Unit 2

Common Core Mathematical Practices (MP)

Prisms and Pyramids

Domains t
Operations and Algebraic Thinking (OA) t
Number and Operations in Base Ten (NBT) t
Measurement and Data (MD)

INVESTIG ATION 1

Finding the Volume of Boxes Day 1 1.1

Session How Many Cubes? SESSION FOLLOW-UP

Daily Practice and Homework

2

1.2

Common Core Adaptation Family Letter: Make copies of C6–C7, Family Letter, as a replacement for M7–M8 Family Letter.

Strategies for Finding Volume DISCUSSION

Strategies for Finding Volume

Teaching Note Volume Formulas Many students will share the strategy of finding out the number of cubes in one layer and then multiplying that amount by the number of layers. When this strategy is shared, write V = b × h on the board and say: Sometimes this strategy is represented with this equation: V = b × h. V stands for volume; b stands for the area of a base of a rectangular prism, which is like the size of one layer or the number of cubes in one layer; h stands for height, which is the number of layers or how tall the rectangular prism is. Some students will share the strategy of using the dimensions of the cube to find the volume. When this strategy is shared, write V = l × w × h on the board and say: Sometimes this strategy is represented with this equation: V = l × w × h. V stands for volume; l stands for length, which is the length of the prism or the number of cubes in a row; w stands for width, which is the width of the prism or the number of rows in a layer; h stands for height, which is how tall the rectangular prism is or the number of layers. Throughout the unit, as students discuss strategies for finding the volume of rectangular prisms, remind students of these equations (formulas) that represent the strategies they are using.

SESSION FOLLOW-UP

Daily Practice and Homework

CC10

UNIT 2

Prisms and Pyramids

Family Letter: Make copies of C8, Family Letter, as a replacement for M9–M10 Family Letter.

Common Core Standards MP1, MP8 5.MD.3.a, 5.MD.3.b, 5.MD.4 MP1, MP8 5.MD.3.a, 5.MD.3.b, 5.MD.5.a, 5.MD.5.b

INVESTIG ATION 1

Finding the Volume of Boxes, continued Day 3 1.3 4 1.4

Session Doubling the Number of Cubes How Many Packages?

5 1.5A Finding the Volume of Solids 6 1.5 7 1.6

Assessment: Finding the Volume of Rectangular Prisms Finding Volume

8 1.7

Designing Boxes

Common Core Adaptation

See p. CC14.

Common Core Standards MP1, MP8 5.MD.5.a MP1, MP4 5.NBT.5, 5.MD.5.a MP1, MP4 5.OA.1, 5.MD.3.a, 5.MD.3.b, 5.MD.5.b, 5.MD.5.c MP1, MP4 5.NBT.5, 5.MD.5.a MP1 5.MD.5.a MP1 5.MD.5.a

Instructional Plan

CC11

INVESTIG ATION 2

Using Standard Cubic Units Day 9 2.1

Session Finding Cubic Centimeters DISCUSSION

How Many Cubic Centimeters?

Common Core Adaptation Teaching Note Volume Formulas As students share their strategies, relate the strategies to the equations V = b × h and V = l × w × h.

Common Core Standards MP5 5.NBT.5, 5.MD.3.a, 5.MD.3.b, 5.MD.4, 5.MD.5.a, 5.MD.5.b

10 2.2

Building Models of Volume Units

MP5 5.NBT.5, 5.MD.4, 5.MD.5.a

11 2.3

The Space Inside Our Classroom

MP5 5.NBT.6, 5.MD.4, 5.MD.5.a, 5.MD.5.b

DISCUSSION

How We Measured

SESSION FOLLOW-UP

Daily Practice

12 2.4A End-of-Unit Assessment

2.4

CC12

UNIT 2

Assessment: Measuring Volume in Cubic Centimeters

Prisms and Pyramids

Teaching Note Volume Formulas As students share their strategies, relate the strategies to the equations V = b × h and V = l × w × h. Daily Practice: In addition to Student Activity Book page 35, students complete Student Activity Book page 36B or C13 (More Volume Problems) for ongoing review. See p. CC19.

Skip this session. The End-of-Unit Assessment is now Session 2.4A.

MP1, MP2, MP5 5.OA.1, 5.MD.3.a, 5.MD.3.b, 5.MD.4, 5.MD.5.a, 5.MD.5.b

INVESTIG ATION 3

Volume Relationships Among Solids Skip this Investigation. Day 3.1

Session Comparing Volumes

3.2

Finding a Three-to-One Relationship

3.3

Prism and Pyramid Partners

3.4

Using Standard Units of Volume

3.5

End-of-Unit Assessment

Common Core Adaptation

Common Core Standards

The End-of-Unit Assessment is now Session 2.4A.

Instructional Plan

CC13

SESSION 1.5A

Finding the Volume of Solids Math Focus Points Using formulas to find the volume of rectangular prisms

Vocabulary

Finding the volume of a solid composed of two rectangular prisms

cubic feet cubic inches

Today’s Plan

Materials

ACTIVITY

• Student Activity Book, p. 19A or

Volume Formulas

25 MIN CLASS

C9, Volume of Rectangular Prisms Make copies. (as needed)

PAIRS INDIVIDUALS

ACTIVITY

• Student Activity Book, pp. 19B–19C or

Combining Volumes

35 MIN CLASS

SESSION FOLLOW-UP

C10–C11, Volume of Solids Make copies. (as needed)

PAIRS

• Student Activity Book, p. 19D or

Daily Practice

•

C12, Volume Problems Make copies. (as needed) Student Math Handbook, pp. 105–107, 109–110

Ten-Minute Math Order of Operations Write the following equation on the board: 18 − [14 − (3 + 5)] ÷ 3 = Ask students to solve the equation and compare solutions with a partner. Call on volunteers to explain their answers. Record each step of the solution process. Then have students solve each of the following equations and explain their work: (15 − 3) ÷ [5 + 3 − (6 + 1)] × 4 = {(2 + 2) × [9 − (10 − 7)]} ÷ 12 = Answers: 16; 48; 2

CC14

INVESTIGATION 1

Finding the Volume of Boxes

1 Activity

AC TIVIT Y

25 MIN

CLASS

2 Activity

3 Session Follow-Up

PAIRS INDIVIDUALS

Volume Formulas When we discussed strategies for finding the volume of a rectangular prism, many of you said that you started by finding out how many cubes would fit in the bottom layer and then you multiplied that number by the number of layers going up. On the board, write V = b × h. This formula says we can find the volume of a rectangular prism by multiplying the area of a base of the prism by the height. V stands for volume, b stands for area of a base, and h stands for height. Does this formula represent the same strategy as the one I described? Talk to a partner and then we’ll compare your ideas. Students might say: “We think it’s the same. We multiply the length and width of the bottom layer to get the number of cubes. That’s faster than counting. And when you multiply the length times the width, you get the area. So that’s b. Then we multiply the area by the number of layers high, and that’s the same as the height, which is the letter h.” [Rachel] said she multiplies the length times the width, and then she multiplies by the height. So, we could write the volume formula another way, like this. Beneath the first formula, write V = l × w × h. How are these two formulas the same? Students might say: "The area of a base is equal to the length times the width of a base. So they are really the same thing." Work with a partner and find the volume of a rectangular prism that is 4 units by 7 units on the bottom and 10 units high. How does the way you solved the problem relate to the volume formulas: V = l × w × h or V = b × h? Now, picture a closet that is a rectangular prism. The area of the floor is 15 square feet. The height of the closet is 8 feet. Find the volume of the closet. How do the volume formulas relate to how you solved the problem? Session 1.5A

Finding the Volume of Solids

CC15

1 Activity

2 Activity

3 Session Follow-Up

Name

Students might say: “You told us the area of the floor, and the height, so we multiplied 15 and 8 and got 120. We knew the area of a base of a rectangular prism and the height—we multiplied the base and the height, but we don’t know the length and the width. So we multiplied b and h and got 120.”

Date

Prisms and Pyramids

Volume of Rectangular Prisms Find the volume of each rectangular prism. Show your work. 1.

2. 4 ft 5 ft

9 in.

7 ft 4 in. 6 in.

Volume:

cubic inches

Volume:

cubic feet

3. Talisha’s bedroom is a rectangular prism. The area of the floor is 156 square feet, and the height of the bedroom is 9 feet. What is the volume of Talisha’s bedroom? Volume:

cubic feet

4. What is the volume of a shoe box with a length of 14 inches, width of 8 inches, and height of 5 inches? Volume: © Pearson Education 5

cubic inches

Session 1.5A

Unit 2

19A

▲ Student Activity Book, Unit 2, p. 19A; Resource Masters, C9

Name

That’s right. The volume is 120 cubic feet. When the dimensions are in feet, the volume is in units called cubic feet. A cubic foot is a cube that measures a foot along each edge. When the dimensions are in inches, the volume is in units called cubic inches. A cubic inch is a cube that measures an inch along each edge. We’ll talk more about cubic feet and cubic inches later in the unit. Have students solve the volume problems on Student Activity Book page 19A or C9. Then discuss the solutions. Relate the volume formulas to each of the students' solutions.

Date

Prisms and Pyramids

Volume of Solids (page 1 of 2) Find the volume of each solid. Show how you found the volume. 1.

AC TIVIT Y Volume:

2.

Combining Volumes

cubic units

CLASS

5 ft

10 ft

12 ft 7 ft

What do you notice about these solids?

3 ft

19B

Unit 2

cubic feet

© Pearson Education 5

10 ft

Volume:

Students might say:

Session 1.5A

▲ Student Activity Book, Unit 2, p. 19B; Resource Masters, C10

“They’re not rectangular prisms, but they’re made up of rectangular prisms.” That’s right. So far we have found the volume of rectangular prisms. How can we find the volume of these solids? Students might say: “You can find the volume of each of the individual prisms and then add them together.”

CC16

INVESTIGATION 1

PAIRS

Ask students to look at the solids on Student Activity Book page 19B or C10.

20 ft

3 ft

35 MIN

Finding the Volume of Boxes

1 Activity

Ask partners to solve Problem 1. Then ask for a student to share how he or she found the volume of the solid. Solutions offered will likely include the following:

2 Activity

3 Session Follow-Up

Name

Date

Prisms and Pyramids

Volume of Solids (page 2 of 2) Find the volume of each solid. Show how you found the volume. 3.

4.

• Dividing the solid into one rectangular prism on top and one rectangular prism on the bottom, finding the volume of each, and adding

6 in.

6 in.

4 in.

15 in.

5 in.

Volume:

• Dividing the solid into one rectangular prism on the left and one rectangular prism on the right, finding the volume of each, and adding

5.

3 ft

cubic units

Volume:

4 in.

cubic inches

6.

2 ft

22 ft 20 ft

2 ft

9 ft

19 ft

20 ft 8 ft

4 ft

10 ft

20 ft 30 ft

© Pearson Education 5

Discuss various solutions. Then have a brief discussion with the class as to why the answers are the same. Highlight the idea that volume is additive, and that as long as you accurately account for all of the “pieces” of the solid, the partial volumes can be combined to find the volume of the solid.

17 in. 5 in.

Volume:

cubic feet

Volume:

cubic feet

Session 1.5A

Unit 2

19C

▲ Student Activity Book, Unit 2, p. 19C; Resource Masters, C11

Ask partners to solve Problem 2 and have them share their solutions. Discuss how students divided the solid. Have students complete Student Activity Book page 19C or C11. ONGOING ASSESSMENT: Observing Students at Work Students find the volume of solids composed of rectangular prisms.

• Do students find the volume of each rectangular prism in the solid and add the volumes? Are they using some other strategy?

• How do the students divide the solid into rectangular prisms?

• How do students find the volume of each of the prisms? DIFFERENTIATION: Supporting the Range of Learners If students are having trouble seeing the rectangular prisms within the solids, help them find one rectangular prism and ask them to draw lines to delineate it. You might want to have some students use cubes to build the solid in Problem 1. Students who finish early can be challenged to find more than one way to determine the volumes of the solids.

Session 1.5A

Finding the Volume of Solids

CC17

1 Activity

2 Activity

Name

3 Session Follow-Up

Date

Prisms and Pyramids

Daily Practice

Volume Problems 1.

SESSION FOLLOW-UP

NOTE Students find the volume l off solids l d that h are made d de up of rectangular prisms. sms.

Find the volume of each solid. Show how you found the volume.

Daily Practice

2. 12 in. 4 in.

Daily Practice: For reinforcement of this unit’s content, have students complete Student Activity Book page 19D or C12.

9 in.

9 in. 5 in.

2 in. 3 in.

Volume:

cubic units

3. Side-by-side refrigerator

Volume:

cubic inches

4. Deon’s L-shaped bedroom

3 ft

Student Math Handbook: Students and families may use Student Math Handbook pages 105–107, 109–110 for reference and review. See pages 141–142 in the back of Unit 2.

20 ft 8 ft

7 ft

8 ft 10 ft 4 ft

19D

10 ft

2 ft

cubic feet

Volume:

Unit 2

cubic feet

© Pearson Education 5

1 ft

Volume:

12 ft

Session 1.5A

▲ Student Activity Book, Unit 2, p. 19D; Resource Masters, C12

CC18

INVESTIGATION 1

Finding the Volume of Boxes

SESSION 2.4A

End-of-Unit Assessment Math Focus Points Determining the volume, in cubic centimeters, of a small prism Considering how the dimensions of a box change when the volume is changed (doubled, halved, or tripled)

Today’s Plan

Materials

ASSESSMENT ASSE AS SESS SSME MENT NT AACTIVITY CTIV CT IVIT ITYY

End-of-Unit Assessment

30 MIN INDIVIDUALS

• C14–C15, End-of-Unit Assessment Make copies. (1 per student) C16, • Assessment Checklist: Measuring • • • • •

ACTIVITY

Boxes for Centimeter Cubes

Volume in Cubic Centimeters Make copies. (1 per 6 students) Centimeter cubes Centimeter rulers Calculators Scissors Tape

• Student Activity Book, pp. 38B–38C or 30 MIN

PAIRS

• SESSION FOLLOW-UP

Daily Practice

C17–C18, More Boxes for Centimeter Cubes Make copies. (as needed) M18 (Make copies, 1–2 per student)

• Student Activity Book, p. 38D or •

C19, Volume Puzzles Make copies. (as needed) Student Math Handbook, pp. 106–108

Ten-Minute Math Order of Operations Write the following expression on the board: 36 ÷ [(21 – 4) – (3 + 5)] ÷ 2 Ask students to evaluate the expression and compare solutions with a partner. Call on volunteers to explain their answers. Record each step of the solution. Then have students evaluate each of the following expressions and explain their work: 35 – [16 ÷ (7 + 1)] × 9 100 – {(5 + 1) × [(5 – 2) × 4]} Answers: 2; 17; 28 Session 2.4A

End-of-Unit Assessment

CC19

1 Assessment Activity

2 Activity

3 Session Follow-Up

Teaching Notes 1

2

Photocopier Distortion Be aware that some photocopiers may slightly distort the dimensions of the box on C14. Therefore, check the dimensions after you make copies and, if necessary, let students know that they should round their measurements to the nearest centimeter. Computation and Measurement Errors Some students may demonstrate understanding of the structure of rectangular prisms and how to determine their volume, but may make computation or measurement errors as they do so. These students can self-correct when asked to demonstrate their strategies.

Professional Development 3

Teacher Note: End-of-Unit Assessment, pp. 128–130. For the problem on C15, refer to the notes for Problem 1.

Name

Date

Prisms and Pyramids

End-Of-Unit Assessment (page 1 of 2) 1. Cut out, fold, and tape to make a closed box.

On a separate sheet of paper, write your answer and explain how you found the volume of this box.

How many cubic centimeters will it take to fill this box? Find out without filling the box with cubes.

A SSESSMENT AC TIVIT Y

End-of-Unit Assessment

30 MIN INDIVIDUALS

This End-of-Unit Assessment (C14–C16) consists of two problems. The first problem (C14) is an observed assessment. Students work individually to first cut out and tape together an unmarked 6 cm × 8 cm × 3 cm box. They find the volume of the box and record their answer and the strategies they used to find the volume on a separate sheet of paper. This assessment addresses Benchmark 2 for this unit: Use standard units to measure volume. 1 C16 is provided to enable you to record your observations of each student’s approach to this first task. Each checklist has space to record observations for six students. Continue the observed assessment while students work on Activity 2. You are likely to see different approaches to measuring the dimensions of the box in the first problem. Some students will use centimeter rulers, and others will use centimeter cubes lined up along the edges of the box. Similarly, some students may use a layer approach, multiplying the number of cubes that form the bottom layer by the number of layers in the box, and others will simply multiply length × width × height. 2 In the second problem (C15), students are given the dimensions of a box and are asked to find the dimensions of two new boxes containing three times the volume of the original box. This assessment addresses Benchmark 3 for this unit: Identify how the dimensions of a box change when the volume is changed. 3 ONGOING ASSESSMENT: Observing Students at Work Students are assessed on their ability to determine volume using a standard unit of measure and on their understanding of how the dimensions of a box change when the volume is changed.

• Do students accurately measure the dimensions of the box Unit 2 Session 2.4A

C14

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

▲ Resource Masters, C14

using a centimeter ruler or by placing cubes along each outside edge?

• Are students able to use their measurements to determine

how many cubic centimeters will fill the bottom layer of the box? Can they use the measurements to determine how many layers there will be?

CC20

INVESTIGATION 2

Using Standard Cubic Units

1 Assessment Activity

• How do students compute the volume of the box? Are their answers accurate?

2 Activity

3 Session Follow-Up

Name

Date

Prisms and Pyramids

End-Of-Unit Assessment (page 2 of 2)

• Do they record the volume in cubic centimeters? • Do students understand how changing the dimensions of a

2. a. You have a box that is 3 units by 6 units by 3 units. (A cube is one cubic unit.) The factory wants you to design a box that will hold three times as many cubes. How many cubes will this new box hold? Explain how you know.

box changes the volume (e.g., tripling one dimension triples the volume)?

• How do students determine the dimensions of the boxes

that have three times the volume of a given box? Do they triple the number of cubes (in this case from 54 to 162) and then figure out what the dimensions could be? Do they triple one or more of the dimensions of the original box and see how many cubes would be needed?

b. Write the dimensions of 2 new boxes that hold three times as many cubes as the original box. Explain how you figured out the new dimensions.

Unit 2 Session 2.4A

C15

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

▲ Resource Masters, C15

AC TIVIT Y

Students explore the relationship between the dimensions of a box and doubling or halving the number of centimeter cubes in the box.

Determines Volume by Multiplying . . . cubes in bottom length × width layer by number × height of layers using centimeter cubes

Measures Dimensions Accurately by . . .

using centimeter ruler

ONGOING ASSESSMENT: Observing Students at Work

Student

Using Student Activity Book pages 38B and 38C or C17 and C18, students use the dimensions (in centimeters) of a box to find two boxes to hold double the number and two boxes to hold half the number of centimeter cubes as the original box. They draw patterns for the new boxes on centimeter grid paper (M18).

Determines Volume Accurately

30 MIN PAIRS

Assessment Checklist: Measuring Volume in Cubic Centimeters

Boxes for Centimeter Cubes

Unit 2 Session 2.4A

• Do students understand the relationship between

C16

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

▲ Resource Masters, C16

changing the dimensions of a box and how many cubes will fit in the box?

• How do students solve these problems? Do they double or

halve the number of cubes and then figure out what the dimensions could be? Do they double or halve one or more of the dimensions to see how many cubes would be needed?

Session 2.4A

End-of-Unit Assessment

CC21

1 Assessment Activity

2 Activity

Name

3 Session Follow-Up

Date

DIFFERENTIATION: Supporting the Range of Learners

Prisms and Pyramids

More Boxes for Centimeter Cubes (page 1 of 2)

Ask students who need a challenge to find a box to hold four times the number of centimeter cubes as a 3 cm × 4 cm × 5 cm box.

You have a box that is 3 centimeters by 4 centimeters by 6 centimeters. 1. How many centimeter cubes does it hold? How do you know?

2. Find two boxes that will hold twice as many centimeter cubes as the box above. a. What are the dimensions of each new box? Dimensions of first box: Dimensions of second box:

SESSION FOLLOW-UP

b. Explain how you found the answers.

Daily Practice 3. Draw the designs for the new boxes on centimeter grid paper.

© Pearson Education 5

Daily Practice: For reinforcement and enrichment of this unit’s content, have students complete Student Activity Book page 38D or C19.

Session 2.4A

Unit 2

38B

▲ Student Activity Book, Unit 2, p. 38B; Resource Masters, C17

Name

Student Math Handbook: Students and families may use Student Math Handbook pages 106–108 for reference and review. See pages 141–142 in the back of Unit 2.

Date

Prisms and Pyramids

More Boxes for Centimeter Cubes (page 2 of 2) Find two boxes that will hold half as many centimeter cubes as the 3 centimeters by 4 centimeters by 6 centimeters box.

Name

Date

4. a. What are the dimensions of each new box?

Volume Puzzles

Prisms and Pyramids

Dimensions of first box:

Daily Practice

NOTE Students find the NO volume of solids that are made up of rectangular prisms by firsst identifying dimensions not ot lab labeled abeled in the diagram diagrams rams. ms

Each solid below is made up of rectangular prisms. Find the volume of each solid. You may need to first determine some of the missing dimensions.

Dimensions of second box: b. Explain how you found the answers.

1.

2.

9 ft

10 in. 3 in. 3 in.

13 ft

5. Draw the designs for the new boxes on centimeter grid paper.

3 in.

6 ft 12 in. 2 ft

14 ft

2 in.

Challenge: Find a box that will hold four times as many centimeter cubes as the 3 centimeters by 4 centimeters by 6 centimeters box. Write the dimensions of the new box and explain how you found your answer.

3.

3 ft

4.

11 ft

4 in.

4 in.

12 in.

12 ft © Pearson Education 5

15 in.

Unit 2

Session 2.4A

▲ Student Activity Book, Unit 2, p. 38C; Resource Masters, C18

2 ft

2 ft 5 in.

© Pearson Education 5

38C

7 ft

6 in.

Session 2.4A

2 in.

Unit 2

38D

▲ Student Activity Book, Unit 2, p. 38D; Resource Masters, C19

CC22

INVESTIGATION 2

Using Standard Cubic Units

Name

Date

Prisms and Pyramids

Family Letter

About the Mathematics in This Unit (page 1 of 2) Dear Family, Our class is starting a new mathematics unit about geometry and measurement called Prisms and Pyramids. During this unit, students study volume—the amount of space a 3-D object occupies. They use paper boxes and cubes to develop a strategy for finding the volume of any rectangular prism. Throughout the unit, students work towards these goals: BENCHMARK/ GOAL

EXAMPLES

9 10 11 12 13 14

What is the volume of this cube?

centimeters 0 1 2 3

4

5

6

7

8

Find the volume of rectangular prisms.

Use standard units to All the edges of the cube are the same length: 6 cm. The base of the cube is 6 × 6, so 36 centimeter cubes measure volume. would fit on the bottom of the box. Since the cube is 6 centimeters high, there are 6 layers in the box. 6 × 36 = 216. The volume of the cube is 216 cubic centimeters (216 cm3).

(continued)

Unit 2 Session 1.1

C6

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Family Letter

About the Mathematics in This Unit (page 2 of 2) BENCHMARK/ GOAL

EXAMPLES

Identify how the dimensions of a box change when the volume is changed.

Compare the volumes and the dimensions of these rectangular prisms. Box B:

Box A:

Box B has twice the volume of Box A; you can see that Box A was doubled to build Box B. The dimensions of Box A are 3 × 2 × 4. The dimensions of Box B are 6 × 2 × 4. Only the dimension across the front is different.

In our math class, students spend time discussing problems in depth and are asked to share their reasoning and solutions. It is important that children solve math problems in ways that make sense to them. At home, encourage your child to explain his or her math thinking to you. Please look for more information and activities about Prisms and Pyramids that will be sent home in the coming weeks.

Unit 2 Session 1.1

C7

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Family Letter

Related Activities to Try at Home (page 1 of 2) Dear Family, The activities below are related to the mathematics in the geometry and measurement unit, Prisms and Pyramids. You can use the activities to enrich your child’s mathematical learning experience. How Many Packages in a Box? Many household items are packaged and sold in boxes. You and your child can take a large cardboard box and predict how many bars of soap (toothpaste, pudding, cereal boxes) would fit in that box. You might try a variety of boxes at home or explore the way things are packaged when you visit grocery stores or other stores. Volume of a Room Another activity for exploring volume is to compare the amount of space in different rooms. At school, students will find the volume of their classroom in cubic meters. At home, your child can find the volume of various rooms. Which room do you think has the largest volume? Which room has the smallest volume? Why? Discuss how to compare rooms of unusual shapes (a slanted ceiling or an L-shape).

Math and Literature You and your child can explore more math topics in this book. Look for a copy at your local library. Juster, Norton. The Phantom Tollbooth.

Unit 2 Session 1.2

C8

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Volume of Rectangular Prisms Find the volume of each rectangular prism. Show your work. 1.

2. 4 ft 5 ft

9 in.

7 ft 4 in. 6 in.

Volume:

cubic inches

Volume:

cubic feet

3. Talisha’s bedroom is a rectangular prism. The area of the floor is 156 square feet, and the height of the bedroom is 9 feet. What is the volume of Talisha’s bedroom? Volume:

cubic feet

4. What is the volume of a shoe box with a length of 14 inches, width of 8 inches, and height of 5 inches? Volume:

cubic inches

Unit 2 Session 1.5A

C9

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Volume of Solids (page 1 of 2) Find the volume of each solid. Show how you found the volume. 1.

Volume:

2.

cubic units

20 ft

3 ft

5 ft

10 ft

12 ft 7 ft

3 ft 10 ft

Volume:

Unit 2 Session 1.5A

C10

cubic feet

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Volume of Solids (page 2 of 2) Find the volume of each solid. Show how you found the volume. 3.

4.

17 in. 5 in. 6 in.

6 in.

4 in.

15 in.

5 in.

Volume:

5.

3 ft

cubic units

Volume:

cubic inches

6.

2 ft

4 in.

22 ft 20 ft

2 ft

9 ft

19 ft

20 ft 8 ft 10 ft

4 ft

20 ft 30 ft

Volume:

cubic feet

Unit 2 Session 1.5A

Volume:

C11

cubic feet

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Daily Practice

Volume Problems

NOTE Students find the volume of solids that are made up of rectangular prisms.

Find the volume of each solid. Show how you found the volume. 1.

2. 12 in. 4 in. 9 in.

9 in. 5 in.

2 in. 3 in.

Volume:

cubic units

Volume:

3. Side-by-side refrigerator

cubic inches

4. Deon’s L-shaped bedroom

3 ft

20 ft 8 ft 8 ft

7 ft

10 ft 4 ft

1 ft

Volume:

12 ft 10 ft

2 ft

cubic feet

Unit 2 Session 1.5A

Volume:

C12

cubic feet

© Pearson Education, Inc., or its affiliates. All Rights Reserved. 5

Name

Date

Prisms and Pyramids

Daily Practice

More Volume Problems Find the volume of each solid. Show how you found the volume. 1.

NOTE Students find the volume of solids that are made up of rectangular prisms.

2. 10 in. 3 in. 4 in.

2 in. 6 in. 12 in.

Volume:

cubic units

3.

Volume:

cubic inches

15 ft

10 ft

15 ft

10 ft 15 ft 40 ft

Volume:

Unit 2 Session 2.3

cubic feet

C13

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Name

Date

Prisms and Pyramids

End-Of-Unit Assessment (page 1 of 2) 1. Cut out, fold, and tape to make a closed box.

On a separate sheet of paper, write your answer and explain how you found the volume of this box.

How many cubic centimeters will it take to fill this box? Find out without filling the box with cubes.

Unit 2 Session 2.4A

C14

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Name

Date

Prisms and Pyramids

End-Of-Unit Assessment (page 2 of 2) 2. a. You have a box that is 3 units by 6 units by 3 units. (A cube is one cubic unit.) The factory wants you to design a box that will hold three times as many cubes. How many cubes will this new box hold? Explain how you know.

b. Write the dimensions of 2 new boxes that hold three times as many cubes as the original box. Explain how you figured out the new dimensions.

Unit 2 Session 2.4A

C15

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Unit 2 Session 2.4A

C16

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Student

using centimeter ruler

using centimeter cubes

Measures Dimensions Accurately by . . .

Determines Volume by Multiplying . . . cubes in bottom length × width layer by number × height of layers

Assessment Checklist: Measuring Volume in Cubic Centimeters Determines Volume Accurately

Name

Date

Prisms and Pyramids

More Boxes for Centimeter Cubes (page 1 of 2) You have a box that is 3 centimeters by 4 centimeters by 6 centimeters. 1. How many centimeter cubes does it hold? How do you know?

2. Find two boxes that will hold twice as many centimeter cubes as the box above. a. What are the dimensions of each new box? Dimensions of first box: Dimensions of second box: b. Explain how you found the answers.

3. Draw the designs for the new boxes on centimeter grid paper.

Unit 2 Session 2.4A

C17

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Name

Date

Prisms and Pyramids

More Boxes for Centimeter Cubes (page 2 of 2) Find two boxes that will hold half as many centimeter cubes as the 3 centimeters by 4 centimeters by 6 centimeters box. 4. a. What are the dimensions of each new box? Dimensions of first box: Dimensions of second box: b. Explain how you found the answers.

5. Draw the designs for the new boxes on centimeter grid paper.

Challenge: Find a box that will hold four times as many centimeter cubes as the 3 centimeters by 4 centimeters by 6 centimeters box. Write the dimensions of the new box and explain how you found your answer.

Unit 2 Session 2.4A

C18

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Name

Date

Prisms and Pyramids

Daily Practice

Volume Puzzles Each solid below is made up of rectangular prisms. Find the volume of each solid. You may need to first determine some of the missing dimensions.

1.

2.

9 ft

NOTE Students find the volume of solids that are made up of rectangular prisms by first identifying dimensions not labeled in the diagrams.

10 in. 3 in. 3 in.

13 ft

3 in.

6 ft 12 in. 2 ft

14 ft

2 in.

3.

3 ft

4.

11 ft

4 in.

4 in.

12 in.

12 ft

15 in. 7 ft

2 ft

2 ft 5 in. 6 in.

Unit 2 Session 2.4A

C19

2 in.

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Nombre

Fecha

Prismas y pirámides

Carta a la familia

HOGAR

Sobre las Matemáticas de esta unidad (página 1 de 2) Estimada familia: Nuestra clase de Matemáticas está comenzando una nueva unidad sobre geometría y medición llamada Prismas y pirámides. Durante esta unidad, los estudiantes estudiarán el volumen, es decir, la medida del espacio ocupado por un cuerpo tridimensional. Usarán cajas de papel y cubos para desarrollar una estrategia que les permita hallar el volumen de cualquier prisma rectangular. A lo largo de esta unidad, los estudiantes trabajarán para lograr los siguientes objetivos: PUNTOS DE REFERENCIA/ OBJETIVOS

EJEMPLOS

0

centímetros 1 2 3

4

5

6

7

8

9 10 11 12 13 14

Hallar el volumen de ¿Cuál es el volumen de este cubo? prismas rectangulares.

Todas las aristas del cubo tienen la misma longitud: 6 cm. Usar unidades estándar para medir La base del cubo es 6 × 6, por tanto, 36 cubos de un centímetro caben en el fondo de la caja. el volumen. Ya que el cubo tiene 6 centímetros de altura, hay 6 capas en la caja. 6 × 36 = 216. El volumen del cubo es 216 centímetros cúbicos (216 cm3 ).

(continúa)

Unidad 2 Sesión 1.1

C6

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Nombre

Fecha

Prismas y pirámides

Carta a la familia

HOGAR

Sobre las Matemáticas de esta unidad (página 2 de 2) PUNTOS DE REFERENCIA/ OBJETIVOS Identificar cómo cambian las dimensiones de una caja al cambiar el volumen.

EJEMPLOS Compara los volúmenes y las dimensiones de estos prismas rectangulares. Caja B:

Caja A:

La Caja B tiene el doble del volumen de la Caja A; puedes ver que la Caja A se duplicó para construir la Caja B. Las dimensiones de la Caja A son 3 × 2 × 4. Las dimensiones de la Caja B son 6 × 2 × 4. Solo la dimensión del frente de la caja es diferente.

En nuestra clase de Matemáticas, los estudiantes comentan problemas a fondo y se les pide que expliquen su razonamiento y sus soluciones a los demás. Es importante que los estudiantes resuelvan problemas de matemáticas de maneras que tengan sentido para ellos. En el hogar, anime a su hijo/a a que le explique su razonamiento matemático. En las próximas semanas le enviaremos más información y actividades sobre Prismas y pirámides.

Unidad 2 Sesión 1.1

C7

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Nombre

Fecha

Prismas y pirámides

Carta a la familia

HOGAR

Actividades relacionadas para hacer en el hogar Estimada familia: Las actividades que siguen están relacionadas con las Matemáticas de la unidad de geometría y medición, Prismas y pirámides. Puede usar estas actividades para enriquecer la experiencia del aprendizaje de las matemáticas de su hijo/a. ¿Cuántos paquetes caben en una caja? Muchos artículos domésticos son empacados y vendidos en cajas. Junto con su hijo/a busque una caja de cartón grande y traten de predecir cuántas barras de jabón (cajas de pasta dental, de pudín, o de cereal) caben en la caja. Puede probar con una variedad de cajas en casa o explorar la manera en que las cosas están empaquetadas cuando visiten una tienda de abarrotes u otras tiendas. Volumen de un cuarto Otra actividad para explorar el volumen es comparar el espacio entre dos cuartos diferentes. En la escuela, los estudiantes hallarán el volumen de su clase en metros cúbicos. En el hogar, su hijo/a puede hallar el volumen de varios cuartos. ¿Qué cuarto crees que tiene el volumen más grande? ¿Qué cuarto crees que tiene el volumen más pequeño? ¿Por qué? Comente y practique con su hijo/a cómo comparar cuartos de formas inusuales (un techo inclinado o un cuarto en forma de L).

Matemáticas y literatura Usted y su hijo/a pueden explorar más temas relacionados con las matemáticas en este libro. Búsquelo en la biblioteca de su vecindario. Juster, Norton. La cabina mágica

Unidad 2 Sesión 1.2

C8

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Nombre

Fecha

Prismas y pirámides

Volumen de prismas rectangulares Halla el volumen de cada prisma rectangular. Muestra tu trabajo. 1.

2. 4 pies

9 pulgs.

5 pies 7 pies 6 pulgs.

Volumen:

4 pulgs.

pulgadas cúbicas

Volumen:

pies cúbicos

3. El cuarto de Talisha es un prisma rectangular. El área del suelo es 156 pies cuadrados, y la altura del cuarto es 9 pies. ¿Cuál es el volumen del cuarto de Talisha? Volumen:

pies cúbicos

4. ¿Cuál es el volumen de una caja de zapatos que mide 14 pulgadas de largo, 8 pulgadas de ancho y 5 pulgadas de altura? Volumen:

pulgadas cúbicas

Unidad 2 Sesión 1.5A

C9

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Nombre

Fecha

Prismas y pirámides

Volumen de sólidos (página 1 de 2) Halla el volumen de cada sólido. Muestra cómo hallaste el volumen. 1.

Volumen:

2.

unidades cúbicas

20 pies 3 pies 5 pies

10 pies 7 pies

12 pies

3 pies 10 pies

Volumen:

Unidad 2 Sesión 1.5A

C10

pies cúbicos

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Nombre

Fecha

Prismas y pirámides

Volumen de sólidos (página 2 de 2) Halla el volumen de cada sólido. Muestra cómo hallaste el volumen. 3.

4.

17 pulgs. 5 pulgs. 4 pulgs. 6 pulgs.

6 pulgs. 15 pulgs.

5 pulgs.

Volumen:

5.

3 pies

unidades cúbicas

2 pies

Volumen:

4 pulgs.

pulgadas cúbicas

6.

22 pies 20 pies

2 pies

9 pies

19 pies

20 pies 8 pies

4 pies

10 pies

20 pies 30 pies

Volumen:

pies cúbicos

Unidad 2 Sesión 1.5A

Volumen:

C11

pies cúbicos

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Nombre

Fecha

Prismas y pirámides

Práctica diaria

Problemas de volumen

NOTA Los estudiantes hallan el volumen de sólidos formados por prismas rectangulares.

Halla el volumen de cada sólido. Muestra cómo hallaste el volumen. 1.

2. 12 pulgs. 4 pulgs. 9 pulgs.

9 pulgs. 5 pulgs.

2 pulgs. 3 pulgs.

Volumen:

unidades cúbicas

3. Refrigerador de lado a lado

Volumen:

pulgadas cúbicas

4. La habitación en forma de L de Deon

3 pies

20 pies 8 pies 8 pies

7 pies

10 pies 4 pies

1 pies

Volumen:

12 pies 10 pies

2 pies

pies cúbicos

Unidad 2 Sesión 1.5A

Volumen:

C12

pies cúbicos

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Nombre

Fecha

Prismas y pirámides

Práctica diaria

Más problemas de volumen

NOTA Los estudiantes hallan el volumen de sólidos formados por prismas rectangulares.

Halla el volumen de cada sólido. Muestra cómo hallaste el volumen. 1.

2. 10 pulgs. 3 pulgs. 2 pulgs. 4 pulgs.

6 pulgs.

12 pulgs.

Volumen:

unidades cúbicas

3.

Volumen:

pulgadas cúbicas

15 pies 10 pies

15 pies

10 pies 15 pies 40 pies

Volumen:

Unidad 2 Sesión 2.3

pies cúbicos

C13

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Nombre

Fecha

Prismas y pirámides

Evaluación final de la unidad (página 1 de 2) 1. Recorta, dobla y pega para armar una caja cerrada.

En una hoja aparte, escribe tu respuesta y explica cómo hallaste el volumen de esta caja.

¿Cuántos centímetros cúbicos se necesitan para llenar esta caja? Averigua la respuesta sin llenar la caja con cubos.

Unidad 2 Sesión 2.4A

C14

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Nombre

Fecha

Prismas y pirámides

Evaluación final de la unidad (página 2 de 2) 2. a. Tienes una caja que mide 3 unidades por 6 unidades por 3 unidades. (Un cubo es una unidad cúbica). La fábrica quiere que diseñes una caja donde quepa el triple de la cantidad de cubos. ¿Cuántos cubos caben en la caja nueva? Explica cómo lo sabes.

b. Escribe las dimensiones de 2 cajas nuevas donde quepa el triple de la cantidad de cubos que caben en la caja original. Explica cómo averiguaste las nuevas dimensiones.

Unidad 2 Sesión 2.4A

C15

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Unidad 2 Sesión 2.4A

C16

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Estudiante

una regla de centímetros

cubos de 1 centímetro

Mide las dimensiones con exactitud usando…

Determina el volumen multiplicando… los cubos en la capa del fondo longitud × por el número de ancho × altura capas

Lista de comprobación: Medir volumen en centímetros cúbicos Determina el volumen con exactitud

Nombre

Fecha

Prismas y pirámides

Más cajas para cubos de 1 centímetro (página 1 de 2) Tienes una caja que mide 3 centímetros por 4 centímetros por 6 centímetros. 1. ¿Cuántos cubos de 1 centímetro caben en la caja? ¿Cómo lo sabes?

2. Halla dos cajas en las que quepa dos veces la cantidad de cubos de 1 centímetro que caben en la caja de arriba. a. ¿Cuáles son las dimensiones de cada una de las nuevas cajas? Dimensiones de la primera caja: Dimensiones de la segunda caja: b. Explica cómo hallaste las respuestas.

3. Dibuja los diseños de las nuevas cajas en papel cuadriculado de 1 centímetro.

Unidad 2 Sesión 2.4A

C17

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Nombre

Fecha

Prismas y pirámides

Más cajas para cubos de 1 centímetro (página 2 de 2) Halla dos cajas en las que quepa la mitad de cubos de 1 centímetro que caben en la caja de 3 centímetros por 4 centímetros por 6 centímetros. 4. a. ¿Cuáles son las dimensiones de cada una de las nuevas cajas? Dimensiones de la primera caja: Dimensiones de la segunda caja: b. Explica cómo hallaste las respuestas.

5. Dibuja los diseños de las nuevas cajas en papel cuadriculado de 1 centímetro.

Desafío: Halla una caja en la que quepa cuatro veces la cantidad de cubos de 1 centímetro que caben en la caja de 3 centímetros por 4 centímetros por 6 centímetros. Escribe las dimensiones de la nueva caja y explica cómo hallaste tu respuesta.

Unidad 2 Sesión 2.4A

C18

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Nombre

Fecha

Prismas y pirámides

Práctica diaria

Rompecabezas de volumen Cada sólido de abajo está formado por prismas rectangulares. Halla el volumen de cada sólido. Es posible que primero tengas que determinar algunas de las dimensiones que faltan.

1.

NOTA Los estudiantes hallan el volumen de sólidos formados por prismas rectangulares identificando primero las dimensiones no rotuladas en los diagramas.

2.

9 pies

10 pulgs. 3 pulgs. 3 pulgs.

3 pulgs.

13 pies 6 pies 12 pulgs. 2 pies

14 pies

2 pulgs.

4 pulgs.

3.

3 pies

4 pulgs.

4.

11 pies

12 pulgs. 15 pulgs.

12 pies 7 pies

2 pies

2 pies 5 pulgs. 6 pulgs.

Unidad 2 Sesión 2.4A

C19

2 pulgs.

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