dots for each number. Then order the numbers from least
to greatest.
5.
5, 7, 2
6.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
7.
4, 2, 1, 0
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
0, 3, 4, 1
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
8.
4, 4, 2
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
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RW127
LESSON 23.4
Name
Problem Solving Skill
Make Generalizations
Thomas and Tina are planning a trip. A travel agent has
given them some information about two locations.
Temperature
(in °F)
Average Monthly Temperature
Month
Thomas and Tina can make some generalizations about the
weather in each location. A generalization is a broad
statement made after seeing some specific information.
Thomas makes the generalization that Skitown is a colder
location than Beachville.
Use the graph to answer the questions.
Number of Points
Ted’s and Al’s Bowling Scores
Ted
1.
What was Ted’s bowling score
in Week 3?
2.
What was Al’s bowling score in
Week 1?
3.
What generalizations can you
make about Ted’s and Al’s
bowling abilities?
4.
What generalization can you
make about Al’s bowling scores
over the four-week period?
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© Harcourt
Al
LESSON 24.1
Name
Use a Coordinate Grid
You can find the point (3,4) on the grid.
Step 2 Go straight across to number 3.
Step 3 Go straight up to the line labeled 4.
The path to (3,4) is traced on the grid.
Always go straight across first,
and then go straight up.
y-axis
Step 1 Start at 0.
8
7
6
5
4
3
2
1
(3,4)
0 1 2 3 4 5 6 7 8
x-axis
Remember: Across begins with an A,
and A comes first in the alphabet.
Complete.
1.
The numbers in an
represent
the number of spaces you move across and up to
locate a point on the grid.
Trace the path from zero to each of the following
points on the grid. Plot each point.
(3,4)
3.
(5,6)
4.
(2,3)
5. (6,7)
6.
(4,5)
7. (7,8)
10
9
8
7
6
5
4
3
2
1
y-axis
2.
0 1 2 3 4 5 6 7 8 9 10
x-axis
Plot each ordered pair on the grid.
Connect the points to make a figure.
(2,3), (4,1), (7,1), (9,3), (7,5), (4,5)
9.
Name the figure you drew.
10
9
8
7
6
5
4
3
2
1
y-axis
© Harcourt
8.
0 1 2 3 4 5 6 7 8 9 10
x-axis
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RW129
LESSON 24.2
Name
Length on the Coordinate Grid
You can find the distance between two points on a
coordinate grid by counting units or by using subtraction.
When finding the horizontal distance between two points,
look at the x-coordinates. When finding the vertical distance
between two points, look at the y-coordinates.
What is the horizontal distance from point (2,3) to (9,3)?
y-axis
Step 1 Graph the ordered pairs (2,3) and (9,3) on the
coordinate plane to the right. Connect the two points.
Step 2 Count the units between the two points or
subtract the x-coordinates.
9 2 7
So, the horizontal distance between the two
points is 7 units.
10
9
8
7
6
5
4
3
2
1
(2, 3)
1 2 3 4 5 6 7
0 1 2 3 4 5 6 7 8 9 10
x-axis
Find the length of each line segment.
5
4
3
2
1
RW130
10
9
8
7
6
5
4
3
2
1
(4, 8)
(4, 3)
(5, 6)
(5, 2)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
x-axis
x-axis
10
9
8
7
6
5
4
3
2
1
4.
10
9
8
7
6
5
4
3
2
1
y-axis
y-axis
3.
2.
y-axis
10
9
8
7
6
5
4
3
2
1
y-axis
1.
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
x-axis
x-axis
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(9, 3)
LESSON 24.3
Name
Use an Equation
An equation may have 2 variables.
The equation y 5x can be written y 5 x.
You can think of this equation as a rule that says:
To find the value of y, you must multiply x by 5.
If the value of x is 3, the value of y 5 3 or 15.
If you need to find several values of y, you can use a
function table to organize the data. You choose any
number as the value of x. This is called the input.
Then you use the equation as a rule to find the output.
The output is the value of y.
Example: y 5x 2
You choose 1 as the value of x.
You use the rule to find y 7.
You choose 3 as the value of x.
You use the rule to find y 17.
You choose 6 as the value of x.
You use the rule to find y 32.
Find the value of y when x 4.
1. y 6x
3.
y 7x 10
Input
x
Rule
y 5x 2
Output
y
1
y (5 1) 2
7
3
y (5 3) 2
17
6
y (5 6) 2
32
2.
y 9x 3
4.
y 3x 8
Use the equation to complete each function table.
5. y 4x
6. y 3x 41
Input
x
1
2
3
Rule
y 4x
Output
y
Input
x
4
5
6
Rule
Output
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RW131
LESSON 24.4
Name
Graph an Equation
To graph an equation on a coordinate grid, you first make a
function table to find values of x and y. Then you write the
values of x and y as ordered pairs and graph the pairs on the
grid. Finally, you draw a line to connect the points.
Graph the equation y 2x 1.
x
y
1
1
2
3
3
5
4
7
5
9
y-axis
Make a function table.
Write the values of x and y as ordered pairs.
(1,1), (2,3), (3,5), (4,7), (5,9)
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x-axis
Graph the ordered pairs on the coordinate grid. Draw a line to
connect the points.
Graph the equation y x 3.
1.
Complete the function table.
x
y
1
2
3
4
5
Write the values of x and y as ordered pairs.
3.
Graph the ordered pairs on the coordinate grid.
Draw a line to connect the points.
y-axis
2.
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x-axis
Graph the equation y 6 x.
Complete the function table.
1
2
3
4
5
5.
Write the values of x and y as ordered pairs.
6.
Graph the ordered pairs on the coordinate grid.
Draw a line to connect the points.
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10
9
8
7
6
5
4
3
2
1
0
© Harcourt
x
y
y-axis
4.
1 2 3 4 5 6 7 8 9 10
x-axis
LESSON 24.5
Name
Problem Solving Skill
Identify Relationships
You can identify relationships between x and y by looking at a function table and
writing a rule for the values of x and y.
Look at the function table below and identify the relationship
between the input x and the output y.
Input, x
Output, y
1
3
2
6
3
9
4
12
5
15
6
18
7
21
8
24
9
27
10
30
Rule: Multiply x by 3 to get y.
You can also identify relationships by looking at a coordinate grid.
y-axis
Look at the grid below. What is the relationship
between the x values and the y values?
Step 1 Write the ordered pairs in a table.
Input, x
Output, y
1
3
2
4
3
5
4
6
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x-axis
Rule: Add 2 to the value of x to get y.
1.
Describe the relationship
between x and y.
2.
Use the graph to complete the function table.
Then describe the relationship between x and y.
Input, x
Output, y
1
2
Input, x
Output, y
3
4
1
6
2
12
3
18
y-axis
© Harcourt
Step 2 Write a rule.
4
24
5
30
10
9
8
7
6
5
4
3
2
1
0
1 2 3 4 5 6 7 8 9 10
x-axis
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RW133
LESSON 25.1
Name
Lines, Rays, and Angles
F
G
•
A line is straight. It continues in both directions.
It does not end.
•
line FG
A
A ray is part of a line and has one endpoint.
You may think of the sun’s rays, which start at
the sun (endpoint) and go on forever into space.
B
•
•
ray AB
An angle is formed when two rays have the
same endpoint.
E
•
There are three types of angles:
•
D
angle C
•
C
A right angle forms a square corner.
An acute angle is an angle with a measure less than a right angle.
An obtuse angle is an angle with a measure greater than a right angle.
Obtuse begins
with an O. Think of
an Open angle.
HINT:
•
•
•
right angle
acute angle
obtuse angle
Write the name of each figure.
1.
E
F
•
2.
Y
Z
•
•
3.
B
4.
•
•
•
C
•
D
What kind of angle is each? Write acute, right, or obtuse.
•
6.
7.
•
© Harcourt
•
5.
8.
9.
•
•
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10.
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•
LESSON 25.2
Name
Line Relationships
•
D
F
•
Lines that intersect to form 4 right angles are
perpendicular lines. Lines AB and CD are
perpendicular lines.
G
•
•E
•
A
C
•
Lines that never intersect are parallel lines.
Lines GH, and LM, and RS are
parallel lines.
D
•
•
B
•
All lines that cross each other are
intersecting lines. Lines DE and
FG intersect.
H
•
M
S•
G
•L
•
R
•
•
For 1–3, draw perpendicular lines.
2.
•
•
N
•
•
B
Y
•
•
M
•
•
A
3.
•
1.
Z
•
For 4–5, use the following map.
way
© Harcourt
eS
t.
St.
chuset
Lak
Massa
Foster
Tufts S
Adams
S
t.
t.
d
Broa
ts Ave
.
4.
Name the streets parallel to Adams Street.
5.
Name the streets perpendicular to Massachusetts Avenue.
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RW135
LESSON 25.3
Name
Congruent Figures and Motion
Congruent figures have the same size and shape. They can be
in different positions and still be congruent. Three ways to
move a figure are flip, slide, and turn.
• You can slide a figure in any direction.
The figure looks exactly the same but
is in a new position.
• You can flip a figure over an imaginary
line. The flipped figure is a mirror image.
•
• You can turn a figure around a point. The
point holds the corner of the figure while
the rest of the figure rotates around the point.
Tell how each figure was moved. Write slide,
flip, or turn.
1.
2.
3.
4.
Use each original figure to draw figures to show a slide, flip, and turn.
Original
Slide
Flip
Turn
© Harcourt
5.
6.
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LESSON 25.4
Name
Symmetric Figures
A figure has rotational symmetry if it can be turned
about a central point and still look the same in at least
two different positions.
This figure has rotational symmetry.
Original
Position
Figure
Matches
Turning
Figure
Matches
Turning
Trace and cut out the figure below. Fold it along the dotted line.
When you unfold the figure, you can
see that one side is a reflection of the
other. When a figure can be folded
about a line so that its two parts are
identical, the figure has line symmetry.
Trace the following figures and turn them to determine whether
each figure has rotational symmetry. Write yes or no.
1.
2.
3.
© Harcourt
Is the dotted line a line of symmetry? Trace and cut out each figure.
Fold to check. Write yes or no.
4.
5.
6.
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RW137
Name
LESSON 25.5
Problem Solving Strategy
Make a Model
You can make a model to solve some problems.
Read the following problem.
1.
Underline what you are asked to find.
2.
What strategy can you use to solve the problem?
3.
How can you use the strategy to solve the problem?
4.
Solve the problem with the grid at the right.
Make a model and solve.
5.
Mike wants to make a larger
picture of the figure at the right.
Use the grid to help Mike
make a larger picture.
RW138 Reteach
© Harcourt
Hannah wants to make a poster
for her room. She wants to make
a larger picture of the geometric
design at the right. How can she
make a model to help her?
LESSON 26.1
Name
Perimeter of Polygons
A polygon is a closed plane figure with straight sides.
Polygons are named by the number of sides or number of
angles.
Here are some polygons.
TRIANGLE
QUADRILATERAL
PENTAGON
HEXAGON
OCTAGON
3 sides
3 angles
4 sides
4 angles
5 sides
5 angles
6 sides
6 angles
8 sides
8 angles
2 cm
The perimeter is
the distance
around a polygon.
Add the lengths of
all the sides to find
the perimeter.
P 4 cm 6 cm 2 cm 5 cm
P 17 cm
5 cm
4 cm
So, the perimeter is 17 cm.
6 cm
Find the perimeter. Name the polygon.
1.
2.
2 cm
2 cm
3.
2 cm
2 cm
2 cm
2 cm
4 cm
3 cm
4 cm
2 cm
2 cm
4 cm
4.
1 cm
5.
1 in.
4 in.
3 cm
6.
5 in.
7 in.
8 in.
© Harcourt
3 in.
7 in.
3 in.
5 in.
6 in.
5 in.
3 in.
1 in.
12 in.
10 in.
7 in.
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RW139
LESSON 26.2
Name
Estimate and Find Perimeter
Perimeter is the distance around a figure. You already
know that you can find the perimeter by adding the
lengths of the sides of a figure.
7 in.
3 in. 2 in.
3 in.
1 in.
1 3 2 7 3 10 = 26
The perimeter is 26 inches.
10 in.
Rectangles and squares have two pairs of sides that are
the same length. You can find the perimeter of a rectangle
by multiplying and adding. For squares, multiply the
length of one side by 4 to find the perimeter.
10 in.
9 ft
3 ft
3 ft
10 in.
10 in.
9 ft
(2 3) (2 9) =
6 18 = 24 ft
10 in.
4 10 = 40 in.
Find the perimeter.
1.
2.
3 cm
3 cm
3cm
5m
3m
3 cm
3.
3m
4.
4 ft
4 ft
3 in.
4 in.
4 ft
5m
3 in.
4 ft
5.
6.
3m
3m
7.
9 ft
9 ft
9 ft
3 yd
5 yd
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9 ft
8.
10 in.
3 in.
2 yd 3in.
8 yd
3m
RW140
4 in.
10 in.
LESSON 26.3
Name
Estimate and Find Area
Area is the number of square units needed to cover a flat surface.
You can find the area of each figure on the geoboard by
counting the number of square units.
1 2
3 4
5 6
1
unit
2
1
unit
2
1
The area is
6 square units.
The area is
2 square units.
1
2 3 4
5 6 7
8
The area is
8 square units.
You can find the area of a rectangle by using multiplication.
You multiply the length times the width of a rectangle to
find the area. Write your answer in square units, such as
sq ft or sq yd.
l
w
=
A
length
width
=
area
7 ft
3 ft
=
21 sq ft
7 ft
3 ft
Find the area.
1.
2.
5.
3.
6.
6 in.
3 in.
4.
7.
4 yd
3m
8m
4 yd
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RW141
LESSON 26.4
Name
Relate Area and Perimeter
Perimeter is the distance around a figure. Perimeter is
measured in linear units.
Area is the number of square units needed to cover a flat
surface. Area is measured in square units.
Figures with the same area can have different perimeters.
Area 20 sq units
Perimeter 22 units
Area 20 sq units
Perimeter 24 units
Figures with the same perimeter can have different areas.
Area 25 sq units
Perimeter 28 units
Area 16 sq units
Perimeter 28 units
Find the area and perimeter of each figure.
1.
2.
3.
Find the area of each figure. Then draw another figure
that has the same area but a different perimeter.
5.
© Harcourt
4.
area RW142
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area LESSON 26.5
Name
Relate Formulas and Rules
You can use formulas, or rules, to find unknown lengths when
you know the area or perimeter and one or more sides.
Formula: Perimeter a b c
EXAMPLE A
EXAMPLE B
Find the
unknown length.
The perimeter
is 22 inches.
6 in.
c
b
7 in.
Pabc
22 a 6 7
22 a 13
22 13 a
9a
The length is 9 inches.
Find the
unknown length.
The perimeter
is 94 meters.
22 m
a
34 m
c
b
a
Pabc
94 a 22 34
94 a 56
94 56 a
38 a
The length is
38 meters.
Formula: Area l w
EXAMPLE C
EXAMPLE D
Find the
unknown width.
The area is 64
square yards.
16 yd
Alw
64 16 w
64 16 w
4w
The width is 4 yards.
Find the
unknown length.
The area is 44
square miles.
4 mi
w
Alw
44 l 4
44 4 w
11 w
The length is
11 miles.
l
Find the unknown length in each triangle.
1.
2.
5 ft
16 cm
?
3.
4 ft
19 cm
Perimeter 47 cm
© Harcourt
?
?
3 mi
3 mi
Perimeter 12 ft
Perimeter 9 mi
Complete for each rectangle.
4.
Area 144 sq. m
Length 12 m
Width m
5.
Area 84 sq. in.
Width 6 in.
Length 6.
in.
Area 360 sq. cm
Length 15 cm
Length Reteach
cm
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LESSON 26.6
Name
Problem Solving Strategy
Find a Pattern
4 cm
3 cm
You can find patterns to solve long problems quickly.
Read the following problem.
Kyle has several different computer chips. Some of his
computer chips are twice as long as others. He wants to
know how the area changes as the length of the computer
chips are doubled. How do the areas change?
1.
Underline what you are asked to find.
2.
What strategy can you use to solve the problem?
3.
Complete.
Width
Length
Computer
chip A
2
2
Computer
chip C
3
4
Computer
chip B
2
4
Computer
chip D
3
8
Width
Length
Computer
chip E
4
4
Computer
chip G
6
4
Computer
chip F
4
8
Computer
chip H
6
8
Width Length
Area
Width Length
Area
Area
Use find a pattern and solve.
© Harcourt
4.
Area
8 cm
3 cm
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LESSON 27.1
Name
Faces, Edges, and Vertices
Polygons are two-dimensional
figures because they have length
and width.
Solid figures are three-dimensional
figures because they have length,
width, and height.
vertex
edge
face
You can identify solid figures by the
plane figures that are their faces,
and by the number of faces, edges,
and vertices they have.
Cube
Rectangular
Prism
Triangular
Prism
Triangular
Pyramid
Square
Pyramid
Some solid figures have curved surfaces.
Cylinder
Cone
Sphere
© Harcourt
Which solid figure do you see in each?
1.
2.
3.
4.
5.
6.
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RW145
LESSON 27.2
Name
Patterns for Solid Figures
You can make a three-dimensional figure
from a two-dimensional pattern called a net.
Look at the net at the right. Can you see how it
forms the three-dimensional rectangular prism?
Take another look at the net, and count the faces.
4
Since the three-dimensional rectangular prism has
6 faces, the net must also have 6 faces.
1 2 3 5
6
Draw a line from the three-dimensional figure to its net.
2.
3.
4.
5.
6.
7.
8.
© Harcourt
1.
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LESSON 27.3
Name
Estimate and Find Volume of Prisms
Cubic unit
The measure of the space that a solid figure occupies
is called volume. Volume is measured in cubic units.
You can use what you know about unit cubes to
estimate volume.
These are two ways to find the volume of this
rectangular prism.
One way: You can count the number of cubic units.
Another way: You can find the length, width, and height
of the rectangular prism in cubes. Then multiply to find
the volume in cubic units.
Step 2
Step 3
Step 4
Find the length.
Count the number
of cubes in one
row.
Find the width.
Count the number
of rows in one
layer.
Find the height.
Count the number
of layers in the
prism.
Multiply length
width height
to find the volume.
↔
Step 1
2
4 →
The length is 4
cubes.
2
→
The width is 2
cubes.
The height is 2
cubes.
4 cubes 2 cubes
2 cubes 16
cubes.
So, the volume is
16 cubic units.
Find the volume.
2.
3.
© Harcourt
1.
4.
What is the volume of a solid figure that is 4 cubes by
7 cubes, by 9 cubes?
Reteach
RW147
LESSON 27.4
Name
Problem Solving Skill: Too Much/Too Little Information
The cost of a box of 400 cotton swabs is $4.35. Each box 10cm
has a height of 4 cm, a length of 18 cm, and a width of 4cm
10 cm. What is the volume of the box of cotton swabs?
18cm
Step 1
Step 2
What does the problem ask you to find?
What information do you need to solve
the problem?
Step 3
Step 4
Decide if there is too much or not
enough information. Explain.
Solve the problem, if possible.
1.
There are 12 boxes of paper
in a case. Each case is 6 feet
in length and 4 feet in height.
What is the volume of each case?
2.
There are 24 tops in a package.
Each package has a height of 5
inches, a width of 6 inches, and
a length of 10 inches. What is
the volume of one package?
3.
Ms. Maguire has a cardboard
box that is 1 meter wide and 4
meters long. What is the volume
of Ms. Maguire’s cardboard box?
4.
There are 3 people in the pool.
The pool is 10 feet deep, 50 feet
long, and 20 feet wide. What is
the volume of the pool?
RW148
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© Harcourt
Decide if the problem has too much information or too little information.
Then solve the problem, if possible.
LESSON 28.1
Name
Turns and Degrees
The unit used to measure an angle is a degree (°).
There are 360° in a circle. The circle at the right shows
and XB
. The angle between them is 20°.
two rays, XA
A
X
B
Turning XB around the circle results in angles of different
sizes. A complete turn around the circle is 360°.
Angle measurements can be related to a complete turn (360°).
3
4
a turn (270°)
1
2
1
4
a turn (180°)
a turn (90°)
1
4
1
2
3
4
Tell whether the rays on the circle show a , , or turn.
1.
2.
3.
© Harcourt
Tell whether the rays have been turned 90°, 180°, 270°, or 360°.
4.
5.
6.
7.
8.
9.
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RW149
LESSON 28.2
Name
Measure Angles
Use a protractor to measure the size of the opening of
an angle. The scale on a protractor is marked from 0°
to 180°.
Measure angle ABC with a protractor.
Step 1
Step 2
Step 3
Place the center of the
protractor on the
vertex of the angle.
Line up the center point
and the 0° mark on the
protractor with one ray
of the angle.
Read the measure
of the angle
where the other ray
passes through
the scale of
the protractor.
A
0
12
80
70
60
50
20
C
13
0
0
12
80
70
Write angle
measure in (°).
60
50
14
40
0
20
180 170
160
30
vertex
10
13
0
B
C
0
0
12
110
100
90
A
80
70
60
50
14
40
0
B
90
0
10
180 170
160
30
15
0
40
A
100
110
15
0
0
90
14
0
13
100
110
20
180 170
160
30
15
0
ray
10
0
B
The measure of
ABC 60°.
Use a protractor to measure the angle between the two rays.
Remember to line up the center mark and the 0° mark on the
protractor with one of the rays.
1.
2.
3.
A
C
B
A
B
ABC A
B
C
ABC 4.
C
ABC 5.
6.
A
A
A
B
ABC RW150 Reteach
C
B
ABC C
B
ABC C
C
LESSON 28.3
Name
Circles
A circle is named by its center point. All the
points on the circle are the same distance
from the center point.
A chord is a line segment that has its endpoints
on the circle.
•F
Chord EF
E
C
•
•
diameter CD
A
•D
•
radius
AB
A radius of a circle is a line segment from the
center to any point on the circle.
B•
Circle A
A diameter of a circle is a chord
that passes through the center.
For 1–4, use Circle L and a centimeter ruler.
1.
2.
The center of the circle is point
.
Name each radius of the circle shown.
Each radius measures
cm.
M
•
L
N
•
•O
•
Circle L
3.
A diameter of the circle is line segment
.
The diameter measures
A diameter is also a
cm.
.
4.
Name three points on the circle.
5.
Draw a circle. Label the center point R.


Draw a radius RS. Draw a diameter TV.
S
•
R
Measure the diameter.
Reteach
•
•
V
•
Measure the radius.
T
RW151
LESSON 28.4
Name
Circumference
The measure of the distance around a circle is its
circumference. Since a circle is not a polygon, its
circumference cannot be measured easily with a ruler.
The circumference of a circle is about 3 times the
diameter. If you know the diameter of the circle,
you can estimate its circumference.
9 in.
Estimate the circumference of the circle.
diameter 9 inches
The circumference is about 3 9 in., or about 27 in.
Estimate each circumference.
2.
5.
6.
13 ft
12 in.
7.
8.
9.
11.
52 in.
RW152 Reteach
9 cm
6m
22 yd
10.
16 mi
11 m
2 cm
4.
3.
8 cm
12.
63 ft
51 mi
© Harcourt
1.
LESSON 28.5
Name
Classify Triangles
Triangles can be classified according to the lengths of
their sides.
6 in.
5 in.
5 in.
5 in.
5 in.
5 in.
5 in.
7 in.
6 in.
A triangle with
3 sides congruent is
an equilateral triangle.
A triangle with
2 sides congruent is
an isosceles triangle.
A triangle with
no sides congruent is
a scalene triangle.
Classify each triangle by the lengths of its sides. Write isosceles,
scalene, or equilateral.
1.
2.
6 in.
3.
8 ft
6 in.
10 ft
6 in.
4.
3 mi
3 mi
5.
6.
1 ft
1 ft
4 cm
6 cm
3 cm
1 ft
7 cm
© Harcourt
3 mi
8 ft
4 cm
4 cm
11 yd
7.
8.
9.
5 cm
4 cm
10 yd
4 in.
10 yd
2 in.
3 in.
6 cm
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RW153
LESSON 28.6
Name
Classify Quadrilaterals
A quadrilateral has four sides and four angles.
Quadrilaterals are grouped by their angles, by the length
of their sides, and by any parallel sides. The table will
help you compare different quadrilaterals.
Grouping
Side Lengths
Parallel Sides
Angles
General
Quadrilaterals
4 different lengths
0 pairs
parallel sides
angles vary
Trapezoids
lengths can vary
1 pair
parallel sides
angles vary
Parallelograms
opposite sides
congruent
2 pairs
parallel sides
2 acute and 2
obtuse, or 4 right
Rectangles
opposite sides
congruent
2 pairs
parallel sides
4 right
Rhombuses
4 sides congruent
2 pairs
parallel sides
2 acute, 2 obtuse,
or 4 right
Squares
4 sides congruent
2 pairs
parallel sides
4 right
Match the drawing with as many terms as possible.
1.
2.
3.
4.
5.
6.
rhombus
D. rectangle
B.
trapezoid
E. parallelogram
C.
A.
square
F. general quadrilateral
7.
How are rhombuses and squares alike?
8.
How are rhombuses and squares different?
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© Harcourt
For 7–8, use the table above.
LESSON 28.7
Name
Problem Solving Strategy
Draw a Diagram
Ms. Mellot wants her students to sort the figures shown below
according to those that have line symmetry and those that have
rotational symmetry. How should the students sort
the figures?
Figure A
Figure B
Figure C
Figure D
Figure E
1.
What you are asked to do?
2.
What information will you use?
3.
What strategy can you use to solve the problem?
4.
What kind of diagram can you make?
Figure F
A Venn diagram uses circles to show relationships among different
sets of things.
© Harcourt
Use these 2 circles to make a Venn diagram. Label one circle Line
Symmetry and the other Rotational Symmetry. Then sort the figures
by writing the letters A–F in the correct circles.
Line Symmetry
Rotational Symmetry
A, B, D
C, E, F
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RW155
LESSON 29.1
Name
Record Outcomes
When you toss one coin, there are 2 possible
ways the coin can land: heads or tails.
When you toss two coins, there are 4 possible
ways the coins can land.
POSSIBLE WAYS FOR
TWO COINS TO LAND
First Coin
Second Coin
There is 1 way to get 2 heads. →
heads
heads
→
heads
tails
→
tails
heads
There is 1 way to get 2 tails. →
tails
tails
There are 2 ways to get 1 head and 1 tail.
When you spin the pointer on this spinner once, there are 3
possible numbers you can get: 2, 3, or 4.
Complete the table to show all the possible sums you can get if
you spin the pointer twice and add the numbers.
2
3
4
POSSIBLE SUMS FOR TWO SPINS
Second Spin
Sum
1.
2
2
22
2.
2
3
23
3.
2
4.
3
5.
3
6.
3
7.
4
8.
4
9.
4
10.
2
Look at the sums. Which sum is found most often?
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© Harcourt
First Spin
LESSON 29.2
Name
Tree Diagrams
Making a tree diagram can help you determine the possible
outcomes for a probability experiment.
Sharon has a coin and a spinner divided into two sections: red
and yellow. She will toss the coin and spin the pointer on the
spinner. What are the possible outcomes? How many are there?
Spinner
red
yellow
Coin
heads
tails
heads
tails
Outcomes
red and heads
red and tails
yellow and heads
yellow and tails
So, there are 4 possible outcomes.
Complete the tree diagram to solve.
1.
Jerome is conducting a probability experiment with a
coin and a bag of marbles. He has 3 marbles in the
bag: 1 red, 1 purple, and 1 brown. He will replace the
marble after each turn. How many possible outcomes
are there for this experiment? What are they?
Marbles
Coin
Outcomes
heads
red and heads
red
tails
purple
© Harcourt
brown
There are
2.
possible outcomes.
Sarah has 10¢. How many different outcomes of
coins could she have? List the outcomes.
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RW157
LESSON 29.3
Name
Problem Solving Strategy
Make an Organized List
You can use a table to help you make an organized list of all the
possible outcomes of an experiment.
Juanita is making airplane reservations. She can choose a
window seat, a middle seat, or an aisle seat. She can choose
regular or vegetarian meals.
In how many different ways can she plan her flight?
Make a table to help you see all the possible outcomes.
AIRPLANE RESERVATIONS
SEAT
MEAL
window
regular
window
vegetarian
middle
regular
middle
vegetarian
aisle
regular
aisle
vegetarian
1.
Name the choices Juanita has.
2.
How many outcomes are there?
3.
Charisse is choosing new
eyeglasses. She can choose
metal or plastic frames, and can
have clear or tinted lenses. List
all the possible choices.
RW158 Reteach
4.
Scott is redecorating his
room. He can either paint or
wallpaper, and have carpeting,
tile, or rugs. In how many
different ways can he decorate
his room?
© Harcourt
Make an organized list to solve.
LESSON 29.4
Name
Predict Outcomes of Experiments
Look at the spinner.
0
30
Is spinning a number with 2 as a digit a likely or
an unlikely event?
6
24
To find out if an event is likely or unlikely, list all
the numbers you could spin.
18 13
• If the event describes half or more than half of
the numbers, it is likely.
• If the event describes fewer than half of the
numbers, it is unlikely.
Numbers you could spin:
Does it have 2 as a digit?
0
no
6
no
13
no
18
no
24
yes
30
no
The event describes fewer than half of the numbers. So, it is
unlikely you will spin a number with 2 as a digit.
For Problems 1–4, use the spinner above. Decide whether the
event describes each number. Write yes or no.
© Harcourt
0
1.
a number greater than zero
2.
a 2-digit number
3.
a number less than 10
4.
a number divisible by 3
6
13
18
24
30
For Problems 5–8 use the spinner above. Tell whether each
event is likely or unlikely.
5.
a number greater than zero
6.
a 2-digit number
7.
a number less than 10
8.
a number divisible by 3
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RW159
LESSON 30.1
Name
Probability as a Fraction
Probability can be described by a number written as
a fraction.
Probability the number of favorable outcomes
total possible outcomes
Using the spinner at the right, what is the probability of
the pointer stopping on a circle?
• List all the
4 outcomes
possible outcomes.
• Identify the outcomes
1 circle
that are circles.
• Record the probability
as a fraction.
number of circles
Probability of circle 1
4
total number of shapes
Use the spinner at the right. Circle the favorable
outcomes. Then write the probability as a fraction.
The first one has been done for you.
What is the probability of the
pointer stopping on a ball?
2.
What is the probability of the
pointer stopping on something
that is not a ball?
3.
What is the probability
of the pointer stopping
on a football?
4.
What is the probability
of the pointer stopping
on a ball or a baseball bat?
RW160 Reteach
3
5
© Harcourt
1.
LESSON 30.2
Name
More About Probability
A box contains 4 black marbles and
4 white marbles.
Experiment: Shake the box, and pull a marble.
Record the color; then replace the marble.
There are two possible outcomes for this
experiment.
• The marble is black.
• The marble is white.
Tom conducts the experiment 16 times. He predicts that the
outcomes will be 8 black and 8 white marbles. He records
the actual results in a table.
MARBLE EXPERIMENT
Possible outcomes
Predicted frequency
Black
White
8
8
Actual frequency
1. What is the mathematical probability of drawing a white marble?
2. Based on the experiment, what is the probability of drawing
a white marble?
© Harcourt
3. How does this compare to the mathematical probability?
4. If Tom were to pull a marble 20 times, how many white
marbles would he predict if he used mathematical
probability?
Reteach
RW161
LESSON 30.3
Name
Test for Fairness
Fairness in a game means that one player is as likely to win as
another. To see if a game is fair, compare the probabilities of winning
and not winning. A game is fair if the probabilities are equal.
Game Description
12 8
• One player tries to spin even numbers.
• One player tries to spin odd numbers.
Is the game fair if this spinner is used?
probability of spinning an even number 3
5
2
probability of spinning an odd number 5
The probabilities are not equal, so the game is not fair.
11
10
9
Use the spinner above. Write the probabilities. Write fair or not
fair to tell if the game is fair.
Game Description
• One player wins if the pointer
stops on 9 or 12.
• One player wins if the pointer
stops on 8 or 12.
1.
The probability of spinning 9 or 12 is
.
2.
The probability of spinning 8 or 12 is
.
3.
The game is
.
Write the letter of the spinner that makes the game fair.
4.
Game Description
B
C
5.
Game Description
• One player tries to spin
a rectangle.
• One player tries to spin
a triangle.
• One player tries to spin a
shape with 4 or fewer sides.
• One player tries to spin a
shape with 4 or more sides.
Spinner
Spinner
RW162 Reteach
© Harcourt
A
LESSON 30.4
Name
Problem Solving Skill
Draw Conclusions
Daryl, Tat, Judy, and Nora have written their first initials in a
spinner with 8 equal sections to play a game. Read each clue
to help you write the letters on the spinner.
ANALYZE
CONCLUDE
The probability of the pointer stopping
on a D is 38.
3 out of 8 sections have a D in them.
The sections with a T take up 14 of the
spinner.
1
4
of 8 is 2, so 2 sections have a T in
them.
There are the same number of sections
with J as with T.
sections have J in them.
Each remaining section has an N.
sections have N in them.
Use what you concluded to write the letters on the spinner.
1.
Is the spinner fair? Explain.
© Harcourt
ge
purple
oran
ge
d
re
d
re
purple
gre
en
blue
2.
or
an
blue
Kelsey made a spinner with 10 equal sections. She colored
2 opposite sections blue. She colored the section to the left
of each blue section orange. She made 120 of the spinner
green and 120 of the spinner red. The rest of the sections
she colored purple. Did Kelsey make a fair spinner?
gree
n
Read the problem. Draw conclusions from each fact and then
color the spinner to solve. Use crayons.
How many sections are each color?
blue
orange
red
purple
green
Write is or is not to complete the sentence.
3.
The spinner
fair.
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RW163