Upper Saddle River, New Jersey Grade 2: Step Up to Grade 3

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Glenview, Illinois • Boston, Massachusetts Chandler, Arizona • Shoreview, Minnesota Upper Saddle River, New Jersey

Grade 2: Step Up to Grade 3Teacher’s Guide• Teacher Notes and Answers

for Step-Up Lessons

• Practice

• Answers for Practice

• Test

• Answers for Test

45094_SLPSHEET_FSD 145094_SLPSHEET_FSD 1 6/6/08 3:56:52 PM6/6/08 3:56:52 PM

A42 elohW a fo straP lauqE

A43 noigeR a fo straP

A47 snoitcarF erapmoC ot sledoM gnisU

A48 tnelaviuqE dniF ot sledoM gnisU Fractions

A74 Repeating Patterns

B45 Using Multiplication to Compare

B48 Multiplying by 9

B56 Multiplying Three Numbers

B62 Dividing by 8 and 9

B63 0 and 1 in Division

C26 ddA ot htaM latneM gnisU

C27 Using Mental Math to Subtract

C28 srebmuN tigiD-owT gniddA

C29 srebmuN tigiD-owT gnitcartbuS

C37 Adding Three Numbers

D59 Solid Figures

D62 Acute, Right, and Obtuse Angles

D63 Polygons

D64 Classifying Triangles Using Sides and Angles

D65 Quadrilaterals

45094_SLPSHEET_FSD 2 6/6/08 3:57:09 PM

Intervention Lesson A42

Math Diagnosis and Intervention SystemIntervention Lesson A42

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. 2Equal 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 A42 177


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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

178 Intervention Lesson A42

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Equal Parts of a Whole (continued)

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. 2

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.


Parts of a Region

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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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. 2Using Models to Compare Fractions

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.Math Diagnosis and Intervention SystemIntervention Lesson A47

Using Models to Compare Fractions

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Materials fraction strips

Use �, �, or � to compare 4 __ 5 and 2 __ 3 by answering 1 to 3.

1. Show 1, 4 __ 5 , and 2 __ 3 with 1

1 __ 5 1 __ 5 1 __ 5 1 __ 5

1 __ 3 1 __ 3

fraction strips.

2. Compare. Which is greater

in total length, 4 __ 5 or 2 __ 3 ?

4__5

3. Since 4 __ 5 is longer than 2 __ 3 ,

4 __ 5 is greater than 2 __ 3 . Write �, �, or �. 4 __ 5 � 2 __ 3

Compare 1 ___ 10 and 1 __ 4 by answering 4 to 6.

4. Show 1, 1 ___ 10 , and 1 __ 4 with fraction strips. 1

1 ___ 10

1 __ 4


in total length, 1 ___ 10 or 1 __ 4 ?

1__4

6. Since 1 ___ 10 is shorter than 1 __ 4 ,

1 ___ 10 is less than 1 __ 4 . Write �, �, or =. 1 ___ 10 � 1 __ 4

Compare 2 __ 5 and 4 ___ 10 by answering 7 to 9.

7. Show 1, 2 __ 5 , and 4 ___ 10 with fraction strips. 1

1 __ 5 1 __ 5

1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10


in total length, 2 __ 5 or 4 ___ 10 ?

They are the same length.

9. Since 2 __ 5 and 4 ___ 10 are the same length,

2 __ 5 is equal to 4 ___ 10 . Write �, �, or �. 2 __ 5 � 4 ___ 10

Compare. Write <, >, or =.

10. 1 __ 4 � 3 __ 4 11. 3 __ 4 � 2 __ 8

1 __ 4

1 __ 4 1 __ 4 1 __ 4

1 __ 4 1 __ 4 1 __ 4

1 __ 8 1 __ 8

12. 2 __ 3 � 4 __ 6 13. 1 __ 5 � 5 ___ 10

1 __ 3 1 __ 3

1 __ 6 1 __ 6 1 __ 6 1 __ 6

1 __ 5

1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10

14. 1 __ 2 � 1 __ 5 15. 7 __ 8 � 3 __ 4

1 __ 5

1 __ 2

1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8

1 __ 4 1 __ 4 1 __ 4

16. 2 __ 6 � 1 __ 2 17. 3 __ 5 � 1 __ 4

1 __ 6 1 __ 6

1 __ 2

1 __ 5 1 __ 5

1 __ 4

1 __ 5

18. Reasoning Give 3 fractions with different

denominators that are less than 4 __ 6 . Answers will vary.

19. Reasoning Two students are writing stories.

Eric’s story is 2 __ 3 of a page. Alba’s story is 4 __ 6 of

a page. Whose story is longer? They are equal.


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Using Models to Compare Fractions (continued)

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Teacher Notes

Ongoing AssessmentMake sure children can translate simple number sentences such as 7 � 8, 4 � 2, and 6 � 6 into words.

Error InterventionIf children have difficulty with the concepts of greater than and less than, or with the symbols,

then use A27: Using �, �, and � to Compare Numbers.

If You Have More TimeHave students pretend they are each painting a board. Have the students say how much they have stained. Then use fraction strips to show and compare who has stained more. Have each student write the comparison on paper.



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Using Models to Find Equivalent Fractions


Using Models to Find Equivalent Fractions

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Materials fraction strips

Find a fraction equivalent to 3 __ 4 by answering 1 to 3.

1. Show a 1 and 3 __ 4 with fraction strips. 1

1 __ 4 1 __ 4 1 __ 4 2. How many 1 __ 8 strips does

it take to equal 3 __ 4 ? 6

3 __ 4 � 6 _______ 8

3. So, 3 __ 4 is equal to six 1 __ 8 strips.

1

1 __ 4 1 __ 4 1 __ 4

1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8 1 __ 8

Find the missing number in 1 __ 2 � ? ____ 10 ,

by answering 4 to 7.

The denominators of the fractions tell which fraction strips to use.

4. Show 1 and 1 __ 2 with fraction strips. 1

1 __ 2 5. What is the denominator of the second fraction? 10

6. Since the denominator of the second 1

1 __ 2

1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10

fraction is 10, find the number

of 1 ___ 10 strips equal to 1 __ 2 . 5

7. So, 1 __ 2 is equal to five 1 ___ 10 strips.

1 __ 2 � 5 ____ 10

Complete each number sentence.

8. 1

1 __ 4

1 __ 8 1 __ 8

9. 1

1 __ 6 1 __ 6 1 __ 6 1 __ 6

1 __ 3 1 __ 3

1 __ 4 � 2 _______ 8 2 __ 3 �

4 _______ 6

10. 1

1 __ 2

1 __ 8 1 __ 8 1 __ 8 1 __ 8

11. 1

1 __ 5 1 __ 5

1 ___ 10 1 ___ 10 1 ___ 10 1 ___ 10

1 __ 2 � 4 _______ 8 2 __ 5 �

4 _______ 10

12. 1

1 __ 3 1 __ 3

1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12 1 ___ 12

13. 1

1 __ 6 1 __ 6 1 __ 6

1 __ 4 1 __ 4

2 __ 3 � 8 _______ 12 2 __ 4 �

3 _______ 6

14. Reasoning On Tuesday, 2 __ 3 of the class time was spent

on math projects. How many sixths of the class time was spent on math projects?

4__6


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Using Models to Find Equivalent Fractions (continued)

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Teacher Notes

Ongoing AssessmentAsk: How many twelfths are equal to 1 __

3 ? 4 ___

12

Error InterventionIf students have trouble finding the equivalent fraction for Exercise 14,

then encourage them to use fraction strips.

If You Have More Time

Have students use fraction strips to find fractions

equivalent to 1 __ 2 . Have them record their findings. If

time allows, do the same for 1 __ 4 and 3 __

4 .



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. 2Repeating Patterns

Teacher Notes

Ongoing AssessmentAsk: Can a pattern be formed by using only circles? Sample answer: Yes, if different size circles are used. The pattern could be small circle, small circle, big circle.

Error InterventionIf students can recognize the numerical patterns, but have trouble recognizing the geometric patterns,

then have students assign/label each type of shape with a different number or letter. Have them look for a pattern with the numbers. For example, the squares could be “1”, triangles “2”, and trapezoids “3”.

If students can not spell the names of the shapes,

then have students draw the shapes instead of naming them.

If You Have More TimeHave one student in each pair use the shapes or index cards to make a pattern for their partner to extend. Change roles and repeat.


Repeating Patterns

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Materials pattern blocks or shapes cut out of colored paper (10 orange squares, 10 green triangles, 10 red trapezoids) for each pair of students; 24 index cards (eight labeled 2, eight labeled 3, and eight labeled 4) for each pair of students

Look at the pattern of shapes.

1. Work with your partner to show the pattern. What is the next shape? square

2. Continue the pattern. What is the 14th shape? square

3. What is the 16th shape? trapezoid

4. Work with your partner and use the shapes to make a new pattern. Draw the pattern below. Draw the next four shapes.

Answers will vary. Check that patterns repeat consistently.

Look at the pattern of numbers.

3

3

2

4

3 3

2

4

3

5. Work with your partner to show the pattern. What is the next number? 3

6. Continue the pattern. What is the 12th number? 4

7. What is the 15th number? 2

8. Work with your partner and use the numbers to make a new pattern. Write the pattern below. Write the next four numbers. Answers will vary. Check that patterns

repeat consistently.

Draw the next three shapes to continue each pattern.

9.

10.

11.

Write the next three numbers to continue each pattern.

12. 1, 4, 6, 7, 1, 4, 6, 7, 1, 4, 6 , 7 , 1

13. 8, 8, 9, 8, 8, 9, 8, 8, 9, 8, 8 , 9 , 8

14. 3, 2, 0, 0, 3, 2, 0, 0, 3, 2, 0, 0 , 3 , 2

15. 4, 4, 6, 6, 8, 8, 4, 4, 6, 6, 8, 8, 4, 4 , 6 , 6

16. Create a pattern using all the shapes shown below.

Answers will vary. Sample answer: square, circle, square, circle, square, circle, square, circle

17. Create a pattern using all the letters shown below.

T T T L L W W L WAnswers will vary. Sample answer: T LW T LW T LW


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Repeating Patterns (continued)

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Intervention Lesson B45

Math Diagnosis and Intervention SystemIntervention Lesson B45

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Using Multiplication to Compare

Teacher Notes

Ongoing AssessmentAsk: 12 is twice as many as what number? 6

Error InterventionIf students have trouble figuring out how to draw the unknown amount,

then encourage students to show the first group n times. For example, since Wayne has 3 times as many as Alicia, show what Alicia has 3 times.

If You Have More TimeHave students cut out 9 small squares and label them 1 through 9. Have one partner pick a square. The other partner calculates 3 times the number picked. Do not return the number to the pile. Continue until all squares have been picked. Repeat the activity by having students calculate 5 times each number picked.


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Intervention Lesson B45 153

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Using Multiplication to Compare

Materials 12 counters per student

Alicia has 2 stickers. Pedro has 3 times as many stickers as Alicia. How many stickers does Pedro have?

1. Show Alicia’s stickers with counters.

2. Show Pedro’s stickers with counters.

3. Write a multiplication sentence.

3 times as many as Alicia has equals number Pedro has

3 � 2 � 6

4. How many stickers does Pedro have? 6

Mia has 4 yo-yos. Flo has twice as many as Mia. How many yo-yos does Flo have?

The word twice in a word problem means 2 times as many.

5. Show Mia’s yo-yos with counters.

6. Show Flo’s yo-yos with counters.

7. Write a multiplication sentence.

2 times as many as Mia has equals number Flo has

2 � 4 � 8

8. How many yo-yos does Flo have? 8

154 Intervention Lesson B45

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Using Multiplication to Compare (continued)

Solve. You may use drawings or counters to help.

9. Janos has 3 stickers. Lucy has twice as many stickers as Janos. How many stickers does Lucy have?

6 stickers

10. Rob has 4 model airplanes. Julio has 3 times as many model airplanes as Rob. How many model airplanes does Julio have?

12 model airplanes

11. Mr. King has 5 apples left in his store. Ruth needs twice as many apples to bake apple pies. How many apples does Ruth need?

10 apples

Use the recipe to answer Exercises 12–15.

12. The recipe serves 5 people. Joan wants to make the recipe for 15 people. How many times more is this?

3 times more

13. How many bananas will Joan need to make the recipe for 15 people?

3 � 3 � 9 bananas

14. How many cups of strawberries will Joan need to make the recipe for 15 people?

6 cups

15. Reasoning If Joan wants to make twice as much as the recipe in the chart, what will she need to do to all of the ingredients?

double them

Fruit Smoothie

3 large bananas2 cups strawberries1 cup orange juice1 cup cranberry juice1 cup ice cubes

Blend until smooth.Makes 5 servings.



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Teacher Notes

Ongoing AssessmentAsk: Larry said that 6 � 9 � 45. Why is this incorrect? The tens digit of the product must be 1 less than the number by which 9 is being multiplied. 6 � 1 � 5, so the tens digit must be 5. The product is 54, not 45.

Error InterventionIf students have trouble while using the method described in the lesson,

then show the students how to use their fingers to multiply by 9. Put both hands on your desk, palms down. Mentally number your fingers and thumbs from left to right, starting with 1. To find 3 � 9, bend down finger number 3. The number of fingers to the left of the bent finger shows the number in the tens place of the product. (2) The number of fingers to the right of the bent finger shows the number of ones in the product. (7) So, 3 � 9 � 27.

If You Have More TimeHave pairs play I’m Thinking of a Number. One partner writes down a number from 0 to 9, such as 7, and says: I’m thinking of a number. When it is multiplied by 9, the product is 63. What is the number? The other partner says the number. Then, students change roles and repeat.

Multiplying by 9


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Multiplying by 9

Learn how to multiply by 9 by answering 1 to 5.

1. Complete the table.

Fact ProductTwo Digits in the

ProductSum of the Two Digits

in the Product

0 � 9 � 0 0 and 0 0 � 0 � 0

1 � 9 � 9 0 and 9 0 � 9 � 9

2 � 9 � 18 1 and 8 1 � 8 � 93 � 9 � 27 2 and 7 2 � 7 � 9

4 � 9 � 36 3 and 6 3 � 6 � 95 � 9 � 45 4 and 5 4 � 5 � 96 � 9 � 54 5 and 4 5 � 4 � 97 � 9 � 63 6 and 3 6 � 3 � 98 � 9 � 72 7 and 2 7 � 2 � 99 � 9 � 81 8 and 1 8 � 1 � 9

2. Reasoning Besides the product of 0 � 9, what pattern do you see in the sums of the digits of each product?

The sum of the digits is always 9.

3. Look at the number being multiplied by 9 in each product and the tens digit of that product.

When 3 is multiplied by 9, what is the tens digit of the product? 2 .

When 6 is multiplied by 9, what is the tens digit of the product? 5 .


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Multiplying by 9 (continued)

4. Reasoning Complete to describe the pattern you see in the tens digits of the products when a factor is multiplied by 9.

The tens digit of the product is 1 less than the other factor.

5. Complete the following to find 7 � 9.

The tens digit is 7 � 1 � 6 .

The ones digit is 9 � 6 � 3 .

So, 7 � 9 � 63 and 9 � 7 � 63 .

Find each product.

6. 1 7. 9 8. 9 9. 9 _� 9 _� 2 _� 4 _� 0

9 18 36 0 10. 6 11. 9 12. 8 13. 5 _� 9 _� 9 _� 9 _� 9

54 81 72 45 14. 9 15. 3 16. 2 17. 9 _� 7 _� 9 _� 9 _� 6

63 27 18 54

18. Reasoning Joshua and his sister have each saved $9. They wish to buy a new game that costs $20. If they put their savings together, do they have enough money to buy the game?

No, they only have $18; they are $2 short.

19. Reasoning Jane said that 7 � 9 � 62. Explain how you know this is incorrect.

The sum of the digits in the product does not equal 9.



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Multiplying Three Numbers

Teacher Notes

Ongoing AssessmentAsk: What are three ways to find 1 � 2 � 3? (1 � 2) � 3; 1 � (2 � 3); and (1 � 3) � 2

Error InterventionIf students forget to multiply the third factor,

then encourage them to write either “ � 4” or “4 � ”. Where the blank shows the product of the first two factors and the number is the third factor.

If You Have More TimeHave students write problems involving products with 3 factors, for a partner to solve.


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Multiplying Three Numbers

Does it matter how you multiply 5 � 2 � 3? Answer 1–8 to find out.

To show the factors you are multiplying first, use parentheses as grouping symbols.

1. Group the first two factors together. ( 5 � 2 ) � 3

2. Multiply what is in the parentheses first. 5 � 2 � 10

3. Then, multiply the product of what isin parentheses by the third factor. 10 � 3 � 30

4. So, (5 � 2) � 3 � 30 .

5. Start again and group the last two factors together. 5 � ( 2 � 3 )

6. Multiply what is in the parentheses first. 2 � 3 � 6

7. Then, multiply 5 by the product of what is in parentheses. 5 � 6 � 30

8. So, 5 � (2 � 3) � 30 .

It does not matter how the factors are grouped; the product will be the same.

9. 5 � (2 � 3) � ( 5 � 2 ) � 3

Find 3 � 2 � 4 two different ways.

10. Do the 3 � 2 first.

3 � 2 � 6 6 � 4 � 24 So, (3 � 2) � 4 � 24 .

11. Do the 2 � 4 first.

2 � 4 � 8 3 � 8 � 24 So, 3 � (2 � 4) � 24 .


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Multiplying Three Numbers (continued)

Find each product two different ways.

12. (1 � 3) � 6 � 18 13. (5 � 2) � 4 � 40

1 � (3 � 6) � 18 5 � (2 � 4) � 40

14. (2 � 4) � 1 � 8 15. (2 � 2) � 5 � 20

2 � (4 � 1) � 8 2 � (2 � 5) � 20

Find each product.

16. 2 � 4 � 3 � 24 17. 7 � 1 � 3 � 21 18. 3 � 3 � 2 � 18

19. 3 � 2 � 6 � 36 20. (4 � 2) � 2 � 16 21. 3 � (0 � 7) � 0

22. 1 � 7 � 9 � 63 23. 8 � (2 � 3) � 48 24. (2 � 5) � 6 � 60

25. 9 � 0 � 3 � 0 26. 4 � 5 � 1 � 20 27. (3 � 6) � 1 � 18

28. Reasoning When multiplying three numbers, if one of the factors is zero, what will the answer be? Zero

29. A classroom of students is getting ready to takea test. There are 5 rows of desks in the room and4 students are in each row. Each student is required to have 2 pencils. How many pencils are needed? 40 pencils



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. 2Dividing by 8 and 9

Teacher Notes

Ongoing AssessmentAsk: What two division facts can be written using 8 � 9? 72 � 8 � 9 and 72 � 9 � 8

Error InterventionIf students have trouble remembering the multiplication facts for 8 or 9,

then use G26: Multiplying by 9 and G31: Multiplying by 8.

If You Have More TimeHave partners make a game like Memory. The partners write the expression on one card and the quotient on another card for the 8 and 9 division facts. Have partner A write the eight 8s facts beginning with 16 � 8 � 2 and ending with 72 � 8 � 9. Have partner B write the eight 9s facts beginning with 18 � 9 � 2 and ending with 81 � 9 � 9. Shuffle all cards and then place face down in a 4 by 8 array. Partner A turns over two cards, if they go together the player keeps the two cards and goes again. When a match is not made the cards are turned back over, and it is the other partner’s turn. The game is finished when all cards are matched. The partner with the most matches wins.


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Dividing by 8 and 9

Materials Have counters available for students to use.

You can use multiplication facts to help you divide.

At the museum, 32 students are divided into 8 equal groups. How many students are in each group?

Find 32 � 8.

1. To find 32 � 8, think about the related multiplication problem.

8 times what number equals 32? 8 � 4 � 32

2. Since you know 8 � 4 � 32, then you know 32 � 8 � 4 .

3. How many students are in each group at the museum? 4 students

Find 36 � 9.

4. To find 36 � 9, think about the related multiplication problem.



Find 8 � � 80 .

6. To find 8 � � 80 , think about the related multiplication problem.


7. Since you know 8 � 10 � 80, then you know 8 � � 80 � 10 .

8. Reasoning Explain how to find 56 � 8.

Think: 8 times what number equals 56. Since 8 � 7 � 56, 56 � 8 � 7.


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Dividing by 8 and 9 (continued)

Use the multiplication fact to find each quotient.

9. 8 � 2 � 16 10. 9 � 5 � 45 11. 8 � 3 � 24

16 � 8 � 2 45 � 9 � 5 24 � 8 � 3

12. 9 � 6 � 54 13. 8 � 4 � 32 14. 8 � 6 � 48

54 � 9 � 6 32 � 8 � 4 48 � 8 � 6

15. 9 � 3 � 27 16. 9 � 10 � 90 17. 8 � 9 � 72

27 � 9 � 3 90 � 9 � 10 72 � 8 � 9

Find each quotient.

7 4 4 18. 9 � � 63 19. 8 � � 32 20. 9 � � 36

8 9 2 21. 8 � � 64 22. 9 � � 81 23. 8 � � 16

5 7 5 24. 9 � � 45 25. 8 � � 56 26. 8 � � 40

27. Reasoning If you know that 8 � 12 � 96,then what is 96 � 8? 12

28. Nine friends go to lunch and split the $54 ticket evenly. How much does eachfriend pay? $6



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0 and 1 in Division

Teacher Notes

Ongoing AssessmentAsk: What is 0 � 245? 0 What is 245 � 1? 245

Error InterventionIf students have trouble understanding that division is actually taking place,

then encourage them to use counters or draw pictures to “see” what is being done. For example, 5 � 1 can be 5 counters put into 1 group to find how many are in the group. And 0 � 7 can be 0 counters put into 7 groups to find how many are in each group.

If You Have More TimePlace students in groups of 3. One student acts as a referee. The referee says a number from 2 to 9 and then says, “On your mark, get set, go.” On go, the referee holds out a fist for 0 or a hand with 1 finger up for one. The other two students race to say the product of 0 or 1 and the number between 2 and 9. The student who says the product first gets a point. The first student to 5 wins. Let students play again, until each one has a turn as referee.


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Think about related multiplication facts to help you divide.

Find 5 � 1.

1. Think: 1 times what number equals 5? 1 � 5 � 5


3. If Karina had 5 oranges to put equally in 1 basket, how many oranges would go in each basket? 5 oranges

Find 9 � 1.

4. 1 � 9 � 9 So, 9 � 1� 9 .

5. What is the result when any number is divided by 1? The number

Find 0 � 7.



8. If Karina had 0 oranges to put equally in 7 baskets,how many oranges would go in each basket? 0 oranges

Find 0 � 2.

9. 2 � 0 � 0 So, 0 � 2 � 0 .

10. What is the result when zero is dividedby any number (except 0)? 0

Find 5 � 0.

11. Reasoning If Karina had 5 oranges to put equally in 0 baskets, how many oranges would go in each basket? Explain.

Karina can not put 5 oranges into 0 baskets.You cannot divide a number by 0.

0 and 1 in Division


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Find 4 � 4.



14. If Karina had 4 oranges to put equally in 4 baskets, how many oranges would go in each basket? 1 orange

Find 8 � 8.

15. 8 � 1 � 8 So, 8 � 8 � 1 .

16. What is the result when any number (except 0) is divided by itself? 1

Find each quotient.

17. 4 � 1 � 4 18. 0 � 5 � 0 19. 6 � 6 � 1

0 1 1 20. 3 � � 0 21. 9 � � 9 22. 5 � � 5

6 1 0 23. 1 � � 6 24. 1 � � 1 25. 8 � � 0

26. Reasoning Use the rule for division by 1 to find 247 � 1. Explain.

A number divided by 1 equals the same number, so 247 � 1 � 247.

27. Larry has 3 friends who would like some cookies but he has no cookies to give them. How many cookies can Larry give each friend?

Each friend gets zero cookies.

0 and 1 in Division (continued)

Intervention Lesson C26

Math Diagnosis and Intervention SystemIntervention Lesson C26

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Teacher Notes

Ongoing AssessmentAsk: How would you break apart 36 to add 48 � 36? Change 36 into 2 � 34. Why? To make a ten, 48 � 2 � 50.

Error InterventionIf students have trouble remembering what they broke each number into,

then encourage them to write the addition problem above each addend. For example, in 23 � 45, have students write 20 � 3 above the 23 and 40 � 5 above the 45. Then they can see the tens and ones that need to be added.

If You Have More TimeHave students describe situations when they might want to add mentally, such as shopping.


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Intervention Lesson C26 107

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Using Mental Math to Add

Materials place-value blocks: 6 tens and 12 ones per pair

Find the sum of 26 and 42 by breaking apart each addend.

1. Show 26 with place value blocks.

2 tens � 6 ones �



3. Add the tens. 20 � �

Add the ones. 6 � �

4. Add the tens and the ones together. � 8 �

So, 26 � 42 � .

Find the sum of 18 and 34 by breaking apart the second addend.


1 ten � 8 ones �



7. Take 2 ones from the 34 and add them to 18. What sum do you have now?

18 � 34 � �

8. Add. 20 � 32 �

So, 18 � 34 � .

20 6

40

20 32

2

40 60

2 8

6860

68

52

52

10 8

30 4

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Using Mental Math to Add (continued)

Find each sum using mental math.

9. 22 � 56 � 10. 37 � 24 � 11. 43 � 36 �

12. 55 � 32 � 13. 23 � 21 � 14. 43 � 44 �

15. 44 � 34 � 16. 52 � 32 � 17. 45 � 4 �

18. 45 � 34 � 19. 37 � 51 � 20. 23 � 46 �

21. 64 � 23 � 22. 26 � 73 � 23. 35 � 63 �

24. 88 � 26 � 25. 39 � 45 � 26. 57 � 16 �

Fill in the blanks to show how to add mentally.

27. 35 � 12 � 40 � � 28. 83 � 46 � � 9 �

29. 49 � 16 � 50 � � 30. 78 � 24 � 80 � �

31. Reggie has 25 crayons. Brett gives him 14 more. How many crayons does he have now?

32. Darla bought 32 stickers on Monday. Two days later she bought 46 more. How many stickers does she have altogether?

33. Rafael has 41 rocks in his rock collection. His friend gave him 18 more rocks. How many rocks did he have then?

34. Reasoning To add 59 and 16, Juan took one from the 16to make the 59 a 60. What number should he add to 60?

35. Reasoning To add 24 and 52, Ashley first added 24 and 50. What numbers should she add next?

78 61 79

87 44 87

78 84 49

79 88 69

87 99 98

114 84 73

7 47 120 129

15 65 22 102

39

78

59

15

74 � 2



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Using Mental Math to Subtract

Teacher Notes

Ongoing AssessmentAsk: When solving 38 � 23, why wouldn’t you round the 38 to 40 and then subtract 23? Sample answer: Subtracting 40 � 23 is much more difficult. It is much easier to subtract a ten from a number.

Error InterventionIf students continually forget to add the extra to the number being subtracted from,

then encourage the students to say to themselves, “What I do to one I have to do to the other.” Then have them put the number being added above the original numbers. For example, in 23 � 17, the student would write a � 3 above both the 23 and the 17. That way they know their new problem is 26 � 20.

If You Have More TimeHave students tell which method they prefer to use and why.


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Using Mental Math to Subtract

20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Find the difference of 46 � 27 one way, by doing the following.

1. Round the number being subtracted.

27 rounded to the nearest ten is .

2. Solve the new problem.

46 � 30 �

3. Since you rounded 27 to 30, did you subtract too much or too little from 46?

4. How much more is 30 than 27?

5. Since 30 is 3 more than 27, you subtracted too much. You must now add 3 to the difference in Question 2.

16 � 3 �

6. So, 46 � 27 � .

Find the difference of 46 � 27 another way, by doing the following.

7. How much needs to be added to the 27 so that it forms a ten? 27 � � 30

8. Since you added 3 to 27, you need to add 3 to 46. 46 � 3 �

9. Solve the new problem. 49 � 30 �

10. So, 46 � 27 � .

11. How can you change 52 � 18 to make it easier to subtract mentally?

52 � 18 � � 20 �

3

30

19

54 34

too much

19

19

16

49

3

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Using Mental Math to Subtract (continued)

Find each difference using mental math.

12. 57 � 38 � 13. 32 � 17 � 14. 61 � 26 �

15. 85 � 29 � 16. 43 � 28 � 17. 67 � 42 �

18. 32 � 18 � 19. 52 � 46 � 20. 41 � 18 �

21. 28 � 16 � 22. 55 � 33 � 23. 86 � 23 �

24. 39 � 26 � 25. 57 � 28 � 26. 93 � 34 �

27. 62 � 47 � 28. 33 � 16 � 29. 84 � 35 �

30. Reasoning To find 56 � 48, add the same amount to both numbers to make it easier to subtract. Explain what you did to solve the problem.

56 � 48

31. Lupe has $32. She buys a present for her mother and gets $9 in change. How much money did she spend on the present?

32. Reasoning Becca subtracts 73 � 26 mentally by thinking: “73 � 30 � 43, and 43 � 4 � 39. The answer is 39.” What did she do wrong? Explain.

19

Add 2 to both numbers: 56 � 2 � 58; 48 � 2 � 50. 58 � 50 � 8; So, 56 � 48 � 8.

15 35

56 15 25

14 6 23

12 22 63

13 29 59

15 17 49

Sample answer: Becca added 4 to the 26 to get 30. Since she subtracted 4 too much, she should have added 4 to the difference. The answer should be 43 � 4 � 47.

$23



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. 2Adding Two-Digit Numbers

Teacher Notes

Ongoing AssessmentAsk: Does every addition problem need regrouping? No, only if there is more than 9 ones.

Error InterventionIf students can not remember the addition facts,

then use some of the addition fact lessons B8 to B15 and B26 to B30.

If You Have More TimeHave students write all the two-digit numbers that can be added to 46 where regrouping is not needed. (10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 40, 41, 42, 43, 50, 51, 52, 53, 60, 61, 62, 63, 70, 71, 72, 73, 80, 81, 82, 83, 90, 91, 92, 93)


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Adding Two-Digit Numbers


There are 25 boys and 38 girls at the library. How many children total?

1. Show 25 using place-value blocks.

2. Show 38 using place-value blocks.

3. Add 25 � 38 to find the total children.

Add the ones. 5 � 8 �

4. Do you have more then 10 ones? Tens Ones

2 5

� 3 8

5. Since you have 13 ones, regroup them into tens and ones

13 ones � ten and ones

6. Record the 3 ones at the bottom of the ones column of the Tens and Ones chart. Record the 1 ten at the top of the tens column.

7. Add the tens. Add the 1 ten that you regrouped, the 2 tens from the 25, and the 3 tens from the 38.

1 ten � 2 tens � 3 tens � tens

8. Record the tens at the bottom of the tens column of the Tens and Ones chart.

9. So, 25 � 38 �

How many children are at the library? .

10. Use place value-blocks and the Tens and Ones Tens Ones

4 6

� 2 9

chart to add 46 � 29.

13

yes

1 3

6

63

1

6 3

1

7 5

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Adding Two-Digit Numbers (continued)

Add.

11. Tens Ones

1 3

� 2 8

12. Tens Ones

2 4

� 2 9

Add. Use a tens and ones chart if you like.

113. 58 14. 56 15. 18 16. 20 _� 17 _� 11 _� 19 _� 28 75

17. 46 18. 36 19. 17 20. 45 _� 45 _� 17 _� 49 _� 14

21. 32 22. 26 23. 22 24. 33 _� 66 _� 37 _� 65 _� 33

25. 21 26. 17 27. 36 28. 64 _� 39 _� 29 _� 16 _� 27

29. A puppy weighs 15 pounds. His mother weighs 65 pounds. How much do the puppy and his mother weigh together?

30. Reasoning What number do you add to 19 to get 30?

1

4 1

1

5 3

67 37 48

53 66 5991

63 87 6698

80 pounds

11

46 5260 91



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Subtracting Two-Digit Numbers

Teacher Notes

Ongoing AssessmentAsk: What can you do if you forget a subtraction fact like 12 � 8? Think of the related addition fact, such as what plus 8 equals 12?

Error InterventionIf students do not recognize the need to regroup and simply subtract the smaller ones value from the larger ones value such as 4 � 1 in 31 � 14,

then encourage students to circle the greater ones value. If the circled number is on the bottom, then they need to regroup.

If students can not remember the subtraction facts,

then use some of the subtraction fact lessons B19 to B24 and B34 to B39.

If You Have More TimeHave students write a real-world subtraction problem for a partner to solve.


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Subtracting Two-Digit Numbers


There are 34 kittens and 16 puppies. How many more kittens than puppies?

1. Show 34 with place-value blocks.

2. Do you have enough ones to take away 6 ones?

3. Regroup 1 ten into 10 ones. Show this with your place-value blocks.

3 tens and 4 ones � tens and 14 ones.

4. Cross out the 3 tens in the Tens and Ones chartand write 2 above it. Cross out the 4 ones and write 14 above it.

Tens Ones

3 4

� 1 6

5. Now, take away 6 ones and write the difference at the bottom of the ones column.

14 ones � 6 ones � ones

6. Subtract the tens and write the difference at the bottom of the tens column.

2 tens � 1 ten � ten

7. So, 34 � 16 �

How many more kittens than puppies are there?

8. Use place-value blocks and the Tens and Ones Tens Ones

5 6

� 2 7

chart to subtract 56 � 27.

no

2

8

1

2 14

1 8

4 16

2 9

18

18


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Subtracting Two-Digit Numbers (continued)

Subtract.

9. Tens Ones

4 2

� 1 9

10. Tens Ones

5 0

� 2 4

Subtract. Use a Tens and Ones chart if you like.

11. 35 12. 80 13. 45 14. 61 _� 17 _� 38 _� 39 _� 13

15. 74 16. 22 17. 50 18. 48 _� 45 _� 18 _� 32 _� 20

19. 95 20. 34 21. 61 22. 90 _� 69 _� 7 _� 26 _� 74

23. Thompson has 32 flowers. If he plants 18 flowers in the front yard, how many will he have left?

24. Reasoning In which problem do you need to regroup to subtract, 53 � 28 or 58 � 23? Explain.

3

2 3

4

2 6

42 6 48

4 18 2829

27 35 1626

53 � 28; There are not enough ones in 53 to take away the 8 ones in 28. However, there are enough ones in 58 to take away the 3 ones in 23.

12 10

18

14



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. 2Adding Three Numbers

Teacher Notes

Ongoing AssessmentAsk: When adding three numbers, do you always have to regroup? No, you only have to regroup if you have more than 9 ones or 9 tens.

Error InterventionIf students are having problems with their basic facts,

then use one of the lessons on addition facts, B27, B28, or B29.

If students are having problems regrouping tens.

then use G9: Adding Two-Digit Numbers.

If You Have More TimeHave each student write a two- or three-digit number on paper. Then place students into groups of 3 and find the sum of the 3 numbers. Continue to have students form random groups three more times. Have each student share with the class their highest and lowest sums.


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Adding Three Numbers

Materials place-value blocks: 2 hundreds, 6 tens, and 14 ones per pair or group

How many total pieces of fruit are in a box containing 45 apples, 107 oranges, and 112 bananas?

1. Show 45, 107, and 112 using place-value blocks.

2. Add 45 � 107 � 112 to find the total pieces of fruit in the box.

3. Do you have more then 10 ones? Add the ones.

5 ones � 7 ones � 2 ones � ones

4. Since you have 14 ones, regroup them into tens and ones.

14 ones � ten and ones

5. Record the 4 ones at the bottom of the Hundreds Tens Ones

4 5

1 0 7

� 1 1 2

ones column of the Hundreds, Tens, and Ones chart. Record the 1 ten at the top of the tens column.

6. Add the tens.

1 ten � 4 tens � 1 ten � tens

7. Do you have more than 10 tens?

8. Record the tens at the bottom of the tens column of the chart.

9. Add the hundreds and record the value at the bottom of the hundreds column.

1 hundred � 1 hundred � hundreds

10. So, 45 � 107 � 112 � 264

How many total pieces of fruit are in the box?

1 4

6

2

no

1

6 42

yes

264

14


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Adding Three Numbers (continued)

Add.

11. Hundreds Tens Ones

2 5 4

1 2 9

� 6 2

12. Hundreds Tens Ones

1 1 7

1 0 6

� 7 4

13. 123 14. 211 15. 23 16. 322 365 423 45 43 _� 50 _� 23 _� 14 _� 16

17. 335 18. 543 19. 613 20. 851 125 144 205 32 _� 32 _� 46 _� 64 _� 40

21. There were 234 books returned to the library on Monday, 109 books returned on Tuesday, and 41 books returned on Wednesday. How many books were retuned to the library in the three days?

22. Reasoning Write the smallest 2-digit number that when added to 345 and 133 would require regrouping of both the ones and the tens.

657 82 381538

384

1

4 5

1

4

1

9 72

733 882 923492

22

Intervention Lesson D59

Math Diagnosis and Intervention SystemIntervention Lesson D59

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Solid Figures

Teacher Notes

Ongoing AssessmentAsk: Which two solids are the most alike? Cube and rectangular prism; they have the same number of faces, edges, and vertices.

Error InterventionIf students have trouble naming the shapes of the faces or counting the number of faces, edges, and vertices,

then use D50: Flat Surfaces of Solid Figures, D57: Flat Surfaces and Corners, and D58: Faces, Corners, and Edges.

If You Have More TimeHave a “Solid Bee.” Put solids into a bag, including at least one of each discussed in the lesson. Mix in real life objects like a ball, piece of chalk, eraser, and number cube. Have students stand in line. Say: “I need to know the name of this solid.” Then pull a solid out of the bag. The first student in line names the solid. If the name is correct, the student goes to the end of the line. If the name is incorrect, the student sits down. Give each student a turn naming a solid. Each round, ask a different question such as: I need to know how many faces (or flat surfaces) this solid has;I need to know how many edges this solid has;I need to know how many vertices this solid has.


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Intervention Lesson D59 207

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Materials power solids arranged in stations around the room

Find each solid to complete the tables below.

SolidNumber of Faces

Number of Edges

Number of Vertices

Shapes of Faces

1. Pyramid

5 8 5

1 square

4 triangles

2. Rectangular Prism

6 12 86

rectangles

3. Cube

6 12 8 6 squares

Objects that roll do not have faces, edges, or vertices.

SolidNumber of Flat

SurfacesShape of Flat Surfaces

4. Cone

1 1 circle

SolidNumber of Flat

SurfacesShape of Flat Surfaces

5. Cylinder

2 2 circles

6. Sphere

0

Name the solid figure that each object looks like.

7. 8. 9.

sphere cylinder

rectangleprism

Use the solids in the table above to answer Exercises 10–12.

10. Which solid figure has 2 flat surfaces that are circles?

sphere 11. Which of the 6 solid figures has 6 rectangular faces?

rectangular prism 12. Which 3 figures have no vertices?

cylinder, cone, sphere 13. Reasoning How are the sphere and cone alike?

Sample answer: They both can roll.208 Intervention Lesson D59

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. 2Acute, Right, and Obtuse Angles

Teacher Notes

Ongoing AssessmentAsk: What type of angle is formed by the hands on the clock when it shows the time school starts? Answer will vary by school start times.

Error InterventionIf students confuse acute and obtuse,

then help students by telling them that people often say to a baby “Look how little you are. You are so cute.” So, a little baby is “acute”. This will help them remember that acute is smaller than a right angle. You can also say the word “acute” with a small, squeaky voice and the word “obtuse” with a big, burly voice.

If You Have More TimeHave students play a math version of “Simon Says”. Have a student be Simon, stand in the front of the class room, and say statements such as the following: “Simon says make an obtuse angle.” Students can show acute, right, and obtuse angles with both arms. They can also show a ray by pointing with one arm extended in any direction.Those who correctly make an obtuse angle continue. Those who do not must sit down. Students who make the figure when Simon doesn’t say “Simon says” must also sit down.


Acute, Right, and Obtuse Angles

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Materials 1 inch square piece of paper for each student, crayons or markers

A ray is part of a line. The endpoint is the beginning rayof the ray, and the arrow shows it goes on forever.

An angle is made by two rays that have the sameendpoint. That endpoint is called the vertex. angle

vertex

1. Color each ray of the angle at theright, a different color.

Check student’s coloringPlace a side of your square on one ray, and the corner on the vertex for each angle in 2 to 4.

2. Reasoning Right angles are shown below. What do you notice about the openings of right angles?

Sample answer: They are the same size as the corner of a piece of paper.

3. Reasoning Obtuse angles are shown below. What do you notice about the openings of obtuse angles?

Sample answer: They are all larger than the corner of a piece of paper.

4. Reasoning Acute angles are shown below. What do you notice about the openings of acute angles?

Sample answer: They are all smaller than the corner of a piece of paper.

Write ray, vertex, right angle, acute angle, or obtuse angle to name each.

5. 6. 7.

obtuse angle right angle acute angle

8. 9. 10.

vertex right angle ray

What kind of angle do the hands of each clock show?

11. 12. 13.

acute angle right angle obtuse angle

214 Intervention Lesson D62

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Polygons

Teacher Notes

Ongoing AssessmentAsk: Why is a circle not a polygon? A polygon must have sides that are line segments. A circle has no line segments.

Error InterventionIf students count the same vertex or side twice,

then have them put an x on each side or vertex as they count it. This will help students to avoid counting a vertex or side more than once.

If You Have More TimeHave students make polygon books. Give each student 3 half-sheets of white paper. With all 3 sheets together, have them fold the papers to make a book. Students should title their book with something having to do with polygons. The first two-page spread should have the heading “Triangles.” Let the students use crayons or markers to draw examples of different types of triangles. Also, let them print pictures from the internet or cut out pictures from magazines. Make other two-page spreads for quadrilaterals, pentagons, hexagons, and octagons. The cover, made with construction paper, can be a picture drawn by using only polygons.


Polygons

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Box A Box B

1. The figures in Box A are polygons. The figures in Box B are not. How are the figures in Box A different from those in Box B?

Answers will vary.

To be a polygon:

• All sides must be made of straight line segments. • Line segments must only intersect at a vertex. • The figure must be closed.

Polygons are named by the number of sides each has. Complete the table.

Shape Number of Sides Number of Vertices Name2.

3 3 Triangle

3.4 4 Quadrilateral

4.

5 5 Pentagon

5.

6 6 Hexagon

6.

8 8 Octagon

Tell if each figure is a polygon. Write yes or no.

7. 8. 9.

no yes no

Name each polygon. Then tell the number of sides and the number of vertices each polygon has.

10. 11.

hexagon; 6, 6 pentagon; 5, 5

12. 13.

triangle; 3, 3 quadrilateral; 4, 4

14. 15.

octagon; 8, 8 quadrilateral; 4, 4

16. Reasoning What is the least number of sides a polygon can have? 3 sides

17. Reasoning A regular polygon is a polygon with all sides the same length. Circle the figure on the right that is a regular polygon.


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Classifying Triangles Using Sides and Angles

Teacher Notes

Ongoing AssessmentAsk: What type of angles are in an equilateral triangle? acute

Error InterventionIf students have trouble identifying right angles, acute angles, and obtuse angles,

then use I4: Acute, Right, and Obtuse Angles.

If You Have More TimeHave students draw a triangle. Trade with a partner and have the partner identify the triangle by its sides and then by its angles.


Classifying Triangles Using Sides and Angles

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Materials 2 yards of yarn, scissors, 6 sheets of construction paper, markers for each student and glue

Create a book about triangles by following 1 to 7.

1. Put the pieces of construction paper together and fold them in half to form a book. Punch two holes in the side and use yarn to tie the book together. Write “Triangles” and your name on the cover.

Each two-page spread will be about one type of triangle. For each two page spread:

• Write the definition on the left page. • Write the name of the triangle near the top of the right page.• Create a triangle with yarn pieces and glue the yarn pieces under the name of the triangle to illustrate the triangle.

2. Pages 1 and 2 should be about an equilateral triangle. This triangle has 3 sides of equal length. So, your 3 yarn pieces should be cut to the same length.

3. Pages 3 and 4 should be about an isosceles triangle. This triangles has at least two sides the same length. Cut 2 pieces of yarn the same length and glue them on the page at an angle. Cut and glue a third piece to complete the triangle.

4. Pages 5 and 6 should be about a scalene triangle. This triangle has no sides the same length. So your 3 yarn pieces can be cut to different lengths.

5. Pages 7 and 8 should be about a right triangle. This triangle has exactly one right angle. Two ofyour yarn pieces should be placed so that they form a right angle. Cut and glue a third piece to complete the triangle.

6. Pages 9 and 10 should be about an obtuse triangle. This triangle has exactly one obtuse angle. Two pieces of yarn should be placed so that it forms an obtuse angle. Cut and glue down a third yarn piece to complete the triangle.

7. Pages 11 and 12 should be about an acute triangle. This triangle has three acute angles. Your 3 yarn pieces should be placed so that no right or obtuse angles are formed.

Tell if each triangle is equilateral, isosceles, or scalene.

8. 9. 10.

isosceles scalene equilateral

Tell if each triangle is right, acute, or obtuse.

11. 12. 13.

acute obtuse right

14 How many acute angles does an acute triangle have? 3

15. Reasoning How many acute angles does a right triangle have? 2

16. Describe this triangle by its sides and by its angles.(Hint: Give it two names.)

acute isosceles


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Classifying Triangles Using Sides and Angles (continued)

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Quadrilaterals

Teacher Notes

Ongoing AssessmentAsk: Are all squares rectangles? yes Are all rectangles squares? no

Error InterventionIf students list only one name for rectangles, squares, or rhombuses,

then ask students leading questions so they can discover that other quadrilateral name(s) can also be used.

If You Have More TimePut students in pairs. Each pair needs five index cards labeled square, rectangle, rhombus, trapezoid, and parallelogram. Have one student shuffle and draw a card. Both students then need to draw an example of the quadrilateral. Students should compare drawings. Tell them to describe the different ways a quadrilateral can be drawn. Help students to discover that quadrilaterals may be different sizes, but they will always have their specific characteristics.

In items 8–13, each quadrilateral should be circled with the color listed below

8. red, blue, green and orange

9. purple

10. green

11. blue, green

12. orange, green

13. purple


Quadrilaterals

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Materials Have quadrilateral power shapes available for students who want to use them.

For 1 to 5 study each quadrilateral with your partner. Identify the types of angles. Compare the lengths of the sides. Then draw a line to match the quadrilateral with the best description. Descriptions can be used only once.

1. Trapezoid Four right angles and all four sides the same length

2. Parallelogram

3. Rectangle

All sides are the same length

4. Square

Exactly one pair of

parallel sides

Two pairs of parallel sides

5. Rhombus

Four right angles and opposite sides

the same length

6. Reasoning What quadrilateral has four right angles and opposite sides the same length, and can also be called a rectangle? square

7. Reasoning What quadrilaterals have two pairs of parallel sides, and can also be called parallelograms?

rectangle, rhombus, square

For Exercises 8–13, circle squares red, rectangles blue, parallelograms green, rhombuses orange and trapezoids purple. Some quadrilaterals may be circled more than once.

See teachers note page. 8. 9. 10.

11. 12. 13.

14. I have two pairs of parallel sides, and all of my sides are equal, but I have no right angles. What quadrilateral am I? rhombus

15. I have two pairs of parallel sides and 4 right angles, but all 4 of my sides are not equal. What quadrilateral am I? rectangle

16. Name all of the quadrilaterals in the picture at the right.

rectangle, rhombus, parallogram, trapezoid

17. Reasoning Why is the quadrilateral on the right a parallelogram, but not a rectangle?

Sample answer: Both a rectangle and a parallelogram have opposite sides parallel. A rectangle must also have four right angles. This quadrilateral does not have four right angles, so it is a parallelogram, but not a rectangle.


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Quadrilaterals (continued)

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Practice A42

Practice

A42Equal 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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Practice A43

Practice

A43Parts of a RegionWrite the fraction of each figure that is shaded.

1. 2. 3. 4.


5. 3 __ 12 6. 1 __ 4 7. 4 __ 5

In 8 and 9, use the information below.

Three parts of a rectangle are red. Two parts are blue.

8. What fraction of the rectangle is red?

10. A banner is made of 8 equal parts. Five of the parts contain stars. Three of the parts contain hearts. Draw the banner.

9. What fraction of the rectangle is blue?

11. How can you write the fraction 4 __ 6 in word form?

fourth sixth four sixes four sixths fourth six

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Practice A47

Practice

A47Using Models to Compare FractionsCompare. Write >, <, or =.

1. 2.

2 __ 4 1 __ 3 3 __ 8 1 __ 2

3. 4.

3 __ 4 6 __ 8 1 __ 5 2 __ 8

5. 6.

4 __ 6 2 __ 3 3 __ 10 1 __ 6

7. 8.

1 __ 5 1 __ 6 2 __ 6 1 __ 3

9. Give 3 fractions with denominators that are less than 6 __ 8 .

10. Which fraction is the same as 1 __ 2 ?

1 __ 4 3 __ 6 3 __ 8 3 __ 4

14

14

13

12

18

18

18

15

18

18

14

14

14

18

18

18

18

18

18

110

110

110

16

16

16

16

16

13

13

16

16

13

16

15

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Practice A48

Practice

A48Using Models to Find Equivalent FractionsComplete each number sentence.

1. 1

110

110

15

2. 1

112

112

112

112

112

112

112

112

112

14

14

14

1 __ 5 � _____ 10 3 __ 4 � _____ 12

3. 1

16

16

16

110

110

110

110

110

4. 1

112

112

112

14

3 __ 6 � _____ 10 1 __ 4 � _____ 12

5. 1

110

110

110

110

110

110

110

110

15

15

15

15

4 __ 5 � _____ 10

Complete each pattern.

6. 1 __ 3 , 2 __ 6 , 3 __ 9 , 4 _____ 7. 1 __ 2 , 2 __ 4 , 3 __ 6 , 4 __ 8 , 5 _____ , 6 _____

8. Samuel has read 5 __ 6 of his assignment. Judy has read 10 __ 12 of her assignment. Which sentence is true?

Samuel read more than Judy. Judy read more than Samuel.

They read the same amount. They will both finish the assignment at the same time.

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Practice A74

Practice

A74Repeating PatternsDraw the next three shapes to continue the pattern.

1.

2.

Write the next three numbers to continue the pattern.

3. 4, 6, 2, 8, 4, 6, 2, 8, 4, 6, 2, 8 4. 3, 3, 5, 3, 3, 5, 3, 3, 5

5. What is the 16th shape in the pattern below?

6. Mrs. Washington placed students in a line. The order was 1 boy, 2 girls, 2 boys, 2 girls, 3 boys, 2 girls and continued. Was the 15th student a boy or a girl?

7. Create a pattern using your favorite letters.

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Practice B45

Practice

B45Using Multiplication to CompareFind each amount.

1. 2 times as many as 5 2. 3 times as many as 7

× � × �


5. twice as many as 8 6. 5 times as many as 3



11. John has 5 computer games. Julian has twice as many computer games as John. How many computer games do they have in all?

12. George Washington is on the $1 bill. Abraham Lincoln is on the bill that is worth 5 times as much as the $1 bill. What bill is Abraham Lincoln on?

13. Paula has twice as many guests this week as she did last week. Last week she had 7 guests. How many guests does she have this week?

14. John F. Kennedy is on the coin that is worth 5 times as much as a dime. What coin is John F. Kennedy on?

nickel quarter half-dollar dollar

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Practice B48

Practice

B48Multiplying by 9Find each product.

1. 4× 9

2. 7× 9

3. 9× 9

4. 8× 9

5. 3× 9

6. 9× 5

7. 2× 9

8. 6× 9

9. 2× 7

10. 8× 9

11. Multiply 4 and 9. 12. Find 3 times 9.

13. Find the product of 14. Multiply 9 and 1.

9 and 10.

15. Paula’s hair was put into 9 braids. Each braid used 3 beads. How many beads were used in all?

16. A baseball game has 9 innings. A doubleheader is 2 games in the same day. How many innings are there in a doubleheader?

17. Write a multiplication story for 9 × 8. Include the product in your story.

18. Which number below is a multiple of 9?

35 46 54 65

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Practice B56

Practice

B56Multiplying Three NumbersFind each product two diffeent ways.

1. 2 × 3 × 3 2. 2 × 2 × 4 3. 8 × 2 × 2 4. 6 × 2 × 3

5. 3 × 3 × 4 6. 5 × 2 × 5 7. 5 × 4 × 2 8. 4 × 2 × 3

Find the missing number.

9. 4 × 5 × 2 10. 5 × 2 × 8 11. 2 × 2 × 5

12. 2 × 5 × 6 13. 3 × 2 × 5 14. 4 × 9 × 0

15. Which number makes this number sentence true?

8 × 2 × 4 = 8 × (☐ × 4)

2 4 8 64

16. Which number makes this sentence true?

(5 × 3) × 4 = 5 × (☐ × 3)

2 3 4 5

17. Write three ways to find 3 × 2 × 4.

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Practice B62

Practice

B62Dividing by 8 and 9Use the multiplication fact to find each quotient.

1. 6 � 8 � 48 2. 9 � 2 � 18 3. 7 � 7 � 49

48 � 8 � 18 � 9 � 49 � 7 �

4. 8 � 8 � 64 5. 9 � 5 � 45 5. 6 � 7 � 42

64 � 8 � 45 � 9 � 42 � 7 �

6. 9 � 8 � 72 7. 9 � 4 � 36 8. 5 � 3 � 15

72 � 8 � 36 � 9 � 15 � 3 �

Find each quotient.

10. 8��56 11. 9��81 12. 8��40

13. 9��90 14. 9��63 15. 8��32

16. 3��90 17. 11��99 18. 5��45

19. Adam made 19 paper cranes Monday and 8 more Tuesday. He gave 9 friends an equal number of cranes. How many cranes did each friend receive? Explain how you found your answer.

20. A short story consists of 81 pages. Andrea will read 9 pages each day. How many days will it take Andrea to finish the story?

6 7 8 9

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Practice B63

Practice

B630 and 1 in DivisionFind each quotient.

1. 0 � 6 2. 8 � 8 3. 6 � 16 � 0 � 0 8 � 1 � 8 6 � 1 � 6

So, 0 � 6 � So, 8 � 8 � So, 6 � 1 �

4. 5. 6. 7. 8. 1��5 4��0 6��6 1��8 1��3

9. 10. 11. 12. 13. 3��24 6��42 8��72 5��30 7��63

14. Find 0 divided by 2. 15. Divide 7 by 1. 16. Find 4 divided by 4.

Write �, �, or � to compare.

17. 6 � 6 8 � 8 18. 0 � 5 5 � 5 19. 9 � 1 7 � 1

20. Tickets for rides cost $1 each at the fair. Bob has $6 to buy tickets. How many tickets can Bob buy?

21. Nikki is the goalie on her soccer team. She has allowed 0 goals in 8 games. How many goals has she allowed in each game?

22. Why is 10 � 0 � 10, but 0 � 10 � 0? Explain.

23. Which has the greatest quotient?

6 � 6 5 � 1 0 � 3 8 � 8

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Practice C26

Practice

C26Using Mental Math to AddFind the sum by breaking apart each addend.

1. 53 + 34

Add the tens. 50 + =

Add the ones. 3 + =

Add the tens and the ones together.

80 + 7 =

So, 53 + 34 =

2. 41 + 28


Add the ones. 1 + =


60 + 9 =

So, 41 + 28 =

Find the sum by breaking apart the second addend

3. 27 + 24

Take 3 ones from the 24 and add them to the 27.What sum do you have now?

27 + 3 =

Add 30 + 21 = So, 27 + 24 = 51

4. 54 + 19

Take 1 one from the 54 and add it to the 19.What sum do you have now?

19 + 1 =

Add 53 + 20 = So, 54 + 19 = 73


5. 52 + 26 6. 47 + 8 7. 32 + 17 8. 28 + 31

9. 43 + 38 10. 72 + 7 11. 42 + 33 12. 36 + 14

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Practice C27

Practice

C27Using Mental Math to SubtractFind each difference using mental math.

1. 38 − 14 2. 42 − 13 3. 55 − 12 4. 62 − 17

5. 72 − 19 6. 94 − 11 7. 32 − 15 8. 85 − 18

9. 43 − 16 10. 66 − 15 11. 53 − 19 12. 72 − 16

13. Gillian started solving 88 − 29. This is what she did.

88 − 29 = ? 88 − 30 = 58

What should Gillian do next?

14. Tell how to find 81 – 16 using mental math.

15. Tiffany needs 63 tiles for her art project. She only needs 17 more tiles. Use mental math to find how many tiles she has already.

16. To solve 35 – 19, Jack used 35 – 20 and then

added 1. subtracted 1. subtracted 9. added 9.

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Practice C28

Practice

C28Adding 2-Digit NumbersUse place-value blocks and the Tens and Ones chart to add.

1. 2.

3. 4.

Add. Use a Tens and Ones chart if you like.

5. 53+ 45

6. 37+ 21

7. 63+ 24

8. 59+ 76

9. 29+ 44

10. There are 72 people on a train when 25 more people enter. How many people are on the train now?

79 87 97 98

Tens Ones

5 2

� 1 9

Tens Ones

1 6

� 4 8

Tens Ones

4 7

� 3 4

Tens Ones

2 8

� 2 5

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Practice C29

Practice

C29Subtracting 2-Digit NumbersSubtract.

1. 2.

3. 4.


5. 34− 16

6. 43− 27

7 76− 28

8. 65− 38

9. 82− 47

10. The tree farm had 65 shade trees for sale. It sold 39 of the trees. How many trees did the farm have left?

26 36 94 104

Tens Ones

8 2

� 4 7

Tens Ones

6 3

� 3 5

Tens Ones

8 2

� 6 5

Tens Ones

7 3

� 3 5

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Practice C37

Practice

C37Adding Three NumbersAdd.

1.

2.

Find each sum.

3. 7536

+ 58

4. 142297

+ 116

5. 52497

+ 176

6. 27318764

+ 249

7. 31948

136+ 347

8. Kyle has 378 pennies, 192 nickels, and 117 dimes. How many coins does he have all together?

495 570 677 687

Hundreds Tens Ones

1 4 3

2 1 9

� 4 7

Hundreds Tens Ones

3 5 6

1 7 1

� 6 3

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Practice D59

Practice

D59Solid FiguresName the solid figure.

1. 2. 3.

4. 5. 6.


7. 8. 9. 10.

11. Reasoning What solid figures would youget if you cut a cube as shown?

12. What solid figure does this figure most resemble?

Cylinder Cone Pyramid Sphere

2

43

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Practice D62

Practice

D62Acute, Right, and Obtuse AnglesWrite ray, vertex, right angle, acute angle, or obtuse angle to name each.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. At what time do the hands of a clock form an acute angle? 2:00 4:00 6:00 8:00

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Practice D63

Practice

D63

11. How many more sides does an octagon have than a pentagon?

1 2 3 4

9. Juan said that the two figures below are quadrilaterals. Is he correct? Explain.

10. If two of the line segments of a polygon are parallel, what is the least number of sides it could have?

PolygonsName the polygon.

1. 2. 3. 4.

Is each figure a polygon? If it is not, explain why.

5. 6. 7. 8.

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Practice D64

Practice

D64Classifying Triangles Using Sides and AnglesTell if each triangle is equilateral, isosceles, or scalene.

1. 2. 3. 4.


5. 6. 7. 8.

9. Can a triangle have 2 right angles? Explain.

10. What is the least number of acute angles that a triangle can have?

11. Which pair of triangle namesidentifies the figure?

Equilateral triangle, acute triangle Equilateral triangle, right triangle Scalene triangle, acute triangle Isosceles triangle, obtuse triangle

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Practice D65

Practice

D65QuadrilateralsWrite as many special names as possible for each quadrilateral.

1. 2. 3. 4. 5.

In 6–9, write the name that best describes the quadrilateral. Draw a picture to help.

10. Can a rectangle also be a rhombus?

11. Which of the following correctly names the figure?

Rhombus Trapezoid Parallelogram Rectangle

6. A parallelogram with 4 equal sides, but no right angles.

8. A figure that is not a parallelogram, with one pair of parallel sides.

7. A rectangle with 4 right angles and all sides the same length.

9. A parallelogram with 4 right angles and sides of different lengths and widths.

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A42 A43

A47 A48Answers: A42, A43, A47, A48

Answers for Practice

A42, A43, A47, A48

equalhalves

unequal equaleighths

equalfourths

eighths thirds fifths sixths

Answers will vary.

4

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Practice A42

Practice

A42Equal 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

45094_Practice_A42-D65.indd A42 6/30/08 12:00:45 PM

1 _ 6 2 _ 5 1 _ 2 7 __ 12

Sample Answers.

3 _ 5 2 _ 5

Answers will vary.

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Practice A43

Practice

A43Parts of a RegionWrite the fraction of each figure that is shaded.

1. 2. 3. 4.


5. 3__12 6. 1__

4 7. 4__5

In 8 and 9, use the information below.

Three parts of a rectangle are red. Two parts are blue.

8. What fraction of the rectangle is red?

10. A banner is made of 8 equal parts. Five of the parts contain stars. Three of the parts contain hearts. Draw the banner.

9. What fraction of the rectangle is blue?

11. How can you write the fraction 4__6 in word form?

fourth sixth four sixes four sixths fourth six


> <

�

� >

<

Answers will vary.

�>

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Practice A47

Practice

A47Using Models to Compare FractionsCompare. Write >, <, or =.

1. 2.

2 __ 4 1 __ 3 3 __ 8 1 __ 2

3. 4.

3 __ 4 6 __ 8 1 __ 5 2 __ 8

5. 6.

4 __ 6 2 __ 3 3 __ 10 1 __ 6

7. 8.

1 __ 5 1 __ 6 2 __ 6 1 __ 3

9. Give 3 fractions with denominators that are less than 6__

8.10. Which fraction is the same as 1__

2?

1 __ 4 3 __ 6 3 __ 8 3 __ 4

14

14

13

12

18

18

18

15

18

18

14

14

14

18

18

18

18

18

18

110

110

110

16

16

16

16

16

13

13

16

16

13

16

15


2 9

5 3

8

12 10 12

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Practice A48

Practice

A48Using Models to Find Equivalent FractionsComplete each number sentence.

1. 1

110

110

15

2. 1

112

112

112

112

112

112

112

112

112

14

14

14

1 __ 5 � _____ 10 3 __ 4 � _____ 12

3. 1

16

16

16

110

110

110

110

110

4. 1

112

112

112

14

3 __ 6 � _____ 10 1 __ 4 � _____ 12

5.1

110

110

110

110

110

110

110

110

15

15

15

15

4 __ 5 � _____ 10

Complete each pattern.

6. 1 __ 3 , 2 __ 6 , 3 __ 9 , 4 _____ 7. 1 __ 2 , 2 __ 4 , 3 __ 6 , 4 __ 8 , 5 _____ , 6 _____

8. Samuel has read 5 __ 6 of his assignment. Judy has read 10 __ 12 of her assignment. Which sentence is true?

Samuel read more than Judy. Judy read more than Samuel.

They read the same amount. They will both finish the assignment at the same time.


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A74 B45

B48 B56Answers: A74, B45, B48, B56


A74, B45, B48, B56

4, 6, 2 3, 3, 5

boy

Answers will vary. Students should show at least 4 repetitions of the pattern.

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Practice A74

Practice

A74Repeating PatternsDraw the next three shapes to continue the pattern.

1.

2.

Write the next three numbers to continue the pattern.

3. 4, 6, 2, 8, 4, 6, 2, 8, 4, 6, 2, 8 4. 3, 3, 5, 3, 3, 5, 3, 3, 5

5. What is the 16th shape in the pattern below?

6. Mrs. Washington placed students in a line. The order was 1 boy, 2 girls, 2 boys, 2 girls, 3 boys, 2 girls and continued. Was the 15th student a boy or a girl?

7. Create a pattern using your favorite letters.


2 5 10 3 7 21

24 27

15 games

28 30

16 15

$5 bill

14 guests

12 48

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Practice B45

Practice

B45Using Multiplication to CompareFind each amount.


× � × �


5. twice as many as 8 6. 5 times as many as 3



11. John has 5 computer games. Julian has twice as many computer games as John. How many computer games do they have in all?

12. George Washington is on the $1 bill. Abraham Lincoln is on the bill that is worth 5 times as much as the $1 bill. What bill is Abraham Lincoln on?

13. Paula has twice as many guests this week as she did last week. Last week she had 7 guests. How many guests does she have this week?

14. John F. Kennedy is on the coin that is worth 5 times as much as a dime. What coin is John F. Kennedy on?

nickel quarter half-dollar dollar

45094_Practice_A42-D65.indd B45 6/30/08 12:00:53 PM

36 63 81 72 27

45 18 54 14 72

36 27

27 beads

18 innings

Answers will vary.

909

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Practice B48

Practice

B48Multiplying by 9Find each product.

1. 4× 9

2. 7× 9

3. 9× 9

4. 8× 9

5. 3× 9

6. 9× 5

7. 2× 9

8. 6× 9

9. 2× 7

10. 8× 9

11. Multiply 4 and 9. 12. Find 3 times 9.

13. Find the product of 14. Multiply 9 and 1.

9 and 10.

15. Paula’s hair was put into 9 braids. Each braid used 3 beads. How many beads were used in all?

16. A baseball game has 9 innings. A doubleheader is 2 games in the same day. How many innings are there in a doubleheader?

17. Write a multiplication story for 9 × 8. Include the product in your story.

18. Which number below is a multiple of 9?

35 46 54 65


(2 � 3) � 3 � 18

(3 � 2) � 4, 3 � (2 � 4), (3 � 4) � 2

40 80 20

2 � (3 � 3) � 18

(2 � 2) � 4 � 16

2 � (2 � 4) � 16

(8 � 2) � 2 � 32

8 � (2 � 2) � 32

(6 � 2) � 3 � 36

6 � (2 � 3) � 36

(3 � 3) � 4 � 36

3 � (3 � 4) � 36

(5 � 2) � 5 � 50

5 � (2 � 5) � 50

(5 � 4) � 2 � 40

5 � (4 � 2) � 40

(4 � 2) � 3 � 24

4 � (2 � 3) � 24

60 30 0

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Practice B56

Practice

B56Multiplying Three NumbersFind each product two diffeent ways.

1. 2 × 3 × 3 2. 2 × 2 × 4 3. 8 × 2 × 2 4. 6 × 2 × 3

5. 3 × 3 × 4 6. 5 × 2 × 5 7. 5 × 4 × 2 8. 4 × 2 × 3

Find the missing number.

9. 4 × 5 × 2 10. 5 × 2 × 8 11. 2 × 2 × 5

12. 2 × 5 × 6 13. 3 × 2 × 5 14. 4 × 9 × 0

15. Which number makes this number sentence true?

8 × 2 × 4 = 8 × (☐ × 4)

2 4 8 64

16. Which number makes this sentence true?

(5 × 3) × 4 = 5 × (☐ × 3)

2 3 4 5

17. Write three ways to find 3 × 2 × 4.


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B62, B63, C26, C27

B62 B63

C26 C27Answers: B62, B63, C26, C27

6 2 7

8 5

7 9 5

10 7 4

3 cranes; 19 � 8 � 27; 27 � 9 � 3

6

9 4 5

30 9 9

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Practice B62

Practice

B62Dividing by 8 and 9Use the multiplication fact to find each quotient.

1. 6 � 8 � 48 2. 9 � 2 � 18 3. 7 � 7 � 49

48 � 8 � 18 � 9 � 49 � 7 �

4. 8 � 8 � 64 5. 9 � 5 � 45 5. 6 � 7 � 42

64 � 8 � 45 � 9 � 42 � 7 �

6. 9 � 8 � 72 7. 9 � 4 � 36 8. 5 � 3 � 15

72 � 8 � 36 � 9 � 15 � 3 �

Find each quotient.

10. 8��56 11. 9��81 12. 8��40

13. 9��90 14. 9��63 15. 8��32

16. 3��90 17. 11��99 18. 5��45

19. Adam made 19 paper cranes Monday and 8 more Tuesday. He gave 9 friends an equal number of cranes. How many cranes did each friend receive? Explain how you found your answer.

20. A short story consists of 81 pages. Andrea will read 9 pages each day. How many days will it take Andrea to finish the story?

6 7 8 9


6 tickets

0 goals

0 1 65 0 1 8 3

8 7 9 6 9

0 7 1

� < >

Answers will vary.

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Practice B63

Practice

B630 and 1 in DivisionFind each quotient.

1. 0 � 6 2. 8 � 8 3. 6 � 16 � 0 � 0 8 � 1 � 8 6 � 1 � 6

So, 0 � 6 � So, 8 � 8 � So, 6 � 1 �

4. 5. 6. 7. 8. 1��5 4��0 6��6 1��8 1��3

9. 10. 11. 12. 13. 3��24 6��42 8��72 5��30 7��63

14. Find 0 divided by 2. 15. Divide 7 by 1. 16. Find 4 divided by 4.

Write �, �, or � to compare.

17. 6 � 6 8 � 8 18. 0 � 5 5 � 5 19. 9 � 1 7 � 1

20. Tickets for rides cost $1 each at the fair. Bob has $6 to buy tickets. How many tickets can Bob buy?

21. Nikki is the goalie on her soccer team. She has allowed 0 goals in 8 games. How many goals has she allowed in each game?

22. Why is 10 � 0 � 10, but 0 � 10 � 0? Explain.

23. Which has the greatest quotient?

6 � 6 5 � 1 0 � 3 8 � 8


78 55 49 59

81 79 75 50

30 804 7

8787

20 608 9

6969

30 2051 73

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Practice C26

Practice

C26Using Mental Math to AddFind the sum by breaking apart each addend.

1. 53 + 34


Add the ones. 3 + =


80 + 7 =

So, 53 + 34 =

2. 41 + 28


Add the ones. 1 + =


60 + 9 =

So, 41 + 28 =

Find the sum by breaking apart the second addend

3. 27 + 24

Take 3 ones from the 24 and add them to the 27.What sum do you have now?

27 + 3 =

Add 30 + 21 = So, 27 + 24 = 51

4. 54 + 19

Take 1 one from the 54 and add it to the 19.What sum do you have now?

19 + 1 =

Add 53 + 20 = So, 54 + 19 = 73


5. 52 + 26 6. 47 + 8 7. 32 + 17 8. 28 + 31

9. 43 + 38 10. 72 + 7 11. 42 + 33 12. 36 + 14

45094_Practice_A42-D65.indd C26 6/30/08 12:01:02 PM

24 29 43 45

53 83 17 67

27 51 34 56

Sample answer: 58 � 1 � 59

Answers will vary. Sample answer: Change 81 to 80; So, 80 � 16 � 64; 64 � 1 � 65.

46 tiles

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Practice C27

Practice

C27Using Mental Math to SubtractFind each difference using mental math.

1. 38 − 14 2. 42 − 13 3. 55 − 12 4. 62 − 17

5. 72 − 19 6. 94 − 11 7. 32 − 15 8. 85 − 18

9. 43 − 16 10. 66 − 15 11. 53 − 19 12. 72 − 16

13. Gillian started solving 88 − 29. This is what she did.

88 − 29 = ? 88 − 30 = 58

What should Gillian do next?

14. Tell how to find 81 – 16 using mental math.

15. Tiffany needs 63 tiles for her art project. She only needs 17 more tiles. Use mental math to find how many tiles she has already.

16. To solve 35 – 19, Jack used 35 – 20 and then

added 1. subtracted 1. subtracted 9. added 9.


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C28, C29, C37, D59

C28 C29

C37 D59Answers: C28, C29, C37, D59

98 58 87 135 73

17

1

46

1

18

1

35

1

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Practice C28

Practice

C28Adding 2-Digit NumbersUse place-value blocks and the Tens and Ones chart to add.

1. 2.

3. 4.

Add. Use a Tens and Ones chart if you like.

5. 53+ 45

6. 37+ 21

7. 63+ 24

8. 59+ 76

9. 29+ 44

10. There are 72 people on a train when 25 more people enter. How many people are on the train now?

79 87 97 98

Tens Ones

5 2

� 1 9

Tens Ones

1 6

� 4 8

Tens Ones

4 7

� 3 4

Tens Ones

2 8

� 2 5


18 16 48 27 35

53

7 12

82

5 13

71

7 12

83

6 13

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Practice C29

Practice

C29Subtracting 2-Digit NumbersSubtract.

1. 2.

3. 4.


5. 34− 16

6. 43− 27

7 76− 28

8. 65− 38

9. 82− 47

10. The tree farm had 65 shade trees for sale. It sold 39 of the trees. How many trees did the farm have left?

26 36 94 104

Tens Ones

8 2

� 4 7

Tens Ones

6 3

� 3 5

Tens Ones

8 2

� 6 5

Tens Ones

7 3

� 3 5


169 555 797 773 850

0 94

9 05

11

11

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Practice C37

Practice

C37Adding Three NumbersAdd.

1.

2.

Find each sum.

3. 7536

+ 58

4. 142297

+ 116

5. 52497

+ 176

6. 27318764

+ 249

7. 31948

136+ 347

8. Kyle has 378 pennies, 192 nickels, and 117 dimes. How many coins does he have all together?

495 570 677 687

Hundreds Tens Ones

1 4 3

2 1 9

� 4 7

Hundreds Tens Ones

3 5 6

1 7 1

� 6 3


rectangular prism

cone cylinder

cube square pyramid

sphere

sphere cylinder cuberectangular prism

rectangular prisms

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Practice D59

Practice

D59Solid FiguresName the solid figure.

1. 2. 3.

4. 5. 6.


7. 8. 9. 10.

11. Reasoning What solid figures would youget if you cut a cube as shown?

12. What solid figure does this figure most resemble?

Cylinder Cone Pyramid Sphere

2

43

45094_Practice_A42-D65.indd D59 6/30/08 12:01:12 PM

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D62, D63, D64, D65

D62 D63

D64 D65Answers: D62, D63, D64, D65

acute angle ray obtuse angle

vertex right angle acute angle

obtuse angle vertex ray

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Practice D62

Practice

D62Acute, Right, and Obtuse AnglesWrite ray, vertex, right angle, acute angle, or obtuse angle to name each.

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. At what time do the hands of a clock form an acute angle? 2:00 4:00 6:00 8:00


pentagonhexagon octagon triangle

No; not a closed figure

Yes No; some sides are curves

Yes

Yes. Quadrilaterals have 4 sides and 4 angles

4

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Practice D63

Practice

D63

11. How many more sides does an octagon have than a pentagon?

1 2 3 4

9. Juan said that the two figures below are quadrilaterals. Is he correct? Explain.

10. If two of the line segments of a polygon are parallel, what is the least number of sides it could have?

PolygonsName the polygon.

1. 2. 3. 4.

Is each figure a polygon? If it is not, explain why.

5. 6. 7. 8.


isoscelesequilateral scalene scalene

right acute rightobtuse

No. Sample Answers: 2 right angles � 180° (90° � 90°). A triangle must have 3 angles which equal 180°

2

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Practice D64

Practice

D64Classifying Triangles Using Sides and AnglesTell if each triangle is equilateral, isosceles, or scalene.

1. 2. 3. 4.


5. 6. 7. 8.

9. Can a triangle have 2 right angles? Explain.

10. What is the least number of acute angles that a triangle can have?

11. Which pair of triangle namesidentifies the figure?

Equilateral triangle, acute triangle Equilateral triangle, right triangle Scalene triangle, acute triangle Isosceles triangle, obtuse triangle


trapezoid rectangle,

parallelogram

parallelogram square,

rectangle,

rhombus,

parallelogram

rhombus,

parallelogram

rhombus

trapezoid

square

rectangle

Yes, if it has 4 equal sides, and 2 pairs of equal angles.Answers will vary.

Answers will vary.

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Practice D65

Practice

D65QuadrilateralsWrite as many special names as possible for each quadrilateral.

1. 2. 3. 4. 5.

In 6–9, write the name that best describes the quadrilateral. Draw a picture to help.

10. Can a rectangle also be a rhombus?

11. Which of the following correctly names the figure?

Rhombus Trapezoid Parallelogram Rectangle

6. A parallelogram with 4 equal sides, but no right angles.

8. A figure that is not a parallelogram, with one pair of parallel sides.

7. A rectangle with 4 right angles and all sides the same length.

9. A parallelogram with 4 right angles and sides of different lengths and widths.


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T1

Name Grade 2

Step Up to Grade 3 Test

1. fi fths sixths

eighths tenths

2. 4 _ 8 5 _ 8

6 _ 8 7 _ 8

3.

1 _ 5 3 __ 10

� � � $

4. 1 __

3 � ____

6

5 _ 6 4 _ 6 3 _ 6 2 _ 6

5. 5, 5, 7, 7, 8, 9, 5, 5, 7, 7, 8, 9, _ , _ , _ ,

5, 5, 7 9, 5, 5 7, 7, 9 5, 7, 7

Directions Mark the best answer. 1. Which word names the equal parts of the whole? 2. What is the fraction for the shaded part of this region? 3. Which sign completes this number sentence? 4. 1 _ 3 is equal to _____ (how many) sixths? 5. What are the next three numbers in this pattern?

110

110

110

15

16

16

13

Name Grade 2


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T2

Directions Mark the best answer. 6. What is four times as many as three? 7. Find the product. 8. Find the product. 9. Find the quotient. 10. Find the quotient 11. Which shows the best way to find the sum with mental math?

6. 12 14 18 28

7. 4 � 9 13 27 36 45

8. 8 � 3 � 3 72 24 14 9

9. 48 � 8 24 12 6 3

10. 8 � 1 1 8 18 81

11. 53 � 24

53 � 24 50 � 3 � 24

50 � 20 � 3 � 4 50 � 10 � 3 � 4

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T3

Name Grade 2


Directions Mark the best answer. 12. Todd has 50 dollars. He spends 37 dollars on a used bike. How much does he have left? 13. Add. Use the workspace to help you solve. 14. Subtract. Use the workspace to help you solve. 15. Add.

12.43 dollars 27 dollars

17 dollars 13 dollars

13.

67 77

81 91

14.

26 36

44 56

15. 736 763

836 863

Tens Ones

�

7 91 2

Hundreds Tens Ones

�

4 1 23 2 1 3 0

Tens Ones

�

6 22 6

Name Grade 2


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T4

Directions Mark the best answer. 16. Name this figure. 17. What kind of angle is shown? 18. Which polygon is not a quadrilateral? 19. Which triangle is not isosceles? 20. What is not a name for this quadrilateral?

16. cube sphere

square pyramid cone

17. acute obtuse

right straight

18.

19.

20. rectangle rhombus

parallelogram trapezoid

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Answers for Test

T1, T2, T3, T4

T1 T2

T4T3Answers: T1, T2, T3, T4

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T1

Name Grade 2


1.fi fths sixths

eighths tenths

2. 4_8 5_8

6_8 7_8

3.

1_5 3__10

�� $

4. 1 __

3 �� ____

6

5_6 4_6 3_6 2_6

5. 5, 5, 7, 7, 8, 9, 5, 5, 7, 7, 8, 9, _ , _ , _ ,

5, 5, 7 9, 5, 5 7, 7, 9 5, 7, 7

Directions Mark the best answer. 1. Which word names the equal parts of the whole? 2. What is the fraction for the shaded part of this region? 3. Which sign completes this number sentence? 4. 1_3 is equal to _____ (how many) sixths? 5. What are the next three numbers in this pattern?

110

110

110

15

16

16

13

45094_T1-T4.indd T1 7/1/08 2:05:11 PM

Name Grade 2


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T2

Directions Mark the best answer. 6. What is four times as many as three? 7. Find the product. 8. Find the product. 9. Find the quotient. 10. Find the quotient 11. Which shows the best way to find the sum with mental math?

6. 12 14 18 28

7. 4 �� 9 13 27 36 45

8. 8 � 3 �� 3 72 24 14 9

9. 48 �� 8 24 12 6 3

10. 8 �� 1 1 8 18 81

11. 53 �� 24

53 �� 24 50 � 3 �� 24

50 � 20 � 3 �� 4 50 � 10 � 3 � 4

45094_T1-T4.indd T2 7/1/08 2:05:12 PM

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T3

Name Grade 2


Directions Mark the best answer. 12. Todd has 50 dollars. He spends 37 dollars on a used bike. How much does he have left? 13. Add. Use the workspace to help you solve. 14. Subtract. Use the workspace to help you solve.15. Add.

12.43 dollars 27 dollars

17 dollars 13 dollars

13.

67 77

81 91

14.

26 36

44 56

15. 736 763

836 863

Tens Ones

�

7 91 2

Hundreds Tens Ones

�

4 1 23 2 1 3 0

Tens Ones

�

6 22 6

45094_T1-T4.indd T3 7/1/08 2:05:13 PM

Name Grade 2


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T4

Directions Mark the best answer. 16. Name this figure. 17. What kind of angle is shown? 18. Which polygon is not a quadrilateral? 19. Which triangle is not isosceles? 20. What is not a name for this quadrilateral?

16.cube sphere

square pyramid cone

17.acute obtuse

right straight

18.

19.

20. rectangle rhombus

parallelogram trapezoid

45094_T1-T4.indd T4 7/1/08 2:05:14 PM

Documents

Upper Saddle River, New Jersey Grade 2: Step Up to Grade 3