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UNIT 1: USE WHOLE NUMBERS
Chapter 1: Place Value, Addition, and Subtraction
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Place Value Through Millions ........... RW1
Understand Billions ........................... RW2
Compare and Order Whole
Numbers............................................. RW3
Round Whole Numbers .................... RW4
Estimate Sums and
Differences ........................................ RW5
Add and Subtract Whole
Numbers............................................. RW6
Problem Solving Workshop
Strategy: Work Backward ................. RW7
Chapter 4: Expressions and Equations
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Write Expressions ............................ RW24
Evaluate Expressions ....................... RW25
Properties......................................... RW26
Mental Math: Use the
Properties......................................... RW27
Write Equations............................... RW28
Solve Equations ............................... RW29
Functions.......................................... RW30
Inequalities ...................................... RW31
Problem Solving Workshop
Strategy: Predict and Test ............... RW32
UNIT 2: USE DECIMALS
Chapter 2: Multiply Whole Numbers
2.1
2.2
2.3
2.4
2.5
2.6
Mental Math: Patterns in
Multiples ............................................ RW8
Estimate Products .............................. RW9
Multiply by 1-Digit Numbers .......... RW10
Multiply by Multi-Digit
Numbers........................................... RW11
Problem Solving Workshop
Strategy: Find a Pattern .................. RW12
Choose a Method ............................ RW13
Chapter 3: Divide by 1- and 2-Digit Divisors
3.1
Estimate with 1-Digit
Divisors ............................................. RW14
3.2 Divide by 1-Digit Divisors................ RW15
3.3 Problem Solving Workshop
Skill: Interpret the Remainder ........ RW16
3.4 Zeros in Division .............................. RW17
3.5 Algebra: Patterns in
Division ............................................ RW18
3.6 Estimate with 2-Digit
Divisors ............................................. RW19
3.7 Divide by 2-Digit Divisors................ RW20
3.8 Correcting Quotients ...................... RW21
3.9 Practice Division .............................. RW22
3.10 Problem Solving Workshop
Skill: Relevant or
Irrelevant Information .................... RW23
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Chapter 5: Understand Decimals
5.1
5.2
5.3
5.4
Decimal Place Value ........................ RW33
Equivalent Decimals ........................ RW34
Compare and Order Decimals ........ RW35
Problem Solving Workshop
Skill: Draw Conclusions ................... RW36
Chapter 6: Add and Subtract Decimals
6.1
6.2
6.3
6.4
6.5
Round Decimals ............................... RW37
Add and Subtract Decimals ............ RW38
Estimate Sums and Decimals ......... RW39
Choose a Method ............................ RW40
Problem Solving Workshop Skill:
Estimate or Find Exact Answer....... RW41
Chapter 7: Multiply Decimals
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Model Multiplication by
a Whole Number ............................. RW42
Algebra: Patterns in Decimal
Factors and Products ....................... RW43
Record Multiplication by
a Whole Number ............................. RW44
Model Multiplication by
a Decimal ......................................... RW45
Estimate Products ............................ RW46
Practice Decimal Multiplication ..... RW47
Problem Solving Workshop
Skill: Multistep Problems ................ RW48
6/29/07 11:18:09 AM
Chapter 8: Divide Decimals by Whole Numbers
8.1
8.2
8.3
8.4
Decimal Division .............................. RW49
Estimate Quotients.......................... RW50
Divide Decimals by Whole
Numbers........................................... RW51
Problem Solving Workshop Skill:
Evaluate Answers for
Reasonableness ............................... RW52
UNIT 3: DATA AND GRAPHING
Chapter 9: Data and Statistics
9.1
9.2
9.3
9.4
9.5
Collect and Organize Data ............. RW53
Mean, Median, and Mode .............. RW54
Compare Data ................................. RW55
Analyze Graphs ............................... RW56
Problem Solving Workshop
Strategy: Draw a Diagram .............. RW57
Chapter 10: Make Graphs
10.1 Make Bar Graphs and
Pictographs ...................................... RW58
10.2 Make Histograms ................... .........RW59
10.3 Algebra: Graph Ordered Pairs ........ RW60
10.4 Make Line Graphs ........................... RW61
10.5 Make Circle Graphs ......................... RW62
10.6 Problem Solving Workshop
Strategy: Make a Graph.................. RW63
10.7 Choose the Appropriate Graph...... RW64
UNIT 4: NUMBER THEORY AND
FRACTION CONCEPTS
Chapter 11: Number Theory
11.1 Multiples and the Least
Common Multiple ........................... RW65
11.2 Divisibility ........................................ RW66
11.3 Factors and Greatest
Common Factor ............................... RW67
11.4 Prime and Composite Numbers...... RW68
11.5 Problem Solving Workshop
Strategy: Make an
Organized List ................................. RW69
11.6 Introduction to Exponents ............. RW70
11.7 Exponents and Square Numbers .... RW71
11.8 Prime Factorization ......................... RW72
G5-NL TOC_TEnew.indd 2
Chapter 12: Fraction Concepts
12.1
12.2
12.3
12.4
12.5
Understand Fractions ...................... RW73
Equivalent Fractions ........................ RW74
Simplest Form .................................. RW75
Understand Mixed Numbers .......... RW76
Compare and Order Fractions
and Mixed Numbers........................ RW77
12.6 Problem Solving Workshop
Strategy: Make a Model ................. RW78
12.7 Relate Fractions and Decimals........ RW79
UNIT 5: FRACTION OPERATIONS
Chapter 13: Add and Subtract Fractions
13.1 Add and Subtract Like Fractions .... RW80
13.2 Model Addition of Unlike
Fractions........................................... RW81
13.3 Model Subtraction of Unlike
Fractions........................................... RW82
13.4 Estimate Sums and Differences ...... RW83
13.5 Use Common Denominators .......... RW84
13.6 Problem Solving Workshop
Strategy: Compare Strategies......... RW85
13.7 Choose a Method ............................ RW86
Chapter 14: Add and Subtract Mixed Numbers
14.1 Model Addition of Mixed
Numbers........................................... RW87
14.2 Model Subtraction of Mixed
Numbers........................................... RW88
14.3 Record Addition and Subtraction .. RW89
14.4 Subtraction with Renaming ........... RW90
14.5 Practice Addition and Subtraction . RW91
14.6 Problem Solving Workshop
Strategy: Use Logical Reasoning .... RW92
Chapter 15: Multiply and Divide Fractions
15.1 Model Multiplication of Fractions . RW93
15.2 Record Multiplication of Fractions RW94
15.3 Multiply Fractions and Whole
Numbers........................................... RW95
15.4 Multiply with Mixed Numbers ....... RW96
15.5 Model Fraction Division .................. RW97
15.6 Divide Whole Numbers
by Fractions...................................... RW98
15.7 Divide Fractions ............................... RW99
15.8 Problem Solving Workshop
Skill: Choose the Operation.......... RW100
6/29/07 11:18:24 AM
UNIT 6: RATIO, PERCENT, AND
PROBABILITY
Chapter 16: Ratios and Percent
16.1 Understand and Express Ratios .... RW101
16.2 Algebra: Equivalent Ratios and
Proportions ........................................RW102
16.3 Ratios and Rates ............................ RW103
16.4 Understand Maps and Scales........ RW104
16.5 Problem Solving Workshop
Strategy: Make a Table ................. RW105
16.6 Understand Percent ...................... RW106
16.7 Fractions, Decimals, and Percents RW107
16.8 Find Percent of a Number..............RW108
Chapter 17: Probability
17.1
17.2
17.3
17.4
Outcomes and Probability ............ RW109
Probability Experiments ................ RW110
Probability and Predictions .......... RW111
Problem Solving Workshop
Strategy: Make an
Organized List ............................... RW112
17.5 Tree Diagrams................................ RW113
17.6 Combinations and
Arrangements................................ RW114
UNIT 7: GEOMETRY AND ALGEBRA
Chapter 18: Geometric Figures
18.1
18.2
18.3
18.4
Points, Lines, and Angles .............. RW115
Measure and Draw Angles ........... RW116
Polygons......................................... RW117
Problem Solving Workshop
Skill: Identify Relationships .......... RW118
18.5 Circles ............................................. RW119
18.6 Congruent and Similar Figures..... RW120
18.7 Symmetry ....................................... RW121
Chapter 19: Plane and Solid Figures
19.1
19.2
19.3
19.4
19.5
Classify Triangles ........................... RW122
Classify Quadrilaterals................... RW123
Draw Plane Figures ....................... RW124
Solid Figures .................................. RW125
Problem Solving Workshop
Strategy: Compare Strategies....... RW126
19.6 Nets for Solid Figures .................... RW127
19.7 Draw Solid Figures from
Different Views ............................. RW128
G5-NL TOC_TEnew.indd 3
Chapter 20: Patterns
20.1
20.2
20.3
20.4
20.5
Transformations ............................ RW129
Tessellations ................................... RW130
Create a Geometric Pattern.......... RW131
Numeric Patterns ........................... RW132
Problem Solving
Workshop Strategy:
Find a Pattern ................................ RW133
Chapter 21: Integers and the Coordinate Plane
21.1 Algebra: Graph Relationships ...... RW134
21.2 Algebra: Equations and
Functions........................................ RW135
21.3 Problem Solving
Workshop Strategy:
Write an Equation ......................... RW136
21.4 Understand Integers ..................... RW137
21.5 Compare and Order Integers ....... RW138
21.6 Algebra: Graph Integers on
the Coordinate Plane .................... RW139
UNIT 8: MEASUREMENT
Chapter 22: Customary and Metric Measurements
22.1
22.2
22.3
22.4
22.5
22.6
22.7
22.8
Customary Length ......................... RW140
Metric Length ................................ RW141
Change Linear Units...................... RW142
Customary Capacity
and Weight .................................... RW143
Metric Capacity and Mass ............. RW144
Problem Solving Workshop
Skill: Estimate or Actual
Measurement ................................ RW145
Elapsed Time.................................. RW146
Temperature .................................. RW147
Chapter 23: Perimeter
23.1 Estimate and Measure
Perimeter ....................................... RW148
23.2 Find Perimeter ............................... RW149
23.3 Algebra: Perimeter
Formulas ........................................ RW150
23.4 Problem Solving Workshop Skill:
Make Generalizations ................... RW151
23.5 Circumference ............................... RW152
6/29/07 11:18:42 AM
Chapter 24: Area and Volume
24.1 Estimate Area ............................... RW153
24.2 Algebra: Area of Squares
and Rectangles ............................. RW154
24.3 Algebra: Relate Perimeter
and Area ....................................... RW155
24.4 Algebra: Area of Triangles .......... RW156
24.5 Algebra: Area of
Parallelograms ............................. RW157
24.6 Problem Solving Workshop
Strategy: Solve a Simpler
Problem ........................................ RW158
24.7 Surface Area ................................. RW159
24.8 Algebra: Estimate and
Find Volume ................................. RW160
24.9 Relate Perimeter, Area,
and Volume .................................. RW161
24.10 Problem Solving Workshop
Strategy: Compare Strategies ..... RW162
G5-NL TOC_TEnew.indd 4
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0
Ones
1
2
Hundreds
Hundreds
5
6
8
7
8
Ones
5.
617,008,235
4.
6.
3.
48,227,304
38,507
6/12/07 10:34:40 AM
© Harcourt • Grade 5
Reteach
8. 60,000,000 1 30,000 1 9,000 1 20 1 4
RW1
153,709
347,254,901
Write each number in two other forms.
7. 803,154
2.
234,621,889
1.
Write the value of the underlined digit.
To find the value of a underlined digit, multiply the digit by its place value.
In 301,256,878 the digit 3 is equal to 3 3 100,000,000 5 300,000,000.
Word Form:
Write the word name for the numbers in each period followed by the name of each
period and a comma.
three hundred one million, two hundred fifty-six thousand, eight hundred seventy-eight
Expanded Form:
Multiply each digit by its place value to write expanded form.
300,000,000 1 1,000,000 1 200,000 1 50,000 1 6,000 1 800 1 70 1 8
Standard Form: 301,256,878
Each period is separated by a comma.
Tens
3
Tens
Ones Period
8
3
5
Hundreds
Hundreds
7
4
2
9
1
0
Hundreds
Ones Period
5
0
809,237,228,771
433,173,983,021
6.
2.
MXENL08AWK5X_RT_CH01_L2.indd 1
7.
3.
51,906,200,141
25,283,998,060
RW2
6,000 1 500 1 20
© Harcourt • Grade 5
Reteach
8. 5,000,000,000 1 200,000,000 1 800,000 1
621,389,007,718
275,487,601,035
Write each number in two other forms.
7. 3,209,003,812
5.
1.
Write the value of the underlined digit.
To find the value of an underlined digit, multiply the digit by its place value.
In 38,752,491,050 the digit 4 is equal to 4 3 100,000 5 400,000.
Word Form:
Write the word name for the numbers in each period followed by the name of each period
and a comma.
thirty-eight billion, seven hundred fifty-two million, four hundred ninety-one thousand, fifty
30,000,000,000 1 8,000,000,000 1 700,000,000 1 50,000,000 1
2,000,000 1 400,000 1 90,000 1 1,000 1 50
Expanded Form:
Multiply each digit by its place value to write expanded form.
Each period is separated by a comma.
Standard Form: 38,752,491,050
Tens
Hundreds Period
Ones
Millions Period
Hundreds
Billions Period
Tens
Hundreds Period
Ones
Understand Billions
Ones
Millions Period
Hundreds
You can use a place-value chart to read larger numbers. The four
periods shown in the place-value chart below are the ones,
hundreds, millions, and billlions.
Tens
You can use a place-value chart to help you read and write whole
numbers and find the value of a digit. A period is a group of
three digits. The three periods shown in the chart below are
ones, hundreds, and millions.
Ones
Place Value Through Millions
Tens
Name
Tens
Name
Ones
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RW1-RW2
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6/12/07 10:35:37 AM
0
0
8
8
2
625,100
962,338
4. 525,100
7. 962,338
8. 18,181
5. 670,430
2. 130,870
.
18,818
640,470
130,870
9. 72,345,995
6. 13,275,104
3. 5,266,918
14. 324,060; 326,040
13. 3,541,320; 3,541,230
RW3
11. 270,908; 270,608
10. 3,218; 3,208
72,345,795
13,276,819
5,264,613
4
1
Ones
6/12/07 10:36:16 AM
© Harcourt • Grade 5
Reteach
15. 12,452,671; 12,543,671
12. 8,306,722; 8,360,272
Name the greatest place-value position where the digits differ.
Name the greater number.
2,815
1. 2,518
Compare. Write ,, ., or 5 for each
Since 0 ten thousand is less than 1 ten thousand, then, 2,306,821 , 2,310,084
0,1
1
6
Compare the digits in the ten thousands place.
3
2
0
Ones Period
353
Tens
Compare the digits in the hundred thousands place.
3
Ones
252
Hundreds
2
Tens
Thousands Period
Ones
Millions Period
Hundreds
Compare the digits in the one millions place.
Hundreds
Compare 2,306,821 and 2,310,084. Write ,, ., or 5.
A place-value chart can help you compare whole numbers.
Compare and Order Whole Numbers
Name
Tens
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MXENL08AWK5X_RT_CH01_L3.indd 1
RW3-RW4
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10. 67,704,257
6. 35,118,247
MXENL08AWK5X_RT_CH01_L4.indd 1
17. 345,591 to 346,000
16. 216,593 to 200,000
RW4
14. 736,147 to 740,000
13. 52,398 to 52,000
12. 487,293,618
8. 849,207,284
4. 355,264,319
© Harcourt • Grade 5
Reteach
18. 3,517,004 to 4,000,000
15. 6,234,581 to 6,234,600
11. 517,218,137
7. 341,618,915
3. 83,445,182
Name the place to which each number was rounded.
9. 888,999,211
5. 6,024
2. 35,211
13,000,000
12,694,022
The 2 increases by 1 to a 3.
Round each number to the place of the underlined digit.
1. 136,237,015
12,694,022
6 . 5, so you round up.
So, 12,694,022 rounded to the nearest million is 13,000,000.
Step 3: Rewrite all digits to the right of the
underlined digit as zeros.
Is the digit to the right 5 or greater?
Increase by 1.
Think: Is the digit to the right
less than 5? The underlined digit
stays the same.
Step 1: Look at the digit to the right of
the underlined digit.
Round to the place value of the underlined digit. 12,694,022
You can round whole numbers by using the rounding rules.
Round Whole Numbers
Name
6/12/07 10:37:11 AM
237,150
2
__
529,617
1__
163,582
294,322
4.
72,543
29,583
1
__
1
5.
$63,895
10.
8.
47,738
78,905
1
__
223,873
221,559
__
11.
56,108
42,336
1
__
2
RW5
1,825
2
__
12. $8,423
9.
37,228
1
__
Estimate by using compatible numbers.
3.
1.
2.
773,645
95,223
1
103,229
1
__
1 613,886
13.
2
135,710
1
__
263,776
6.
2__
173,509
925,461
14.
7.
6/12/07 10:41:00 AM
© Harcourt • Grade 5
Reteach
125,318
2
__
554,903
132.881
2
__
745,556
You can estimate to find an answer that is close to the exact answer.
Use compatible numbers to estimate.
Compatible numbers are easy to compute mentally.
Estimate using compatible numbers.
103,883 1 71,852
Think: 104 1 72 is easy to add mentally, so
103,883
104,000
104,000 1 72,000 are good compatible
1 71,000
1 72,000
numbers to use for an estimate.
176,000
Estimate by rounding.
Add and Subtract Whole Numbers
You can use a place-value chart to help you add or subtract.
Estimate Sums and Differences
millions
1
1
3
1
2
thousands
4
5
9
1
1
0
hundreds
1
9
5
3
4
5
9
5,382
241,393
1
__
259,562
487,018
1 8,723
__
6.
2.
MXENL08AWK5X_RT_CH01_L6.indd A
9. 18,275 + 5,225 + 3,093
5.
1.
7.
3.
306,657
2
227,242
__
319,007
182,322
2
__
RW6
10. 2,705,243 – 1,192,013
2
2,119,625
___
4,678,128
1 29,218
__
33,617
Estimate. Then find each sum or difference.
1
617,634
__
603,438
1 647,273
__
© Harcourt • Grade 5
Reteach
11. 500,601– 74,581
8.
4.
1,114,184 is close to the estimate of 1,000,000, so the answer is reasonable.
Regroup 11 hundred thousands as 1 millions and 1 hundred thousands.
129,336
Add the thousands. Regroup 14 thousands
as 1 ten thousands and 4 thousands.
Then ad the hundreds.
Then add then tens.
First add the ones. Regroup 14 ones as 1
tens and 4 ones.
Start adding from right to left. Regroup as
needed.
Estimate: 800,000 1 300,000 5 1,100,000.
Then add the ten thousands. Regroup 11 ten thousands as 1 hundred
thousands and 1 ten thousand. Last add the hundred thousands.
+
1
8
hundred
thousands
1
ten
thousands
7
tens
6/12/07 10:48:37 AM
Inverse operations are operations that undo each other. The inverse relationship
allows you to check addition by using subtraction and to check subtraction by using
addition.
Estimate. Then find the sum of 789,039 1 325,155
When you add, you find the sum of two or more numbers. When you subtract, you
find the difference of two numbers.
Name
Name
ones
Grade5.indd 3
MXENL08AWK5X_RT_CH01_L5.indd 1
RW5-RW6
6/18/07 5:51:32 PM
Grade5.indd 4
Write your answer in a complete sentence.
6/12/07 10:51:10 AM
© Harcourt • Grade 5
Reteach
students signed up for summer sports
increased by 635. From 2007 to 2008 the
number increased by 224. In 2008, 1,783
students were signed up for summer
sports. How many students were signed
up in 2006?
7. From 2006 to 2007, the number of
RW7
supplies to decorate the school. They
also spent $85 to print banners to put
above the bleachers. They now have
$183 left in their budget. How much
money did they start with in their budget?
6. The school pep club spent $326 on
Work backward to solve.
Check
5. Is there another strategy you could use to solve the problem?
4.
worked backward to solve the problem.
Solve
3. Solve the problem. Use the space below to show how you
Plan
2. What strategy can you use to solve the problem?
Read to Understand
1. What does the problem ask you to find?
On Saturdays, Samantha has stretching class for 45 minutes and ballet
for an hour and a half. After a 30-minute break, she has jazz class for 1
hour, which is over at 1:45 P.M. At what time does she begin?
Mental Math: Patterns in Multiples
Problem Solving Workshop Strategy:
Work Backward
5 80 3 1 hundreds 5 80 hundreds
80 3 100
2 3 80
10. 3 3 600
MXENL08AWK5X_RT_CH02_L01.indd 1
26. 20 3 5,000
25. 8 3 300
24. 7 3 6,000
RW8
21. 800 3 80
16. 100 3 10
20. 5 3 200
15. 50 3 5,000
27. 800 3 400
22. 900 3 100
17. 7,000 3 40
12. 20 3 700
© Harcourt • Grade 5
Reteach
28. 70 3 100
23. 300 3 500
18. 9,000 3 20
13. 9 3 4,000
7 3 100 5
2 3 900 5
11. 60 3 50
7 3 10 5
6. 7 3 1 5
2 3 90 5
19. 80 3 6,000
14. 10 3 60
9.
Find the product.
4 3 500 5
4 3 50 5
5. 2 3 9 5
6 3 200 5
9 3 400 5
5 3 300 5
4. 4 3 5 5
6 3 20 5
3. 6 3 2 5
basic fact times 100
basic fact times 10
9 3 40 5
2. 9 3 4 5
5 8,000
5 800
5 80
basic fact
5 3 30 5
1. 5 3 3 5
Find the missing numbers.
5 8 3 1 hundreds
8 3 100
5 80 hundreds
5 8 3 1 tens
8 3 10
5 8 tens
You can use basic multiplication facts and patterns to find the
product when you multiply by a multiple of 10.
Find the product.
80 3 100
5 8 ones
58
831
5 8 3 1 ones
Name
Name
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Grade5.indd 5
5
5
10. 44 3 260
11. 489 3 706
Estimate the product.
5. 456 3 76
6. 79 3 61
3
3. 63 3 43
3
1. 52 3 31
RW9
12. 3,485 3 59
7. 53 3 1,299
Round each factor and estimate the product.
30 3 700 5 21,000
32 3 723
13. 45 3 914
8. 26 3 725
3
4. 512 3 49
3
2. 731 3 47
STEP 2: Use basic multiplication facts and patterns to find the
product of the rounded factors.
723 is closer to 700 than 800.
723 rounds to 700.
32 is closer to 30 than 40.
32 rounds to 30.
6/12/07 11:29:02 AM
© Harcourt • Grade 5
Reteach
14. 38 3 4,118
9. 71 3 $9.58
5
5
Record the 4 in the
ones place. Write
the 2 above the
tens place.
4
4
3
3
464
7
42
3 3
MXENL08AWK5X_RT_CH02_L03.indd 1
6.
1.
7.
2.
3
764
8
65
3 4
2
5
2
T
4
4
6
O
8.
3.
3
604
5
RW10
1,208
9
3
Record the 2 in
the tens place.
Write the 2 above
the hundreds
place.
9.
4.
3
532
6
3
2
5
2
H
2
5
2
T
4
4
6
O
10.
5.
3
3
© Harcourt • Grade 5
Reteach
4,365
6
745
3
Record the 2 in the
hundreds place and
the other 2 in the
thousands place.
6/12/07 11:31:06 AM
4 3 5 hundreds 5 20 hundreds
20 hundreds 1 2 hundred 5
22 hundreds
2
T
Step 3: Multiply the
hundreds.
3,045
8
3
4 3 5 tens 5 20 tens
20 tens 1 2 ten 5 22 tens
5
H
5
T
2
6
O
2
T
4 3 6 ones 5 24 ones
24 ones 5 2 ten 4 ones
5
H
Step 2: Multiply the tens.
Estimate. Then find the product.
3
T
Step 1: Multiply the ones.
Estimate: 600 3 9 5 2,400
You can use rounding numbers and use multiplication facts to
estimate products.
Remember:
If the digit to the right of
the greatest place is 0– 4,
round down.
If the digit is 5– 9, round up.
Multiply by 1-Digit Numbers
You can use a place-value chart to help you multiply by
1-digit numbers.1
Estimate. Then find the product
556 3 4
Estimate Products
Estimate the product.
32 3 723
STEP 1: Round both factors to the greatest place.
Name
Name
MXENL08AWK5X_RT_CH02_L02.indd 1
RW9-RW10
7/19/07 5:55:35 PM
Grade5.indd 6
1
6
3
8
5
8
6
T
6
H
Think:
7 3 688 5 4,816
4
Th
6
7
8
O
4
Tt
287
3 38
6.
11. 237 3 16 5
253
3 17
1.
7.
2.
392
3 81
439
3 56
5
6
2
8
3
8
1
6
8
4
5
6
T
H
466
3 29
$324
3 45
RW11
12. 407 3 28 5
8.
3.
9.
4.
0
6
7
8
O
Think:
60 3 688 5 41,280
1
4
Th
Step 2: Multiply the tens.
Estimate. Then find the product.
Tt
Step 1: Multiply the ones.
688 3 67
Estimate: 700 3 70 5 49,000
Estimate. Then find the product.
6
4
0
2
9
8
1
6
3
8
8
6
5
4
5
6
T
H
O
6
0
6
7
8
10.
5.
6/29/07 3:09:20 PM
© Harcourt • Grade 5
Reteach
189
3 86
805
3 62
13. 683 3 53 5
507
3 54
576
3 43
Think: 46,096 is close
to the estimate of
49,000. So the answer
is reasonable.
1
4
Th
4
Tt
Step 3: Add the partial
products.
MXENL08AWK5X_RT_CH02_L05.indd 1
11 3 35 5 385, 11 3 42 5 462, and
11 3 63 5 693, what is the middle digit
in the product of 11 and 81?
7. If 11 3 23 5 253, 11 3 33 5 363,
Find a pattern to solve.
6. How can you check your answer?
Check
numbers?
© Harcourt • Grade 5
Reteach
8. What is the sum of the first 8 odd
RW12
5. Write your answer in a complete sentence.
3. Solve the problem. Describe the strategy you used.
Solve
2. How can finding a pattern help you solve the problem?
Plan
1. Write the question as a fill-in-the-blank sentence.
Read to Understand
A large art museum is planning a show of famous oil paintings.
The show will last for 4 days. If 38,888 people attend the show
each day, how many people will attend the show in all?
Problem Solving Workshop Strategy:
Find a Pattern
Multiply by Multi-Digit Numbers
You can use a place-value chart and regrouping to help you multiply by
2-digit numbers.
Name
Name
MXENL08AWK5X_RT_CH02_L04.indd 1
RW11-RW12
7/19/07 5:55:52 PM
6/12/07 11:33:25 AM
Grade5.indd 7
11. 7 3 2,394 5
6.
450
3 18
3
7.
384
3 258
8.
RW13
12. 54 3 37 3 19 5
9,000
30
3
6/12/07 11:36:31 AM
© Harcourt • Grade 5
Reteach
8
9
10. 702
3
5. 143
13. 40 3 400 3 10 5
423
3 12
9.
Find the product. Choose mental math, paper and pencil, or a calculator.
2. 453
3. 5,000
4. 285
1. 305
3 24
3 627
3 123
3
30
Multiply. 3 3 75 3 216
This is a good problem
to solve using a
calculator because it
requires two
calculations and uses
greater numbers.
This is a good problem to solve using
paper and pencil since a 3-digit
number is multiplied by a 2-digit
number.
You can use a calculator to solve problem with greater
numbers or problems with more than one step.
3,432
429
8
3
__
27
You can use paper and pencil to solve problems
when mental math is too difficult.
Multiply. 42 3 60
42 3 60 5 (40 1 2) 3 60
5 (40 3 60) 1 (2 3 60)
5 2,400 1 120
5 2,520
6qw
216
12
18
24
30
180 4 6 5 30
216 4 6
MXENL08AWK5X_RT_CH03_L1.indd 1
10. 499 4 7 5
7. 265 4 4 5
5qw
314
3qw
252
Estimate the quotient.
So, 180 4 6 is about 30
4.
42
48
157
2qw
546
6qw
RW14
11. 345 4 6 5
8. 344 4 8 5
5.
2.
So, 216 4 6 is about 40
240 4 6 5 40
216 4 6
8qw
289
254
4qw
54
12. 189 4 8 5
9. 372 4 5 5
6.
3.
18 and 24 are both close to 21. You can use either number, or
both numbers to estimate the quotient.
Step 2: Estimate using compatible numbers.
1.
36
Find multiples that are close to the first 2 digits of the dividend.
6
Step 1: Think of the multiples of 6:
Estimate the quotient.
Compatible numbers are numbers that are easy to work with mentally. In
division, one compatible number divides evenly into the other. Think of
the multiples of a number to help you find compatible numbers.
You can use mental math, paper and pencil, or a calculator to find a product.
This is a good problem for
mental math because you can
use the Distributive Property
to compute mentally.
Estimate with 1-Digit Divisors
Choose a Method
You can use mental math to solve problems with numbers
that are easy to compute.
Name
Name
MXENL08AWK5X_RT_CH02_L06.indd 1
RW13-RW14
7/19/07 5:56:10 PM
Reteach
© Harcourt • Grade 5
6/12/07 11:39:38 AM
Grade5.indd 8
22
Divide. 4qw
Multiply. 4 3 5 5 20
Subtract. 22 2 20 5 2
Compare. 2 , 4
8qw
136
3qw
741
Divide.
1.
7qw
297
7qw
456
2.
5qw
8,126
RW15
8,659
4qw
3.
6qw
5,238
256
4qw
1,027
2 8
22
2 20
27
2 24
3
4.
6/12/07 11:44:03 AM
© Harcourt • Grade 5
Reteach
4qw
9,449
4,973
7qw
27
Divide. 4qw
Multiply. 4 3 6 5 24
Subtract. 27 2 24 5 3
Compare. 3 , 4
Write the remainder.
Step 4:
Bring down the 7 ones. Then divide the 27
ones.
Name the position of the first digit of the quotient.
Then find the first digit.
25
4qw
1,027
2 8
22
2 20
27
Step 3:
Bring down the 2 tens. Then divide the 22
tens.
I will only use the remainder
B
C
MXENL08AWK5X_RT_CH03_L3.indd 1
© Harcourt • Grade 5
Reteach
8 miles per day along a 125-mile long
trail. How many days will Jessie and her
family hike exactly 8 miles?
7. Hannah and her family want to hike
RW16
troop. They will canoe a total of 75 miles,
and want to travel 8 miles each day. How
many days will they need to travel the
entire distance?
6. Harry goes on a canoe trip with his scout
Solve the following problems and then tell how you would interpret the remainder.
5. How can you check to see if your answer is reasonable?
4. How many minivans are needed for the field trip to the park?
D I will only use the quotient
I will add 1 to the quotient
I will use the quotient and write the remainder as a fraction
A
3. How would you interpret the remainder? Circle your answer then explain.
9qw
75
2. What is the quotient? What is the remainder, if there is one?
1. Write the question as a fill-in-the-blank sentence.
A total of 75 fifth-grade students are going on a field trip to a
local park. The school is providing minivans to take the
students to the park. If each minivan holds 9 students, how
many minivans are needed?
Problem Solving Workshop Skill:
Interpret the Remainder
Divide by 1-Digit Divisors
You can use estimation to help you place the first digit in the quotient.
Then, you can follow steps to divide.
Name the position of the first digit of the quotient. Then find the first digit.
Divide. 4qw
1,027
Step 1:
Step 2:
Use estimation to place the first digit.
Divide the 10 hundreds.
10
Divide. 4qw
2
1,027 4 4
4qw
1,027
Multiply. 4 3 2 5 8
2 8
800 4 4 5 200
Subtract. 10 2 8 5 2
2
So, the first digit, 2, is in the hundreds place.
Compare. 2 , 4
Name
Name
MXENL08AWK5X_RT_CH03_L2.indd 1
RW15-RW16
6/18/07 5:52:29 PM
6/12/07 11:45:07 AM
Grade5.indd 9
9. 8qw
330
13. 846 4 7
RW17
8. 7qw
843
6/12/07 11:47:53 AM
© Harcourt • Grade 5
Reteach
14. 5,420 4 5
1,423
10. 7qw
5. 3qw
6,024
1 0 2 Multiply: 7 3 2 = 14
714
7qw
27
Subtract: 14 – 14 = 0
014
214 Compare: 0 < 7
0
4. 5qw
5,412
1 < 7, so there are
not enough tens to
divide. Write 0 in
the tens place of
the quotient.
3. 3qw
927
12. 3,291 4 3
7. 4qw
8,126
807
6. 2qw
11. 605 4 3
2. 3qw
624
10
7qw
714
27
_____
01
1. 8qw
872
Divide.
So, 714 4 7 5 102.
Compare: 0 < 7
Subtract: 7 – 7 = 0
Multiply: 7 3 1 = 7
Divide the ones.
Bring down the 1 ten.
Divide the tens.
Divide the hundreds.
7
7qw
714
20
_____
0
Step 4:
Step 3:
Step 2:
So, the first digit is in the hundreds place.
700 4 7 = 100
714 4 7
Step 1: Use estimation to place the first digit.
MXENL08AWK5X_RT_CH03_L5.indd 1
17. 8,000 4 10
13. 6,300 4 7
9. 210 4 3
5. 500 4 50
1. 40 4 2
18. 1,400 4 7
14. 6,000 4 2
10. $300 4 10
6. 120 4 40
2. 160 4 8
RW18
19. $2,400 4 30
15. 3,000 4 30
11. 630 4 90
7. 480 4 6
3. $270 4 90
Use basic facts and patterns to find the quotient.
36,000 4 60= 600
Step 3: Divide.
Use the basic fact.
36,000 4 60
Step 2: There are zeros in the dividend
and the divisor.
Cancel out one zero each.
The basic fact is 36 4 6 = 6.
36,000 4 60
Step 1: Find the basic fact.
Use basic fact patterns to find the quotient.
36,000 4 60.
You can use basic facts and patterns to find quotients.
714
Divide. 7qw
Algebra: Patterns in Division
Zeros in Division
Name
When you have zeros in division, you treat them the same way you would treat any other
digit when dividing.
Name
MXENL08AWK5X_RT_CH03_L4.indd 1
RW17-RW18
6/18/07 5:53:21 PM
© Harcourt • Grade 5
Reteach
20. 5,600 4 8
16. $4,500 4 50
12. 540 4 60
8. 560 4 70
4. 420 4 6
6/12/07 11:50:04 AM
Grade5.indd 10
157
42qw
622
11. 12qw
Estimate the quotient.
6. 409 4 63
1.
73q 268
12.
RW19
34qw
293
7. 478 4 19
2.
13.
738
81qw
8. 7,145
3.
Write two sets of compatible numbers for each. Then give two
possible estimates.
So, 8 and 9 are reasonable estimates of the quotient.
or 500 4 50 5 9
427 4 49
400 4 50 5 8
427 4 49
Step 3: Divide the compatible numbers to find the estimates.
427 is between 400 and 500
Step 2: Find two numbers close to the dividend that are compatible
with the rounded divisor.
Step 1: Round the divisor to the nearest ten.
49 rounds to 50.
Reteach
6/12/07 11:51:44 AM
© Harcourt • Grade 5
2,369
Divide. 53qw
Estimate the quotient.
343
54qw
You can use estimation to help you place the first digit in the quotient. Then, you can follow
steps to divide.
Compatible numbers are numbers that are easy to work with mentally. In
division, one compatible number divides evenly into the other. Think of the
multiples of a number to help you find compatible numbers.
MXENL08AWK5X_RT_CH03_L7.indd 1
6,413
4. 43qw
612
1. 52qw
Divide. Check your answer.
So, 2,369 4 53 5 44 r37.
44 r37
53qw
2,369
2 212
249
2 212
37
RW20
4,684
5. 27qw
917
2. 63qw
Step 3: Bring down the 9 ones.
Then divide the 249 ones.
4
53qw
2,369
2 212
24
Step 2: Divide 236.
40
2,000
50qw
1,597
6. 89qw
608
3. 24qw
© Harcourt • Grade 5
Reteach
6/12/07 11:52:11 AM
Write the remainder to the right of the whole
number part of the quotient.
Compare: 37 , 53
Subtract: 249 2 212 5 37
Multiply: 53 3 4 5 212
Think:
Multiply: 53 3 4 5 212
Subtract: 236 2 212 5 24
Compare: 24 , 53
Think:
So, the first digit is in the tens place.
Step 1: Use estimation to place the first digit.
Remember to use compatible numbers to estimate.
Divide by 2-Digit Divisors
Estimate with 2-Digit Divisors
49qw
427
Name
Name
MXENL08AWK5X_RT_CH03_L6.indd 1
RW19-RW20
6/18/07 5:53:31 PM
Grade5.indd 11
20
400
6. 16qw
845
Divide.
1. 58qw
1,325
20
7. 24qw
217
6
2. 37qw
241
80
RW21
8. 37qw
4,819
3. 29qw
2,276
Write low, high, or just right for each estimate.
So, for 16qw
416 the estimated quotient, 20, is too low.
96 is much greater than 16.
96 > 16
20
416
16qw
2 32
96
2 0
96
9. 71qw
488
10
4. 82qw
910
Then divide using your compatible dividend and divisor.
20
20qw
400
2 40
00
Next, use the estimated quotient to check the degree of accuracy.
16
416
Write too high, too low, or just right for the estimate below.
20
16qw
416
First, use compatible numbers for the dividend and divisor.
6/12/07 11:57:09 AM
© Harcourt • Grade 5
Reteach
10. 43qw
9,189
60
5. 63qw
3,784
Write 1 as the remainder.
Compare: 1 < 4
Subtract: 33 – 32 = 1
Multiply: 8 3 4 = 32
Compare: 3 < 4
Subtract: 35 – 32 = 3
Multiply: 8 3 4 = 32
6qw
115
MXENL08AWK5X_RT_CH03_L9.indd 1
6. 219 4 7
1.
7. 935 4 4
2. 9qw
326
RW22
8. 6,121 4 5
3. 7qw
2,198
Divide. Multiply to check your answer.
So, 353 4 4 5 88 r1
88 r1
4qw
353
2 32
33
2 32
1
Step 3: Divide the ones.
8
4qw
353
2____
32
3
Step 2: Divide the tens.
So, the first digit is in the tens place.
350 4 4 = 8
353 4 4
Step 1: Use estimation to place the first digit.
353 . Multiply to check your answer.
Divide 4qw
3,504
9qw
9. 9,217 4 7
4.
3,167
6qw
© Harcourt • Grade 5
Reteach
10. 8,032 4 4
5.
When you divide, it helps to remember that division is an operation that tells the number of
equal groups, or the number in each equal group.
Practice Division
Correcting Quotients
Estimates can help you identify the first digit in the quotient, but sometimes you will need to
correct the quotient.
Name
Name
MXENL08AWK5X_RT_CH03_L8.indd 1
RW21-RW22
6/18/07 5:53:40 PM
6/12/07 11:59:23 AM
Grade5.indd 12
7 1 (72 4 9)
RW23
elderly neighbor with chores. Her goal is to
earn $1,300. She saves $25 per week of
her earnings. She spends $10 at the
shopping mall with her friends. Her
brother, Germaine, is saving $15 per week
to buy an MP3 player. How many weeks
must Shonda save $25 to reach her goal?
6/12/07 12:02:24 PM
© Harcourt • Grade 5
Reteach
1,890 miles. He paid an average of
$2.57 for gas and his car got 15 miles
to the gallon. On average, how many
miles did Mr. Greene drive each day?
2. 52 more than 24
3. 6 plus the quotient of 56
and 7
MXENL08AWK5X_RT_CH04_L1.indd 1
He grew out of 1 pair and
bought some more jeans.
4. Kelly has 4 pairs of jeans.
© Harcourt • Grade 5
Reteach
more markers than
crayons.
He then hiked for 20
more minutes.
RW24
6. Quinn has three times
5. Adam hiked for a while.
Write an algebraic expression. Tell what the variable represents.
week and 17 miles the
next week.
Write a numerical expression. Tell what the expression represents.
1. Hank ran 14 miles one
Tell which information is relevant and irrelevant to solve the problem. Then solve.
6. Over a 45-day period, Mr. Greene drove
Think: It is a good
idea to use a
variable that helps
you remember
what it represents.
In this case,
n = books.
n19
let n 5 the number of books Dee had.
19
Then, “given 9 more books”
So, “Dee had some books.
She was given 9 more books.”
can be represented as
variable n
First, “some books”
Dee had some books.
She was given 9 more books.
Write an algebraic expression. Tell what the variable represents.
An algebraic expression is an expression with at least one variable.
A variable is a letter or symbol that stands for one or more numbers.
So, 7 plus the quotient of 72 divided by 9
represents a sum.
Use clue words to
help you write
Write a numerical expression. Tell what the expression represents. expressions. For
example: more,
7 plus the quotient of 72 divided by 9
sum, added, and
plus indicate
First, “7 plus”
71
addition.
Then, “the quotient of 72 divided by 9”
72 4 9
A numerical expression has only number and operation signs.
5. Shonda earns $35 per week helping her
4. What was the price for each student ticket?
information in the space on the
right to write an equation to solve?
3. How could you use the relevant
A total of 48 fifth graders and 4 teachers went on a field trip
to the. The total cost for the students’ tickets
was $576. The total cost for the teachers’ tickets was $60.
What was the price for each student ticket?
cross out irrelevant information? What would you circle, if you were
told to circle relevant information?
2. What would you cross out in the problem below, if you were told to
1. How would you write the problem as a fill in the blank sentence?
A total of 48 fifth graders and 4 teachers went on a field trip to the museum. The
total cost for the students’ tickets was $576. The total cost for the teachers’ tickets
was $60. What was the price for each student ticket?
Write Expressions
Problem Solving Workshop Skill: Relevant or
Irrelevant Information
An expression has numbers, operation signs, and sometimes variables.
An expression does not have an equal sign.
Name
Name
MXENL08AWK5X_RT_CH03_L10.indd 1
RW23-RW24
6/18/07 5:53:50 PM
6/12/07 12:09:43 PM
Grade5.indd 13
10. 38 2 (18 4 6)
9. 3 3 (54 4 9)
11. 64 2 (24 4 3)
7. (20 2 13) 3 5
3. (75 1 5) 2 8
2
12. (13 2 7) 3 4
8. (21 1 42) 2 6
4. (16 3 2) 2 10
3418
8 1 8 5 16
(6 4 3) 3 4 1 8
14. (16 3 w) 2 5
if w 5 4
18. (22 1 k) 3 5
if k 5 3
13. 2 1 (15 4 n)
if n 5 3
17. (8 1 4) 3 n
if n 5 2
RW25
if z 5 185
19. z 1 (8 3 1)
if r 5 7
15. (r 1 9) 1 6
Reteach
6/12/07 12:13:34 PM
© Harcourt • Grade 5
6
if m 5 30
__ 1 14
20. m
if x 5 5
16. (96 4 12) 5 x
Evaluate the algebraic expression for the given value of the variable.
6. (81 1 11) 2 8
2. 33 2 9 1 14
5. 6 3 (63 4 7)
1. 15 1 6 1 3
Evaluate each expression.
So, (6 4 3) 3 4 1 8 5 16.
• Perform the operation in parenthesis first:
• Multiply or divide, from left to right:
• Add or subtract, from left to right:
Remember to follow the order of operations.
(6 4 3) 3 4 1 8
Evaluate the expression.
Commutative Property
Zero Property of Multiplication
MXENL08AWK5X_RT_CH04_L3.indd 1
4. 12 3 (6 3 8) 5 (12 3 n) 3 8
1. 27 3 (n 3 8) 5 (27 3 9) 3 8
RW26
© Harcourt • Grade 5
Reteach
3. 6 3 n 5 85 3 6
5. (4 1 n) 1 3 5 4 1 (7 1 3) 6. n 3 120 5 0
2. 61 1 33 5 33 1 n
Find the value of n. Identify the property used.
So, this follows the Identity Property of Multiplication, n 5 1.
Notice that the product and the factor other than n are the same.
15 3 0 5 0
The product of any number and zero is zero.
Find the value of n. Identify the property used.
81 3 n 5 81
4 3 11 5 11 3 4
8325238
If the order of the addends or the factors
is changed, the sum or product stays the
same.
(7 3 4) 3 5 5 7 3 (4 3 5)
(9 1 3) 1 2 5 9 1 (3 1 2)
51055
33156
The way addends are grouped on factors
are grouped does not change the sum or
product.
Associative Property
The sum of zero and any number equals
that number. The product of one and any
number equals that number.
Identity Property
Expressions using addition and multiplication follow certain properties.
Properties
Evaluate Expressions
To evaluate an expression, or to find the value of an
expression, you have to perform each operation separately
to determine the answer.
Name
Name
MXENL08AWK5X_RT_CH04_L2.indd 1
RW25-RW26
6/18/07 5:54:01 PM
6/12/07 12:18:21 PM
Grade5.indd 14
RW27
14. 68 1 81 1 42
13. 73 3 3
8. 3 3 360
7. 3 3 9 3 2
11. 37 3 5
5. 21 1 39 1 38
4. 7 3 5 3 3
10. 7 3 3 3 9
2. 4 3 27
1. 27 1 26 1 33
Use properties and mental math to find the value.
So, 5 3 29 5 145.
5 145
5 100 1 45
5 (5 3 20) 1 (5 3 9)
5 3 29 5 5 3 (20 1 9)
15. 12 3 4 3 5
Reteach
6/12/07 12:19:17 PM
© Harcourt • Grade 5
12. 43 1 (47 1 46)
9. 62 1 28 1 17
6. 6 3 45
3. (24 1 19) 1 16
Use the Distributive Property and mental math to work out 5 3 29.
5 3 29
Use properties and mental math to find the value.
You can use addition or subtraction to break apart a factor.
The Distributive Property states that you can break apart a factor to multiply.
Let k stand for the number
of keys Omar has.
Choose a variable.
Step 2
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seedlings and some cucumber
seedlings. She planted 31 seedlings
altogether. How many cucumber
seedlings did Mrs. Greene plant?
4. Mrs. Greene planted 18 tomato
After he bought the jacket he had $18
left. How much did the jacket cost?
2. Jerrod saved $85 to buy a new jacket.
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test. He studied 3 times longer for his
science test than for his spelling test.
How long did Ryan study for his spelling
test?
3. Ryan studied 45 minutes for his science
muffins. Beth’s family ate some. Now
there are 16 muffins left. How many
muffins did Beth’s family eat?
1. Beth’s mother made 24 blueberry
k25=7
Write an equation.
Step 3
Write an equation for each. Tell what the variable represents.
So, the equation is
k2557
Let k 5 number of keys Omar has.
A number decreased by
5 is 7.
Write a representative
sentence.
Step 1
Tia has 5 fewer keys than Omar. If Tia has 7 keys, how many does Omar have?
Write an equation. Tell what the variable represents.
An equation is a number sentence that shows that two quantities are
equal. Like an expression, an equation has numbers, operation signs,
and sometimes variables. An equation is different from an expression
because an equation does have an equal sign.
Write Equations
Mental Math: Use the Properties
You can use properties and mental math to help you solve problems.
Name
Name
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A function is a relationship between two quantities. One quantity
depends upon the other. You can show a function using a function table.
To solve an equation, you find a value for the variable that makes the equation true.
That value is the solution.
15 4 3 5 5
Try 3
Yes
No
Yes
6. 7 3 v 5 42
5. f 1 17 5 20
7. 48 4 k 5 8
3. 5 3 u 5 60
10. 32 5 p 1 9
14. 18 5 a 1 2
9. 45 5 5 3 t
13. 24 5 6 3 z
RW29
15. x 2 11 5 23
11. 35 2 n 5 14
Use mental math to solve each equation. Check your solution.
2. 12 4 e 5 4
1. t 2 5 5 1
Which of the numbers 3, 6, or 12 is the solution of the equation?
So, for 15 4 r 5 5, r 5 3.
3 3 5 5 15
Check: Use an inverse operation to check your work.
15 4 5 5 3
Try 5
Test some possibilities by replacing r with your predictions.
Think: 15 divided by what number equals 5?
15 4 r 5 5
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16. 18 4 v 5 6
12. 77 4 y 5 11
8. 41 2 g 5 29
4. 6 1 p 5 12
5
30
6
36
42
8
48
0
y
1
MXENL08AWK5X_RT_CH04_L7.indd 1
k
j
3. k 5 7j 1 1
0
x
1. y 5 4x
3
1
5
2
Complete each function table.
7
3
64
9
4
0
b
a
2
4. b 5 6a 2 8
n
m
2. n 5 5m 1 4
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So, when x 5 42 in the function table, y 5 7.
Rule: Divide by 6
Equation: x 4 6 5 y
Replace: 42 4 6 5 y
75y
Think of a rule. Write it as an equation.
Look at the pattern.
Outputs
Inputs
16
4
3
6
29
5
Write an equation to represent each function. Then complete the table.
Functions
Solve Equations
Use mental math to solve the following equation. Check your solution.
Name
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8
8
© Harcourt • Grade 5
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10
10
6/15/07 3:14:17 PM
Grade5.indd 16
The capacity of a large aquarium is 12 gallons more than the
capacity of a small aquarium. The aquariums hold a total of
52 gallons of water. What is the capacity of each aquarium?
Read to Understand
1. What are you asked to find?
An inequality is a number sentence that shows that two amounts are not equal.
925<6
or
4 < 6 is true
10 2 5 < 6
or
5 < 6 is true
6. x 1 9 , 18
7. x 1 5 . 13
8. x . 15 2 6
13. x 2 14 . 3
9. x . 16
14. x 1 5 . 11
10. x , 11 1 7
RW31
15. x 1 3 , 22
11. x . 24 2 8
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16. x 2 9 , 8
12. x 1 1 , 18
Which of the numbers 16, 17, and 18 are solutions of each inequality?
5. x 2 4 , 6
Which of the numbers 8, 9, and 10 are solutions of each inequality?
1. x . 8
2. x , 10
3. x , 12 2 3
4. x . 6 1 1
So, 8, 9, and 10 are solutions to the inequatity x 2 5 < 6.
825<6
or
3 < 6 is true
Replace x with each of the numbers that are possible solutions.
(
1
2 12)
2 12)
5
5
52
52
MXENL08AWK5X_RT_CH04_L9.indd 1
and bettas cost $5 each. Andy spent
$19 at the store. How many of each
type of fish did Andy buy?
6. Catfish cost $3 each at the pet store
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tetras in her aquarium. The product of
the numbers of each type of fish is 63. If
Charlotte has more goldfish than tetras,
how many of each type does she have?
7. Charlotte has a total of 16 goldfish and
How can you use the equation c + (c + 12) = 52 to check your answer?
Predict and test to solve.
5.
(
1
Small Aquarium
What is the capacity of each aquarium?
Check
4.
Large Aquarium
Solve
3. What might 2 of your predictions be? Test these in the equations below.
Plan
2. What do you know about the capacities of the two aquariums?
Problem Solving Workshop Strategy: Predict and Test
Inequalities
Which of the numbers 8, 9, and 10 are solutions of the inequality, x – 5 < 6?
Name
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3.
2
2
2
2
Hundredths Thousandths
5.
4.
7. 3.01
8.
3.01
9.
Find the value of the underlined digit.
2.
1.
RW33
9.814
10. 54.236
Write the decimal shown by the shaded part of each model.
So, the decimal represented by the model is 0.08.
Tenths
Ones
There are 100 squares in a hundredths model.
In this model 8 squares are shaded.
Use a place value chart to help write the decimal.
6.
11. 54.236
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© Harcourt • Grade 5
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Equivalent decimals are different name for the same number or amount. You can use
decimal squares to find equivalent decimals.
You can use models to write decimals.
This model shows 420 thousandths, or 0.420.
There are 420 shaded parts.
The model shows 1,000 equal parts. Each
part represents 1 thousandth, or 0.001 of
the model.
Now divide each of the 100 parts into 10
equal parts. The model at the right shows
what each small square would look like.
4.87 and 4.870
0.23 and 0.230
5.
2.
9.87 and 9.78
0.51 and 0.500
0.830
0.803
0.83
MXENL08AWK5X_RT_CH05_L2.indd template1
7.
8.
RW34
0.93
0.093
0.930
Write the two decimals that are equivalent.
4.
1.
9.
6.
3.
1.007
1.070
1.07
© Harcourt • Grade 5
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1.11 and 1.111
0.680 and 0.68
Write equivalent or not equivalent to describe each pair of decimals.
Each model shows the same amount shaded.
So, 0.42 and 0.420 are equivalent.
This model shows 42 hundredths,
or 0.42.
42 of the parts are shaded.
The model shows 100 equal parts.
Each part is 1 hundredth of the model.
0.42 and 0.420.
Write equivilant or not equivilant to describe the pair at decimals below.
Equivalent Decimals
Decimal Place Value
Write the decimal shown by the shaded part of the model below.
Name
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8.106
0.6
8.16
0.603
4.
7.
11.
14.
0.614, 0.641, 0.64
3.08, 3.801, 3.8
13.
8.
5.
2.
10.
Order from least to greatest.
9.9
9.39
1.
Compare. Write <, >, or = for each
So, 0.28 > 0.208.
7.8
0.89
0.30
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0.159, 0.154, 0.14
1.576, 1.765, 1.567
0.78
0.69
0.308
.
4.71
1.83
7.245
7.254
4.071
1.833
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15. 7, 6.99, 6.099, 7.001
12. 3.971, 3.9, 4, 3.901
9.
6.
3.
Analyze
Complete the table.
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Julia has three plants. One plant receives 1 tablespoon of plant food each month and
grows to a height of 7 cm. A second plant receives 3 tablespoons of plant food each
month and grows to a height of 8.5 cm. A third plant is not given any plant food and
grows to a height of 5.75 cm. What conclusion can you draw about how the amount
of food given affects plant growth?
MXENL08AWK5X_RT_CH05_L4.indd template1
5.
Conclusion
What conclusion can Mr. Hall draw about the number of weeks students spent on their
plant experiments?
Solve by drawing a conclusion.
4.
How much time did the least number
of students spend on the experiment?
How much time did the greatest number
of students spent on the experiment?
3.
students spent 4 to 6 weeks.
students spent 2 to 4 weeks.
Since 8 hundredths is greater than 0 hundredths, 0.28 is greater than 0.208.
students spent less than 2 weeks.
8>0
How would you complete the statements below using the details from the problem?
2=2
2.
Compare the digits in the hundredths place.
0
Compare the digits in the tenths place.
8
What does the problem ask you to find?
0=0
2
0
Hundredths Thousandths
Compare the digits in the ones place.
Tenths
Ones
Use a place value chart.
1.
Mr. Hall discovers that 5 of his students spent less than
2 weeks on their plant experiment, 16 students spent
2 to 4 weeks, and 3 students spent 4 to 6 weeks.
What conclusion can Mr. Hall draw about the number of weeks
students spent on their plant experiment?
A place-value chart can help you compare decimals. You may need to add zeros so you can
compare the same number of digits in each decimal. Compare the digits from left to right.
.
Problem Solving Workshop Skill: Draw Conclusions
Compare and Order Decimals
Compare 0.28 and 0.208. Write <, >, or = for the
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ROUNDING RULES:
• If the digit to the right
is less than 5, the
underlined digit stays
the same.
Rewrite: 0.130 or 0.13
Step 2: Rewrite all digits to
the right of the
underlined digit as zeros.
An equivalent decimal can
be written by leaving off the
trailing zeros.
6. 11.323
10. 20.595
5. 12.63
9. 0.964
11. 6.89
7. 4.289
3. 108.108
13. 12.35 to 12.4
RW37
14. 0.428 to 0.43
Name the place to which each number was rounded.
2. 9.028
1. 7.325
Round each number to the place of the underlined digit.
Compare: 4 < 5
4 is less than 5, so
the digit stays the same.
Step 1: Compare the digit
to the right of the
underlined digit to 5 using the
rounding rules.
hundredths place
15. 9.462 to 9.46
Reteach
6/12/07 1:46:43 PM
© Harcourt • Grade 5
12. 32.514
8. 7.547
4. 26.199
• If the digit to the right
is greater than or equal
to 5, the underlined
digit increases by 1.
$ 1 3. 0 4
2 $ 0. 9 5
$ 1 2. 0 9
$13
$1
$12
0. 4 5
1 0. 7
1. 5 8
1 4. 5 3
MXENL08AWK5X_RT_CH06_L2.indd 1
9. 3.5 1 2.89 1 0.4
5.
1.
1
4
6. 8
$1 8. 5 2
1 $3. 7 3
11.
7.
3.
RW38
10. $15 2 $1.27
6.
2.
Find the sum or difference.
21.05 2 2.65
2. 9
2 0. 6 3
6. 3 9
2. 1 8
1 7. 8 5
So, $13.04 2 $0.95 5 $12.09 is close to the estimate, $12.
So the answer is reasonable.
$13.04
2 $0.95
2
$0.95
__
12 19 14
You can use lined notebook paper to add and subtract decimals.
You can use the same rules you learned for rounding
whole numbers to round decimals.
Subtract. $13.04
Add and Subtract Decimals
Round Decimals
Round 0.134 to the place of the underlined digit.
Name
Name
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$2 1. 40
2 $1. 33
0. 3 2 1
2 0. 1 2 3
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12. $7.00 2 $1.05
8.
4.
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Grade5.indd 20
5.4
3.4
3.4
+
0.8
0.82
0.305
0.78
0.8
0.5
2 0.3
6.17
2 3.5
51.234
2 28.4
11. 0.427 + 0.711
6.
1.
7.
2.
1.73
1.4
1 3.17
8.
3.
3.28
2 0.86
7.6
2 2.15
0.78
2 0.305
RW39
9.
4.
10.
5.
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© Harcourt • Grade 5
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$23.07
2 $ 7.83
2.083
0.56
1 0.41
13. 40.512 + 30.399
15.27
1 41.8
0.443
1 0.207
• greater than or equal to 5, round up.
If the digit to the right of the place you are
rounding to is:
• less than 5, round down.
Remember the rounding rules:
12. 61.05 – 18.63
$29.38
1 $42.75
Estimate by rounding.
So, the difference is about 0.5.
2
Estimate the difference by rounding.
1
1.2
1.247
Round and add to estimate.
First, decide what place to round to. Since two of the
addends have digits to the hundredths, tenths would be a good
place to round to.
8
.
4
4
5
_
5
.
8
71.4
1__
11.5
1
73.9
1 4.37
MXENL08AWK5X_RT_CH06_L4.indd 1
5.
1.
6.
2.
90.4
1 88.76
127.35
1 928.527
3.3
1_
5.6
1
RW40
7.
1__
2.25
1
3. 10 10
1
7 13 1
38.445
2 25.86
12.585
2
.
5
• Use a calculator for difficult
numbers or very large numbers.
$ 48.60
1 32.81
__
$ 15.79
0.4
• Use paper and a pencil for
larger numbers.
1.2
2
0.8
_
Choose a method. Find the sum or difference.
So, 3
2
25.86
__
Choose a method. Find the difference. 2
Estimate the sum by rounding.
These are very large numbers,
so use a calculator.
There is more than one way to find the sums and differences of whole
numbers and decimals. You can use mental math, a calculator, or paper
and pencil.
38.445
You can estimate to find an answer that is close to the exact answer. You
can use the rounding rules to help you estimate sums or differences.
8.
4.
8
• Use mental math for problems with
fewer digits or rounded numbers.
Choose a Method
Estimate Sums and Differences
1.247
0.82
1 3.4
Name
Name
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14.219
1.793
1 15.881
1
0.364
1
1__
1.558
5
38.445
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Grade5.indd 21
How long does her last throw need to be for her to advance to the final round?
4.
5.
A waitress charged Hannah $6.75 for
lunch. Hannah wants to tip the
waitress about 20% of $6.75. How
much should she leave?
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© Harcourt • Grade 5
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car. Each tire costs $110.60. What is
the total cost?
6. Lance is purchasing 4 new tires for his
Tell whether you need an estimate or an exact answer.
Then solve the problem.
Show how you solve the problem in the space below.
Will you estimate or find an exact answer to solve this problem? Explain your choice.
2.
3.
What are you asked you to find?
1.
In a baseball-throwing contest, a score of 50 meters or more is
needed to advance to the final round. Jenna’s first two throws were
16.64 meters and 15.33 meters. How long does her last throw need
to be for her to advance to the final round?
Model Multiplication by a Whole Number
Problem Solving Workshop Skill:
Estimate or Find Exact Answer
MXENL08AWK5X_RT_CH07_L1.indd 1
7. 4 3 0.12
4. 8 3 0.05
1. 9 3 0.23
Find the product.
So, 2 3 0.96 5 1.92
RW42
8. 0.09 3 6
5. 2 3 0.84
2. 7 3 0.25
Step 1
Shade 0.96 two times using a different
color each time.
Find the product.
2 3 0.96
9. 3 3 0.32
6. 6 3 0.52
3. 4 3 0.71
© Harcourt • Grade 5
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Step 2
Count how many squares are shaded.
There are 192 hundredths squares or
1 whole and 92 hundredths. Place the
decimal after the whole number.
You can use the hundredths models, to show multiplication of
decimals and whole numbers.
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6.37 3 1,000 5
0.185 3 1,000 5
$3.75 3 1,000 5
0.008 3 1,000 5
$0.25 3 10,000 5
1.313 3 10,000 5
3.94 3 1,000 5
3.94 3 10,000 5
2.002 3 1,000 5
2.002 3 10,000 5
RW43
3.94 3 100 5
2.002 3 100 5
11. 3.94 3 10 5
$0.25 3 1,000 5
1.313 3 1,000 5
10. 2.002 3 10 5
$0.25 3 100 5
1.313 3 100 5
8. $0.25 3 10 5
$3.75 3 100 5
0.008 3 100 5
7. 1.313 3 10 5
$3.75 3 10 5
0.008 3 10 5
5. $3.75 3 1 5
6.37 3 100 5
0.185 3 100 5
4. 0.008 3 1 5
6.37 3 10 5
2. 6.37 3 1 5
0.185 3 10 5
1. 0.185 3 1 5
Use patterns to find the product.
0.005 3 1
0.005 3 10
0.005 3 100
0.005 3 1,000
Answer
6/13/07 1:52:33 PM
© Harcourt • Grade 5
0.05 3 10,000 5
0.05 3 1,000 5
0.05 3 100 5
12. 0.05 3 10 5
0.6 3 10,000 5
0.6 3 1,000 5
0.6 3 100 5
9. 0.6 3 10 5
89.36 3 1,000 5
89.36 3 100 5
89.36 3 10 5
6. 89.36 3 1 5
9.999 3 1,000 5
9.999 3 100 5
9.999 3 10 5
Reteach
0.005
0.05
0.5
5
3. 9.999 3 1 5
Number of Zeros in Number of Places to
Whole Number
Move Decimal Point
0
0
1
1
2
2
3
3
Use the patterns to find the product.
When multiplying a decimal by 10, 100, 1,000 or 10,000, first
count the number of zeros in the whole number. Then move the
decimal point one place to the right in your answer for every
zero that you counted.
Record Multiplication by a Whole Number
Algebra: Patterns in Decimal Factors
and Products
MXENL08AWK5X_RT_CH07_L3.indd 1
32
21. 237.89
3 32
16. 25.68
11. 0.04 3 86
6. 6.31 3 53
1. 0.933 3 6
3 39
22. 77.42
37
17. 159.46
12. 33.33 3 72
7. 8.492 3 10
8. 0.688 3 2
3. 32.96 3 44
RW44
3 70
23. 3.043
3 44
18. 621.3
3 54
24. 0.333
32
19. 736.07
14. 1.917 3 41
9. 121.3 3 5
4. 379.4 3 5
25.
© Harcourt • Grade 5
Reteach
39.83
3 66
3 51
20. 800.9
15. 5.585 3 28
10. 9.57 3 34
5. 0.007 3 26
Start at the right end of your answer
and count to the left the same
number of places as are in the
decimal factor.
There are two digits to the
right of the decimal point.
13. 290.6 3 6
126.88
1 1
5 4
4.88
3
26
_
2928
1 9760
__
2. 5.27 3 18
Find and record the product.
Write the problem
vertically and multiply
like whole numbers.
Find and record the product.
4.88 3 26
When given a horizontal multiplication problem, rewrite the
problem vertically. Multiply and put the decimal point in the answer the
same number of places to the right as the decimal point in the problem.
Name
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7.
4.
1.
0.8 3 0.3
0.4 3 0.2
0.6 3 0.8
8.
5.
2.
RW45
0.7 3 0.8
0.3 3 0.9
0.5 3 0.5
Use the model to find the product.
So, 1.6 3 0.5 5 0.80
9.
6.
3.
1.4 3 0.1
1.6 3 0.8
1.1 3 0.7
First, use a color pencil to shade 16
columns. Next, use a different color
pencil to shade 5 rows. Then, count the
squares in the area in which the color
shading overlaps.
Make a model to find the product of 1.6 3 0.5
Reteach
6/13/07 2:57:12 PM
© Harcourt • Grade 5
8
2
3
4
_
2.15
3
3.92
__
MXENL08AWK5X_RT_CH07_L5.indd 1
RW46
11. 300.59 3 0.3
8. 6.6 3 8.2
7. $0.39 3 291
10. 487.66 3 2.12
5. 29.3 3 0.31
2. $26.83 3 11
4. 6.71 3 4.22
1. 4.25 3 7.82
Estimate the product.
12
3
3
4
_
6
Round both numbers up.
Get the range
Round both numbers down.
2.15
2
3
3.92
3
3
__
_
Does 8 fall between 6 and 12?
Yes, so this estimate is reasonable.
Round
2.15
3
3.92
__
Estimate the product.
2.15 3 3.92
12. 0.409 3 1.47
9. 7.6 3 9.217
6. $8.54 3 9
3. 3.3 3 9.4
© Harcourt • Grade 5
Reteach
Your orginal
estimated answer
should fall between
the other two
estimates in the
range.
When estimating to find the product of two decimals, use a range to
determine if your answer is reasonable.
Estimate Products
Model Multiplication by a Decimal
When you use models to multiply decimals, remember that each square in the
hundredths grid represents 0.01. So, 60 squares represent 60. 0.1 represented in the
hundredths model is the entire first column or 10 squares. One full grid represents
the whole number, one.
Name
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Grade5.indd 24
.4
0.41
8.
3.43
3 9.3
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15. 0.08 3 2
12.4
3 8.5
14. $6.95 3 5.3
7.
0.5
3 0.7
12. 0.381 3 14
7.9
3 6.6
3.
11. 194.6 3 0.2
6.
Estimate. Then find the product.
1.
72
2.
0.9
3 1.3
3 0.4
You know your actual answer should be close to 0.24.
So, the actual answer is 0.2296.
2,296
Multiply as whole numbers.
41
3 56
__
0.41
3
0.56
__
9.
4.
3
3
5
.24
10.
5.
16. 1.11 3 1.1
13. $4.50 3 9.5
60.2
3 2.6
3.6
3 0.8
.6
0.56
Now estimate the factors.
4.1
3 5.3
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© Harcourt • Grade 5
Reteach
$5.94
3 0.07
A fishing boat caught a total of 4,012 lobsters in one
season, selling the lobsters to fish markets for $0.05 each.
How much money did the boat earn for the season?
When you mulitply decimals, first mulitiply the factors as whole numbers.
Use estimation to figure out where to place the decimal point in your
answer. You can do this by estimating your factors. Multiply your
estimated factors and place the decimal point in the answer. Your
estimate should show that your actual answer is reasonable.
the season?
4. How much money did the boat earn for
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Describe the steps required to solve. Then solve the problem.
6. The U.S. population in 1990 was 248.7
American ate 92.4 lb of red meat that
million people. The 2000 population was
year. They ate 1.21 times that amount in
1.13 times that. The 2010 population is
2003. In 1940, Americans ate an average
projected to be 1.1 times the 2000
of 12.3 lb of poultry for the year, and 5.79
population. How many people, to the
times that in 2003. How many pounds of
nearest million, are projected to live in
poultry and red meat did the average
the United States in 2010?
American eat per year?
3
$0.05
__
4,012
5. FAST FACT in 1940, the average
3. Multiply.
2. What step or steps are needed to solve problems?
1. What are you asked to find?
Problem Solving Workshop Skill: Multistep Problems
Practice Decimal Multiplication
Estimate. Then find the product.
Name
Name
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There are four groups of
7 squares. Since we are
using hundredths models,
the answer must be in
hundredths.
Now divide the squares
into four groups of the
same size.
5. $7.32 4 6
8. 8.56 4 4
12. 4.88 4 8
4. 6.03 4 9
7. 0.94 4 2
11. $9.99 4 3
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2. 1.95 4 3
1. 0.36 4 6
Use decimal models or play money to model the quotient.
Record your answer.
So, 0.28 4 4 5 0.07.
Show 0.28 using a
decimal model.
13. 5.53 4 7
9. $9.18 4 9
6. 7.63 4 7
3. $9.75 4 5
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© Harcourt • Grade 5
8. 91.7 4 18
5. 6.4 4 9
2. 46.8 4 5
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10. 47.8 4 59
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11. 8.91 4 27
Find two estimates for the quotient.
7. 65.8 4 22
4. 5.3 4 8
1. 23.7 4 9
Estimate the quotient.
So, an appropriate estimate is 0.04.
• 3.4 and 80 are closer to 3.409 and 83.
Which compatible numbers are closer to the
original problem?
• 3.6 and 60 are compatible.
3.6 4 60 = 0.03
• 3.2 and 80 are compatible.
3.2 4 80 = 0.04
Think of some compatible numbers
near 3.409 and 83:
3.409 4 83
12. 7.42 4 35
9. 45.43 4 36
6. 0.312 4 5
3. 94.2 4 6
When dividing a decimal by a two-digit number, use compatible numbers
for the dividend and divisor. Compatible numbers are
numbers that are easy to divide.
Just as you did with whole numbers, you can use decimal
models to divide decimals into groups.
Estimate the quotient.
Estimate Quotients
Decimals Division
Use decimal models to model the quotient.
Record your answer.
0.28 4 4
Name
Name
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6/13/07 3:41:46 PM
Grade5.indd 26
5 3.15
046
84 38.64
10.
14.
13. 82 459.2
6.
9. 5 34.15
Find the quotient.
5.
77 $24.64
2 85.12
$092
58 $53.36
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15.
11.
7.
4 1.74
17 $99.28
0063
64 4.032
Copy the quotient and correctly place the decimal point.
0285
876
35
1.
2.
3.
6 1.71
9 78.84
47 164.5
So, 3.15 4 5 5 0.63
Step 3. Divide.
Step 2. Place a zero in the ones place above the
dividend if the division is greatest then the
divdend.
Step 1. On the quotient, put the decimal point above
the decimal point of the dividend.
Find the quotient.
.
5 3.15
16.
12.
8.
4.
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6/13/07 3:45:11 PM
© Harcourt • Grade 5
61 1.159
38 78.66
0008
7 0.056
$241
39 $93.99
0.63
5 3.15
30
15
0.
5 3.15
+ 0.57
1.05
0.97
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her average speed was faster than Danielle’s
total average speed. Susie drove 96.4 mph,
88.5 mph, and 83.9 mph. Is Susie’s answer
reasonable? Explain.
6. For the first three laps of the race, Susie says
average speed of 19.18 mph. Brent says
Buddy was faster by 9.18 mph. Whose
answer is reasonable?
5. Buddy says he was faster than Brent by an
USE DATA For 5-6, use the table.
4. Whose answer is reasonable? Explain.
45
73.25 mph
89.47 mph
Susie
Brent
© Harcourt • Grade 5
Reteach
50
50
43
99.10 mph
Rico
21
98.65 mph
91.43 mph
Number
of Laps
Finished
Average
Speed per
Lap
Middleville Auto Race
Danielle
Buddy
Driver
3. Which answer, Brad’s or Brittany’s, is closer to your estimate?
rounding them to whole numbers?
2. How can you estimate the three decimal parties of fruit,
1. What are you asked to find?
Britney bought 0.97 kilogram of apples, 1.05 kilograms of
bananas, and 0.57 kilogram of oranges. Britney says she
bought 25.9 kilograms of fruit. Brad says that Britney bought
2.59 kilograms of fruit. Whose answer is reasonable?
Problem Solving Workshop Skill: Evaluate Answers
for Reasonableness
Divide Decimals by Whole Numbers
When dividing decimals, placing the decimal point in your
answer is an important step to finding the correct value.
Name
Name
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people who live in
California.
4. a random sample of 200
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adults who live in Los
Angeles.
5. a random sample of 200
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© Harcourt • Grade 5
Reteach
people who live in Los
Angeles.
6. a random sample of 200
A radio station wants to find out the favorite type of music of people that live in
Los Angeles, California. Tell whether each sample represents the population. If it
does not, explain.
A playground maker wants to find out if children in grades 4–6 like their new
playground equipment. Tell whether each sample represents the population. If it
does not, explain.
2. a random sample of
3. a random sample of
1. a random sample of 400
400 teachers
children in grades 4–6
boys in grades 4–6
So, the survey does not represent the population because the
sample includes children at only one school.
• The survey does involve children ages 10–14.
• But, it is a survey of only one school.
Does the random sample fairly represent the population?
A fruit juice company wants to survey children ages 10–14. Tell
whether the sample below represents the population.
If it does not, explain.
A random sample of 100 children at one school.
The median is 8.
(8 1 8) 4 2 5 8
Step 2: Find the middle
number. Since there is an
even number of data, the
median is between the two
middle numbers.
6. 21, 24, 22, 24, 21, 31, 25
8. 43, 53, 63, 53
5. $25, $36, $28, $27
7. 350, 378, 350, 252, 275
9.8, 10.2, 11.6,
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15. 10.5, 12.5,
13.
11. $64, $48, $40,
12. 75, 62,
, 10, 13; mean: 11
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; mean: 73
© Harcourt • Grade 5
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, $7; mean: $10.60
, 45, 32; mean: 33
16. $12, $8, $17,
, 9.7; mean: 10.3 14. 21,
; mean: $51
Use the given mean to find the missing number in each data set.
10. 1.3, 1.55, 2.75, 1.3, 2.6
4. 164, 215, 174, 174, 193
3. 7.8, 9.4, 10.6, 7.8, 7.8, 9.4
9. 873, 954, 896, 941
2. 641, 874, 614, 755
Sometimes there is only
one mode or no mode.
There are two modes
5 and 8.
Step 2: Find the number
that occurs most often.
1. 21, 15, 17, 21, 16
Find the mean, median, and mode for each set of data.
The mean is 13.
78 4 6 5 13
Step 2: Divide the sum by the
number of addends.
5, 5, 8, 8, 14, 38
Step 1: Order the data from
least to greatest.
Step 1: Order the data from
least to greatest.
Step 1: Add to find the sum.
8 1 8 1 14 1 5 1 38 1
5 5 78
The mode is the number
that occurs most often in a
set of data.
The median is the middle
number when a set of data
is arranged in order.
Find the mean, median, and mode for the data set:
8, 8, 14, 5, 38, 5
A survey is a way to gather information about a group.
When you are gathering information about a group,
• the whole group is called the population,
• the people surveyed are called the sample.
The mean is the average of
a set of data.
Mean, Median, and Mode
Collect and Organize Data
A sample must fairly represent the population.
In a random sample, everyone in the population has
an equal chance of being surveyed.
Name
Name
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10
24
68
52
65
52
31
68
12
69
53
72
Range: 72 2 12 5 60
2.
15
12
20
15
9
17
13
16
5
8
3
2
3
2
4
1
7
0
6
9
2
3
A: Songs Students Heard
13
11
A: Weights of Boxes
4
1
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18
13
21
15
14
16
B: Weights of Boxes
5
10
7
5
2
7
0
9
37
68
9
6
15
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© Harcourt • Grade 5
Reteach
52
22
18
B: Songs Students Heard
11
22
Compare the mean, median, and range of the data sets.
1.
12
52
Median: (52 1 52) 4 2 5 52
The median for data set B is the same as the median for data set A.
Median: (52 1 52) 4 2 5 52
The range for data set A is greater than the range for data set B.
Range: 79 2 10 5 69
68
60
Mean: 543 4 12 5 45.25
32
15
The mean for data set A is greater than the mean for data set B.
Mean: 573 4 12 5 47.75
43
79
A: Pages Students Read
31
Duke
27
23
Women
25
26
33
Tennessee Connecticut Michigan
State
School
14
Men
30
Key:
Poetry
Biography
Mystery
Fantasy
= four books
Types of Books in Mr. Li’s Class
A pictograph displays countable data using
pictures and symbols. Pictographs have a
key to show how many each picture or
symbol stands for.
Birds, 20
wins?
MXENL08AWK5X_RT_CH09_L4.indd 1
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basketball had?
4. What if football had 20 wins. How would this
amount be shown on the pictograph?
2. How many wins did baseball have?
3. How many more wins did soccer have than
Tues.
Wed.
Day
Thurs.
Fri.
Key:
Hockey
Basketball
Baseball
Soccer
= 8 wins
© Harcourt • Grade 5
Reteach
Sport Team Wins
The temparure increases steadily.
. Which sport had this number of
1. Twenty-eight wins would be shown as
Mon.
Daily Temperatures
How can you describe the trend in
temperature from Wednesday to Friday?
100
80
60
40
20
0
A line graph shows how data changes
over a period of time.
USE DATA For 1–4, use the pictograph at the right.
What kinds of animals does the pet store
have the same number of?
cats and dogs
Dogs, 10
Cats, 10
Reptiles, 5
Pet Store Population
A circle graph shows how parts of data are
related to each other and to the whole.
Which team had the most wins?
How many fantasy books are in Mr. Li’s
Duke: 27 1 31 5 58 Connecticut: 23 1 25 5 48 class?
Tennessee: 14 1 30 5 44 Michigan: 26 1 33 5 59 The key shows each symbol stands for 4
books. A half symbol stands for 2 books.
(5 3 4) 1 2 5 22 fantasy books
So, Michigan had the most wins.
40
35
30
25
20
15
10
5
0
NCAA Basketball Wins 2004–2005
A bar graph uses bars to display
countable data. A bar graph is useful
when comparing data by groups.
Graphs can help you draw conclusions, answer questions, and
make predictions about data.
You can compare data sets using the mean, range, and median.
B: Pages Students Read
Analyze Graphs
Compare Data
Compare the mean, median, and range of the data sets.
Name
Name
Number of Wins
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3
5
people used the Internet and 4 people
used an atlas. Two of these people
used the Internet and an atlas. How
many people used the Internet or an
atlas during the research period?
6. During a one-hour research period, 8
Draw a Venn diagram to solve.
RW57
Check
5. What is one way you could check your solution.
4. How would you write your answer as a complete sentence.
2
U.S. Presidents First Ladies
Solve
3. Solve the problem using the Venn diagram.
Plan
2. How can drawing a diagram help you solve the problem?
1. How would you write the question as a fill-in-the blank sentence?
Read to Understand
Five students wrote a report about U.S. Presidents, 8 wrote a report
about U.S. first ladies, and 3 wrote a report about both U.S.
Presidents and first ladies. How many students wrote reports?
Reteach
6/14/07 8:13:40 AM
© Harcourt • Grade 5
13
11
Boots
MXENL08AWK5X_RT_CH10_L01.indd 1
6
19
Stoves
Backpacks
/VNCFS4PME
(FBS
Tents
May Camping Sales
1. Make a bar graph for the data set.
• Add a key for Week 1 and Week 2.
• Use one color for Week 1 and a
different color for Week 2.
• Add the second set of data using the
key for Week 2.
56
48
40
32
24
16
8
0
24
16
8
0
40
32
56
48
Name
Emma
Name
5JOB
8FFL
&NNB
Biking Record
Tina
8FFL
.BSDVT
Marcus
Week 1 Biking Record
Philip
1IJMJQ
© Harcourt • Grade 5
Reteach
from exercise 1 and this data: June Sales:
tents, 20; stoves, 9; boots, 11;
backpacks, 15.
RW58
Describe how to make a double-bar graph of
the data in the table and this data: Week 2:
Marcus, 56 miles; Tina, 44 miles; Emma, 32
miles; Phillip, 48 miles.
A double-bar graph is used to
compare similar kinds of data.
• Draw a bar to show 40 miles for Tina
• Draw a bar to show miles for Phillip
Use the data at the right to complete
the bar graph.
Week 1 Biking Record
Name
Distance (in miles)
Macus
48
Tina
40
Emma
28
Phillip
52
2. Make a double-bar graph for the data set
Make Bar Graphs and Pictographs
Problem Solving Workshop Strategy:
Draw a Diagram
Pictographs use pictures or symbols to
display or organize data. Bar graphs
display data using vertical or horizontal
bars.
Name
Name
Distance (in miles)
Distance (in miles)
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2
8
7
6
5
4
3
2
1
0
1-3
List the intervals.
1. Use 10 inches for each interval.
Height (in inches) of Students
52
48
47
41
54
60
42
46
39
57
44
49
46
47
61
43
63
49
56
52
51
60
54
42
62
USE DATA For 1–2, use the table.
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10-12
4
10-12
6
Height (in Inches) of Students
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© Harcourt • Grade 5
Reteach
2. Make a histogram of the data.
4-6
7-9
Ages (in years)
6
11
Frequency Table
1-3
4-6
7-9
3
8
3
Ages of Students Taking
Swimming Lessons
Interval
Frequency
5
6
12
12
10
3
6
11
4
7
3
• So, 8 children ages 4-6 take swimming lessons the most.
• Label each bar with the frequency
table interval and the axis.
• Make each bar the same width.
• Draw the bars touching, but not
overlapping.
• Use the frequency table to find how
long to make each bar.
• Make the histogram by first choosing
an appropriate scale for the vertical
axis. Label the axis.
• Make a frequency table to determine
the number of times each age appears
in the data.
• A reasonable interval would be 3
years, starting at 1.
• The data in the table shows ages
from 2 to 12.
• To make a histogram, you need to
find a reasonable interval.
8
4
6
11
9
6. F (6, 2)
4. D (5, 5)
2. B (1, 9)
12. M
11. L
MXENL08AWK5X_RT_CH10_L03.indd 1
10. K
8. H
9. J
7. G
Use the coordinate grid.
Write an ordered pair for each point.
5. E (9, 3)
3. C (3, 7)
1. A (1, 6)
Graph and label the following points
on a coordinate grid.
(x , y)
The x-coordinate is the 1st number
in the ordered pair. The y-coordinate
is the 2nd number in an ordered pair.
(10, 4)
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• Start at 0.
• Since x comes before y in the
alphabet, remember to move across
the x-axis first.
• Move across the x-axis to the number 10.
• Then, go straight up to the line
labeled 4 on the y axis.
• Draw a dot at the intersection of the
two lines, and label the ordered pair
(10, 4).
An ordered pair contains two numbers,
x and y. To graph an ordered pair:
Graph and label (10, 4) on the coordinate grid.
0
1
2
3
4
5
6
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
1
K
1
2
2
A coordinate grid is like a sheet of graph paper with two borders,
the x-axis and y-axis. The x-axis is the horizontal line at the bottom
of the grid. The y-axis is the vertical line on the left side of the grid.
An ordered pair is the location of a point on the grid.
A histogram shows the number of times, or frequency,
an event or data item occurs.
Ages of Children
Taking Swimming Lessons
Algebra: Graph Ordered Pairs
Make Histograms
Swimming Lessons
Make a histogram of the data.
How many children ages 4-6 take
swimming lessons?
Name
Name
ZBYJT
ZBYJT
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3
G
J
3
4
4
5
L
8
M
9 10
9 10 11
8
© Harcourt • Grade 5
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H
6 7
YBYJT
6 7
YBYJT
5
6/15/07 1:19:20 PM
(10, 4)
Age
Height
4
41
5
43
Jim’s Check-Up Heights
3
39
Jim’s Height (inches)
6
46
7
48
1
3
54
2
3
.POUI
4
5
4
62
5
70
Reteach
July
66
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© Harcourt • Grade 5
Sacramento, CA Average Lows
San Diego, CA Average Lows
Month
Mar. April May June
Temperature (oF).
53
53
60
62
R61
2.
2
44
Tupelo, MS Average Highs
1
40
se
ar
ch
ar
se
Re
ch
Research
Re
ail
E-m
E-m
ail
Tom’s Weekly Computer Usage
Activity
Hours
E-mail
3
Games
1
Research
6
MXENL08AWK5X_RT_CH10_L05.indd 1
1.
Use the data to make a circle graph.
2.
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• Shade and label one section each as guitar,
violin, and drums.
• Shade and label two of the sections piano.
• The circle should be divided into five equal
sections.
• Use this number to divide the circle into equal
sections.
• There are 10 students who take music lessons.
• To make a circle graph, you need to determine
the total number of parts of the data.
arch
MXENL08AWK5X_RT_CH10_L04.indd 1
1.
Make a line graph for each the data set.
• Connect the points to show the
average temperature rising from
month 1 to month 5.
• Use the data to find where to graph
each point.
• Write the months along the horizontal
axis using even spacing.
• The scale should go from 08 to 808
using intervals of 208.
• The greatest temperature in the table
is 708.
Month
o
Temperature (in F)
P
• To make a line graph, you need to
determine the greatest number on
the vertical scale.
5FNQFSBUVSFJO '
Use the data to make a circle graph.
ail
Average Monthly Temperature in Tupelo, MS
A circle graph shows how parts of the data
are related to the whole and to each other.
A line graph shows how a set of data changes over time.
E-m
Make a line graph of the data
in the table.
Make Circle Graphs
Make Line Graphs
Name
violin
drums
Cat
Cat
Pet
Fish
Dogs
Cats
Rabbits
3
4
2
1
Fish
Reteach
© Harcourt • Grade 5
Dog
Number of Votes
Favorite Pets in CJ’s Class
guitar
piano
Music Lessons
Students
4
2
2
2
piano
Instrument
Piano
Guitar
Violin
Drums
b
Rese
Ra
Dog
Name
D
bit
g
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Do
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og
Fis
h
Fish
s
me
Ga
Research
Rese
arch
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Grade5.indd 32
Leaves
Did the basketball team score least often in the 50’s, 60’s or 70’s?
Stem
6.
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The number of people eating lunch at a
diner for the past 12 days was 41, 17,
28, 12, 37, 44, 32, 26, 18, 24, 36, and
25. Should the diner be more prepared
to serve customer numbers in the
10’s, 20’s, 30’s or 40’s range?
Make a graph to solve.
Check
5. What other graph might you have used to solve this problem?
4.
Solve
3. How would you fill out the
stem–and–leaf plot for this
problem in the space at
the right?
Plan
2. How can you organize the numbers to better answer the question?
Read to Understand
1. What are you asked to find?
Hampton High’s basketball team had the following scores
this season 63, 67, 73, 55, 61, 53, 60, 63, 52, 61, and 64. Did
the basketball team score least often in the 50s, 60s, or 70s?
Reteach
6/15/07 1:26:33 PM
© Harcourt • Grade 5
Choose the Appropriate Graph
Problem Solving Workshop Strategy:
Make a Graph
Stem Leaves
A circle graph shows how
parts of the data are
related to the whole and
to each other.
4
3
MXENL08AWK5X_RT_CH10_L07.indd 1
season.
hours.
© Harcourt • Grade 5
Reteach
4. The temperature every hour for 24
inches.
2. The daily growth of a sunflower in
RW64
3. The scores of a basketball team for one
6 major cities.
1. The populations of males and females in
Choose the best type of graph or plot for the data. Explain your choice.
5
6/15/07 1:28:12 PM
A stem-and-leaf plot is
used to organize data
by place value.
Stem-and-leaf Plot
A line graph shows
change over time.
Circle Graph
Line Graph
A line plot is used to
record data as it is
collected.
Double-bar Graph
A double-bar graph is
used to compare two
sets of data.
Line Plot
A circle graph will best display how Jen spends the hours in one
day because it will show each hour spent in relation to the whole.
4
3
1
2
Bar Graph
A bar graph is used to compare
data by category.
How Jen spends the hours of the day.
Choose the best type of graph or plot for the data.
Explain your choice.
Bar and double-bar graphs, line graphs, line plots and
stem-and-leaf plots organize numerical data.
Name
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9 3 3 = 27
9 3 8 = 72
9 3 4 = 36
9 3 9 = 81
9 3 5 = 45
9 3 10 = 90
2. 10
3. 6 and 7
RW65
4. 4, 5, and 10
Write the least common multiple for each set of numbers.
1. 6
List the first eight multiples of each number.
So, 24 is the least common multiple.
Multiples of 12: 12 24 36 48 60 72 84 96 ...
8 16 24 32 40 48 56 64 ...
Multiples of 8:
8 12 16 20 24 28 32 ...
4
Multiples of 4:
Write the least common multiple for: 4, 8, and 12.
6/14/07 8:18:00 AM
© Harcourt • Grade 5
Reteach
The least common multiple, or (LCM), of two or more numbers is the least number
that is a multiple of all of the numbers.
When a number is a multiple of two or more numbers it is a common multiple.
So, the first 10 multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, and 90.
9 3 2 = 18
9 3 7 = 63
931=9
9 3 6 = 54
To find the first eight multiples, multiply 9 by 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
5, 6, 7, 5
51617155 23
5, 6 7 5
51617155 23
5, 6 7 5
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3. 6,420
1. 489
RW66
4. 8,703
2. 364
Test each number to determine whether it is divisible by 2, 3, 5, 6, 9, or 10.
So, 5,675 is divisible by 5. It is not divisible by 2, 3, 6, 9, or 10.
A number is divisible by 10 if the last digit is 0.
5,675 is not divisible by 10.
A number is divisible by 9 if the sum
of its digits is divisible by 9.
5,675 is not divisible by 9.
A number is divisible by 6 if is divisible by 2 AND by 3.
5,675 is not divisible by 3, so it is not divisible by 6.
A number is divisible by 5 if the last
digit is 0 or 5.
5,675 is divisible by 5.
A number is divisible by 3 if the sum
of its digits is divisible by 3.
5,675 is not divisible by 3.
A number is divisible by 2 if the last
number is an even number.
5,675 is not divisible by 2.
Test the following number to see if it is divisible by 2, 3, 5, 6, 9, or 10.
5,675
24 is divisible by 4.
24 is not divisible by 5.
A number is divisible by another number if the quotient is a whole number
and the remainder is 0.
A multiple of a number is the product of that number and any other number.
24 4 4 = 6
24 4 5 = 4 R4
Divisibility
Multiples and the Least Common Multiple
List the first 10 multiples of 9.
Name
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© Harcourt • Grade 5
Reteach
not a 0
sum not
divisible
by 9
5
sum not
divisible
by 3
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odd numbe
Grade5.indd 34
Reteach
A composite number is a number that has more
than two factors.
List the factors of 21.
4
8
10
5
40
Factors of 40
2. 27
3. 16
6. 35, 28
7. 18, 24
9. 20, 35
10. 18, 42
RW67
11. 30, 35
Write the greatest common factor for each set of numbers.
5. 21, 30
Write the common factors for each set of numbers.
1. 14
List the factors of each number.
12. 49, 35
8. 45, 15
4. 32
Identify the greatest common factor in the overlap section, which in this case is 8.
So, the greatest common factor of 16 and 40 is 8.
12
16
Factors of 16
Use a Venn diagram to find the common factors.
Write the greatest common factor for 16 and 40.
The greatest common factor, or GCF, is the greatest factor that two or more
numbers have in common.
A common factor is a factor that two or more numbers share.
So, the factors of 21 are 1,3,7, and 21.
3 3 7 = 21
6/14/07 8:26:59 AM
© Harcourt • Grade 5
A prime number has exactly two factors, 1 and itself.
A factor is a number multiplied by another number to find a product.
438
32 3 1
834
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7. 67
4. 38
1. 15
8. 44
5. 45
2. 19
RW68
9. 31
6. 24
3. 11
Write prime or composite. You may use counters or draw arrays.
So, the number 32 is composite.
Since 32 can already be represented by more than
2 arrays there is no reason to keep going.
Draw arrays to represent 32.
• Set an array of 1 row of 32.
Label it. “1 3 32”
• Set an array with 8 rows of 4.
Label it “8 3 4”.
• Set an array of 32 rows at 1.
Label it. “32 3 1”
• Set an array with 4 rows of 8. with Label it. “4 3 8”
Write prime or composite 32.
You may use couners or draw arrays.
32
You can make arrays to find if a number is prime
or composite. An array is an arrangement of objects
in rows and columns. A number with exactly two
arrays is prime. A number with more than two arrays
is composite.
The number 1 is neither prime nor composite.
Prime and Composite Numbers
Factors and Greatest Common Factor
1 3 21 = 21
Name
Name
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© Harcourt • Grade 5
Reteach
1 3 32
6/14/07 8:41:41 AM
Grade5.indd 35
.
Introduction to Exponents
Problem Solving Workshop
Strategy: Make an Organized List
6.
; red and
; green and
Look back at the problem. Does the answer make sense? Explain.
RW69
Anna can have a sandwich on
white, pumpernickel, or wheat bread.
She can have it grilled, baked, or cold.
How many sandwich choices does
Anna have?
Make an organized list to solve.
5.
; white and
How many ways can Toni and Hector combine two colors?
white and
red and white; red and
Color Combinations
How can you complete the organized list below to help solve
the problem? Do not list any combination more than once.
For example, red and white is the same as white and red.
Check
4.
3.
Solve
Plan
2. How can an organized list help you solve the problem?
Reteach
6/14/07 8:48:11 AM
© Harcourt • Grade 5
100
1,000
What is the value of 10 3 10?
What is the value of 100 3 10?
MXENL08AWK5X_RT_CH11_L06.indd 1
5. 10 4
Find the value.
6. 10 9
3. 10 3 10 3 10 3 10 3 10 3 10 3 10
1. 10 3 10 3 10 3 10 3 10
7. 10 6
8. 10 8
© Harcourt • Grade 5
Reteach
4. 10 3 10 3 10 3 10 3 10 3 10 3 10
RW70
Write in exponent form. The find the value.
2. 10 3 10
103.
So, 10 3 10 3 10 written in exponent form is
So, the value of 103 5 1,000.
3 times
How many times is the base being used as a factor?
What is the base? 10
10 3 10 3 10
How would you write the question as a fill-in-the blank sentence?
Read to Understand
1.
Write in exponent form.
Then find the value.
Toni and Hector are weaving potholders using the colors
red, white, green, and blue. If they use only two colors for
each potholder, how many ways can they combine two colors?
The exponent is a number that tells how many times another
number, the base, is used as a factor.
Name
Name
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base
4. 27
Find the value.
1. 5 3 5 3 5 3 5 3 5 3 5 3 5
5. 123
RW71
2. 8 3 8 3 8
6. 192
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© Harcourt • Grade 5
Reteach
3. 2 3 2 3 2 3 2 3 2
Exponent form: 63
Words: the third power of six
or six cubed
Write in exponent form and then write in words.
Exponent form: 42
Words: the second power or four
or four squared
Write 4 3 4 and 6 3 6 3 6 in exponent form and in words.
There are two ways to write the word form for an exponent of 2 or 3.
A perfect square or a square number
is the product of a number and itself.
A square number can be represented
with the exponent 2.
36 in words is “the sixth power of three.”
The exponent 6 means “the sixth power.”
Write 36 in words.
A base with an exponent can be written in words.
3 3 3 3 3 3 3 3 3 3 3= 36
The base is repeated 6 times, so 6 is the exponent.
3 is the repeated factor, so 3 is the base.
exponent
A factor tree is a diagram that shows the prime factorization of a composite number.
In the last lesson, the base was always 10. The base does not always have to be 10, though.
36
Prime factorization is a way to show a composite number as the product of prime factors.
You already know that an exponent is a number that tells how many times
the base is used as a factor.
3
3 2 3
3
3
2. 20
5. 3 3 5 3 3 3 5
MXENL08AWK5X_RT_CH11_L08.indd 1
7. 52 3 7
RW72
8. 2 3 3 3 5 3 2 3 7 3 3
Find the number for each prime factorization.
4. 2 3 3 3 5 3 5
Rewrite the prime factorization using exponents.
1. 42
Find the prime factorization. You may use a factor tree.
2 3 2 3 3 = 22 3 3
Rewrite 2 3 2 3 3 using exponents.
You can use exponents in prime factorization for factors that
appear two or more times.
9. 23 3 3 3 112
© Harcourt • Grade 5
Reteach
6. 2 3 5 3 3 3 5 3 2 3 5
3. 90
Write each composite number as a product of two factors.
Write the number as a product of two factors.
Write the number being factored at the top.
So, the prime factorization of 12 is 2 3 2 3 3.
2
4
12
Find the prime factorization of 12. You may use a factor tree.
Prime Factorization
Exponents and Square Numbers
Write 3 3 3 3 3 3 3 3 3 3 3 in exponent form.
Name
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6.
5.
7.
3.
12.
9.
0
0
A
B
1
1
13.
10.
0
0
RW73
D
E
Write a fraction to name the point on the number line.
2.
1
1
14.
11.
0
0
8.
4.
F
Write a fraction for the shaded part. Write a fraction for the unshaded part.
1.
3
1
1
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© Harcourt • Grade 5
Reteach
C
4
3
3. What is the fraction? __
2. Unshaded parts?
1. Parts in the group? 4
1. How many parts make up the group? 4
2. How many parts are shaded? 1
1
3. What is the fraction? __
4
Unshaded Part:
Shaded Part:
Equivalent fractions are fractions that name the same number or amount.
1 are equivalent fractions because they are equal in value.
__ and __
For example, 3
6
2
2
Write an equivalent fraction for __ .
5
A fraction describes the number of parts of a whole or a group.
__ shows that 5 parts make a whole, and 4 parts are being used.
For example, 4
5
Numerator
4
__
5
Denominator
__ , which have 1 as the numerator, are called unit fractions.
Fractions such as 1
7
Write a fraction for the shaded part. Write a fraction for the unshaded part.
MXENL08AWK5X_RT_CH12_L02.indd 1
8
2
21. __
10
4
16. ___
8
3
11. __
3
__
6. 2
10
8
22. ___
9
__
17. 5
7
__
12. 4
6
__
7. 4
Write an equivalent fraction.
1
__
1. __
2. 1
2
6
RW74
11
3
23. ___
12
6
18. ___
6
3
13. __
9
__
8. 3
5
__
3. 2
12
2
24. ___
10
7
19. ___
8
6
14. __
4
2
9. __
7
__
4. 2
6
6
25. __
5
4
20. __
12
1
15. ___
7
5
10. __
4
__
5. 3
© Harcourt • Grade 5
Reteach
“What you do to the top
(numerator) you must do to do
the bottom (denominator).”
Equivalent Fraction Rule:
Equivalent Fractions
Understand Fractions
__ :
To write an equivalent fraction for 2
5
1. Choose a number to multiply with.
Let’s use 2.
2. Multiply the top (numerator) by 2.
232=4
3. Multiply the bottom (denominator) by 2.
5 3 2 = 10
4 .
__ is 4 over 10, or ___
So, an equivalent fraction for 2
10
5
Name
Name
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90
20
___
35
7. 21
___
9
2. 6
__
RW75
27
18. 18
___
9
30
17. 25
___
8
16. __
18
13. ___
5
12. 3
__
4
9
45
8. 30
___
12
3. ___
3
11. __
2
Write each fraction in simplest form.
6.
8
1. 4
__
Name the GCF of the numerator and denominator.
10 is 2.
The GCF of the numerator and denominator for ___
14
• Divde the numerator and denominator by the GCF 2.
__
10
4255
_______
14 4 2 7
10 in simplest for is 5
__ .
So, ___
14
7
2. What is the greatest common
factor? 2
49
28
19. ___
12
14. ___
4
7
4
9. __
16
4. ___
8
18
13
20. 13
___
24
15. ___
16
10
10. ___
5
5. ___
6
Another way to find the greatest
common factor (GCF) of the
numerator and denominator is to
list the factors of the lesser
number and eliminate the ones
that aren’t common.
Reteach
6/14/07 9:54:09 AM
© Harcourt • Grade 5
A mixed number is made up of a whole number and a fraction.
A fraction greater than 1 is sometimes called an improper fraction.
A fraction is in simplest form when the numerator and
denominator have 1 as their only common factor.
10
Write ___
14 in simplest form.
1. List the factors of 10 and 14.
10: 1, 2, 5, 10
14: 1, 2, 7, 14
Understand Mixed Numbers
Simplest Form
2 1 20 5 22
__
42
5
Step 2. Add
Step 3. Remember to keep
the same denominator, 5.
5 3 4 5 20.
2
4 __
5
MXENL08AWK5X_RT_CH12_L04.indd 1
10
11. 17
___
6
6. ___
19
4
12.
7.
4
6 __
5
3
3 __
8
2
2. ___
11
13.
8.
RW76
7
4 ___
12
2
5 __
5
5
3. ___
12
3
14. ___
23
9
5
1 __
6
9. ___
29
4.
15.
Reteach
© Harcourt • Grade 5
1
7 __
8
7
1
2 __
3
10. ___
18
5.
Write each mixed number as a fraction. Write each fraction as a mixed number.
1. 7
__
3
2 ___
10
Step 1. Multiply
22.
2 written as a fraction is ___
So, the mixed number 4 __
5
5
2 as a fraction.
Write 4 __
5
2
To change 4 __ to a
5
fraction, multiply the
denominator by the whole
number, then add the
numerator to the sum.
The denominator stays the
same for the answer.
23
3
___
So, the fraction ___
10 written as a mixed number is 2 10.
23
___
Write the fraction 10 as a mixed number.
23
___
Step 1. Divide 10
To change 10 to a mixed
23 .
number, divide the
Step 2. See where the following go:
numerator over the
2r3
whole number
denominator.
23
10
remainder
220
divisor
3
Name
Name
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1
1
4
1
__
3
5 __
7
9
7 ___
12
4
18
10. 5 ___
7
24
9. 7 3
__
7
6. ___
15
3
__
5
5. 5
__
12
2. ___
7
4
__
9
1.
5
__
9
Compare. Write <, >, or = for each
.
11.
4
4 __
7
5
7. __
4
2
3. 1
__
RW77
__
54
9
7
___
12
5
___
12
7
7
7
7
5 ___
5 ___
5 ___
5 ___
21
21
21
21
3
3 3 5 ___
9
9
3
______
5 ___
5 __
7
733
21
21
Then, compare.
9.
7 ,53
7 , 5 ___
__ because 5 ___
5 ___
7
21
21
21
6
4 ___
11
10
___
11
5
__
6
__
22
3
6
5
12. __
8.
8
4. __
2
6/14/07 10:24:26 AM
© Harcourt • Grade 5
Reteach
13
___
16
__
25
6
1
__
4
12 12 12 12 12 12 12 12 12
Next, rename the fractions using the common denominator.
Multiples of 7: 7, 14, 21
Multiples of 21: 21, 42, 63
7
Compare. Write ,, ., or 5 for 5 ___
21
First, list the multiples of denominators
4
1
__
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1
___
4
1
__
The two rows are of the same length.
Or you can look for the common denominators.
9 .
__ 5 1 ___
So, 1 3
4
12
Use fraction bars to compare.
What is the order of the four people?
2
3
Bernice threw the shotput
10 5_9 feet. Terry threw the shotput
10 4_7 ft, and Carla threw the
shotput 10 2_5 ft. Who threw the
shotput the longest distance?
Who threw for the shortest distance?
MXENL08AWK5X_RT_CH12_L06.indd 1
6.
Make a model to solve.
RW78
Check
5. What other strategy could you use to solve the problem?
4.
1
Paul
Solve
3. How can you use the strategy to solve the problem? Fill out the model.
Plan
2. What strategy can you use to solve the problem?
Read to Understand
1. What are you asked to find?
Amber, Marcus, Paul, and Shelly line up to make their
jumps. Shelly is not first. Amber has at least two people
ahead of her. Paul is third. Give the order of the four.
4
© Harcourt • Grade 5
Reteach
Problem Solving Workshop Strategy: Make a Model
Compare and Order Fractions and Mixed Numbers
Fractions and mixed numbers can be compared by using
fraction bars.
9
Compare. Write <, >, or = for 1 _3_
1 ___
4
12 .
Name
Name
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7. 1.7
6. 0.004
9. 5.58
8. 3.92
9
17. 3 ____
1000
100
20
437
16. _____
4
1
12. ___
__
11. 1
1000
RW79
667
18. 1 _____
8
3
13. __
4
Reteach
Step 2
(
('
)
('
*
('
+
('
,
('
('
.
('
/
('
0
('
(
'
(
('
)
('
*
('
+
('
,
('
('
.
('
/
('
0
('
(
'
(
('
(
*
)
*
(
'
(
*
)
*
(
6
6
10
RW80
10
3 1 ___
5
2. ___
Find the sum or difference. Write it in simplest form.
115
__
1. __
('
.
('
/
('
0
('
(
5
5
423
__
3. __
'
(
*
)
*
(
6/15/07 1:30:01 PM
© Harcourt • Grade 5
Reteach
__ .
The subtracted fraction is 1
3
1.
Jump back 1 to __
3
,
('
Step 3
+
('
2.
The first fraction is __
32
.
Start at 0. Jump 2 to __
3
*
('
Step 2
2 1 1 1
You land at 1 third, so __ 2 __ = __ . __ is written in simplest form.
3 3 3 3
'
The denominators are 3.
Divide a number line into
3 equal parts.
Step 1
)
('
1
The second addend is ___
10 .
7
___
and jump
Start at
10
1 more tenth.
Step 3
8
8 54
8 , so ___
7 1 ___
1 5 ___
__ .
. Written in simplest form, ___
You land at ___
10
10 5
10 10 10
You can also use a number line to subtract fractions.
1
2 2 __
Find the difference, write it in simplest form. __
3 3
'
7 .
The denominator of each
The first addend is ___
10
fraction is 10. Divide a number Start at 0 and jump 7.
7 .
line into 10 equal parts.
This brings you to ___
10
Step 1
6/14/07 10:33:37 AM
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© Harcourt • Grade 5
3
20. 5 __
7
19. 9 ___
35
7
15. __
8
10. 4.35
5. 0.365
22
14. ___
25
4. 0.45
3. 0.78
Write each fraction or mixed number as a decimal.
2. 0.32
1. 0.8
Write each decimal as a fraction or mixed number in simplest form.
51 .
So, 0.255 written as a fraction is ____
200
Simplify.
255 4 5
1,000 4 5
Step 2.
51
5 ____
200
0.255 has three digits to the right of the
decimal point, so the denominator will have
three zeros.
255
0.255 5 ______
1,000
You can use a number line to add fractions.
1
7 1 ___
Find the sum, write it in simplest form. ___
10 10
You can represent a fraction as a decimal and vice versa. For
1.
example, 0.5 is the decimal equivalent to __
2
Write the decimal 0.255 as a fraction or mixed number in simplest form.
Step 1.
Add and Subtract Like Fractions
Relate Fractions and Decimals
To write a decimal as a
fraction, remove the decimal
point and place the number
over the appropriate power
of 10.
Name
Name
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12
12
12
3 1 ___
5
2 5 ___
___
Add the fractions with
like denominators.
__ bar for
Trade each 1
4
1 bars.
three ___
12
3 1 ___
1
2 5 ___
2
__ 1 ___
4 12 12 12
12
12
1
12
1
12
1
12
1
12
1
12
1
12
MXENL08AWK5X_RT_CH13_L02.indd 1
RW81
Find each sum using fraction bars. Write it in simplest form.
3
3
5 1 ___
2
7
__ 1 1
__
__ 1 __
__
4
4
12
1. 3
2. 8
3. 6
4
5 .
2 5 ___
1 1 ___
So, __
Step 3
Step 2
1
12
Reteach
© Harcourt • Grade 5
10
5
___
2
10
1
___
5
10
4
___
Subtract the fractions with
like denominators.
5 2 ___
1
1 5 ___
1
__ 2 ___
2 10 10 10
__ bar for
Trade the 1
2
1 bars.
five ___
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
MXENL08AWK5X_RT_CH13_L03.indd
6/15/07 1:30:55 PM
1
3
4
221
__
1. __
8
2
1
__ 2 __
2. 7
RW82
6
3
__ 2 1
__
3. 5
Use fraction bars to find the difference. Write it in simplest form.
1 5 ___
4 . Written in simplest form, ___
4 5 __
2.
1 2 ___
So, __
2 10 10
10 5
Step 3
Step 2
1
2
1
__ 2 ___
Subtract. 1
2 10
Step 1
__ fraction bar to
Use a 1
2
model the first fraction.
2
1 1 ___
Write the sum using fraction bars. Write it in simplest form. __
4 12
1
4
You can use fraction bars to help you subtract fractions with unlike
denominators. Trace fraction bars of fractions with unlike denominators
for equivalent bars of fractions with like denominators.
You can use fraction bars to help you add fractions with unlike denominators.
Trade fractions bars of fractions with unlike denominators for equivalent bars
of fractions with like denominators.
__ bar and two
Use a 1
4
1 bars to model fractions
___
12
with unlike denominators.
Model Subtraction of Unlike Fractions
Model Addition of Unlike Fractions
Step 1
Name
Name
Reteach
6/15/07 1:31:13 PM
© Harcourt • Grade 5
RW83-RW84
Grade5.indd 42
7/19/07 6:00:15 PM
2
4
__
6
6
4
__
1
1
(
-
,
-
(
-
8
7
5. __
+
-
3
__
8
(
)
*
-
2. __
2
8
)
-
1
__
MXENL08AWK5X_RT_CH13_L04.indd 1
4.
1.
'
'
-
Estimate each sum or difference.
6
6 9
add the two rounded numbers.
411
__ ,
To estimate the sum __
__ on the number line.
Find 1
9
1 ,or 1?
Is it closest to 0, __
2
__ is closest
to 0.
The fraction 1
9
4 on the number line.
Find __
6
1 ,or 1?
Is it closest to 0, __
2
4
1.
__
The fraction is closest to __
6
2
4 1 __
1
__
6 9
1.
4 1 __
1 is about __
So, __
6 9
2
Step 3
Step 2
Step 1
Estimate the sum.
5
__
6
7
__
8
(
-
)
0
*
0
)
-
+
0
(
)
'
(
)
*
-
(
)
,
0
6.
0
+
-
.
0
,
-
/
0
6
1
__
6
1
2
(
(
0
0
7
__
8
3
__
8
(
, - . / 0
/ / / / /
3. 5
__
' ( ) * +
/ / / / /
__
1051
2
RW83
2
1
2
1
__
'
' (
0 0
'
'
-
Reteach
© Harcourt • Grade 5
3
12
12
12
2
10
10
10
MXENL08AWK5X_RT_CH13_L05.indd
6/15/07 1:31:39 PM
1
7 1
__
__
5. 8 2 4
3 2
__
__
6. 4 2 3
RW84
9
4
___
__
7. 10 2 5
Find the sum or difference. Write the answer in simplest form.
1 2
1 1
3 1
__
__
__
__
1. __ 1 __
2. 2 1 5
3. 4 1 6
5 3
10
10
12
5
8.
8
__ 2 5
__
9 6
3
1
__
__
4. 5 1 4
6/15/07 1:32:08 PM
© Harcourt • Grade 5
Reteach
Stop when you find
2 fractions with
like denominators.
4
9 2 ___
5 5 ___
9 2 __
1 becomes ___
4 . Written in simplest form, ___
4 5 __
2.
___
9 2 __
1
Example 2: Subtract ___
10 2
9 , 18
9 .
___
___, 27
___, 36
___
Step 1 Write equivalent fractions for ___
10
10 20 30 40
1
1.
5
2, 3
4, ___
__, __
__, __
Step 2 Write equivalent fractions for __
2
2 4 6 8 10
Step 3 Rewrite the problem using the equivalent fractions.
Then subtract.
12
5 1 __
5 1 ___
9 . Written in simplest form, ___
9 5 __
3.
1 becomes ___
4 5 ___
___
To add or subtract unlike fractions, you need to rename them as like
fractions. You can do this by making a list of equivalent fractions. When
you find two fractions with the same denominator, they are like fractions.
5 1 __
1
Example 1: Add. ___
12 3
5 , 10
5 .
___
___, 15
___, 20
___
Step 1 Write equivalent fractions for ___
Stop when you find
12
12 24 36 48
1
1
3
2
4
__
__
__
__
___
2 fractions with
, , ,
Step 2 Write equivalent fractions for .
3
3 6 9 12
like denominators.
Step 3 Rewrite the problem using the equivalent fractions.
Then add.
Use Common Denominators
Estimate Sums and Differences
1 , or to 1 estimate sums and differences.
You can round fractions to 0, to __
Name
Name
RW85-RW86
Grade5.indd 43
7/19/07 6:00:31 PM
What information will you use?
MXENL08AWK5X_RT_CH13_L06.indd 1
the morning. He spends 15 minutes
getting dressed, 15 minutes eating
breakfast, 5 minutes brushing his
teeth, and 20 minutes riding to school.
What time should he get out of bed?
7. Jeff needs to be at school at 8:00 in
Compare strategies to solve.
RW85
© Harcourt • Grade 5
Reteach
1 are
__ are 6th-graders, __
8th-graders. If 3
8 __
4
are 8th-graders,
7th-graders, and 1
6
then what fraction are 5th-graders?
8. Central School has 5th-, 6th-, 7th-, and
How much money did David have before his purchases?
Check
6. How can you check your answer?
5.
Solve
4. You can write an equation. Let n 5 the amount of money David had before his
purchase. Solve n 5 3.99 1 1.24 1 4.55 1 1.57.
Plan
3. What are two different strategies you could use to solve this problem?
2.
Read to Understand
1. What are you asked to find?
David bought some supplies for the science project. He spent $3.99 for poster board, $1.24
for a glue stick, and $4.55 for color pencils. If David had $1.57 when he left the store, how
much money did he have before his purchases?
Choose a Method
Problem Solving Workshop Strategy:
Compare Strategies
X
X
paper
and
pencil
X
calculator
MXENL08AWK5X_RT_CH13_L07.indd
6/15/07 1:32:47 PM
1
9
6
1. __
512
__
12
RW86
12
2. ___
5
7 2 ___
Choose a method. Find the sum or difference.
Write it in simplest form.
4
8
3. 3
__ 2 3
__
The chart states that if one denominator is a multiple of the other,
use paper and pencil.
2 1 ___
1 1 ___
9 .
7 5 ___
7 becomes ___
Rewrite using equivalent fractions: __
5 10
10 10 10
9 .
7 5 ___
__ 1 ___
So, 1
5 10 10
5 is a multiple of 10.
Choose a method. Find the sum. Write it in simplest form.
7
1 1 ___
__
5
10
1 1 ___
7 5 10
The fractions do not have like denominators: __
5 10
Is one denominator a multiple of the other? Yes
Denominators are not alike
and are not multiples of one
another.
One denominator is a multiple
of the other.
Denominators are alike.
mental
math
You can use this chart to decide whether to use mental math,
paper and pencil, or a calculator to add or subtract fractions.
Name
Name
Reteach
6/15/07 1:34:37 PM
© Harcourt • Grade 5
Grade5.indd 44
1
1
1__
4
4.
10
4
3
__
1 2 10
2
_
3
3
_
12
4
1. 5
1
__
5.
11
5
8
3
_
1
_
4
RW87
5
_
8
1
_
14
2
2. 2
Use fraction bars to find the sum. Write it in simplest form.
12 12 12
6.
2
1
_
14
2
5
_
6
1
_
13
6
3. 3
1
STEP 3 Count like bars to find the sum.
1 bars is 2___
7 .
Two 1 bars and seven ___
12
12
7.
__ 1 11
__ 5 2___
So, 11
4
3
12
1
1__
3
1 __
1
1 __
__
1
12 12 12 12
1
1__
4
1
1__
3
1 __
1 __
1 __
1
__
1.
1 and 1__
STEP 2 Draw like fraction bars to model 1__
4
3
1
_
3
1
_
4
STEP 1 Draw a picture to show the addends.
1
__ 1 11
__
11
4
3
5
1
_
Reteach
6/13/07 9:31:29 AM
© Harcourt • Grade 5
1
4
_
_
52 .
2
8
1
3
_
_
2 1
2
8
3
1
1
_
_
_
21 51 .
2
8
8
4
3
1
_
_
_
21 51 .
8
8
8
2
3
_
__
2 1 10
5
MXENL08AWK5X_RT_CH14_L02.indd 1
4. 4
5. 5
RW88
1
5
_
_
22
6
4
Use fraction bars, or draw a picture to find the difference.
Write it in simplest form.
1
1
4
1
_
_
_
__
1. 6 2 4
2. 3 2 2
3
6
5
10
So, 2
2
3. Cross out bars to subtract 1__
3.
STEP 3 Subtract 1__
8
8
Count the bars that are left.
2
1 bars.
STEP 2 Rename the bars with __
4
2
Fraction bars can help you subtract mixed numbers.
Fraction bars can help you add mixed numbers.
Draw a picture to find the difference.
Write the answer in simplests form.
1
STEP 1 Draw a picture to show 2 _ .
2
Model Subtraction of Mixed Numbers
Model Addition of Mixed Numbers
Add. Write the answer in simplest form.
Name
Name
MXENL08AWK5X_RT_CH14_L01.indd 1
RW87-RW88
6/18/07 6:03:40 PM
6. 2
3. 6
5
2
_
__
21
3
12
1
3
_
_
24
4
8
4
_
8
1
_
2
1
_
2
© Harcourt • Grade 5
Reteach
2
2
2
6/13/07 9:41:07 AM
Grade5.indd 45
4 13
9
__
__
__
11
5 .
12
12 12
3
1
1
_
_
__
.
1 1 57
4
3
12
1
__
.
12
1. 2
1
2
_
_
14
9
6
2. 10
3
5
_
_
15
6
4
RW89
3. 11
5
7
_
_
29
8
6
Find the sum or difference. Write it in simplest form.
So, 5
Write the answer in simplest form 7
13
13
__
__
1656 .
12
12
4
9
__
__
11
56
12
12
Add the sums:
5
Step 4 Add the whole numbers.
5
Step 3 Add the fractions.
4. 18
Reteach
6/13/07 9:46:00 AM
© Harcourt • Grade 5
1
3
_
_
2 14
5
2
3
3
1
_
_
_
2 1 5 .
2
4 4
6
3
1
3
3
_
_
_
_
_
2 1 or 1 2 1 5 .
2
4
4
4
4
5. 5
1
3
_
_
2245
3
4. 6
MXENL08AWK5X_RT_CH14_L04.indd 1
2. 5
5
1
_
_
21 5
8
5
1. 3
6. 4
5
1
_
_
22 5
6
6
RW90
3. 4
3
7
__
__
2 3 10 5
10
Use fraction bars to find the difference. Write it in simplest form.
So, 2
Subtract, 2
4
_
Rename one whole bar as .
4
1
3
1 _2
_
_
_
and . Rename as .
2
4
2 4
1
1
_
_
using two whole bars and one bar.
2
2
3
1
_
_
2 1
2
4
Think of the LCD for
Model 2
2
3
1
_
_
21 5
5
5
5
1
_
_
21 5
3
6
When you subtract mixed numbers you may need to rename the whole numbers.
When you add or subtract mixed numbers, you may need to
rename the fractions as fractions with a common denominator.
3
1
_
_
Find the sum. Write the answer in simplest form. 5 1 1 .
4
3
3
1
_
_
Step 1 Model 5 and 1 .
4
3
1
9
4
3
3
1
_
__
_
__
Step 2 The LCD for _ and _ is twelfths, so rename 5 as 5
and 1 as 1
.
4
12
3
12
4
3
Subtraction with Renaming
Record Addition and Subtraction
Use fraction bars to find the difference. Write it in simplest form.
Name
Name
MXENL08AWK5X_RT_CH14_L03.indd 1
RW89-RW90
7/19/07 6:00:50 PM
Reteach
© Harcourt • Grade 5
6/13/07 9:48:11 AM
Grade5.indd 46
STEP 2
STEP 3
STEP 4
35 5 7 15
3573
____
___
7__
9
9 35
45
39 5 15 ___
9
__ 5 15 1
____
151
5
5 39
45
STEP 2
Write equivalent
fractions using the LCD.
Estimate.
45
9
15___
5
___
739
45
___
2 715
45
__
45
54
14___
1.
___
1 8 11
15
__
2
4 __
9
2.
6
_
5
2 8__
3
12__
8
RW91
3.
2
_
__
1 91
2
2__
7
4.
6/13/07 9:54:14 AM
© Harcourt • Grade 5
Reteach
7
2 7 ___
12
__
1
15 __
3
13
43 5 7___
______
5 7 39
4543
15
STEP 3
Rename so you can subtract. Subtract
the whole numbers. Subtract the fractions.
Write the answer in simplest form.
15 2 7 5 8
___
1114
15
Add the whole Write the answer
numbers. Add in simplest form.
the fractions.
10
4 5 11___
4
___ 5 10 1 15
___ 1 ___
1019
2___
15
15
15 15
15
9
__
So,
22
1 8___
3
15
__
3
19
1 8__
10___
5
15
_
Estimate. Find the sum or difference. Write it in simplest form.
5: 5,10,15, ..., 45
9: 9,18,27, ..., 45
STEP 1
Find the LCD.
Subtract.
358 333
9
8__
5 8___
5
533
15
358
__ 2 7__
151
5
9
Find the LCD.
Write equivalent
fractions using the
LCD.
10
2 3 5 5 2___
2
__
2
3: 3, 6, 9, 12, 15 ... 3 5 2 3 3 5
15
5: 5, 10, 15, 20 ...
STEP 1
4
7 ft farther than Raul
not the shortest hit; 1 __
8
Sonya
MXENL08AWK5X_RT_CH14_L06.indd 1
© Harcourt • Grade 5
Reteach
grade watch baseball and football. If
132 students watch football and 119
watch baseball, how many students
watch both baseball and football?
7. In October, 200 students in Fiona’s
RW92
down the total distances they ran last
week. The farthest distance ran was
5 miles farther
1 miles. Deidre ran 8 __
19 __
8
8
3
__
than Amy and 3 miles farther than
8
17 miles less than
Brad. Clarence ran 6 ___
20
Brad. Who ran the shortest distance,
and what was that distance?
6. Amy, Brad, Clarence, and Deidre wrote
Use logical reasoning to solve.
5. How did the strategy help you solve the problem?
Check
4. Who hit the ball the farthest, and what distance was the ball hit?
not the shortest hit
__ ft farther than Raul; 3 1
__ ft less than Stanley
not the shortest hit; 71
2
8
Lenny and Sonya hit the ball farther.
Distance Clues
Stanley
Raul
Lenny
Player
Solve
3 feet.
3. Complete the table. Remember that the shortest hit was 100 __
2. How can you organize the data to solve the problem?
Plan
Read to Understand
1. How could you write the question as a fill-in-the blank sentence.
Lenny, Raul, Stanley, and Sonya wrote down their longest hits during batting practice. The
3 feet. Lenny hit the ball 7 __
1 feet farther than Raul, but 3 __
1
shortest of the 4 hits was 100 __
4
2
8
__ feet farther than Raul did. Of Stanley, Raul,
feet less than Stanley. Sonya hit the ball 1 7
8
Sonya, and Lenny, who hit the ball the farthest, and what distance was the ball hit?
Problem Solving Workshop
Strategy: Use Logical Reasoning
Practice Addition and Subtraction
Estimate. Then write the sum or difference in simplest form.
3
__
22
3
Estimate. 1 9
Add.
_
__
1 83
12
5
Name
Name
MXENL08AWK5X_RT_CH14_L05.indd 1
RW91-RW92
7/19/07 6:01:08 PM
6/15/07 1:35:42 PM
Grade5.indd 47
Y
3
6
__
__ 3 5
9. 2
5
Find the product.
1 3 __
25
5. __
5
3
1.
4
5
5
4
__ 3 3
__ 5
10. 1
3 3 __
25
6. __
2.
8
5
4
3
__ 3 2
__ 5
11. 3
5 3 __
45
7. __
3.
RW93
Write the product each model represents.
• The area of the blue shaded part is 1 3 1, or 1 square units.
__ .
• The fraction of the part that is shaded blue is 1
4
1
1
1 5 __
__ 3 __
So, 2
4.
2
Use the denominator of the first factor, 2, to make rows.
1 of the yellow column blue.
Shade __
2
• The area of the rectangle is 2 3 2, or 4 square units.
Use the denominator of the second factor, 2, to make columns.
Shade the left column yellow.
1 3 __
1
__
2
2
3
5
8
3
6/13/07 11:33:00 AM
© Harcourt • Grade 5
Reteach
__ 3 2
__ 5
12. 7
35
1 3 __
8. __
4.
B
Y
You can use an area model to record multiplication of fractions.
You can use grid paper to model multiplication of fractions.
You will use the grid to make and shade a rectangle.
5
MXENL08AWK5X_RT_CH15_L02.indd 1
3
10
7. 1
__ 3 2
__ 5
3
6
4. 2
9 5
__ 3 ___
5
1. __
3 3 5
__ 5
10
8
5
__
RW94
12
2 5
3 __
3
8
5 5
9 3 ___
8. ___
5.
9
3 5
2. 4
__ 3 __
Find the product. Write in simplest form.
6 5 ______
6 4 6 5 __
1
__ 3 3
__ 5 ___
So, in simplest form, 2
3 4
12 12 4 6 2
3 ___
6
2
__
__
So, 3 3 4 5 12 .
area
of rectangle shaded twice ___
____________________________
5 6
12
area of big rectangle
Use the two areas to write a fraction.
• The rectangle has an area of 12 square units.
2
7
8
9
5
4 5
1 3 __
9. __
3
6. __
4 5
2 3 __
6 5
1 3 __
3. __
• The area of the rectangle that is shaded twice is 6 square units.
Then use the denominator of
the first factor, 3, to make
2 of the
rows. Shade __
3
shaded columns.
Use the denominator of the second
factor, 4, to make columns.
__ of the columns.
Shade 3
4
2
__ 3 3
__
3
4
Find the product. Write the answer in simplest form.
Record Multiplication of Fractions
Model Multiplication of Fractions
Find the product:
Name
Name
MXENL08AWK5X_RT_CH15_L01.indd 1
RW93-RW94
6/18/07 6:04:09 PM
Reteach
© Harcourt • Grade 5
6/13/07 10:01:07 AM
Grade5.indd 48
1365
7. __
3
4
__ 5
4. 2 3 1
4
__ 5
1. 5 3 3
Find the product.
40 5 8
4 5 ___
So, 10 3 ___
5
5
8.
5.
2.
Step 4 Count the shaded parts.
There are 40 shaded parts.
RW95
1 3 10 5
__
4
2355
__
3
2345
__
5
4
Step 3 The numerator of the fraction __
5 is 4.
So shade 4 parts of each rectangle.
__ is 5.
Step 2 The denominator of the fraction 4
5
This means there are 5 equal parts. So
divide the rectangles into 5 equal parts.
Step 1 Draw 10 rectangles.
9.
6.
3.
2
1
__
335
__ 5
235
6
__ 5
931
5
Reteach
6/13/07 10:02:45 AM
© Harcourt • Grade 5
You can use paper and pencil to multiply mixed numbers.
2
1 3 2 __
Find the product.
1 __
2
3
You can use a model to multiply a fraction and a whole number.
131
__
1 __
3 2
4 3 2 __
1
1 __
5
5
1
__ 3 3 __
21
5
2
MXENL08AWK5X_RT_CH15_L04.indd 1
7.
4.
1.
Find the product.
2
1
So, 1 __ 3 2 __ 5 4.
2
3
5
6
RW96
3
1
__ 3 1 __
8. 3 5
4
3
__ 3 2 2
__
8
3
2
__ 3 1 __
5. 1 3
2.
2
8
3
3
1
__ 3 3 __
9. 1 5
3
2322
__
6. 2 __
8
3. 2 5
__ 3 1 1
__
54
24
___
Step 3
Write the product in simplest form
6
3
8 5 24
__ 3 __
___
2 3
6
© Harcourt • Grade 5
Reteach
(3 3 2) 1 2 8
2 5 ___________
2__
5 __
3
3
3
3 1) 1 1 3
__ 5 (2
___________
5 __
11
2
2
2
Step 2
Multiply the fractions.
Step 1
Write each mixed number as a fraction.
First multiply the denominator by the whole
number, then add the numerator.
Multiply with Mixed Numbers
Multiply Fractions and Whole Numbers
Find the product.
__
10 3 4
5
Name
Name
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2
10 10 10 10 10 10 10 10 10 10
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1 ___
1
1 ___
___
5.
1.
3
1
6. 3 4 __
12
__
141
5
4
1
1 4 ___
2. __
141
__
__
2 4
Use fraction bars to find the quotient.
1 5 2.
__ 4 ___
So, 1
5 10
5
1.
There are 2 tenths in __
2
5
__
7. 2 4 1
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Step 3 Count the number of shaded tenths.
Step 2 Divide the parts into tenths.
Step 1 Draw one whole rectangle and shade
one fifth of it.
141
__
3. __
4
10
__
8. 3 4 1
2
1
__ 4 ___
4. 1
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__
447
8
Divide.
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5
3
__
6. 4 4 2
__
5. 4 4 2
5
3
2. 3 4 __
4
3
1. 1 4 __
7
4 __
8
8 5 ___
32
3 __
7
7
8
__
8. 1 4 5
6
7. 3 4 __
7
1
4. 4 4 __
2
© Harcourt • Grade 5
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32
4
___ 54__
7
7
1
4
__
4
__ 3 8
__
1 7
1
4
__
2
3. 6 4 __
3
Think: The reciprocal
8.
__ is __
of 7
8 7
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Find the quotient. Write it in simplest form.
__ 5 4 4
__ .
So, 4 4 7
7
8
Step 4
Rewrite product as a mixed fraction, if needed.
Step 3
Multiply
Step 2
Use the reciprocal of the divisor to write a
multiplication problem.
4.
Think: Write 4 as __
1
Find the quotient. Write it in simplest form.
You can use pictures to model division of fractions.
Step 1
Rewrite 4 as a fraction.
Divide Whole Numbers by Fractions
Model Fraction Division
1
1
__ 4 ___
5 10
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Name
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3
8
__
3
1 4 __
3 __
3 8
__ 5 4
14 1
4
8
4
__ 4 3
__
3. 2 1
9
6
545
__
4. __
5.
2.
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Divide. Write the answer in simplest form.
1.
Write a division sentence for each model.
Write a division sentence for
3 41
__
4 ___
10 5
Step 2
Write the reciprocal of the
divisor.
To write the reciprocal, switch
the numerator and
denominator of the divisor.
80 5 8 8
__ 4 3
__ 5 ___
__
So, 3 1
.
3 8
9
9
We can use a model to write a division sentence.
143
__
3 __
3 8
The divisor is always the
second number.
Step 1
Locate the divisor.
Step 3 Write the
multiplication problem.
6.
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80
__ 5 ___
38
3
9
5
__ 4 ___
32
7 14
3
10
___
reciprocal and change 4 to
__
3. The mixed number 3 1
3
10
__
written as a fraction is
3
143
__
3 __
3 8
Replace the divisor with its
How can you check to see if your answer is reasonable?
How many fish are goldfish?
What equation can you write to solve?
What operation can you use to solve the problem? Explain your choice.
What are you asked to find?
Twelve students in Danita’s
cooking class play a musical
__ of the entire
instrument. This is 3
4
class. How many students are in
Danita’s cooking class?
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6.
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7.
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1
Lance spends __
30 of each year
12
studying on weekends and ___
60
of each year studying on weekdays.
What fraction of the year does
Lance study?
Tell which operation you would use to solve the problem.
Then solve.
5.
4.
3.
2.
1.
2 are goldfish.
Tanya has 10 fish in her aquarium. Of these fish, __
5
How many fish are goldfish?
Problem Solving Workshop Skill: Choose the Operation
Divide Fractions
We can rewrite all division problems as multiplication problems.
3.
1 4 __
Write the answer in simplest form.
Divide 3 __
3 8
Write the division problem as a multiplication problem.
Name
Name
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Red
3 blue stripes
8 stripes in all
Red
5. triangles to hearts
4. hearts to all shapes
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2. all shapes to triangles
1. circles to triangles
Write each ratio in three ways. Then name the type of ratio.
__
The ratio can be written: 3 to 8, 3:8, or 3
8
.
The type of ratio is: part to whole
3 stripes
What is the ratio of blue stripes to all stripes?
The ratio can be written as:
5 to 3
5:3
5
__
3
The type of ratio is: part to part.
Red
5 red stripes
Count the stripes of each color.
Blue
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6. circles to all shapes
3. circles to hearts
Red
Blue
(632)
12
6
12
3
3. 5:25
13
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8
__ and 10
___
5. 5
7
21
12
_ and ___
6. 4
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40
8
7
___ and _
7. 35
Tell whether the ratios form a proportion. Write yes or no.
Write two equivalent ratios for each ratio.
Use multiplication and division.
___
1. 15
2. 12 to 16
20
3
8 5 __
254
2 are proportions.
__
__ and ___
A proportion is an equation that shows that 2 ratios are equal.
6 (642) 3
So, two equivalent ratios for 4 to 6 are 8 to 12 and 2:3.
2 = 2 to 3
4 = (442)
__
_____ = __
Use division to write an equivalent fraction.
Divide the numerator and the denominator
by the same number.
6
8 = 8 to 12
4 = (432)
__
______ = ___
Use multiplication to write an equivalent
fraction. Multiply the numerator and the
denominator by the same number.
4
4 to 6 becomes __
6
42
6
© Harcourt • Grade 5
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5
28
8. ___ and __
4. 9 to 18
Write to equivalent ratios for 4 to 6. Use multiplication and division.
Write the ratio as a fraction with the first
number being the numerator and the second
number being the denominator.
Write the ratio in three ways. Then name the type of ratio.
Blue
Equivalent ratios make the same comparison. You can use pictures to model
equivalent ratios.
A ratio is the comparison of two quantities. Ratios can compare part to part, part to whole, or
whole to part.
Red
Algebra: Equivalent Ratios and Proportions
Understand and Express Ratios
Red stripes:blue stripes.
Name
Name
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hours
miles
______
The denominator is 14.
Divide the numerator and
denominator by 14.
14
770
____
2. 65 miles in 5 hours
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8. 624 miles in 13 hours
7. 24 paintings in 12 days
$42 for 6 hours of work
5.
4. 72 dog toys in 12 boxes
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9. $1.14 for 6 ears of corn
6. 27 plants in 3 rows
3. $2.70 for 6 fruit bars
55 5 55
770
4 14 5 ___
_________
14 4 14
1
Unit Rate
Write each ratio in fraction form. Then find the unit rate.
1. 231 miles in 3 hours
Write each ratio in fraction form.
The unit rate, 55, is the number of miles traveled per hour.
14
770
____
Rate
Write the rate as fraction
Write the ratio in fraction form. Then find the unit rate.
770 miles in 14 hours
A unit rate is a rate that when written as a fraction has a 1 as its denominator.
3
3
1
3.6
Map Distance (cm)
Actual Distance (km)
1
7
720
200
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2. 6.5 in.
3. 1.25 in.
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4. 3 in.
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5. 4.5 in.
28
200
200
3
actual distance
1:200
1 inch
0
1 inch 5 200 miles
*
Georgetown
COLORADO
scale
The map distance is given. Find the actual distance.
The scale is 1 in. = 300 mi.
1.
Complete the ratio table.
So, the actual distance is 720 miles.
3
distance on map
2
*
Northglen
Write ratios for the map scale and the actual distance.
The map distance is given. Find the
actual distance of 3.6 in. The scale
is 1 in. 5 200 mi.
A map scale is the ratio that compares the |
distance on a map to the actual distance.
A scale drawing is a reduced or enlarged
drawing whose shape is the same as an
actual object and whose size is
determined by the scale.
Understand Maps and Scales
Ratios and Rates
A rate is a ratio that compares two quantities having different units of measure.
Name
Name
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200
2 Multiply. 2.5 3 77 5 192.5
77
80
2.5
grouped by age:
21–30, 31–40, 41–50, and 51+.
The ages of the students are 25, 68,
33, 61, 46, 59, 46, 29, 35, 28, 21,
51, 47, and 79. Make a table to
show how many students are in
each age group.
6. In aerobics class, the students are
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Check
5. How can you check to see if your answer is reasonable? Explain.
4. How could you write your answer in a complete sentence.
Weight on Venus
Weight on Earth
1 Divide. 200 4 80 5
3. Make and complete the table to solve the problem.
Solve
Plan
2. How can making a table help you answer the question?
1. How could you write the question as a fill-in-the-blank sentence?
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Percent is the fraction part of a number to 100. Fractions and
decimals can be written as percents. The symbol for percent is %.
This symbol means “per hundred.”
Tania weighs 80 pounds on Earth and 77 pounds on Venus. Tania’s father weighs 200
pounds. How much would Tania’s father weigh on Venus?
2.
3.
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4.
5.
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6.
Write a decimal and a percent to represent the shaded part.
1.
Write a fraction and a percent to represent the shaded part.
Decimal: 0.54.
Divide. 54 4100 5 0.54
54
Fraction: ____
100
Percent: 54%
54 of the hundred squares are shaded.
60
Decimal: ____
100
Write a decimal and a percent to represent
the shaded part.
Percent: 60%
60 squares are shaded.
shaded 5 ____
60 .
Write the fraction as parts: _______
100
whole
60
____
Fraction:
100
The grid is divided into 100 squares.
Write a fraction and a percent to represent
the shaded part.
Understand Percent
Problem Solving Workshop Strategy: Make a Table
Read to Understand
Name
Name
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5. 76%
4. 32%
8. 0.45
11. 0.10
8
10. ___
20
5
3
7. __
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Write each fraction or decimal as a percent.
2. 35%
1. 80%
12. 0.62
10
9
9. ___
6. 51%
3. 40%
Write each percent as a decimal and as a fraction in simplest form.
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© Harcourt • Grade 5
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To write a fraction in simplest form, divide the numerator and the denominator by the same
number. Keep doing this until 1 is the only common factor.
98 4 2 5 49
98 5 ________
____
___
100 100 4 2 50
So, 98% as a decimal 5 0.98
49
98% as a fraction 5 ___
50
98
• 98% also means ___
100
• 98% means ninety-eight hundredths, or 0.98.
Write 98% as a decimal and as a fraction in simplest form.
20%
30%
40%
50%
60% 70%
2
3
4
20%
5
6
7
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7. 25% of 20
5. 40% of 35
6. 50% of 26
3. 50% of 12
8
80%
Find the percent of each number.
2. 90% of 20
1. 40% of 80
So, 25% of 24 is 6.
Change the color of the counters in 1 of the 4 groups.
Count the counters whose color you changed.
1
_
Since 25% 5 , separate the counters into
4
4 equal groups.
Show 24 counters.
What is 25% of 24?
1
2
So, 90% of 10 is 9.
Count by 1s to find 90% of 10.
10%
Flip over 90% of the counters.
1
Each counter also represents 1, since 10 3 1 5 10.
10%
What is 90% of 10?
Show 10 counters. Each counter represents 10%.
You can use two-color counters to model percent of a number.
Find Percent of a Number
Fractions, Decimals, and Percents
Percents can be written as decimals or as fractions and vice versa.
Name
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10
100%
8. 75% of 28
4. 75% of 32
9
90%
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6/13/07 10:57:42 AM
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R
B
6 equal sections.
3
2
4 sections
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1. red
2. blue
G
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Y
© Harcourt • Grade 5
Reteach
3. a color other than red
Use the spinner to find the probability of spinning each event.
• Place the number of favorable outcomes in the numerator of the probability
formula:
__
P(color other than blue) 5 4
6
4
1
• Next, count the number of favorable outcomes
number of favorable outcomes
P(color other than blue) 5 ____________________________
6
• Then, place the number of equally likely outcomes in the probability formula:
• First, find the number of equally likely outcomes
Use the spinner to write the probability of spinning a color
other than blue.
B
Probability is the likelihood that an event will happen.
A prediction is a reasonable guess about the possible
outcome, or result, of a probability experiment.
An event is the set of one or more outcomes in a probability
experiment. The sample space of an event is the set of all
possible outcomes. If the sample space is divided into equal
parts, each part is equally likely to be selected. The
theoretical probability can be expressed in words, as a
ratio, or as a fraction:
number of favorable outcomes .
P(event) 5 _______________________________
number of equally likely outcomes
The probability of an event can range from impossible to
certain, or from 0 to 1.
Y
Probability Experiments
Outcomes and Probability
Total
Number of Pulls
12
Red
4
Blue
Marble Experiment
7
3
10
Blue
Green
Total
Red
Tally
Tile Experiment
Color
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1
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predict that Sarah would pull a blue tile or green tile more
often if she pulled tiles from the bag 60 more times?
Explain.
3. Based on the experimental probabilities, would you
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© Harcourt • Grade 5
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2. What is the experimental probability of Sarah pulling a blue tile? a red tile? a green tile?
and put the tile back in the bag 20 times. Predict
how many times out of 40 pulls that Sarah would
pull a red tile from the bag.
1. Sarah pulled a tile from a bag, recorded its color,
For 1–3, use the table.
So, you can predict that a red or blue marble will be pulled 32 times out of 50.
16 3 2 5 32
Multiply by 2 since 25 goes into 50 2 times.
9
Green
25
Add the times a red marble was pulled to how many times a blue marble was pulled.
12 1 45 16
Count the total number of pulls/tally marks.
Predict the number of times out of 50 pulls
that Dylan would get a marble that is either
red or blue.
A box contains 6 black marbles and 3 white marbles.
Toby shakes the box and pulls a marble without
looking. He records the color marble on a tally
table, replaces the marble, and repeats the experiment.
An experimental probability of an event is the ratio
of the number of times an event occurs to the total
number of times the activity is performed.
Name
Name
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14 coin tosses
pulls; 30 more pulls
1. 12 blue marbles in 20
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cube tosses; 45 more
tosses
2. 8 evens in 12 number
more spins
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© Harcourt • Grade 5
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3. 4 purple in 24 spins; 60
Express the experimental probability of as a fraction in simplest form. Then predict
the outcomes of future trials.
• Place the number of favorable outcomes in the numerator of the probability
formula and simplify:
6 53
__
P(color other than blue) 5 ___
14 7
• Finally, to predict the outcome of future trials, multiply the probability by the
number of future trials.
3 3 35
___ 5 15
P(color other than blue) 5 __
7
1
3
__
So, 6 heads in 14 coin tosses is in simplest form.
7
You can expect 15 heads in 35 coin tosses.
• Then, place the number of trials as a denominator in the probability formula:
number of times an event occurs
P(heads) 5 ______________________________
14
• Next, count the number of times an event occurs
4 sections
• First, find the number of trials
Use the spinner to write the probability of spinning a color
other than blue.
Express the experimental probability of 6 heads in 14 coin
tosses as a fraction in simplest form. Then predict the
outcome of 35 more tosses.
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cereal, fruit, yogurt, egg
© Harcourt • Grade 5
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chooses 3 items each morning instead of
two. How many different breakfasts can
he make from Joshua’s list.
8. Suppose Joshua’s brother Steven
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Joshua make if he choose exactly two
different items from this list each morning?
7. How many different breakfasts can
Make a list to solve.
6. How do you know your answer is correct?
Check
5. What is the answer to the question?
4. How can you use the strategy to solve the problem?
Solve
3. What strategy can you use to solve the problem?
Plan
2. What are you asked to find?
1. What math details can you identify?
Read to Undertand
Nia, Sam, and Toni are waiting in line to be seated
at Kaye’s Restaurant. How many different ways can
they line up?
Problem Solving Workshop Strategy:
Make an Organized List
Probability and Predictions
Experimental probability is a number taken from an actual situation. To find
experimental probability of an event, write the ratio of the number of times
the event occurs to to total number of trials.
of times an event occurs.
______________________________
Experimental probability 5 number
total number of trials
Name
Name
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A combination is a selection of different items in which the
order is not important. An example of a combination is a table
place setting that consists of a cup, saucer, plate, and bowl.
Listing them in that order or as bowl, plate, cup, and saucer
does not change the combination of dishes.
To find all possible outcomes of an event, you can use either of
the following:
3
Another event has n
possible outcomes
5
Total outcomes of
both events
occurring together
Spinner 2
red
blue
yellow
red
blue
yellow
red
blue
yellow
Outcomes
red, red
red, blue
red, yellow
blue, red
blue, blue
blue, yellow
yellow, red
yellow, blue
yellow, yellow
green, blue, or red shirt,
and white or tan shorts
1. choosing outfits with a
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tossing a number cube
labeled 1 to 6.
2. tossing a coin and
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© Harcourt • Grade 5
Reteach
labeled 1 to 6 and
spinning a spinner
labeled 1 to 4.
3. tossing a number cube
Draw a tree diagram or use the Fundamental Counting Principle
to find the total number of outcomes.
So, there are 9 possible outcomes.
yellow
blue
red
Spinner 1
First, Make a column for each event. Next, list all possible
Outcomes for the 1st event and the second event.
Draw a tree diagram.
Draw a tree diagram or use the Fundamental Counting Principle
to find the total number of outcomes for using two spinners, both
with 3 equal sections colored red, blue, and yellow.
One event has m
possible outcomes
• Fundamental Counting Principle. In Fundamental Counting Principle, the
total possible outcomes of two events can be stated as a multiplication formula.
Topping
cheese
pepperoni
cheese
pepperoni
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letters T, E, A.
1. ways to arrange the
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A, T so they form a real
word
2. ways to arrange letters C,
© Harcourt • Grade 5
Reteach
and Martin can stand in a
ticket line
3. ways that Amy, Greg, Lisa
Make a list or draw a tree diagram to find the total possible choices.
Crust
thin
thin
thick
thick
So, there are four possible combinations.
Step 3 Count the number of pizza choices.
Step 2 List all possible outcomes each event. A table can help
you. Make sure each topping is listed with each type of crust.
The pizza is the same if you order the crust either
before or after the topping. So, this is a combination.
Order does not matter.
Step 1 Decide if it is a combination or an arrangement.
pizzas with choices of thick or thin crust and either cheese or
pepperoni topping
Make a list or draw a tree diagram to find the total number of possibilities.
An arrangement is a group of items that are ordered. An
example of an arrangement is assigned seats in a math class.
Only one person is assigned any particular seat.
Combinations and Arrangements
Tree Diagrams
• Tree diagram. A tree diagram lists all possible choices or outcomes.
Name
Name
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A
Z
X
6
7
4
8
9
10
5
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© Harcourt • Grade 5
MXENL08AWK5X_RT_CH18_L02.indd 1
2. /KXN
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3. /MXN
4. /KXL
J
K
N
M
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X
Z
© Harcourt • Grade 5
L
obtuse angle
5
6.
4
13
50 0
1. /LXN
3
12
0
60
right angle
2
110
70
5.
2
Y
100
80
Estimate the measure of each angle. Then use a protractor to find the measure.
Z
T
1
90
acute angle
X
Y
mm
cm
80
70
100
110
4.
T
U
W
U
1
60
20
50 0 1
13
W
Step 3 Measure /UXY. So, /UXY is 105°.
intersecting lines
Reteach
X
Y
Step 2 Place the center point of the
protractor _
on the vertex, point
›
X, so that UX passes through 0˚.
T
U
W
Point
Step 1 Think:
_›
_› X is the vertex. Rays
XY and XU from the sides of an angle.
3.
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G
Z
perpendicular lines
E
Y
180
0
MXENL08AWK5X_RT_CH18_L01.indd 1
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2.
C
F
X
W
180°
170
10
parallel lines
H
D
T
U
135°
Use a protractor to measure /UXY. A protractor is a tool for measuring angles.
The measure of /UXY is about 90˚.
Compare /UXY to the benchmark 90˚.
Think: /UXY appears to be greater than 45˚.
/UXY appears to be less than 135˚.
90°
160
20
1.
B
45°
Estimate the measure of /UXY.
0°
You can use benchmarks to estimate angle measures.
Measure and Draw Angles
0
30
Draw the following on the grid. Label your drawings.
Straight angle 5 180°
Obtuse angle . 90°
Acute angle , 90°
Right angle 5 90°
‹_›
Look for lines that cross at exactly one point.
‹_›
AD and BC cross at exactly one point.
‹_›
‹_›
So, AD and BC are intersecting lines.
Use the figure. Name an example of intersecting lines.
Parallel lines are lines in a plane that are always the same distance apart.
Perpendicular lines are two lines that intersect to form right angles.
Intersecting lines are lines that cross at exactly one point.
A line segment is part of a line between two endpoints and all of the
points between them. Use both endpoints to name a line segment.
A line is a straight path in a plane. It has no endpoints and can be
named by any two points on the line.
A plane is an endless flat surface.
A point is an exact location.
In geometry, objects have special names.
Points, Lines, and Angles
Name
4
14 0
0
30
15
0
Name
15
20
160
0
40
0
10
180 170
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14
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4 sides and
angles
3 sides and
angles
5 sides and
angles
6 sides and
angles
Regular Polygons
Pentagon
Hexagon
8 sides and
angles
Octagon
10 sides and
angles
Decagon
4 sides and
angles
5 sides and
angles
6 sides and
angles
1.
2.
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3.
Name each polygon and tell if it is regular or not regular.
The polygon has 10 sides.
It is a decagon.
Measure the sides to see if they are each the same length.
The sides are not each the same length.
So, the figure is a pentagon that is not regular.
Octagon
4.
8 sides and
angles
Name the polygon below and tell if it is regular or not regular.
3 sides and
angles
Triangle
Polygons that Are Not Regular
Quadrilateral
Pentagon
Hexagon
6/13/07 11:06:22 AM
© Harcourt • Grade 5
Reteach
10 sides and
angles
Decagon
A polygon that has sides and angles that are not the same measure is not regular, but it is
still named by the number of sides and angles.
Quadrilateral
Triangle
MXENL08AWK5X_RT_CH18_L04.indd 1
page, predict how many lines will be
drawn for 7-, 8-, and 9-sided figures if
you connect their vertices.
6
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© Harcourt • Grade 5
Reteach
with 1-inch sides is 6 inches. The distance
around a regular hexagon with 2-inch
sides is 12 inches, and the distance
around a regular hexagon with 3-inch
sides is 18 inches. What is the distance
around a regular hexagon with 6-inch
sides?
7. The distance around a regular hexagon
5
2
+3
5
4
6. Using the information from the top of the
Identify the relationship. Then solve.
Number of Sides
Number of Lines
Connecting Vertices
out the missing parts of the table?
3. Using the relationships already demonstrated by the table, how would you fill
How many lines did you draw?
2. Connect the vertices of the hexagon at the right.
1. What are you asked to do?
Connect the vertices within a square, and a regular pentagon.
A square and a regular pentagon are shown at the right.
Count the lines within each figure. How many lines
would you draw within a regular hexagon?
Problem Solving Workshop Skill:
Identify Relationships
Polygons
A polygon is a closed plane figure formed by three or more line segments. Polygons are
named by the number of sides and angles. A regular polygon has sides that are all the same
length and angles that are all the same measure.
Name
Name
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The length of a diameter is always twice the length of a radius.
N
M
B
13.5 cm
4. Name a radius.
V
H
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Complete 5–6. Then use a compass to draw each circle.
Draw and label the measurements.
5. radius 5 1.5 cm
6. radius 5
diameter 5
diameter 5 2.5 cm
3. Name a chord.
Use data for 1–4, use the circle at the right.
1. Name the circle.
2. Name a diameter.
So, the diameter is twice this length or 7 cm.
V
F
X
You know that the radius is 3.5 cm.
Use a compass set at 3.5 cm to draw this circle around a center point.
Label your center point, point V.
Complete. Then use a compass to draw the circle. Label the measurements.
Radius: 3.5 cm
Diameter:
A chord has its endpoints on the circle. NO and MP are chords.
•
A diameter of a circle passes through the center and has it
endpoints on the circle. MP is a diameter.
A radius of a circle connects the center of the circle with any
point on the circle. BL is a radius. BM and BP are also radii.
O
P
L
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© Harcourt • Grade 5
E
R
T
S
W
V
2.
6. GH
MXENL08AWK5X_RT_CH18_L06.indd 1
5. EF
7. /G
3.
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8. /M
Identify the corresponding side or angle.
1.
G
E
L
J
4.
Write whether the two figures appear to be congruent, similar, or neither.
/R corresponds to /U
/S corresponds to /V
So, /T corresponds to /W.
Figures RST and UVW are congruent.
Identify the angle that corresponds to /T
So, the two figures are neither.
Think, do the figures appear to be the same size?
• No
Think, do the figures appear to be the same shape?
• No
The figure on the left appears to have greater height
than the figure on the right.
Tell whether the two figures at the right appear to be congruent,
similar, or neither.
Figures can be congruent, similar, both, or neither.
Congruent figures have the same size and the
same shape. Similar figures have the same shape,
but may or may not be the same size. Corresponding
angles and corresponding sides are in the same
related position in different angles.
A circle is round. All of its points are the same distance from its
center. A circle is named by its center point.
Look at circle B.
Congruent and Similar Figures
Circles
Circle B
Name
Name
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F
K
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M
H
U
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À
6.
2.
7.
3.
9.
10.
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11.
Draw lines of symmetry. Tell whether each figure has
rotational symmetry. Write yes or no.
5.
1.
Tell whether the parts on each side of the line match. Is the
line a line of symmetry?
So, the line on this figure is a line of symmetry.
Fold the cut-out figure along the dashed line.
Use a piece of paper to trace the figure and line.
Cut out the figure and line of you piece of paper.
Tell whether the parts on each side of the line match.
Is the line a line of symmetry?
12.
8.
4.
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© Harcourt • Grade 5
2 cm
15 cm
14 cm
5 in.
2.
5 in.
5 in.
MXENL08AWK5X_RT_CH19_L01.indd 1
4.
5.
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Classify each triangle. Write acute, right, or obtuse.
9 cm
1.
6.
3.
Classify each triangle. Write isosceles, scalene, or equilateral.
It is an acute triangle.
It has three angles, each less than 90°.
Classify the triangle. Write acute, right, or obtuse.
It is a scalene triangle.
It has no equal side lengths.
Classify the triangle. Write isosceles, scalene, or equilateral.
10 cm
10 cm
8 cm
7 cm
right triangle: One angle is right
and the other two angles are acute.
© Harcourt • Grade 5
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4 cm
obtuse triangle: One angle is obtuse
and the other two angles are acute.
isosceles triangle: Two sides
are the same length.
scalene triangle: All sides
are of different lengths.
Use the corner of a paper to classify angles.
acute triangle: All three angles
are acute.
equilateral triangle: All sides
are the same length.
You can classify triangles either according to their sides or according to their angles.
If you fold the picture of the light
bulb in half along the dashed line,
the two parts will match exactly.
The light bulb has line symmetry.
Use a ruler to compare side lengths.
Classify Triangles
Symmetry
A figure has rotational symmetry
if it can be rotated less than 360˚
around a center point and still
match the orginal figure.
Name
Name
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trapezoid
quadrilateral
one pair of parallel sides
You can draw a plane figure based on a description.
You can use this chart to classify quadrilaterals.
rhombus
parallelogram
4 congruent sides
2.
3.
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4.
Describe each quadrilateral using parallel, perpendicular, and congruent.
1.
Classify each figure in as many ways as possible.
Write quadrilateral, parallelogram, square, rectangle, rhombus, or trapezoid.
1 in.
1 in.
1 in.
1 in.
1 in.
1 in.
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pair of angles measuring 60°, the other
pair measuring 120°
2. quadrilateral: four 3-centimeter sides; 1
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side between the angles that measures 4
centimeters
1. triangle: 2 angles that measure 30°; one
Use a protractor and a ruler to draw each figure. Classify each figure.
It has 4 congruent sides and 4 right angles, so the figure is a square.
1 in.
1 in.
Use a protractor to make sure the other 3
angles are right angles.
Draw two more 1 in. lines to connect
your figure.
1 in.
Step 4
Step 3
1 in.
Use a protractor to make sure the angle formed
by the two lines you drew is a right angle.
Draw:
1 in.
Step 2
Step 1
1 in.
A quadrilateral with 4 right angles; 4 congruent sides each measuring 1 inch.
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It has 4 sides so it is a quadrilateral.
Its opposite sides are parallel and congruent so it is a
parallelgram.
It has 4 right angles with opposite sides that are parallel and
congruent. However, not all of its sides are congruent, so it
is a rectangle.
Classify the figure in as many ways as possible.
Write quadrilaterals, parallelogram, square, rectangle, rhombus, or trapezoid.
square
rhombus
rectangle
rectangle
parallelogram
4 right angles
parallelogram
quadrilateral
opposite sides are parallel
opposite sides congruent
Draw the figure on a coordinate plane. Classify the figure.
Draw Plane Figures
Classify Quadrilaterals
quadrilateral
4 sides
Name
Name
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The one base
is a hexagon.
hexagonal
pyramid
hexagon
prism
pentagonal
prism
shape of a
ball and has
no base
The two
bases are
circles.
The two
bases are
hexagon.
The two
bases are
pentagons.
All faces are
squares.
1.
2.
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3.
4.
Classify each solid figure. Write prism, pyramid, cone, cylinder, or sphere.
So, the figure is a pyramid.
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prism. How many faces does the prism
have? How many vertices does the
prism have?
7. Mackenzie used 15 straws to make a
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does it have? How many vertices does it
have?
6. A pyramid has 9 faces. how many edges
Solve by comparing strategies.
Check
5. How can you check your answer?
The figure has only one base.
4. Write your answer in a complete sentence.
3. Solve the problem. Describe the strategy used.
Solve
2. What are two strategies you can use to solve the problem?
Plan
1. Write the quesiton as a fill-in-the-blank sentence.
Read to Understand
The figure is a polygon.
Classify the solid figure. Write prism, pyramid, cone, cylinder, or sphere.
sphere
A solid figure with curves is not a polyhedron.
cone
The one base
cylinder
is a circle.
The one base
is a pentagon.
pentagonal
pyramid
The one base
is a square.
square
pyramid
cube
rectangular
prism
The one base
is a rectangle.
rectangular
pyramid
All faces are
rectangles.
A prism has two congruent and parallel
polygons as bases.
triangular
The two
prism
bases are
triangles.
A pyramid is a polyhedron
with only one polygon base.
triangular
All faces are
pyramid
triangles.
Mackenzie used 12 straws as edges to build a pyramid. How many sides did the base
of her pyramid have? How many vertices did her pyramid have?
Problem Solving Workshop Strategy: Compare
Strategies
Solid Figures
You can identify a polyhedron by the shape of its faces.
They are solid figures with faces that are polygons.
Name
Name
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c.
2.
b.
1.
a.
Match each solid figure with its net.
So, the solid figure matches net b.
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c.
3.
Net b has 3 rectangular and 2 triangular faces.
Nets a and b each have 5 faces.
Look for the net with 5 faces.
It has 5 faces; 3 rectangular and 2 triangular.
The figure is a triangular prism.
Name the solid figure for the net.
d.
4.
d.
b.
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© Harcourt • Grade 5
shape of base
shape of base with
a point in the center
circle
top view
top
top
MXENL08AWK5X_RT_CH19_L07.indd 1
3.
1.
front
front
side
side
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4.
2.
top
top
top
circle
triangle
rectangle
front view
Identify the solid figure that has the given views.
So the figure is a square pyramid.
The top view is a square.
All inside segments meet at a point.
The front and side views are triangles,
so the figure is a pyramid or cone.
Identify the solid figure that has the given views.
sphere
pyramid or cone
prism or cylinder
side
© Harcourt • Grade 5
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side
side
circle
triangle
rectangle
side view
front
front
front
You can use this chart to help you identify solids from different views.
You can identify solid figures by their nets. A net can be cut out, folded, and taped together
to form a polyhedron.
a.
Draw Solid Figures from Different Views
Nets for Solid Figures
Match the solid figure with its net.
Name
Name
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Can you pick it up and turn it over?
Yes, so it is a reflection.
•
2.
5.
1.
4.
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6.
3.
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Can you turn it about a point?
No, so it is not a rotation.
•
Name each transformation. Write translation, reflection, or rotation.
So, the transformation is a reflection.
Can you push it along a straight line?
No, so it is not a translation.
•
Ask how the figure changes position.
Name the transformation. Write translation, reflection, or
rotation.
A rotation is a turn. An example is the way the hands on an
analog clock move.
A reflection is a transformation that flips the figure over a line.
An example is turning a page in a book.
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4.
1.
5.
2.
and
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6.
3.
© Harcourt • Grade 5
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6/13/07 10:49:49 AM
If you try to cover a surface with closed figures, an
you have gaps or overlaps, then you have not
made a tessellation.
Predict whether the figure or figures will or will not tessellate.
Write yes or no.
So, yes, the figure will tessellate.
The pair of figures
do not tessellate.
Example – Not a Tessellation
A tessellation is a design that uses closed figures. In a
tessellation, one or more figures are put together to cover a
surface without any gaps or overlaps.
A transformation is a change in position of a figure that does
not change its size or shape.
Example – Tessellation
Predict whether the figure will tesselate.
Test your prediction.
Tessellations
Transformations
A translation is a transformation that slides the figure along a
straight line. An example is a sled sliding downhill.
Name
Name
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3.
Rotation
A reflection is a flip over a
line. It must be picked up for
this transformation.
1.
2.
RW131
Tell how each pattern might have been created.
This pattern might have been created by translating
the figure to the right.
The triangles are all facing the same way. Each triangle
can slide to the next position without being picked up.
Describe how the pattern might have been created.
This pattern might have been created by
reflecting the figure over a vertical line.
This figure must be picked up and flipped
to form this pattern.
It is not a translation because the figure turns in
two different ways.
Describe how the pattern might have been created.
A translation is a slide that
occurs without turning or
picking the figure up.
189 4 63 5 3
5. 3888, 648, 108, 18, ...
8. 828, 818, 808, 798, ...
4. 101, 120, 139, 158, ...
7. 7, 35, 63, 91, ...
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2. 313, 307, 301, 295, ...
63 4 21 5 3
1. 15, 45, 135, 405, ...
Identify the rule of each pattern.
Step 3 Write the rule.
So the rule is divide by 3.
• Are the quotients the same?
Yes, so the rule is division.
567 4 189 5 3
Then, test division, the other remaining operation.
© Harcourt • Grade 5
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9. 12, 48, 192, 768, ...
6. 612, 598, 584, 570, ...
3. 240, 120, 60, 30, ...
567 2 378 5 189
189 2 378 fi 63
• Are the differences the same as the pattern in the problem?
No. So the rule subtract 378 doesn’t work.
Step 2 Test subtraction, one of the remaining operations.
Think of a number to subtract from 567 to get 189.
• Does the pattern decrease?
Yes. So the rule is either subtraction or division.
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A rotation is a turn about
a point.
Step 1 See if you can eliminate two operations
• Does the pattern increase?
No. So eliminate addition and multiplication.
567, 189, 63, 21, ...
Identify the rule for the pattern.
You can find a rule for a number pattern and use it to extend
the pattern.
You can use reflections, rotations, or translations to create a
pattern.
Reflection
Numeric Patterns
Create a Geometric Pattern
Translation
Name
Name
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?
?
What are the next two figures in his
pattern?
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© Harcourt • Grade 5
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What will the next three figures in Amelia’s
design be?
Look for a pattern to solve.
7. Amelia uses parallelograms to decorate a
6. Nate uses a design made of figures to
picture frame. Her design so far is shown
decorate a ceramic bowl he made in
below.
pottery class. His design so far looks like
this:
What colors will the next two squares be?
Check
5. How did finding the pattern help you solve the problem?
4. Answer the question.
Then draw a line under the part that repeats.
3. Color the first 10 squares to show Suri’s pattern so far.
Solve
a sequence?
2. What kind of pattern does Suri use, a repeating pattern or
Plan
Read to Understand
1. Write the question as a fill-in-the blank sentence.
Suri is painting a design that is a pattern of squares.
She paints the squares black, white, black, gray, black,
white, black, gray. What color could the next two
squares be?
Algebra: Graph Relationships
Problem Solving Workshop
Strategy: Find a Pattern
10
5
Number of sides, y
15
3
20
4
6
3
Number of Vertices, y
9
3
3
18
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2
1
2
1
Number of cubes, y
Number of Triangles, x
12
6
Number of faces, x
MXENL08AWK5X_RT_CH21_L1.indd 1
2.
1.
Write the ordered pairs. Then graph them.
For this relationship, the y-coordinate is always
5 times the x-coordinate.
You can graph the ordered pairs for the
relationship shown in the table.
Each column can be written as an ordered pair (x,y):
1 pentagon, 5 sides: (1,5)
2 pentagons, 10 sides: (2,10)
3 pentagons, 15 sides: (3,15)
4 pentagons, 20 sides: (4,20)
2
1
Number of pentagons, x
A table can show the relationship.
Write the ordered pairs and then graph them.
12
4
0
2
4
6
y
2
You can graph ordered pairs to show relationships between two
amounts. For example, you can show the relationship between a
number of pentagons and the number of sides.
Name
Name
4
Number of Sides
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0
2
4
6
8
10
12
0
2
4
6
8
10
12
14
16
18
20
22
y
6
y
2
8
4
6
8
4
10
6
12
14
x
x
x
Reteach
10
18
10
© Harcourt • Grade 5
8
16
Number of Pentagons
2
(1,5)
(2,10)
(3,15)
(4,20)
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0
y
1
4
2
8
+8
3
12
4
4
16
3
5
2
4
1
3
output, y
2. input, x
2
10
1
5
15
3
4
25
5
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3.
+6
x
x
4
16
Plan
2. What strategy can you use to solve the problem?
0
+2
+4
+6
0
+4
+8
+2
+4
output, y
input, x
1
0
4
1
+6
2
+8
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13
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© Harcourt • Grade 5
10
3
MXENL08AWK5X_RT_CH21_L3.indd 1
lunches. How much does he spend for
lunches in one year?
6. Richard spends $150 each month on
Write an equation to solve.
sense for the problem.
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© Harcourt • Grade 5
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paycheck. She has $360 in her savings
account. How much will she have in the
acount 12 weeks from now?
7. Mary saves $20 each week from her
5. Look back at the problem. Explain why your answer makes
Check
5. How many more miles does Nancy need to travel?
4. Solve the equation for n.
84 5
Solve
+4
3
12
3. Write an equation. Let n represent the number of miles Nancy still must travel.
+2
2
8
+16
y
1
4
1. What information is given?
Read to Understand
+12
+18
+8
0
0
4 3 4 5 16
3 3 4 5 12
x
y
23458
13454
x
y
Find the rule to complete the function table. Then write an equation.
output, y
input, x
graph the ordered pair.
Rule: Subtract 3.
Equation: y 5 x 2
1. Complete the function table. Then
Graph the ordered pairs.
Use the function table to write the ordered pairs.
(0,0), (1,4), (2,8), (3,12), (4,16)
Equation: 4x
Rule: multiply by 4.
• Use the rule to write an equation.
So, the value of y when x 5 0 is 0.
03450
• Complete the function table.
Use the rule to find the value of y
when x 5 0.
• Find a rule. Each value of y is 4
times the value of x.
So, the rule is multiply 4.
Find a rule and complete the function
table. Then write an equation.
Nancy has traveled 39 miles on a train to visit her
grandmother. The total distance of the trip is 84 miles.
How many more miles will Nancy travel to reach her
grandmother’s house.
Problem Solving Workshop Strategy:
Write an Equation
Algebra: Equations and Functions
You can write an equation and make a function table
to show the relationship between two amounts.
Name
Name
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-6
-4
-2
-8
-6
-4
-2
0
+2
+4
+6
+8
+10
2
2, 1, and 5.
0
+2
+4
+6
+8
+10
No profit or loss is at $0, and profit is a positive word.
-8
-6
-4
-2
0
+2
+4
+6
+8
+10
2. a gain of 12 yards
RW137
3. 7 degrees below freezing
Write an integer to represent each situation.
-10
Identify the integers graphed on the number line.
So, the integer for a profit of $800 is 1800, or just 800.
1.
positive integers
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© Harcourt • Grade 5
Reteach
4. 8 floors up
Some words used with Some words used with
negative integers
positive integers
loss
gain or profit
decrease
increase
behind
ahead
backward
forward
below
above
down
up
under
over
withdrawal
deposit
to the left
to the right
Write an integer that describes a profit of $800.
So, the interger that represents 79
degrees below 0 is 279.
“Below” is a word used with negative
integers.
Use the chart at the right to see which
category “below” falls under.
Notice how the word “below” is in
the situation.
Write an integer to represent 79 degrees below 0.
So, the integers are
• The point that is 2 units left of 0 represents 22.
•The point that is one place to the right of 0 represents 1.
• The point that is 5 units right of 0 represents 5.
-10
Identify the integers graphed on the number line.
negative integers
-8
-8
-6
-4
-2
0
+2
+4
-8
-6
-4
-2
0
-8
-6
-4
-2
0
+2
+4
-8
-6
-4
-2
0
+2
+4
+6
+6
+6
+6
+8
+8
+8
+8
+10
+10
+10
+10
1
1
6
2
7
1
2
1
2
1
6
1
MXENL08AWK5X_RT_CH21_L5.indd 1
7.
4.
1.
Compare. Write ,, ., or 5.
8.
5.
2.
1
2
2
5
2
9
6
10
8
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2
2
2
9.
6.
3.
2
2
1
3
2
4
2
1
3
6
0
Two integers are equal only when they have the same sign and same number.
-10
+4
Since 24 is to the left of 19, 24,19
Compare 24 and 19. Use ,, ., or 5.
-10
+2
Since 14 is to the left of 19, 14,19
Compare 14 and 19. Use ,, ., or 5.
-10
Since 14 is to the right of 28, 14.28
Compare 14 and 28. Use ,, ., or 5.
-10
Since 24 is right of 28, 24.28
Compare 24 and 28. Use ,, ., or 5.
You can use a number line to compare and order integers
by graphing the integers on a number line.
Integers are the set of whole numbers and their opposites.
-10
Compare and Order Integers
Understand Integers
Integers can be graphed on a number line.
Positive integers are always greater than 0.
Numbers with no sign in front of them are
positive integers.
Negative integers are always less than 0.
The absolute value of an integer is its distance
from 0 on a number line.
Name
Name
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Reteach
© Harcourt • Grade 5
6/13/07 11:38:57 AM
Grade5.indd 70
to 23 and
to 12
To graph (23, 12), you would start at (0,) and the go
U
H
6.
9.
E
Q
7.
8.
Z
5.
D
4.
N
3.
S
2.
RW139
For 2–9, identify the ordered pair for each point.
1.
So, the ordered pair (4, 24) names the location of point F.
• Point F is 4 units below the x-axis, so the
y-coordinate is 24.
• Point F is 4 units right of the y-axis, so the
x-coordinate is 4.
What ordered pair names the location of point F?
So, the ordered pair (21, 3) names the location of point C.
• Point C is 3 units above the
x-axis, so the y-coordinate is 3.
• Point C is 1 unit left of the
y-axis, so the x-coordinate is 21.
What ordered pair names the location
of point P on the coordinate plane?
The y-coordinates above the x-axis are positive integers.
The y-coordinates below the x-axis are negative integers.
+5
+4
C +
3
+2
+1
E
N
-4
Q
E
-2
Z
D
+2
H
y- axis
-4
-2
0
+U2
+4
y- axis
S
x -axis
x -axis
6/13/07 11:50:21 AM
© Harcourt • Grade 5
Reteach
+4
B
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
-1
A
-2
-3
F
-4
-5
The x-coordinates to the right of the y-axis are positive integers.
The x-coordinates to the left of the y-axis are negative integers.
The origin is the place where the x- and y-axis intersect.
The horizontal line is the x-axis. The vertical line is the y-axis.
Just as a coordinate grid is formed by two perpendicular rays,
a coordinate plane is formed by two intersecting and number perpendicular lines.
Customary Length
Algebra: Graph Integers on the
Coordinate Plane
2
MXENL08AWK5X_RT_CH22_L1.indd 1
7.
RW140
8.
__ inches
5. 3 feet or 34 1
1 inch.
Measure the length to the nearest __
8
8
7 inches or 6 inches
4. 5 __
2
1
2. nearest __ inch.
Tell which measurement is more precise.
1. nearest inch
4
© Harcourt • Grade 5
Reteach
1 inches
6. 7 inches or 7 __
8
1
3. nearest __ inch.
Estimate the length of the caterpillar in inches. Then measure the length to the
nearest inch.
5 in. mark.
The crayon is almost exactly on the 2__
8
1 inch, the crayon is 2__
5 inches long.
So, to the nearest __
8
8
__ inch using a ruler.
Now measure the crayon to the nearest 1
8
Estimate: About 3 inches long
Estimate the length of the crayon in inches. Then measure the length to the
nearest 1_8 inch.
When you measure with precision, you use the smallest unit possible.
Measuring to the nearest 1_2 inch is more precise than measuring to the nearest inch.
Measuring to the nearest 1_4 inch is even more precise.
Name
Name
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Grade5.indd 71
1 cm
1 cm 5 10 mm
CM kilometer (km)
1 km 5 1,000 m
1 m 5 100 cm
1 m 5 1,000 cm
5
6
2. to the nearest millimeter.
inches
smaller
15 in.
10 ft
ft
4.
3 ft
1 5 ft
5 in.
4 in.
Find the sum or difference.
1. 33 yd 5
Change the unit.
5.
RW142
5 yd
1 8 yd
2. 500 cm 5
Rename, because 15 in. > 1 ft.
15 in. 5 1 ft 3 in.
10 ft 15 in. = 10 ft 1 1 ft 3 in. 5 11 ft 3 in.
So, 6 ft 7 in. 1 4 ft 8 in. 5 11 ft 3 in.
7 in.
8 in.
6 ft
1 4 ft
4 ft.
2 ft.
2 ft.
8 yd
2 2 yd
6 yd
m
6.
10 ft
2 3 ft
3. 13 cm 5
Reteach
© Harcourt • Grade 5
4 in.
11 in.
mm
So, 9 yd 1 ft 2 2 yd 2 ft 5 6 yd 2 ft
Subtract.
Subtract. 9 yd 1 ft 2 2 yd 2 ft
There are not enough feet to subtract
2 feet from 1 foot, so rename.
9 yd 1 ft 5 8 yd 4 ft
Add feet. Add inches.
2 ft.
2 ft.
100 m 5 1 m,
so 800 m 5 cm
Metric Units
of Length
10 mm 5 1
100 cm 5 1 m
1,000 m 5 1 km
12in. 5 1 ft
3 ft 5 1 yd
5,280 ft 5 1 mi
1,760 yd 5 1 mi
Customary Units
of Length
Add. 6 ft 7 in. 1 4 ft 8 in.
So, 800 centimeters 5 8 m.
m
larger
Divide.
800 4 100 5 8
Think:
cm
smaller
STEP 3
STEP 2
Multiply.
3 3 12 5 36
STEP 3
Decide:
Multiply or Divide
m
1 ft 5 12 in., so
3 ft 5 in
Think:
STEP 2
inches
STEP 1
Change the unit: 800 cm 5
So, 3 feet 5 36 inches.
feet
larger
Decide:
Multiply or Divide
STEP 1
Change the unit: 3 feet 5
Divide to change from smaller to larger linear units
6/13/07 12:01:10 PM MXENL08AWK5X_RT_CH22_L3.indd 1
© Harcourt • Grade 5
Reteach
6. width of the North American Continent
5. height of a skyscraper
RW141
4. distance from Austin to Phoenix
3. width of a paperclip wire
Write the appropriate metric unit for measuring each.
1. to the nearest centimeter.
Estimate the length of the string in centimeters. Then measure the length
These 2 cities are in different states, so there is a considerable amount of distance
between them.
So, the most appropriate unit for measuring the distance from Boston to Dallas
is the kilometer.
Write the appropriate metric unit for measuring the distance from Boston to Dallas.
So, to the nearest centimeter, the paperclip is 4 cm long.
The paperclip is exactly
4 centimeters long.
Notice where the paperclip
ends on the ruler.
4
Align the left side of the
paperclip with the zero
mark on the ruler. This is
already shown in the picture.
3
STEP 2
2
STEP 1
cm 1
How long is the paperclip to the nearest centimeter?
You can use a centimeter ruler to measure length.
1 mm
CM meter (m)
Some common metric units of length are:
centimeter (cm)
Change Linear Units
Multiply to change from larger to smaller linear units.
Metric Length
millimeter (mm)
Name
Name
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Grade5.indd 72
oz.
1 lb 5 16 oz,
__ lb 5 oz
so 21
2
qt
oz
gal
1. 14 pt =
4. 7.5 lb =
7. 20 qt 5
Change the unit.
8. 3 T 5
RW143
5. 9 gal =
2. 12,000 lb =
1 pounds is equal to 40 ounces.
So, 2__
2
ounces
smaller
Think:
Decide:
Multiply or Divide
pounds
larger
STEP 2
STEP 1
__ lb 5
Change the unit: 21
2
lb
qt
T
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© Harcourt • Grade 5
Reteach
pt
qt
fl oz
2,000 lb 5 1 T
16 oz 5 1 lb
9. 16 fl oz 5
6. 16 c =
3. 7 c =
Multiply.
__ 3 16 5
21
2
8
5 ___
16 5 40
__
23 1
STEP 3
Customary Units
of Weight
4 qt = 1 gal
2 c = 1 pt
2 pt = 1 qt
So, there are 2 qt in 8 c.
4 c 5 1 qt,
so 8 c 5 oz
8 fl oz = 1 c
4 c = 1 qt
8 c 4 4 5 2 qt
Think:
Decide:
Multiply or Divide
cups
quarts
so divide to change units
smaller
larger
Divide.
STEP 2
STEP 1
kilograms
larger
9,000 g 5 1 kg,
so 9,000 g 5 kg
Think:
STEP 2
milligrams
smaller
1 g = 1,000 mg, so
10 g =
mg
Think:
STEP 2
mg.
MXENL08AWK5X_RT_CH22_L5.indd 1
7. 20 metric cups 5
4. 10,000 mL 5
1. 6,000 L 5
Change the unit.
kL
L
L
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8. 9 g 5
5. 4 kg 5
2. 350 mg 5
So, there are 10,000 milligrams in 10 grams.
grams
larger
Decide:
Multiply or Divide
STEP 1
Change the unit: 10 g 5
So, there are 9 kilograms in 9,000 grams.
grams
smaller
Decide:
Multiply or Divide
STEP 1
Change the unit: 9,000 g 5
Change the unit: 8 c 5
STEP 3
Divide to change from smaller to larger units.
Divide to change from smaller to larger units.
kg
Multiply to change from larger to smaller units.
Multiply to change from larger to smaller units.
Customary Units
of Capacity
Customary Capacity and Mass
Customary Capacity and Weight
qt.
Name
Name
MXENL08AWK5X_RT_CH22_L4.indd 1
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mg
g
g
9. 2,500 ml 5
6. 4 L 5
metric cu
Reteach
© Harcourt • Grade 5
kL
6/13/07 1:07:46 PM
metric cups
1,000 g = 1 kg
1,000 mg = 1 g
Metric Units of
Mass
1,000 liters = 1 kL
4 metric cups = 1 L
250 mL = 1 metric cup
1,000 mL = 1 L
Metric Units of Capacity
3. 1,750 mL 5
10 3 1,000
510,000 mg
Multiply.
STEP 3
Divide.
4,000
______
1,000
5 9 kg
STEP 3
Grade5.indd 73
329-cm-long piece of corduroy.
He needs to use 146-cm-long pieces
for two pillows and an 83-cm-long
piece for the third. Does Colin have
enough corduroy?
7. Colin wants to make pillows from a
RW145
6/6/07 11:16:38 AM
© Harcourt • Grade 5
Reteach
costumes for the school play. She
doesn’t want any left-over fabric. She
needs 1.2 meters for three costumes
and 1.9 meters for two costumes. How
much fabric should Julie buy?
8. Julie needs to buy fabric to make
Tell whether you need an estimate or an actual measurement. Then solve.
6. How can you check your answer?
5. How far did Jennifer jog last week?
4. What equations can you use to solve the problem?
3. What operations will you use to solve the problem?
measurement? Why?
2. Can you estimate, or do you need to find an actual
1. How could you write the question as a fill-in-the blank sentence.
Jennifer went jogging last week. She jogged 2.65
kilometers on Tuesday and twice as far on Wednesday.
How far did Jennifer jog last week?
Problem Solving Workshop
Skill: Estimate or Actual Measurement
Name
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Count the number on
minutes in 5 minute intervals.
11 12 1
2
10
9
3
4
8
7 6 5
3
T
4
W
Jan.
5
R
6
F
7
S
3
F
4
S
Count days.
Step 2
March 2 to March 5 is 3 days.
January 12 to March 5 is 7 weeks.
MXENL08AWK5X_RT_CH22_L7.indd 1
End: 9:52 P.M.
End:
RW146
Elapsed time: 3 hr 29 min
3. Start:
Elapsed time: 2 hr 5 min
2. Start: 4:35 P.M.
Write the time for each.
June 30. How many weeks away is Larry’s trip?
1. On June 2, Larry planned a whitewater rafting trip for
6
M
7
T
8
1
W
March
3
F
4
S
9 10 11
2
R
5
M
6
T
7
W
June
3
S
9 10
2
1
8
F
R
25 26 27 28 39 30
18 19 20 21 22 23 24
11 12 13 14 15 16 17
4
S
26 27 28 29 30 31
19 20 21 22 23 24 25
12 13 14 15 16 17 18
5
S
5 minute interval:
ending on the 6
11 12 1
2
10
9
3
4
8
7 6 5
© Harcourt • Grade 5
Reteach
End: June 25, 5:30 P.M.
Elapsed time:
4. Start: June 21, 3:20 P.M.
So, the elapsed time from January 12 to March 5 is 7 weeks and 3 days.
Count weeks.
Step 1
26 27 28
9 10 11
2
R
19 20 21 22 23 24 25
8
1
W
Feb.
12 13 14 15 16 17 18
7
T
29 30 31
6
M
22 23 24 25 26 27 28
5
S
15 16 17 18 19 20 21
9 10 11 12 13 14
2
1
8
M
S
What is elapsed time from January 12 to March 5?
You can use a calendar to find elapsed time in weeks and days.
So, the elapsed time is 2 days 1 hr, and 30 min.
Count hours until
you reach 2 P.M.
11 12 1
2
10
9
3
4
8
7 6 5
Count the hours and minutes.
Step 2
May 5 – May 3 5 5 2 3 5 2 days
Count the days.
Start: May 3, 1 P.M.
End: May 5, 2:30 P.M.
Step 1
Write the elapsed time.
You can use a clock to find elapsed time in hours and minutes.
Elapsed Time
Name
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Grade5.indd 74
212
°F
32
49
7. -20 ºF to 45 ºF
6. 111 ºF to 77 ºF
8. 8 ºF to 103 ºF
4. 75 ºC to 39 ºC
RW147
3. -35 ºC to 50 ºC
2. 7 ºC to 61 ºC
Use the thermometer to find the change in temperature.
By how many degrees Fahrenheit did the temperature change?
105
°C
–45
–35
–25
–15
–5
5
15
25
35
45
55
65
75
85
95
95ºC
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© Harcourt • Grade 5
Reteach
9. 69 ºF to -13 ºF
5. 52 ºC to -44 ºC
1. The temperature is 49 ºF. Seven hours later, the temperature is 39 ºF.
Use the thermometer to solve.
So, the change in temperature from -25 ºC to 10 ºC is 35 ºC.
• Equation: 25 1 10 5 35
• Change: Increase
• Think: -25 ºC is lower than 10 ºC
Find the change in temperature from -25 ºC to 10 ºC.
So, the change in temperature from 102 ºF to 95 ºF is 7 ºF.
• Equation: Subtraction
102 2 95 5 7
• Change: Decrease
• Think: 102 ºF is higher than 95 ºF
Find 102 ºF to 95 ºF
Use the thermometer to calculate changes in temperature.
Temperatures below zero are written as negative numbers. For
example, 9 degrees below zero is written -9 ºC or -9 ºF.
The thermometer at the right measures degrees Fahrenheit and
degrees Celsius.
2
3
4
5
6
7
8
9
10
MXENL08AWK5X_RT_CH23_L01.indd 1
3.
1.
RW148
4.
2.
Find the perimeter of each polygon in centimeters.
The string is about 13 centimeters long, so the perimeter of
the polygon is about 13 centimeters.
centimeters
1
Step 3 Lay the string in a straight line and
measure its length with a centimeter ruler.
Step 2 Cut the string where it meets itself.
Step 1 Lay a piece of string around the figure.
Find the perimeter of the polygon in centimeters.
11
12
13
The perimeter of a figure is the distance around the figure. You can use a
piece of string to estimate the perimeter of a figure.
Estimate and Measure Perimeter
Temperature
Degrees Fahrenheit (°F) are the customary units for measuring
temperature. Degrees Celsius (°C) are the metric units for
measuring temperature.
Name
Name
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14
© Harcourt • Grade 5
Reteach
15
6/13/07 1:23:49 PM
Grade5.indd 75
4.
1.
5 in.
7 in.
4 cm
4 in.
2 cm
3 in.
4 in.
5.
2.
7 ft
6 ft
RW149
7 ft
4 ft
3 ft
5 ft
6.
3.
6/13/07 1:28:39 PM
© Harcourt • Grade 5
Reteach
3m
7 cm
So, the perimeter of the polygon is 24 ft.
So, the perimeter of the polygon is 18 in.
Find the perimeter of each polygon.
8 1 4 1 8 1 4 5 24
4 ft
Each of the 3 sides are 6 in. long.
Multiply. 6 3 3 5 18
6 in.
8 ft
6 ft
6 in.
8 ft
6 ft
MXENL08AWK5X_RT_CH23_L03.indd 1
4.
1.
3 in.
5.
2.
3 yd
8 yd
5 yd
RW150
4.5 m
Find the perimeter of each polygon by using a formula.
So, the perimeter of the square is 36 in.
P 5 36
P5439
Perimeter (P) 5 (number of sides) 3 S
So, the perimeter of the pentagon is 40 cm.
P 5 40
P5538
Perimeter (P) 5 (number of sides) 3 S
Find the perimeter of each polygon by
using a formula.
Since the sides of a regular polygon
are equal, you can use a formula to find the
perimeter of a regular polygon.
Find the perimeter of each polygon.
Since opposite sides of a parallelogram are
equal, you can use addition to find the
perimeter of a parallelogram.
Algebra: Perimeter Formulas
Find Perimeter
Since the sides of a regular polygon
are equal, you can use multiplication
to find the perimeter of a regular polygon.
Name
Name
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2 yd
6.
3 in.
5.7 cm
3. 3.6 cm
10 in.
3 in.
© Harcourt • Grade 5
Reteach
5.7 cm
3.6 cm
10 in.
9 in.
8 cm
6/13/07 1:35:08 PM
Grade5.indd 76
shape of a regular hexagon. The
perimeter is 84 inches. What is the
length of each side of the lid?
6. The lid of a designer box is in the
Make generizations to solve.
5. How can you check your answer?
Check
RW151
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© Harcourt • Grade 5
Reteach
rectangular shaped pen in her
backyard. The width of the pen is
15 feet. What is the length of the pen?
7. Jen has 96 feet of fencing to make a
4. What is the length of one side of the base of the other tissue box?
3. Which formula would you use to solve the problem?
Solve
2. How does knowing that the two tissue boxes are congruent help you?
MXENL08AWK5X_RT_CH23_L05.indd 1
4. a radius of 9 ft
1. a diameter of 4 m
RW152
5. a diameter of 20 cm
2. a radius of 10 in.
1 in.
6. a radius of 2 yd
© Harcourt • Grade 5
Reteach
3. a diameter of 10 in.
To the nearest tenth, find the circumference of a circle that has
So, the circumference of the circle is about 6.28 in.
C = 6.28
C = 3.14 3 2
C=p3d
Use the formula now that you have the length of the diameter
Plan
To the nearest hundredth, find the circumference of a circle that has a radius of 1 in.
If you know the diameter, d, of the circle, you can use the formula C = p 3 d to find the
circumference, C.
The distance across a circle is called the diameter.
Half the distance across a circle is called the radius.
The distance around, or circumference, of a circle is p (about 3.14) mulitplied by the
distance across the circle.
Circumference
Name
The radius is half of the diameter so multiply the measurement
at the radius by 2.
1 3 2 5 2 in.
1. What are you asked to find?
Read to Understand
Two tissue boxes are congruent cubes. If the perimeter of the base of one
tissue box is 16 in., what is the length of one side of the base of the other
tissue box?
Problem Solving Workshop Skill:
Make Generalizations
Name
MXENL08AWK5X_RT_CH23_L04.indd A
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Grade5.indd 77
4 1 18 5 22
Add the values from Steps 2 and 3.
Step 4
2.
5.
1.
4.
RW153
6.
3.
Estimate the area of the shaded figure. Each square on the grid is 1 cm2.
So, the estimated area of the shaded figure is about 22 in.2
Total number of full
squares: 18
Count the number of full squares,
including those missing only a tiny
corner.
84254
Divide the total by 2.
Step 2
Step 3
Total number of partial
squares: 8
Count the number of partial squares.
Skip squares with only a tiny corner.
Step 1
2
Each square on the grid is 1 in.
Estimate the area of the shaded figure.
Reteach
6/13/07 1:38:22 PM
© Harcourt • Grade 5
12.4 km
9.6 km
8 cm
MXENL08AWK5X_RT_CH24_L2.indd 1
1.
8 cm
Find the area of each figure.
2.
So, the area of the figure is 211.36 km2.
153.76 1 57.6 5 211.36
RW154
4 yd
10 yd
3.
5 in.
© Harcourt • Grade 5
Reteach
12 in.
5 in.
5 in.
6 km
Area of a polygon.
To find the total area, add the area of the square to the area of the rectangle.
A 5 9.6 3 6 or A 5 57.6
To find the area of the
rectangle, use the formula A 5 1 3 w.
Area of a rectangle.
The other part of the figure is a 6 km-by-9.6 km rectangle.
A 5 12.42 or A 5 153.76
To find the area of the square,
use the formula
A 5 S2.
12.4 km
You can find the area of a square or rectangle by using formulas.
Find the area of the figure at the right.
The area of a figure is the number of square units needed to cover it.
Area of a square.
One part of the figure is a
12.4 km-by-12.4 km square.
Algebra: Area of Squares and Rectangles
Estimate Area
You can use centimeter grid paper to estimate the area of a figure.
Name
Name
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Grade5.indd 78
Make as many different rectangles as possible.
Find the area of each.
Complete a table showing possibilities.
Stop when you realize you have found the
greatest area.
Divide the perimeter by 2. The length and
width of a rectangle whose perimeter is 44
have a sum of 22.
1. 20 in.
2. 38 km
RW155
3. 32 cm
4. 26 in.
Area
21
40
57
72
85
96
105
112
117
120
121
120
117
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© Harcourt • Grade 5
Reteach
5. 8 yd
You can use factors of
a given area to find the
length and width of
rectangles.
Length Width Perimeter
1
21
44
2
20
44
3
19
44
4
18
44
5
17
44
6
16
44
7
15
44
8
14
44
9
13
44
10
12
44
11
11
44
12
10
44
13
9
44
For the given perimeter, find the length and width of the
rectangle with the greatest area. Use whole numbers only.
For a given perimeter,
the square has the
greatest area.
So , the rectangle with the greatest area for a
rectangle with a perimeter of 44 yards is
11 yd 3 11 yd.
Step 2
Step 1
2
8 in.
10.5 in.
MXENL08AWK5X_RT_CH24_L4.indd 1
1.
2.
7m
Find the area of each triangle.
So the area is 17.5 m .
Find the area of
the triangle.
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7 ft
5m
__ ft
51
8
3.
__ bh
A51
2
__ 3 7 3 5
A51
2
A 5 17.5
If b is the base and h is the height, you can use the formula
1 bh to find the area, A, of a triangle.
A 5 __
2
How does the area of the triangle relate to the area of the rectangle?
Repeat steps 1 through 4 with the same-size rectangle,
but place the dot in Step 3 on a different side.
• What does that say about the area of the triangle?
Fold the rectangle along two sides of the triangle.
• Does it cover the triangle without overlapping?
© Harcourt • Grade 5
Reteach
5.6 m
7.6 m
Place a dot on one side of the rectangle.
Use a ruler to draw a line from the dot to each opposite corner.
You have made a triangle.
Step 3
Step 4
Cut out the rectangle.
Step 2
Draw a rectangle on grid paper.
Make each side greater than 10 units.
Find and record the area of the rectangle.
You can use grid paper to find a relationship between the areas of
triangles and rectangles.
You can use grid paper to relate perimeter and area.
Step 1
Algebra: Area of Triangles
Algebra: Relate Perimeter and Area
For the perimeter 44 yards, find the length and width of the rectangle
with the greatest area. Use whole numbers only.
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1.
2.
Find the area of each parallelogram.
A 5 15 units2
A5335
A 5 bh
So, the area of the parallelogram is:
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The formula for the area of a parallelogram is base 3 height
Count the grid squares to find the area of the parallelogram.
The base is 3 units and the height is 5 units.
Step 4
Cut out the triangle on the bottom and move it to
the top of the parallelogram to form a rectangle.
Step 3
Draw a line segment at the bottom row to form
a right triangle.
Step 2
Draw a parallelogram on grid paper and cut it out.
Step 1
Write the base and height of the parallelogram.
Then find its area in square units.
You can use grid paper and the base and height of a
parallelogram to find its area.
Algebra: Area of Parallelograms
Name
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3.
3
b
h
5
Reteach
6/6/07 1:47:08 PM
© Harcourt • Grade 5
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2m
8m
5m
7m
6. Find the area of the figure.
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© Harcourt • Grade 5
Reteach
long and one of its sides is 8 cm while
its perimeter is 25 cm, then what is
the length of its other 3 sides
combined?
7. If the area of a rectangle is 36 cm
Use the solve a simpler problem strategy to solve.
5. What is another way to solve this problem?
Check
sun catcher?
4. Multiply your answer from question 3 by 6. How many small squares are in Fran’s
How many small squares are in this 1st row?
3. The first row of the sun catcher is shown.
Solve
2. How can pictures help you break the problem into simpler parts?
Plan
1. What are you asked to find?
Read to Understand
Fran’s sun catcher has 6 rows of 5 squares. Each square
has 3 rows of 3 small squares. How many small squares are
in Fran’s sun catcher?
Problem Solving Workshop Strategy:
Solve a Simpler Problem
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1.
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2.
Use the net to find the surface area of each prism
in square units.
So, the surface area of the square pyramid is 48 square units or 48 units2.
Find the area of the square
A5s3s
A5434
A 5 16
Find the area of one triangle
A 5 1_2 3 b 3 h
A 5 1_2 3 4 3 4
A58
Add the areas of the square and the 4 triangles;
16 1 8 1 8 1 8 1 8
E
C
D
Reteach
6/13/07 1:44:20 PM
© Harcourt • Grade 5
Volume is measured in cubes, or in cubic units.
Use the net to find the surface area of the figure in square units.
B
Area is measured in squares, or in square units.
Another way to find the surface area is to use a net.
5 in
5 in
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1.
4 in
8 cm
2.
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10 cm
8 cm
3.
Find the volume of each rectangular prism in cubic centimeters.
So, the volume of the prism is 27 cubic feet, or 27 ft3.
The formula for volume is
Volume 5 length 3 width 3 height
V 5 3ft 3 3ft 3 3ft
V 5 27ft3
The volume will tell you how much
space can fit inside the box, the
cube, or any other figure.
Find the volume of the rectangular prism.
3 ft
Just as you used squares to find the area of a rectangle,
you can use cubes to find the volume of a rectangular prism.
You can find the surface area, the total area of the surface of a solid figure,
by adding the area of each face.
A
Algebra: Estimate and Find Volume
Surface Area
Each face of a square is one square unit.
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3.1 m
3 ft
Reteach
4.8 m
© Harcourt • Grade 5
2.5 m
3 ft
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Grade5.indd 81
5 in2
ft 3
ft 3
ft 5 ft
4 ft
1. area of this parallelogram
5 cm
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3 cm
triangular prism
2. volume of this
Write the units you would use for measuring each.
7 cm
Use cubic units or cm3, yd3, in3, mi3, m3, or km3.
So, the unit for volume 5 ft3
3
5 3 ft 3 3 ft 3 7 ft 5 63 ft3
Volume 5 l 3 w 3 h
Volume
Use the unit or cm., ft, yd, mi, in, or k.
So, unit for perimeter 5 m
5 5 m 1 5 m 1 5 m 1 5 m 1 5 m 1 5 m 5 30 m
Perimeter 5 side 1 side 1 side1 side 1 side 1 side
Perimeter
Use unit square or cm2, ft2, yd2, mi2, m2, or km2.
So, unit for area 5 in2
in 3 in
5 8 in 3 4 in 5 32 in2
Area
A513w
4 in.
5m
3 ft
3 ft
7 ft
8 in
pentagon
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© Harcourt • Grade 5
Reteach
3. perimeter of this regular
8 in.
Write the units you could use for measuring the area of this triangle
Volume is the amount of space a solid figure takes up.
Area is the space on a flat surface of a flat figure.
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© Harcourt • Grade 5
Its floor is a 10 ft-by-10 ft square.
What is its height?
6. The volume of an elevator is 1200 ft3.
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pizza. What is the most number of pieces
possible? [Hint: Pieces do not have to be
equal in size or shape.]
5. Three straight slices are made across a
Make a model or write an equation to solve.
4. How can you check that the missing length is 12 feet?
Check
Which is the better method? Explain.
Another way to solve the problem is to Write an Equation and solve it.
Since A 5 l 3 w, 156 5 13 3 w.
a 13 3 1 rectangle. Then see if it has 156 squares. If it does not, draw a 13 3 2 and then
a 13 3 3 rectangle. Continue until you have 156 squares inside the rectangle.
3. One way to solve is to use Make a Model. The length is 13 feet, so use grid paper to draw
Solve
2. Name two different strategies you can use to solve this problem.
Plan
1. What is the shape and size of the kitchen floor?
Read to Understand
Tashi is repairing tiles on a kitchen floor. The floor is in
2
the shape of a rectangle. The area of the floor is 156 ft .
The length is 13 feet. What is the width of the floor?
Problem Solving Workshop Strategy:
Compare Strategies
Relate Perimeter, Area, and Volume
Perimeter is the distance around a flat figure.
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