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Copyright © by Pearson Education, Inc., or its affi liates. All rights reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. The publisher hereby grants permission to reproduce Practice Pages and Tests, in part or in whole, the number not to exceed the number of students in each class. For information regarding permissions, write to Pearson School Rights and Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458.
Pearson © is a trademark, in the U.S. and/or other countries, of Pearson plc or its affi liates.
Scott Foresman® and enVisionMATH™ are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc., or its affi liates.
Glenview, Illinois • Boston, Massachusetts Chandler, Arizona • Shoreview, Minnesota Upper Saddle River, New Jersey
Grade 3: Step Up to Grade 4Teacher’s Guide• Teacher Notes and Answers
for Step-Up Lessons
• Practice
• Answers for Practice
• Test
• Answers for Test
45095_SLPSHEET_FSD 145095_SLPSHEET_FSD 1 6/6/08 3:57:25 PM6/6/08 3:57:25 PM
F8 Rounding Numbers Through Thousands
F9 Comparing and Ordering Numbers Through Thousands
F10 Place Value Through Millions
F11 Rounding Numbers Through Millions
F29 Patterns and Equations
G7 Estimating Sums
G42 Dividing with Objects
G59 Factoring Numbers
G66 Mental Math: Multiplying by Multiples of 10
G67 Estimating Products
H1 Equal Parts of a Whole
H2 Parts of a Region
H3 Parts of a Set
H14 Equivalent Fractions
H18 Mixed Numbers
I8 Congruent Figures and Motions
I10 Solids and Nets
I11 Views of Solid Figures
I13 Congruent Figures
I45 More Perimeter
45095_SLPSHEET_FSD 245095_SLPSHEET_FSD 2 6/6/08 3:57:30 PM6/6/08 3:57:30 PM
Math Diagnosis and Intervention SystemIntervention Lesson F8
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Rounding Numbers Through Thousands
Teacher Notes
Ongoing AssessmentAsk: What does 99,249 round to when rounded to the nearest hundred thousand? 100,000
Error InterventionIf students have trouble remembering place value,
then have the students make a table across the top of their page that looks like the following:
Hun
dre
d
Thou
sand
s
Ten
Thou
sand
s
Thou
sand
s
Hun
dre
ds
Tens
One
s
If You Have More TimeHave students list all the four-digit numbers that stay the same when rounded to the nearest thousand (the multiples of 1,000), the five-digit numbers that stay the same when rounded to the nearest ten thousand (the multiples of 10,000), and the six-digit numbers that stay the same when rounded to the nearest hundred thousand (the multiples of 100,000).
Math Diagnosis and Intervention SystemIntervention Lesson F8
Rounding Numbers Through Thousands
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Intervention Lesson F8 71
Name
To round 34,853 to the nearest ten thousand, answer 1 to 6.
Materials 8 inches of yarn per pair
1. Plot 34,853 on the number line below.
2. Use the yarn to help you decide whether 34,853 is closer to 30,000 or 40,000. Which is it closer to?
3. So, what is 34,853 rounded to the nearest ten thousand?
4. Plot 37,421 on the number line above.
5. Use the yarn to help you decide whether 37,421 is closer to 30,000 or 40,000. Which is it closer to?
6. So, what is 37,421 rounded to the nearest ten thousand?
Round 739,218 to the nearest hundred thousand without using a number line. Answer 7 to 11.
7. Which digit is in the hundred thousands place?
8. What digit is to the right of the 7?
9. Is that digit less than 5, or is it 5 or greater?
If the digit to the right of the number is 5 or more, the number rounds up. If the digit is less than 5, the number rounds down.
10. Do you need to round up or down?
11. Keep the 7 and change the other digits to 0s. So, 739,218 rounds to .
30,000
30,000
40,000
40,000
7
3
less than 5
down
700,000
Round each number to the nearest ten.
12. 94,519 13. 537,681
Round each number to the nearest hundred.
14. 968,458 15. 58,326
Round each number to the nearest thousand.
16. 318,512 17. 21,236
Round each number to the nearest ten thousand.
18. 285,431 19. 184,032
Round each number to the nearest hundred thousand.
20. 735,901 21. 472,992
Use the information in the table for Exercises 22–23.
22. What is the price of the house rounded to the nearest hundred thousand?
23. What is the cost of repairs rounded to the nearest thousand?
24. Reasoning Write three numbers that would round to 44,000 when rounded to the nearest thousand.
25. Reasoning Explain how to round 496,275 to the nearest ten thousand.
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Rounding Numbers Through Thousands (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson F8
Real Estate
Sale
Amount
in Dollars
Price of House 169,256
Advertising Fee 7,177
Repairs 5,873
94,520 537,680
968,500 58,300
319,000 21,000
290,000 180,000
700,000 500,000
200,000
6,000
Any numbers between 43,500 and 44,499
Sample answer: The digit in the ten thousands place is 9. The digit to the right of 9 is 6, which is 5 or greater. So, round up. Think of adding 1 to 49 to make 50. To the nearest ten thousand, 496,275 rounds to 500,000.
Intervention Lesson F8
Math Diagnosis and Intervention SystemIntervention Lesson F9
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Comparing and Ordering Numbers Through Thousands
Teacher Notes
Ongoing AssessmentAsk: Why do you work from left to right instead of right to left when comparing or ordering numbers? Sample answer: The digits on the left have a higher value than the digits on the right.
Error InterventionIf students have trouble ordering numbers that are written horizontally,
then encourage them to write one number above the other, making sure they line up corresponding place values. Then compare the numbers in each column.
If You Have More TimeHave students write ordering problems for a partner to solve.
Math Diagnosis and Intervention SystemIntervention Lesson F9
Comparing and Ordering Numbers Through Thousands
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Intervention Lesson F9 73
Name
In a recent county election, Henderson received 168,356 votes. Juarez received 168,297 votes. Determine who received more votes by answering 1 to 7.
1. Write 168,356 and 168,297 in the place-value chart.
hundredthousands
tenthousands thousands hundreds tens ones
For Exercises 2–5, write �, �, or �.
2. Start with the left column in the chart. 100,000 100,000
3. Since the hundred thousands are equal, compare the ten thousands. 60,000 60,000
4. Since the ten thousands are equal, compare the thousands. 8,000 8,000
5. Since the thousands are equal, compare the hundreds. 300 200
6. Since 300 � 200, compare 168,356 and 168,297.
�
7. So, which candidate received more votes?
Order 346,217; 319,304; and 348,862 from least to greatest by answering 8 to 12.
8. Write 346,217; 319,304; and 348,862 in the place-value chart on the next page.
�
�
�
�
168,356 168,297
Henderson
1 6 8 3 5 61 6 8 2 9 7
hundredthousands
tenthousands thousands hundreds tens ones
9. Start on the left. Write �, �, or �. 300,000 300,000 300,000
10. Since the hundred thousands are all equal, compare the ten thousands. Since 10,000 � 40,000, what is the least number?
11. Since 6,000 _____ 8,000, compare the thousands place
of the other two numbers. �
12. The numbers in order from least to greatest are:
Use � or � to compare each pair of numbers.
13. 8,112 8,221 14. 418,412 481,930
15. 321,159 312,147 16. 20,657 21,687
17. 118,111 118,147 18. 914,146 904,168
Order the numbers from least to greatest.
19. 8,200; 820; 7,980 20. 12,984; 12,875; 11,987
21. 200; 12,945; 2,309 22. 321,984; 345,879; 323,490
23. Reasoning When comparing 17,834 and 17,934, can you start by comparing hundreds? Explain.
No; Always compare the left-most digit first.74 Intervention Lesson F9
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Comparing and Ordering Numbers Through Thousands (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson F9
� �
319,304
348,862346,217
<
348,862346,217319,304
3 4 6 2 1 73 1 9 3 0 43 4 8 8 6 2
< <
> <
< >
820; 7,980; 8,200 11,987; 12,875; 12,984
200; 2,309; 12,945 321,984; 323,490; 345,879
Intervention Lesson F9
Intervention Lesson F10
Math Diagnosis and Intervention SystemIntervention Lesson F10
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Place Value Through Millions
Teacher Notes
Ongoing AssessmentAsk: How does the number of zeros in the place value chart change as you move each place to the left? One zero is added to each place as you move left away from the ones.
Error InterventionIf students are having trouble remembering the number of zeros each value has,
then encourage students to write #00,000,000 above the heading hundred millions, #0,000,000 above the heading ten millions, #,000,000 above the heading millions, and so on. That way they know to place the number (#) and then the appropriate number of zeros.
If You Have More TimeHave students look up the population of the United States and write the number in standard form, expanded form, and word form.
Math Diagnosis and Intervention SystemIntervention Lesson F10
Place Value Through Millions
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Intervention Lesson F10 75
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1. Write 462,397,158 in the place-value chart below.
Millions Thousands Ones
Hu
nd
red
M
illi
on
s
Ten
M
illi
on
s
Mil
lio
ns
Hu
nd
red
Th
ou
san
ds
Ten
Th
ou
san
ds
Tho
usa
nd
s
Hu
nd
red
s
Ten
s
On
es
2. Complete the table to find the value of each digit in 462,397,158.
Digit Place Value
4 hundred millions 400,000,000
6 ten millions
2
3
9
7
1
5
8
3. Use the table above to help you write 462,397,158 in expanded form.
400,000,000 � � � �
90,000 � � 100 � � .
4. Write the short word form of 462,397,158.
462 million, thousand,
5. Write 462,397,158 in word form.
4
millionshundred thousandsten thousandsthousandshundredstensones
60,000,000
6 2 3 9 7 1 5 8
60,000,0002,000,000
300,00090,000
7,000100
508
2,000,000 300,000
7,000 50 8
397 158
Four hundred sixty-two million, three hundred ninety-seven thousand, one hundred fifty-eight
Write the value of the underlined digit.
6. 4,562,398 7. 15,347,025 8. 37,814,956
9. 526,878,953 10. 782,354,065 11. 918,403,760
Write each number in word form and in short word form.
12. 2,160,500
13. 91,207,040
14. 510,200,450
15. An underground rail system in Osaka, Japan carries 988,600,000 passengers per year. Write this number in expanded form.
16. Reasoning What number would make the numbersentence below true?
3,589,000 � 3,000,000 � ■ � 80,000 � 9,000
17. Reasoning What number can be added to 999,990 to make 1,000,000?
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Place Value Through Millions (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson F10
Two million, one hundred sixty thousand, five hundred; 2 million, 160 thousand, 5 hundred
Ninety-one million, two hundred seven thousand, forty; 91 million, 207 thousand, 40
Five hundred ten million, two hundred thousand, four hundred fifty; 510 million, 200 thousand, 4 hundred, 50
4,0005,000,00060,000
10,000,00060500,000,000
900,000,000 � 80,000,000 � 8,000,000 � 600,000
500,000
10
Intervention Lesson F11
Math Diagnosis and Intervention SystemIntervention Lesson F11
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Rounding Numbers Through Millions
Teacher Notes
Ongoing AssessmentAsk: What is the smallest number that will round to 1,000,000 when rounded to the nearest million? 950,000
Error InterventionIf students are having trouble identifying the place values for rounding,
then have the students make a place value chart across the top of their page.
If students do not know the place values,
then use F10: Place Value Through Millions.
If You Have More TimeHave students round 5,830,957 to the nearest ten, hundred, thousand, ten thousand, hundred thousand, and million. Repeat with 9,999,999.
Round 4,307,891 to the nearest million by answering 1 to 5.
1. What digit is in the millions place?
2. What digit is to the right of the 4?
3. Is the digit to the right of 4 less than 5, or is it 5 or greater?
If the digit to the right of the number is 5 or more, the number rounds up. If the digit is less than 5, the number rounds down.
4. Do you need to round up or down?
5. Keep the 4 and change the other digits to 0s. What is 4,307,891 rounded to the nearest million?
Round 6,570,928 to the nearest hundred thousand by answering 6 to 11.
6. Which digit is in the hundred thousands place?
7. What digit is to the right of the 5?
8. Is the digit to the right of 5 less than 5, or is it 5 or greater?
9. Do you need to round up or down?
10. Change the 5 to the next highest digit and change the other digits to 0s. What is 6,570,928 rounded to the nearest hundred thousand?
11. What is 6,570,928 rounded to the nearest thousand?
Intervention Lesson F11 77
Math Diagnosis and Intervention SystemIntervention Lesson F11
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Rounding Numbers Through Millions
4
3
less than 5
Down
4,000,000
5
7
5 or greater
Up
6,600,000
6,571,000
Round 1,581,267 to each place.
12. ten 13. hundred
14. thousand 15. ten thousand
16. hundred thousand 17. million
Round each number to the nearest ten.
18. 3,194,764 19. 8,967,001
Round each number to the nearest hundred.
20. 1,265,906 21. 6,906,294
Round each number to the nearest thousand.
22. 8,070,126 23. 9,264,431
Round each number to the nearest ten thousand.
24. 7,514,637 25. 2,437,894
Round each number to the nearest hundred thousand.
26. 1,395,384 27. 3,992,460
Round each number to the nearest million.
28. 4,578,952 29. 5,022,121
30. 2,439,019 31. 8,888,888
32. Reasoning A number rounded to the nearest million is 4,000,000. One less than the same number rounds to 3,000,000 when rounded to the nearest million. What is the number?
Rounding Numbers Through Millions (continued)
78 Intervention Lesson F11
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Name
Math Diagnosis and Intervention SystemIntervention Lesson F11
3,194,760 8,967,000
1,265,900 6,906,300
8,070,000 9,264,000
7,510,000 2,440,000
1,400,000 4,000,000
5,000,000 5,000,000
2,000,000 9,000,000
1,581,270 1,581,300
1,581,000
1,600,000
1,580,000
2,000,000
3,500,000
Intervention Lesson F29
Math Diagnosis and Intervention SystemIntervention Lesson F29
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Patterns and Equations
Teacher Notes
Ongoing AssessmentAsk: If a rule for a table is x � 7, what equation represents the relationship between the inputs x and the outputs y? y � x � 7
Error InterventionIf students have trouble writing an expression for the pattern in the table,
then use F26: Expressions with Addition and Subtraction or F27: Expressions with Multiplication and Division.
If You Have More TimeHave students think of a real-world situation which can be represented by an equation, write the equation, and create a table.
Math Diagnosis and Intervention SystemIntervention Lesson F29
Patterns and Equations
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Intervention Lesson F29 113
Name
1. Complete the table at the right. Cost of Food Order
c
Cost with Delivery
d
$8.50 $10.50
$9.25 $11.25
$10.47 $12.47$10.98 $12.98
$12.06 $14.06
2. What expression describes the cost with delivery for a food order costing c dollars?
c � 2
3. Set d equal to the expression to get an equation representing the relationship between the cost of the food order cand the cost with delivery d.
d � c � 2
4. Use the equation to find d, when c � $14.52. $16.52
5. Complete the table at the right. Number of Pretzels
p
Total Cost
c
2 $11.00
3 $16.50
5 $27.506 $33.00
7 $38.50
6. Write an equation representing the relationship between the number of pretzels p and the total cost c.
c � 5.5p
7. Use the equation to find c, when p � 4.
$22.00
8. A field goal in football is worth 3 points. Complete the table below to show the relationship between the number of field goals f and the number of points p.
Sample answers are shown in the table.
Field Goals (f) 1 2 3 4 5Points (p) 3 6 9 12 18
9. Write an equation to represent the relationship between the number of field goals f and points p. p � 3f
Write an equation to represent the relationship between x and y in each table. Then use the equation to complete the table.
10. x y
2 8
3 9
8 1414 20
21 27
11.x y
12 2
18 3
24 4
30 542 7
12. x y
4 32
5 40
7 568 64
10 80
y � x � 6 y � x � 6 y � 8x
13. x y
10 1
14 5
22 13
26 17
30 21
14.x y
2 4.8
3 7.2
4 9.6
6 14.47 16.8
15. x y
50 10
40 835 7
25 5
15 3
y � x � 9 y � 2.4x y � x � 5
16. x 2 2.5 3.2 3.8 4.7
y 9.8 10.3 11 11.6 12.5
y � x � 7.8
17. Reasoning If a rule is c � p � 5, what is c when p � 20? 25
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Patterns and Equations (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson F29
Intervention Lesson G7
Math Diagnosis and Intervention SystemIntervention Lesson G7
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Estimating Sums
Teacher Notes
Ongoing AssessmentAsk: When estimating 124 � 138, which would give a closer estimate, rounding to the nearest ten or to the nearest hundred? Rounding to the nearest ten.
Error InterventionIf students are having trouble with rounding the addends correctly,
then use F2: Rounding to Nearest Ten and Hundred.
If You Have More TimeHave students work with a partner to list ten different sums which are about 300 when estimated by rounding each addend to the nearest hundred.
Math Diagnosis and Intervention SystemIntervention Lesson G7
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Intervention Lesson G7 91
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Estimating Sums
When Joseppi added 43 and 28, he got a sum of 71. To check that this answer is reasonable, use estimation.
1. Round each addend to the nearest ten.
43 rounded to the nearest ten is .
28 rounded to the nearest ten is .
2. Add the rounded numbers.
40 � 30 �
Since 71 is close to 70, the answer is reasonable.
When Ling added 187 and 242, she got a sum of 429. To check that this answer is reasonable, use estimation.
3. Round each addend to the nearest hundred.
187 rounded to the nearest hundred is .
242 rounded to the nearest hundred is .
4. Add the rounded numbers.
200 � 200 �
Since 429 is close to 400, the answer is reasonable.
40
70
200
30
200
400
92 Intervention Lesson G7
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Estimating Sums (continued)
Estimate by rounding to the nearest ten.
5. 71 � 36 6. 24 � 81 7. 43 � 91 8. 54 � 66
9. 68 � 27 10. 19 � 93 11. 89 � 75 12. 54 � 33
Estimate by rounding to the nearest hundred.
13. 367 14. 791 15. 506 16. 458 _� 141 _� 632 _� 249 _� 891
17. 940 � 190 18. 675 � 460 19. 531 � 776
20. 369 � 481 21. 151 � 260 22. 705 � 936
23. Reasoning Jaimee was a member of the school chorus for 3 years. Todd was a member of the school band for 2 years. The chorus has 43 members and the band has 85 members. About how many members do the two groups have together?
24. Luis sold 328 sport bottles and Jorge sold 411. About how many total sport bottles did the two boys sell?
25. Reasoning What is the largest number that can be added to 46 so that the sum is 70 when both numbers are rounded to the nearest ten? Explain.
110
24; Since 46 rounds to 50, and 20 � 50 � 70, you need the largest number that rounds to 20, which is 24.
100 130 120
100 110 170 80
1,3001,2001,100
1,600500900
130
500 1,400 700 1,400
700
Intervention Lesson G42
Math Diagnosis and Intervention SystemIntervention Lesson G42
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Dividing with Objects
Teacher Notes
Ongoing AssessmentAsk: Why should the remainder always be less than the divisor? For example, when dividing 10 by 3, why can you not have a remainder of 4? If the remainder is equal to or greater than the divisor, then the quotient should be larger. For example, when dividing 10 counters into 3 equal groups, if 4 are left, there are enough to put another counter into each group.
Error InterventionIf students have trouble completing problems using counters or drawings,
then encourage them to use repeated subtraction to solve the problem. Continue to subtract until it can no longer be done. What is left is the remainder.
If You Have More TimeHave partners find all the division sentences that can be written if 8 is the dividend and the divisor is 1 to 8.
Math Diagnosis and Intervention SystemIntervention Lesson G42
Dividing with Objects
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Intervention Lesson G42 161
Name
Materials 7 counters and 3 half sheets of paper for each student or pair
Andrew has 7 model cars to put on 3 shelves. He wants to put the same number of cars on each shelf. How many cars should Andrew put on each shelf? Answer 1 to 8.
Find 7 � 3.
1. Show 7 counters and 3 sheets of paper.
2. Put 1 counter on each piece of paper.
3. Are there enough counters to putanother counter on each sheet of paper? yes
4. Put another counter on each piece of paper.
5. Are there enough counters to putanother counter on each sheet of paper? no
6. How many counters are on each sheet? 2
7. How many counters are remaining, or left over? 1
So, 7 � 3 is 2 remainder 1, or 7 � 3 � 2 R1.
8. How many cars should Andrew put on each shelf? How many cars will be left over?
Andrew can put 2 cars on each shelf with 1 car left over.
Use counters or draw a picture to find each quotient and remainder.
9. 8 � 3 � 2 R2 10. 17 � 3 � 5 R2 11. 14 � 3 � 4 R2
12. 11 � 4 � 2 R3 13. 22 � 4 � 5 R2 14. 34 � 4 � 8 R2
15. 13 � 5 � 2 R3 16. 27 � 5 � 5 R2 17. 46 � 5 � 9 R1
18. 14 � 6 � 2 R2 19. 26 � 6 � 4 R2 20. 38 � 6 � 6 R2
21. 17 � 7 � 2 R3 22. 27 � 7 � 3 R6 23. 45 � 7 � 6 R3
24. 18 � 8 � 2 R2 25. 28 � 8 � 3 R4 26. 37 � 8 � 4 R5
27. 14 � 9 � 1 R5 28. 28 � 9 � 3 R1 29. 39 � 9 � 4 R3
30. 12 � 8 � 1 R4 31. 68 � 7 � 9 R5 32. 59 � 8 � 7 R3
33. 9 � 4 � 2 R1 34. 26 � 5 � 5 R1 35. 34 � 6 � 5 R4
36. 14 � 5 � 2 R4 37. 21 � 6 � 3 R3 38. 3 � 2 � 1 R1
39. Reasoning Grace is reading a book for school. The book has 26 pages and she is given 3 days to read it. How many pages should she read each day? Will she have to read more pages on some days than on others? Explain.
She should read 8–9 pages each day; she will read 9 pages on 2 days and 8 pages on 1 day.
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Dividing with Objects (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson G42
Intervention Lesson G59
Math Diagnosis and Intervention SystemIntervention Lesson G59
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Factoring Numbers
Teacher Notes
Ongoing AssessmentAsk: What is the only even prime number? 2
Error InterventionIf students do not find all of the factors of a number,
then have them use manipulatives such as color tiles or counters to create arrays. Have them find all arrays methodically. Start with 1 row. See if the given number of counters can be put into 2 rows. Then see if they can be put into 3 rows and so on.
If You Have More TimeHave students write any number from 1 to 1,000 on a note card. Collect all of the cards and shuffle them. Have a student draw a card and explain why the number on the cards is either prime or composite. Repeat until all students have a turn.
Answers: Exercise 22 from page 196
6 � 3 � 18 2 � 9 � 18 3 � 6 � 18
1 � 18 � 18
9 � 2 � 18
Math Diagnosis and Intervention SystemIntervention Lesson G59
Factoring Numbers
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Intervention Lesson G59 195
Name
Materials color tiles or counters, 24 for each student
The arrays below show all of the factors of 12.
1 � 12
2 � 6 3 � 4 4 � 3 6 � 2
1. What are all the factors of 12?
1 , 2 , 3 , 4 , 6 , 12
2. Create all the possible arrays you can with 17 color tiles.
3. What are the factors of 17? 1 , 17
Numbers which have only 2 possible arrays and exactly 2 factors are prime numbers.
4. Is 17 a prime number? yes
5. Is 12 a prime number? no
Numbers which have more than 2 possible arrays and more than 2 factors are composite numbers.
6. Is 17 a composite number? no
7. Is 12 a composite number? yes
8. What are all the factors of 24? 1, 2, 3, 6, 8, 12, 24
9. Is 24 a prime number or a composite number? composite
12 � 1
The order factors are listed may vary.
The order factors are listed may vary.
196 Intervention Lesson G59
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Name
Math Diagnosis and Intervention SystemIntervention Lesson G59
Find all the factors of each number. Tell whether each is prime or composite.
10. 7 11. 8 12. 21
1, 7; 1, 2, 4, 8; 1, 3, 7, 21;
prime composite composite
13. 48 14. 51 15. 9
1, 2, 3, 4, 6, 8, 1, 3, 17, 51; 1, 3, 9;
12, 16, 24, 48; composite composite
16. 13 17. 26 18. 40
1, 13; 1, 2, 13, 26; 1, 2, 4, 5, 8,
prime composite 10, 20, 40;
19. 55 20. 70 21. 83
1, 5, 11, 55; 1, 2, 5, 7, 10, 1, 83;
composite 14, 35, 70; prime
22. Mr. Lee has 18 desks in his room. He would like them arranged in a rectangular array. Draw all the different possible arrays and write a multiplication sentence for each.
See answers on page 59.
23. Reasoning Lee says 53 is a prime number because it is an odd number. Is Lee’s reasoning correct? Give an example to prove your reasoning. No; 53 is a prime number not because it is an odd number, but because it only has 2 factors: 1 and 53. A counterexample is 15 is an odd number but it is not a prime number because 15 has more than 2 factors: 1, 3, 5, 15.
composite
composite
composite
18 � 1 � 18
Intervention Lesson G66
Math Diagnosis and Intervention SystemIntervention Lesson G66
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Mental Math: Multiplying by Multiples of 10
A publishing company ships a particular book in boxes with 6 books each. How many books are in 20 boxes? How many in 2,000 boxes?
Find 20 � 6 and 2,000 � 6 by filling in the blanks.
1. 20 � 6 � (10 � 2) � 6 2. 2,000 � 6 � (1,000 � 2) � 6
� 10 � ( 2 � 6) � 1,000 � ( 2 � 6)
� 10 � 12 � 1,000 � 12
� 120 � 12,000
3. How many books are in 20 boxes? 120 books
4. How many books are in 2,000 boxes? 12,000 books
The same publishing company ships a smaller book in boxes with 40 books each. How many books are in 50 boxes? How many are in 500 boxes?
Find 50 � 40 and 500 � 40 by filling in the blanks.
5. 50 � 40 � (5 � 10 ) � 6. 500 � 40 � (5 � 100 ) �
(4 � 10 ) (4 � 10 )
� 5 � 4 � 10 � 10 � 5 � 4 � 100 � 10
� 20 � 100 � 20 � 1,000
� 2,000 � 20,000
7. How many books are in 50 boxes? 2,000 books
8. How many books are in 500 boxes? 20,000 books
Intervention Lesson G66 209
Math Diagnosis and Intervention SystemIntervention Lesson G66
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Mental Math: Multiplying by Multiples of 10
Notice the pattern when multiplying multiples of 10.
9. 7 � 80 � 560 10. 4 � 60 � 240
70 � 80 � 5,600 40 � 60 � 2,400
70 � 800 � 56,000 40 � 600 � 24,000
Multiply.
11. 30 � 40 � 12. 10 � 600 � 13. 70 � 20 �
1,200 6,000 1,400
14. 50 � 400 � 15. 700 � 30 � 16. 40 � 800 �
20,000 21,000 32,000
17. 600 � 30 � 18. 40 � 90 � 19. 90 � 500 �
18,000 3,600 45,000
20. 70 � 500 � 21. 30 � 800 � 22. 200 � 70 �
35,000 24,000 14,000
23. 800 � 80 � 24. 30 � 600 � 25. 40 � 300 �
64,000 18,000 12,000
26. A class of 30 students is collecting pennies for a school fundraiser. If each of them collects 400 pennies, how many have they collected all together? 12,000 pennies
27. Reasoning Raul multiplied 60 � 500 and got 30,000. Since there are 4 zeros in the answer, he thought his answer was incorrect? Do you agree? Why or why not?
No, one of the zeros is from 6 � 5 which is 30.
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Mental Math: Multiplying by Multiples of 10 (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson G66
Teacher Notes
Ongoing AssessmentAsk: Will the number of zeros in the product always be the same as the sum of the number of zeros in the factors? No Why not? Sometimes the leading numbers in the factors will multiply to make another zero such as 4 � 5 � 20.
Error InterventionIf students are making mistakes with multiplication facts,
then use some of the intervention lessons on basic multiplication facts, G25 to G32.
If You Have More TimeHave students write and solve a word problem that involves multiplying multiples of ten.
Intervention Lesson G67
Math Diagnosis and Intervention SystemIntervention Lesson G67
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Estimating Products
Math Diagnosis and Intervention SystemIntervention Lesson G67
Estimating Products
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Intervention Lesson G67 211
Name
Mrs. Wilson’s class at Hoover Elementary School is collecting canned goods. Their goal is to collect 600 cans. There are 21 students in the class and each student agrees to bring in 33 cans. Answer 1 to 7 to find if the class will meet their goal.
Estimate 21 � 33 and compare the answer to 600.
Round each factor to get numbers you can multiply mentally.
1. What is 21 rounded to the nearest ten? 20
2. What is 33 rounded to the nearest ten? 30
3. Multiply the rounded numbers. 20 � 30 � 600
The answer is the same as the number needed to meet the goal.
4. 21 was rounded to 20. Was it rounded up or down? down
5. 33 was rounded to 30. Was it rounded up or down? down
6. Is 21 � 33 more or less than 20 � 30? more
7. Will the goal be reached? yes
Hoover Elementary School had a goal to collect 12,000 canned goods. There are 18 classes and each class collects 590 cans. Answer 8 to 13 to find if the school will meet their goal.
Estimate 18 � 590 and compare the answer with 12,000.
Round each factor to get numbers you can multiply mentally.
8. What is 18 rounded to the nearest ten? 20
9. What is 590 rounded to the nearest hundred? 600
10. Multiply the rounded numbers. 20 � 600 � 12,000
The answer is the same as the number needed to meet the goal.
11. 18 was rounded to 20. Was it rounded up or down? up
590 was rounded to 600. Was it rounded up or down? up
12. Is 18 � 590 more or less than 20 � 600? less
13. Will the goal be reached? no
Round each factor so that you can estimate the product mentally.
14. 71 � 382 15. 27 � 62 16. 45 � 317
70 � 400 30 � 60 50 � 300
28,000 1,800 15,000
17. 58 � 176 18. 831 � 42 19. 16 � 768
60 � 200 800 � 40 20 � 800
12,000 32,000 16,000
20. 87 � 67 21. 373 � 95 22. 57 � 722
90 � 70 400 � 100 60 � 700
6,300 40,000 42,000
23. Debra spends 42 minutes each day driving to work. About how many minutes does she spend driving to work each month? 1,200 minutes
24. Reasoning If 64 � 82 is estimated to be 60 � 80, would the estimate be an overestimate or an underestimate? Explain.
It would be an underestimate because you are rounding both factors down.
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Estimating Products (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson G67
Teacher Notes
Ongoing AssessmentAsk: Why is it preferable to round 591 to the nearest hundred rather than the nearest ten? It is easier to multiply 600 than 590.
Error InterventionIf students have problems rounding numbers,
then use F2: Rounding to Nearest Ten and Hundred and F4: Numbers Halfway Between and Rounding.
If You Have More TimeHave students name situations when an estimate if sufficient and situations when an exact answer is needed.
Intervention Lesson H1
Math Diagnosis and Intervention SystemIntervention Lesson H1
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Equal Parts of a Whole
Teacher Notes
Ongoing AssessmentAsk: Looking at the names for shapes divided into 4, 5, 6, 8, 10, and 12 equal parts, what might be the name of a shape divided into seven equal parts? sevenths
Error InterventionIf children have trouble understanding the concept of equal parts,
then use A35: Equal parts.
If You Have More TimeHave students fold other rectangular sheets of paper and circular pieces of paper to find and name other equal parts.
Intervention Lesson H1 85
Math Diagnosis and Intervention SystemIntervention Lesson H1
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Equal Parts of a Whole
Materials rectangular sheets of paper, 3 for each student; crayons or markers
1. Fold a sheet of paper so the two shorter edges fold
are on top of each other, as shown at the right.
2. Open up the piece of paper. Draw a line down the fold. Color each part a different color.
The table below shows special names for the equal parts. All parts must be equal before you can use these special names.
3. Are the parts you colored equal in size? yes
4. How many equal parts are there? 2 Number of Equal Parts
Name of Equal Parts
2 halves
3 thirds
4 fourths
5 fifths
6 sixths
8 eighths
10 tenths
12 twelfths
5. What is the name for the parts you colored?
halves
6. Fold another sheet of paper like above. Then fold it again so that it makes a long slender rectangle as shown below.
7. Open up the piece of paper. Draw lines down the folds. Color each part a different color.
8. Are the parts you colored equal in size? yes
9. How many equal parts are there? 4
10. What is the name for the parts you colored?
Newfold
Oldfold
fourths
11. Fold another sheet of paper into 3 parts that are not equal. Open it and draw lines down the folds. In the space below, draw your rectangle and color each part a different color.
Check that students draw unequal parts.
Tell if each shows parts that are equal or parts that are not equal. If the parts are equal, name them.
12. 13.
equal
not equal
fourths
14. 15.
equal
equal
thirds eighths
16. 17.
equal
not equal
twelfths
18. 19.
not equal
equal
fifths
20. 21.
equal
not equal
halves
22. 23.
equal
not equal
sixths
24. Reasoning If 5 children want to equally share a large pizza and each gets 2 pieces, will they need to cut the pizza into fifths, eighths, or tenths? tenths
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Equal Parts of a Whole (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson H1
Intervention Lesson H2
Math Diagnosis and Intervention SystemIntervention Lesson H2
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Parts of a Region
Teacher Notes
Ongoing AssessmentAsk: Janet said she ate 4 __
4 of an orange. Explain
why Janet could have said she ate the whole
orange. Sample answer: The orange would be cut
in 4 pieces and she ate 4 pieces, so she ate the
whole thing.
Error InterventionIf children have trouble writing fractions for parts of a region,
then use A36: Understanding Fractions to Fourths and A38: Writing Fractions for Part of a Region.
If You Have More TimeHave students design a rectangular flag (or rug, placemat, etc.) that is divided into equal parts. Have them color their flag and then on the back write the fractional parts of each color.
Math Diagnosis and Intervention SystemIntervention Lesson H2
Parts of a Region
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Intervention Lesson H2 87
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Materials crayons or markers
1. In the circle at the right, color 2 of the equal parts blue and 4 of the equal parts red.
Write fractions to name the parts by answering 2 to 6.
2. How many total equal parts does the circle have? 6
3. How many of the equal parts of the circle are blue? 2
4. What fraction of the circle is blue?
2 ____
6 �
number of equal parts that are blue _____________________________ total number of equal parts � (numerator) ____________ (denominator)
Two sixths of the circle is blue.
5. How many of the equal parts of the circle are red? 4
6. What fraction of the circle is red?
4 ____
6 �
number of equal parts that are red ____________________________ total number of equal parts � (numerator) ____________ (denominator)
Four sixths of the circle is red.
Show the fraction 3 __ 8 by answering 7 to 9.
7. Color 3 __ 8 of the rectangle at the right.
8. How many equal parts doesthe rectangle have? 8
9. How many parts did youcolor? 3
Write the fraction for the shaded part of each region.
10. 11. 12.
2__3
1__4
4__5
13. 14. 15.
1__2
2__8
2__5
16. 17. 18.
1__3
3__5
5__8
Color to show each fraction.
19. 3 __ 4 20. 5 __ 6 21. 7 ___ 10
22. Math Reasoning Draw a picture to show 1 __ 3 . Then divide
each of the parts in half. What fraction of the parts does
the 1 __ 3 represent now? Check students’ drawings. 2__6
23. Ben divided a pie into 8 equal pieces and ate 3 of them. How much of the pie remains?
5__8
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Parts of a Region (continued)
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Math Diagnosis and Intervention SystemIntervention Lesson H2
Intervention Lesson H3
Math Diagnosis and Intervention SystemIntervention Lesson H3
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Parts of a Set
Math Diagnosis and Intervention SystemIntervention Lesson H3
Parts of a Set
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Intervention Lesson H3 89
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Materials two-color counters, 20 for each pair; crayons or markers
1. Show 4 red counters and 6 yellow counters.
Write a fraction for the part that is red and the part that is yellow by answering 2 to 6.
2. How many counters are there in all? 10
3. How many of the counters are red? 4
4. What fraction of the group of counters is red?
4 ____
10 � number of counters that are red __________________________ total number of counters � (numerator) ____________ (denominator)
Four tenths of the group of counters is red
5. How many of the counters are yellow? 6
6. What fraction of the counters are yellow?
6 ____
10 �
number of counters that are yellow _____________________________ total number of counters � (numerator) ____________ (denominator)
Six tenths of the group of counters is yellow.Answer can vary on 7 and 8. Another correct set of answers is 12 and 10.
Show a group of counters with 5 __ 6 red by answering 7 to 9.
7. How many counters do you need in all? 6
8. How many of the counters need to be red? 5Check that students color 5 circles red.
9. Show the counters and color below to match.
Write the fraction for the shaded parts of each set.
10. 11.
2__3
1__2
12. 13.
3__4
4__5
14. 15.
2__5
4__6
Draw a set of shapes and shade them to show each fraction.
16. 5 __ 9 17. 6 ___ 10
Check students’ drawings.
18. Reasoning If Sally has 5 yellow marbles and 7 blue marbles. What fraction of her marbles are yellow? Draw a picture to justify your answer.5__12
; Students should draw 12 marbles with 5 blue.
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Parts of a Set (continued)
Name
Math Diagnosis and Intervention SystemIntervention Lesson H3
Teacher Notes
Ongoing AssessmentAsk: What does it mean that 3 ___
12 of a group of
marbles are green? There are 3 green marbles out
of a total of 12 marbles.
Error InterventionIf children have difficulty describing parts of a set,
then use A37: Fractions of a Set and A39: Writing Fractions for Part of a Set.
If You Have More Time
Have students look around the classroom and
name different fractions they see. For example: 6 ___ 18
of the students have their math book out; 2 __ 5 of the
computers are turned off; 5 __ 9 of the boys have on
shorts.
Intervention Lesson H14
Math Diagnosis and Intervention SystemIntervention Lesson H14
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Equivalent Fractions
Math Diagnosis and Intervention SystemIntervention Lesson H14
Equivalent Fractions
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Intervention Lesson H14 111
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Materials crayons or markers 1
1 __ 3 1 __ 3 1 __ 3
1 __ 6 1 __ 6 1 __ 6 1 __ 6 1 __ 6 1 __ 6
1. Show 2 __ 3 by coloring 2 of the 1 __ 3 strips.
2. Color as many 1 __ 6 strips as it takes
to cover the same region as the 2 __ 3 .
How many 1 __ 6 strips did you color? 4
3. So, 2 __ 3 is equivalent to four 1 __ 6 strips. 2 __ 3 � 4 ____ 6
You can use multiplication to find a fraction equivalent to 2 __ 3 . To do this, multiply the numerator and the denominator by the same number.
4. What number is the denominator � 2
2 __ 3 � 4 ____ 6
� 2
of 2 __ 3 multiplied by to get 6?
2
5. Since the denominator was multiplied by 2, the numerator must also be multiplied by 2. Put the product of 2 � 2 in the numerator of the second fraction above.
Multiply the numerator and denominator of each fraction by the same number to find a fraction equivalent to each.
6. � 3
2 __ 3 � 6 ____ 9
� 3
7. � 4
1 __ 2 � 4 ____ 8
� 4
8. Show 9 ___ 12 by coloring 9 of the 1 ___ 12 strips. 1
1 __ 4 1 __ 4 1 __ 4 1 __ 4
1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12
9. Color as many 1 __ 4 strips as it takes
to cover the same region as 9 ___ 12 .
How many 1 __ 4 strips did you color? 3
10. So, 9___12 is equivalent to three 1__
4 strips. 9___12 �
3____4
You can use division to find a fraction equivalent to 9___12 . To do this,
divide the numerator and the denominator by the same number.
11. What number is the denominator of � 3
9 ___ 12 � 3 ____ 4
� 3
9___12 divided by to get 4? 3
12. Since the denominator was divided by 3, the numerator must also be divided by 3. Put the quotient of 9 � 3 in the numerator of the second fraction above.
Divide the numerator and denominator of each fraction by the same number to find a fraction equivalent to each.
13. � 2
8 ___ 10 � 4 ____ 5
� 2
14. � 5
10 ___ 15 � 2 ____ 3
� 5
If the numerator and denominator cannot be divided by anything else, then the fraction is in simplest form.
15. Is 5___12 in simplest form? yes 16. Is 6__
8 in simplest form? no
Find each equivalent fraction.
17. 1__5 �
3____15 18. 8___
10 �4____5 19. 2__
8 �1____4
20. 7___10 �
14____20 21. 6___
14 �3____7 22. 8___
11 �16____22
Write each fraction in simplest form.
23. 6__8
3__4 24. 8___
12
2__3 25. 7___
35
1__5 26. 16___
24
2__3
27. Reasoning Explain why 4__6 is not in simplest form.
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Equivalent Fractions (continued)
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Math Diagnosis and Intervention SystemIntervention Lesson H14
Teacher Notes
Ongoing Assessment
Ask: Explain how you can tell 3 __ 7 is in simplest
form. 3 and 7 have only 1 as a common factor.
Error InterventionIf students do not understand how two fractions can be equivalent,
then use H7: Using Models to Find Equivalent Fractions.
If You Have More Time
Have students create and number a cube 2, 2, 3, 3,
6, and 6. Label 8 index cards: 1 __ 2 , 2 __
3 , 4 __
6 , 6 ___
18 , 6 ___
24 , 6 ___
10 , 6 ___
12 ,
and 3 __ 4 . Shuffle the index cards. One student rolls the
number cube and the other draws an index card.
Both students find a fraction equivalent to the one
on the index card by either multiplying or dividing
by the number rolled. Have students check each
others’ work. Sometimes either operation can be
done, sometimes only one can be used.
Intervention Lesson H18
Math Diagnosis and Intervention SystemIntervention Lesson H18
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Mixed Numbers
Teacher Notes
Ongoing AssessmentAsk: What is 15 ___
4 written as a mixed number? 3 3 __
4
Error InterventionIf students have difficulty dividing when converting from an improper fraction to a mixed number,
then use G42: Dividing with Objects and H15: Fractions and Division.
If students are changing a mixed number to an improper fraction and the students multiply the whole number and the numerator,
then, have the students write the word DoWN on their paper to remind them of the correct procedure: (D � W) � N.
If You Have More TimeHave students work in pairs. Have each student write 5 mixed numbers on index cards, one number per card. Then have students each write an improper fraction on an index card to match each mixed number written by the partner. Then the pair can play a memory game. Have the students shuffle the cards and lay them face down in an array. Have one student flip over two cards. If the cards are a match, then that player keeps the cards and can take another turn. If the cards are not a match, then the cards are turned back over and the other student has a chance to find a match. The game continues until all of the matches are found. The player with the most matches wins.
Math Diagnosis and Intervention SystemIntervention Lesson H18
Mixed Numbers
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Intervention Lesson H18 119
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Materials fraction strips
A mixed number is a number written with a whole number and a fraction. An improper fraction is a fraction in which the numerator is greater than or equal to the denominator.
1. Circle the number that is a mixed number. 3 4 1 __ 3 8 __ 5
2. Circle the number that is an improper fraction. 3 __ 4 2 3 __ 5 9 __ 4
Write the improper fraction 7 __ 3 as a mixed number by answering 3 to 6.
3. Show seven 1 __ 3 fraction strips.
4. Use fraction strips. How many 1 strips can you make with
seven 1 __ 3 strips? 2
5. How many 1 __ 3 strips do you
have left over? 1
6. Write 7 __ 3 as a mixed
number. 21__
3
Write 7 __ 3 as a mixed number without fraction strips by answering 7 to 9.
7. Divide 7 by 3 at the right. 2 R1 3 � � 7
� 6
1
8. Fill in the missing numbers below.
Quotient 1
2 3
Remainder
Divisor
Notice, the quotient 2 tells how many one strips you can make.
The remainder 1 tells how many 1 __ 3 strips are left over.
9. Write 7 __ 3 as a mixed number. 21__
3
10. Since 14 � 5 � 2 R4, what is 14___5 as a mixed number?
24__5
Write 2 1__5 as an improper fraction by answering 11 to 13.
11. Show 2 1__5 with
fractions strips.
12. Use fraction
strips. How many 1__5 strips does it
take to equal 2 1__5?
11
13. What improper fraction equals 2 1__5 ?
11__5
Write 2 1__5 as an improper fraction without using fraction strips by
answering 14 to 16.
14. Fill in the missing numbers.
Whole Number � Denominator � Numerator
2 � 5 � 1 � 10 � 1 � 11
15. Write the 11 you found above over the denominator, 5.
11__5
16. What improper fraction equals 2 1__5 ?
11__5
Notice, 2 � 5 tells how many 1__5 strips equal 2 wholes. The 1 tells
how many additional 1__5 strips there are.
17. Since 6 � 4 � 3 � 27, what is 6 3__4 as an improper fraction?
27__4
Change each improper fraction to a mixed number or a whole number and change each mixed number to an improper fraction.
18. 3 2__7
23__7 19. 7__
3
21__3 20. 7__
2
31__2 21. 1 1___
10
11__10
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Mixed Numbers (continued)
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Math Diagnosis and Intervention SystemIntervention Lesson H18
Intervention Lesson I8
Math Diagnosis and Intervention SystemIntervention Lesson I8
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Congruent Figures and Motions
Teacher Notes
Ongoing AssessmentAsk: Why are all squares not congruent? Congruent figures must have the same size and shape. Squares can be different sizes.
Error InterventionIf students have trouble understanding congruency,
then use D54: Same Size, Same Shape.
If students have trouble differentiating between slides, flips, and turns,
then use D55: Ways to Move Shapes.
If students understand congruency, but have trouble deciding if two figures are congruent,
then have students trace one of the figures and place the tracing over the other figure to see if it has the same size and shape.
If You Have More TimeHave students write their name using letters that have been flipped, turned, or slid. Exchange with a partner and have the partner identify what motion was used on each letter. Show the students that some letters can look like a slide and a flip. For example, the letter “I” looks the same when it is flipped and slid to the right.
Math Diagnosis and Intervention SystemIntervention Lesson I8
Congruent Figures and Motions
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Intervention Lesson I8 105
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Materials construction paper, markers, and scissors
Follow 1–10.
1. Cut a scalene triangle out of construction paper.
2. Place your cut-out triangle on the bottom left side of another piece of contruction paper. Trace the triangle
Slide
with a marker.
3. Slide your cut-out triangle to the upper right of the same paper and trace the triangle again.
4. Look at the two triangles that you just traced. Are the two triangles the same size and shape? yes
When a figure is moved up, down, left, or right, the motion is called a slide, or translation.
Figures that are the exact same size and shape are called congruent figures.
5. On a new sheet of paper, draw a straight dashed line as Flipshown at the right. Place your cut-out triangle on the left side of the dashed line. Trace the triangle with a marker.
6. Pick up your triangle and flip it over the dashed line, like you were turning a page in a book. Trace the triangle again.
7. Look at the two triangles that you just traced. Are the two triangles congruent? yes
When a figure is picked up and flipped over, the motion is called a flip, or reflection.
8. On a new sheet of paper, draw a point in the middle of Turn
the paper. Place a vertex of your cut-out triangle on the point. Trace the triangle with a marker.
9. Keep the vertex of your triangle on the point and move the triangle around the point like the hands on a clock. Trace the triangle again.
10. Look at the two triangles you just traced. Are the two triangles congruent? yes
When a figure is turned around a point, the motion is a turn, or rotation.
Write slide, flip, or turn for each diagram.
11. 12. 13.
slide flip turn
14. 15. 16.
turn flip slide
For Exercises 17 and 18, use the figures to the right.
17. Are Figures 1 and 2 related by a slide, a flip, or a turn? slide
18. Are Figures 1 and 3 related by a slide, a flip, or a turn? flip
19. Reasoning Are the polygons at the right congruent? If so, what motion could be used to show it?
Yes; one is a turn of the other.
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Math Diagnosis and Intervention SystemIntervention Lesson I8
Intervention Lesson I10
Math Diagnosis and Intervention SystemIntervention Lesson I10
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Solids and Nets
Teacher Notes
Ongoing AssessmentAsk: What solid best represents a can of corn? cylinder
Error InterventionIf students have trouble identifying solids,
then use I1: Solid Figures.
If You Have More TimeIn pairs, have students play Guess My Solid. One student should select a solid made from the nets; make sure the other student cannot see which solid the partner picks. The second student asks yes-or-no questions such as, Does it have more than 5 vertices? Does it have at least one square face? and then tries to guess what the solid is using the clues.
Math Diagnosis and Intervention SystemIntervention Lesson I10
Solids and Nets
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Materials tape, scissors, copy of nets for all prisms, square and rectangular pyramids from Teaching Tool Masters
Cut out and tape each net to help complete the tables. Each group should make 7 solids.
Solid Faces Edges Vertices Shapes of Faces1. Pyramid
5 8 5
1 square
4 triangles
2. Rectangular Pyramid
5 8 51 rectangle, 4 triangles
3. Cube
6 12 8 6 squares
4. Rectangular Prism
6 12 86
rectangles
5. Triangular Prism
5 9 62 triangles,
3 rectangles
What solid will each net form?
6. 7.
cylinder cube
8. 9.
rectangular prism triangular prism
10. 11.
cone square pyramid
12. Reasoning Is the figure a net for a cube? Explain.
No; the net only has five faces and a cube has six faces.
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Intervention Lesson I11
Math Diagnosis and Intervention SystemIntervention Lesson I11
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Views of Solid Figures
Teacher Notes
Ongoing AssessmentAsk: Which views would change if a cube were added behind the far, back, left cube in a solid? The top view and the side view would change. The front view would not change.
Error InterventionIf students have trouble seeing the views using blocks or small cubes,
then use larger boxes to illustrate the solids in front of the class.
If You Have More TimeHave student work in pairs. One student draws a top, side, or front view and the other student constructs a possible solid having the given view with blocks or small cubes.
Math Diagnosis and Intervention SystemIntervention Lesson I11
Views of Solid Figures
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Materials 6 blocks or small cubes from place-value blocks for each pair or group, crayons or markers
Stack blocks to model the solid shown at the Top View
Side View
Front View
right. Assume that there are only 6 cubes in the solid so that none are hidden.
The top view of the solid is the image seen when looking straight down at the figure.
Draw the top view of the solid at the right by answering 1 and 2.
1. How many cubes can you see when you look straight down at the solid? 3
2. Color in squares on the grid to indicate the blocks seen from the top view.
The front view is the image seen when looking straight at the cubes.
Draw the front view of the solid above by answering 3 and 4.
3. How many cubes can you see when you look straight at the solid? 6
4. Color in squares on the grid to indicate the blocks seen from the front view.
The side view is the image seen when looking at the side of the cubes.
Draw the side view of the solid above by answering 5 and 6.
5. How many cubes can you see when you look at the solid from the side? 3
6. Color in squares on the grid to indicate the blocks seen from the side view.
Draw the front, right, and top views of each solid figure. There are no hidden cubes.
7. 8.
TopFront Side
TopSideFront
9. 10. 11.
Top
Front
Side
Top
Front
Side
Top
Front
Side
12. Reasoning If a cube is added to the top of the solid in Exercise 11, what views would change? What view would not change?
The front and side views would change but the top view would not.
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Intervention Lesson I13
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Congruent Figures
Teacher Notes
Ongoing AssessmentAsk: Do figures have to be facing the same way in order to be considered congruent? No, they have to be the same size and the same shape but they can be turned in different directions and still be considered congruent.
Error InterventionIf students have trouble identifying shapes that are the same size,
then have students trace one figure and move the tracing over the other figure.
If You Have More TimeHave student work in pairs to find congruent objects in the classroom.
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Congruent Figures
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Materials tracing paper and scissors
Two figures that have exactly the same size and shape are congruent.
1. Place a piece of paper over Figure A and trace the shape. Is the figure you drew congruent to Figure A? yes
Cut out the figure you traced and use it to answer 2 to 10. Figure A
Figure B Figure C Figure D
2. Place the cutout on top of Figure B. Is Figure B the same size as Figure A? no
3. Is Figure B congruent to Figure A? no
4. Place the cutout on top of Figure C. Is Figure C the same shape as Figure A? no
5. Is Figure C congruent to Figure A? no
6. Place the cutout on top of Figure D. Is Figure D the same size as Figure A? yes
7. Is Figure D the same shape as Figure A? yes
8. Is Figure D congruent to Figure A? yes
9. Circle the figure that is congruent to the figure at the right.
Tell if the two figures are congruent. Write Yes or No.
10. 11. 12.
yes no yes
13. 14. 15.
yes no no
16. 17. 18.
yes no no
19. Divide the isosceles triangle shown at the right into 2 congruent right triangles.
20. Divide the hexagon shown at the right into 6 congruent equilateral triangles.
21. Divide the rectangle shown at the right into 2 pairs of congruent triangles.
22. Reasoning Are the triangles at the right congruent? Why or why not?
No; the triangles are the same shape, but they are not the same size.
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Math Diagnosis and Intervention SystemIntervention Lesson I45
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More Perimeter
Teacher Notes
Ongoing AssessmentAsk: Why is the formula for the perimeter of a square P � 4s? Because all of the sides of a square are equal, you can just multiply one side by 4 instead of adding each side.
Error InterventionIf students do not know the properties of rectangles,
then use I7: Quadrilaterals.
If students are having trouble remembering the formula for the perimeter of a square and the formula of the perimeter of a rectangle,
then have students create formula cards on note cards including examples of how to use the formula correctly.
If You Have More TimeHave students work in pairs to create a stack of 16 index cards with numbers 2 to 9 each written on cards. Have students draw two cards from the deck. The first card represents the length and the second card represents the width. The students work together to find the perimeter of a rectangle with the given dimensions. Have the students draw from the deck at least five different times and record their work.
Math Diagnosis and Intervention SystemIntervention Lesson I45
More Perimeter
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Jonah’s pool is a rectangle. The pool is 15 feet long and 10 feet wide. What is the perimeter of the pool?
Find the perimeter of the pool by answering 1 to 3.
1. Write in the missing measurements on the pool 15 ft
10 ft 10 ft
15 ft
shown at the right.
2. Add the lengths of the sides.
10 ft � 10 ft � 15 ft � 15 ft � 50 ft
3. What is the perimeter of the pool? 50 ft
Find a formula for the perimeter of a rectangle by answering 4 to 10.
8 in.
3 in.Rectangle A
5 ft
4 ftRectangle
B
4. Write the side lengths of the rectangle. 7. Write the side lengths of the rectangle.
8 � 3 � 8 � 3 � 22 in. 5 � 4 � 5 � 4 � 18 ft
5. Rearrange the numbers. 8. Rearrange the numbers.
8 � 8 � 3 � 3 � 22 in. 5 � 5 � 4 � 4 � 18 ft
6. Rewrite the number sentence. 9. Rewrite the number sentence.
2(8) � 2 ( 3 ) � 22 in. 2(5) � 2( 4 ) � 18 ft
16 � 6 � 22 in. 8 � 10 � 18 ft
10. Complete the table.
Rectangle Length Width Perimeter
A 8 3 2( 8 ) � 2(3)
B 5 4 2(5) � 2(4)
Any � w 2� � 2 w
The formula for the perimeter of a rectangle is P � 2� � 2w
11. Reasoning Use the formula to find the perimeter of Jonah’s pool.
P � 2� � 2w
P � 2 ( 15 ) � 2( 10 ) � 30 � 20 � 50 ft
12. Is the perimeter the same as you found on the previous page? yes
A square is a type of rectangle where all of the side lengths are equal.
Find a formula for the perimeter of the square by answering 13 to 15.
13. Add to find the perimeter of the square shown at the right. 5 cm
5 � 5 � 5 � 5 � 20
14. What could you multiply to find the perimeter of the square? 4 � 5
15. If s equals the length of a side of a square, how could you find the perimeter? P � 4s
Find the perimeter of the rectangle with the given dimensions.
16. � � 9 mm, w � 12 mm 17. � � 13 in., w � 14 in.
42 mm 54 in.
18. � � 2 ft, w � 15 ft 19. � � 17 cm, w � 25 cm
34 ft 84 cm
Find the perimeter of the square with the given side.
20. s � 2 yd 21. s � 10 in. 22. s � 31 km 23. s � 11 m
8 yd 40 in. 124 km 44 m
24. Reasoning Could you use the formula for the perimeter of a rectangle to find the perimeter of a square? Explain your reasoning.
Yes; the formula for the perimeter of a rectangle can be used to find the perimeter of a square because a square is a rectangle.
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Intervention Lesson I45
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Practice F8
Rounding Numbers Through ThousandsRound each number to the nearest ten.
1. 326 2. 825 3. 162 4. 97
Round each number to the nearest hundred.
5. 1,427 6. 8,136 7. 1,308 8. 3,656
Round each number to the nearest thousand.
9. 18,366 10. 408,614 11. 29,430 12. 63,239
Round each number to the nearest ten thousand.
13. 12,108 14. 70,274 15. 33,625 16. 17,164
17. What is 681,542 rounded to the nearest hundred thousand?
A 600,000 B 680,000 C 700,000 D 780,000
18. Mrs. Kennedy is buying pencils for each of 315 students at Hamilton Elementary. The pencils are sold in boxes of tens. How can she use rounding to decide how many pencils to buy?
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Practice F9
Practice
F9Comparing and Ordering Numbers Through ThousandsUse the chart for help if you need to.
hundredthousands
tenthousands thousands hundreds tens ones
Compare. Write � or � for each .
1. 54,376 45,763 2. 6,789 9,876
3. 9,635 9,536 4. 4,125 3,225
Order the numbers from least to greatest.
5. 959,211 936,211 958,211
6. 456,318 408,405 459,751
7. Write three numbers that are greater than 43,000 and less than 44,000.
8. Which number has the greatest value?
A 86,543,712 B 86,691,111 C 86,381,211 D 86,239,121
9. Tell how you could use a number line to determine which of two numbers is greater.
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Practice F10
Place Value Through MillionsWrite the number in standard form and in word form.
1. 300,000,000 � 70,000,000 � 2,000,000 � 500,000 � 10,000 � 2,000 � 800 � 5
Write the word form and tell the value of the underlined digit for each number.
2. 4,600,028
3. 488,423,046
4. Number Sense Write the number that is one hundred million more than 15,146,481.
5. The population of a state was estimated to be 33,871,648. Write the word form.
86. Which is the expanded form for 43,287,005?
A 4,000,000 � 300,000 � 20,000 � 8,000 � 700 � 5
B 40,000,000 � 3,000,000 � 200,000 � 80,000 � 7,000 � 5
C 400,000,000 � 30,000,000 � 2,000,000 � 8,000 � 500
D 4,000,000 � 30,000 � 2,000 � 800 � 70 � 5
7. In the number 463,211,889, which digit has the greatest value? Explain.
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Practice F11
Practice
F11Rounding Numbers Through MillionsRound each number to the nearest thousand.
1. 6,326 2. 2,825 3. 2,162 4. 4,097
Round each number to the nearest ten thousand.
5. 31,427 6. 68,136 7. 76,308 8. 93,656
Round each number to the nearest hundred thousand.
9. 618,366 10. 409,614 11. 229,930 12. 563,239
Round each number to the nearest million.
13. 12,108,219 14. 7,570,274 15. 9,333,625 16. 4,307,164
17. What is1,486,428 rounded to the nearest million?
A 1,000,000 B 1,250,000 C 1,500,000 D 2,000,000
18. What is 681,542 rounded to the nearest hundred thousand?
A 600,000 B 680,000 C 700,000 D 780,000
19. Round 2,672,358 to each place value?
ten hundred
thousand ten thousand
hundred thousand million
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Practice F29
Practice
F29Patterns and EquationsFor 1 through 6, complete each table. Find each rule.
1. 2. 3. 4.
Rule: Rule: Rule: Rule:
5. 6.
Rule: Rule:
7. Lucas recorded the growth of a plant. How tall will the plant be on Day 5?
Day 1 2 3 4 5 Height in inches 6 12 18 24 ?
8. Danielle made a table of the rental fees at a video store. What is the rule?
x 1 2 3 4 y $3 $6 $9 $12
A y � 3x C y � x � 3 B y � 6x D y � 4x
9. Curtis found a rule for a table. The rule he made is y � 7x. What does the rule tell you about every y-value?
x y81 963 745 527 ?
x y6 428 5610 7012 ?
x y36 642 748 854 ?
x y10 3015 4520 6025 ?
x 2 4 6 8y 22 44 66 ?
x 9 12 15 18y 3 4 5 ?
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Practice G7
Estimating SumsEstimate by rounding to the nearest ten.
1. 51 � 46 2. 34 � 71 3. 33 � 81 4. 53 � 69
5. 58 � 47 6. 39 � 64 7. 88 � 76 8. 33 � 23
Estimate by rounding to the nearest hundred.
9. 478 10. 680 11. 617 12. 569_ � 252 _ � 743 _ � 358 _ � 780
13. 840 � 501 14. 764 � 571 15. 421 � 887
16. 458 � 681 17. 141 � 370 18. 605 � 825
19. Corey was a member of the baseball team for 4 years. Dan was a member of the football team for 3 years. The baseball team has 51 members and the football team has 96 members. About how many members do the two groups have together?
20. Karen sold 534 magazines and Carly sold 308. About how many total magazines did the two girls sell?
21. What is the largest number that can be added to 28 so that the sum is 50 when both numbers are rounded to the nearest ten? Explain.
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Practice G42
Practice
G42Dividing with ObjectsDivide. You may use counters or pictures to help.
1. 27 � 4 2. 32 � 6 3. 17 � 7 4. 29 � 9
5. 27 � 8 6. 27 � 3 7. 28 � 5 8. 35 � 4
9. 19 � 2 10. 30 � 7 11. 17 � 3 12. 16 � 9
If you arrange these items into equal rows, tell how many will be in each row and how many will be left over.
13. 26 shells into 3 rows
14. 19 pennies into 5 rows
15. 17 balloons into 7 rows
16. Ms. Nikkel wants to divide her class of 23 students into 4 equal teams. Is this reasonable? Why or why not?
17. Which is the remainder for the quotient of 79 � 8?
A 7 B 6 C 5 D 4
18. Pencils are sold in packages of 5. Explain why you need 6 packages in order to have enough for 27 students.
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Practice G59
Practice
G59Factoring NumbersIn 1 through 12, find all the factors of each number. Tell whether each number is prime or composite.
1. 81 2. 43 3. 572 4. 63
5. 53 6. 87 7. 3 8. 27
9. 88 10. 19 11. 69 12. 79
13. 3,235 14. 1,212 15. 57 16. 17
17. Mr. Gerry’s class has 19 students, Ms. Vernon’s class has 21 students, and Mr. Singh’s class has 23 students. Whose class has a composite number of students?
18. Every prime number larger than 10 has a digit in the ones place that is included in which set of numbers below?
A 1, 3, 7, 9 C 0, 2, 4, 5, 6, 8
B 1, 3, 5, 9 D 1, 3, 7
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Practice G66
Practice
G66Mental Math: Multiplying by Multiples of 10Multiply. Use mental math.
1. 4 � 30 � 2. 5 � 90 �
3. 9 � 200 � 4. 6 � 500 �
5. 3 � 600 � 6. 0 � 600 �
7. 90 � 70 � 8. 70 � 400 �
9. 50 � 800 � 10. 30 � 800 �
11. 90 � 500 � 12. 30 � 4,000 �
13. Number Sense How many zeros are in the product of 60 � 900? Explain how you know.
Truck A can haul 400 lb in one trip. Truck B can haul 300 lb in one trip.
14. How many pounds can Truck A haul in 9 trips?
15. How many pounds can Truck B haul in 50 trips?
16. How many pounds can Truck A haul in 70 trips?
A 280 B 2,800 C 28,000 D 280,000
17. There are 9 players on each basketball team in a league. Explain how you can find the total number of players in the league if there are 30 teams.
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Practice G67
Practice
G67Estimating ProductsRound each factor so that you can estimate the product mentally.Use rounding to estimate each product.
1. 38 � 29 2. 71 � 47
3. 54 � 76 4. 121 � 62
5. 548 � 28 6. 823 � 83
7. 67 � 289 8. 183 � 34
Use compatible numbers to estimate each product.
9. 28 � 87 10. 673 � 85
11. 54 � 347 12. 65 � 724
13. 81 � 643 14. 44 � 444
15. 72 � 285 16. 61 � 761
17. Vera has 8 boxes of paper clips. Each box has 275 paper clips. About how many paper clips does Vera have?
A 240 B 1,600 C 2,400 D 24,000
18. Ana can put 27 stickers on each page of her scrapbook. The scrapbook has 112 pages. About how many stickers can Ana put in the scrapbook?
A 6,000 B 4,000 C 3,000 D 2,000
19. A wind farm generates 330 kilowatts of electricity each day. About how many kilowatts does the wind farm produce in a week? Explain.
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Practice H1
Practice
H1Equal Parts of a WholeTell if each shows equal or unequal parts. If the parts are equal, name them.
1. 2. 3. 4.
Name the equal parts of the whole.
5. 6. 7. 8.
Use the grid to draw a region showing the number of equal parts named.
9. tenths 10. sixths
11. Geometry How many equal parts does this figure have?
12. Which is the name of 12 equal parts of a whole?
halves tenths sixths twelfths
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cheesegreenpeppers
mushrooms
Region BRegion A
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Practice H2
Practice
H2Parts of a RegionWrite a fraction for the part of the region below that is shaded.
1. 2.
Shade in the models to show each fraction.
3. �24� 4. 7
__ 10
5. What fraction of the pizza is cheese?
6. What fraction of the pizza is mushroom?
7. Number Sense Is �14� of 12 greater than �
14� of 8? Explain your
answer.
8. A set has 12 squares. Which is the number of squares in �13� of
the set?
A 3 B 4 C 6 D 9
9. Explain why �12� of Region A is not larger than �
12� of Region B.
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Practice H3
Practice
H3Parts of a Set
1. Draw a group of squares with 9 ___ 10 shaded.
2. How many squares do you need to draw altogether.
3. How many should be shaded?
Write the fraction for each shaded model.
4. 5.
6. 7.
8. 9.
Draw a model and shade it in to show each fraction.
10. 8 ___ 12 11. 15 ___ 20
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Practice H14
Practice
H14Equivalent FractionsTo find an equivalent fraction by To find the equivalent fraction by multiplying, use this example. dividing, use this example.
� 2
2 __ 3 � ____ 6 � 2
� 5
10 ___ 15 � ____ 3 � 5
Find the equivalent fraction.
1. 1 __ 6 � ____ 18 2. 6 ___ 10 � ____ 5 3. 6 __ 8 � ____ 4
4. 6 ___ 10 � ____ 20 5. 8 ___ 14 � ____ 7 6. 9 ___ 11 � ____ 22
Write each fraction in simplest form.
7. 9 ___ 12
8. 5 ___ 10
9. 5 ___ 25
10. 24 ___ 36
Multiply or divide to find an equivalent fraction.
11. 11 ___ 22 12. 6 ___ 36 13. 9 ___ 10 14. 5 ___ 35 15. 7 ___ 12
16. At the air show, 1 _ 3 of the airplanes were gliders. Which fraction is not an equivalent fraction for 1 _ 3 ?
A 5 ___ 15 B 7 ___ 21 C 6 ___ 24 D 9 ___ 27
17. In Missy’s sports-cards collection, 5 _ 7 of the cards are baseball. In Frank’s collection, 12
__ 36 are baseball. Frank says they have the same fraction of baseball cards. Is he correct?
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Practice H18
Practice
H18Mixed NumbersWrite 5 _ 2 as a mixed number without using fraction strips.
1. Divide 5 by 2 at the right. 2 � � 5
�
2. Fill in the missing numbers below.
Quotient
Remainder
Divisor
3. Write 5 _ 2 as a mixed number.
Write each improper fraction as a mixed number.
4. 12 ___ 7
5. 7 __ 3
6. 9 __ 7
7. 9 __ 4
8. 29 ___ 13
9. 34 ___ 8
10. Write 2 4 _ 5 as an improper fraction.
Whole Number � Denominator � Numerator __________________________________________ Denominator �
2 � � 4 � 10 � ____________________________________________
� _______
Change each mixed number to an improper fraction.
11. 2 4 __ 5
12. 8 7 __ 9
13. 3 6 __ 7
14. 7 1 __ 8
15. 4 3 __ 7
16. 5 1 __ 4
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Practice
I8Congruent Figures and MotionsCongruent figures have the same size and shapes, although they may face different directions.
Write yes or no to tell if the figures are congruent.
1. 2. 3.
4. 5. 6.
Slide
Flip
Turn
Write slide, flip, or turn for each diagram.
6. 7. 8.
10. 11. 12.
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Practice I10
Practice
I10Solids and NetsFor 1 and 2, predict what shape each net will make.
1. 2.
For 3–5, 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 5. A net that has 2 circles and a rectangle 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. Draw a net for a triangular pyramid. Explain how you know your diagram is correct.
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Practice
I11Views of Solid FiguresDraw the front, right, and top views of each solid figure.
1. 2.
Top Front Side Top Side Front
3. 4.
Top Front Side Top Side Front
5.
Top Front Side
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Practice
I13Congruent FiguresCongruent figures have the same size and shape, although they may face different directions.
Tell if the figures are congruent.
1. 2. 3.
4. 5. 6.
7. Divide this shape into 8. Divide this shape into 4 congruent shapes. 2 congruent shapes.
9. Divide this shape into 10. Divide this shape into 3 congruent shapes. 8 congruent shapes.
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Practice
I45PerimeterLook at Rectangle A and Rectangle B. Complete the table.
9 in.
3 in.Rectangle A
6 ft
4 ftRectangle
B
Rectangle Length Width Perimeter
A
B
So, Any � w 2� � 2w
Find the perimeter of these rectangles.
1. � � 9 w � 2 2. � � 6 w � 7
3. � � 11 w � 12 4. � � 10 w � 5
Find the perimeter of the square with the given side.
5. s � 4 6. s � 2
7. s � 9 8. s � 6
9. s � 20 10. s � 5
11. s � 8 12. s � 10
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F8 F9
F10 F11Answers: F8, F9, F10, F11
Answers for Practice
F8, F9, F10, F11
330 830 160 100
1,400 8,100 1,300 3,700
18,000 409,000 29,000 63,000
10,000 70,000 30,000 20,000
Round to the nearest 10.
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Practice F8
Rounding Numbers Through ThousandsRound each number to the nearest ten.
1. 326 2. 825 3. 162 4. 97
Round each number to the nearest hundred.
5. 1,427 6. 8,136 7. 1,308 8. 3,656
Round each number to the nearest thousand.
9. 18,366 10. 408,614 11. 29,430 12. 63,239
Round each number to the nearest ten thousand.
13. 12,108 14. 70,274 15. 33,625 16. 17,164
17. What is 681,542 rounded to the nearest hundred thousand?
A 600,000 B 680,000 C 700,000 D 780,000
18. Mrs. Kennedy is buying pencils for each of 315 students at Hamilton Elementary. The pencils are sold in boxes of tens. How can she use rounding to decide how many pencils to buy?
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Practice
F8
45095_Practice_F8-I45.indd F8 6/30/08 12:01:54 PM
<>>
936,211 958,211
Answers will vary.
>
959,211
408,405 456,318 459,751
Answers will vary.
Name
Practice F9
Practice
F9Comparing and Ordering Numbers Through ThousandsUse the chart for help if you need to.
hundredthousands
tenthousands thousands hundreds tens ones
Compare. Write � or � for each .
1. 54,376 45,763 2. 6,789 9,876
3. 9,635 9,536 4. 4,125 3,225
Order the numbers from least to greatest.
5. 959,211 936,211 958,211
6. 456,318 408,405 459,751
7. Write three numbers that are greater than 43,000 and less than 44,000.
8. Which number has the greatest value?
A 86,543,712 B 86,691,111 C 86,381,211 D 86,239,121
9. Tell how you could use a number line to determine which of two numbers is greater.
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372,512,805; three hundred seventy-two million, five hundred twelve thousand, eight hundred five
Four million, six hundred thousand, twenty-eight; six hundred thousand
Thirty-three million, eight hundred seventy-one thousand, six hundred forty-eight
4; it is in the hundred millions place.
Four hundred eighty-eight million, four hundred twenty-three thousand, forty-six; forty
115,146,481
Name
Practice F10
Place Value Through MillionsWrite the number in standard form and in word form.
1. 300,000,000 � 70,000,000 � 2,000,000 � 500,000 � 10,000 � 2,000 � 800 � 5
Write the word form and tell the value of the underlined digit for each number.
2. 4,600,028
3. 488,423,046
4. Number Sense Write the number that is one hundred million more than 15,146,481.
5. The population of a state was estimated to be 33,871,648. Write the word form.
86. Which is the expanded form for 43,287,005?
A 4,000,000 � 300,000 � 20,000 � 8,000 � 700 � 5
B 40,000,000 � 3,000,000 � 200,000 � 80,000 � 7,000 � 5
C 400,000,000 � 30,000,000 � 2,000,000 � 8,000 � 500
D 4,000,000 � 30,000 � 2,000 � 800 � 70 � 5
7. In the number 463,211,889, which digit has the greatest value? Explain.
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Practice
F10
45095_Practice_F8-I45.indd F10 6/30/08 12:01:57 PM
6,000 3,000 2,000 4,000
30,000 70,000 80,000 90,000
600,000 400,000 200,000 600,000
9,000,00012,000,000 8,000,000 4,000,000
2,672,360 2,672,4002,672,000 2,670,000
2,700,000 3,000,000
Name
Practice F11
Practice
F11Rounding Numbers Through MillionsRound each number to the nearest thousand.
1. 6,326 2. 2,825 3. 2,162 4. 4,097
Round each number to the nearest ten thousand.
5. 31,427 6. 68,136 7. 76,308 8. 93,656
Round each number to the nearest hundred thousand.
9. 618,366 10. 409,614 11. 229,930 12. 563,239
Round each number to the nearest million.
13. 12,108,219 14. 7,570,274 15. 9,333,625 16. 4,307,164
17. What is1,486,428 rounded to the nearest million?
A 1,000,000 B 1,250,000 C 1,500,000 D 2,000,000
18. What is 681,542 rounded to the nearest hundred thousand?
A 600,000 B 680,000 C 700,000 D 780,000
19. Round 2,672,358 to each place value?
ten hundred
thousand ten thousand
hundred thousand million
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45095_Practice_F8-I45.indd F11 6/30/08 12:01:59 PM
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F29 G7
G42 G59Answers: F29, G7, G42, G59
Answers for Practice
F29, G7, G42, G59
x � 9 7x x � 6 3x
11x x � 3
30 inches
The y-value is 7 times greater than the x-value.
3 84 9 75
88 6
Name
Practice F29
Practice
F29Patterns and EquationsFor 1 through 6, complete each table. Find each rule.
1. 2. 3. 4.
Rule: Rule: Rule: Rule:
5. 6.
Rule: Rule:
7. Lucas recorded the growth of a plant. How tall will the plant be on Day 5?
Day 1 2 3 4 5 Height in inches 6 12 18 24 ?
8. Danielle made a table of the rental fees at a video store. What is the rule?
x 1 2 3 4y $3 $6 $9 $12
A y � 3x C y � x � 3 B y � 6x D y � 4x
9. Curtis found a rule for a table. The rule he made is y � 7x.What does the rule tell you about every y-value?
x y81 963 745 527 ?
x y6 428 5610 7012 ?
x y36 642 748 854 ?
x y10 3015 4520 6025 ?
x 2 4 6 8y 22 44 66 ?
x 9 12 15 18y 3 4 5 ?
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45095_Practice_F8-I45.indd F29 6/30/08 12:02:00 PM
100
24; 28 rounds to 30, so the second number must round to 20, 30 � 20 � 50
100 110 120
110 100 170 50
1,3001,4001,300
1,4005001,200
150members
800 1,400 1,000 1,400
800magazines
Name
Practice G7
Estimating SumsEstimate by rounding to the nearest ten.
1. 51 � 46 2. 34 � 71 3. 33 � 81 4. 53 � 69
5. 58 � 47 6. 39 � 64 7. 88 � 76 8. 33 � 23
Estimate by rounding to the nearest hundred.
9. 478 10. 680 11. 617 12. 569_� 252 _� 743 _� 358 _� 780
13. 840 � 501 14. 764 � 571 15. 421 � 887
16. 458 � 681 17. 141 � 370 18. 605 � 825
19. Corey was a member of the baseball team for 4 years. Dan was a member of the football team for 3 years. The baseball team has 51 members and the football team has 96 members. About how many members do the two groups have together?
20. Karen sold 534 magazines and Carly sold 308. About how many total magazines did the two girls sell?
21. What is the largest number that can be added to 28 so that the sum is 50 when both numbers are rounded to the nearest ten? Explain.
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Practice
G7
45095_Practice_F8-I45.indd G7 6/30/08 12:02:02 PM
6 R3 5 R2 2 R3 3 R2
3 R3 9 5 R3 8 R3
9 R1 4 R2 5 R2 1 R7
No. The teams will not be even.23 � 4 � 5R3
6 � 5 � 30 5 � 5 � 25 (not enough)
2 rows; 3 left over3 rows; 4 left over8 rows; 2 left over
Name
Practice G42
Practice
G42Dividing with ObjectsDivide. You may use counters or pictures to help.
1. 27 � 4 2. 32 � 6 3. 17 � 7 4. 29 � 9
5. 27 � 8 6. 27 � 3 7. 28 � 5 8. 35 � 4
9. 19 � 2 10. 30 � 7 11. 17 � 3 12. 16 � 9
If you arrange these items into equal rows, tell how many will be in each row and how many will be left over.
13. 26 shells into 3 rows
14. 19 pennies into 5 rows
15. 17 balloons into 7 rows
16. Ms. Nikkel wants to divide her class of 23 students into 4 equal teams. Is this reasonable? Why or why not?
17. Which is the remainder for the quotient of 79 � 8?
A 7 B 6 C 5 D 4
18. Pencils are sold in packages of 5. Explain why you need 6 packages in order to have enough for 27 students.
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45095_Practice_F8-I45.indd G42 6/30/08 12:02:04 PM
Ms. Vernon’s
1, 3, 9, 27, 81 1, 43 1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572
1, 3, 7, 9,
21, 63
1, 53 1, 3, 29, 87 1, 3 1, 3, 9, 27
1, 2, 4, 8, 11,
22, 44, 88
1, 19 1, 3, 23, 69 1, 79
compositecomposite composite
prime
compositeprime compositeprime
compositeprimecomposite
prime
composite compositecomposite prime
1, 5, 647,
3235
1, 2, 4, 6, 12, 101, 202, 303, 606, 1212
1, 3, 19, 57 1, 17
Name
Practice G59
Practice
G59Factoring NumbersIn 1 through 12, find all the factors of each number. Tell whether each number is prime or composite.
1. 81 2. 43 3. 572 4. 63
5. 53 6. 87 7. 3 8. 27
9. 88 10. 19 11. 69 12. 79
13. 3,235 14. 1,212 15. 57 16. 17
17. Mr. Gerry’s class has 19 students, Ms. Vernon’s class has 21 students, and Mr. Singh’s class has 23 students. Whose class has a composite number of students?
18. Every prime number larger than 10 has a digit in the ones place that is included in which set of numbers below?
A 1, 3, 7, 9 C 0, 2, 4, 5, 6, 8
B 1, 3, 5, 9 D 1, 3, 7
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45095_Practice_F8-I45.indd G59 6/30/08 12:02:06 PM
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Answers for Practice
G66, G67, H1, H2
G66 G67
H1 H2Answers: G66, G67, H1, H2
1201,8001,8006,300
40,00045,000
4503,000
028,00024,000120,000
3,600 lb15,000 lb
Answers will vary. Sample answer: Multiply 3 � 9 � 27; 27 � 10 � 270
3 Answers will vary.
Name
Practice G66
Practice
G66Mental Math: Multiplying by Multiples of 10Multiply. Use mental math.
1. 4 � 30 � 2. 5 � 90 �
3. 9 � 200 � 4. 6 � 500 �
5. 3 � 600 � 6. 0 � 600 �
7. 90 � 70 � 8. 70 � 400 �
9. 50 � 800 � 10. 30 � 800 �
11. 90 � 500 � 12. 30 � 4,000 �
13. Number Sense How many zeros are in the product of 60 � 900? Explain how you know.
Truck A can haul 400 lb in one trip. Truck B can haul 300 lb in one trip.
14. How many pounds can Truck A haul in 9 trips?
15. How many pounds can Truck B haul in 50 trips?
16. How many pounds can Truck A haul in 70 trips?
A 280 B 2,800 C 28,000 D 280,000
17. There are 9 players on each basketball team in a league. Explain how you can find the total number of players in the league if there are 30 teams.
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45095_Practice_F8-I45.indd G66 6/30/08 12:02:07 PM
21,00015,0004,000
3,500
6,00064,0006,000
1,200
30 � 90; 2,700 700 � 90; 6,30050 � 300; 15,00080 � 600; 48,00070 � 300; 21,000
70 � 700; 49,00040 � 400; 16,00060 � 800; 48,000
7 � 300 � 2,100 kilowatts Multiply 7 days in one week � 300.(300 is incompatible to 330)
Answersmay vary.
Name
Practice G67
Practice
G67Estimating ProductsRound each factor so that you can estimate the product mentally.Use rounding to estimate each product.
1. 38 � 29 2. 71 � 47
3. 54 � 76 4. 121 � 62
5. 548 � 28 6. 823 � 83
7. 67 � 289 8. 183 � 34
Use compatible numbers to estimate each product.
9. 28 � 87 10. 673 � 85
11. 54 � 347 12. 65 � 724
13. 81 � 643 14. 44 � 444
15. 72 � 285 16. 61 � 761
17. Vera has 8 boxes of paper clips. Each box has 275 paper clips. About how many paper clips does Vera have?
A 240 B 1,600 C 2,400 D 24,000
18. Ana can put 27 stickers on each page of her scrapbook. The scrapbook has 112 pages. About how many stickers can Ana put in the scrapbook?
A 6,000 B 4,000 C 3,000 D 2,000
19. A wind farm generates 330 kilowatts of electricity each day. About how many kilowatts does the wind farm produce in a week? Explain.
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45095_Practice_F8-I45.indd G67 6/30/08 12:02:09 PM
Equal,halves
Unequal Equal,eighths
Equal,fourths
eighths thirds fifths sixths
Answers will vary.
4
Name
Practice H1
Practice
H1Equal Parts of a WholeTell if each shows equal or unequal parts. If the parts are equal, name them.
1. 2. 3. 4.
Name the equal parts of the whole.
5. 6. 7. 8.
Use the grid to draw a region showing the number of equal parts named.
9. tenths 10. sixths
11. Geometry How many equal parts does this figure have?
12. Which is the name of 12 equal parts of a whole?
halves tenths sixths twelfths
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45095_Practice_F8-I45.indd H1 6/30/08 12:02:10 PM
cheesegreenpeppers
mushrooms
Region BRegion A
�48��
34�
or
Yes. Answers will vary. �14� of 12 is 3 and �
14� of 8 is 2; 3 > 2.
They each have the same region shaded for
1_2
3_8
2_8
1_2
Name
Practice H2
Practice
H2Parts of a RegionWrite a fraction for the part of the region below that is shaded.
1. 2.
Shade in the models to show each fraction.
3. �24� 4. 7__
10
5. What fraction of the pizza is cheese?
6. What fraction of the pizza is mushroom?
7. Number Sense Is �14� of 12 greater than �
14� of 8? Explain your
answer.
8. A set has 12 squares. Which is the number of squares in �13� of
the set?
A 3 B 4 C 6 D 9
9. Explain why �12� of Region A is not larger than �
12� of Region B.
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45095_Practice_F8-I45.indd H2 6/30/08 12:02:12 PM
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Answers for Practice
H3, H14, H18, I8
H3 H14
H18 I8Answers: H3, H14, H18, I8
Answers will vary.
Answers will vary.
109
4_5
2_3
6_8
2_5
7_9
4__16
Name
Practice H3
Practice
H3Parts of a Set
1. Draw a group of squares with 9___10 shaded.
2. How many squares do you need to draw altogether.
3. How many should be shaded?
Write the fraction for each shaded model.
4. 5.
6. 7.
8. 9.
Draw a model and shade it in to show each fraction.
10. 8___12 11. 15___
20
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45095_Practice_F8-I45.indd H3 6/30/08 12:02:14 PM
4 3
3 3 3
12 4 18
3_4
1_2
1_5
14__24
1_7
18__20
1_6
1_2
In simplest form, 12__36 � 1_
3. The fraction 5_7
is already in simplest form. Since 1_3 � 5_
7,Frank and Missy do not have the same fraction of baseball cards.
2_3
Sample answers given
Name
Practice H14
Practice
H14Equivalent FractionsTo find an equivalent fraction by To find the equivalent fraction by multiplying, use this example. dividing, use this example.
� 2
2__3 � ____
6 � 2
� 5
10___ 15 � ____ 3 � 5
Find the equivalent fraction.
1. 1__6 � ____
18 2. 6___10 � ____
5 3. 6__8 � ____
4
4. 6___10 � ____
20 5. 8___14 � ____
7 6. 9___11 � ____
22
Write each fraction in simplest form.
7. 9___12 8. 5___
10 9. 5___25 10. 24___
36
Multiply or divide to find an equivalent fraction.
11. 11___22 12. 6___
36 13. 9___10 14. 5___
35 15. 7___12
16. At the air show, 1_3 of the airplanes were gliders. Which fraction
is not an equivalent fraction for 1_3?
A 5___15 B 7___
21 C 6___24 D 9___
27
17. In Missy’s sports-cards collection, 5_7 of the cards are baseball.
In Frank’s collection, 12__36 are baseball. Frank says they have
the same fraction of baseball cards. Is he correct?
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45095_Practice_F8-I45.indd H14 6/30/08 12:02:16 PM
15_7 2 1_7 1 2_7
21_4 2 3__
13 4 2_8; 4 1_4
14__5
79__9
27__7
57__8
31__7
21__4
221
2
41
21_2
55
4 1415
Name
Practice H18
Practice
H18Mixed NumbersWrite 5_
2 as a mixed number without using fraction strips.
1. Divide 5 by 2 at the right.
2 � � 5
�
2. Fill in the missing numbers below.
Quotient
Remainder
Divisor
3. Write 5_2 as a mixed number.
Write each improper fraction as a mixed number.
4. 12___7 5. 7__
3 6. 9__7
7. 9__4 8. 29___
13 9. 34___8
10. Write 2 4_5 as an improper fraction.Whole Number � Denominator � Numerator__________________________________________
Denominator �
2 � � 4 � 10 �____________________________________________ � _______
Change each mixed number to an improper fraction.
11. 24__5 12. 87__
9 13. 36__7
14. 71__8 15. 43__
7 16. 51__4©
Pea
rson
Ed
ucat
ion,
Inc.
3
45095_Practice_F8-I45.indd H18 6/30/08 12:02:18 PM
No Yes No
Yes No Yes
slide turn fl ip
turn slide fl ip
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Practice I8
Practice
I8Congruent Figures and MotionsCongruent figures have the same size and shapes, although they may face different directions.
Write yes or no to tell if the figures are congruent.
1. 2. 3.
4. 5. 6.
Slide Flip Turn
Write slide, flip, or turn for each diagram.
6. 7. 8.
10. 11. 12.
45095_Practice_F8-I45.indd I8 7/1/08 11:40:17 AM
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Answers for Practice
I10, I11, I13, I45
I10 H11
I13 I45Answers: I10, I11, I13, I45
rectangular prism square pyramid
cubetriangular pyramid
cylinder
Answers will vary.
Name
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Practice I10
Practice
I10Solids and NetsFor 1 and 2, predict what shape each net will make.
1. 2.
For 3–5, 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
5. A net that has 2 circles and a rectangle
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. Draw a net for a triangular pyramid. Explain how you know your diagram is correct.
45095_Practice_F8-I45.indd I10 6/30/08 12:02:23 PM
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Practice I11
Practice
I11Views of Solid FiguresDraw the front, right, and top views of each solid figure.
1. 2.
TopFront Side TopSideFront
3. 4.
TopFront Side TopSideFront
5.
TopFront Side
45095_Practice_F8-I45.indd I11 6/30/08 12:02:25 PM
7–10: Answers will vary.
yes no no
no yes yes
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Practice I13
Practice
I13Congruent FiguresCongruent figures have the same size and shape, although they may face different directions.
Tell if the figures are congruent.
1. 2. 3.
4. 5. 6.
7. Divide this shape into 8. Divide this shape into 4 congruent shapes. 2 congruent shapes.
9. Divide this shape into 10. Divide this shape into 3 congruent shapes. 8 congruent shapes.
45095_Practice_F8-I45.indd I13 6/30/08 12:02:26 PM
9 3 246 4 20
16
P � 22 P � 26
P � 30P � 46
36
8
24
80 20
32 40
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Practice I45
Practice
I45PerimeterLook at Rectangle A and Rectangle B. Complete the table.
9 in.
3 in.Rectangle A
6 ft
4 ftRectangle
B
Rectangle Length Width Perimeter
A
B
So, Any � w 2� � 2w
Find the perimeter of these rectangles.
1. � � 9 w � 2 2. � � 6 w � 7
3. � � 11 w � 12 4. � � 10 w � 5
Find the perimeter of the square with the given side.
5. s � 4 6. s � 2
7. s � 9 8. s � 6
9. s � 20 10. s � 5
11. s � 8 12. s � 10
45095_Practice_F8-I45.indd I45 6/30/08 12:02:29 PM
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Name Grade 3
Step Up to Grade 4 Test
T1
1. Round 43,864 to the nearest ten
thousand.
A 40,000
B 43,000
C 43,900
D 44,000
2. Which is the expanded form of
4,321,189?
A 4,000,000 � 300,000 � 20,000 �
1,000 � 100 � 80 � 9
B 4,000,000 � 300,000 � 21,000 �
100 � 89
C 4,000,000 � 300,000 � 20,000 �
1,000 � 90 � 9
D 400,000 � 300,000 � 21,000 �
1,000 � 100 � 80 � 9
3. Which digit is in the thousands place?
128,698,473
A 1
B 2
C 6
D 8
4. Round this number to the nearest
ten million. 15,679,841
A 5,000,000
B 11,000,000
C 15,000,000
D 16,000,000
5.
If x = 35, what is the value of y?
A 30
B 27
C 25
D 23
6. Estimate by rounding to the nearest
hundred. 689 � 228
A 628
B 800
C 889
D 900
x 20 25 30 35y 12 17 22 ?
Name Grade 3
Step Up to Grade 4 Test
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T2
7. Find the quotient and the remainder.
26 � 4
A 4 R 2
B 6 R 2
C 6 R 4
D 8 R 2
8. What are all the factors of 12?
A 1, 6, 12
B 1, 2, 6, 12
C 1, 2, 3, 6, 12
D 1, 2, 3, 4, 6, 12
9. The comic book store packs 30 comic
books into one envelope. How many
comic books will be packed into
30 envelopes?
A 90
B 300
C 600
D 900
10. Estimate the product by rounding
each factor.
28 � 23
A 900
B 600
C 500
D 400
11. Name the equal parts of the whole.
A fifths
B fourths
C thirds
D halves
12. Write the fraction for the shaded part
of the region.
A 4 _ 5
B 5 _ 8
C 6 _ 8
D 7 _ 8
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Name Grade 3
Step Up to Grade 4 Test
T3
13. Find the fraction for the shaded parts
of this set.
A 3 _ 5
B 4 _ 5
C 5 _ 6
D 4 _ 7
14. Which fraction is not equivalent to 2 _ 3 ?
A 4 _ 6
B 8 __ 12
C 9 __ 12
D 10 __ 15
15. What is the mixed number for this
improper fraction? 15 __ 8
A 2 7 _ 8
B 2 1 _ 8
C 1 7 _ 8
D 1 5 _ 8
16. What motion does this shape show?
A slide
B fl ip
C turn
D motion
17. Which solid can be made from this net?
A square
B cube
C triangular prism
D rectangular prism
18. What would the top view of this solid
look like?
A B
C D
6 ft
3 ft 4 ft
3 ft3 ft 1 ft
Name Grade 3
Step Up to Grade 4 Test
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T4
19. Which set of fi gures is not congruent?
A
B
C
D
20. What is the perimeter of this garden?
A 12 ft
B 20 ft
C 22 ft
D 24 ft
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Answers for Test
T1, T2, T3, T4
T1 T2
T4T3Answers: T1, T2, T3, T4
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Name Grade 3
Step Up to Grade 4 Test
T1
1. Round 43,864 to the nearest ten
thousand.
A 40,000
B 43,000
C 43,900
D 44,000
2. Which is the expanded form of
4,321,189?
A 4,000,000 � 300,000 � 20,000 �
1,000 � 100 � 80 � 9
B 4,000,000 � 300,000 � 21,000 �
100 � 89
C 4,000,000 � 300,000 � 20,000 �
1,000 � 90 � 9
D 400,000 � 300,000 � 21,000 �
1,000 � 100 � 80 � 9
3. Which digit is in the thousands place?
128,698,473
A 1
B 2
C 6
D 8
4. Round this number to the nearest
ten million. 15,679,841
A 5,000,000
B 11,000,000
C 15,000,000
D 16,000,000
5.
If x = 35, what is the value of y?
A 30
B 27
C 25
D 23
6. Estimate by rounding to the nearest
hundred. 689 � 228
A 628
B 800
C 889
D 900
x 20 25 30 35y 12 17 22 ?
45095_T1-T4.indd T1 7/1/08 2:09:36 PM
Name Grade 3
Step Up to Grade 4 Test
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T2
7. Find the quotient and the remainder.
26 � 4
A 4 R 2
B 6 R 2
C 6 R 4
D 8 R 2
8. What are all the factors of 12?
A 1, 6, 12
B 1, 2, 6, 12
C 1, 2, 3, 6, 12
D 1, 2, 3, 4, 6, 12
9. The comic book store packs 30 comic
books into one envelope. How many
comic books will be packed into
30 envelopes?
A 90
B 300
C 600
D 900
10. Estimate the product by rounding
each factor.
28 � 23
A 900
B 600
C 500
D 400
11. Name the equal parts of the whole.
A fifths
B fourths
C thirds
D halves
12. Write the fraction for the shaded part
of the region.
A 4 _ 5
B 5 _ 8
C 6 _ 8
D 7 _ 8
45095_T1-T4.indd T2 7/2/08 1:20:15 PM
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Name Grade 3
Step Up to Grade 4 Test
T3
13. Find the fraction for the shaded parts
of this set.
A 3 _ 5
B 4 _ 5
C 5 _ 6
D 4 _ 7
14. Which fraction is not equivalent to 2 _ 3 ?
A 4 _ 6
B 8 __ 12
C 9 __ 12
D 10 __ 15
15. What is the mixed number for this
improper fraction? 15 __ 8
A 2 7 _ 8
B 2 1 _ 8
C 1 7 _ 8
D 1 5 _ 8
16. What motion does this shape show?
A slide
B fl ip
C turn
D motion
17. Which solid can be made from this net?
A square
B cube
C triangular prism
D rectangular prism
18. What would the top view of this solid
look like?
A B
C D
45095_T1-T4.indd T3 7/1/08 2:09:38 PM
6 ft
3 ft 4 ft
3 ft3 ft 1 ft
Name Grade 3
Step Up to Grade 4 Test
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T4
19. Which set of fi gures is not congruent?
A
B
C
D
20. What is the perimeter of this garden?
A 12 ft
B 20 ft
C 22 ft
D 24 ft
45095_T1-T4.indd T4 7/1/08 2:09:39 PM