EnVisionMATH Grade 5 Workbook Answer
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EnVisionMATH Grade 5...
Description
enHanced Workbook
Taken from: enVisionMATH™ Teacher Resource Masters, Topics 1-20, Grade 5 by Scott Foresman-Addison Wesley Scott Foresman-Addison Wesley Mathematics, Grade 5: Review From Last Year Masters by Scott Foresman-Addison Wesley
Cover Art: by Luciana Navarro Powell
Taken from:
enVisionMATH ™ Teacher Resource Masters, Topics 1–20, Grade 5 by Scott Foresman-Addison Wesley Copyright © 2009 by Pearson Education, Inc. Published by Scott Foresman-Addison Wesley Glenview, Illinois 60025 Scott Foresman-Addison Wesley Mathematics Grade 5: Review From Last Year Masters by Scott Foresman-Addison Wesley Copyright © 2004 by Pearson Education, Inc. Published by Scott Foresman-Addison Wesley All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. This special edition published in cooperation with Pearson Learning Solutions. All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only.
Pearson Learni ng Solutions, 501 Boylston Street, Suite 900, Boston, MA 02116 A Pearson Education Company www.pearsoned.com
000200010270667033 JH
ISBN 10: 0-558-98645-5 ISBN 13: 978-0-558-98645-2
Table of Contents
Review from Last Year ...................................................................................................................... I
Reteaching and Practice Topic 1 Numeration............................................................................................................................21
Topic 2 Adding and Subtracting Whole Numbers and Decimals ..................................................... 31 Topic 3 Multiplying Whole Numbers ................................................................................................ 47 Topic 4 Dividing by 1-Digit Divisors ................................................................................................ 63 Topic 5 Dividing by 2-Digit Divisors ................................................................................................ 81 Topic 6 Variables and Expressions.................................................................................................... 97 Topic 7 Multiplying and Dividing Decimals ................................................................................... 109 Topic 8 Shapes ................................................................................................................................ 127 Topic 9 Fractions and Decimals ...................................................................................................... 139 Topic 10 Adding and Subtracting Fractions and Mixed Numbers ...................................................161 Topic 11 Multiplying Fractions and Mixed Numbers ......................................................................175 Topic 12 Perimeter and Area ...........................................................................................................185 Topic 13 Solids ................................................................................................................................ 201 Topic 14 Measurement Units, Time, and Temperature ....................................................................215 Topic 15 Solving and Writing Equations and Inequalities .............................................................. 233 Topic 16 Ratio and Percent.............................................................................................................. 243 Topic 17 Equations and Graphs....................................................................................................... 253 Topic 18 Graphs and Data ............................................................................................................... 263 Topic 19 Transformations, Congruence, and Symmetry ................................................................. 281 Topic 20 Probability ........................................................................................................................ 293 Credits ............................................................................................................................................ 301
Review From Last Year
Corresponds with Student enHanced Workbook pages 1–20
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I
5
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II
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y le s e W n o is d d A n a m s e r o F tt o c S ©
III
Reteaching and Practice
Corresponds with Student enHanced Workbook pages 21–300
Reteaching
Name
1-1
Place Value
R e te a c h i n g
1 1
Place-value chart: Billions period
Millions p eriod Thousands p eriod
Ones p eriod
s s d d d ds s d d d s s re re s re n n n s s n n d ns d on d a a a n n re o o s s s n io n io n n o i i i n n d u u s l l u u l u u li n o n il h ill te ill h il te il h ho te ho il u b b b m m m t t th h te
6,
3
9,
2
5
80 , 1
0
o
e n
s
1
Expanded form: 6,000,000,000 300,000,000 90,000,000 2,000,000 500,000 80,000 100 1 Standard form: 6,392,580,101 Word form: six billion, three hundred ninety-two million, five hundred eighty thousand, one hundred one Write the word name for each number and tell the value of the underlined digit.
1. 3,552,308,725
Three billion, five hundred fifty-two million, three hundred eight thousand, seven hundred twenty-five; fifty million
2. 843,208,732,833
Eight hundred forty-three billion, two hundred eight million, seven hundred thirty-two thousand, eight hundred thirty-three; 800 billion
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3. Write 2,000,000,000 70,000,000 100,000 70,000 3,000 800 10 in standard form.
2,070,173,810
4. Number Sense What number is 100,000,000 more than 5,438,724,022?
5,538,724,022 Topic 1
21
Practice
Name 1 1 e c i t c a r P
1-1
Place Value Write the word form for each number and tell the value of the underlined digit.
1. 34,235,345
Thirty-four million, two hundred thirtyfive thousand, three hundred forty-five; five thousand Nineteen billion, six hundred seventythree million, eight hundred ninety thousand, four; nine billion
2. 19,673,890,004
3. Write 2,430,090 in expanded form.
2,000,000 400,000 30,000 90 Write each number in standard form.
4. 80,000,000 4,000,000 100 8
84,000,108
5. twenty-nine billion, thirty-two million
29,032,000,000
6. Number Sense What number is 10,000 less than 337,676?
327,676
7. Which number is 164,502,423 decreased by 100,000? A.
164,402,423 B. 164,501,423 C. 164,512,423 D. 264,502,423
8. Explain It Explain how you would write 423,090,709,000 in word form.
Start at the left and write four hundred twenty-three. Write billion for the place, ninety, million for the place, seven hundred nine, and thousand for the place 22 Topic 1
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Reteaching
Name
1- 2
Comparing and Ordering Whole Numbers
R e te a c h i n g
1 2
Order these numbers from least to greatest: 4,752,213 5,829,302 4,234,295 4,333,209. Step 1: Write the numbers, lining up places. Begin at the left to find the greatest or least number. 4,752,213 5,829,302 4,234,295 4,333,209
Step 2: Write the remaining numbers, lining up places. Find the greatest and least of these. 4,752,213 4,234,295 4,333,209
Step 3: Write the numbers from least to greatest.
4,234,295 4,333,209 4,752,213 5,829,302
greatest least
4,752,213 is the greatest of these. 4,234,295 is the least.
5,829,302 is the greatest.
Complete. Write , , or in each
1. 7,642 7,843
.
2. 2,858,534
2,882,201
Order these numbers from least to greatest.
3. 768,265 769,205 739,802
739,802; 768,265; 769,205
4. Write the areas of each country in order from greatest to least.
28,748; 28,450; 27,830; 27,750 5
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Country
Area in Square Kilometers
Albania
28,748
Burundi
27,830
Solomon Islands
28,450
Haiti
27,750
Topic 1
23
Practice
Name 2 1 e c i t c a r P
1- 2
Comparing and Ordering Whole Numbers Complete. Compare the numbers. Use or for each
1. 23,412
23,098
. 9,421,090
2. 9,000,000
Order these numbers from least to greatest.
3. 7,545,999
7,445,999
7,554,000
7,445,999 7,545,999 7,554,000 4. Number Sense What digit could be in the ten millions place of a number that is less than 55,000,000 but greater than 25,000,000?
2, 3, 4, or 5
5. Put the trenches in order from the least depth to the greatest depth.
Depths of Major Ocean Trenches
Kermadec Trench, Philippine Trench,
Trench
Tonga Trench, Mariana Trench
Depth(infeet)
Philippine Trench
32,995
Mariana Trench
35,840
Kermadec Trench TongaTrench
32,963 35,433
6. These numbers are ordered from greatest to least. Which number could be placed in the second position? 2,643,022
A
2,743,022
1,764,322
927,322
B 1,927,304
C 1,443,322
D 964,322
7. Explain It Explain why 42,678 is greater than 42,067.
Although the digits in the thousands period are the same, the hundreds digits differ, and since 6 is greater than 0, 42,678 is the greater number. 24 Topic 1
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Reteaching
Name
1-3
Decimal Place Value
R e te a c h i n g
1 3
Here are different ways to represent 2.753149.
Place-value chart: Ones
2
Tenths
.
TenHundredths Thousandths thousandths
7
5
3
Hundredthousandths
Millionths
4
9
1
Expanded Form: 2
0.7
0.05
0.003
0.0001
0.00004
0.000009
Standard form: 2.753149 Word Form: Two and seven hundred fifty-three thousand one hundred forty-nine millionths Complete the place-value chart for the following number. Write its word form and tell the value of the underlined digit.
1. 6.324657 Ones
6
Tenths
.
3
TenHundredths Thousandths thousandths
2
4
Hundredthousandths
Millionths
5
7
6
Six and three hundred twenty-four thousand six hundred fifty-seven millionths; Five hundred-thousandths Write each number in standard form.
2. 5 0.1 0.03 0.006 0.0007 0.00002 0.000004 5
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5.136724 3. Two and seven hundred twenty-four thousand three hundred sixty-five millionths
2.724365 Topic 1
25
Practice
Name 3 1 e c i t c a r P
1-3
Decimal Place Value Write the word form of each number and tell the value of the underlined digit.
1. 3.100
Three and one hundred thousandths; one tenth
2. 5.267
Five and two hundred sixty-seven thousandths; six hundredths 3. 2.778
Two and seven hundred seventy-eight thousandths; eight thousandths
Write each number in standard form.
4. 8 0.0 0.05 0.009 0.0006
8.0596 5. 1 0.9 0.08 0.001 0.00002
1.98102 Write two decimals that are equivalent to the given decimal.
6. 5.300
5.3, 5.30
7. 3.7
3.700, 3.7000 0.90, 0.90000
9. The longest stem on Eli’s geranium plant is 7.24 inches. Write 7.24 in word form.
Seven and twentyfour hundredths
26 Topic 1
8. 0.9
10. Explain It The number 4.124 has two 4s. Why does each 4 have a different value?
The first 4 is in the ones place; the second 4 is in the thousandths place.
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Reteaching
Name
1-4
Comparing and Ordering Decimals
R e te a c h i n g
1 4
List the numbers in order from least to greatest: 6.943, 5.229, 6.825, 6.852, 6.779 Step 1: Write the numbers, lining up places. Begin at the left to find the greatest or least number.
Step 2: Write the remaining numbers, lining up places. Find the greatest and least. Order the other numbers. 6.943 6.825 6.852 6.779
6.943 5.229 6.825 6.852 6.779 5.229 is the least.
7.344
6.825 6.852 least
5.229 6.779 6.825 6.852 6.943
6.779 is the least. 6.943 is the greatest. 6.852 is greater than 6.825.
Complete. Write , , or for each
1. 7.539
greatest
Step 3: Write the numbers from least to greatest.
2. 9.202
. 9.209
0.750
3. 0.75
Order these numbers from least to greatest.
4. 3.898 3.827 3.779
3.779, 3.827, 3.898
5. 5.234 5.199 5.002 5.243
5.002, 5.199, 5.234, 5.243
Which had the faster speed?
Driver
Average Speed (mph)
Driver A
Driver A
145.155
7. Driver C or Driver A
Driver B Driver C
145.827 147.956
Driver D
144.809
6. Driver A or Driver D
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Car Racing Winners
Driver C
Topic 1
27
Practice
Name 4 1 e c i t c a r P
1-4
Comparing and Ordering Decimals Write , , or for each
1. 5.424
5.343
4. 21.012
.
2. 0.33
21.01
0.330
5. 223.21
9.479
3. 9.489
223.199
5.432
6. 5.43
Order these numbers from least to greatest.
7. 8.37, 8.3, 8.219, 8.129 8. 0.012, 0.100, 0.001, 0.101
8.129, 8.219, 8.3, 8.37 0.001, 0.012, 0.100, 0.101
9. Number Sense Name three numbers between 0.33 and 0.34.
Sample answer: 0.334, 0.335, 0.336 10. Which runner came in first place?
Liz
Half-Mile Run
11. Who ran faster, Amanda or Steve?
Amanda 12. Who ran for the longest time?
Steve
Runner Amanda
Time (minutes) 8.016
Calvin
7.049
Liz
7.03
Steve
8.16
13. Which number is less than 28.43? A
28.435
B 28.34
C 28.430
D 29.43
14. Explain It Explain why it is not reasonable to say that 4.23 is less than 4.13.
The number 4.23 is greater than 4.13, because there is a 2 in the tenths place, and 2 is greater than 1. 28 Topic 1
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Reteaching
Name
1-5
Problem Solving: Look for a Pattern Mr. Nagpi works in a machine shop. In the shop, the drill bits are kept in a cabinet with drawers. The drawers are marked with the diameter of the bits as shown on the right. Some of the labels are missing. Help Mr. Nagpi complete the drawer labels.
R e te a c h i n g
1 5
Drill Bits
0.10 in. 0.20
0.12 in. 0.22
0.14 in. 0.24
0.16 in. 0.26
0.18 in. 0.28
in. 0.30 in.
in. 0.32 in.
in. 0.34 in.
in.
in.
0.36 in.
0.38 in.
Read and Understand
What do you know?
Some drawers are labeled with decimals.
What are you trying to find?
A way to find the values of the missing labels
Plan and Solve
Find a pattern for the decimals.
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1. Look for a pattern to the change in the tenth-values across a row or down a column.
1. The tenth-values are not increasing across a row. They are increasing by 1 down a column.
2. Look for a pattern to the change in the hundredthvalues across a row or down a column.
2. The hundredth-values are increasing by 2 across a row. They are not increasing down a column.
3. Use the patterns to complete the table.
3. The missing labels in the third row are 0.36 in. and 0.38 in.
Find pattern in theintable. Then fill in thethe missing values the table.
0.20
0.21
0.22
0.23
0.24
0.50
0.51
0.52
0.53
0.54
0.80
0.81
0.82
0.83
0.84
Topic 1
29
Practice
Name 5 1 e c i t c a r P
Problem Solving: Look for a Pattern
1-5
Determine the pattern and then complete the grids.
1.
0.87
0.88
0.89
0.90
2.
0.12 0.22
0.32 3.
0.22
0.23
0.24
0.25
4.
0.56
0.66 0.76
5. Critical Thinking In a list of numbers, the pattern increases by 0.001 as you move to the right. If the third number in the list is 0.064, what is the first number in the list? Explain how you know.
0.062; subtract 0.001 twice to find the first number in the list. 6. If 5 school buses arrive, each carrying exactly 42 passengers, which expression would you use to show how many people in all arrived on the school buses? A
42 5
B 42 5
C 42 5
D 42 5
7. Explain It Mishell arranged her coins in the following pattern: $0.27, $0.29, $0.31, $0.33. Explain what her pattern is, and then tell what the next amount of coins would be.
Pattern increases by $0.02; the next amount would be $0.35.
30 Topic 1
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Name
Reteaching 2-1
Mental Math
R e te a c h i n
There are several ways that you can add and subtract numbers mentally to solve a problem.
Associative Property of Addition
You can add two numbers in any order.
You can change the groupings of addends. 17 (13 10) (17 13) 10
Compatible numbers are numbers that are easy to compute mentally. 25 93 75
With compensation, you adjust one number to make computations easier and compensate by changing the other number.
25 and 75 are compatible because they are easy to add.
320 190 10 10
⇓
25 93 75 (25 75) 93
⇓
330 200
130
100 93 193
Add or subtract mentally.
1. 265 410 335 3. 2,500 1,730 70
1,010 4,300
4. 1,467
did Pitcher A have than Pitcher C?
152 more strikeouts 6. How many strikeouts did Pitcher B and Pitcher E have altogether?
498 strikeouts
730 1,070 397
2. 885 155
5. Number Sense How many more strikeouts
5
2 1
Commutative Property of Addition
15 27 27 15
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g
Strikeout Data Pitcher Number of Strikeouts A 372 B 293 C 220 D 175 E 205
7. How many strikeouts were recorded by all five pitchers?
1,265 strikeouts Topic 2
31
Name
Practice 2-1
Mental Math 1 2 e c i t c a r P
Show how you can use mental math to add or subtract.
1. 70 90 30
2. 350 110
Sample answers:
350 110 50 50
70 90 30
(70 30) 90 100 190
⇓
⇓
400 160 240
National Monuments N a me
State
George Washington Carver Navajo
Acres
Missouri
210
Arizona
360
FortSumter
SouthCarolina
200
RussellCave
Alabama
310
3. How many more acres are there at Navajo monument than at George Washington Carver monument?
4. How many acres are there at Fort Sumter and Russell Cave combined?
150 more acres 510 acres
5. Fresh Market bought 56 lb of apples in August from a local orchard. In September, the market purchased an additional 52 lb of apples and 32 lb of strawberries. How many pounds of fruit did the market buy? A 108 lb
B 140 lb
C 150 lb
D 240 lb
6. Explain It Write the definition and give an example of the
Sample answer: The Commutative Property of Addition says you can add numbers in any order. 30 15 2 15 30 2
Commutative Property of Addition.
32 Topic 2
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Name
Reteaching 2-2
Rounding Whole Numbers and Decimals
R e te a c h i n
g
2 2
You can use the number line below to help you round 8,237,650 to the nearest million. Is 8,237,650 closer to 8,000,000 or 9,000,000? 8,237,650
8,000,000
halfway
8,500,000
9,000,000
8,237,650 is less than halfway to 9,000,000. 8,237,650 is closer to 8,000,000. The number line can also help you round 7.762 to the nearest tenth. Is 7.762 closer to 7.7 or 7.8? halfway 7.762 7.7
7.75
7.8
7.762 is more than halfway to 7.8. 7.762 is closer to 7.8. Round each number to the place of the underlined digit.
1. 4,725,806
4,700,000 3. 165,023,912
200,000,000 5. Round the number of connected computers in Year 2 to the nearest ten million.
40,000,000 5
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2. 7.049
7 4. 18.692
18.7 Number of Computers Connected to the Internet Year1
30,979,376
Year2
42,199,279
Year3
63,592,854
6. Number Sense Marc earned $9.37 per hour working at the library. Round his wage to the nearest ten cents.
$9.40 Topic 2
33
Name
2 2 e c i t c a r P
Practice
Rounding Whole Numbers and Decimals
2-2
Round each number to the place of the underlined digit.
1. 32.60 2. 489,334,209 3. 324,650 4. 32.073
32.6 489,000,000 325,000 32.1
5. Reasoning Name two different numbers that round to 30 when rounded to the nearest ten.
Sample answer: 29 and 31 In 2000, Italy produced 7,464,000 tons of wheat, and Pakistan produced 21,079,000 tons of wheat. Round each country’s wheat production in tons to the nearest hundred thousand.
6. Italy 7. Pakistan
7,500,000 tons 21,100,000 tons
The price of wheat in 1997 was $3.38 per bushel. In 1998, the price was $2.65 per bushel. Round the price per bushel of wheat for each year to the nearest tenth of a dollar.
8. 1997 9. 1998
$3.40 per bushel $2.70 per bushel
10. Number Sense Which number rounds to 15,700,000 when rounded to the nearest hundred thousand? A 15,000,000 B 15,579,999 C 15,649,999 D 15,659,999 11. Explain It Write a definition of rounding in your own words.
Sample answer: Rounding helps you adjust a number to make it easier to use. 34 Topic 2
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Name
Reteaching 2-3
Estimating Sums and Differences
R e te a c h i n
g
2 3
During one week, Mr. Graham drove a truck to five different towns to make deliveries. Estimate how far he drove in all.
Mr. Graham’s Mileage Log Cities MansleytoMt.Hazel Mt.HazeltoPerkins PerkinstoAlberton Alberton to Fort Maynard Fort Maynard to Mansley To estimate the sum, you can round each number to the nearest hundred miles.
243 303 279 277
352
200 300 300 300
⇒ ⇒ ⇒ ⇒ ⇒
400 mi 1,500
Mr. Graham drove about 1,500 mi.
Mileage 243 303 279 277 352 You can estimate differences in a similar way. Estimate 7.25
4.98.
You can round each number to the nearest whole number.
7.25
4.98
⇒ ⇒
7
5 2
The difference is about 2.
Estimate each sum or difference.
1. 19.7 6.9
About 13 3. 582 169 23
About 770 5
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2. 59 43 95
About 200 4. 87.99 52.46
About 36
5. Estimation Brigid worked 16.75 h. Kevin worked 12.50 h. About how many more hours did Brigid work than Kevin?
About 4 h Topic 2
35
Name
Practice 2-3
3 2 e c i t c a r P
Estimating Sums and Differences Sample answers are Estimate each sum or difference. given for 1– 4. 3,300 20 1. 5,602 2,344 2. 7.4 3.1 9.8 3,000 52 3. 2,314 671 4. 54.23 2.39
5. Number Sense Wesley estimated 5.82 4.21 to be about 2. Is this an overestimate or an underestimate? Explain.
It is an overestimate, because 5.82 was rounded up and 4.21 was rounded down. Average Yearly Precipitation of U.S. Cities 6. Estimate the total precipitation in inches and the total number of days with precipitation for Asheville and Wichita.
About 77 in.; about 210 days
City Asheville, North Carolina Wichita,Kansas
Inches
D a ys
47.71
124
28.61
85
7. Reasonableness Which numbers should you add to estimate the answer to this problem: 87,087 98,000? A 88,000 98,000
C 87,000 98,000
B 85,000 95,000
D 80,000 90,000
8. Explain It Estimate the total weight of two boxes that weigh 9.4 lb and 62.6 lb using rounding and compatible numbers. Which estimate is closer to the actual total weight? Why?
The rounded numbers are closer to the srcinal numbers, so the rounded estimate will be closer to the actual total weight.
36 Topic 2
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Name
Reteaching 2-4
Problem Solving: Draw a Picture and Write an Equation
R e te a c h i n
g
A community center is raising funds to buy a computer. Here is a picture of the sign they put outside the center. How much more money must the center raise?
New Computer Fund Please contribute!
2 4
$1,600 $1,400 $1,200 $1,000 $ 800 $ 600 $ 400 $ 200 $ 0
How to write an equation number sentence for a problem:
One Way
Another Way
The goal is $1,600.
The goal is $1,600.
So far, $1,000 has been raised.
So far, $1,000 has been raised.
The amount yet to be raised is the unknown.
The amount yet to be raised is the unknown.
Think: The amount raised so far and the amount yet to be raised will reach the goal.
Think: The difference between the goal and what has been raised so far is the amount yet to be raised.
Write an equation. 1,000 x 1,600
Write an equation. 1,600 1,000 x
Think: What number added to 1,000 will result in 1,600?
Think: What number will result if 1,000 is subtracted from 1,600?
1,000 600 1,600 The amount yet to be raised is $600.
1,600 1,000 600 The amount yet to be raised is $600.
A mason needs 22 bricks to make a stoop. So far he has carried 15 to the site. How many more bricks must he carry?
22 bricks in all
Draw a picture. Write an equation. Write a number sentence. Solve. 5
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15
x
Sample: 22 15 x; 22 15 7; 7 bricks Topic 2
37
Name
4 2 e c i t c a r P
Practice 2-4
Problem Solving: Draw a Picture and Write an Equation 1. Dayana picked apples for 2 hours. She picked 28 apples in the first hour, and at the end of two hours, she had 49. How many apples did she pick during the second hour?
28 a 49 49 28 a 21 apples
2. Dixon bought a pack of pencils and then gave 12 away. He now has 24 left. How many pencils were in the pack of pencils that Dixon bought?
p 12 12 24 24 p
Write two different equations; then solve each problem.
36 pencils
Copy and complete the picture. Then write an equation and solve.
3. Rumina is baking 25 muffins for the bake sale. She has already baked 12. How many more does she need to bake? 25 muffins in all 12
n
12 n 25; 13 muffins 4. Estimation Janet saved 22 dollars one month and 39 dollars the next month. She wants to buy a bicycle that costs $100. About how much more money does she need? A about $40
B about $50
C about $60
D about $70
5. Explain It Stefany ran 2 miles each day for 14 days. How many miles did she run in 14 days? Explain two different ways to solve this problem, and then solve.
Sample: Draw a picture or write an equation; 28 miles 5
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38 Topic 2
Name
Reteaching 2-5
Adding and Subtracting
R e te a c h i n
Find 35,996 49,801. Step 1: Write the numbers, lining up places. Add the ones and then the tens.
11
s d n a s s d u n
s d e o a r h s t u d s s n n o n e u e n e h t t h t o
Step 2: Continue adding hundreds, thousands and ten thousands. Regroup as needed.
Reminder: When the sum of a column is greater than 10, write the ones of the sum and regroup the tens onto the next column.
35,996 49,801 85,797
35,996 49,801 97
So 35,996 49,801 85,797. Find 35,996 17,902. Step 1: Write the numbers, lining up places. Subtract the ones, tens, and hundreds.
Step 2: Continue by subtracting thousands. Regroup as needed.
Reminder: The 3 in the ten thousands place is regrouped to 2 ten thousands and 10 thousands.
2 15
s d n a s s s d u n d e o a r h s t u d s s n n e n o u e n e h t t h t o
3/ 5/ , 9 9 6 1 7, 9 0 2 1 8, 0 9 4
35,996 17,902 094
So 35,996 17,902 18,094. Add or subtract
1.
7,502 9,909
17,411
2.
64,782 33,925
3.
30,857
835,029 26,332
80 8, 697
Myronville School District has 23,081 students, and Saddleton School District has 45,035 students. 5
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4. Number Sense How many more students are there in Saddleton than in Myronville?
21,954 more students
Topic 2
39
g
2 5
Name
Practice 2-5
Adding and Subtracting 5 2 e c i t c a r P
Add or subtract.
1.
29,543 13,976
2.
43, 519 5. 971,234 55,423
3.
93,210 21,061
4.
369,021 325,310
893,887 22,013
7 2, 149 43, 71 1 1,026,657
9 15, 90 0
6. Number Sense Is 4,000 a reasonable estimate for the difference of 9,215 5,022? Explain.
Yes, because 9,000 5,000 4,000. For questions 7 and 8, use the table at right.
7. How many people were employed as public officials and natural scientists?
1,319,000 people 8. How many more people were employed as university teachers than as lawyers and judges?
People Employed in U.S. by Occupation in 2000 Occupation
Workers
Public officials
753,000
Natural scientists
566,000
University teachers
961,000
Lawyers and judges
926,000
35,000 more people 9. Which is the difference between 403,951 and 135,211? A 200,000
B 221,365
C 268,740
D 539,162
10. Explain It Issac is adding 59,029 and 55,678. Should his answer be greater than or less than 100,000? Explain how you know.
Greater than 100,000, because an estimate of 60,000 60,000 120,000 5
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40 Topic 2
Name
Reteaching 2-6
Adding Decimals
R e te a c h i n
In February, Chantell ran a 5K race in 0.6 hour. She ran another 5K race in May in 0.49 hour. What was her combined time for the two races? Step 1: Write the numbers, lining up the decimal points. Include the zeros to show place value.
Step 2: Add the hundredths.
0.60 0.49
0.49 0.60
g
2 6
Step 3: Add the tenths. Remember to write the decimal point in your answer. 1
9
0.60 0.49 1.09
You can use decimal squares to represent this addition problem.
Chantell’s combined time for the two races was 1.09 hours.
Add.
1. 2.97 0.35 3. 39.488
3.32 26.7 66.188
2.
13.88 7.694
4. 88.8 4.277
21.574 78.95 172.027
5. Number Sense Is 16.7 a reasonable sum for 7.5 9.2? Explain.
Yes; 7.5 rounds to 8.0; 9.2 rounds to 9.0; 8.0 9.0 17.0. 6. How much combined snowfall was there in Milwaukee and Oklahoma City? 5
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105.1 in.
City
Snowfall (inches) in 2000
Milwaukee,WI Baltimore,MD
87.8 27.2
OklahomaCity,OK
17.3
Topic 2
41
Name
Practice 2-6
Adding Decimals 6 2 e c i t c a r P
Add.
1.
58.0 3.6
2.
61.6 5. 16.036 7.009
40.5 22.3
62.8 23. 04 5
3.
4.
34.587 21.098
55. 68 5 6.
43.1000 8.4388
92.30 0.32
51.5388 92. 62
7. Number Sense Reilly adds 45.3 and 3.21. Should his sum be greater than or less than 48? Tell how you know.
The sum should be greater than 48, because the whole numbers have a sum of 48 without adding the decimals.
In science class, students weighed different amounts of tin. Carmen weighed 4.361 g, Kim weighed 2.704 g, Simon weighed 5.295 g, and Angelica weighed 8.537 g.
8. How many grams of tin did Carmen and Angelica have combined?
12.898 g of tin 9. How many grams of tin did Kim and Simon have combined?
7.999 g of tin 10. In December the snowfall was 0.03 in. and in January it was 2.1 in. Which was the total snowfall? A 3.2 in.
B 2.40 in.
C 2.13 in.
D 0.03 in.
11. Explain It Explain why it is important to line up decimal numbers by their place value when you add or subtract them.
Sample answer: When adding or subtracting, you must always add or subtract digits that have the same place value.
42 Topic 2
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Name
Reteaching 2-7
Subtracting Decimals
R e te a c h i n
Mr. Montoya bought 3.5 lb of ground beef. He used 2.38 lb to make hamburgers. How much ground beef does he have left?
g
2 7
Step 1: Write the numbers, lining up the decimal points. Include the zeros to show place value. 3.50 2.38
You can use decimal squares to represent this subtraction problem.
Step 2: Subtract the hundredths. Regroup if you need to. 4 10
3.50 . 2.38 2
Step 3: Subtract the tenths and the ones. Remember to write the decimal point in your answer. 4 10
3.50 . 2.38 1.12
Mr. Montoya has 1.12 lb of ground beef left over. Subtract.
1. 82.7 5.59 5
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77.11
2.
43.3
3.
7.28
12.82
4.928
30. 48
2. 352
Topic 2
43
Name
Practice 2-7
Subtracting Decimals 7 2 e c i t c a r P
Subtract.
1.
92.1 32.6
59. 5 5. 8.7 0.3
2.
8.4
52.7 36.9
3.
15.8
4.
85.76 12.986
32.7 2.328
7 2. 774
30. 37 2 21.98
6. 23.3 1.32
Kelly subtracted 2.3 from 20 and got 17.7. Explain why this 7. Number answer isSense reasonable.
A good estimate is 20 2 18, which is close to 17.7. At a local swim meet, the second-place swimmer of the 100-m freestyle had a time of 9.33 sec. The first-place swimmer’s time was 1.32 sec faster than the second-place swimmer. The thirdplace time was 13.65 sec.
8. What was the time for the first-place swimmer?
8.01 sec
9. What was the difference in time between the second- and third-place swimmers?
4.32 sec
10. Miami’s annual precipitation in 2000 was 61.05 in. Albany’s was 46.92 in. How much greater was Miami’s precipitation than Albany’s? A 107.97 in.
B 54.31 in.
C 14.93 in.
D 14.13 in.
11. Explain It Explain how to subtract 7.6 from 20.39.
Sample answer: Write the numbers, lining up the decimal points. Write zeros to show place value. Subtract the hundredths, the tenths, and the ones. Place the decimal point in the answer. 44 Topic 2
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Name
Reteaching
Problem Solving: Multiple-Step Problems
2-8 R e te a c h i n
g
2 8
Kim has a $10 bill, a $20 bill, and 2 $5 gift certificates. She uses the gift certificates toward the purchase of a CD for $14.00. How much money does Kim have left after buying the CD? Read and Understand
What do you know?
Kim has a ten-dollar bill, a twenty-dollar bill, and two five-dollar gift certificates. She uses the 2 certificates toward the purchase of a CD that costs $14.00.
What are you trying to find?
How much money does Kim have left after she buys the CD?
Plan and Solve
Answer these hidden questions. How much money does Kim have?
$20.00 $10.00 $30.00
How much are the two certificates worth?
$5.00 $5.00 $10.00
How much cash will Kim need to buy the CD?
$14.00 $10.00 $4.00
Solve the problem.
Money cash paid for CD Money left $30.00 $4.00 $26.00
Write the answer in a complete sentence.
Kim has $26 left after buying the CD.
Look Back and Check
Is your answer correct?
Yes, $4.00 $26.00 $30.00
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1. You can also find how much money Kim has left by completing the following expression. $10.00 $20.00 $5.00 $5.00
$14.00
Topic 2
45
Name
8 2 e c i t c a r P
Problem Solving: Multiple-Step Problems
Practice 2-8
Solve.
1. Theater tickets for children cost $5. Adult tickets cost $3 more. If 2 adults and 2 children buy theater tickets, what is the total cost?
$26 2. Luis has a $10 bill and three $5 bills. He spends $12.75 on the entrance fee to an amusement park and $8.50 on snacks. How much money does he have left?
$3.75 3. Number Sense Alexandra earns $125 from her paper route each month, but she spends about $20 each month on personal expenses. To pay for a school trip that costs $800, about how many months does she need to save money? Explain.
About 8 months, since she can save about $100 per month 4. Critical Thinking Patty is a member of the environmental club. Each weekday, she volunteers for 2 hours. On Saturday and Sunday, she volunteers 3 hours more each day. Which expression shows how to find the number of hours she volunteers in one week? A 25 B 2222255 C 22233 D 233 5. Explain It Marco’s goal is to eat only 2,000 calories each day. One day for breakfast he consumed 310 calories, for lunch he consumed 200 more calories than breakfast, and for dinner he consumed 800. Did he make his goal? Explain.
Yes: 310 510 800 1,620 calories. 46 Topic 2
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Name
Reteaching 3-1
Multiplication Properties You can use multiplication properties to help you multiply more easily.
R e te a c h i n g
Associative Property of Multiplication You can change the grouping of the factors. The product stays the same. (3 4) 4 48 Factors
3 (4 4)
Product
Factors
12 48 4
3
3 1
48
Product 16
48
Commutative Property of Multiplication You can change the order of the factors. The product stays the same. 7 428 Factors
4
Product
Factors
7
28
Product
Zero Property of Multiplication When one of the factors is 0, the product is always 0. 30 Factors
0
03
Product
Factors
0
Product
Identity Property of Multiplication When one of the factors is 1, the product is always the other factor. Identify the multiplication property or properties used in each equation.
zero 55 identity
commutative 6 (7 9) associative
1. 100 0 0
2. 7 2 2 7
3. 1 55
4. (6 7) 9
Reasoning Use the multiplication properties to determine what number must be in the box. 5
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5. 5 4
4
7. (3 12) 9.
50
8
6. 99
5
3 (12 8)
2 2 50
8.
0
10. (16
1
99
10
33 )
25 16 (33 25) Topic 3
47
Name
Practice 3-1
Multiplication Properties In 1 through 5, write the multiplication property used in each equation. 1 3 e c i t c a r P
1. 53 6 6 53 2. 0 374,387 0 3. 5 (11 4) (5 11) 4
commutative zero associative identity commutative
4. 42 1 42 5. 14 5 5 14
6. Reasoning Chan bought 2 large frozen yogurts at $1.50 each and 1 small bottle of water for $1.00. How much did she pay in total?
$4.00 7. Dan has 4 shelves. He has exactly 10 books on each shelf. Judy has 10 shelves. She has exactly 4 books on each shelf. Who has more books? Explain.
They have the same number of books. 4 10 10 4 8. Algebra If 3 8 12 8 3 n, what is the value of n? A 3
B 8
C 12
D 18
9. Explain It Write a definition for the Associative Property of Multiplication in your own words and explain how you would use it to compute 4 25 27 mentally.
Sample answer: Using the Associative Property of Multiplication you can group (4 25) and multiply the product by 27 to get 2,700. 48 Topic 3
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Name
Reteaching 3- 2
Using Mental Math to Multiply You can also use patterns to multiply mentally.
R e te a c h i n g
Fact: 6 8 48 60 8 480
6 80 480
600 8 4,800
60 80 4,800
6,000 8 48,000
600 80 48,000
60,000 8 480,000
6,000 80 480,000
3 2
Pattern: Notice that the product is always the digits 48 followed by the total number of zeros that are in the factors. Find 30
3 50.
Use the Associative Property of Multiplication to regroup. (30 50) 3 1,500 3 4,500
Commutative Property of Multiplication You can multiply factors in any order. 15
9
91
5
Associative Property of Multiplication You can change the grouping of factors. (820)
5
8
Find each product. Use patterns and properties to compute mentally.
1. 80 90
7,200 1,000
3. 5 10 20
2. 40 800 4. 4 30 25
(20
5)
32,000 3,000
5. Number Sense You know that 6 7 42. How can you find 60 700?
5
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Sample answer: Multiply 6 7, then place the total number of zeros in both factors after the product. So, 60 700 42,000. Topic 3
49
Name
Practice 3- 2
Using Mental Math to Multiply Use mental math to find each product. 2 3 e c i t c a r P
1. 150 20
2. 0 50 800
3, 000
0
4. 120 50
20,000
5. 60 70 1
6,000
4,200
7. 100 10 1
8. 1,800 20 0
1,000
0
10. 1,400 2,000
6. 9,000 80
720,000 9. 30 20
600
11. 7,000 50 1
2,800,000
3. 500 40
350,000
12. 1,000 200 30
6,000,000
13. Number Sense A googol is a large number that is the digit one followed by one hundred zeros. If you multiply a googol by 100, how many zeros will that product have?
102 zeros 14. Gregorios drives 200 miles per day for 10 days. How many miles did he drive in all?
2,000 miles
15. Algebra If a b c 0, and a and b are integers greater than 10, what must c equal? A 0
B 1
C 2
D 10
16. Explain It SungHee empties her piggybank and findsthat she has 200 quarters, 150 dimes, and 300 pennies. How much money does she have? Explain.
(200 25) (150 10) (300 1) 5,000 1,500 300 6,800 cents, so she has 68 dollars. 50 Topic 3
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Name
Reteaching 3-3
Estimating Products
A bus service drives passengers between Milwaukee and Chicago every day. They travel from city to city a total of 8 times each day. The distance between the two cities is 89 mi. In the month of February, there are 28 days. The company’s budget allows for 28,000 total miles for February. Is 28,000 mi a reasonable budget mileage amount? One Way to Estimate
Estimate 28 8 Use rounding.
R e te a c h i n g
3 3
Another Way to Estimate
89.
Estimate 28 8 89. Use compatible numbers.
You can round 89 to 100 and 8 to 10. Then multiply. 28 10 100 280 100 28,000 Because this is an overestimate, there are enough miles.
Replace 28 with 30, 89 with 90, and 8 with 10. 30, 90, and 10 are compatible numbers because they are close to the actual numbers in the problem and they are easier to multiply. Now the problem becomes 30 90 10. 30 90 = 2,700
Multiply 3 × 9, then place two zeros after the product.
2,700 10 = 27,000 Multiply 27 1 using the Identity Property of Multiplication, then place three zeros after the product. In the estimate, we used numbers greater than the srcinal numbers, so the answer is an overestimate.
28,000 total miles is a reasonable budget amount. Estimate each product. Use rounding or compatible numbers.
1. 42 5 90
1 8, 000
2. 27 98 4
12, 00 0
Mrs. Carter ordered new supplies for Memorial Hospital.
3. About how much will it cost to purchase 48 electronic thermometers? Supplies 5
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About $1,000
4. About how much will it cost to purchase 96 pillows?
About $2,000
Electronic thermometers
Pulse monitors
$19 each $189 each
Pillows
$17each
Telephones
$19each Topic 3
51
Name
Practice 3-3
Estimating Products Estimate each product. 3 3 e c i t c a r P
1. 68 21
2. 5 101
1, 40 0 4. 99 99
50 0
6. 19 718
36,000 8. 47 29 11
2,000 10. 69 21 23
3,000
5. 87 403
10,000 7. 39 51
3. 151 21
14,000 9. 70 27
15,000 11. 7 616
28,000
2,100 12. 8,880 30
4,200
270,000
13. Number Sense Give three numbers whose product is about 9,000.
Possible answer: 99 x 10 x 9 Electronics Prices 14. About how much would it cost to buy 4 CD/MP3 players and 3 MP3 players?
About $1,100
CD player MP3 player
$ 74.00 $ 9 9.00
CD/MP3 player $199.00 AM/FM radio
$ 29.00
15. Which is the closest estimate for the product of 2 19 5? A 1,150
B 200
C 125
D 50
16. Explain It Explain how you know whether an estimate of a product is an overestimate or an underestimate.
Sample answer: If the numbers are rounded down, it is an underestimate. If the numbers are rounded up, it is an overestimate.
52 Topic 3
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Name
Reteaching 3-4
Multiplying by 1-Digit Numbers
R e te a c h i n g
Mr. McGuire drives his truck 275 miles each day. How far does he drive in 3 days?
3 4
Find 275 3. Step 1: Multiply the
What You Think 3 5 ones 15 ones
ones. Regroup if necessary.
Regroup 15 ones as 1 ten and 5 ones.
Step 2: Multiply the tens. Regroup if necessary.
What You Think 3 7 tens 21 tens 21 tens 1 ten 22 tens Regroup as 2 hundreds and 2 tens.
What You Write
Step 3: Multiply the hundreds. Regroup if
What You Think 3 2 hundreds 6 hundreds 6 hundreds 2 hundreds 8 hundreds No need to regroup.
What You Write
necessary.
What You Write 1
275 3 5 21
275 3 25
21
275
3 825
Mr. McGuire drives 825 miles in 3 days. Find each product. Estimate to check that your answer is reasonable.
1. 31 7 4. 25 9 7. 552 3 10. 266 8 5
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217 2. 225 5. 1,656 8. 2,128 11.
29 4 102 8 471 9 390 2
116 816 4,239 780
3. 88 6 6. 211 7 9. 73 4 12. 514 6
528 1,477 292 3,084
13. Estimation Estimate the product of 48 and 7. Do you have an underestimate or overestimate?
Sample answer: 50 7 350; overestimate Topic 3
53
Name
Practice 3-4
Multiplying by 1-Digit Numbers 4 3 e c i t c a r P
Find each product. Estimate to check that your answer is reasonable.
1. 58 3 3. 83 5 5. 273 4 7. 789 6 9.
68 2
13 6
174 415 1,092 4,734 10.
582 5
2. 49 8 4. 95 6 6. 35 8 8. 643 7 11.
392 570 280 4,501
84 4
12.
2,910
33 6
13. Xavier painted five portraits and wants to sell them for 36 dollars each. How much money will he make if he sells all five?
$180
14. A farmer wants to build a square pigpen. The length of one side of the pen is 13 ft. How many feet of fencing should the farmer buy?
52 ft
15. Jasmine wants to buy 4 green bags for 18 dollars each and 3 purple bags for 15 dollars each. She has 100 dollars. How much more money does she need?
$17
926 7
6, 48 2
16. Geometry A regular octagon is a figure that has eight sides with equal lengths. If one side of a regular octagon is 14 inches long, what is the perimeter of the entire octagon? A 148 in.
B 140 in.
C 112 in.
D 84 in.
17. Explain It Why is 2,482 not a reasonable answer for 542
6? Sample answer: 542 6 is about 3,000; 3,000 is an underestimate, so 2,482 is not a reasonable answer.
54 Topic 3
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Name
Reteaching 3-5
Multiplying 2-Digit by 2-Digit Numbers
R e te a c h i n g
Find 43 26. Step 1: Multiply by the ones. Regroup if necessary.
What You Think 6 3 ones 18 ones Regroup 18 ones as 1 ten and 8 ones.
What You Write
3 5
1
43
6 4 tens 24 tens 24 tens 1 ten 25 tens Regroup 25 tens as 2 hundreds and 5 tens.
Step 2: Multiply by the tens. Regroup if necessary.
26 258
1
43 26 258 860
What You Think 20 3 ones 60 ones Regroup 60 ones as 6 tens. 20 4 tens 80 tens Regroup 80 tens as 8 hundreds.
Step 3: Add the partial products.
1
What You Think 6 43 258 20 43 860
43 26 258 860 1,118
partial products
Find the product.
1.
38 12
4 56 5.
26 21
5 46
2.
5
2, 11 2 6.
49 27
1,323 7.
73 19
1,387
9. Number Sense In the problem 62 . c In , n o it a c u d E n o s r a e P ©
3.
64 33
57 28
1,596
4.
85 15
1, 275 8.
91 86
7, 826
45, what are the partial products?
310 and 2,480
Topic 3
55
Name
Practice 3-5
Multiplying 2-Digit by 2-Digit Numbers 5 3 e c i t c a r P
Find each product. Estimate to check that your answer is reasonable.
1.
56 34
1, 90 4 5.
64 51
3, 264
2.
45 76
3.
3, 420 6.
47 30
1, 41 0
4.
35 15
525 7.
4, 41 8 8.
56 19
1,064
9. To pay for a sofa, Maddie made a payment of 64 dollars each month for one year. How much did the sofa cost ? 10. Geometry To find the volume of a box, you multiply the length times the width times the height. What is the volume, in cubic feet, of a box that is 3 ft long, 8 ft wide, and 16 ft high?
47 94
92 49
4, 50 8
$768 384 cubic feet
11. Estimation Katie is in charge of buying juice for the teachers’ breakfast party. If one teacher will drink between 18 and 22 ounces of juice, and there are 32 teachers, which is the best estimate for the amount of juice Katie should buy? A B C D
about 200 ounces about 400 ounces about 600 ounces about 800 ounces
12. Explain It Is 7,849 a reasonable answer for 49
49? Why or why not?
Sample answer: No; 49 49 is about 2,500; 2,500 is an overestimate, and 7,849 is not close to 2,500.
56 Topic 3
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Name
Reteaching 3-6
Multiplying Greater Numbers Find 128 23.
Estimate: 100 Step 1 Multiply by the ones. Regroup as needed.
Step 2 Multiply by the tens. Regroup as needed.
2
128 23 1 384 2,560 2,944
20 2,000
R e te a c h i n g
Step 3 Add the products.
3 6
1
128 3 384
128 20 2,560
Because the answer is close to the estimate, the answer is reasonable. Find the product. Estimate to check if your answer is reasonable. Problem
Multiply by the Ones
Multiply by the Tens
71
1.
282 19 2,538
2,820
282 9 2,538
Add the Products
1
282 10
2,538 2,820
2,820
5,358
5,358 2.
538 46
24,748
3. Reasonableness Is 2,750 a reasonable answer for 917 5
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33? Explain. No; if you round 917 to 900 and 33 to 30, the product is 900 30 or 27,000, so 2,750 is not reasonable. Topic 3
57
Name
Practice 3-6
Multiplying Greater Numbers Find each product. Estimate to check that your answer is reasonable. 6 3 e c i t c a r P
1.
556 34
2.
234 75
3.
395 76
4.
483 57
18,904
17,550
30, 02 0
27,531
5.
6.
7.
8.
628
33
20,724
154
35
5, 39 0
643
49
31, 50 7
536
94
5 0, 384
9. Number Sense In a class of 24 students, 13 students sold over 150 raffle tickets each, and the rest of the class sold about 60 raffle tickets each. The class goal was to sell 2,000 tickets. Did they reach their goal? Explain.
Sample answer: Yes; 13 150 1,950 and the rest of the class sold more than 50, so they went over their goal of 2,000. 10. Player A’s longest home run distance is 484 ft. If Player A hits 45 home runs at his longest distance, what would the total distance be?
21,780 feet
11. Player B’s longest home run distance is 500 ft. There are 5,280 ft in 1 mi. How many home runs would Player B need to hit at his longest distance for the total to be greater than 1 mi?
11 home runs
12. Algebra Which equation shows how you can find the number of minutes in one year? A 60 24 365 B 60 60 24 C 60 365 D 60 60 365 5
13. Explain It Write a real-world problem where you would have to multiply 120 and 75.
Check students’ work. 58 Topic 3
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Name
Reteaching 3-7
Exponents You can use exponential notation to write a number that is being multiplied by itself.
R e te a c h i n g
There are two parts in exponential notation. The base tells you what factor is being multiplied. The exponent tells you how many of that factor should be multiplied together. The exponent is not a factor.
3 7
exponent 82 = 8 8 The base is 8, so 8 is the factor to be multiplied. The exponent is 2, so 2 factors of 8 should be multiplied together. base You can write 82 in two other forms. In expanded form, you write out your factors. Since 8 2 means you multiply two factors of 8, 82 in expanded form is 8 8. In standard form, you write down the product of the factors. Since 8 8 = 64, 64 is the standard form of 82.
Write in exponential notation.
1. 2 2 2
23
2. 6 6 6 6 6
65
Write in expanded form.
3. 14
1111
4. 53
555
6. 83
512
Write in standard form.
5. 2 2 2 2 5
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16
7. A used car lot has 9 lanes for cars and 9 rows for cars in each lane. What is the exponential notation for the number of spaces on the lot? Can the owner fit 79 cars on the lot?
92; Yes. 9 9 81. There are 81 spaces on the lot. Topic 3
59
Name
Practice 3-7
Exponents For questions 1–4, write in exponential notation. 7 3 e c i t c a r P
1. 13 13 13 3. 64 64
133
642
2. 8 8 8 8 8 8 4. 4 4 4 4 4444
86
48
For questions 5–8, write in expanded form.
2 2 2 2 2 6. 20 squared 20 20 11 11 11 11 11 8. 9 cubed 9 9 9
5. 25 7.
4
For questions 9–12, write in standard form.
64 6 7,776
9. 4 4 4 11.
5
10. 14 squared
196 9 6,561
12. 9 9 9
13. Number Sense Which of these numbers, written in expanded form, is equal to 625? A B C D
5555 55 555 55555
14. Number Sense Find the number equal to 6 raised to the second power. A B C D
18 36 6 12
15. Explain It Explain what 4 raised to the fourth power means.
4 raised to the fourth power means four factors of 4 multiplied together: 4 4 4 4. 60 Topic 3
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Name
Reteaching 3-8
Problem Solving: Draw a Picture and Write an Equation
R e te a c h i n g
A hardware store ordered 9 packs of screws from a supplier. Each pack contains 150 screws. How many screws did the store order?
3 8
Read and Understand What do you know?
The store ordered nine packs of screws.
What are you trying to find?
Each pack contained 150 screws. The total number of screws ordered
Plan and Solve Draw a picture of what you know.
Write an equation.
Let x the total number of screws. 9 150 x
Multiply.
4
150 9 1,350
The store ordered 1,350 screws.
Look Back and Check Is your answer reasonable?
Yes, 150 10 1,500.
A state aquarium has display tanks that each contains 75 fish. Three of these tanks 5
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are at the entrance. How many fish are on display at the entrance? Draw a picture. Write an equation. Solve.
Sample: 75 3 x; 225 fish
x sh in all 75
75
75 Topic 3
61
Name
Practice 3-8
Problem Solving: Draw a Picture and Write an Equation 8 3 e c i t c a r P
Draw a picture and write an equation. Then solve.
1. When Mary was born, she weighed 8 pounds. When she was 10 years old, she weighed 10 times as much. How much did she weigh when she was 10 years old?
8 10 n; n 80 lb 2. Sandi is 13 years old. Karla is 3 times Sandi’s age. How old is Karla?
13 3 n; n 39 3. Reasoning Hwong can fit 12 packets of coffee in a small box and 50 packets of coffee in a large box. Hwong has 10 small boxes and would like to reorganize them into large boxes. Which boxes should he use? Explain.
4.
Since 10 small boxes is 120 packets of coffee, he can use 2 large boxes and 2 small Number Senseboxes. Daniel has 12 tennis balls. Manuel has twice as many tennis balls as Daniel. Kendra has twice as many balls as Manuel. How many tennis balls do they have in all?
A 24
B 36
C 84
D 96
5. Explain It William travels only on Saturdays and Sundays and has flown 400 miles this month. Jason travels every weekday and has flown 500 miles this month. Who travels more miles per day? Explain.
William; he travels only two days per week but has flown almost the same number of miles this month as Jason. 62 Topic 3
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Name
Reteaching 4-1
Dividing Multiples of 10 and 100 You can use math facts and patterns to help you divide mentally. What is 480 6? You already know that 48
What is 60,000 6?
6 8.
60 6 10
480 has one more zero than 48, so 60,000 has three more zeros than place one more zero in the quotient. 6, so place three zeros in the quotient. 480 6 80.
60,000 6 10,000.
Find each quotient. Use mental math.
1. 32 8
2. 320 8
3. 3,200 8
4. 32,000 ÷ 8 =
5. 56 7
6. 560 7
7. 5,600 7
8. 56,000 ÷ 7 =
9. 15 3
10. 150 3
11. 1,500 3
12. 15,000 ÷ 3 =
4 8 5
40 80
400 800
50
500
4,000 8,000 5,000
13. Number Sense Explain how dividing 720 by 9 is like dividing 72 by 9.
You divide 72 by 9. For 720, you add a zero for the ones place. Arlo has a newspaper delivery job. He wants to wrap each of his newspapers in a plastic bag to protect them from the rain. The newspapers are in bundles.
Arlo’s Newspaper Delivery
Number of bundles Number of newspapers per bundle
12 9
Use mental math to answer the following questions. 5
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14. How many bags will he use for 5 bundles? 15. How many bags will he use for 7 bundles? 16. How many bags will he use for all 12 bundles?
45 bags 63 bags 108 bags Topic 4
63
R e te a c h i n g
4 1
Name
Practice 4-1
Dividing Multiples of 10 and 100 Use mental math to find each quotient.
1. 27 9 1 4 e c i t c a r P
2. 270 9
3
3. 2,700 9
30
4. 24 4
300
5. 240 4
6
6. 2,400 4
60
7. 720 9
600
8. 140 7
80
9. 2,100 3
20
700
10. If a bike race covers 120 mi over 6 days and the cyclists ride the same distance each day, how many miles does each cyclist ride each day?
20 mi
Use mental math to answer the following questions.
11. If the vehicles are divided evenly between the sections, how many vehicles are in each section?
Dealership Vehicle Storage Sections of vehicles ............. 4 Vehicles for sale ............ 1,200 Rows per section ................10
300 vehicles 12. If the vehicles are divided evenly between the rows in each section, how many vehicles are in each row? 13. Algebra If 160,000 n 4, find n.
30 Vehicles 40,000
14. Find 32,000 8 mentally. A 4,000
B 400
C 40
D 4
15. Explain It Solve the equation n 50 5,000. Explain your solution.
n 100; Sample answer: Divide each
side by 50. 64 Topic 4
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Name
Reteaching 4-2
Estimating Quotients There are several ways to adjust whole numbers to estimate quotients. Example:
R e te a c h i n g
There are 216 students. The school has 8 classrooms. How many students will be in each classroom? Estimate 216
4 2
8.
Rounding
Compatible Numbers
Multiplication
You can use rounding to estimate a quotient.
You can use compatible numbers to estimate a quotient.
You can use multiplication to estimate a quotient.
Round 216 to the nearest hundred.
Change 216 to a compatible number for 8.
Think: 8 times what number is about 216?
In this case, 216 rounds to 200.
Compatible numbers for 8 are numbers divisible by 8, such as 160, 240, and 320. Choose 240, because it is the closest compatible number to 216.
8 25 200
200 8 25 25 students per room is an underestimate, because 216 was rounded down to 200.
240 8 30
8 30 240 216 is between 200 and 240. So a good estimate is a little more than 25 and a little less than 30 students per classroom.
30 students per class is an overestimate, because 216 was rounded up to 240.
Estimate each quotient. You may use any method.
1. 411 2
400 2 200
2. 162 4
160 4 40
3. Number Sense If you estimate 342 7 by using 350 7 50, is 50 greater than or less than the exact answer? How did you decide? Is 50 an overestimate or an underestimate?
It is greater, because you rounded 342
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up to 350. So, 50 is an overestimate.
Topic 4
65
Name
Practice 4-2
Estimating Quotients
Sample answers for 1–8. Multiplication Rounding Rounding Compatible numbers Rounding Compatible numbers Multiplication
Estimate each quotient. Tell which method you used.
2 4 e c i t c a r P
1. 195 4 2. 283 5 3. 766 8 4. 179 2 5. $395.20 5 6. $31.75 8 7. $247.80 5
50 60 90 90 $80 $4 $50
8. Reasoning If you use $63.00 9 to estimate $62.59 than or less than the exact answer? Explain.
9, is $7.00 greater
Greater than; 9 7 63, which is greater than $62.59. 9. A band played 3 concerts and 10. At a department store, a woman’s earned a total of $321.00. The band total was $284.00 for 7 items. earned about the same amount for Estimate the average cost per item. each concert. Estimate how much the band earned each night.
About $40.00
About $100 each night 11. Which is the closest estimate for 213 A 50
B 40
4?
C 30
D 20 5
12. Explain It Explain how to estimate 524
9.
Sample answer: Round 524 to 500, 9 to 10, and divide: 500 10 50. 66 Topic 4
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Name
Reteaching
Problem Solving: Reasonableness
4-3
After you solve a problem, check to see if your answer is reasonable. Also check that you answered the right question.
R e te a c h i n g
Example: 74 students are going to a special class dinner where they will be seated 8 to a table. Will 9 tables be enough?
4 3
Reasonableness 76 8 9 R2
The answer is close to 9 tables.
Answering the right question
All students must have seats, so there must be one more table to hold the remaining 2 students, making 10 tables in all.
Tell whether each answer is reasonable.
1. Kendra wants to paste 500 photographs into an album, 6 photos to a page. She figures that she will need about 100 pages.
Six times 100 is 600, much more space than Kendra needs. Since 500 is between 480 (6 80) and 540 (6 90), she needs between 80 and 90 pages. 2. Hwong has 39 muffins. If each of his guests will eat 2 muffins, Hwong figures that he can serve muffins to 19 guests.
This is reasonable, since 2 19 38. Tell whether each answer answers the right question.
3. Ivan has a piece of lumber 104 inches long. He is sawing it into 12-inch lengths to make fence posts. He says he can get about 9 fence posts out of the board. 5
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Al thoupost gh 10must 4 12 9 thanand to 8, each beisa closer full 12 to inches, therefore any remainder must be thrown away. Ivan can only make 8 posts. Topic 4
67
Name
Practice 4-3
Problem Solving: Reasonableness Solve. 3 4 e c i t c a r P
1. One tray holds eight sandwiches. If there are 30 sandwiches in all, how many trays are needed?
4 trays
Mrs. Favicchio has 72 students in her science class. The table shows how many students can use each item of lab supplies she is ordering.
3. How many packets of pH paper does she need to order?
8 4. How many cases of test tubes does she need to order?
2. There are 53 students on a field trip. One chaperone is needed for every 6 students. How many chaperones are needed?
9
Lab Supplies Item
Number of Students
PacketofpHpaper
10
Caseoftesttubes
5
Caseofpetridishes
4
15 5. Algebra A loaf of banana bread serves 6 guests. There will be 47 guests attending the faculty breakfast. Which expression shows how many loaves are needed to serve them all? A 47 divided by 6 is 7 R 5, so 7 loaves are needed. B 47 divided by 6 is 7 R 5, so 8 loaves are needed. C 47 plus 6 is 53, so 53 loaves are needed. D 47 minus 6 is 41, so 41 loaves are needed. 6. Explain It You are in line at an amusement park. You count 34 people in front of you. Each rollercoaster fits 11 people. How many rollercoasters must run before you can get on? Explain.
3 rides must go by, and you will get on the fourth one. 68 Topic 4
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Name
Reteaching 4-4
Connecting Models and Symbols Divide 138 equally into 3 groups.
R e te a c h i n g
4 4
Step 1: You can model 138 as 13 groups of 10 plus 8 ones. 130 3 40, so each group will get 4 groups of 10. 40 3 120 130 120 10, so there is 1 group of 10 left. Step 2: There is 1 group of 10 plus 1 group of 8 ones left. You can model 18 as 18 ones. 18 3 6, so each group will also get 6 ones. There is nothing left.
What You Think
What You Write
4 3 138 1 2 1
What You Think
What You Write
46 3 138 1 2 18 1 8 0 138 3 46
Use models to help you divide.
19
1. 4 76 5
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34
4. 7 238
47
2. 2 94
71
5. 6 426
7. Algebra If n 3 57, what is the value of n?
26
3. 5 130
88
6. 3 264
171 Topic 4
69
Name
Practice 4-4
Connecting Models and Symbols After mowing lawns for one week, John put the money he earned on the table. There were four $100 bills, three $10 bills, and five $1 bills. 4 4 e c i t c a r P
1. If John’s brother borrowed one of the $100 bills and replaced it with ten $10 bills, a. how many $100 bills would there be? b. how many $10 bills would there be?
Three $100 bills Thirteen $10 bills
2. If John needed to divide the money evenly with two other workers, how much would each person receive?
$145
3. If John needed to divide the money evenly with four other workers, how much would each person receive?
$87
Complete each division problem. You may use play money or draw diagrams to help.
4.
5.
3 4
5 4
4 1 3 6
3 1 6 2
1 2 1 6 1 6 0
1 5 1 2 1 2 0
6. If $644.00 is divided equally among 7 people, how much will each person receive? A $82.00
B $92.00
C $93.00
D $103.00
7. Explain It Write a story problem using two $100 bills, nine $10 bills, and seven $1 bills.
Sample answer: Karl borrowed $297 from his sister. He pays her $33 each week. How many weeks will it take Karl to pay $297?
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Name
Reteaching 4-5
Dividing by 1-Digit Divisors Find 362 5. Step 1: To decide where to place the first digit in the quotient, compare the first digit of the dividend with the divisor.
Step 2: Divide the tens. Use multiplication facts and compatible numbers. Think 5 ? 35.
Step 3: Divide the ones. Use multiplication facts and compatible numbers. Think 5 ? 10.
3 5, so the first digit in the quotient will not go in the hundreds place.
Write 7 in the tens place of the quotient. Multiply. 5 7 35
Write 2 in the ones place of the quotient. Multiply. 5 2 10
Now, compare the first two digits of the dividend with the divisor.
7 5 36 3 5 1
7 2R2 5 362 3 5 12 1 0 2
36 5, so the first digit in the quotient will go in the tens place.
Subtract. 36 35 1 Compare. 1 5 Bring down the ones.
R e te a c h i n g
Step 4: Check by multiplying. 5 72 360 2 362
Subtract. 12 10 2 Compare. 2 5 There are no more digits to bring down, so 2 is the remainder.
Divide. Check by multiplying.
107 R7
1. 8 863
56 4. 8 448
35 R4 2. 7 249
2R 41 9 5. 2 499
73
3. 5 365
66
6. 6 396
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7. Number Sense How can you tell before you divide 425 by 9 that the first digit of the quotient is in the tens place?
4 0.7 mm.
5
10. Algebra If 4n 3.60, which is the value of n? A 0.09 112 Topic 7
B 0.9
C 9
D 90
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Name
Reteaching
Estimating the Product of a Decimal and a Whole Number
7-3
You can estimate when you are multiplying a decimal by a whole number to check the reasonableness of your product. Zane needs to buy 27 lb of roast beef for the company party. The roast beef costs $2.98 per pound. About how much will the roast beef cost? There are two ways to estimate. Round both numbers
R e te a c h i n g
Adjust your factors to compatible numbers you can multiply mentally.
$2.98 27 $3 30 $90 The roast beef will cost about $90.
$2.98 27
7 3
$3 25 $75 The roast beef will cost about $75.
Sample answers are given. 22 20 2. 19.3 6 100 5.79 1,800 4. 966 0.46 500
Estimate each product.
1. 0.8
3. 345
Number Sense Use the chart to answer questions 5 through 7. 5. About how much would it cost for Angelina and her 4 sisters to get a shampoo and a haircut?
(5 $8) (5 $13) about $105
Treatment Cost Shampoo
$7.95
Haircut
$12.95
Coloring
$18.25
Perm
$22.45
6. Could 3 of the sisters get their hair colored for less than $100? 5
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Yes; $20 3 $60. 7. Angelina gets 9 haircuts per year. About how much does she spend on haircuts for the year?
10 $13 about $130
Topic 7
113
Name
Practice
Estimating the Product of a Decimal and a Whole Number
7-3
Estimate each product using rounding or compatible numbers.
1. 0.97 312
300 5. 0.33 104
3 7 e c ti c a r P
2. 8.02 70
3. 31.04 300
4. 0.56 48
6. 0.83 12
7. 0.89 51
8. 4.05 11
5 60
9,000
25
30
10
50
44
9. 0.13 7
10. 45.1 5
11. 99.3 92
12. 47.2 93
9,000
4, 50 0
0.7
2 25
13. Critical Thinking Mr. Webster works 4 days a week at his office and 1 day a week at home. The distance to Mr. Webster’s office is 23.7 miles. He takes a different route home, which is 21.8 miles. When Mr. Webster works at home, he drives to the post office once a day, which is 2.3 miles from his house. Which piece of information is not important in figuring out how many miles Mr. Webster drives per week to his office? A B C D
the number of days at the of ce the distance to his of ce the distance to the post of ce the distance from his of ce
14. Strategy Practice Mrs. Smith bought her three children new snowsuits for winter. Each snowsuit cost $25.99. How much did Mrs. Smith pay in all? A $259.90
B $77.97
C $51.98
D $25.99
15. Explain It How can estimating be helpful before finding an actual product?
If your estimate is very different from your actual product, you know that you need to check your work. 114 Topic 7
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Name
Reteaching 7-4
Multiplying Two Decimals Caroline earns $2.40 per hour for babysitting her brother. She babysat last night for 3.25 hours. How much did she earn? First, estimate your product so you can check for reasonableness. $2.40 3.25 $2 3 6
Caroline earned about $6.00.
Step 1: Multiply as with whole numbers.
Step 2: Count the total number of decimal places in both factors.
3.25 2.40 000 13,000 65,000 78,000
3.25 2 decimal places
7 4
2.40 2 decimal places So, 4 decimal places total.
Step 3: Place the decimal point in the product with the correct total number of decimal places following it.
7.8000 $7.80 Caroline $7.80 last night. Because $7.80 is close to your estimate earned of $6, your answer is reasonable. Find each product. Check by estimating.
0.92 2. 3.98 3.40485. 9.3 6.4
1. 0.2 4.6 4. 0.532
28.258 3. 8.54 5.86 54.498 6. 0.37
7.1
1. 19 56 4.4 1.628 0.14
7. Critical Thinking Jackie wants to buy a new CD player. It costs $32.95. She has saved $26 and has a coupon for 30% off the price. Does Jackie have enough money to buy the CD player?
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R e te a c h i n g
Y es.CD Theplayer coupon reduces the price of the to $23.07.
Topic 7
115
Name
Practice 7-4
Multiplying Two Decimals Find each product.
1.
1.11
3.7 0.3
5. 0.79
2.
4.4 0.2
0.88 3.397 4.3
3.
4.148
0.61 6.8
6. 0.79
4.
1.9 0.005
0.0095 0.00395 0.005
7. Number Sense The product of 4.7 and 6.5 equals 30.55. What is the product of 4.7 and 0.65? 4.7 and 65? 4 7 e c i t c a r P
3.055, 305.5 8. What would be the gravity in relation to Earth of a planet with 3.4 times the gravity of Mercury?
Relative (to Earth) Surface Gravity
1.326
Planet 9. The gravity of Venus is 0.35 times that of Jupiter. What Mercury Neptune is the gravity of Venus in relation to Earth’s gravity? Jupiter
Gravity 0.39 1.22 2.6
0.91 10. How many decimal places are in the product of a number with decimal places to the thousandths multiplied by a number with decimal places to the hundredths? A 2
B 3
C 4
D 5
11. Explain It Explain how you know the number of decimal places that should be in the product when you multiply two decimal numbers together.
The total number of decimal places in the numbers being multiplied is the number of decimal places in the product.
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Name
Reteaching 7-5
Dividing Decimals by 10, 100, or 1,000 You can use place-value patterns when you divide a decimal by 10, 100, or 1,000. Sanjai has 27.5 lb of clay. If he uses the clay to make 10 bowls, how much clay will he use for each bowl? What if he makes 100 bowls from the clay? What if he makes 1,000 bowls?
Dividing a number by 10 moves the decimal point one place to the left. R e te a c h i n g
27.5 10 2.75 Dividing a number by 100 moves the decimal point two places to the left.
7 5
27.5 100 0.275 Dividing a number by 1,000 moves the decimal point three places to the left. 27.5 1,000 0.0275 Sanjai will use 2.75 lb for each of 10 bowls, 0.275 lb for each of 100 bowls, and 0.0275 lb for each of 1,000 bowls. Remember: When you divide a number by 10, 100, or 1,000, your quotient will be smaller than that number.
For questions 1 through 6, find the quotient. Use mental math.
1.64 540.9 ÷ 10 54.09
1. 16.4 10 4. 7.
2. 38.92 100
2.971 10 0.033
3. 297.1 100
0.38926. 0.33 5. 41.628 1,000 0.041628 The city has a section of land 3,694.7 ft long. The city wants
to make 100 equal-sized gardens with this land. How long will each garden be?
36.947 ft
8. Reasonableness Connor divided 143.89 by 100. He said his answer was 14.389. Is this a reasonable answer?
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No. Connor moved the decimal point only one place to the left. He should have moved it two places to the left. Topic 7
117
Name
Practice 7-5
Dividing Decimals by 10, 100, or 1,000 Find each quotient. Use mental math.
8.66 0.0199 0.22855
1. 86.6 10 3. 1.99 100 5. 228.55 1,000
2. 192.5 100 4. 0.87 10 6. 0.834 100
7. 943.35 1,000 8. 1.25 10 Algebra Write 10, 100, or 1,000 for each n.
0.94335
5 7 e c i t c a r P
9. 78.34 n 0.7834
n
100
10. 0.32 n 0.032
n
10
12. There are 145 children taking swimming lessons at the pool. If 10 children will be assigned to each instructor, how many instructors need to be hired?
1.925 0.087 0.00834 0.125
11. (75.34 25.34) n 5
n
10
15 instructors
13. Ronald ran 534.3 mi in 100 days. If he ran an equal distance each day, how many miles did he run per day? A 5
B 5.13
C 5.343
D 6.201
14. Explain It Carlos says that 17.43 100 is the same as 174.3 0.01. Is he correct? Explain.
No, he is not correct. 17.43 100 is 0.1743 and 174.3 0.01 is 1.743.
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118 Topic 7
Name
Reteaching 7-6
Dividing a Decimal by a Whole Number Jia needs to cut a board into 3 equal pieces to make bookshelves. The board is 1.5 yd long. How long will each bookshelf be?
0
0.5
1.0
1.5
Laying a measuring tape next to the board, Jia sees that her board can be cut into 3 pieces that are each 0.5 yd long. Jia also writes out her problem to make sure her answer is correct. Sometimes you will need to add a zero to the right of the dividend so that you can continue dividing. Example: 8.1 18
Step 1. Write a decimal point in the quotient directly above the decimal point in the dividend. Step 2. Divide as you would with whole numbers.
0.45 18 8.10 72
0.5 3 1.5 15
90 90
0
0
Step 3. Check your quotient by multiplying. 0.5 3 1.5 Since the quotient matches the measurement that Jia saw on her measuring tape, she knows that her answer is correct. For 1 through 6, find each quotient. Check by multiplying.
1. 14 6.3
0.45 34 3.62
4. 123.08 5
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0.03 30 0.019
2. 77 2.31
3. 89 2.492
5. 0.57
6. 562.86
0.028 59 9.54
7. A family of five people attends a theme park. They purchase 2 adult tickets for $27.50 each and 3 student tickets for $12.50 each. If the 5 tickets are purchased with a $100 bill, how much change do they receive?
$7.50
Topic 7
119
R e te a c h i n g
7 6
Name
Practice 7-6
Dividing a Decimal by a Whole Number Find each quotient.
5.3
1. 13 68.9
5. 123.08 34 7. 562.86 59
11.78
2. 35 412.3
3.62 9.54
0.16
0.89
3. 90 14.4
4. 60 53.4
0.019 0.305
6. 0.57 30 8. 24.4 80
6 7 e c i t c a r P
9. John paid $7.99 for 3 boxes of cereal. The tax was $1.69. Excluding tax, how much did John pay for each box of cereal if they all were the same price?
$2.10
10. If a package of granola bars with 12 bars costs $3.48, how much does each granola bar cost? A 29¢
B 31¢
11. Estimation 64.82 11 is A a little more than 6. B a little more than 60.
C 44¢
D $1.00
C a little less than 6. D a little less than 60.
12. Explain It Explain how to divide 0.12 by 8.
Sample answer: Write zeros in the ones and tenths places in the quotient. Annex a zero in the dividend and continue dividing. The quotient is 0.015.
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Name
Reteaching 7-7
Estimation: Decimals Divided by Whole Numbers
The merry-go-round in the Smithson Town Park took in $795.60 in the last six months. The ticket taker says that 442 people rode the merrygo-round during that time. About how much does it cost to ride? You can estimate to find the quotient of $795.60 442 either by rounding or using compatible numbers and mental math. To round your numbers, cover the digit in the hundreds place with your hand. Look at the number next to your hand. Since 9 5, you round $795.60 up to $800. Since 4 5, you round 442 down to 400.
R e te a c h i n g
7 7
800 400 2
It costs about $2 to ride.
Or you can use compatible numbers and mental math. 800 400 2
It costs about $2 to ride.
Sample answers are given. 81.2 19 4 2. 376.44 22 20 3. 62.91 16 4 763.85 82 10 5. 550.8 9 60 6. 486.5 3 150
Estimate each quotient.
1. 4.
7. Number Sense Explain how you would know an error had been made if you find the quotient for 231.68 16 to be 144.8.
Possible answer: by estimating the division as 200 20 10, which is 5
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not close to 144.8
8. Reasonableness Is 61.5 15 a little less than 4, a little more than 4, a little less than 40, or a little more than 40?
a little more than 4 Topic 7
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Name
Practice 7-7
Estimation: Decimals Divided by Whole Numbers Estimate each quotient.Sample answers are given. 1. 73.5 10 7 2. 246.78 83 3 3. 185.7 3 60
15 4. 535.6
7. 35.6 7 7 7 e c i t c a r P
65. 553.9
35
5
10. 839.7 90
8. 86.4 4
9
11. 93.26
13. Joseph is saving $23 a week to buy a graphing calculator that costs $275.53. About how many weeks will it take before he can buy the calculator?
about 12 weeks
30
90
6. 366.6 12
20 3 30
9. 270.53 3 12. 77.3 11
90 7
14. Juan works at a health food store two hours a day, three days a week. His weekly pay is $73.50. About how much does Juan make per hour?
about $12 per hour
15. Reasonableness Which of the following is a reasonable estimate for the operation 566.3 63? A about 16
B about 9
C about 4
D about 6
16. Explain It When would you estimate a quotient instead of finding the exact quotient?
Sample answer: Estimating a quotient could help you quickly figure out how many items you can buy with your money or how long a trip will take. 5
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Reteaching 7-8
Dividing a Decimal by a Decimal Sebastian is shipping a box to his cousin. The box weighs 23.6 lb. The cost to ship the box is $88.50. How much does it cost per pound to ship the box? To divide a decimal by a decimal, you need to change the problem into a simpler type of problem that you already know how to solve.
Step 1: To change this problem, you multiply the divisor by a power of 10 until it becomes a whole number. Step 2: You must also multiply the dividend by the same power of 10. Sometimes, you may have to add zeros as placeholders.
23.6 10 236
88.50 10 885.0
Step 3: Place a decimal point in the quotient and divide as you would with whole numbers.
3.75 236 885.00 708 1770 1652
1180
1180 0
The box costs $3.75 per pound to ship. For questions 1 through 6, find each quotient.
1. 0.104 ÷ 0.08 4. 0.427 ÷ 61
1.3 2. 5.49 ÷ 0.9 6.1 0.007 5. 0.8449 ÷ 0.711.19
42 9.483 ÷ 8.7 1.09
3. 50.4 ÷ 1.2 6.
7. Miriam needs to buy new notebooks for school. The notebooks cost $0.98 each including tax. Miriam’s mother gave her $6.45 and told her to buy 8 notebooks. Did her mother give her enough money?
No. She can only buy 6 notebooks 5
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with $6.45.
8. Miriam finds a coupon in the store that reduces the price of notebooks to $0.75 each including tax. How many notebooks can Miriam buy now?
She can buy 8 notebooks.
Topic 7
123
R e te a c h i n g
7 8
Name
Practice 7-8
Dividing a Decimal by a Decimal Find each quotient.
2.3
1. 0.8 1.84
0.9 6.39 5. 7.1 8 7 e c i t c a r P
11
9. 0.07 0.77
3
2. 0.9 2.7
0.81 0.648 6. 0.8
0.9
10. 4.8 4.32
13. Chan paid $4.75 for trail mix that costs $2.50 a pound. How many pounds of trail mix did he buy?
1.9 pounds
1.9
1.1
3. 2.5 4.75
4. 1.1 1.21
8.25
2.55
10.725 7. 1.3
12
11. 0.7 8.4
0.51 8. 0.2
3
12. 2.3 6.9
14. Max’s family car has a gas tank that holds 12.5 gallons of gas. It cost $40.62 to completely fill the tank yesterday. What was the price of gas per gallon?
$3.25
15. Strategy Practice Strawberries cost $5.99 per pound, and bananas cost $0.59 per pound. How many pounds of bananas could you buy for the cost of one pound of strawberries? A 101.5 pounds B 10.15 pounds C 5.99 pounds
D .59 pounds
16. Explain It When dividing a decimal by a decimal, why is it sometimes necessary to add a zero to the right of the decimal point in the quotient?
Sample answer: Often, after moving decimal points to make the numbers ready for division, you must add a zero so that a smaller number can be divided by a larger number. 124 Topic 7
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Reteaching 7-9
Problem Solving: Multiple-Step Problems A multiple-step problem is a problem where you may need more than one step to find your answer. Marcie was in a 3-day charity walk. Her friend Gayle said she would give the charity $1.50 for each mile that Marcie walked. The first day, Marcie walked 26.42 miles. The second day, Marcie walked 32.37 miles. The third day, Marcie walked 28.93 miles. How much money did Gayle give?
Step 1.
R e te a c h i n g
Read through the problem again and write a list of what you already know.
7 9
Marcie walked 26.42, 32.37, and 28.93 miles. Gayle gave $1.50 for each mile.
Step 2.
Write a list of what you need to know. Total amount Gayle gave
Step 3.
Write a list of the steps to solve the problem. Find the total number of miles Marcie walked. Find the amount Gayle gave.
Step 4.
Solve the problem one step at a time. 26.42 32.37 28.93 87.72
total number of miles Marcie walked
87.72 $1.50 $131.58
total amount Gayle gave
Use the information above to answer Exercise 1.
1. Marcie’s brother Tom was also in the charity walk. He only walked 0.8 as far as Marcie on the first day, 0.7 as far on the second day, and 0.9 as far on the third day. How many miles did Tom walk, rounded to the nearest hundredth of a mile? 5
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69.83 miles 2. Diego is buying fruit at the store. Which costs less: 1 pound of each fruit or 4 pounds of peaches?
4 pounds of peaches
Fruit Apples Oranges Peaches Grapes
Cost per pound $0.89 $1.29 $0.99 $1.09 Topic 7
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Name
Practice 7-9
Problem Solving: Multiple-Step Problems Write and answer the hidden question or questions in each problem and then solve the problem. Write your answer in a complete sentence.
1. Sue bought 2 pairs of jeans and a belt that cost $6.95. The tax on the items was $5.85.
Storewide Sale
Jeans
$29.95 for 1 pair OR 2 pairs for $55.00
T-shirts $9.95 for 1 OR 3 T-shirts for $25.00
Sue paidreceive the cashier $70.00. How much money did Sue in change? 9 7 e c i t c a r P
What was the total cost of Sue’s items? $67.80; Sue received $2.20 in change. 2. A recreation department purchased 12 T-shirts for day camp. The department does not have to pay sales tax. It paid with a $100.00 bill. How much change did it receive?
What was the cost for 12 T-shirts? $100.00; The department received no change, because it paid the exact amount. 3. When Mrs. Johnson saw the sale, she decided to get clothes for each child in her family. She bought each of her 6 children a pair of jeans and a T-shirt. She paid $14.35 in sales tax. How much was Mrs. Johnson’s total bill?
A $94.35
B $119.70
C $229.35
D $253.35
4. Write a Problem Write a two-step problem that contains a hidden question about buying something at the mall. Tell what the hidden question is and solve your problem. Use $8.95 somewhere in your equation. Write your answer in a complete sentence.
Check students’ problems. 5. Explain It What are hidden questions and why are they important when solving multiple-step problems?
Sample answer: Hidden questions are not directly stated in the problem. The answers to the hidden questions give information 126 Topic 7 needed to solve the problem.
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Reteaching 8-1
Basic Geometric Ideas Points and lines are basic geometric ideas. Lines are sometimes described by relationships to other lines.
Draw
Write
Sa y
J
point J
J
Description J is a point. It shows an exact location in space. ___
‹
N
O
___
‹
›
NO ___
line NO ray PQ
›
Q
PQ
P
›
NO is a line. It is a straight path of points that goes on forever in two directions. ___ PQ is a ray. It goes on forever in only one direction. ›
___
‹
F E
___
___
‹
H
‹
›
›
EF GH
G
Line EF is parallel to line GH.
›
T
S
U
V
‹
___
›
SU intersects ___ VT ›
›
EF and GH are parallel lines. They are the same distance apart and will not cross each other.
___
‹
Line SU intersects line VT.
___ SU
‹
›
‹
›
and VT are intersecting lines. They pass through the same point.
Use the diagram on the right. Name the following.
____ MN, OP
1. two parallel lines ___ ‹
›
‹
›
2. two intersecting rays
__ _ ___ ›
5
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›
8 1
___
‹
H K
E F M J
R e te a c h i n g
O
G N
P
GH, JK
Topic 8
127
Name
Practice 8-1
Basic Geometric Ideas
Sample answers given for 1–7.
Use the diagram at the right. Name the following.
1. three points
points P, R, and A
P
J Q
___
2. a ray
R
›
E
EJ
I N
F A
3. two perpendicular ___intersecting lines but not__ ‹
1 8 e c ti c a r P
›
‹
›
RF intersects NI
__
‹
___
›
‹
›
NI ___ AF
4. two parallel lines 5. a line segment
JE ___
6. two perpendicular lines
PR
‹
___
›
‹
›
RF
7. Explain It Can a line segment have two midpoints? Explain.
No, a midpoint is a point halfway between the endpoints of a line segment. There cannot be two midpoints. 8. Which type of lines are shown by the figure? A Congruent
C Perpendicular
B Parallel
D Curved
A B
9. Draw a picture Draw and label ___ two ____ perpendicular line segments KL and MN. 5
Sample drawing:
M
N L
128 Topic 8
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Reteaching 8-2
Measuring and Classifying Angles The chart below can help you describe and identify an angle. MeasureEFG.
90°
180°
Remember to place the 0 mark on one side of the angle.
E
EFG
measures 45°.
0° F
90°
Place the center of the protractor on S. Line up SW with the 0 mark. Place a point at 65. Label it J. Draw SJ.
J
180°
Acute between 0 and 90 Right exactly 90
G
Draw an angle of 65°.
Classifying Angles
Obtuse between 90 and 180 Straight exactly 180 R e te a c h i n g
0° S
W
8 2
Classify each angle as acute, right, obtuse, or straight. Then measure each angle.
1.
2.
Acute; 60˚
3.
Obtuse; 120˚ Straight; 180˚
Draw an angle with each measure.
4. 100°
5. 170°
6. Reasoning ABC measures less than 180° but more than 90°. Is ABC a right, an acute, an obtuse, or a straight angle?
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Obtuse; it has a measure between 90˚ and 180˚.
Topic 8
129
Name
Practice 8-2
Measuring and Classifying Angles Classify each angle as acute, right, obtuse, or straight. Then measure each angle. (Hint: Draw longer sides if necessary.)
2.
R i g h t a n g l e ; 90˚ 2 8 e c ti c a r P
Acute angle; 30˚
Draw an angle with each measure.
3. 120°
4. 180°
5. Draw an acute angle. Label it with the letters A, B, and C. What is the measure of the angle?
Check students’ drawings and measures. 6. Which kind of angle is shown in the figure below? A Acute
C Obtuse
B Right
D Straight
7. Explain It Explain how to use a protractor to measure an angle.
Place center of protractor on angle’s vertex so the 0˚ mark is on one side of angle. Read measure where other side of angle crosses protractor. 130 Topic 8
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Reteaching 8-3
Polygons A polygon is a closed plane figure made up of line segments. Common polygons have names that tell the number of sides the polygon has.
Triangle 3 sides
Pentagon 5 sides
Octagon 8 sides
A regular polygon has sides of equal length and angles of equal measure.
60
3in. 60
Hexagon 6 sides
Open Figure
Quadrilateral 4 sides
3 in.
3in. 60
Each side is 3 in. long. Each angle is 60. R e te a c h i n g
Name each polygon. Then tell if it appears to be a regular polygon.
8 3
1.
Quadrilateral; notregular
2.
Hexagon; regular
3.
Pe n t a g o n ; notregular
4.
Triangle; regular
5. Reasoning Shakira sorted shapes into two different groups. Use geometric terms to describe how she sorted the shapes.
Group A
Group B
The shapes in
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Group A are all open and are not polygo ns. The shapes in Group B are closed and are polygons. Topic 8
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Name
Practice 8-3
Polygons Name each polygon. Then tell if it appears to be a regular polygon.
1.
2.
Octagon; regular 3 8 e c ti c a r P
Quadrilateral; not regular H
3. Name the polygon. Name the vertices.
Quadrilateral; B, E, H , T
T B E
4. Which polygon has eight sides? A quadrilateral
B pentagon
C hexagon
D octagon
5. Explain It Draw two regular polygons and two that are irregular. Use geometric terms to describe one characteristic of each type.
Sample answer: The regular polygons have congruent sides. The irregular polygons have different angle measures. 132 Topic 8
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Reteaching 8-4
Triangles You can classify triangles by the lengths of their sides and the sizes of their angles. acute all angles less than 90
60
8cm 60
8cm 60
equilateral all sides the same length
This triangle is both equilateral and acute.
isosceles two sides the same length
This triangle is both isosceles and right.
8 cm
right one right angle
16in.
16in.
7.5 ft
obtuse one obtuse angle
120
scalene no sides the same length
4 ft
5 ft
Not all acute triangles are equilateral.
Not all right triangles are isosceles.
R e te a c h i n g
8 4
This triangle is both scalene and obtuse. Not all obtuse triangles are scalene.
Remember that the sum of the measures of the angles of a triangle is 180°. Classify each triangle by its sides and then by its angles.
1.
cm 8 26
8 128 cm 26 14 cm
2.
5 in.
3. 3 in.
60
3ft
4 in.
60
3ft 60
3 ft
Isosceles; obtuse
Scalene; right
Equilateral; acute
The measures of two angles of a triangle are given. Find the measure of the third angle. 5
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4. 40°, 100°,
40°
5. 14°, 98°,
68°
6. 38°, 38°,
104 °
Topic 8
133
Name
Practice 8-4
Triangles Classify each triangle by its sides and then by its angles.
2. 60 12.42 ft
4m
9.5 ft
4m
60
Scalene triangle; right triangle 4 8 e c ti c a r P
60 4m
8 ft
Equilateral triangle; acute triangle
The measures of two angles of a triangle are given. Find the measure of the third angle.
61° 6. 54º, 36°, 90°
71° 5. 75°, 75°, 30° 3. 47°, 62°,
4. 29°, 90°,
7. Judy bought a new tent for a camping trip. Look at the side of the tent with the opening to classify the triangle by its sides and its angles.
4 ft 4 ft 3 ft
Isosceles triangle; acute triangle 8. Reasonableness Which describes a scalene triangle? A 4 equal sides
B 3 equal sides
C 2 equal sides
D 0 equal sides
9. Explain It The lengths of two sides of a triangle are 15 in. each. The third side measures 10 in. What type of triangle is this? Explain your answer using geometric terms.
It is an isosceles triangle, because two sides are the same length.
134 Topic 8
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Reteaching 8-5
Quadrilaterals Quadrilateral
Definition
Example
5 in.
Parallelogram A quadrilateral with both pairs of opposite sides parallel and equal in length Rectangle
Rhombus
in. 2
in. 2 5 in. 5 ft
A parallelogram with four right angles
ft2
ft2 5 ft 4 in.
A parallelogram with all sides the same length
4in.
4in. 4 in.
Square
R e te a c h i n g
1 ft
A rectangle with all sides the same length
1ft
1ft
8 5
1 ft Trapezoid
2 in.
A quadrilateral with only one pair of parallel sides
in. 2
in. 3 6 in.
Remember that the sum of the measures of the angles
of a quadrilateral is 360 .
Classify each quadrilateral. Be as specific as possible.
1.
2.
6 ft
ft 3
ft 3 6 ft
Rectangle
3.
4 in.
7 ft 9 ft
in. 4
in. 4
3 ft 6 ft
4 in.
Rhombus
Trapezoid
The measures of three angles of a quadrilateral are given. Find the measure of the fourth angle. 5
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56° 6. 82°, 78°, 90°, 110° 4. 65°, 150°, 89°,
5. 100°, 80°, 100°,
80 °
Topic 8
135
Name
Practice 8-5
Quadrilaterals Classify each quadrilateral. Be as specific as possible.
2.
6 cm 18 cm
18 cm
15 cm cm8
cm 8 15 cm
20 cm
Trapezoid 8 in.
3. 5 8 e c ti c a r P
Rectangle 17 cm
4.
4 in.
4 in.
17cm
17cm 17 cm
8 in.
Parallelogram
R h om bu s
For 5 and 6, the measures of three angles of a quadrilateral are given. Find the measure of the fourth angle.
5. 90°, 145°, 78°,
47°
6. 110°, 54°, 100°,
7. Name the vertices of the square to the right.
F
G
L
A
96°
F, G, A , L 8. Three of the angles of a quadrilateral measure 80°, 100°, and 55°. Which is the measure of the fourth angle? A 115°
B 120°
C 125°
D 130°
9. Explain It Can a trapezoid have four obtuse angles? Explain.
Sample answer: The figure would not be able to close the fourth side if all of the angles were obtuse. 136 Topic 8
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Reteaching 8-6
Problem Solving: Make and Test Generalizations Here is a generalization to be tested: any square can be cut in half through a diagonal. The result is always two isosceles triangles, each with a 90 angle. Test one example of this generalization: Draw a square, ABCD. A
B
Draw a diagonal, AC. A
Inspect the triangles, ABC and CDA.
B
A
D
C
D
Triangle ABC:
C
B
A
C
D
R e te a c h i n g
C
Triangle CDA:
1. AB BC
All sides of a square are equal length .
1. CD DA
2. Angle B 90
All angles of a square are 90.
2. Angle D
All sides of a square are equal length.
90
All angles of a square are 90.
Conclusion: Each triangle has two equal sides and contains a right angle. The generalization is true for the square ABCD. Repeat for more squares. If for each square the conclusion is the same, the generalization appears to be correct.
Explain It Show that the triangles ABC and CDA are congruent.
Sample answer: AB CD and BC DA (All sides of a square are equal length.) 5
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AC
AC. Two triangles are congruent if
all three sides have matching lengths.
Topic 8
137
8 6
Name
Practice
Problem Solving: Make and Test Generalizations
8-6
In 1 through 6, test the generalization and state whether it appears to be correct or incorrect. If incorrect, give an example to support why.
1. All triangles have right angles.
Incorrect; some triangles have one right angle and some triangles don’t have any. 2. All rectangles have right angles. Correct. 6 8 e c ti c a r P
3. Any two triangles can be joined to make a rhombus.
Incorrect; only two congruent isosceles triangles can be joined to make a rhombus. 4. All rectangles can be cut in half vertically or horizontally to make two congruent rectangles.
Correct.
5. Intersecting lines are also parallel.
Incorrect; parallel lines will never intersect. 6. How many whole numbers have exactly three digits? Hint: 999 is the greatest whole number with three digits. A 890
B 900
C 990
D 999
7. Explain It How can you show that a generalization is correct?
Sample answer: test it twice to show that it is correct both times.
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Reteaching 9-1
Meanings of Fractions What fraction of the set of shapes are squares?
Step 1: Find the denominator.
Step 2: Find the numerator.
Step 3: Write the fraction.
How many shapes are there in the set?
How many squares are there in the set?
Write the numerator over the denominator.
There are 5 shapes in the set.
There are 3 squares in the set.
The denominator is the total number of shapes. So, the denominator is 5.
The numerator is the number of squares in the set. So, the numerator is 3.
3 __ 5
Numerator Denominator
3 _ of the set are squares. 5
R e te a c h i n g
Write the fraction that names the shaded part.
1.
3 __
4 ___ 16
4 3.
4.
5 __ 8 5
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9 1
2.
4 ___ 14
5. Number Sense If _15 of a region is not shaded, what part is shaded? 6. Alex has 7 dimes and 3 nickels. What fraction of the coins are dimes?
__ 4
5 7 ___ 10 Topic 9
139
Name
Practice 9-1
Meanings of Fractions Write the fraction that names the shaded part.
1 __
1.
6 3 __
2.
8 In 3 and 4, draw a model to show each fraction.
3. _48 as part of a set
4.
5 __ 10
Sample answers given for 3 and 4. as part of a region
1 9 e c ti c a r P
5 of a region is shaded, what part 5. Number Sense If __ 17 is not shaded?
6. Camp Big Trees has 3 red canoes and 4 blue canoes. What fraction of the canoes are red?
12 ___ 17 3 __ 7
7. In a class of 24 students,13 students are girls. What fraction of the students are boys? 11 A ___ 13
11 B ___ 24
13 C ___ 24
24 D ___ 11
8. Explain It Trisha says that if 5_7 of her pencils are yellow, then _27 are not yellow. Is she correct? Explain.
Sample answer: Yes, Trisha is correct because if 5 out of 7 are yellow, 2 out of 7 must not be yellow. 140 Topic 9
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Reteaching 9-2
Fractions and Division Fractions can represent division. You can write a division expression as a fraction. For example: Write a fraction for 5
7.
The first number in the division expression is the numerator of the fraction. The second number in the division expression
5 Numerator __
57
7 Denominator
is the denominator of the fraction. So, 5 7 5_7 . Give each answer as a fraction.
1. 3 10 4. 8 9 7. 6 10
3 ___ 10 8 __ 9 6 ___ 10
2. 7 12 5. 2 5 8. 9 13
7 ___ 12 2 __ 5 9 ___ 13
3. 2 3 6. 1 6 9. 14 16
2 __
R e te a c h i n g
3 1 __ 6 14 ___ 16
9 2
Reasoning Three congruent circles are each divided into three equal parts. Use these three circles for 10 through 12.
10. What part of a whole circle is shown by the white, or unshaded, area of one circle? 11. What part of a whole circle is shown by the white, or unshaded, area of two circles? 5
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12. What part of a whole circle is shown by the white, or unshaded, area of three circles?
1 __ 3 2 __ 3 3 or 1 __ 3 Topic 9
141
Name
Practice 9-2
Fractions and Division Give each answer as a fraction. Then graph the answer on the number line.
1. 3 7 2. 4 9 3. 1 5
3 __
74 __ 91
0 0
5
1
4 9
__ 0
1
3 7
1
1
6 ___ 16
5
4. Use the number line to name the fraction at point A. 0
2 9 e c ti c a r P
1
A
At a golf course, there are 18 holes. Of the 18 holes, 3 are par threes, 8 are par fours, and 7 are par fives. What fraction of the holes are
5. par fives?
7 ___ 18
6. par threes?
3 ___ 18
7. par fours?
8 ___ 18
8. Number Sense Explain how you know that 7 9 is less than 1.
Sample answer: Because 7 is less than 9, 7 cannot be divided by 9 one time. 9. After school, Chase spends 20 min reading, 30 min practicing the piano, 15 min cleaning his room, and 40 min doing his homework. Chase is busy for 105 min. What fraction of the time does he spend cleaning his room?
15 ____ 105
10. Venietta read 4 books in 7 weeks. How many books did she read each week? 6 A __ 7
4 B __ 7
3 C __ 7
11. Explain It In 5 min, Peter completed 2 math problems. Yvonne says he did 3_5 of a problem each minute. Is she correct? Explain.
2 D __ 7
Sample answer: No, Peter completed 2 _ of a problem each minute. 5
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Reteaching 9-3
Mixed Numbers and Improper Fractions 13 Example 1: Write __ as a mixed number. 4
To write an improper fraction as a mixed number, draw a model. Draw squares divided into 4 equal parts until you have 13 parts. 1
2
1
2
1
2
3
4
3
4
3
4
4 4 1 whole
4 4 1 whole
The 13 parts make up 3 wholes and 13 1 4 34
Example 2:
1 4
1
4 4 1 whole
1 4 R e te a c h i n g
of a whole.
9 3
Write 5_13 as an improper fraction.
To write a mixed number as an improper fraction, draw a model. Draw circles divided into 3 equal parts to represent 51. 3
1
2
1
3 1 whole
2
1
3 3 3
2
1
3
1 whole
3 3
1 whole
2
1
3 3 3
1 whole
2
1
3 3 3
1 whole
3 3
1 3
3 3 3 3 3 1 16 333333 3
Write each improper fraction as a mixed number.
1.
8 __ 3
2 2__ 3
2.
3 __ 1 7
10 ___
7
3.
5 __
1 2__ 2
2
Write each mixed number as an improper fraction. 5
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7 5
2 __ 4. 1__ 5
6 5. 4__ 7
34 ___ 7
5 6. 2__ 8
21 ___ 8 Topic 9
143
Name
Practice 9-3
Mixed Numbers and Improper Fractions Write an improper fraction and a mixed number for each model.
1.
7, __ 3
2.
1 2__
10 , ___ 4
3
2 2__ 4
Write each improper fraction as a mixed number. 3 9 e c ti c a r P
12 3. ___ 7
9 6. __ 4
5 1__ 7 1 2__ 4
1 2__
4. 7__ 3
29 7. ___ 13
3 3 2___ 13
1 2__
5 5. __ 2
34 8. ___ 8
2 2 4__ 8
Write each mixed number as an improper fraction. 4 9. 2__ 5 1 12. 7__ 8
14 ___ 5 57 ___ 8
7 10. 8__ 9 3 13. 4__ 7
79 ___ 9 31 ___ 7
6 11. 3__ 7 1 14. 5__ 4
27 ___ 7 21 ___ 4
15. Number Sense Jasmine has 41 lb of dog food to pour into 5 dishes. How many pounds of dog food should she pour in each dish? 1 lb A 4__ 5
1 lb B 8__ 5
C 10 lb
1 lb D 11__ 8
16. Explain It Hank needs 3 quarters to play one video game each time. If he has 14 quarters, how many times can he play? Explain.
Sample answer: Hank can play the video game 4 times. He cannot play the game 4_23 times because the game will not work with only 2 quarters.
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Name
Reteaching 9-4
Equivalent Fractions The fractions shown below are equivalent. They all describe the same part of a whole.
_3
6 __
5
12 __
10
20
To find equivalent fractions, multiply or divide the numerator and denominator by the same number. 3 __ 6 2 ___ __ 5
2
10
6 __ 2 ___ 12 ___
6 12 __ 2 ___ ___
6 __ 3 2 __ ___
3 12 __ 4 __ ___
10 10
2 2
20
20
5
2
20
4
10 5
Name two equivalent fractions for each fraction.
1.
1 __ 3
4 3. ___ 20
3 2 , __ __
2 2. ___
6 9 2 , __ 1 ___ 10 5
Sample answers given for 1–4. 1 ___ 4 __
,
6 __ 1, 8
12
2 4. ___ 16
24 4 ___ 32
Find the missing number to make the fractions equivalent. 8 4 ___ 5. __ 7
14
6.
3 ___ 7. __
4 ___ __ 18
6
4
12
9
12
15 __ 3 8. ___ 4
20
12 9. Number Sense Are _34 and __ equivalent fractions? Explain. 16
Yes: If you multiply the numerator and 3 _
5
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denominator of 4 by 4, you get
12 __ 16 .
Topic 9
145
R e te a c h i n g
9 4
Name
Practice 9-4
Equivalent Fractions
Sample answers given for 1–6. 3 1 ; ___ 1 ; ___ 4 __ __ 3. 2 12
Name two equivalent fractions for each fraction. 5 1. ___ 15
4 4. ___ 28
10 1 ; ___ __ 3 30 2 ; __ 1 ___ 14 7
6 2. ___ 36
3 5. ___ 21
___
6 18 6 ; __ 1 ___ 42 7
2 6. ___ 11
6 24 6 4 ; ___ ___ 22 33
Find the missing number to make the fractions equivalent. 8 4 __ 7. ___ x 13 q 9. ___ 54 4 9 e c ti c a r P
2 __ 9
x 26 q 12
n 12 ___ 8. ___
30
90
14 ___ 7 10. ___ h
11. Renie gave each of six people left. Explain how this is true.
1 __ 10
20
n h
36 40
of a veggie pizza. Renie has _25 of the pizza
6 Sample answer: Renie gave away __ , 10 4 or _35, of the pizza, so _25 , or __ , of it is left. 10
12. Which fraction is equivalent to _37? 3 6 A __ B ___ 6
14
3 C ___ 17
7 D __ 7
13. Explain It Jacqueline had four $5 bills. She bought a shirt for $10. Explain what fraction of her money Jacqueline has left. Use equivalent fractions.
Sample answer: Four $5 bills equal 10 $20. After she spent $10, she had __ , 20 or _12, of her money. 5
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146 Topic 9
Name
Reteaching 9-5
Comparing and Ordering Fractions and Mixed Numbers You can compare fractions by finding a common denominator. Samantha and her brother Jacob went out for pizza. Samantha ate _12 of her pizza. Jacob 4 ate __ of his pizza. Who ate more pizza? 12 Because these fractions have different denominators, you need to find a common denominator. Then you can compare them.
Step1.
Jacob’s pizza
Write multiples of the two denominators until you get a common multiple.
2: 2, 4, 6, 8, 10, 12 12: 12, 24, 36, 48, 60 Step 2.
Samantha’s Pizza
R e te a c h i n g
Use 12 as the common denominator.
Since you multiply 2 6 to get 12, you must multiply 1
6.
9 5
6 1 ___ __ 2
Step 3.
12
Compare the fractions with common denominators. 6 ___ 4 ___ 12
So, Samantha ate more pizza.
12
Remember: If you don’t know the multiples of the denominators, you can multiply the denominators together to get a common denominator. Compare. Write , , or for each
1.
>
2 __ 3
1 __
3 __
2.
6
4
.
>
2
Order the numbers from least to greatest. 3 , __ 3 4 , __ 4. __
,
5 5 4
5 , 1__ 3 , 1___ 2 5. 1__ 6
6
12
, ,
5 3. __
1 __
6
<
21 ___ 24
3 , __ 3 , __ 4 __ 5 4 5
,
2 1__ , 1_36, 1_56 12
6. Geometry Sofia baked three kinds of pie. Sofia’s Mom told her to 8 __ 5
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_4
3 _
bring 16 of the apple pie, 8 of the pecan pie, and 6 of the pumpkin pie to school to share with her friends. Draw the pies and show which pie will have the greatest amount brought to school.
All three pies will have equal amounts 8 3 4 brought to school, because __ _ _. 16 8 6 Topic 9
147
Name
Practice 9-5
Comparing and Ordering Fractions and Mixed Numbers Compare the numbers. Write , , or for each
1. 4.
>8 5 4 2__ 2__ 5 < 6 6 __
6 __
7
2. 5.
.
< 3 6 2 3__ 3__ 9 = 3 4 __ 9
2 __
2 6. __ 5
4 5 4 3 4 8 , __ 8 , __ 6 , __
6 8 4 8
8.
4
1, 4__ 8
10 , 5___ 11
2 4___ 12
3 , 1__ 3 , 1__ 8 2 , 1___ 9. 1__ 7
5 9 e c ti c a r P
4
___
10
Order the numbers from least to greatest. __ 3 , __ 5 4 , __ 4 , __ 7. __ 1, 4__
> 1121 > __28
1 3. 1___
4
14
8 1_37, 1_24, 1_34, 1__ 14
10 2 1 _ __ 4_18 , 4__ , 4 , 5 4 11 12
4? 10. Number Sense How do you know that 5_14 is less than 5___
5_14
5 5__ ; 20
4 5__ 10
8 5__ ; 20
5 5__ 20
<
8 5__
10
20
fix Mrs. Aaron’s car. 7 11. A mechanic uses four wrenches to__ 5 The wrenches are different sizes: 16 in., _12 in., _14 in., and __ in. 16 Order the sizes of the wrenches from greatest to least.
_1 2
5 7 __ in., __ in., in., _14 in. 16 16
12. Which is greater than 6_13? A 6_16
B 6_15
C 6_14
D 6_12
3 2 13. Explain It Compare 3__ and 3__ . Which is greater? 22 33 How do you know?
3 9 3__ 3__, which is greater than 22 66 2 3__ 33
148 Topic 9
4 3__ 66
5
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Name
Reteaching 9-6
Common Factors and Greatest Common Factor The greatest common factor (GCF) of two numbers is the greatest number that is a factor of both. Find the greatest common factor of 12 and 18. You can use arrays to find the factors of 12. 1 1212
6 12
2
3
4 12
Factors of 12 are 1, 2, 3, 4, 6, and 12. R e te a c h i n g
You can use arrays to find the factors of 18. 1 1818
2
18 9
3
6 18
9 6
Factors of 18 are 1, 2, 3, 6, 9, and 18. You can see that the common factors of 12 and 18 are 2, 3, and 6. The greatest common factor of 12 and 18 is 6. Find the GCF of each pair of numbers.
1. 9, 27 3. 7, 36
9 1
2. 25, 40 4. 40, 48
5 8
5. Number Sense Can the GCF of 18 and 36 be greater than 18? Explain.
No. No number greater than 18 can 5
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be a factor of 18.
Topic 9
149
Name
Practice 9-6
Common Factors and Greatest Common Factor Find the GCF of each pair of numbers.
1. 15, 50 4. 18, 32 7. 25, 55
5 2
2. 6, 27
5
8. 14, 48
5. 7, 28
3 7
3. 10, 25
2
9. 81, 135
6. 54, 108
5 54 27
10. Number Sense Can the GCF of 16 and 42 be less than 16? Explain.
6 9 e c ti c a r P
Yes; Sample answer: The greatest common factor will be less than 16, because 42 is not evenly divisible by 16. 11. A restaurant received a shipment of 42 gal of orange juice and 18 gal of cranberry juice. The juice needs to be poured into equal-sized containers. What is the largest amount of juice that each container can hold of each kind of juice?
12. At a day camp, there are 56 girls and 42 boys. The campers need to be split into equal groups. Each has either all girls or all boys. What is the greatest number of campers each group can have?
6 gal 14 campers
13. Which is the GCF of 24 and 64? A 4
B 8
C 14
D 12
14. Explain It Do all even numbers have 2 as a factor? Explain.
Sample answer: Yes, because all even numbers are divisible by 2
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Name
Reteaching 9-7
Fractions in Simplest Form There are two different ways to write a fraction in simplest form. 20 in simplest form. Write ___ 24
Divide by Common Factors • Divide by common factors until the only common factor is 1. • You can start by dividing by 2, since both numbers are even.
Divide by the GCF • First find the GCF of 20 and 24. 20: 1, 2, 24: 1, 2,
4, 5, 10, 20 3, 4, 6, 8, 12, 24
20 2 ___ 10 ______ 24 2 12
• The common factors of 20 and 24 are 1, 2, and 4. The GCF of 20 and 24 is 4.
But both 10 and 12 are also even, so they can be divided by 2.
• Divide both the numerator and the denominator by 4.
10 2 __ 5 ______ 12 2 6
R e te a c h i n g
20 4 __ 5 ______ 24 4 6
9 7
• Since 5 and 6 do not have any common factors, _56 is the simplest form. 20 ___ 24
5. written in simplest form is __ 6
Write each fraction in simplest form. 16 1. ___ 20
4.
8 ___
7.
18 ___
32 21
4 _ 5 1 _ 4 6 _ 7
8 2. ___ 16
5.
18 ___
8.
24 ___
42 40
1 _ 2 3 _ 7 3 _ 5
_1
5 3. ___ 10
2
6.
15 ____
3 __
9.
55 ___
11 __
100
20
70
14
31 is in simplest form. 10. Number Sense Explain how you can tell that ___ 33
5
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Sample answer: 31 is a prime number; 31 and 33 do not have any common factors 31 except 1, so __ is in simplest form. 33 Topic 9
151
Name
Practice 9-7
Fractions in Simplest Form Write each fraction in simplest form.
1 2 1 __ _____ 5 1 __ _____ 9
1 4 5 __ _____ 6 5 ___ _____ 12
1 3 3 __ _____ 5 1 __ _____ 6
5 _____ __ 1. ___
6 _____ __ 2. ___
9 _____ __ 3. ___
3 4. ___
10 5. ___
9 6. ___
10
15
2 7. ___ 18
24
12
25 8. ___ 60
27
15
12 9. ___ 72
10. Number Sense Explain how you can tell _45 is in simplest form.
7 9 e c ti c a r P
Sample answer: The numerator and the denominator have no common factors other than 1.
Write in simplest form.
Math Test
11. What fraction of the problems on the math test will be word problems?
➡ ➡
20 Multiple-choice problems 10 Fill in the blanks
➡
__ 1 7
5 Word problems
12. What fraction of the problems on the math test will be multiple-choice problems? 13. Which is the simplest form of 1 A __ 8
14. Explain It
1 B ___ 22
4 __ 7
10 __ ? 82
10 C ___ 82
5 D ___ 41
100 Explain how you can find the simplest form of ____ . 1,000
Sample answer: The GCF of 100 and 1,000 The simplest form 100 is 100. 100 1 is ________ __. 1,000100 10 152 Topic 9
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Name
Reteaching 9-8
Tenths and Hundredths Fractions can also be named using decimals.
8 out of 10 sections are shaded. The fraction is 8 . 10 The word name is eight tenths.
Write 2 as a decimal. 5 Sometimes a fraction can be rewritten as an equivalent fraction that has a denominator of 10 or 100. 2 5
4 10
0.4
So,
2 5
The decimal is 0.8. Remember: the first place to the right of the decimal is tenths.
22 52
3 Write 3 as a 5 decimal.
Write 0.07 as a fraction.
First write the whole number.
The word name for 0.07 is seven hundredths.
3 Write the fraction as an equivalent fraction with a denominator of 10.
4 10
0.4.
“Seven” is the numerator, and “hundredths” is the denominator. 7 . 100
Change the fraction to a decimal.
So, 0.07
3 5
Remember: the second place to the right of the decimal is hundredths.
3 5
2 2
6 10
0.6
Write the decimal next to the whole number
9 8
3.6 3 So, 3 5
3.6.
Write each fraction or mixed number as a decimal. 1 1. __ 5
0. 2
6 2. ___ 25
0. 24
3 3. 2__ 4
2. 75
9 4. 3___ 10
3.9
Write each decimal as a fraction or mixed number.
25 1____
5. 1.25
7. 0.65
9. Number Sense Dan says 5
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29 3____
100 65 ____ 100
6. 3.29
8. 5.6 3 _ 5
R e te a c h i n g
100 6 5___ 10
is the same as 3.5. Is he correct? Explain.
5 1; _ 3 No. 3.5 is the same as 3__ 10 or 3_ 2 5 is the same as 0.6.
Topic 9
153
Name
Practice 9-8
Tenths and Hundredths Write a decimal and fraction for the shaded portion of each model.
1.
2.
7 __
4 __
0.16;
25
0.7;
10
Write each decimal as either a fraction or a mixed number.
6 __
3. 0.6 8 9 e c ti c a r P
5. 6.9
10 9 6__ 10
73 ___ 4. 0.73 6. 8.57
100 57 8___ 100
Write each fraction or mixed number as a decimal. 7 7. ___ 10
2 9. 7___
0. 7 7.2
10
33 8. ____ 100
9 10. 3____
0. 33 3.09
100
Use division to change each fraction to a decimal. 4 11. __ 5
1 13. ___ 50
0. 8 0. 02
12 12. ___ 25
11 14. ___ 20
0. 48 0. 55
15. Strategy Practice When you convert 0.63 to a fraction, which of the following could be the first step of the process? A Since there are 63 hundredths, multiply 0.63 and 100. B Since there are 63 tenths, divide 0.63 by 10. C Since there are 63 tenths, place 63 over 10. D Since there are 63 hundredths, place 63 over 100.
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Name
Reteaching 9-9
Thousandths Example 1: Write 0.025 as a fraction. Ones 0
. .
Tenths 0
Hundredths 2
Thousandths 5
You can use a place-value chart to write a decimal as a fraction. Look at the place-value chart above. The place farthest to the right contains a digit tells you denominator theplacefraction. In thisthat case, it is thousandths. The the number written inofthe value chart tells you the numerator of the fraction. Here, it is 25. 25 0.025 1,000 11 Example 2: Write 1,000 as a decimal.
Ones
. .
Tenths
Hundredths
R e te a c h i n g
Thousandths
9 9
You can also use a place-value chart to write a fraction as a decimal. The denominator tells you the last decimal place in your number. Here, it is thousandths. The numerator tells you the decimal itself. Write a 1 in the hundredths place and a 1 in the thousandths place. Fill in the other places with a 0. 11 1,000
0.011
Write each decimal as a fraction.
1. 0.002
2 ____
1,000
2. 0.037
Write each fraction as a decimal. 5
4. 1,000
0.005
76
5. 1,000
37 ____ 1,000
0.076
7. Explain It Matt reasoned that he can write your answer. 5
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3. 0.099
9 1,000
40
6. 1,000
99 ____ 1,000
0.040
as 0.9. Is he correct? Explain
9 is read as “nine thousandths,” 1,000 No: ____ which is the decimal 0.009. 0.9 is read 9 as “nine tenths,” which is the fraction __ . 10 Topic 9
155
Name
Practice 9-9
Thousandths Write each decimal as either a fraction or a mixed number. 7 1,000 38 1,000 20 1,000
1. 0.007 3. 0.038
2. 0.052 4. 0.259
3
5. 3.020
6. 4.926
4
52 1,000 259 1,000 926 1,000
Write each fraction as a decimal.
0.073 0.854 0.005
73
7. 1,000 854
9. 1,000 9 9 e c ti c a r P
5
11. 1,000
593
8. 1,000 11
10. 1,000 996
12. 1,000
0.593 0. 011 0.996
Write the numbers in order from least to greatest. 5 9 13. 1,000 , 0.003, 1,000
14. 0.021, 0.845,
99
5 9 ____ 0.003, ____ , 1,000 1,000 99 ____ 0.021, 1,000 , 0.845
1,000
15. Look at the model at the right. Write a fraction and a decimal that the model represents.
11 ____
1,000
and 0.011
16. Reasoning In Tasha’s school, 0.600 of the students participate in a school sport. If there are one thousand students in Tasha’s school, how many participate in a school sport? A 6,000
B 600
17. Explain It Explain how knowing that 5 decimal for 4_58.
C 60
D 6
8 0.625 helps you write the
Sample answer: Knowing that 5 8 0.625 gives the digits to write to the right of the decimal point.
156 Topic 9
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Name
Reteaching 9-10
Fractions and Decimals on the Number Line 6 Show 0.8, __ , and 0.519 on the same number line. 20
0
0.3
0.519
0.8
1
Step 1: Starting at 0, count 8 tenths to the right. This point is
6 Step 2: Change __ to a decimal. 20 6 __ can be thought of as 6 20. 20
8 0.8 or __ 10.
6 20 0.3 Starting at 0, count 3 tenths to 6 the right. This point is __ or 0.3. 20
Step 3: Estimate the location of 0.519. You know that 0.5 is the same as 0.500. You also know that 0.6 is the same as 0.600. So, 0.519 is between 0.5 and 0.6. You know that 0.519 0.550. So, 0.519 is between 0.500 and 0.550 and closer to 0.500 than to 0.550.
Show each set of numbers on the same number line. 6 1. 0.1, __ , 0.5 10
2.
9 __ , 10
0 0.1
4 0.7, __ 20
0
3. 0.25, 0.40, and
5 __ 50
1
0.5 0.6 0.2
0.9 1
0.7
0 0.1 0.25 0.4
1
Name the fraction and decimal for each point.
4. Point A
3 0.3 or __ 10
5. Point B
55 0.55 or ___ 100
6. Point C
89 0.89 or ___ 100
0
A
B
C 1
5
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Topic 9
157
R e te a c h i n g
9 1 0
Name
Practice 9-10
Fractions and Decimals on the Number Line Draw a number line to show the set of numbers. Then order the numbers from least to greatest. 8 , 0.2, __ 2 1. 0.75, ___ 5
10
8 0.2, _25 , 0.75, __ 10 2
0 .2
8
0 .7 5
5
10
0
1
Write a fraction or mixed number in simplest form and a decimal that name each point. 0 1 9 e c i t c a r P
Q
R
0
2. Point Q
S
1
3 __ 0;.3
2
_40;.8
3. Point R
10
5
1
4. Point S
_2 ; 5
1.4
5. Uma recorded the distances that volunteers walked in the charity event. Grace walked 1_35 mi, Wendell walked 1.3 mi, 1 mi. Show these distances on a 10 and Simon walked 1__ number line. Who walked the farthest?
Sample drawing:
1
10
1.3
1
Grace 5
1
2
6. Number Sense Which is a decimal that could go between the 9 mixed numbers 4_35 and 4__ on a number line? 10 A 4.45
B 4.5
C 4.75
D 4.92
7. Explain It Explain how you know that 5.5 is to the right of 5_41 on the number line.
Sample answer: 5_14 5.25; 5.25 5.5; So, 5.5 must be to the right of 5 1_4 . 158 Topic 9
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Name
Reteaching 9-11
Problem Solving: Writing to Explain An environmental scientist is studying an old apple orchard. The orchard is shown on the right. Some of the trees are infected with mold. Other trees are infested with beetles. Some trees are normal. LEGEND normal apple tree apple tree infected with mold apple tree infested with beetles
R e te a c h i n g
The scientist knows that pictures and symbols can be used to write a good math explanation. So she decides to organize her findings in the chart on the right.
9 1 1
Use this with chartmold, to estimate fractional fraction part of the is infected using athe benchmark thatorchard is closethat to the actual amount. A little more than half the grid is covered by trees that are infected with mold.
Use this chart to estimate the fractional part of the orchard that is infested with beetles. Explain how you decided.
One of the eight columns is filled with trees infested with beetles. 5
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This column represents _18 of the orchard.
Topic 9
159
Name
Practice 9-11
Problem Solving: Writing to Explain Estimate the fractional part of the shaded portions below. Explain how you decided.
1.
The shaded area is about half _1
of 2, so it is about 4 . About _23 , because it is more than _12, but less than _34
2.
1 1 9 e c i t c a r P
1 _
3. Draw a square and shade about _18 of it. How did you decide how much to shade?
Check students’ work. Sample: I shaded half of _14 .
4. Draw two rectangles that are different sizes. Shade about _1 of each. Are the shaded parts the same amount? Explain. 2
Check students’ drawings. No; the shaded parts are not the same amount, because the rectangles are not the same size. 5. Explain It Look at a picture of the American flag. Approximately what part of the flag is blue? Explain.
3 __
_1
16 ; since less than 4 of the flag About is blue, and white stars are also in the blue area. 160 Topic 9
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Name
Reteaching
Adding and Subtracting Fractions with Like Denominators
10-1
Aisha cut her birthday cake into 10 slices. She and her friends ate 6 slices. Her parents ate 2 slices. What fraction of cake was eaten? 6 ___
amount of cake Aisha and her friends ate.
10 2 amount of cake Aisha’s parents ate. ___ 10
When two fractions have the same denominator, their sum or difference will also have the same denominator.
Step 1. Add the numerators.
Step 2. Write the sum over the denominator. 8 ___
628
Step 3. Simplify the fraction if possible.
10
8 __ 4 ___ 10
4 of the cake was eaten. So, __
5
5
You can do the same when subtracting fractions with like denominators.
R e te a c h i n g
How much more cake did Aisha and her friends eat than Aisha’s parents? 6 ___ 2 ___ 10
10
=
Step 1. Subtract the numerators.
Step 2. Place the difference over the denominator and simplify if possible. 4 __ 2 ___
624
10
5
Find the sum or difference. Simplify your answer.
1.
3 __
5
11 ___ 12 5 ___ 12
_3
1 _
4
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2.
8 3 __ 8
2
3.
9 ___
4.
10 4 ___ 10
7 __ 9
2 __ 9
3 1__ 10
5 _ 9
5. Joachim has 15 medals he won at swim meets. Of the medals, 6 are first place, 4 are second place, 2 are third place, and 3 are fourth place. What fraction of Joachim’s medals are first or second place? _2
3
6. Algebra Mr. Lucero’s cornbread recipe calls for 3 cups of cornmeal. He put in 1 _23 cups from one open bag and _23 cup 2 from another open bag. How many cups of cornmeal does _ 3 Mr. Lucero need to add when he opens a third bag?
cups Topic 10
161
1 0 1
Name
Practice 10-1
Adding and Subtracting Fractions with Like Denominators Add or subtract. Simplify if possible.
1.
2.
10 ___ 12 8 ___ 12
6 __ 5 __ 3 5. __ 8
8
3 1 __ 2 __ 7. __ 4
4
4
3 2 __ 2 __ 9. __ 5
1 0 1 e c i t c a r P
5
3.
7 ___
9 5 __ 9
1 1_2
8
8 __
5
__ 1 3
1_34
6.
1_12
8.
1_25
4.
10 2 ___ 10
10.
2 __ 3 1 3
__
9 __ 1 10 3 __ 3 ___ 1 8 ___ ___ 10 10 28 ___ 9 ___ 1 ___ 11 11 11 1 __ 3 7 __ __ 8 8 2
11. What fraction could you add to _47 to get a sum greater than 1?
any fraction greater than _37 12. Reasoning Write three fractions, using 10 as the denominator, whose sum is 1.
3 __ 6 1 1 __ Possible answer: __ 10 10 10 13. Which of the following represents the difference between two equal fractions? A 1
1 B __ 2
1 C __ 4
D 0
14. Explain It In one night, George reads 3 chapters of a book with 27 chapters. After the second night, he has read a total 8 of __ of the book. Explain how you would determine the number 27 of chapters George read the second night. Solve the problem. 5
8 __ 3 __ 5 __ ; on the second night, 27 27 27 5 George __ 27 5 chapters 27 162 Topic 10
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Name
Reteaching 10-2
Common Multiples and Least Common Multiple To find the least common denominator (LCD) of two fractions with different denominators, you need to find the least common multiple (LCM). Find the least common multiple and LCD of _34 and _56 . Step 1. Find the least common multiple of 4 and 6. 4 XXXX
6 Are there the same number of X’s XXXXXX in each column? Since the 1st column has fewer, add another set of 4 X’s.
4 XXXX XXXX
6 Are there the same number of X’s XXXXXX in each column? Since the 2nd column has fewer, add another set of 6 X’s.
4 XXXX
Are there the same number of X’s 6 XXXXXX in each column?
XXXX
XXXXXX Since the 1st column has fewer, add another set of 4 X’s.
4 XXXX XXXX XXXX
Are there the same number of X’s 6 XXXXXX in each column? Yes. XXXXXX Since the columns are equal, the number of X’s is the LCM. The LCM 12.
Step 2. Make the least common multiple the least common denominator of both fractions.
3
9 12
3 4
2 10 12
5 6
3
2
1 0 2
Find the least common multiple of each number pair.
1. 2 and 3
6
2. 6 and 9
18
3. 5 and 6
30
4. 8 and 3
24
13 5. Reasonableness Can the LCD of _49 and __ be less than 17? Explain. 17
5
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R e te a c h i n g
No. The LCD for two fractions must be the LCM of the denominators. The LCM of two numbers cannot be less than either of the numbers. Topic 10
163
Name
Practice
Common Multiples and Least Common Multiple Find the LCM of each pair of numbers. 1. 3 and 6 3. 8 and 12 5. 4 and 6 7. 5 and 8 9. 6 and 7 11. 5 and 20
2 0 1 e c i t c a r P
6 24 12 40 42 20
2. 7 and 10 4. 2 and 5 6. 3 and 4 8. 2 and 9 10. 4 and 7 12. 6 and 12
10-2
70 10 12 18 28 12
13. Rosario is buying pens for school. Blue pens are sold in packages of 6. Black pens are sold in packages of 3, and green pens are sold in packages of 2. What is the least number of pens she can buy to have equal numbers of pens in each color?
6 of each 14. Jason’s birthday party punch calls for equal amounts of pineapple juice and orange juice. Pineapple juice comes in 6-oz cans and orange juice comes in 10-oz cans. What is the least amount he can mix of each kind of juice without having any left over?
30 oz of each 15. Reasonableness Dawn ordered 4 pizzas each costing between 8 and 12 dollars. What is a reasonable total cost of all 4 pizzas? A less than $24
C between $32 and $48
B between $12 and $24
D about $70
16. Explain It Why is 35 the LCM of 7 and 5?
Sample answer: It is the smallest number divisible by both 7 and 5. 164 Topic 10
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Name
Reteaching 10-3
Adding Fractions with Unlike Denominators Danisha ate _23 cup of yogurt at breakfast. She ate _14 cup of yogurt at lunch. How much yogurt did she eat today? You can add fractions with unlike denominators. Step 1: Find the least common denominator of the two fractions.
Step 2: Once you have equivalent Step 3: Place the sum over fractions with the same the common denominator and denominator, add the numerators. simplify your fraction if possible.
multiples of 3: 3, 6, 9, 12, 15 multiples of 4: 4, 8, 12, 16, 20
11 Danisha ate __ cup of yogurt 12 today.
8 3 11
8 3 _2 __ and _14 __ 3 12 12
R e te a c h i n g
For questions 1 through 5, find the sum. Simplify if possible.
1.
_4_
2.
5 1 _ _ 2
3 1___ 4. _1_ _1_ _3_ 6
3.
9 5 _ _ 6
13 1___
10
2
_8_
4
5 1___ 12
1 0 3
9 3 __ 4
7 1___
18
5. _2_ _1_ _5_ 3
_4_
9
6
11 1___
36
18
6. Kevin and some friends ordered several pizzas and cut them into different numbers of slices. Kevin ate _16 of one pizza, 5 _1 of another, __ of another, and _13 of another. Did Kevin eat 4 12 the equivalent of a whole pizza? __
yes: 11 pizzas
3 7. Cathy spent __ of an hour on her math homework, _25 of an 10 hour on her science homework, and _34 of an hour on her
6
9 hours 1___
reading homework. How long did Cathy work on homework? 5
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20
Topic 10
165
Name
Practice 10-3
Adding Fractions with Unlike Denominators Find each sum. Simplify if necessary.
9
3 1 ___ 2. __ 7 21
12 7 12 __ 11 1__ 20
1 __ 4 5. ___
2 __ 1 1. __ 9
3
1 __ 2 4. __ 4
3
5 1 ___ 7. __ 6
12
3 __ 4 10. __ 4
3 0 1 e c i t c a r P
5
11 __
12
6
4 __ 1 8. __ 6
3
11 __ 1 11. ___ 12
13 __
_2
5 _
3
4
3 1 __ 6. __
1 1_14
2 __ 1 9. __
15 1 1__ 10 21 40 __
4 __ 1 12. __
1
7
3 _
2 __ 1 3. __ 3 2 5 8
5 5 8 2
Jeremy collected nickels for one week. He is making stacks of his nickels to determine how many he has. The thickness of one 1 nickel is __ in. 16
13. How tall is a stack of 16 nickels?
1 in.
is the__combined height of 3 nickels, 2 nickels, and 14. What 1 nickel?
3 in. 8
15. Number Sense Which fraction is greatest? 5 A __ 6
7 B __
2 C __
9
3
D
9 ___ 12
16. Explain It Which equivalent fraction would you have to use 21 in order to add _35 to __ ? 25
15 ___ 21 ___ 25
25
5
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166 Topic 10
Reteaching
Name
10-4
Subtracting Fractions with Unlike Denominators You can subtract fractions with unlike denominators by using the least common multiple (LCM) and the least common denominator (LCD). Beth wants to exercise for _45 hour. So far, she has exercised for _2 hour. What fraction of an hour does she have left to go? 3
Step 1: Find the LCM of 5 and 3. 12 15
11 10
12 1 3
8
4 7
10 15
6
3
2
9
2 15
Step 2: Using your LCD, write the Step 3: Subtract the numerators. equivalent fractions. Place the difference over the LCD. Simplify if possible.
12 15
4 5
5
10 15
2 3
12 15
11 10
5
3
5
2 15
multiples of 3: 3, 6, 9, 12, 15
2
9
3
8
4 7
10 15
multiples of 5: 5, 10, 15, 20
12 1
12 10 2
6
2 ___ 15
5
hour left
Since 15 is the LCM, it is also your LCD.
In 1 through 5, find each difference. Simplify if possible.
1.
3 __
2.
4 2 __ 5
7 ___ 10 1 __ 5
7 ___
1 __
20 7 3. ___ 12
5
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1 __ 4
1 __ 3
2 5 __ 3 4. __ 6
8
11 ___ 24
23 __ 7 5. ___ 24
8
1 ___ 12
6. Natasha had _78 gallon of paint. Her brother Ivan took _14 gallon to paint his model boat. Natasha needs at least _1 gallon to 2 paint her bookshelf. Did Ivan leave her enough paint?
5 gallon left. Yes. She has __ 8
Topic 10
167
R e te a c h i n g
1 0 4
Practice
Name
10-4
Subtracting Fractions with Unlike Denominators Find the difference. Simplify if necessary.
7 __
10 __ 1 1. ___ 12
4
7 __ 1 4. ___ 12
4
4 __ 1 7. __ 8
4
9 __ 1 10. ___ 15
3
12 1 _ 3 1 4 _ 4 __ 15
9 __ 3 2. ___ 10
5
4 __ 1 5. __ 5
3
4 __ 1 8. ___ 10
5
4 __ 1 11. ___ 12
6
13 __
3 __ 10 7 __ 15 1 _5 _1 6
7 __ 2 3. __ 8
6
2 __ 1 6. __ 3
6
7 __ 2 9. __ 9
3
3 14 __ 12. ___ 20
5
24 1 _ 2 1 _9 1 __ 10
13. The pet shop owner told Jean to fill her new fish tank _34 full 9 with water. Jean filled it __ full. What fraction of the tank does 12 Jean still need to fill? 4 0 1 e c i t c a r P
0
14. Paul’s dad made a turkey pot pie for dinner on Wednesday. The family ate _48 of the pie. On Thursday after school, Paul 2 of the pie for a snack. What fraction of the pie remained? ate __ 16
3 __ 8
15. Algebra Gracie read 150 pages of a book she got for her birthday. The book is 227 pages long. Which equation shows how to find the amount she still needs to read to finish the story? A 150 n 227
C n 150 227
B 227 150 n
D n 150 227
16. Explain It Why do fractions need to have a common denominator before you add or subtract them?
The denominator tells you what size parts you are adding or subtracting. The parts must be the same size when you add or subtract. 168 Topic 10
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Reteaching
Name
10-5
Adding Mixed Numbers Randy talks on the telephone for 2_56 hours, and then surfs the Internet for 3_34 hours. How many hours did he spend on the two activities? Step 1. Write equivalent Step 2. Add the fractions. Then fractions with the least common add the whole numbers. denominator. You can use fraction 9 ___ 10 ___ 19 ___
strips to show the equivalent 3 fractions. 9 3 3 4
1
12
1
1
5 26 1
12
12
3
12
19 ___
2
Step 3. Simplify the sum if possible.
5
512
7 ___
612 hours
3 2__ 5 5___ 19 So, 3__ 4
6
12
10 2 12
1
In 1 through 6, find each sum. Simplify if possible.
1.
5 2__
2.
6 1 __ 3 4
1 12 6___ 4.
1 __ 7 10__ 3
9
1 11__ 9
3 1__
1 8 8__
5.
3.
8 3 __ 6 4
1 3__ 4
2 6__ 3
11 9___ 12
2 5__
5 1 4__ 2
5
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1 0 5
9 9___ 10 6.
5 1__ 7
7. Geometry Tirzah wants to put a fence around her garden. She has 22 yards of fence material. Does she have enough to go all the way around the garden?
No. She needs 5 yards to 22__ go6around the garden.
R e te a c h i n g
Tirzah’s garden
1 3__ 2
3 5___ 14
2 4 3 yards
3 6 4 yards
Topic 10
169
Practice
Name
10-5
Adding Mixed Numbers Estimate the sum first. Then add. Simplify if necessary.
16_12 19 __ 11 9 3 1 14 20 10 20 7 __ 8 1 9 18 5__ 3__ 9 2
5 2 8__ 1. 7__ 3
3. 5.
6
___
___
3 7__ 20 1 _ 6 2 13 7__ 5__ 7 7 7 7 __ 2 39 21 11 17__ 12 12 3
3 2__ 2 2. 4__ 5
4
4. 6.
___
7. Number Sense Write two mixed numbers with a sum of 3.
33 1 1__ Sample answer: 1__ 4
5 0 1 e c i t c a r P
8. What is the total measure of an average man’s brain and heart in kilograms?
7 1__ kg 10
4
Vital Organ Measures
Average woman’s brain Average man’s brain Average human heart
9. What is the total weight of an average woman’s brain and heart in pounds? 10. What is the sum of the measures of an average man’s brain and an average woman’s brain in kilograms?
3 kg 1 ___ 10 2 kg 1 __ 5 3 kg ___ 10
__ lb 24
5
3lb 7 ___ 10
lb
3_12 lb 7 2__ kg 10
11. Which is a good comparison of the estimated 11 sum and the actual sum of 7 _78 2__ ? 12 A Estimated actual
C Actual estimated
B Actual estimated
D Estimated actual
12. Explain It Can the sum of two mixed numbers be equal to 2? Explain why or why not. No; Sample answer: It is impossible for two mixed numbers to equal 2 because every mixed number is greater than 1. 170 Topic 10
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Reteaching
Name
10-6
Subtracting Mixed Numbers The Plainville Zoo has had elephants for 12_23 years. The zoo has had zebras for 5_12 years. How many years longer has the zoo had elephants? Step 1. Write equivalent fractions with the least common denominator. You can use fraction strips. 2 12 3 1
1
1
1
1
1
1
1 5 2 1
1
12
4 6
1
3 __ 4 __ 1 __ 1
1
1
1
1 6
1 6 1 6
1 6
1 1 3 3
3 5 6
1
Step 2. Subtract the fractions. Then subtract the whole numbers. Simplify the difference if possible.
1
1
1 6
6
6
12 5 7
6
2 5__ 1 7__ 1 years. So, 12__ 3
2
6
Tip: Sometimes you may have to rename a fraction so you can subtract.
1 6 1 6
1 2
6
rename 3 2__
8 __
5
8
8
3 2__
R e te a c h i n g
8
5 3__ 8
1 0 6
For questions 1 through 4, find the difference. Simplify if possible. Remember: You may have to rename a fraction in order to subtract.
1.
3 4__
5 1 __ 2 3
4 2___ 15
2.
6 5__
7 1 1__ 2
5 4___ 14
3.
4.
3
5 6__
3 1__ 4
6 1 5__ 2
1 1__
1 1__ 3
4
5. Number Sense Rename the number 7 so that you would be 5 able to find the difference of 7 3__ . 12
12 6___ 12
1 6. Robyn ran 5 _34 miles last week. She ran 4 __ miles this week. 10 How many more miles did she run last week? 5
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13 1___ 20
Topic 10
171
Practice
Name
10-6
Subtracting Mixed Numbers Estimate the difference first. Then subtract. Simplify if necessary.
1.
3 10__
2.
4 1 __ 7 4
3_12 5
5
9 6__ 6 5. 9__ 1 3__ 1 7. 6__ 4
3
2 7__ 1 9. 8__ 7
3
3 7__
7 8 ___ 2 21
3.
3
_1 3
3
2
12
13 1__ 40
1 3__ 7 8. 5__ 5
8
17 __ 30
9 2__ 1 10. 2___ 10
5_58
1 2__
4 2__ 3 6. 4__
18 11 2__ 12 20 __ 21
8 3 12___ 12
2 2__
1 5__ 21
13 2__
7 17__
4.
3
3
Strategy Practice The table shows the length and width of several kinds of bird eggs. 6 0 1 e c i t c a r P
Egg Sizes
11. How much longer is the Canada goose egg than the raven egg?
12.
1 _ 12
in. longer How much wider is the turtledove egg than the robin egg?
3 __ 10
Bird
Length
Canada goose
__ in. 32 __ 3
Robin Turtledove
in. wider
5
in.
4 __ in. 11 5 9 in. 1 ___
Raven
10
Width 3 in. 2 ___ 10
5 in. 9 in. ___ 10 3 in. 1 ___ 10 __ 3
15 3 13. Which is the difference of 21 __ 18_ ? 16 4
7 A 2___ 16
9 B 2___ 16
3 C 3___
9 D 3___
16
14. Explain It Explain why it is necessary to rename subtract _34 from it.
16
4_14
if you
Sample answer: You cannot subtract _34 from _14 , so you must borrow 1 whole from the 4 and rename 4_14 as 3_54.
172 Topic 10
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Reteaching
Name
10-7
Problem Solving: Try, Check, and Revise Can you use similar square tiles to cover a floor without cutting the tiles? Try
2 ft
Look at the square tile and the floor.
8 ft
2 ft
4 ft
Ask yourself:
8 ft
1. Can I divide the floor into a whole number of rectangles 4 ft that are 2 feet wide?
1 2
AND
2. Can I divide the floor into a whole number of rectangles that are 2 feet long?
R e te a c h i n g
8 ft 1
2
3
4
1 0 7
If the answers are both yes, you can use the tiles to cover the floor without cutting them. Check.
Answer the following questions remembering that you cannot cut the tiles.
5
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1. Can you completely cover a 4 ft by 8 ft area with 3 ft by 3 ft tiles?
No
2. Can you completely cover a 3 ft by 6 ft area with 3 ft by 3 ft tiles?
Yes
3. Can you completely cover a 3 ft by 6 ft area with 2 ft by 2 ft tiles?
No Topic 10
173
Practice
Name
10-7
Problem Solving: Try, Check, and Revise For questions 1 and 2, suppose you have 2 2 ft, 3 4 4 ft, and 5 5 ft tiles.
3 ft,
1. Which tiles can be used to cover a 12 12 ft floor?
2 2, 3 3, and 4 4
used to 2. Which cover atiles 9 can 9 ft be floor?
33
3. What size rectangular floor can be completely covered by using only 3 3 ft tiles OR 5 5 ft tiles? Remember, you can’t cut tiles or combine the two tile sizes.
Sample: 15 ft 45 ft, 30 ft 60 ft
7 0 1 e c i t c a r P
4. Adult tickets cost $6 and children’s tickets cost $4. Mrs. LeCompte says that she paid $30 for tickets, for both adults and children. How many of each ticket did she buy?
3 adults and 3 children OR 1 adult and 6 children 5. Reasoning The sum of two odd numbers is 42. They are both prime numbers, and the difference of the two numbers is 16. What are the two numbers? A 20 and 22 B 17 and 25 C 9 and 33 D 13 and 29 Marcy to putkind tilesofon a bathroom floor that 6. Explain It 10 measures ft wants 12 ft. What square tiles should she buy to tile her floor? Explain.
Sample: She can buy 2 2 ft tiles to tile her floor. 174 Topic 10
5
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Reteaching
Name
11-1
Multiplying Fractions and Whole Numbers You can find the product of a fraction and a whole number. Tran needs __23 yard of fabric to sew a pair of shorts. How many yards of fabric will Tran need to sew 6 pairs of shorts? Step 1: Multiply the numerator by the
Step 2: Place the product over the
whole number.
denominator. Simplify if possible.
2 6 12
12 ___ 3
4 yards of fabric
Remember: In word problems, “of” means “multiply.” 3 of 15 __ 3 15 Example: __ 5
5
In 1 through 4, find each product. Simplify if possible. 1 60 1. __ 3
3 of 32 2. __ 4
7 3. __ 8 40 2 of 35 4. __ 7
20 24
R e te a c h i n
g
1 1 1
35 10
For Exercises 5 through 7, use the table to the right.
5.
What is _17 the speed of a cheetah?
6.
What is _15 the speed of a cat?
7. What is _15 the speed of a jackal?
10 mi/h 6 mi/h 7 mi/h
Animal Cat Cheetah Jackal
Speed (in mi/h) 30 70 35
5
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Topic 11
175
Practice
Name
11-1
Multiplying Fractions and Whole Numbers Find each product. 1 of 96 1. __ 4
3 4. 45 __ 9
1 of 118 7. __ 2
1 10. 84 __ 6
5 of 48 13. __ 8
3 16. 42 ___ 14
5 84 19. __ 6
1 1 1
24 15 59 14 30 9 70
4 of 28 2. __ 7
7 5. 56 __ 8
3 of 56 8. __ 8
5 11. 64 ___ 16
1 of 77 14. __ 7
5 17. 72 __ 8
11 144 20. ___ 12
16 49 21 20 11 45 132
3 72 3. __ 4
3 6. 42 __ 7
9.
1 ___
400
10
11 12. 40 ___ 20
4 90 15. __ 5
2= 18. 18 __ 3
6 21. __ 7
42
54 18 40 22 72 12 36
22. Strategy Practice Complete the table by writing the product of each expression in the box below it. Use a pattern to find each product. Explain the pattern.
e c i t c a r P
1 _ 2
1 _ 4
32
32
1 _ 8
1 __
32
16
32
16 8 4 2 Pattern: divide by 2 23. Reasoning If _12 of 1 is _12 , what is _12 of 2, 3, and 4? 24. Which is
_2 3
A 75
1, and 2 1, 1__ 2
of 225?
B 113
C 150
D 450
25. Explain It Explain why _1 of 2 equals one whole.
__ 1
2
1 2 __ 21 of 2 __
176 Topic 11
2
2
2
5
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Reteaching
Name
11-2
Multiplying Two Fractions Musa and Karen are riding a bike path that is _45 mile long. Karen’s 3 bike got a flat tire __ of the way down the path and she had to 10 stop. How many miles did Karen ride? You can find the product of two fractions by drawing a diagram. Step 1. Draw a diagram using
Step 2. Draw lines vertically using dots to
shading to represent _45 .
3 represent __ . 10
4 5 3 10 Step 3. Count the parts of the diagram that are
Step 4. Count the total number of parts of the
shaded and dotted. This is the product numerator.
diagram. This is the product denominator.
12
50
R e te a c h i n
Step 5. Simplify if possible.
6 12 ___ ___ 50
g
25
1 1 2
Another way to find the product: Step 1. Multiply the numerators: 4
Step 2. Multiply the denominators: 5 12 Step 3. Simplify if possible: ___ 50
3 12.
10 50.
6 ___. 25
In 1 through 6, find the product. Simplify if possible. 1 __ 2 1. __ 3
5
16 4. __ 2
5
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2 ___ 15
3
5 __ 1 2. __ 8
4
5 ___
5 ___ 3 3. __ 6
32
3 5. 14 __ 7
67
10
1 __
4 9 ___ 7 35
3 __ 6 1 __ 6. __
3 __ 1. 7. Draw a picture Using a diagram, show __ 4
5
2
Topic 11
177
Practice
Name
11-2
Multiplying Two Fractions Write the multiplication problem that each model represents then solve. Put your answer in simplest form.
1.
2.
2 __ 1 ___ 2 __ 1 __ 3
6
18
5 __ 5 1 ___ __
9
7
Find each product. Simplify if possible. 7 3. __ 8
4 __ 5
1 __ 2 5. __ 6
7.
5
2 __ 1 __ 9
2
3 __ 4 9. __ 8
2 1 1 e c i t c a r P
9
5 14 2 __ 11. __ 3
6
7 ___
10 ___ 1 1 15 __ 9 1 __ 7 6
7__ 9
13. Algebra If _45 _25, what is ?
4
3 __ 2 4. __ 7 3 2 __ 1 6. __ 7
8.
4
3 __ 1 __ 4
3
5 1 __ 10. __ 5
6
1 __ 1 __ 1 12. __
1 __
2
3
4
28 2 __ 7 ___ 1 1 14 __ 4 1 __ 1 6 ___
24
2
14. Ms. Shoemaker has 35 students in her classroom. Of those students, _67 are right-handed, and _45 of the right-handed students have brown eyes. How many right-handed brown-eyed students are in Ms. Shoemaker’s class?
24
15. Which does the model represent? A _38 _35
C _35 _58
B _78 _25
D _48 _35
16. Explain It Describe a model that represents _33 _44 .
Each fraction represents one whole, so the model would be completely shaded in for both fractions. 178 Topic 11
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Reteaching
Name
11-3
Multiplying Mixed Numbers You can find the product of two mixed numbers. Millwood City is constructing a new highway through town. The construction crew can complete 5 _35 miles of road each month. How many miles will they complete in 6 _12 months? Step 1. Round the mixed numbers to whole
Step 3. Multiply the numerators and the
numbers so you can make an estimate.
denominators. Simplify the product if possible. Remember to look for common factors.
3 6 __ 1 5 __ 5
2
14
6 7 42
28 ___ 13 ____ 182 ___ 5
So, they can complete about 42 miles.
2
5
2 36__ 5
1
Step 2. Write the mixed numbers
Step 4. Compare your product to your estimate to
as improper fractions.
check for reasonableness.
3 6 __ 28 ___ 13 1 ___ 5 __ 5
2
5
36 _25 is close to 42, so this answer is reasonable.
2
The construction crew will complete 36 _25 miles of highway in 6 _12 months.
1.
4. 1_25 2_14
4_38 3 3__ 20
2.
1_15 1_23
5. 2_12 10
2 25
3.
2 2_14
4_12
6. 1_23 _15
1 __ 3
7. Using the example above, the new highway will be a total of 54 miles long. Will the highway be finished in 8 months?
8. 5
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g
1 1 3
For 1 through 6, estimate a product. Then solve for each actual product. Simplify if possible. 1_34 2_12
R e te a c h i n
No. They will have completed only 44_45 miles. Reasonableness Sayed gave an answer of 6_ for the 6
7 problem 4_27 1_35. Using estimates, is this a reasonable answer?
Yes: 4 2 8, and 8 is close to 6_67. Topic 11
179
Practice
Name
11-3
Multiplying Mixed Numbers Estimate the product. Then complete the multiplication.
29
7 1. 5_45 7 ______ __ 5
1
3 40__ 5
11 36
2. 3_23 5_17 ______ ______ 7
3
Estimate. Then find each product. Simplify.
1 3___
3. 4_35 _23 5. 7_45 2_13 7. 2_15 _78 9.
1_45 1_13 1_34
1 18__ 5 1 4__
15 37 1___ 40
5
8. 6_13 1_56 10.
_3 2_2 5_1 5 4 3
7
1 10__ 2 2 10__
e c i t c
11 11___ 18
5
11. Algebra Write a mixed number for p so that 3_14 p is more than 3_14.
3 1 1
7
5 13__
4. 6 2_27 6. 3_34 2_45
6 18__
Answers will vary. Any mixed number greater than 1 will work. _
12. A model house is built on a base that measures 914 in. wide and 8_45 in. long. What is the total area of the model house’s base?
2 81__ 5 in2
a r P
13. Which is 1 _34 of 150_12 ? A 263
B 263_18
C 263_38
D 264_38
14. Explain It Megan’s dog Sparky eats 4_14 cups of food each day. Explain how Megan can determine how much food to give Sparky if she needs to feed him only _23 as much. Solve the problem.
Megan would multiply 4_14 c by _23 to find 2_65 c.
180 Topic 11
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Reteaching
Name
11-4
Relating Division to Multiplication of Fractions How can you divide by a fraction? Dividing a whole number by a fraction
Think: How can I divide two into one-thirds?
1 2 __ 3
1. Two is the sum of one plus one. 2. Each one is the sum of three one-thirds. 3. Count the number of one-thirds.
2
1 1 __ 1 __ 1 __ 3
3
1 1 __ 1 __ 1 __
3
3
3
3
6
Check. To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.
3 __ 3 __ 66 1 2 __ 2 __ 2 __
3 3 __
Think: How can I divide three into three-fourths? 3 1 1 1
4
1. Three is the sum of one plus one plus one.
3
3 __ 1 __ 4 4
4
4
4
1
3 __ 1 __ 4 4
4
4
4
4
4
3 __
4
4
4 3 3 __ 3 __ 4 __ 4 ___ 12 4 3 __ 3
Draw a picture that shows each division and write the answer.
4
3 __ 1 __ 4 4
3 __ 3 __ 3 __
4
2
1
3 __ 3 __ 3 __ 1 __ 1 __ 1 __
3. Count the number of three-fourths.
1 1. 2 __
1
R e te a c h i n
g
2. Each one is the sum of one three-fourths and one one-fourth.
Check. Multiply the whole number by the reciprocal of the fraction.
1
2 2. 2 __ 3
1
3
3
3
5
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Topic 11
181
1 1 4
Practice
Name
11-4
Relating Division to Multiplication of Fractions In 1 and 2, use the picture to find each quotient.
1. How many thirds are in 1?
2. How many thirds are in 7?
3
21
In 3 and 4, draw a picture to find each quotient. 1 3. 3 __ 2
4 1 1 e c ti c
6
1 4. 4 __ 8
32
In 5 and 6, use multiplication to find each quotient.
r a P
1 5. 6 __ 3
6 3 18 7. Julie bought 3 yards of cloth to make holiday napkin rings. If she needs _34 of a yard to make each ring, how many rings can she make?
4
1 6. 5 ___ 10
5 10 50 8. Reasoning When you divide a whole number by a fraction with a numerator of 1, explain how you can find the quotient.
Sample answer: You can multiply the whole number by the denominator of the fraction.
182 Topic 11
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Reteaching
Name
Problem Solving: Draw a Picture and Write an Equation
11-5
Travis earned 3 stickers for each song he played in his piano lesson. He received a total of 24 stickers. How many songs did he play? You can solve a problem like this by drawing a picture and writing an equation.
Step 1. Write out what you already know. Travis earned 3 stickers for each song he played. Travis had 24 stickers at the end of the lesson. Step 2. Draw a picture to show what you know.
Travis’s total stickers
Step 3. Write out what you are trying to find. How many songs did Travis play? Step 4. Write an equation from your drawing. Since you are dividing Travis’s total stickers into groups of 3 (stickers earned per song), this is a division problem. 24 3 s
s number of songs Travis played
R e te a c h i n g
1 1 5
groups of 3 stickers Travis earned per song
Step 5. Solve the equation. 24 3 8 s8 So, Travis played 8 songs during his lesson. Step 6. Check your answer by working backward. 8 3 24: your answer is correct.
Draw a picture, write an equation, and solve. _3 5
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Rudy, Mario each for have 1 4 cups 1. Sasha, flour. Can theyand make a recipe bread that of needs 5 cups of flour?
3 1_34 c; c 5_14 cups of flour; yes
Sasha’s flour
Rudy’s flour
Mario’s flour
Topic 11
183
Practice
Name
Problem Solving: Draw a Picture and Write an Equation
11-5
Solve each problem. Draw a picture to show the main idea for each problem. Then write an equation and solve it. Write the answer in a complete sentence.
Check student’s drawings.
1. Bobby has 3 times as many model spaceships as his friend Sylvester does. Bobby has 21 spaceships. How many model spaceships does Sylvester have?
21 3 x, x 7; Sylvester has 7 spaceships. 2. Dan saved $463 over the 12 weeks of summer break. He saved $297 of it during the last 4 weeks. How much did he save during the first 8 weeks? 5 1 1 e c ti c r a P
$463 $297 x, x $166; Dan saved $166 during the first 8 weeks. 3. Strategy Practice Use a separate sheet of paper to show the main idea for the following problem. Choose the answer that solves the problem correctly. A box of peanut-butter crackers was divided evenly among 6 children. Each child got 9 crackers. How many crackers were in the box?
A 54
B 48
C 39
D 36
4. Explain It Why is it helpful to draw a picture when attempting to solve an equation?
184
Sample answer: Drawing can give you a picture of the problem that will help you create the right equation to solve Topic 11 the problem.
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Reteaching
Name
12-1
Using Customary Units of Length One customary unit of length is the inch. On most rulers it is divided into fractions to help you measure more precisely. How to measure length Measure the length of the pen to the nearest inch, nearest _1 inch, _1 inch, and _1 inch. 2
4
8
012345 INCHES Step 1: Measure to the nearest inch. The length is more than 4 in., but less than 5 in.
Step 2: Measure to the nearest _12 inch.
Step 3: Measure to the nearest _14 inch.
Step 4: Measure to the nearest _18 inch.
The length is more than 4_12 in., but less than 4_22 (5) in.
The length is more than 4_24 (4_12 ) in., but less than 4_34 in.
The length is more than 4_48 (4_12 ) in., but less than 4_58 in.
It is closer to 4 _12 in.
It is closer to 4 _12 in.
It is closer to 4_58 in.
The pen’s length is closest to 4_12 in.
The pen’s length is closest to 4_12 (4_24 ) in.
The pen’s length is closest to 4_58 in.
It is closer to 5 in. The pen’s length is closest to 5 in.
Measure to the nearest inch, _12 inch, _14 inch, and _18 inch.
1.
1 in., 1 in., 1_14 in., and 1_14 (1_28 ) in. 5
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Use your ruler to draw a line segment of each length. 7 inch 2. __ 8
1 inch 3. 2__ 4
Topic 12
185
R e te a
c h i n g
1 2 1
Practice
Name
12-1
Using Customary Units of Length Measure each segment to the nearest inch, _12 inch, _14 inch, and _1 inch. 8
1.
1 in.; 2__ 1 in.; 2__ 1 in. 2 in.; 2__ 4
2
2.
4
1 in.; 1__ 1 in.; 1__ 1 in.; 2 in.; 1__ 2
2
2
3. Reasoning Sarah gave the same answer when asked to round 4_78 in. to the nearest _12 inch and the nearest inch. Explain why Sarah is correct.
Sample answer: 5 in. is the nearest 1 in. to 4__ 7 in. inch and __ 2
1 2 1 e c i t c a r P
8
4. Estimate the length of your thumb. Then use a ruler to find the actual measure.
Answers will vary.
5. Estimation A real motorcycle is 18 times as large as a model 1 motorcycle. If the model motorcycle is 5 __ in. long, about 16 how long is the real motorcycle? A 23 in.
B 48 in.
C 90 in.
D 112 in.
6. Explain It If a line is measured as 1 _48 in. long, explain how you could simplify the measurement.
4 Sample answer: I could simplify 1__ 8
1 to 1__ 2 by dividing the numerator and denominator by 4. 186 Topic 12
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Reteaching
Name
12-2
Using Metric Units of Length Measurements in the metric system are based on the meter.
ExamplA e
ExamplB e
Which metric unit of length would be most appropriate to measure the length of a bumblebee? A bumblebee is very small, so the millimeter is the most appropriate unit of length.
Write mm, cm, m, or km to complete the following sentence. A chair is about 1 tall and a child’s hand is about 8 wide. A chair is about 1 m tall, and a child’s hand is about 8 cm wide.
Example C What is the length of the twig to the nearest centimeter and to the nearest millimeter?
12345678 R e te a
CENTIMETERS
c h i n g
1 2 2
Step 1: Measure to the closest centimeter.
Step 2: Measure to the closest millimeter.
The length of the twig is more than 7 cm but less than 8 cm. The twig’s length is closest to 8 cm.
The length of the twig is more than 70 mm but less than 80 mm. The length of the twig is more than 75 mm but less than 76 mm. The twig’s length is closest to 76 mm.
Which unit would be most appropriate for each measurement? Write mm, cm, m, or km. 5
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1. distance between two cities 2. width of a room 3. Measure the line segment below to the nearest centimeter and to the nearest millimeter.
km
3cm
;
m
31mm Topic 12
187
Practice
Name
12-2
Using Metric Units of Length Measure each segment to the nearest centimeter then to the nearest millimeter.
5 cm, 54 mm 4 cm, 39mm
1. 2. Number Sense Some of the events at an upcoming track and field meet are shown at the right. 3. In which event or events do athletes travel more than a kilometer?
Track and Field Events 50-m dash 1,500-m dash 400-m dash 100-m dash
1,500-m dash 4. In which event or events do athletes travel less than a kilometer?
50-m dash, 100-m dash, 400-m dash 2 2 1 e c i t c a r P
5. Reasonableness Which unit would be most appropriate for measuring the distance from Chicago to Miami? A mm
B cm
C m
D km
6. Explain It List one item in your classroom you would measure using centimeters and one item in the classroom you would measure using meters.
Check students’ work.
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188 Topic 12
Reteaching
Name
12-3
Perimeter The perimeter is the distance around the outside of a polygon. You can find the perimeter in two different ways.
Add the lengths of the sides:
Use a formula:
Find the perimeter of the figure.
Find the perimeter of the rectangle. 11 cm
7 cm
cm
3
cm
3 cm
5
4 cm
3 cm
3 cm
To find the perimeter, add up the sides. 3 3 7 5 3 4 25
Perimeter = (2 length) + (2 width) P = (2 l) + (2 w) P = (2 11) + (2 3) P = 22 + 6 = 28 cm
So, the perimeter of the rectangle is 28 cm.
So, the perimeter of the figure is 25 cm. Find the perimeter of each figure. 8.9 cm
.
R e te a
11 m
2. 4.2 cm
c h i n g
7m 3m
4m
1 2 3
8m
26.2cm 3.
3 mm
33m 4.
4 mm
5 mm
4.7 cm
5 mm
6.5 mm 5
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23.5mm
18.8cm
5. Number Sense The perimeter of a square is 24 in. What is the length of each side?
6 in. Topic 12
189
Practice
Name
12-3
Perimeter Find the perimeter of each figure.
1. 3 cm
3 cm 5 cm
2.
3.
7 km 7 km
7 km
3m
2m 1m
28km
11m
5. Number Sense What is the perimeter of a square if one of the sides is 3 mi? Use the dimensions of the football field shown at the right for 6 and 7.
6. What is the perimeter of the entire football field including the end zones?
25mm 12 mi
e n o z
160 ft
ft30
300 ft
e n o
Playing field
d n E
1,040 ft e c i t c a r P
5 mm
2m
ft30
3 2 1
7.5 mm 5 mm 7.5 mm
7 km
11cm
4.
1m 2m
300 ft
ft30
160 ft z
d n E
ft30
7. What is the perimeter of each end zone?
380 ft 8. What is the perimeter of this figure? A 18 m
C 12 m
B 15 ft
D 10 ft
6m 6m 3m
9. Explain It A rectangle has a perimeter of 12 m. If each side is a whole number of meters, what are the possible dimensions for the length
3m
and width? List them and explain your answer.
1 and 5, 2 and 4, 3 and 3, 4 and 2, 5 and 1. If the perimeter is 12 m, the length and width must add up to 6 m.
190 Topic 12
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Reteaching
Name
12-4
Area of Squares and Rectangles You can use formulas to find the areas of rectangles and squares. Find the area of the rectangle.
Find the area of the square.
5m
42 cm 4m
42 cm
Use this formula for rectangles: Area length width
42 cm 42 cm
Use this formula for squares:
Alw
Area side side
A5m4m A 20 square meters 20 m2
A 42 cm 42 cm A 1,764 square centimeters 1,764 cm2 R e te a
Find the area of each figure.
.
2. 58 ft
3,364 ft2 3. a square with a side of 12.4 m 4. a rectangle with a length of 9.7 cm and a width of 7.3 cm 5. Number Sense If the area of a square is 81 in2, what is the length of one side? 5
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6. What is the area of the tennis court?
c h i n g
5.5 in.
1 2 4
3.2 in.
17.6 in2 153.76 m2 70.81 cm2 9 in.
195.16 m2
8.2 m
23.8 m
Topic 12
191
Practice
Name
12-4
Areas of Squares and Rectangles Find the area of each figure.
1.
4 cm w 3 cm
4. 5.
s 9.5 mi
s 9.5 mi
s 9.5 mi
90.25 mi2 22.1 km2 a rectangle with sides 6.5 km and 3.4 km 104.04 ft2 a square with a side of 10.2 ft 82.8 m2 a rectangle with sides 9 m and 9.2 m 12
3.
s 9.5 mi
2.
cm2
6. Number Sense Which units would you use to measure the area of a rectangle with l 1 m and w 34 cm? Explain. 4 2 1 e c i t c a r P
Sample answer: Centimeters; I would convert the 1 m to centimeters. 7. Which of the following shapes has an area of 34 ft2? A a square with s 8.5 m B a rectangle with l 15 ft, w 2 ft C a square with s 16 ft D a rectangle with l 17 ft, w 2 ft 8. Explain It The area of a square is 49 m2. What is the length of one of its sides? Explain how you solved this problem.
7 m; Sample answer: The length of each side of a square is equal. The factors of 49 are 1, 7, 7, and 49, so each side must be 7 m. 192 Topic 12
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Reteaching
Name
12-5
Area of Parallelograms The formula used to find the area of a parallelogram is similar to the one you used to find the area of a rectangle. Instead of using length width, use base height.
Height (like length) Base (like width)
How to find the area of a parallelogram:
How to find the missing measurement of a parallelogram:
Find the area of the parallelogram.
Area 55 cm2 base 11 cm height ? cm Remember, area base height.
8 cm
55 cm2 11 cm ? cm
10 cm
55 cm2 11 cm 5 cm
Area base height
So, the height of the parallelogram is 5 cm.
Abh A 10 cm 8 cm A 80 cm2
R e te a
c h i n g
Find the area of each parallelogram.
70
1.
1 2 5
m2
5m 14 m
11.96 ft2
2. 2.3 ft 5.2 ft
Find the missing measurement for each parallelogram. 5
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3. A 72 in2, b 9 in., h 4. A 238 ft2, b
17 ft
8 in.
, h 14 ft
Topic 12
193
Practice
Name
12-5
Area of Parallelograms Find the area of each parallelogram.
1.
2.
2 mi
3 cm
9 mi 5 cm
15 cm2
18 mi2
3.
4. 1 mm
1.5 m 6m
2 mm
9 m2
2 mm2
Algebra Find the missing measurement for the parallelogram. 5. A 34 in2, b 17 in., h
2 in.
6. List three sets of base and height measurements for parallelograms with areas of 40 square units. 5 2 1 e c i t c a r P
Sample answer: b 2, h 20; b 4, h 10; b 5, h 8 7. Which is the height of the parallelogram?
A 44 m2
A 55 m B 55.5 m
h?
C 5m D 5.5 m
b8m
8. Explain It What are a possible base and height for a parallelogram with an area of 45 ft 2 if the base and height are a whole number of feet? Explain how you solved this problem.
Sample answer: 9 ft and 5 ft; 9 and 5 are factors of 45, so a base of 9 ft and height of 5 ft would result in an area of 45 ft2. 194 Topic 12
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Reteaching
Name
12-6
Area of Triangles 1 base height Area of a triangle __ 2
How to find the area of a triangle:
How to find the missing measurement of a triangle: Area 100 cm2 base 40 cm height ? cm
4 ft 3 ft
1 base height. Remember, Area __
1bh A __
1 40 cm ? cm 100 cm2 __
2
2
A
1 __ 2
2
3 ft 4 ft
100 cm2 20 cm ? cm
1 12 ft2 A __
100 cm2 20 cm 5 cm
2
So, the height of the triangle is 5 cm.
A 6 ft2 Find the area of each triangle.
1.
2.
3.
4 in.
2.2 m
R e te a
5 cm
c h i n g
5.4 m
7 in.
1 2 6
9 cm
14 in2
5.94 m2
22.5 cm2
Find the missing measurement for each triangle.
4. A 16 in2, b 8 in., h
4 in.
10 m 24.32 ft2 , b
5. A 20 m2, b 6. A
7. A 14 cm2, b 2 cm, h 5
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,h4m
6.4 ft, h 7.6 ft
14 cm
Topic 12
195
Practice
Name
12-6
Area of Triangles Find the area of each triangle.
1.
2.
3.
10 ft
7 mm
3.6 yd
13 mm
6 yd
8 ft
40 ft2
10.8 yd2
4. Number Sense What is the base measurement of a triangle with an area of 30 m2 and a height of 10 m?
45.5 mm2 6m
Algebra Find the missing measurement for each triangle.
6 mi 45 mm2 , b
5. A 36 mi2, b 6. A
, h 12 mi
12 mm, h 7.5 mm
6 2 1 e c i t c a r P
7. List three sets of base and height measurements for triangles with areas of 30 square units.
Sample answer: b 4, h 15; b 6, h 10; b 8, h 7.5 8. Which is the height of the triangle? A 4.5 ft
C 8 ft
B 6 ft
D 9 ft
A 27 ft2
12 ft
9. Explain It Can you find the base and height measurements for a triangle if you know that the area is 22 square units? Explain why or why not.
No; Sample answer: You need to know at least 2 of the measurements of the base, the height, and the area.
196 Topic 12
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Reteaching
Name
12-7
Circles and Circumference All the points on a circle are the same distance from the center. The center of circle C is point C. A chord has both endpoints on a circle. ___ HD is a chord of circle O. A central angle is an angle whose vertex is the center. ACB is a central angle.
A radius goes from the center to any point on the circle. ___ RS is a radius of circle R. A diameter is a chord that passes through
C
O H
D
D
C
A B
S R
D
M T
___ the
center of a circle. DT is a diameter of circle M. The circumference is the distance around a circle. The circumference equals the circle’s diameter multiplied by (3.1416). C D R e te a
Use the terms at the top of the page to identify each figure in circle F. 1. point F
2.
GFS
___
3. RS ___
4. UV ___
5. GF
Center Central angle Diameter Chord Radius
c h i n g
R U
1 2 7
G F
V
S
6. Estimation If a circle has a circumference of three feet, estimate its diameter to the nearest foot. 5
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1 foot
Topic 12
197
Practice
Name
12-7
Circles and Circumference
P
In 1 through 3, use circle X to identify the following.
___
1. a diameter
QP
___ RS and ___ QP
2. two chords
O
3. a central angle
QXP, QXO,
In 4 through 9, find the circumference. Use 3.14 for .
4. d
20 in.
62.8 in.
7. r 3.20 in.
20.1 in.
5. d
X
R
PXO
S
Q
5 yd
15.7 yd
8. r 13 ft
81.64 ft
6. d
9 cm
28.26 cm
9. r 20 yd
125.6 yd
10. A round swimming pool has a radius of 8 meters. What is its circumference?
50.24 meters 7 2 1 e c i t c a r P
11. Estimation The length of the diameter of a circle is 11 centimeters. Is the circumference more or less than 33 centimeters? Explain.
3 11 33, and is greater than 3, so 11 33 12. Which equation can be used to find the circumference of a circle with a radius that measures 10 feet? A C 2 20
C C 1.0
B C 2 10
D C 10 5
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198 Topic 12
Reteaching
Name
12 -8
Problem Solving: Draw a Picture and Make an Organized List In a garden, a landscaper has to make a small patio using twelve 1-foot-square tiles. The patio must have the smallest possible perimeter. What should be the dimensions of the patio? Draw a picture.
A
1. Construct pictures of rectangles using the twelve tiles.
B
2. List the perimeter of each figure in a table.
C
3. Count tile edges or calculate to find the perimeter.
D
E F
Solve. Both the 3 4-tile and the 4 3-tile patios have the same perimeter, 14 feet. This value is the smallest perimeter in the table. The landscaper can use either design.
Rectangle
Length in feet
A B C D E F
12 6 4 3 2 1
Width in feet
Perimeter in feet
1 2 3 4 6 12
26 16 14 14 16 26
c h i n g
1 2 8
The landscaper must build another patio with the smallest perimeter possible using nine 1-foot-square tiles. What should be the dimensions?
3 feet 3 feet
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Topic 12
R e te a
199
Name
Problem Solving: Draw a Picture and Make an Organized List
Practice 12 -8
Draw a picture and make a list to solve.
1. Erica painted a picture of her dog. The picture has an area of 3,600 cm 2 and is square. She has placed the picture in a frame that is 5 cm wide. What is the perimeter of the picture frame?
280 cm 2. The new playground at Middledale School will be enclosed by a fence. The playground will be rectangular and will have an area of 225 yd 2. The number of yards on each side will be a whole number. What is the least amount of fencing that could be required to enclose the playground?
60 yd
8 2 1 e c i t c a r P
3. Reasoning Evan is thinking of a 3-digit odd number that uses the digit 7 twice. The digit in the tens place is less than one. What is the number? A 707
B 717 C 770 D 777 4. Explain It Explain how you solved Exercise 3.
Check students’ work. 5
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200 Topic 12
Reteaching
Name
13-1
Solids The solid’s vertices are: A, B, C, D, E, F, G, and H. ___ ___ ___ ___ ___ ___ The are: AC,___ AB, CD, DB, AH, BG, ___ solid’s ___ ___ ___ ___edges HG, HE, GF, EF, CE, and DF. The solid’s faces are: ACEH, BDFG, ABCD, EFGH, CDEF, and ABHG.
Face
A
H
B C D
Vertex
G
E F
Edge
Here are some common solid figures: C ub e
Cylinder
Rectangular Prism
Cone
J
Use the solid at the right for 1 through 3.
___ ____ ____ ____ OJ , KQ, OQ, ON, NM, ____, JK ___ ___ ___ ___ ___
1. Name edges. __ _ the ___
K
O
Q P
MQ, KL, LM, JP, NP, LP
N
L M
2. Name the faces.
NMQO, KLMQ, JKLP, JONP, JKQO, LMNP
3. Name the vertices.
J, K, L , M , N , O , P , Q
What solid figure does each object resemble?
4.
5
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Ar t Supplies
5.
6.
Soup
Rectangular Cylinder prism
C on e Topic 13
201
R e te a c h i n g
1 3 1
Practice
Name
13-1
Solids For 1 through 3, use the solid at the right.
A
1. Name the vertices.
A, B, C, D
D B
2. Name the faces.
C
ABC, ABD, ACD, BCD
___ ___ ___ ___ ___ ___ AB, AC, AD, BC, BD, CD
3. Name the edges.
For 4 through 6, tell which solid figure each object resembles.
4.
5.
6.
1 3 1 e c i t c a r P
Cube
Cylinder
Rectangular prism
7. Which term best describes the figure at the right? A B C D
Cone Triangular prism Pyramid Rectangular prism
8. Explain It How many vertices does a cone have? Explain.
One; the circular base does not have any vertices but every point on the circle meets at one point, the vertex, outside the base.
202 Topic 13
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Reteaching
Name
Relating Shapes and Solids
13-2
If each net on the left were folded, it would form the figure on the right. Top
Side
Side
Side
Side
Bottom
Side
Side
Bottom
Side R e te a c h i n g
Side
1 3 2
What does each net represent?
1.
2.
5
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Cube
Cylinder Topic 13
203
Practice
Name
Relating Shapes and Solids
13-2
For 1 and 2, predict what shape each net will make.
1.
2.
Rectangular prism
Square pyramid
Reasoning For 3 through 5, tell which solid figures could be made from the descriptions given. 3. A net that has 6 squares 4. A net that has 4 triangles 2 3 1 e c i t c a r P
5. A net that has 2 circles and a rectangle
Cube Triangular pyramid Cylinder
6. Which solid can be made by a net that has exactly one circle in it? A Cone
B Cylinder
C Sphere
D Pyramid
7. Explain It Draw a net for a triangular pyramid. Explain how you know your diagram is correct.
Sample answer: The 4 equilateral triangles fold into a triangular pyramid.
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Name
13-3
Surface Area The surface area of a rectangular prism is the sum of the areas of all of its faces.
8 in.
Danica has a wood block. She wants to paint it. In order to know how much paint to buy, Danica needs to find the surface area of the block.
5 in.
15 in.
5 in.
If you could “unfold” this block, it would look like this: You can find the area of each face by multiplying the length times the width. area of A area of B area of C area of D area of E area of F
E
15 5 75 15 8 120 15 5 75 15 8 120 8 5 40 8 5 40
D
F
C
5 in.
B
8 in.
A
5 in.
8 in.
15 in.
Adding all the areas together will give the total surface area. 75 120 75 120 40 40 470 in2 Another way to find the surface area of a rectangular prism is to use the following formula. surface area surface area surface area surface area
R e te a c h i n g
Remember: l = length w = width h = height
2(l w) 2(l h) 2(w h) 2(15 5) + 2(15 8) + 2(5 8) 150 240 80 470 in2
1 3 3
In 1 through 3, find the surface area of each figure.
1.
2.
1.5 mm
3 cm 3 cm
3.
4 mm 10.5 mm
12 ft
3 cm 5
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5 ft
54 cm2
127.5 mm2
7 ft
358 ft2 Topic 13
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Practice
Name
13-3
Surface Area Find the surface area of each rectangular prism.
1.
2. 3m 12 ft
5m 8m
12 ft
12 ft
158 m2
864 ft2
Strategy Practice Music and computer CDs are often stored in plastic cases called jewel cases. 3. One size of jewel case is 140 mm 120 mm 4 mm. What is the surface area of this jewel case?
35,680 mm2 4. A jewel case that holds 2 CDs is 140 mm 120 mm 9 mm. What is the surface area of this jewel case?
2 3 3 1 e c i t c a r P
38,280 mm 5. What is the surface area of a rectangular prism with the dimensions 3 in. by 4 in. by 8 in.? A 96 in 2
B 112 in2
C 136 in 2
D 152 in2
6. Explain It Explain why the formula for finding the surface area of a rectangular prism is helpful.
Possible answer: the formula gives the total area of a solid rectangular shape that needs to be covered or painted.
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13-4
Views of Solids Here is how each figure on the left would look like from the front, side, and top. In each figure, the number of cubes hidden by the view is given beneath it.
Front
Side
Top
There are no hidden cubes.
Front
Side
Top
There are no hidden cubes.
Front
Side
Top
There is one hidden cube.
R e te a c h i n g
Look at the figure. Label its front, side, and top views.
1 3 4
1.
2.
3.
Side
Front
Top
How many cubes are hidden in each figure? 5
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4.
5.
6.
N on e
O ne
N on e Topic 13
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Practice
Name
13-4
Views of Solids For 1 and 2, draw front, side, and top views of each stack of unit blocks. Top
1. Front
Front
Side
Side
Top
2. 3. Reasoning In the figure for Exercise 2, how many blocks are not visible?
1 4. In the figure at the right, how many unit blocks are being used?
4 3 1 e c i t c a r P
A B C D
8 9 10 11
5. Explain It A figure is made from 8 unit blocks. It is 3 units tall. What is the maximum length the figure could be? Explain.
The maximum length is 6, because 2 are used to give the figure height.
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208 Topic 13
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Name
13-5
Volume Volume is a measure of the space inside a solid figure. It is measured in cubic units. A cubic unit is the volume of a cube which has edges that are 1 unit. How to find the volume of a rectangular prism
Counting unit cubes
Using a formula You know the length, l, the width, w, and the height, h. Calculate the volume, V, using the formula V l w h.
3 cm 4 cm 2 cm
3 cm
Count the cubes in each layer: 8 cubes. h
Multiply by the number of layers. 8 cubes
3 24 cubes
The volume of each cube is 1 cm3. The volume of the prism is 24 cm 3.
l
4 cm
w 2 cm
V 2 cm 4 cm 3 cm V 24 cm3
R e te a c h i n g
1. Find the volume of the rectangular prism using a formula.
1 3 5
1.2 m
2.1m 4m
10.08 m3
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Practice
Name
13-5
Volume Find the volume of each rectangular prism.
336 in3 384 cm3 336 m3
1. base area 56 in 2, height 6 in. 2. base area 32 cm2, height 12 cm 3. base area 42 m2, height 8 m 4. 5 yd
5.
8 cm
10 cm
2 cm
5 yd
5 yd
125 yd3
160 cm3
6. Algebra What is the height of a solid with a volume of 120 m3 and base area of 30 m 2?
4m
Michael bought some cereal at the grocery store.
1
3 2 in.
7. What is the base area of the box? 5 3 1 e c i t c a r P
28 in2
13 in.
8. What is the volume of the box?
364 in3
8 in.
9. What is the base area of this figure? A 3.2 m 2
C 320 m 2
B 32 m2
D 3,200 m 2
V
320 m3
10 m
10. Explain It Explain how you would find the base area of a rectangular prism if you know the volume and the height.
Sample answer: Divide the volume by the height to find the base area. 210 Topic 13
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13-6
Irregular Shapes and Solids To find the area of an irregular shape, separate it into familiar shapes. Find the area of each shape and add the areas.
1. To find the area of the figure at the right, separate the shape into two rectangles. (See the dotted line in the figure.)
10 cm 5 cm
Rectangle A 8 cm 4 cm 3 cm Rectangle B 6 cm
Area of Rectangle A Area of Rectangle B 2. Use the formula A w A 10 5 50 A 6 3 18 to find the area of each rectangle; then add the areas. The area of the shape is 50 18 68 cm2. To find the volume of an irregular solid, separate it into familiar shapes. Find the volume of each shape and add the volumes.
1. To find the volume of the figure at the right, separate the shape into two rectanglar prisms. (See the dotted line in the figure.)
1 ft Prism A
7 ft
5 ft
R e te a c h i n g
Prism B
2 ft 2 ft 3 ft
2. Use the formula V w h to find the volume of each prism; then add the volumes.
1 3 6
4 ft
Volume of Prism A A 1 4 7 28
Volume of Prism B A 2 4 2 16
The volume of the solid is 28 16 44 ft3.
1. Show two ways of dividing the given solid into two regular solids. 5
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Practice
Name
13-6
Irregular Shapes and Solids For 1 and 2, find the area of the irregular shape.
1.
m c 2
2.
4 cm
70 yd
4 c m
m c 5
d y 0 4
70 yd
2 cm
70 yd
2,800 yd2
5 cm
37 cm2 For 3 and 4, find the volume of the irregular solid.
3.
4. 6 ft
20 ft
2m
1m
3m 5m
25 ft
4m
54 m3
16 ft
6 3 1 e c i t c a r P
9,200 ft3 5. What is the area of this irregular shape? 120 mm 60 mm 90 mm 200 mm
A 1,200 mm2 B 12,000 mm2
6. Explain It When do you find an area or volume by separating the figure into smaller figures?
When the figure is irregular
C 13,200 mm 2 D 14,400 mm 2 5
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13-7
Problem Solving: Use Objects and Solve a Simpler Problem At a math fair, Willie saw a puzzle about a giant cube made of identical white smaller cubes. The giant cube was 4 4 4. It contained 64 smaller cubes. Each of the six faces of the giant cube was painted red. The puzzle asked, “If the giant cube were taken apart, how many smaller cubes would have only one face painted red?” Here is how Willie tried to solve the puzzle. corner cube
1. Construct cubes using 8 and 27 smaller cubes. Imagine painting each of the larger cubes.
corner cube
2. Classify the smaller cubes. Think: Where are the cubes located in the larger cube? How are they painted differently from each other?
face cube
3. Make a table.
edge cube
CubAe Location Corner Edge Face
Number
Center
8
CubBe
Painted Faces
Number
3
8
none
12
none
6
none
1
Painted Faces
3 2 1 0
1 3 7
Willie organized the data about the 64 smaller cubes in the giant cube. Use the table above to complete the table below. One set of data has already been completed. Small cubes PaintedSurfaces 5
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Locationingiantcube
3
corner (Think: Same as a 3
2
edge (Think: One more than a 3
1
face (Think: Three more than a 3
0
center (Think: The center is now 2
3
3.)
3
3
Number 8
3 on each edge.)
2
3 on each face.)
2.)
R e te a c h i n g
24 24 8 Topic 13
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Practice
Name
Problem Solving: Use Objects and Solve a Simpler Problem
13-7
Use objects to help you solve a simpler problem. Use the solution to help you solve the srcinal problem.
1. Number Sense Six people can be seated at a table. If two tables are put together, 10 people can be seated. How many tables are needed to make a long table that will seat 22 people?
5 tables 2. Donna is building a large cube that will have 5 layers, each with 5 rows of 5 small cubes. How many small cubes will the larger cube contain?
5 5 5 125 3. Strategy Practice Jerome’s job duties include feeding the fish. There are 5 kinds of fish that he feeds: guppies, zebra danios, betas, platys, and neon tetras. Use the following clues to find the order in which Jerome 7 3 1 e c i t c a r P
feeds them. • Jerome feeds the guppies third. • Jerome does not feed the betas right before or right after the guppies. • Jerome feeds the zebra danios last. • Jerome feeds the platys after the betas.
A B C D
guppies, zebra danios, betas, platys, and neon tetras betas, platys, guppies, neon tetras, zebra danios neon tetras, zebra danios, guppies, platys, betas betas, guppies, platys, neon tetras, zebra danios
4. Explain It Suppose Ann is placing bowling pins in the following manner: 1 pin in the first row, 2 pins in the second row, 3 pins in the third row, and so on. How many pins will she use if she has 5 rows in her placement? Explain.
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14-1
Customary Units of Capacity Capacity is the measure of an amount of liquid. Common units of capacity are the gallon (gal), quart (qt), pint (pt), and cup (c). The chart below shows how capacity units are related.
gallon
1 gallon
4
q u q ar u a t
q u q ar u a t
rt
rt
quarts
q u a r t
1 quart
cup
Pint
pint
cup
Pint
2 pints
1 pint
2 cups
How to estimate vol ume in different units: What is the volume estimated in gallons? in quarts?
gallon
q u q a u r a t r t
Step 1: Think: The gallon container looks about half full.
The volume of the liquid is about half the capacity of the container.
Step 2: Estimate.
The volume of the liquid is about _1 gallon. 2
Step 3: Remember how gallons and quarts are related.
1 gal 4 qt gal
Step 5: Simplify.
1 __
(4 qt) 2 qt
Step 6: State your estimate in quarts.
The volume of the liquid is about 2 quarts.
Step 4:
2 2
1 __
1 4 1
1 __
Relate _12 gallon to quarts.
2
R e te a c h i n g
(4 qt)
_1
1. Number Sense How many cups are in 1 2 quarts of lemonade? 5
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Practice
Name
Customary Units of Capacity
14-1
Which unit(s) of capacity would be most reasonable to measure each?
gal cup or fl oz cup or pt
1. barrel of oil 2. can of applesauce 3. individual carton of orange juice
Give the amount of liquid in each container. Use a fraction if necessary.
4.
5. q u a r t
2
pt
gallon
qt
4
6. Reasoning Do 1 gallon and 8 pints represent the same amount? A No, 8 pints is 8 times the amount of 1 gallon. B No, 1 gallon is equal to 4 quarts. C Yes, 1 gallon is equal to 4 quarts, which is equal to 8 pints. D Yes, 8 pints is equal to 8 cups, which is equal to 1 gallon. 1 4 1 e c i t c a r P
7. Explain It If you only needed 1 c of milk, what is your best choice at the grocery store—a quart container, a pint container, or a _12 gal container? Explain.
A pint container (16 fl oz) is closer to a cup (8 fl oz) than t he other measures.
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14-2
Metric Units of Capacity Two common metric units of capacity are the liter (L) and the milliliter (mL).
1 cm
1 cm
1milliter (mL)
1 cm 1liter (L) 1,000 mL
How to estimate volume in liters and milliliters: What is the volume estimated in liters? in milliliters?
1liter (L)
Step Think. 1:
The liter container looks about half full.
Step Estimate. 2:
The volume of the liquid is about
Step 3: Relate liters to milliliters.
1 L 1,000 mL
Step 4: Relate _12 liter to milliliters. Complete.
1. 2,400 mL 2. 0.9 L
3. 334 mL 4. 0.23 L
2.4 900 mL 0.334 230 mL
_1 liter. 2
_1 L _1 (1,000 mL) 500 mL 2 2
R e te a c h i n g
L
1 4 2
L
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Practice
Name
14-2
Metric Units of Capacity Estimation Which unit of capacity would be most reasonable to measure each?
1L 20 mL 80,000 L
1. bottle of water: 1 mL or 1 L 2. bottle cap: 20 mL or 20 L 3. swimming pool: 80,000 mL or 80,000 L Give the amount of liquid in each container.
5. 150 L
1200 mL
800 mL
400 mL
10L0 2 4 1 e c i t c a r P
20m 0L
6. Reasonableness In a science fair project, you test four different-sized containers of water: 1 L, 2 L, 4 L, and 5 L. Which of the following expresses these capacities in milliliters? A B C D
1 mL, 2 mL, 4 mL, 5 mL 10 mL, 20 mL, 40 mL, 50 mL 100 mL, 200 mL, 400 mL, 500 mL 1,000 mL, 2,000 mL, 4,000 mL, 5,000 mL
7. Explain It Tell whether you would use multiplication or division to convert milliliters to liters. Explain your answer.
Sample answer: I would divide by 1,000 to convert milliliters to liters, because 1,000 mL 1 L.
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14-3
Units of Weight and Mass Weight is a measure of how heavy or light something is.
rose: about 1 oz
man’s shoe: about 1 lb
500 bricks: about 1 ton
Mass is a measure of the quantity of matter, or stuff, in an object.
1 cm
1 cm
M IL
K
1 cm 1L
mL of water: about 1 mg
dollar bill: about 1 g
How Weight Units are Related 16 oz 2,000 lb
liter of milk: about 1 kg
How Mass Units are Related
1 lb 1T
1,000 mg 1,000 g
1g 1 kg R e te a c h i n g
Which customary weight unit would be best for measuring each object?
1. elephant
ton
2. hot dog bun
3. loaf of bread
ounce
pound
1 4 3
Which metric mass unit would be best for measuring the amount of stuff in each object?
4. dust particle 5
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milligram
5. plastic fork
gr a m
6. dictionary
k i l ogr a m
7. Estimation Estimate the weight in pounds of 162 oz of hot dogs.
10 lb Topic 14
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Practice
Name
14-3
Units of Weight and Mass Which customary unit of weight would be best to measure each weight?
pound ounce
1. newborn baby 2. earrings
Which metric unit of mass would be better to measure each mass?
gram kilogram
3. necktie 4. 5 math textbooks
Which mass or weight is most reasonable for each?
5. laptop computer: 5 lb or 5 mg 6. dozen donuts: 1 oz or 1 kg
5 lb 1 kg
7. Reasonableness Which unit would you use to weigh a garbage truck? A ton 3 4 1 e c i t c a r P
B kilogram
C pound
D milligram
8. Explain It Did you know that there is litter in outer space? Humans exploring space have left behind bags of trash, bolts, gloves, and pieces of satellites. Suppose there are about 4,000,000 lb of litter in orbit around Earth. About how many tons of space litter is this? Explain how you found your answer.
2,000 T
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14-4
Converting Customary Units 1 gal 128 fl oz 1 qt
3 2 f l oz
1 qt
3 2 f l oz
1 qt
32 fl o z
1 qt
32 fl oz
1 pt 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt 1 pt (16 fl oz) (16 fl oz) (16 fl oz) (16 fl oz) (16 fl oz) (16 fl oz) (16 fl oz) (16 fl oz) 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c 1c Converting a capacity measurement from a smaller unit to a larger unit: 4 pints to ____ quarts You know 2 pt 1 qt. Think: Quarts are larger than pints. If I convert pints to quarts, there will be a smaller number of quarts than there were pints. Operation: divide.
Find 4 2. 4 pt 2 qt
1 yd 36 in. 1 ft
1i2n.
f1t
12in.
1ft
12 in.
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in. in.
Converting a length measurement from a larger unit to a smaller unit: 2 feet ____ inches
You know 1 ft 12 in.
Think: Inches are smaller than feet. If I convert feet to inches, there will be a larger number of inches. Operation: multiply.
16
3. 8 gal 5. 6 ft
7. 48 oz
32 2 3
2,000 lb 1 T 5 4. 10 c pt 15,840ft 6. 3 mi 3 8. 6,000 lb T 2.
oz 1 lb
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R e te a c h i n g
1 4 4
Complete.
1.
Find 2 12. 2 ft 24 in.
qt yd lb
9. Estimation A vat has a capacity of 624 fl oz. Estimate its capacity in cups.
80 c Topic 14
221
Practice
Name
14-4
Converting Customary Units In 1 through 9, convert each measurement. You may need to convert more than once.
2 mi 4. 4 gal 64 c 13 pt 7. 6 qt 1 pt 1. 3,520 yd
125 in. 256 fl oz 10 gal
2. 10 ft 5 in. 5. 2 gal
8. 40 qt
96,000 oz 6. 6 yd 2 ft 240 in. 9. 48 ft 16 yd 3. 3 T
10. Jenny bought a gallon of orange juice. The recipe for punch calls for 2 quarts of juice. How many quarts will she have left over to drink at breakfast?
2 qts left over 11. Algebra Laura has a roll of ribbon. There are 20 yards of ribbon on the roll. What equation can Laura use to find the number of inches of ribbon on the roll?
n
20 36
12. Ramona drives a truck weighing 1 ton. She picks up 3 cows weighing 3,000 pounds total. How much do the truck and the cows weigh together? 4 4 1 e c i t c a r P
A B C D
50 lb 500 lb 5,000 lb 50,000 lb
13. Explain It If you want to convert pounds to ounces, do you multiply or divide?
Sample answer: every pound contains of pounds 16 ounces.by You 16 multiply to find the thenumber numberof ounces.
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Name
14-5
Converting Metric Units Converting a capacity measurement from a smaller metric unit to a larger metric unit: 2,000 milliliters
liters
You know 1,000 mL 1 L.
Think: Liters are larger than milliliters. If I convert milliliters to liters, there will be a smaller number
Find 2,000 1,000. 2,000 mL 2 L
of liters. Operation: divide. Converting a length measurement from a larger metric unit to a smaller metric unit: 2 kilometers
meters
You know 1 km 1,000 m.
Think: Meters are shorter than kilometers. If I convert kilometers to meters, there will be a larger number of meters. Operation: multiply.
Complete.
1. 1 kg
3. 3 L 5. 5 m 7. 72 g
1,000 g 3,000 mL 500 cm 72,000 mg
Find 2 1,000. 2 km 2,000 m
R e te a c h i n g
1,000 mg 4. 8,000 mL L 8 6. 14,000 mm m 14 8. 8,000 g kg 8 2. 1 g
1 4 5
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Practice
Name
14-5
Converting Metric Units In 1 through 6, convert each measurement.
1. 3,000 mm 3. 6 m
600
5. 20,000 mg
3
8, 000 m 9,800 g 4. 9.8 kg 11 6. 11,000 mL 2. 8 km
m
cm
20
g
L
In 7 through 12, compare the Use . measurements. , , or for each
7.4,000 g 10. 8 kg
5 kg
5,000 g
572 mL
8. 0.72 L 11. 44 L
44,000 mL
9.5.12 m 12.200 m
512 cm
20,000 mm
13. A watering can holds 4.2 liters. If a small cactus requires 310 mL of water once every 3 weeks, how many of the same cactus plant can be watered at once without refilling the watering can?
13 cactus plants
14. Critical Thinking How do you convert kilometers to meters?
Multiply by 1,000
5 4 1 e c i t c a r P
15. Jeremy’s apartment building is 15 meters tall. How many centimeters is this? A 1.5 cm
B 50 cm
C 1,500 cm
D 15,000 cm
16. Explain It Explain how to convert 24 kilograms to grams.
Sample answer: Since 1kg 1,000g, multiply 24 kg by 1,000. 24 1,000 24,000 grams
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14-6
Elapsed Time How to find elapsed time: Iris left her house at 1:15 P.M. and arrived at her grandparents’ house at 3:30 P.M. How long did the trip take? Use a number line to count up. h1
h1
15
Start
min End
1:15 P.M.
1:00 P.M.
2:00 P . M.
3:00 3:30 P.M. P.M.
How to use elapsed time to find when an event began or ended: Omar and his brothers played floor hockey for 1 hour and 9 minutes. They finished playing at 6:30 P.M. At what time did they begin playing? You can subtract to find the start time. End Time Elapsed Time Start Time 6 h 30 min h 9 min 5 h 21 min
1
The trip took 2 h 15 min.
So, they began playing at 5:21 P.M. Find each elapsed time.
1. 8:13 P.M. to 10:00 P.M.
2. 1:24 P.M. to 4:47 P.M.
1h47min
3h23min
3. 3:35 P.M. to 6:09 P.M.
4. 9:55 P.M. to 11:42 P.M.
2h34min
1h47min
R e te a c h i n g
Find each start time or end time using the given elapsed time.
5. Start: 5:49 A.M. Elapsed: 5 h 20 min
1 4 6
6. End: 8:27 P.M. Elapsed: 4 h 13 min
End: 11:09 A.M.
Start: 4:14 P.M.
Add or subtract.
7. 5
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6 h 31 min 7 h 16 min
8.
7 h 12 min 3 h 30 min
1 3 h 4 7 m i n 3 h 42 m i n
9.
3 h 5 min 8 h 55 min
12 h
10.
9 h 5 min 8 h 22 min
4 3 m in
Topic 14
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Practice
Name
14-6
Elapsed Time Find each elapsed time.
46 min 3 h 53 min
1. 9:59 P.M. to 10:45 P.M. 2. 1:45 P.M. to 5:38 P.M. 3.
8:45
11:36
A.M.
A.M.
4. 11 10
12 1
9
3
8
2h51min 5. Start: 3:46 P.M.
12 1
9
4
2 3
8
4
7 6 5
7 6 5
P.M.
P.M.
9h38min
Find the end time using the given elapsed time.
6. Add.
11 10
2
Elapsed: 2 h 20 min
2 h 45 min 3 h 58 min
6 h 43 min
6:06 P.M.
7. Add.
6 h 47 min 5 h 28 min
12 h 15 min
The White House Visitor Center is open from 7:30 A.M. until 4:00 P.M. 6 4 1 e c i t c a r P
8. Tara and Miguel got to the Visitor Center when it opened, and spent 1 hour and 20 minutes there. At what time did they leave?
8:50 A.M.
9. Jennifer left the Visitor Center at 3:30 P.M. after spending 40 minutes there. At what time did she arrive?
2:50 P.M.
10. A football game lasted 2 hours and 37 minutes. It finished at 4:22 P.M. When did it start? A 1:45 P.M.
B 1:55 P.M.
C 2:45 P.M.
D 2:50 P.M. 5
11. Explain It What is 1 hour and 35 minutes before 4:05 P.M.? Explain how you solved this problem.
Sample answer: 2:30 P.M.; I solved this problem using subtraction.
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14-7
Elapsed Time in Other Units
A ship left Houston, Texas, at 7:10 A.M. on Tuesday. It arrived at Progreso, Mexico, at 10:15 P.M. on Wednesday. How long did the voyage take? Method 1 Method 2 1. Represent each time on a 48-hour 1. Represent each time on a 24-hour number line. See the line below. clock. See the chart below. 12 Hour P( .M.) 1:00 24 Hour
2:00
3:00
4:00
5:00
6:00
7:00
8:00
9:00
7:10
10:00 11:00 12:00 0246
13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 00:00
2. Find the time from departing Houston until midnight of Tuesday.
46:15
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
2. Subtract the departing time from the arrival time.
24 h 00 min 23 h 60 min 7:10 min 7 h 10 min 16 h 50 min
46 h 15 min 7 h 10 min 39 h 5 min
3. Find the time the boat sailed on Wednesday until it reached Progreso. 0 h 00 min
22 h 15 min 22 h 15 min
4. Add the two times. 16 h 50 min 22 h 15 min 38 h 65 min
R e te a c h i n g
38 h 65 min 1 h 60 min 39 h 5 min
1 4 7
Use a 12-hour clock to represent time. Complete the following addition to find the sailing time from Houston to Progreso, Mexico. 4h 50 min
12h00min
10h15min
12 h 00 min
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Topic 14
227
Practice
Name
14-7
Elapsed Time in Other Units Find the elapsed time.
1. 9:50 P.M. to 4:00 A.M. 2. 5:15 A.M. to 2:15 P.M. 3. 10:00 P.M. to 4:45 A.M. 4. 6:30 P.M. to 5:10 A.M.
6 h 10 min 9h 6 h 45 min 10 h 40 min
Find the end or start time.
5. Start time: 8:15 A.M. Elapsed time: 30 h 20 min.
End time: 2:35 P.M.
6. End time: 10:30 P.M. Elapsed time: 13 h 30 min.
Start time: 9:00 A.M.
7. Number Sense Krishan wakes up at 7:30 A.M. every morning. He spends 30 minutes getting ready, 6 hours at school, and 2 hours volunteering. Then he takes the bus for 15 minutes and arrives at his house. What time does he arrive at his house? A 2:00 P.M. 7 4 1 e c i t c a r P
B 3:45 P.M.
C 4:15 P.M.
D 5:45 P.M.
8. Explain It Emre left for a trip Sunday night at 10:00 P.M. He returned home Tuesday morning at 6:00 A.M. How many hours was he away on his trip? Explain how you found your answer.
24 hours from Sunday to Monday 10 P.M., and another 8 hours to Tues morning. Total: 24 8 32 hours 5
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228 Topic 14
Reteaching
Name
14-8
Temperature Change How to read temperature:
How to find changes in temperatures:
Miami’s average high temperature in June is 87°F. What is the average high temperature in degrees Celsius?
Find the change in temperature from 57°F to 18°F.
°F 87
31 °C
Step 1: Find 87° on the Fahrenheit scale. Step 2: Read across to find the temperature on the Celsius scale.
Step 1: Find the difference between 57°F and 18°F by subtracting. 57°F 18°F 39°F Step 2: Tell if the difference is an increase or decrease. Since the second temperature is less than the first temperature, there was a decrease. So, the change is a decrease of 39°F.
31°C is the same as 87°F. Write each temperature in Celsius and Fahrenheit.
.
°F
55
13
°C
2.
55°F1, 3°C
°F 92
33 °C
92°F3, 3°C R e te a c h i n g
Find each change in temperature.
3. 34°F to 89°F
55°F increase 5. 86°F to 54°F
32°F decrease 7. 3°F to 81°F 5
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78°F increase
4. 11°C to 26°C
1 4 8
15°C increase 6. 30°C to 8°C
22°C decrease 8. 12°C to 5°C
7°C decrease
9. Number Sense Yesterday’s high temperature was 76°F. The difference between the high and low temperature was 33°F. What was yesterday’s low temperature?
43°F Topic 14
229
Practice
Name
14-8
Temperature Change Write each temperature in Celsius and Fahrenheit.
1. °F
80 70 60 50
2. °F
°C
100 90 80
20 10
70°F 21,°C
,
40
°C
3. °F
30
94°F 34°C ,
60 50 40 30
°C 10 0
5 0° F 10° C
Find each change in temperature.
4. 34°F to 67°F
5. 12°C to 7°C
33°F increase
5°C decrease
6. Number Sense Which is a smaller increase in temperature: a 5°F increase or a 5°C increase?
5°F is a smaller increase. Record High
Information about the record high temperatures
8 4 1 e c i t c a r P
in four states is shown. 7. What is the difference between the record high temperature in Florida and the record high temperature in Alaska in °C?
5°C
Temperature
State
°F
°C
Alaska Florida Michigan Hawaii
100 109 112 100
38 43 44 38
8. What is the difference between the record high temperature in Michigan and the record high temperature in Florida in °F? 9. What is the difference between A 6°C
B 12°C
6°C
3°F
and 12°C?
C 18°C
D 19°C
10. Explain It Which is warmer, 1°F or 1°C? Explain how you found this answer.
1°C is warmer. Sample answer: You can tell by looking at the thermometers. 230 Topic 14
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Reteaching
Name
14-9
Problem Solving: Make a Table A bus company schedules three buses from Elm to Coretown. Starting at 8:05 A.M., the first bus departs Elm with a bus following every 45 minutes. The trip takes 70 minutes. What time is the third bus scheduled to arrive in Coretown? Construct a table of what you know.
Bus 1 2 3
Departure Time ( A.M.) 8:05 8:05 45 min 8:50 45 min 45 min
TravelTime min 70 70 min 70 min
ArrivalTime 8:05 70 min 8:05 45 min 70 min 8:05 45 min 45 min 70 min
Simplify the arrival time of the third bus.
8h
05 min
8:05 45 min 45 min 70 min
8:05 160 min
8h
165 min
10 h Arrival time of third bus in Coretown: Look back and check: is your answer reasonable?
160 min
2 h 120 min
45 min 10:45 A.M. 3 h __ 3 h 1__ 3 h; 1 h is 2__ Yes, __ 4
4
4
4
3 h is 10:45 A.M. 8:00 A.M. 2__ 4
1 4 9
State the following times in standard form.
1. Arrival time of first bus at Coretown 2. Departure time of second bus from Elm
9:15 A.M. 8:50 A.M.
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Topic 14
R e te a c h i n g
231
Name
Problem Solving: Make a Table
Practice 14-9
Make a table to solve the problems.
1. The temperature in the room was 55 F at 8:00 A.M. Ed turned the heat on, and the temperature rose 4F every 30 minutes. What was the temperature of the room at 11:30 A.M.?
83F 2. Charles can type 72 words per minute. He needs to type a paper with 432 words. How many minutes will it take Charles to type the paper?
6 minutes 3. Number Sense Yugita wakes up at 5:00 A.M. on Monday. Each day after that, she wakes up 17 minutes later. What time will she wake up on Friday? A 5:17 A.M. B 5:51 A.M. C 6:08 A.M. D 6:25 A.M. 9 4 1 e c i t c a r P
4. Explain It Train A leaves from New York at 10:15 A.M. and arrives in New Haven at 12:23 P.M. Train B takes 2 hours to go from New York to New Haven, but takes a 20-minute rest break in the middle of the trip. Which train is faster? Explain.
Train A. It takes 2 hours 8 minutes and Train B takes 2 hours 20 minutes.
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232 Topic 14
Reteaching
Name
15-1
Solving Addition and Subtraction Equations
Paige gave 6 peaches to Simon. Paige now has 8 peaches. How many peaches did Paige have to start? The number of peaches Paige now has equals the number of peaches she had at the start minus the number she gave to Simon. So, p 6 8 p number of peaches at the start To solve this equation, you need to get p alone. You can do this by using an inverse operation. An inverse operation is an opposite—addition and subtraction, for example, are inverse operations. To get p alone, add 6 to both sides of the equation.
Tip: If you add or subtract on one side of the equation, you must do the same on the other side. p6686 p 14
So, Paige started out with 14 peaches.
For questions 1 through 4, solve each equation.
a2 8 p 15 p 7
y 23 52 c 18 c 34
1. a 3 5
2. y 14 9
3.
4.
5. Mary’s sister is 4 years older than Mary. Her sister is 12. Use the equation m 4 12 to find Mary’s age.
m 8: Mary is eight years old. 6. Explain It Lyndon solved x 18 4. He said the answer was x 4. Explain why this is correct or incorrect.
This is incorrect because he didn’t add 5
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18 to both sides of the equation. The answer should be x 22. Topic 15
233
R e te a c h i n g
1 5 1
Practice
Name
15-1
Solving Addition and Subtraction Equations Solve and check each equation.
1. x 4 16 3. m 9 81 5. k 10 25 7. f 18 20 9. 75 n 100
x 12 m 90 k 35 f2 n 25
2. t 8 15 4. 7 y 19 6. 15 b 50 8. w 99 100 10. p 40 0
t 23 y 12 b 35 w 199 p 40
11. Jennifer has $14. She sold a notebook and pen, and now she has $18. Solve the equation 14 m 18 to find how much money Jennifer received by selling the notebook and pen.
$4
12. Kit Carson was born in 1809. He died in 1868. Use the equation 1,809 x 1,868 to find how many years Kit Carson lived.
59 years 13. Strategy Practice Which is the solution for y when y 6 19? 1 5 1 e c i t c a r P
A 13
B 15
C 23
D 25
14. Explain It Nellie solved y 3 16. Is her answer correct? Explain and find the correct answer if she is incorrect.
No; Sample answer: Nellie used the wrong operation. She needed to use addition to get the variable alone. The correct answer is y 19. 234 Topic 15
y 3 16 y 13 Subtract 3 5
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Reteaching
Name
Solving Multiplication and Division Equations
15-2
Dan’s computer prints at the rate of 3 pages per minute. How many pages can he print in 15 minutes? The number of minutes is equal to the number of pages divided by the rate the pages are printed. 15 d 3 d number of pages To solve this equation, you can do an inverse operation. An inverse operation is an opposite—multiplication and division are inverse operations. To get d alone, multiply 3 times both sides of the equation.
Remember: If you multiply or divide one side of an equation, you must do the same to the other side. 15 3 d 3 3 45 d Dan can print 45 pages in 15 minutes. In 1 through 6, solve each equation.
a4 250 25x x 10 p 33 3 p 99
1. 12a 48 3. 5.
t 63 4. 75 3c c 25 6. 4 y 16 y 64 2. t 9 7
R e te a c h i n g
1 5 2
7. Number Sense A carton of eggs is on sale for $0.49. Use the equation $0.49x $2.45 to find out how many cartons you can purchase for $2.45.
x 5
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5 cartons
8. Geometry Shanika has a piece of cardboard with an area of 63 square inches; one side is 9 inches long. Write an equation and solve for the value of the other side.
9y 63, y 7 inches
Area 63 square inches y
9 inches Topic 15
235
Practice
Name
15-2
Solving Multiplication and Division Equations Solve each equation.
y5 p6 150 25p
c6 2. __
3.
4. 16
9
54 w w 64 __
1. 11y 55
c
4
k 8 36 5. ___
7.
k
9. 81 11. 20
288
3 9 9t t 5 4a a
30 10x x
13. Reasoning Antoinette divides 54 by 9 to solve an equation for y. One side of the equation is 54. Write the equation.
9y
54
6. 13d 39 m 8. __ 7
d 3 91 m 13
6 e 160 2
10. 5b 30 e 12. ___ 80
b
14. Adam is making a trout dinner for six people. He buys 48 oz of trout. How many ounces of trout will each person get?
6x
48; x
15. Which operation would you use to solve the equation 19x 646? A add 19 2 5 1 e c i t c a r P
B subtract 17
C divide by 19
8 oz each
D multiply by 19
16. Explain It How would you use mental math to find m in the m equation 63 (__ 2? 63 )
Sample answer: Multiplying and dividing m by 63 undo each other. So the equation is m 2. 5
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236 Topic 15
Reteaching
Name
15-3
Inequalities and the Number Line How can you graph an inequality? An inequality is a mathematical sentence that contains one of the symbols , , , or . A solution of an inequality is any number that makes the inequality true.
Graph the inequality x 5. 1. Draw a number line 0–10 with an open circle around 5. The open circle shows that 5 itself is not a solution.
0 1 2 3 4 5 6 7 8 9 10
2. Find and graph three solutions of x 5 on the number line. Draw an arrow from the open circle toward the graphed solutions. The arrow shows that all numbers greater than 5 are solutions to the inequality.
0 1 2 3 4 5 6 7 8 9 10
Determine the inequality represented by the following graph. 0 1 2 3 4 5 6 7 8 9 10
1. The closed circle represents the number 4. One value of x can be 4.
x
4
2. The arrow represents all numbers less than 4. So x can have any value less than 4.
x
4
3. The graph shows that x can have a value equal to or less than 4.
x
4
Graph each inequality.
1. x 6
1 5 3
2. x 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Determine the inequality represented by each graph. 5
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3.
4. 0 1 2 3 4 5 6 7 8 9 10
x
7
0 1 2 3 4 5 6 7 8 9 10
x
R e te a c h i n g
3 Topic 15
237
Practice
Name
15-3
Inequalities and the Number Line Name three solutions of each inequality. Then graph each inequality on a number line.
1. b 5 0
2. s 2 4 2
4
6
8
10
5, 6, 8
0
2
3
4
5
6
2, 1, 0
3. x 3
4. d 1 6 0
012345
0, 1, 2, 3
2
4
6
8
10
8, 9, 10
5. Lizette wants to read more books. Her goal is to read at least 2 books each month. Let m represent the number of books she will read in a year. Use m 24 to graph the number of books she plans to read on a number line. 20 21 22 23 24 25 26 27 3 5 1 e c i t c a r P
1
6. Estimation Marcus is making cookies for his class. There are 26 students, and he wants to bring 3 cookies per student. On Sunday, he runs out of time and decides he must buy at least half the cookies and will make no more than half from scratch. Draw a graph to represent how many cookies he will make himself. 20
32
34
36
38
40
7. Which sentence is graphed on the line below? 70
71
A m 74
72
73
74
75
76
B m 74
C m 74
D m 74
8. Explain It How do you know whether to use an open circle or a closed circle when graphing an inequality?
238
Use an open circle when the inequality symbol is or . Use a closed circle when the inequality symbol is or . Topic 15
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Reteaching
Name
15-4
Patterns and Equations You can find a rule for a pattern and use that rule to fill in a table.
b1st
bnd 2
ox
ox
brd 3
ox
th
4box
How many apples will be in the 8th box? Which box will have 39 apples? Step 1: Draw a table to show the information. Box number (b) 1 2 3 4
Number of apples (a) 3 6 9 12
Step 3: Write an equation to express the rule.
Step 2: Find a relationship between the two columns that is the same for each pair. This relationship is the rule for the pattern. For these two columns the relationship is to multiply the box number by 3.
Step 4: Use this equation to find the missing parts of the table.
number of apples 3 times the box number a 83 39 a 24, 24 apples 39 in the box 8th 13
a 3b
3b 3b b, 13th box will have
39 apples
R e te a c h i n g
For questions 1 through 4, write an equation for each table and solve for the missing values for x and y.
1.
5
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y
5
16
64
3
8
9
36
7
12
6
24
y
0
10 15
y
2.
x
x
x5
16 y 4x
3.
x
y
2
48
4
16
y
1 5 4
4.
x
y
4
30
3
2
40
4
2
50
10
80
5
x6
8
x y ___ 10 Topic 15
239
Practice
Name
15-4
Patterns and Equations For 1 through 3, find a rule for each table. Write an equation for each rule.
1.
y
2.
x
y
5
15
18
2
6
50
11
33
12
6
18
34
3x
x
3.
x
y
9
4
–4
25
8
0
6
12
4
17
16
8
y
x y __
y
2
x
8
4. Reasoning Write an equation that will give the answer y 5 when x 12.
Possible answer: y x 7 5. A farmer sells 200 apples at the market. The next week, he sells 345 apples. How many more apples did he sell the second week?
200 x 345; 145 more apples 6. In the equation y x 5, which numbers would you use for x if you wanted y 0? A 4 5 1 e c i t c a r P
numbers 10
B 1, 2, 3, 4
C 6, 7, 8
D 10, 11, 12
7. Explain It If you know the rule for a table, how can you add pairs of numbers to the table?
Sample answer: Use values for x not already in the table and solve for y.
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240 Topic 15
Reteaching
Name
15-5
Problem Solving: Draw a Picture and Write an Equation Salim collects football cards and baseball cards. He has 38 baseball cards. If he has 25 football cards, how many sports cards does Salim have? Step 1: Write down what you already know.
Step 2: What are you trying to find?
Salim has 25 football cards. He has 38 baseball cards.
How many sports cards Salim has.
Step 3: Draw a diagram to show the information. Step 4: Write an equation and solve. Choose a variable to represent what you are trying 38 25 b to find. b 63 Salim has 63 sports cards.
b
38
25
For questions 1 through 3, draw a picture or diagram, write an equation, and solve the equation to answer the question.
1. Hallie spent $28 buying two figurines at the flea market. If one figurine costs $11.28, how much did Hallie pay for the other one?
$28.00 $11.28
x
$28.00 x $11.28, x $16.72 2. Cody is selling popcorn for his Cub Scout pack. His goal is to sell $350. So far, Cody has sold $284. How much more does Cody need to sell to meet his goal?
p
5
$284
p
1 5 5
$284 $350, p $66
3. In Briana’s class there are 5 fewer students than there were last year. There are 21 students in . c In , n o it a c u d E n o s r a e P ©
R e te a c h i n g
$350.00
x 5
21
Briana’s last year?class. How many students were there
x
5 21, x 26 students last year Topic 15
241
Practice
Name
15-5
Problem Solving: Draw a Picture and Write an Equation Draw a picture, write, and solve an equation to answer the question.
1. Suki and Amy made a total of 15 homemade holiday cards. Amy gave away 7 of them. How many cards did Suki give away?
15 7 p; p 8 cards for Suki to give away 3. Critical Thinking A total of 64 children are going on a field trip. If 14 ofare theboys? children are girls, how many
14 w 64; w 50 boys
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2. Ramon ate 3 more pieces of fruit today than he did yesterday. Today he ate 4 pieces. Write an equation to find out how many pieces of fruit he ate yesterday.
3 p 4; p 1 piece of fruit eaten yesterday 4. Naomi’s class went to the museum. There are 16 students in her class. If the total cost of admission for the class was $96, what does one admission to the museum cost?
$96 16x; $6 per admission
5. Paul is in a 17-kilometer canoe race. He has just reached the 5-kilometer marker. Which of the following equations can you use to find out how many more kilometers he needs to paddle? A k 5 17
B 5 k 17
C 5 k 17
D 17 5 k
6. Explain It How can you use estimation to decide whether 24.50 multiplied by 4 is close to 100?
Sample answer: 24.50 is about 25, and 25 4 is 100. 242 Topic 15
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Reteaching
Name
16-1
Understanding Ratios Coach Sanders has 9 basketballs, 6 footballs, and 5 soccer balls in her equipment bin. What is the ratio of basketballs to footballs? What is the ratio of soccer balls to total balls? Using ratios, you can compare parts of a whole to the whole, some parts to other parts, or the whole to its parts. number of basketballs 9
number of footballs 6
Remember There are three different ways to write the same ratio. 9 to 6
_9_
9:6
6
In 1 through 4, write each ratio. Give each answer in the three different ratio forms.
1. small stars to large stars
7 to 5,
7 7:5, __ 5
3. large unfilled stars to small unfilled stars 5
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2 2 to 4, 2:4, __ 4
2. filled stars to total stars
6 6 to 12, 6:12, ___ 12
4. small filled stars to total stars
3 3 to 12, 3:12, ___ 12
Topic 16
243
R e te a c h i n g
1 6 1
Practice
Name
16-1
Understanding Ratios Use the chart below in 1 through 4 to write each ratio three ways.
Mr. White’s 3rd-Grade Class (24 Students) Gender:
Male
8
Female 16
Eye Color: Blue
6
Brown
4
Hazel
Red
1
Brown 15
Hair Color: Blond 5
12
Green 2 Black
3
8 (or __ 1) 8 to 16, 8:16, ___ 16___ 2 __ 16 (or 2) 16 to 8, 16:8, female students to male students 1 1 8 ___ 1 to 24, 1:24, red-haired students to all students 24 24 12 ) 24 to 2, 24:2, ___ (or ___ all students to green-eyed students
1. male students to female students 2. 3. 4.
2
1
5. Reasonableness Is it reasonable to state that the ratio of male students to female students is the same as the ratio of male students to all students? Explain.
No; numberare of students that the to malethe students being compared is different.
6. George has 2 sons and 1 daughter. What is the ratio of daughters to sons? 1 6 1 e c i t c a r P
A 2 to 1
B 1 to 2
C
3:1
D
2 __ 1
7. Explain It The ratio of blue beads to white beads in a necklace is 3:8. Nancy says that for every 11 beads, 3 are blue. Do you agree? Explain.
Yes; Sample answer: If there are 3 blue beads to 8 white beads, that is 11 total beads, so 3 are blue. 244 Topic 16
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Reteaching
Name
16-2
Understanding Percent A percent is a special kind of ratio that shows the relationship between a number and 100. Percent actually means per hundred. The Hortonville Hoopsters have won 70 games of the last 100 games they have played. What percentage of the games has the team won? Using a 100-square grid can help.
There are 70 squares shaded so the ratio is 70 to 100 or 70 ___ 100
70 ___ . 100
70%.
Remember The % sign is the same thing as saying “out of 100.” In 1 through 4, write the ratio and percent of the shaded section on each grid.
1.
25 ___ , 100
25%
2.
43 ___ , 43% 100
63 ___ ,
3. 100
0 63% 4. ___ , 0% 100
Number Sense
R e te a c h i n g
5. What is “eighty-two percent” written as a fraction in simplest form?
41 __
1 6 2
50
6. Mystery Bookstore has 100 books on its shelves. Forty-one are hardbacks and 59 are paperbacks. What percentage of hardbacks and what percentage of paperbacks are there? 5
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41% hardbacks, 59% paperbacks
Topic 16
245
Practice
Name
16-2
Understanding Percent Write the fraction in lowest terms and the percent that represents the shaded part of each figure.
1.
2.
9 , 36% ___ 25
13 __ , 25
52%
3. Strategy Practice In the square, if part A is 1 _1 of the square and part C is __ of the square, 4 10 what percent of the square is part B?
B
65%
C
4. In Russia, _14 of the land is covered by forests. What percent of Russia is covered by forest? What percent of Russia is not covered by forest?
A
25%; 75%
3 5. In the United States, __ of the land is forests and woodland. 10 What percent of the United States is forest and woodland?
2 6 1 e c i t c a r P
30% 6. If _25 of a figure is shaded, what percent is not shaded? A 20%
B 30%
C 50%
D 60%
7. Explain It Explain how a decimal is related to a percent.
A percent is a decimal with the decimal point moved two places to the right.
246 Topic 16
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Reteaching
Name
16-3
Percents, Fractions, and Decimals At Slider Park, 40% of the children playing were boys. What is this percent expressed as a fraction and a decimal? You can change percents into fractions and decimals just as you can change fractions and decimals into percents. To change a percent into a fraction, drop the percent sign and place the number over 100. Simplify the fraction if possible. 40 __ 2 40% ____ 5
100
To change a fraction to a percent, divide the numerator by the denominator. Make sure your quotient goes to the hundredths place. If your quotient has too few digits, add zeros as placeholders. If your quotient has too many places, round to the hundredths. Then move the decimal point two places to the right and add the percent sign. 2 __ 5
0.40 40%
To change a percent to a decimal, drop the percent sign and move the decimal point two places to the left. 40% 0.40 In 1 through 3, write each percent as a decimal and a fraction in simplest form.
1. 36%
9 0.36, ___ 25
2. 13%
13 3. 0.13, ____
4. Reasonableness Pilar wrote that 3% reasonable answer?
100
59%
59 0.59, ____ 100
0.30. Is this a
No, she forgot to put a zero as a placeholder in the tenths place. It should be 0.03. 5. On the Pennyville Pigskins football team, two quarterbacks are competing for the starting job. Jake completes 87% of 5
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his passes. Brett only has 13% of his passes fall incomplete. Who is the better quarterback?
They are equal. If 87% of the passes are complete, then 13% of the passes are incomplete. Topic 16
247
R e te a c h i n g
1 6 3
Practice
Name
16-3
Percents, Fractions, and Decimals For questions 1 through 3, write the percent, decimal, and fraction in simplest form represented by the shaded part of each 100-grid.
1.
2.
2 40%, 0.4, __ 5
3.
1 50%, 0.5, __ 2
1 5%, 0.05, ___ 20
For questions 4 through 9, write each percent as a decimal and a fraction in simplest form.
3 5. 0.3, ___ 10 11 8. 11% 0.11, ____
4. 30% 7.
100
3 0.6, __ 51 150% 1.5, 1__ 60%
2
8 0.32, ___ 25 9. 100% 1.0, 1 6. 32%
10. Reasoning If 40% of Jeanne’s friends play kickball on weekends, what fraction of her friends don’t play kickball?
3 __ 5
11. If there are 6 eggs in 50% of an egg crate, how many eggs are in the whole crate?
12 eggs 12. What would you do first to order the following numbers from least to greatest? 3 6 1 e c i t c a r P
30%, _23, 0.67, _89, 0.7
A Order the decimals.
B Convert the decimals to percents.
C Order the fractions.
13. Explain It When writing a percent as a decimal, why do you move the decimal point 2 places?
D Convert all numbers to decimals.
Percent means “per hundred.” Each decimal place is a factor of 10; 100 is two factors of 10.
248 Topic 16
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Reteaching
Name
16-4
Finding Percent of a Whole Number Stefani wants to buy a skirt that costs $28. She received a coupon in the mail for a 20% discount from the store. How much will she save? To find the percent of a whole number, multiply the whole number by the decimal equivalent of the percent. 28 0.20 $5.60 discount
In 1 through 6, find the percent of each number.
13.94 4. 50% of 800 400 1. 17% of 82
2. 3% of 115 5. 73% of 280
3.45 204.4
3. 42% of 600 6. 39% of 22
252 8.58
7. In the problem above, Stefani is able to save $6 each week. With the discount, how many weeks will it take for Stefani to save enough to buy the skirt?
4 weeks 8. Geometry Kirk ordered a pizza with 10 slices. How many slices would amount to 30% of the pizza?
3 slices
R e te a c h i n g
9. Jackson High School’s band sells programs at the State Fair. They receive 15% of the total amount sold. The band sold $3,500 worth of programs. How much money did they receive?
1 6 4
$525 5
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10. Mental Math Using mental math, find 50% of 500. How did you get your answer?
1 , and __ 1 250: 50% is the same as __ 2 2 of 500 is 250.
Topic 16
249
Practice
Name
16-4
Finding Percent of a Whole Number Find each using mental math.
1. 20% of 60 3. 25% of 88
12 22
2. 30% of 500 4. 70% of 30
150 21
5. Reasoning Order these numbers from least to greatest. 1 , 72%, __ 5 , 20%, 0.3 0.85, __ 4 8
1 , 0.3, __ 5 , 72%, 0.85 20%, __ 4
8
6. What is 40% of 240? A 48
B 96
C 128
D 960
The table below shows the percent of the population that live in rural and urban areas of each country. Use the table to answer 7 through 9. Rural
4 6 1 e c ti c a r P
Urban
Bermuda
0%
100%
Cuba Guatemala
25% 60%
75% 40%
7. Out of every 300 people in Cuba, how many of them live in a rural area?
75
8. Out of every 1,000 people in Guatemala, how many live in urban areas?
400
9. Explain It If there are 1,241,356 people who live in Bermuda, how many residents of Bermuda live in urban areas? How many live in rural areas? Explain your answer.
1,241,356 in urban; 0 in rural; Sample answer: If 100% are in urban areas, then 0% are in rural areas. 250 Topic 16
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Reteaching
Name
16-5
Problem Solving: Make a Table and Look for a Pattern Sometimes when you are trying to solve a problem, it can help to make a table and look for a pattern. Nasser has 5 hours every day between when school ends and when he goes to bed. He spent 3 hours last night studying. If he continues like that every night, what percent of his time will be spent studying? Step 1. Make a table with the information you already know.
Hours spent studying
3
Total hours
5
Step 2. Fill in the table using equal ratios until you get a comparison with 100. Remember You find equal ratios by multiplying or dividing the top and bottom of the ratio by the same number. Hours spent studying 3 Total hours
12
5
20
30
48
50
80
60 100
So, Nasser spends 60% of his time studying. In 1 and 2, find the percent by completing the table.
1. won 7 out of 10 games Games won Total games 10 2. ran 13 miles out of 20 Miles run 5
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Total miles
20
70% 7 14 20
35
50
R e te a c h i n g
70
100
1 6 5
65% 13 6.5 32.5 65 10
50
100
3. Denise gets 450 minutes each month to use on her cell phone. Last month, she used 360 minutes. What percent of her minutes did Denise use?
80%
Topic 16
251
Practice
Name
Problem Solving: Make a Table and Look for a Pattern
16-5
For exercises 1 through 4, find each percent by completing each table.
30% Rainy days 12 6 3 30 Total days 40 20 10 100 2 out of 8 marbles are blue. 25% Blue 2 4 1 25 Total marbles8 16 4 100 80% 32 out of 40 days were windy. Windy days 32 4 16 80 Total days 40 5 20 100
1. 12 out of 40 days were rainy.
2.
3.
4. 16 out of 40 pets on Jack’s street are dogs. Dogs Total pets
40%
32 4 40 80 10 100
16 40
5. Write a Problem Write a real-world problem that you can solve using a table to find a percent.
Sample answer: Coins as a percent of a dollar or chance of rain 5 6 1 e c i t c a r P
6. Emmy plans to hike 32 miles this weekend. On Saturday, she hiked 24 miles. What percent of her goal has Emmy hiked?
75% 7. Explain It Dave estimated 45% of 87 by finding 50% of 90. Will his estimate be greater than or less than the exact answer?
Greater than: since 50% 45% and 90 87, his estimate will be greater than the exact answer.
252 Topic 16
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Reteaching
Name
17-1
Understanding Integers You can use integers to represent word descriptions. Remember: An integer is a whole number or its opposite (7 and 7 are integers). 0 is its own opposite. Use a positive integer to represent word descriptions that show an increase. up 3 floors 45 degrees above zero
3
5 steps forward
5
45
Use a negative integer to represent word descriptions that show a decrease. 300 feet below sea level
300
cut off 10 inches
10
2 steps backward
2
In 1 through 8, write an integer for each word description.
5 lose 100 points 100 spent $14 14 go down 8 floors 8
3 20
1. win 5 games
2. earned $3
3.
4. grew 20 inches
5. 7.
2,200 15 minutes before test time 15
6. 2,200 feet above sea level 8.
Draw a Picture Draw a number line to show the following: 9.
11
0
10.
R e te a c h i n g
11
1 7 1
2
5
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2
0
Topic 17
253
Practice
Name
17-1
Understanding Integers Write an integer for each word description.
1. a withdrawal of $50 2. a temperature rise of 14° 3. 10° below zero
50
10
14
Use the number line for 4 through 7. Write the integer for each point. A 12 10
8
6
8
4. A
CB 4
D 2
0
2
1
5. B
4
5
>
9.
9
8
10 12
2
6. C
Compare. Use , , or for each
8.
6
7. D
7
. 8
>
10.
12
21
>
26
Write in order from least to greatest.
11.
4, 11, 11, 4
12.
6, 6, 0, 14
13.
11, 8, 7, 4
11 14 8
, , ,
4 6 4
11 0 , 6 , , 7 ,11 ,
4
,
14. Strategy Practice Which point is farthest to the right on a number line? A
6
B
2
C 0
D 2
15. Explain It In Fenland, U.K., the elevation from sea level is 4m. In San Diego, U.S., it is 40 ft. The elevations are given in different units. Explain how to tell which location has a greater elevation. 1 7 1 e c i t c a r P
San Diego: any positive elevation is greater than a negative one. 5
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254 Topic 17
Reteaching
Name
17-2
Ordered Pairs You can locate points on a grid using ordered pairs. Draw a point on the grid for (1, 4). On a grid, the axis that goes across is called the x-axis. The axis that goes up and down is called the y-axis. The point where the x-axis and the y-axis cross is called the srcin. To locate a point on a grid using an ordered pair, start at the srcin.
y-axis
The first number in the ordered pair is called the x-coordinate. It tells how far to move to the right (if positive) or to the left (if negative). Since 1 is positive, move 1 unit to the right.
1
x-axis
4
The second number in the ordered pair is the y-coordinate. It tells how far to move up (if positive) or down (if negative). Since 4 is negative, move 4 units down to the point (1, 4).
(1,4)
Write the ordered pair for each point.
1. B
(3, 4)
2. D
(4, 1)
y E
Name the point for each ordered pair.
3. (3, 4)
F
4. (2, 2)
5
F
4
G
3
G
2
A
1
5. What is the ordered pair for the srcin?
(0, 0)
5
4
3
D
B
2
1
1
2
3
4
5
1
2
3
4
5
x
H
1 7 2
C
5
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Topic 17
R e te a c h i n g
255
Practice
Name
17-2
Ordered Pairs y
Write the ordered pair for each point.
+10
(3, 4) B (2, 3) C (2, 3)
1. A 2. 3.
+8
L
G X
B
+2 H
4. D
6.
A
+4
E
-10 -8
5.
F
+6
(5, 4) E (7, 1) F (8, 6)
-6
-4
-2J 0 -2 -4
D
+2 +4 +6 C
+8 +10
x
Y I
-6 K
-8 -10
Name the point for each ordered pair.
7. (5, 0) 10. (6, 5)
H I
8. (1, 1) 11. (4, 8)
J K
9. (0, 7) 12. (5, 5)
13. Strategy Practice If a taxicab were to start at the point (0, 0) and drive 6 units left, 3 units down, 1 unit right, and 9 units up, what ordered pair would name the point the cab would finish at?
L G
(5, 6)
14. Use the coordinate graph above. Which is the y-coordinate for point X? A
6
B
3
C 3
D 6
15. Explain It Explain how to graph the ordered pair (2, 3). 2 7 1 e c i t c a r P
Begin at the srcin and move 2 places to the left for the x-value of 2. Then move 3 spaces up for the y-value of 3. 256 Topic 17
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Reteaching
Name
17-3
Distances on Number Lines and the Coordinate Plane How do you find a distance on a number line? Finding the distance from 2 to 5 on a number line:
1. Plot the integers on a number line.
3 2 1 0 1 2 3 4 5 6 7
2. Draw a second number line beneath the first. Beneath the 2 on the top number line, start labeling the lower number line 0, 1, 2. Stop labeling when you reach 5 on the top line. Count the units on the lower line.
0 1 2 3 4 5 6 7
The distance between 2 and 5 is 7.
How do you find a distance on a coordinate grid? Finding the distance between (1, 3) and (7, 3) on a grid: y
1. Plot the points named (1, 3) and (7, 3) on a coordinate grid.
4
2. Because the y-coordinate of each
named point is the same, draw a matching number line parallel to the x-axis. Next to the point named (1, 3), start labeling the number line 0, 1, 2, until you reach the point (7, 3). Count the units on the number line. The distance between (1, 3) and (7, 3) is 6.
2
(1,3)
0 1 2 3 4 5 6
0
2
(7,3)
x
2
4
6
8
2
R e te a c h i n g
Find the distance between each pair of integers.
1. 5
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3, 4
7
2.
2, 3
5
1 7 3
5 4 3 2 1 0 1 2 3 4 5
5 4 3 2 1 0 1 2 3 4 5
Find the distance between each pair of points.
3. (1, 6) and (3, 6)
4
4. (3, 2) and (3, 5)
7 Topic 17
257
Practice
Name
17-3
Distances on Number Lines and the Coordinate Plane Find the distance between each pair of integers on a number line.
1.
2.
7, 3
4 units
4. 0, 5
7,
3.
1
8units
5. 2, 8
5units
4,
0
3,
3
4units
6.
6units
6 units
Find the distance between the points named by each set of ordered pairs on the coordinate plane.
6 units 2) 9 units
7. (2, 5), (2, 1) 9. (1, 7), (1,
11. At 2:00 P.M., the temperature was 3°C. By 4:00 P.M., the temperature had dropped to 1°C. What was the amount of the decrease in temperature?
4 units 1, 6) 1 unit
8. (4, 1), (0, 1) 10. (2, 6), (
12. Strategy Practice What integer on a number line is the same distance from 0 as 4?
4
4°C
13. On the coordinate plane, what is the distance between the points named by (2, 6) and (3, 4), if you move only along the lines of the grid? A 11 3 7 1 e c i t c a r P
B 9
C 7
D
7
14. Explain It How can you tell if two points lie along the same grid line just by looking at the ordered pairs?
If two points lie along the same grid line, then the ordered pairs that n ame the points either have the same x-value or the same y-value. 258 Topic 17
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Reteaching
Name
17-4
Graphing Equations An equation can predict data. The data can be plotted on a coordinate grid. If the graph is a straight line, the equation is called a linear equation.
STEP 1
Hours
Earnings
0 1 2 3 4
0 5 10 15 20
How to graph an equation:
Step 1 Use the equation to calculate ordered pairs. Put at least 3 ordered pairs into a table.
Step 2 Choose what to plot on the horizontal axis and what to plot on the vertical axis. Choose the starting point and ending point for each axis. Choose an interval for each axis. Label and number the axes.
Ron’s Earnings
25
)s r lla 20 o d 15 ni ( s 10 gn nri 5 a E
0 STEP 2
STEP 3
1
2
3
4
Hours Worked
Step 3 Graph the data by using the coordinates for each set of data as a point. Connect all the points in a straight line. Title your graph. Graph the equation below to show the cost, c, of buying harmonicas, h. Harmonicas are available online for $3 each, plus a single shipping charge of $2. c 3h 2 h
c
1 2 3
5 8 11
R e te a c h i n g
12 10 5
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1 7 4
8 ost 6 C 4 2 0
1 2 3 Harmonicas Topic 17
259
Practice
Name
17-4
Graphing Equations In 1 through 4, find the value of y when x = 4, 9, and 15.
1. y 2x 3. y 4x
8, 18, 30 16, 36, 60
2. y x 4 4. y x 6
0, 5, 11 10, 15, 21
In 5 through 8, make a table of values for each equation and then graph the equations. Let x 1, 3, 5, and 7.
5. y x 2 7. y x 1
3, 5, 7, 9 0, 2, 4, 6
6. y 3x 8. y x
3, 9, 15, 21 5 6, 8, 10, 12
9. Reasoning Without drawing the graph, describe what the graph of x 100 would look like. Explain.
It would be a vertical line at 100; every point on that line would have an x-coordinate of 100. 10. Reasonableness Which of the following ordered pairs is NOT on the graph of the equation y x 9? A
(7, 16)
B (15, 24)
C (20, 29)
D (28, 36)
11. Explain It A graph contains the ordered pair (2, 4). Write two different equations that would be possible for this graph. Explain how you found your answer.
2x or y x 2 are both possible answers because their solutions both include the ordered pair listed.
y 4 7 1 e c i t c a r P
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260 Topic 17
Reteaching
Name
17-5
Problem Solving: Work Backward Sometimes when you are trying to solve a problem, it is easier to start at the end and work backward. The Rosewood City train makes its first stop 10 minutes from the main station. The second stop is 20 minutes farther. The third stop is 15 minutes after that, and the fourth stop is 15 minutes after the third stop. The end station is 5 minutes from the fourth stop. The train reaches the end station at 9:00 A.M. What time did the train leave the main station? To solve this kind of problem, you need to work backward. Think about if the train were actually going backward.
Since you know it is 5 minutes from the end station back to the fourth stop, subtract 5 from 9:00 A.M. Continue subtracting to keep backing the train up until it is back at the main station.
main station ?
10
st 1stop
min
nd 2 stop
20
9:00 0:05 8:55 8:25 0:208:05
min
rd 3 stop
15
min
4
15
th
stop
min
8:55 – 0:15 8:40 8:05
5
end station 9:00 A.M.
min
8:40
0:15 8:25
0:10 7:55
So, the train left the main station at 7:55 A.M.
1. Zev has a rosebush in his yard. The first year it grew 3 inches. The second year it grew 12 inches. Zev then cut it back 5 inches. The third year it grew twice as much as the first year. The rose bush is now 42 inches tall. How tall was it when Zev planted it?
26 inches 2. Jesse went on a bike ride. He rode from his house to the library. Then he rode 5
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2 miles from the library to his friend’s house. Then he rode 1_21 miles to buy a snack at the store. Finally he rode the 3 miles back to his house. He rode a total of 9 miles. How many miles is it from Jesse’s house to the library?
2_12 miles
Topic 17
261
R e te a c h i n g
1 7 5
Practice
Name
17-5
Problem Solving: Work Backward Solve each problem by working backward. Write the answers in complete sentences. Barbara is refilling her bird feeders and squirrel feeders in her yard.
1. After filling her bird feeders, Barbara has 3 _12 c of mixed birdseed left. The two feeders in the front yard took 4_12 c 3
each. The two feeders in the backyard each took 2_4 c. The two feeders next to the living room window each took 3_14 c. How much mixed birdseed did Barbara have before filling the feeders? __
241 c 2
2. After Barbara fills each of her 4 squirrel feeders with 2_23 c of peanuts, she has 1_34 c of peanuts left. How many cups of peanuts did Barbara start with?
5 c 12___ 12
3. Strategy Practice Clint spends _14 hour practicing trumpet, _1
_1
3 around the house, 1 2 hour doing hour doingand tasks _ homework, hour cleaning his room. He is finished 4 at 7:30 P.M. When did Clint start? 2
A 4:00 P.M.
B 4:15 P.M.
C 4:30 P.M.
D 5:30 P.M.
4. Write a Problem Write a real-world problem that you can solve by working backward.
Answers will vary. See students’ work.
5 7 1 e c i t c a r P
5
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262 Topic 17
Reteaching
Name
18-1
Data from Surveys In a survey, each student was asked this question. How many trees do you have in your yard? Here are the responses: 1, 2, 3, 1, 4, 4, 1, 2, 1, 2, 4.
How to display data collected from surveys: The data can be displayed in a frequency table or a line plot.
FrequencyTable 1. Count the number of times each different response was made. 1: ////; 2: ///; 3: /; 4: ///
LinePlot 1. Count the number of times each different response was made. 1: ////; 2: ///; 3: /; 4: ///
2. Construct a frequency table. The table lists each response and its frequency. (The frequency of a response is how many times it was made.)
2. Construct a line plot. The plot lists each response along a horizontal line. The frequencies are stacked as x’s above each response.
3. Give the frequency table a title that clearly explains what information is in the table.
3. Give the line plot a title that clearly explains what information is in the plot.
Survey How many treesQuestion: are in your yard? Number of Trees 1 2 3 4
Number 4 3 1 3
y c n e u q e r F
Survey Question: How many trees are in your yard?
xx xx
x xx
x
x xx
1234 Number of Trees
1. How many students responded to the tree survey? 2. Which number of trees had the greatest responses?
11 1
3. Explain It Describe how you might pick a sample of 50 minivan 5
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owners that represent the minivan owners of your state. Sample response: Minivan owners living in four different regions of the state might be asked to respond. Topic 18
263
R e te a c h i n g
1 8 1
Practice
Name
18-1
Data from Surveys Ms. Chen’s class took a survey on how many minutes it took each student to get to school. The results are below: 12
14
5
22
18
12
12
6
14
18
12
1. What are the highest and lowest times?
5
11
22 and 5
Students in Ms. Chen’s Class
2. Make a line plot to display the data.
xx x x x
x x x
x x
x x
x
5 6 7 8 9 101 11 21 31 41 51 6 17 18 19 20 21 22 23 Minutes it Takes to Get to School Number of Bags of Groceries Bought in One Week y c n e u q e r F
x x x
x x x
x
x x x x x x
x x x x x
56789 Number of Bags
3. How many people responded to the survey about bags of groceries?
18 students The number of 4. What information was collected about groceries? bags of groceries bought in one week. 5. Use the line plot above. Which number of bags of groceries were bought most often? A 8 1 8 1 e c i t c a r P
B 7
C 6
D 5
6. Explain It Write a survey question that might gather the following information. “In one school there are 6 sets of twins, 2 sets of triplets, and one set of quadruplets.”
Sample answer: Are you one of a set of twins, triplets, or quadruplets? 264 Topic 18
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Reteaching
Name
18-2
Bar Graphs and Picture Graphs A bar graph uses rectangles to show data. A double bar graph uses two different-colored or shaded bars to show two similar sets of data. A picture graph uses symbols to show data. You can use each to compare data. Look at the information about
Conference Baseball Championships
the baseball teams in the frequency table at right. Display this information as a double bar graph.
HighSchool
1. Decide on a scale and its intervals. 2. Draw the graph. Label the axes. 3. Graph the data by drawing bars of the correct length and title the graph.
J u n i or
Varsity
Dominican
4
3
Phillips
6
7
Smith
5
4
Conference Girls’ Baseball Championships
Dominican Step 3
Phillips Step 2 Smith
Junior Varsity 012345678
Step 1
Look at the information about the junior teams in the frequency table above. Use this information to complete the picture graph with the values for Dominican High School.
5
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Number of Championships
Conference Girls’ Junior Baseball Championships s p i h s n o i p m a h C f o r
= 2 championships
e b m u N
Smith
Phillips
R e te a c h i n g
1 8 2
Dominican
Topic 18
265
Practice
Name
18-2
Bar Graphs and Picture Graphs In 1 and 2, answer the questions about the double-bar graph below.
1. How many boys play indoor soccer? How many girls play?
12 boys and 14 girls 2. What is the least popular sport among girls? Among boys?
Sports Students Play Number of Students
0
2
4
6
8 10 12 14 16 18
Indoor Soccer Track Basketball
Boys Girls
Swimming
Track is least popular among girls and swimming among the boys. 3. Frank bought 4 dozen doughnuts for his class. He had 4 left over. Which shows how to find how many doughnuts Frank gave away? A (48 2) 4
C (12 2) 4
B (24 2) 12
D (12 4) 4
4. Explain It Could the data in the bar graph in Exercise 1 be presented in a picture graph? Explain.
2 8 1 e c i t c a r P
Yes, each game could be represented by a symbol like a basketball, water, etc. and a human figure could represent each participating student. The figures would be differently colored for boys or girls.
266 Topic 18
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Reteaching
Name
18-3
Line Graphs How to make a line graph:
STEP 1
Step 1 Put the data into a table so that each set of data has two values.
Months passed 1 2 3 4 5
Step 2 Choose what to plot on the horizontal axis and what to plot on the vertical axis. Choose the starting point and ending point for each axis. (The starting and ending points must include the smallest and largest data values.)
STEP 4 Dave’s Weightlifting
160 STEP 3
140
s) d nu 120 o p 100 in( 80 ed t lift 60 h ige 40 W
Choose an interval for each axis. Label and number the axes.
Step 3 Graph the data by using the coordinates for each set of data as a point. Draw lines to connect the points.
20 0
Step 4 Title your graph.
1 STEP 2
2
Make a line graph of the data. Since the data for the x-axis are from 1 to 5, the scale can go from 0 to Since the data for the y-axis are from 20 to 45, the scale can go from 0 to
5
.
50
.
Blood Drive
50
5
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Weight lifted pounds) 70 100 100 120 140
s onr 40 o D f 30 roe 20 b um 10 N
0
3 4 Months Passed
5
6
Blood Drive Number Week of donors 1 20 2 20 3 30 4 35 5 45
R e te a c h i n g
1 8 3
1
2
3
4
5
Week Topic 18
267
Practice
Name
18-3
Line Graphs Display the data in the table below on the coordinate grid.
Hour 1 2 3 4
Temperature (Celsius)
4° 8° 16° 21°
24 ) 20 C ( er 16 tu a er 12 p m eT 8
4 1234 Hour
1. Which hour had the highest temperature?
Hour 4
2. How much higher was the temperature in Hour 4 than Hour 1?
17 degrees
3. Reasoning If Hour 1 was really 10:00 P.M., do you think the trend on the line graph would keep increasing? Explain.
No, because in the evening, the temperature should drop. 4. Critical Thinking Look at the line graph at the right. What do you know about the trend of the housing prices? A Housing prices increased. B Housing prices decreased. C Housing prices increased, then decreased. 3 8 1 e c i t c a r P
se 500 ci Pr 400 gn i 300 uso H200
100
D Housing prices decreased, then increased. 1234
5. Explain It In the example above, if housing prices had stayed the same for all four years, what would the line graph look like? Explain.
Years
A straight horizontal line; it indicates no change occurred.
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Reteaching
Name
18-4
Stem-and-Leaf Plots A stem-and-leaf plot is a convenient way to organize data.
Number of Students Absent Each Day
A school records the number of students absent each day. The records for a two-week period are shown in the stem-and-leaf plot on the right.
Stem
Leaf
1
2 5 6 9
2
1 4 7 8
3 0 3 KEY: 1|2 12 How do you read a stem-and-leaf plot?
1. Identify the stem for each The stems in this plot represent the tens digits data value. 10, 20, and 30. 2. Identify the leaf for each data value.
The leaves in this plot represent the ones digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.
3. Identify the key.
The key shows that each stem-leaf combination represents a two-digit number.
Read the following and complete the following questions. A small card shop records the number of birthday cards sold each day. The record is displayed as the stem-and-leaf plot shown on the right.
Number of Birthday Cards Sold Each Day Stem
Leaf
2
1 5 6 7 8 9 9
3
0 3
4
2 2 3 4 9
KEY: 2|1 21
1. Using the key, what is the value of 3|0? 2. How many days are represented by the plot? 5
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3. What is the least number of cards sold in one day? 4. What is the most number of cards sold in one day?
30 14 21 49
R e te a c h i n g
1 8 4
Topic 18
269
Practice
Name
18-4
Stem-and-Leaf Plots For 1 through 3, use the stem-and-leaf plot below. It shows the ages of the 17 people who used the outdoor pool from 6:00 A.M. to 7:00 A.M. on a Tuesday morning in the summer.
Ages (years) Stem Leaf 0
1. How many swimmers were younger than 30?
1 1 2 39
3
3
2. Which age group was swimming the most at this hour?
people in their sixties or retirees
4 7 5 5779 6 5566789 7 01 Key: 7 | 1 means 71
3. Why are there two 5’s as leaves next to the stem 6?
the age 65 appears in the table twice
Stem Leaf 0 9 1 0 2 0014 3 7 7 5 Key: 7 | 5 means 75
4. Make a stem-and-leaf plot of the data below. Prices of couch pillows (dollars) 10
75
20
20
37
24
21
9
4 8 1 e c i t c a r P
5. Refer to the stem-and-leaf plot in Exercise 4. Which stem (or stems) have the most leaves? A 70
270 Topic 18
B 9
C 20
D
30
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Reteaching
Name
18-5
Histograms A histogram is a type of bar graph that shows frequency of data with equal intervals. The equal intervals are shown on the horizontal axis. Mr. Sato asked each student in his class which month they were born in. He put the data into a table. Make a histogram to determine the two most common birthday months.
Month Frequency
Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sept. Oct. Nov. Dec. 32
12
Step 1: You want to divide the data by month, so lay out the months along the horizontal axis. Step 2: The number of students born in each month is the frequency data, which goes on the vertical axis.
4
1 6 5 r e 4 b m3 u N2 1 2 P0 E T S
32
5
01
2
STEP 3
J
F M A M
Step 3: Use the data from the
STEP 1
J J A Months
S O N D
table to fill in the histogram. September and May are the most common birthday months.
1. How many people ages 0–19 and 40–59 combined attended the tournament?
95 people 2. What is the total number of 5
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people who attended the tournament?
Spectators at School Tournament 50
r 40 e b 30 m20 u N 10
0
R e te a c h i n g
0–19 20–39 40–59 60–79 Age (in years)
1 8 5
135 people Topic 18
271
Practice
Name
18-5
Histograms This table shows the results of a class survey to find out how many pieces of fruit each student ate that week.
Amount of Fruit
Frequency
0–7
12
8–15
8
16–25
5
1. Complete the histogram below. What percentage of students ate 16–25 pieces of fruit that week? ts 12 n e 10 d u t 8 s f o 6 r e 4 b m 2 u N 0
0–7
8–15
16–25
Pieces of fruit consumed last week
5 out of 25 total students 20% 2. Reasoning Mary says a histogram shows that about 3 times as many people in the 60–79 age group answered a survey as in the 80–99 age group. How does she know this from looking at the histogram?
The bar that represents ages 60–79 is three times as tall as the bar that represents ages 80–99.
5 8 1 e c i t c a r P
3. Explain It A political campaign recorded the ages of 100 callers. In a histogram, which data would go on the horizontal axis and which on the vertical?
The horizontal axis would show equal age intervals. The vertical axis would show the numbers of callers in each age bracket. 272 Topic 18
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Reteaching
Name
18-6
Circle Graphs A circle graph shows 100% of a data set. Each wedge in the circle represents part of the whole amount.
Pet
One hundred people were asked if they owned a pet and if they did, what kind of pet. The table at the right shows the data. From this table, you can create a circle graph.
Each category is part of the whole. You can find out how big a wedge to use for each category by changing the ratios into fractions.
100
50
Cat
25
Bird
10
Other
10
No pet
5
To finish the graph, do the same with the rest of the table. 25 10 1 1 cat ___ _, bird ___ __, 100 4 100 10 10 5 1 1 other ___ __, no pet ___ __ 100 10 100 20
50 people out of 100 had a dog. 50 ___
Dog
No pet
is _12 .
Other Bird
So, the wedge for dog will be _1 of the circle. 2
Dog
Dog Cat
1. The table below shows what kind of books people said they had checked out at the library. Using the data, copy the circle graph and label each section with the correct type of book and a fraction in simplest form. Magazine
Type of book Fiction
60
Magazine
20
Nonfiction
20
No book
20
1 6 Nonfiction
1 6 1 6
Fiction 1 2
R e te a c h i n g
No book 5
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1 8 6
2. How many total people were asked about their book choice?
120 Topic 18
273
Practice
Name
18-6
Circle Graphs 1. A bagel shop offers a variety of bagels. One morning, the following choices were made by the first 20 customers of the day: plain, 10; poppy seed, 5; sesame seed, 3; multigrain, 2. Complete the table below. Fraction
Percent
_1 2
Plain Poppy Seed
1 _ 4 3 __ 20 1 __ 10
Sesame Seed Multigrain
50% 25% 15% 10%
2. Copy and complete the circle graph below with the data in the table above. Label each section with the percent and fraction of each bagel.
Multigrain 1 10 , 10%
Poppy 25% Plain 1 2,
1 4,
50% Sesame 3 20 , 15%
3. Number Sense A circle graph is divided into four sections. One section equals 40%. The other three sections are equal in size. What percent does each of the other three sections represent?
20% 6 8 1 e c i t c a r P
4. Reasoning If 10 out of 30 students in a survey chose ice skating as their favorite sport, what fraction of the circle should be shaded to represent the students who chose ice skating?
274 Topic 18
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Reteaching
Name
18-7
Mean The mean is the sum of all the values in a set divided by the number of items in the set. The mean is also called the average.
How to find the mean of a set of data: Eduardo surveyed 7 of his friends to find out how many books they read during the month. The frequency table shows the data. What is the average number of books read by Eduardo’s friends?
Friend Jean
Book Reading Number ofbooks read 2
Raul Sally Jonathan Haley Kristen Owen
3 8 5 6 3 1
1. Add the number of books read by each friend.
2 3 8 5 6 + 3 1 28
2. Divide the sum by the number of friends.
28 ___
3. Use the average to answer the question.
Eduardo’s friends read an average of 4 books during the month.
7
4
1. Find the mean of this set of data: 241, 563, 829, 755.
597
2. This frequency table shows the number of silver US Silver Medals medals won by American athletes in Summer Summer Olympics Games Olympic Games between 1972 and 2000. What Year Medals is the mean of this set of data? 2000 24 1996 32 1992 34 3. Estimation What is the approximate 1988 31 average of these three 1984 61 numbers: 9, 18, and 31? 1980 0 4. Explain It Explain how you would find the 1976 35 mean of this set of data: 4, 3, 5. 1972 31
31
about 20
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Sample AddThen the divide by three. numbersanswer: in the set.
Topic 18
275
R e te a c h i n g
1 8 7
Practice
Name
18-7
Mean Find the mean of each set of data.
4. 3, 6, 3, 7, 8
5 60 37 5.4
5. 120, 450, 630 6. 4.2, 5.3, 7.1, 4.0, 11.9
400 6.5
1. 2, 5, 9, 4 2. 44, 73, 63 3. 11, 38, 65, 4, 67
Gene’s bowling scores were as follows: 8, 4, 10, 10, 9, 6, 9.
8
7. What was his average bowling score? 8. If Gene gets two more strikes (scores of 10), what is his new average?
8.44
9. Reasoning Krishan wants his quiz average to be at least 90 so that he can get an A in the class. His current quiz scores are: 80, 100, 85. What does he have to get on his next quiz to have an average of 90? A 85
B 90
C 92
D 95
10. Explain It Suppose Krishan’s teacher says that he can drop one of his test scores. Using his test scores of 80, 100, and 85, which one should he drop, and why? What is his new average?
Drop the lowest score: 80. His new average would be 92.5. 7 8 1 e c i t c a r P
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276 Topic 18
Reteaching
Name
18-8
Median, Mode, and Range The median, mode, and range are each numbers that describe a set of data. Here is Eduardo’s survey of how many books his friends read last month. What are the median, mode, and range of Eduardo’s survey?
Median: The median is the middle number in a set of data. To find it: 1. Arrange the data in order from least to greatest. 2. Locate the middle number.
Book Reading Number of Friend books read Jean 2 Raul 3 Sally 8 Jonathan 5 Haley 6 Kristen 3 Owen 1
1, 2, 3, 3, 5, 6, 8 middle number
3
The median number of books read is 3.
Mode: The mode is the data value that occurs most often. To find it: 1. List the data.
1, 2, 3, 3, 5, 6, 8
2. Find the number that occurs most. 3 The mode of the books read by Eduardo’s friends is 3 books.
Range: The range is the difference between the greatest and least values. To find it: 1. Identify the greatest and least values.
8 and 1
2. Subtract the least from the greatest value. 8 1 7 The range of the books read by Eduardo’s friends is 7 books.
1. Find the median of this data set: 12, 18, 25, 32, 67. 2. Find the mode of this data set: 123, 345, 654, 123, 452, 185. 3. Find the range of this data set: 24, 32, 38, 31, 61, 35, 31. 5
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25 123 37
Topic 18
277
R e te a c h i n g
1 8 8
Practice
Name
18-8
Median, Mode, and Range 1. Find the range of this data set: 225 342 288 552 263. 2. Find the median of this data set: 476 234 355 765 470.
327 470
3. Find the mode of this data set: 16 7 8 5 16 7 8 4 7 8 16 7.
7
4. Find the range of this data set: 64 76 46 88 88 43 99 50 55.
56
5. Reasoning Would the mode change if a 76 were added to the data in Exercise 4?
Yes. The mode is now 88. With another 76, the modes would be 76 and 88. The table below gives the math test scores for Mrs. Jung’s fifth-grade class. 76 54 92 88 76 88 75 93 92 68 88 76 76 88 80 70 88 72 Test Scores
6. Find the mean of the data. 7. Find the mode of the data. 8. Find the median of the data. 9. What is the range of the data set?
80 88 78 39
10. Find the range of this data set: 247, 366, 785, 998. A 998 8 8 1 e c i t c a r P
B 781
C 751
D 538
11. Explain It Will a set of data always have a mode? Explain your answer.
No: if every item is different, there is no mode. 278 Topic 18
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Reteaching
Name
18-9
Problem Solving: Make a Graph Choosing which graph to use depends on what kind of data set you are looking at.
Bar graphs and circle graphs are good graphs to use when you are comparing data that can be counted. Line graphs are good at showing changes in the data over time. Histograms divide the data into intervals and record how many data items fall within each interval. There are three ferries that run from Port Shoals to Tiger Island. 187 people were on the 12:30 P.M. ferry, 122 people were on the 3:00 P.M. ferry, and 58 people were on the 5:30 P.M. ferry. Make a graph for this data set and analyze the data. This data set is comparing data that can be counted, so a bar graph or a circle graph would be a good graph choice. Make a bar graph of the data. e l What can you say about the data from this graph? The 12:30 P.M. ferry was the most popular. The 5:30 P.M. ferry was the least popular.
p 200 o e 150 P f o 100 r e b m u N
50 0
12:30 P.M. 3:00 P.M. 5:30 P.M. Ferry Time
1. Each month, Martina measures the height of a tree she planted. Make a line graph to show the tree’s growth.
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Month April
Height (in.) 36
May
42
June
60
July August
75 84
Tree Growth
100 ). 80 in ( t 60 h ig 40 e H 20 0 Apr. May Jun.
R e te a c h i n g
Jul. Aug.
Topic 18
1 8 9
279
Practice
Name
18-9
Problem Solving: Make a Graph 1. In a survey, 100 students from around the country were asked what news source they preferred. Which news source is most popular? Make a circle graph to solve the problem. News S ource
Number of Votes
Students’ Preferred News Source
Radio
Television Internet
50 25
Newspaper
15
Radio
10
NewsPaper Television Internet
2. In a survey, 32 students from around the country were asked how they traveled to and from school. Make a circle graph to show the data. How Students Get to School
Subway Bus Bike
8
16
4
On Foot
On Taxi Foot 4
Subway
Bike
0
Bus
3. If a graph shows that there were 10 people who watched between 7 and 12 movies, what kind of graph could you be looking at? A Circle
B Bar
C Histogram
D Line
4. Explain It Would a line graph be an appropriate graph in Exercise 3? Why or why not?
9 8 1 e c i t c a r P
Sample answer: No; a line graph is used to show data that changes over time. These data do not change over time. 280 Topic 18
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Reteaching
Name
19-1
Translations A translation is the motion of a figure up, down, left, or right.
How to translate a figure: Translate the triangle to the right 10 units and up 7 units.
y
12 11 10 9 8 7 6 5
(13, 11) (11,9)
Slide u p
(17,9)
C 4 (3, 4) Slide right (13, 4) 3 A B 2 (7, 2) (11, 2) (17, 2) 1 (1, 2) x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
1 Part
2Part
Translate the triangle to the right 10 units.
Step 1: Increase the x-coordinates of each vertex of the triangle by 10. (Do not change the y-coordinate of each vertex.)
Step 4: Increase the y-coordinates of each vertex of the triangle at its new location. (Do not change the X-coordinate of each vertex.)
Step 2: Draw the triangle at its new location.
Step 5: Draw the triangle at its final location.
Step 3: Label each vertex with its ordered pair.
Step 6: Label each vertex with its ordered pair.
1. Translate the triangle shown here up 7 units and to the right 10 units.
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Translate the triangle up 7 units.
y
12 (3, 11) (13, 11) 11 10 9 (11,9) (17,9) (7,9) 8 (1,9) 7 6 5 C (3, 4) 4 3 A B 21 (1, 2) (7, 2) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
2. Number sense Vertex A of a figure is located at (18, 6). The rhombus is translated down 5 units and to the left 3 units. What is the new location of vertex A?
x
(15, 1) Topic 19
281
R e te a c h i n
g
1 9 1
Practice
Name
19-1
Translations Give the vertical and horizontal translation in units. y
1.
4 3 2 1 0
6 5 4 3 2 1 0
123456
x
4.
y
8 7 6 5 4 3 2 1 0
y
5 4 3 2 1 0 123456
x
123456789
left 5 units
up 3 units 3.
y
2.
123456
x
x
right 33units, down units
right up 2 units 2 units,
5. Critical Thinking If a vertex of a square moved from (11, 9) to (6, 4), which of the following describes the translation? A 5 left, 5 up
C
5 right, 5 up
B 5 left, 5 down
D
5 right, 5 down
6. Explain It Explain how the x and y coordinates are related to the direction an object moves.
The x-coordinate increases when the
1 9 1 e c ti c a r P
object and ydecreases when itmoves movesright left. The -coordinate increases when an object moves up and decreases when it moves down. 282 Topic 19
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Reteaching
Name
19-2
Reflections A reflection is the mirror image of a figure. The reflection appears as if the image has been flipped.
How to recognize a reflection:
Reflection across a horizontal line
Reflection across a vertical line
The image looks like the figure flipped top to bottom.
The image looks like the figure flipped left to right.
Place a checkmark beneath the correct image for each reflection.
1. Reflection across a vertical line
2. Reflection across a horizontal line
✓
✓ 3. Reflection across a vertical line
✓ 4. Reflection across a horizontal line R e te a c h i n
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g
1 9 2
✓ Topic 19
283
Practice
Name
19-2
Reflections Draw the reflection of the object across the reflection line.
1.
y 10 9 8 7 6 5 4
2.
23 1 0 1 2468 3
5
7
9 10
x
y 10 9 8 7 6 5 4
23 1 0 1 2468 3
5
7
9 10
x
Tell whether the figures in each pair are related by a translation or a reflection.
3.
4.
Reflection
Reflection or
5. Think About the Process Which of the following gives enough information to determine whether the figures A and B have been translated or reflected?
translation
A Figure A is larger than Figure B. B Figure A and Figure B have the same shape and size. C Figure A has an identical shape to Figure B. D Figure A and Figure B are next to each other.
2 9 1
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e c ti c a r P
284 Topic 19
Reteaching
Name
19-3
Rotations A rotation is a turn that moves a figure about a point.
How to recognize a rotation: C A
C
B
D B
A
Rotation of the triangle around point A
Rotation of the triangle around point D
Each vertex rotates the same amount and Each vertex rotates the same amount and in the in the same direction around A. same direction around D.
1. Place a checkmark beneath the image that is NOT a rotation of the figure around K. K
K
K
K
K
✓ 2. Place a checkmark beneath the image that is NOT a rotation of the figure around R. R
R
R
R e te a c h i n
5
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R
g
R
1 9 3
✓ Topic 19
285
Practice 19-3
Rotations Describe whether the figures in each pair are related by a rotation, a translation, or a reflection.
1.
2.
translation or reflection
rotation
3. If the figure at the right is rotated 90° around the point shown, which of the figures below could be the result? A
B
C
D
4. Explain It Mark says that the figure at the right is a reflection. Faith says that it is a rotation. Who is correct? Explain.
They are both correct. It could be a reflection over a vertical line or it could be a 180° rotation.
3 9 1
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e c ti c a r P
286 Topic 19
Reteaching
Name
19-4
Congruence
The change of position of a figure is called a transformation. Figures that are the same size and shape are called congruent figures.
How to recognize a transformation:
2
2 1
1
A translation and a rotation of a figure produce a transformation. The size and shape of the figure do not change.
A rotation and a translation of a figure produce a transformation. The size and shape of the figure do not change.
How to recognize congruent figures:
Congruent with Figure 1.
Congruent Congruent with Figure 1. with Figure 1.
Not Congruent with Figure 1.
Congruent figures have the same size and shape. In each group, cross out the figure that is not congruent with the other figures.
1.
R e te a c h i n
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g
1 9 4
Topic 19
287
Practice
Name
19-4
Congruence Tell whether the following figures are congruent or not.
1.
2.
No
Yes
Identify each transformation. If not a transformation, explain why.
3.
4.
Not; figures are not Rotation the same size or shape
5. Reasonableness Which of the following situations uses an example of congruent figures? A B C D
A baker uses two sizes of loaf pans. The fifth-grade math book is wider than the fourth-grade book. You give five friends each a quarter. Two pencils are sharpened, the other two are not.
6. Explain It If two triangles are congruent, do they both have the same side lengths? Explain.
Yes, if the triangles are congruent, they
4 9 1 e c ti c a r P
are exactly the same size, and all side lengths are equal.
288 Topic 19
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Reteaching
Name
19-5
Symmetry What is line symmetry? A figure has line symmetry if it can be folded in half so that one side reflects onto the other side. The fold line is called a line of symmetry.
This figure has line symmetry. There is only one way to fold it so that two sides match. It has one line of symmetry.
This figure has line symmetry. It can be folded into matching halves two different ways. It has two lines of symmetry.
This figure has no line symmetry. It cannot be folded into matching halves.
What is rotational symmetry? A figure has rotational symmetry if it turns onto itself in less than a full rotation.
This figure has rotational symmetry. The figure and its rotated shape match after half a rotation.
This figure has no rotational symmetry. The figure and its rotated shape match only after a full rotation.
Circle the number of each figure that has line symmetry.
1.
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2.
3.
R e te a c h i n
Circle the number of each figure with rotational symmetry.
4.
5.
g
6.
1 9 5
Topic 19
289
Practice
Name
19-5
Symmetry For 1 through 4, tell if the figure has line symmetry, rotational symmetry, or both. If it has line symmetry, how many lines of symmetry are there? If it has rotational symmetry, what is the smallest rotation that will rotate the figure onto itself?
1.
2.
1 line of symmetry 3.
4.
1 turn 180° or __
4 lines of symmetry; 90º or _14 turn 2 lines of symmetry; 1 turn 180° or __
2
2
For 5 and 6, copy the figure. Then complete the figure so the dashed line is a line of symmetry.
5.
6.
7. Does any rectangle rotate onto itself in less than a half-turn?
A rectangle that is a square rotates onto itself in a 90° or quarter turn. 5 9 1 e c ti c a r P
8. Draw a Picture Draw a quadrilateral that has neither line symmetry nor rotational symmetry.
Answers will vary. 290 Topic 19
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Reteaching
Name
Problem Solving: Use Objects
19-6
A tetramino is an arrangement of four identical squares in a plane. The squares are attached to each other edge to edge. A unique tetramino cannot be matched to another by rotating or reflecting. Use square tiles to show if the two figures are unique.
1. Construct the two figures using tiles. 2. Construct a rotation of the first figure. Try other rotations.
3. Check for a match. Think: If figures match, they are not unique. If figures do not match, figures may be unique.
no match
rotation
match
4. Construct a reflection of the first figure. 5. Repeat Step 3. reflection
6. Write your answer in a complete sentence.
The two figures are not unique, because they match by reflection.
1. Show if these two figures are unique tetraminoes. Explain your answer.
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The two figures do not match either by reflection or rotation. They are unique tetraminoes.
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Topic 19
291
Practice
Name
Problem Solving: Use Objects
19-6
1. Is the following a pentomino? Explain.
Yes; there are 5 identical squares. Tell whether the pentominoes in each pair are related by a reflection or a rotation.
2.
Reflection
3.
Rotation
4. How many possible different pentominoes can be formed? A 3 B 7 C 10 D 12 5. Explain It Use objects to build pentominoes with 1 square in each row. How many of these kinds of pentominoes can be built? Explain.
Only 1; using 5 squares, you can only put one square in each row, and every pentomino after that will be the same!
6 9 1
5
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292 Topic 19
Reteaching
Name
20-1
Outcomes Almond, Keisha, and Mona are running for student council president. Barry, Andy, and Maurice are running for vice-president. Each student has an equal chance of being elected.
You can use a tree diagram to find all the President Vice-President Outcome possible outcomes. The set of all possible Barry Almond, Barry Almond Andy Almond, Andy outcomes is called the sample space. Keisha
There are 9 possible outcomes in the sample space.
Mona
Maurice Barry Andy Maurice Barry Andy Maurice
Almond, Maurice Keisha, Barry Keisha, Andy Keisha, Maurice Mona, Barry Mona, Andy Mona, Maurice
1. Complete the tree diagram to show the possible outcomes when Spinner A and Spinner B are spun. Color
Color
Outcome
Black, Red Black Black
Red Blue
White Green
Black , Blue Black , Green
Spinner A
Red Red Green
Blue
White
Blue Green
White , Red White , Blue White , Green
Spinner B
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2. How many times does the outcome black/green occur in the tree diagram?
1 time
R e te a c h i n g
2 0 1
Topic 20
293
Practice
Name
20-1
Outcomes The coach is trying to decide in what order Jane, Pete, and Lou will run a relay race.
1. Complete the tree diagram below to show the sample space. 1st
Jane
Pete Lou
2nd
3rd
Pete
Lou Pete
Lou Jane
Lou
Lou Jane Pete
Jane Pete Jane
2. How many possible outcomes are there in the sample space?
6
3. After the first runner is chosen, how many choices are there for the second runner?
2
4. Reasonableness Tom, Bill, John, and Ed are running for school president. The person in second place automatically becomes vice-president. How many possible outcomes are there in the sample space? A 6
B 9
C 10
D 12
5. Explain It The weather tomorrow could be sunny, cloudy, rainy, or snowy. Is there a 1 out of 4 chance of the weather being sunny?
You do not know what the probability is, because not all the outcomes are equally likely. 1 0 2 e c ti c a r P
294 Topic 20
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Reteaching
Name
20-2
Writing Probability as a Fraction The probability of an event is a number that describes the chance that an event will occur. Probability can be expressed as a fraction. Probability of an event
number of favorable outcomes number of possible outcomes
If Felice spins the spinner, what is the probability of landing on Orange?
Orange
There are 6 possible outcomes (spaces) and 2 favorable outcomes (Orange spaces). Probability (landing on Orange)
number of Orange spaces number of spaces
The probability of landing on Orange is _26 or _13.
2 __ 6
Black
Green
Orange
Blue Red
The probability of landing on Orange can be written as P(Orange) _13 .
Pencil VCR
Television
Lamp
Radio Cup
1. Find P(object that does not use electricity) 2. Find P(object that uses electricity) 3. Find P(object used for writing) 5
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2 _ 6 4 _ 6 1 _ 6
4. Explain It What does P(radio) mean?
The probability of the pointer stopping on the segment marked radio in one spin. Topic 20
295
R e te a c h i n g
2 0 2
Practice
Name
20-2
Writing Probability as a Fraction Tom put 4 yellow marbles, 2 blue marbles, 6 red marbles, and 5 black marbles in a bag.
4 __ 17 2 __ 17
1. Find P(yellow). 2. Find P(blue).
5 __ 17 6 __
3. Find P(black).
17
4. Find P(red).
A bag contains 12 slips of paper of the same size. Each slip has one number on it, 1–12. 6 __
12 5 __ 12 6 __ 12 4 __ 12
5. Find P(even number). 6. Find P(a number less than 6). 7. Find P(an odd number). 8. Find P(a number greater than 8).
or _12 or _12 or _13
9. Describe an impossible event.
You will pull out a number greater than 12. 10. A cube has 6 sides and is numbered 1 through 6. If the cube is tossed, what is the probability that a 3 will be tossed? 1 A __ 6
2 B __ 6
3 C __ 6
6 D __ 6
11. Explain It Explain the probability of tossing a prime number when you toss the cube with numbers 1 through 6. 2 0 2 e c ti c a r P
Prime numbers would be 2, 3, and 5, so the probability would be _36 or _12. 296 Topic 20
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Reteaching
Name
20-3
Experiments and Predictions You can use an experiment to predict outcomes.
5
2
4
3
Spinner A
Compare the chances of spinning an even number and spinning an odd number. 7
5
5
6
Spinner B
Compare the chances of spinning an even number and spinning an odd number.
Event: Spinning an even number Favorable outcomes: 2, 4 2 out of 4 possible outcomes are favorable, so in 2 out of 4 spins, you can expect an even number.
Event: Spinning an odd number Favorable outcomes: 3, 5 2 out of 4 possible outcomes are favorable, so in 2 out of 4 spins, you can expect an odd number.
Spinning an even number and spinning an odd number are equally likely events.
Event: Spinning an even number Favorable outcomes: 6 1 out of 4 possible outcomes are favorable, so in 1 out of 4 spins, you can expect an even number.
Event: Spinning an odd number Favorable outcomes: 5, 5, 7 3 out of 4 possible outcomes are favorable, so in 3 out of 4 spins, you can expect an odd number.
Spinning an even number is less likely than spinning an odd number. Spinning an odd number is more likely than spinning an even number.
Think about tossing a standard number cube.
1. What are the possible outcome numbers? Are the numbers equally likely? Explain.
2
1 3
The possible outcome numbers are 1, 2, 3, 4, 5, and 6. They are equally likely because there is one of each on the six faces of the number cube. 2. Number Sense In 6 tosses, how many times would you expect to toss the number 6? 5
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1 time
R e te a c h i n g
2 0 3
Topic 20
297
Practice
Name
20-3
Experiments and Predictions Write each of the following as a fraction in lowest terms.
1. 20 out of 60
1 __
2. 16 out of 64
3. 24 out of 60
1 __
2 __
4
3
5
The table below shows data from the woodshop classes at Jones Elementary School. Students had the choice of making a shelf, a chair, or a birdhouse. Project
Numberofstudents
Shelf
15
Chair
25
Birdhouse
20
4. What is the total number of students who were taking a woodshop class? 5. What is the probability that someone will make a chair? a birdhouse? 6. Predict how many students will make a shelf if there are 180 students in the woodshop class.
60
5 __ 1 ___ 12, 3
45
7. Reasonableness Using the letters in the word ELEMENTARY, find the probability of choosing a letter that is not E. 3 A ___ 10
B
5 ___ 10
7 C ___ 10
9 D ___ 10
3 0 2 e c ti c a r P
8. Explain It If _23 of the 45 customers in the past hour bought a cup of coffee, predict the number of cups that will be sold in the next 3 hours if sales continue at the same level.
90; if 30 are sold in 1 hour, 30 3 90 298 Topic 20
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Reteaching
Name
Problem Solving: Solve a Simpler Problem
20-4
A farmer has 4 garden plots in a row. He wants to build paths connecting each pair of plots. He also doesn’t want the paths to cross. How many paths must he build? Read and Understand
What doyouknow?
Thereare four plotsinarow. Each pair of plots must be connected by a path. The paths cannot cross.
What are you trying to find?
The total number of paths
Plan and Solve
Solve a simpler problem.
1. Construct a picture of two garden plots as shown below. 2. Then draw another picture of three plots. Look for a pattern for the design and number of paths. 3. Construct a diagram of the four plots.
A farmer has four garden plots arranged in a square. Look at the following diagrams. Use them to help answer this question. Can the four garden plots be connected by six paths that do not cross?
yes
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R e te a c h i n g
2 0 4
Topic 20
299
Name
Problem Solving: Solve a Simpler Problem
Practice 20-4
Solve the simpler problems. Use the solutions to help you solve the srcinal problem.
1. Reggie is designing a triangular magazine rack with 5 shelves. The top shelf will hold 1 magazine. The second shelf will hold 3 magazines, and the third shelf will hold 5 magazines. This pattern continues to the bottom shelf. How many magazines will the magazine rack hold altogether? Simpler Problem What is the pattern?
Each shelf down holds 2 more magazines than the shelf above. 7 magazines How many magazines will the fourth shelf hold? 9 magazines How many magazines will the bottom shelf hold? Solution:
The magazine rack will hold 25 magazines. 2. At the deli, you receive 1 free sub after you buy 8 subs. How many free subs will you receive from the deli if you buy 24 subs?
3 subs
3. The chef has 5 different kinds of pasta and 3 different flavors of sauce. How many different meals is she able to make?
15 different meals 4 0 2 e c ti c a r P
300 Topic 20
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Credits Pages I-III Taken from Scott Foresman-Addison Wesley Mathematics, Grade 5: Review From Last Year Mastersby Scott Foresman-Addison Wesley Pages 21-300 Taken from enVisionMATH™ Teacher Resource Masters,Topics 1-20, Grade 5 by Scott Foresman-Addison Wesley
Credits
301
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