surface area and volume of solids 15 - my.hrw.com · surface area and volume of solids 416 do not...

ESSENTIAL QUESTION?

Real-World Video

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MODULE

How can a model help you to solve surface area and volume problems?

15Surface Area and Volume of Solids

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you work through practice sets.

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Surface area and volume can be important considerations when constructing or repairing buildings or other structures.

LESSON 15.1

Nets and Surface Area

6.G.4

LESSON 15.2

Volume of Rectangular Prisms

6.G.2

LESSON 15.3

Solving VolumeEquations

6.EE.7, 6.G.2

COMMON CORE

COMMON CORE

COMMON CORE

A net of the figure makes it easier to see the faces and find their areas when finding surface area; a drawing of the figure makes it easier to choose a base and see the height when finding the volume.

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YOUAre Ready?Personal

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Online Assessment and

Interventionmy.hrw.com

8 ft

3 ft

Complete these exercises to review skills you will need for this module.

Use of ParenthesesEXAMPLE 1 _ 2 (14) (12 + 18) = 1 _ 2 (14) (30)

= 7 (30)

= 210

Evaluate.

1. 1 _ 2 (3) (5 + 7)

2. 1 _ 2 (15) (13 + 17)

3. 1 _ 2 (10) (9.4 + 3.6)

4. 1 _ 2 (2.1) (3.5 + 5.7)

Area of Squares, Rectangles, TrianglesEXAMPLE Find the area of the

rectangle. A = bh

= 8 · 3

= 24

Area equals 24 square feet.

Find the area of each figure.

5. a triangle with base 6 in. and height 3 in.

6. a square with sides of 7.6 m

7. a rectangle with length 3 1 _ 4 ft and width 2 1 _ 2 ft

8. a triangle with base 8.2 cm and height 5.1 cm

Use the formula for area of a rectangle.

Substitute for base and height.

Multiply.

Perform operations inside parentheses first.

Multiply left to right.

Multiply again.

18

9 in 2

8 1 _ 8 ft 2

57.76 m 2

20.91 cm 2

225 65 9.66

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6_MFLESE056695_U6MO15.indd 416 13/11/13 10:14 AM

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PROFESSIONAL DEVELOPMENT VIDEOmy.hrw.com

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Professional Development

Are You Ready?Assess ReadinessUse the assessment on this page to determine if students need intensive or strategic intervention for the module’s prerequisite skills.

Interactive WhiteboardsEngage students with interactive whiteboard-ready lessons and activities.

Personal Math Trainer: Online Assessment and InterventionAssign automatically graded homework, quizzes, tests, and intervention activities.Prepare your students with updated practice tests aligned with Common Core.

Online Teacher EditionAccess a full suite of teaching resources online—plan, present, and manage classes and assignments.

ePlannerEasily plan your classes and access all your resources online.

Interactive Answers and SolutionsCustomize answer keys to print or display in the classroom. Choose to include answers only or full solutions to all lesson exercises.

Author Juli Dixon models successful teaching practices as she explores the concepts of finding surface area and volume of solids in an actual sixth-grade classroom.

Intervention Enrichment

Access Are You Ready? assessment online, and receive instant scoring, feedback, and customized intervention or enrichment.

Online and Print Resources

Skills Intervention worksheets

• Skill 50 Use of Parentheses

• Skill 85 Area of Squares, Rectangles, Triangles

Differentiated Instruction

• Challenge worksheets PRE-AP

Extend the Math PRE-AP Lesson Activities in TE

Surface Area and Volume of Solids 416

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Reading Start-Up

Active ReadingBooklet Before beginning the module, create

a booklet to help you learn the concepts in this

module. Write the main idea of each lesson on

each page of the booklet. As you study each

lesson, write important details that support the

main idea, such as vocabulary and important steps

in solving problems. Refer to your finished booklet

as you work on assignments and study for tests.

VocabularyReview Words

area (área) base (base) height (altura) rectangular prism (prisma

rectangular) volume (volumen)

Preview Words

net (plantilla) pyramid (pirámide) surface area (área total)

Visualize VocabularyUse review words to complete the graphic.

A = b ×

A = 1 _ 2

× × h

Shape Area FormulaShape Area Formula

Understand VocabularyComplete the sentences using the preview words.

1. The total area of all the faces of a three-dimensional figure is

called the .

2. A model that looks like an unfolded three-dimensional figure is a

.

3. A three-dimensional shape with a polygon for a base and triangles

for sides is a .

height

base

surface area

net

pyramid

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Reading Start-Up

Have students complete the activities on this page by working alone

or with others.

Visualize VocabularyThis graphic helps students review the vocabulary associated with the

area formulas for parallelograms and triangles. Students should write

the word from the Review List in the box that belongs in the blank.

Understand VocabularyUse the following explanations to help students learn the preview

words.

Finding the total area of all the faces of a three-dimensional

figure is finding the surface area of that solid.

A net is a two-dimensional model [flat] of a solid that

allows you to see all the faces of the solid.

A pyramid is a solid that has faces that are

triangle-shaped.

Active ReadingIntegrating Language ArtsStudents can use these reading and note-taking strategies to help

them organize and understand new concepts and vocabulary.

Additional ResourcesDifferentiated Instruction

• Reading Strategies ELL

In this moduleIn this module, students will:

• identify nets and use nets to find the surface area

of a solid

• calculate the volume of rectangular solids and

use volume equations to solve problems

After

Students will solve real-world and

mathematical problems involving:

• area, volume, and surface area

of two- and three-dimensional

objects composed of triangles,

quadrilaterals, polygons, cubes,

and right prisms

Before

Students will have explored

volume to:

• recognize volume as an attribute

of solid figures and understand

concepts of volume measurement

• measure volumes by counting unit

cubes, using cubic cm, cubic in.,

cubic ft, and improvised units

• relate volume to the operations

of multiplication and addition and

solve real-world and mathematical

problems involving prisms and

composite solids made of prisms

COMMONCORE ELA-Literacy.RST.6-8.7 Integrate quantitative or technical information

expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table).

417 Module 15

4 in.3 in.

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Unpacking the StandardsUnderstanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this module.

What It Means to YouYou will use a net to find the surface area of a square pyramid.

Meg drew a net to find the surface area of a square pyramid.

Square face: A = b × h = 16 square inches

Triangular face: A = 1 _ 2 b × h = 6 square inches

Total of the areas: 16 + (4 × 6) = 40 square inches

The surface area is 40 square inches.

What It Means to YouYou will use the formula for the volume of a rectangular prism.

Jala has an aquarium in the shape of a rectangular prism. The dimensions of the base of the aquarium are 1 1 _ 4 feet by 1 _ 2 foot, and the height is 3 _ 4 foot. Find the volume of the aquarium.

The volume of the aquarium is 15 __ 32 cubic foot.

MODULE 15

v = l · w · h

= 1 1 _ 4 · 1 _ 2 · 3 _ 4

= 5 _ 4 · 1 _ 2 · 3 _ 4

= 15 __ 32

6.G.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. …

COMMON CORE

6.G.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

COMMON CORE

UNPACKING EXAMPLE 6.G.4

Visit my.hrw.com to see all the Common Core Standards unpacked.

UNPACKING EXAMPLE 6.G.2

Unit 6418

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Use the examples on the page to help students know exactly what they are expected to learn in this module.

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Unpacking the Standards

Go online to see a complete unpacking of the Common Core Standards.

Common Core Standards

Content Areas

COMMONCORE Geometry—6.G

Solve real-world and mathematical problems involving area, surface area, and volume.

Common Core Standards Lesson 15.1

Lesson 15.2

Lesson 15.2

6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

COMMONCORE

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

COMMONCORE

COMMONCORE

6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures.

COMMONCORE



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Interactive Whiteboard Interactive example available online

ADDITIONAL EXAMPLE 1Use a net to find the surface area of the square pyramid. 132 cm 2

EngageESSENTIAL QUESTION

How can you use nets to find surface areas? A net shows a three-dimensional shape as two-dimensional shapes, which makes it easier to find the area of each face.

Motivate the LessonAsk: How many of you have every wrapped a present? Have you ever wondered how to figure out how much wrapping paper you need? Begin the Explore Activity to find out.

ExploreEXPLORE ACTIVITY

Focus on Modeling Mathematical PracticesShow students an empty cereal box. Rip the seams so the box will lie flat. Tell the students that the box was a rectangular prism with six faces. Also, tell them that the flattened box can be considered a net and represents the surface area of the box. The surface area determines how much cardboard is needed to make the box.

ExplainEXAMPLE 1

Focus on Modeling Mathematical PracticesPoint out to students that if they draw the net for any figure on graph paper, they can count the squares to estimate the surface area of the figure before they actually do the calculations.

Questioning Strategies Mathematical Practices • How do the triangular faces of a square pyramid compare? All four triangular faces are congruent, so they have the same area.

• How are the triangular faces of a rectangular pyramid different from the triangular faces of a square pyramid? Not all four faces are congruent. Two faces will have a different slant height from the other two faces.

YOUR TURNEngage with a Whiteboard

Have a student volunteer draw and label the net on the whiteboard. Then have another student write expressions for the areas of the four triangles and the area of

the base before performing the calculations. Then have another student write the answer on the whiteboard.

Questioning Strategies Mathematical Practices • What type of pyramid is this? a square pyramid How many faces does this pyramid have? There are 5 faces.

• What shapes are used in the net? There are 4 congruent triangles and 1 square.

15.1L E S S O N

Nets and Surface Area

CC

CC

CC

CC

Common Core StandardsThe student is expected to:

Geometry—6.G.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. Also 6.EE.2c

Mathematical Practices

MP.6 Precision

COMMONCORE

COMMONCORE

6 cm 6 cm

8 cm

419 Lesson 15.1


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17 in.

16 in. 16 in.

17 in.

16 in.

16 in.

20 in.

16 in. 16 in.

20 in.

Animated Math

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Surface Area of a PyramidThe surface area of a three-dimensional figure is the sum of the areas of its

faces. A net can be helpful when finding surface area.

A pyramid is a three-dimensional figure whose base is a polygon and whose

other faces are triangles that meet at a point. A pyramid is identified by the

shape of its base.

Make a net of this square pyramid,

and use the net to find the surface area.

Make a net of the pyramid.

Draw the square base.

Draw a triangle on each side.

Label the dimensions.

Use the net to find the

surface area.

There are four triangles with

base 16 in. and height 17 in.

The area of the 4 triangles

is 4 × 1 _ 2 (16)(17) = 544 in 2 .

The area of the base is 16 × 16 = 256 in 2 .

The surface area is 544 + 256 = 800 in 2 .

EXAMPLE 1

STEP 1

STEP 2

How many surfaces does a triangular pyramid have? What shape

are they?

4. Use a net to find the surface area of the

pyramid.

YOUR TURN

EXPLORE ACTIVITY (cont’d)

3. What shapes will appear in a net for a rectangular prism that is not a

cube? How many of these shapes will there be?

Math TalkMathematical Practices

6.G.4COMMONCORE

A triangular pyramid has 4 triangular surfaces.

896 i n 2

rectangles; 6

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EXPLORE ACTIVITY

Net A Net B

ESSENTIAL QUESTION

Using a NetA net is a two-dimensional pattern of shapes that can be folded

into a three-dimensional figure. The shapes in the net become

the faces of the three-dimensional figure.

Copy each net on graph paper. Cut out each net along the blue lines.

One of these nets can be folded along the black lines to make a cube.

Which net will NOT make a cube?

On your graph paper, draw a different net that you think will make a

cube. Confirm by cutting out and folding your net. Compare results

with several of your classmates. How many different nets for a cube did

you and your classmates find?

ReflectHow do you know that each net cannot be folded into a cube without

actually cutting and folding it?

1.

2.

A

B

How can you use nets to find surface areas?

L E S S O N

15.1Nets and Surface Area

6.G.4

Represent three-dimensional figures using nets ... and use the nets to find...surface area.... Also 6.EE.2c

COMMONCORE

6.G.4COMMONCORE

Net B

Check answers.

The net has only 5 faces,

but a cube has 6.

The net folds into a

“loop” of squares with

no top or bottom.

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PROFESSIONAL DEVELOPMENT

Math BackgroundA net is a two-dimensional pattern of shapes

that can be folded into a three-dimensional

figure. The shapes in the net become the faces of

the three-dimensional figure. You can use the net

to find the surface area of the three-dimensional

figure by adding up the areas of the faces.

CC Integrate Mathematical Practices MP.6

This lesson provides an opportunity to address

the Mathematical Practices standard that calls for

students to attend to precision. Throughout this

lesson, students are drawing nets and calculating

surface area. They need to be careful when

labeling the dimensions of the nets. They also

need to calculate accurately and efficiently when

finding the surface area of a given figure.

Nets and Surface Area 420

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ADDITIONAL EXAMPLE 2Angela wants to wrap the present shown. How much wrapping paper will she need? The wrapping paper costs $0.18 per square foot. How much will the wrapping paper cost? 62 square feet; $11.16

EXAMPLE 2Connect Vocabulary ELL

Point out to students that a prism is a three-dimensional figure named for the shape of its base; rectangular, square, or triangular. The two identical bases are congruent polygons. In this book, all the prisms are right prisms and it is assumed that the faces of a prism that are not its bases are rectangles. In a later geometry course, students will learn about oblique prisms.

Questioning Strategies Mathematical Practices • How many faces does a square prism have? 6 faces How many faces need to be covered with foil? 5 faces Why are only 5 faces being covered? The bottom face will be touching the floor and will not need to be covered with foil.

• How are a rectangular prism and a rectangular pyramid similar? How are they different? Both are three-dimensional shapes, and both have faces that are polygons. Prisms have two bases; pyramids only have one base. Rectangular prisms have 6 faces while rectangular pyramids have 5 faces.

Engage with the WhiteboardIn Step 1, have a student volunteer draw and label the net on the whiteboard. Discuss the net with the class, confirming its shape and the labeling of the

measurements.

YOUR TURNAvoid Common ErrorsSome students may omit faces when finding the surface area of solid figures. Encourage them to either begin by drawing a net or by listing the faces.

ElaborateTalk About ItSummarize the Lesson

Ask: What is a net, and how does it help us find surface area? A net is a two-dimensional representation of a three-dimensional figure. We can use it to find the

area of each face and then use the sum of all the areas to find the surface area of the figure.

GUIDED PRACTICEEngage with the Whiteboard

For Exercise 3, have students draw and label a net on the whiteboard. Then have them write expressions for bases and sides before calculating the areas.

Avoid Common ErrorsExercises 1, 3 Some students may omit faces when finding the surface area of solid figures. Encourage them to either begin by drawing a net or by listing the faces.Exercises 1–3 Remind students to look at the shape of each face and then choose the most appropriate formula to calculate its area.

CC3 ft

2 ft

5 ft

421 Lesson 15.1


My Notes


4 ft

4 ft 4 ft

4 ft

4 ft

4 ft

3 ft

3 ft

3 ft 3 ftTop

Back Rightside

Leftside Front

Surface Area of a PrismA prism is a three-dimensional figure with two identical and parallel bases that are polygons. The other faces are rectangles. A prism is identified by the shape of its base.

A sculpture sits on pedestal in the shape of a square prism. The side lengths of a base of the prism are 3 feet. The height of the prism is 4 feet. The museum director wants to cover all but the underside of the pedestal with foil that costs $0.22 per square foot. How much will the foil cost?

Use a net to show the faces that will be covered with foil.

Draw the top.

Draw the faces of the prism that are connected to the top.

You don’t need to include the bottom of the pedestal.

Use the net to find the area that will be covered with foil.

Area of top = 3 ⋅ 3 = 9 ft 2

The other four faces are identical.

Area of four faces = 4 ⋅ 3 ⋅ 4 = 48 ft 2

Area to be covered = 9 + 48 = 57 ft 2

Find the cost of the foil.

57 ⋅ $0.22 = $12.54

The foil will cost $12.54.

Reflect5. Critical Thinking What shapes would you see in the net of a triangular

prism?

EXAMPLEXAMPLE 2

STEP 1

STEP 2

STEP 3

6.G.4COMMONCORE

two identical triangles and three rectangles

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5 ft

2 ft

3 ft 4 ft

10 in.

6 in.

4 in.


and Intervention

Personal

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4 in.6 in.

6 in.

6 in. 6 in.

4 in.

A square pyramid is shown.

1. The figure has square base

and triangular faces. (Explore Activity)

2. Find the surface area. (Example 1)

The area of the base is square inches.

The area of the four faces is square inches.

The surface area is square inches.

3. Yolanda makes wooden boxes for a crafts fair. She makes 100 boxes like the one shown, and she wants to paint all the outside faces. (Example 2)

a. Find the surface area of one box.

b. Find the total surface area of 100 boxes.

c. One can of paint will cover 14,000 square inches. How many cans of paint will Yolanda need to buy?

Guided Practice

6. The figure shown is a triangular prism. How muchwould it cost to cover the bases and the other threefaces with foil that costs $0.22 per square foot?

YOUR TURN

4. How is a net useful when finding the surface area of prisms and pyramids?

ESSENTIAL QUESTION CHECK-IN???A net shows the three-dimensional figure as flat and

two-dimensional, which makes it easier to visualize the

shapes of the surfaces and add up their areas.

2 cans

24,800 square inches

248 square inches

$7.92

1

4

3648

84

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DIFFERENTIATE INSTRUCTION

ManipulativesSome students may have difficulty interpreting a drawing of a prism. Give these students three-dimensional models to help them identify the bases and sides. You may also wish to have them use the models to trace the faces when drawing their nets.

ModelingGive students the nets for several three-dimensional figures on graph paper. Have them cut out the nets and assemble them to make models of the figures. Students can then use the graph paper squares to estimate the surface area before computing the actual surface area. Comparing their actual answers with the estimates will help them determine whether their answers are reasonable.

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP



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assignment available

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Lesson Quiz available online

15.1 LESSON QUIZ

1. What shape does this net represent? What is its surface area?

2. Find the surface area.

3. Lauren bought a rectangular box that is 6 inches long, 10 inches wide, and 4 inches high. She wants to cover it in fabric. The fabric costs $0.25 per square inch. How much will the fabric cost?

Evaluate

GUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Explore Activity

Using a Net Exercise 1

Example 1

Surface Area of a PyramidExercise 2

Example 2

Surface Area of a PrismExercises 3, 5–9

Additional ResourcesDifferentiated Instruction includes:

• Leveled Practice Worksheets

CC

6.G.4COMMONCORE

COMMONCORE 6.G.4

Exercise Depth of Knowledge (D.O.K.) Mathematical Practices

5–8 2 Skills/Concepts MP.4 Modeling

9 2 Skills/Concepts MP.2 Reasoning

10 2 Skills/Concepts MP.4 Modeling

11 2 Skills/Concepts MP.6 Precision

12–13 3 Strategic Thinking MP.3 Logic

COMMONCORE

Exercise 12 combines concepts from the Common Core cluster “Solve

real-world and mathematical problems involving area, surface area, and

volume.”

3 cm

4 cm

3 cm

7 ft

5 ft

1 ft

Answers1. square pyramid; 33 square centimeters

2. 94 square feet

3. $62

423 Lesson 15.1




12 in.

8 in. 2 in.

8 in.6 in.

5 in.

3 in.

15 in.

3.6 in.

4 in.

3.6 in.

3.6 in.3 in. 4 in. 4 in.

15 in.

3.6 in.

3.6 in.3.6 in.

3.6 in.

Name Class Date

Independent Practice

5. Use a net to find the surface area of the cereal box.

Total surface area:

6. Inez bought a shipping container at a packaging store. She measured the dimensions shown to the nearest tenth.

a. Sketch a net of the container, and label the dimensions.

b. Find the surface area of the shipping container.

7. Raj builds a side table in the shape of a cube. Each edge of the cube measures 20 inches. Raj wants to cover the top and four sides of the table with ceramic tiles. Each tile has an edge length of 5 inches. How many tiles will he need?

8. Santana wants to cover a gift box shaped like a rectangular prism with foil. The foil costs $0.03 per square inch. Santana has a choice between Box A which is 8 inches long, 3 inches wide, and 6 inches high, and Box B which is 10 inches long, 3 inches wide, and 4 inches high. Which box will be less expensive to cover with foil, and by how much?

9. Vocabulary Name a three-dimensional shape that has four triangular faces and one rectangular face. Name a three-dimensional shape that has three rectangular faces and two triangular faces.

10. Victor wrapped the gift box shown with adhesive paper (with no overlaps). How much paper did he use?

15.16.G.4COMMON

CORE

236 in 2

rectangular pyramid; triangular prism

Box B will cost $0.48 less than Box A.

80 tiles

180 square inches

272 in 2

423Lesson 15.1

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Work Area

11. Communicate Mathematical Ideas Describe how you approach a problem involving surface area. What do you do first? What are some strategies you can use?

12. Persevere in Problem Solving A pedestal in a craft store is in the shape of a triangular prism. The bases are right triangles with side lengths of 12 centimeters, 16 centimeters, and 20 centimeters. The store owner wraps a piece of rectangular cloth around the pedestal, but does not cover the identical bases of the pedestal with cloth. The area of the cloth is 192 square centimeters.

a. What is the distance around the base of the pedestal? How do you know?

b. What is the height of the pedestal? How did you find your answer?

13. Critique Reasoning Robert sketches two rectangular prisms, A and B. Prism A’s side lengths are 5 centimeters, 6 centimeters, and 7 centimeters. Prism B’s side lengths were twice those of prism A’s: 10 centimeters, 12 centimeters, and 14 centimeters. Robert says the surface area of prism B is twice the surface area of prism A. Is he correct? If he is not, how many times as great as prism A’s surface area is prism B’s surface area? Show your work.

FOCUS ON HIGHER ORDER THINKING

Robert is incorrect; Surface area of A is 30 + 30 + 42 +

42 + 35 + 35 = 214 square centimeters. Surface area of

B is 120 + 120 + 168 + 168 + 140 + 140 = 856 square

centimeters. B’s surface area is 4 times as great as A’s.

Sample answer: Decide whether the figure is a prism or

a pyramid, and what the shapes of the bases and faces

are. Make a net of the figure, and label the dimensions.

Use appropriate area formulas to find the areas of the

faces, and then find the sum of the areas.

4 centimeters; Sample answer: The cloth is a rectangle

whose longer side is 48 centimeters long; the width is

the height of the pedestal; 192 ÷ 48 = 4 cm.

48 centimeters; the base is a triangle with perimeter

12 + 16 + 20 cm.

Unit 6424

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Activity available online my.hrw.comEXTEND THE MATH PRE-AP

Activity Find the surface area of the rectangular prism shown with a rectangular-prism-shaped hole all the way through it.

20 cm25 cm

12 cm

8 cm

8 cm2,592 cm 2


DO NOT EDIT--Changes must be made through "File info"CorrectionKey=D

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ADDITIONAL EXAMPLE 1Find the volume of the rectangular prism. 77 cubic centimeters


How do you find the volume of a rectangular prism? Use the formula V = lwh, where l is the length, w is the width, and h is the height.

Motivate the LessonAsk: Have you ever wondered how to find the volume of a rectangular aquarium or terrarium? Begin the Explore Activity to find out how to find the volume of a rectangular prism.

ExploreEXPLORE ACTIVITY

Focus on Modeling Mathematical PracticesUse centimeter cubes to build a rectangular prism that is 4 centimeters long, 2 centimeters wide, and 2 centimeters high. Explain to students that the volume of a three-dimensional figure is the number of cubes it can hold. Show them that you can find the volume of the rectangular prism by counting the cubes or by multiplying the lengths of the edges.

EXAMPLE 1Focus on Math Connections Mathematical PracticesPoint out to students that the process of finding the volume of a three-dimensional figure is similar to the process of finding the area of a two-dimensional figure, except that now you are multiplying three dimensions. Also, point out that volume is measured in cubic units and area is measured in square units.

Questioning Strategies Mathematical Practices • Does the order in which you multiply the dimensions of the prism matter? No. Multiplication is commutative which means that you can multiply numbers in any order and get the same result.

• Describe another way to find the volume of the prism. You can first find the area of the base and then multiply by the height, V = Bh.

Connect Vocabulary ELL

Make sure that students understand the distinction between the base of a prism and the base of a rectangle or parallelogram. In the formula V = Bh, B represents the area of a base. In the formula A = bh, b represents the length of the base of a rectangle or parallelogram.

YOUR TURNAvoid Common ErrorsSome students may multiply the fraction parts and whole number parts separately when multiplying. Remind them that this method results in an incorrect product, and emphasize that mixed numbers must first be written as improper fractions before multiplying.

15.2L E S S O N

Volume of Rectangular Prisms

CC

CC

CC


Geometry—6.G.2



MP.4 Modeling

COMMONCORE

COMMONCORE

4 cm

3 cm12

5 cm12

425 Lesson 15.2



3 m

2 m14

4 m12

Finding VolumeA rectangular prism has six faces. Any pair of opposite faces can be called the

bases of the prism.

Find the volume of the rectangular prism.

ℓ = 3 meters w = 2 1 _ 4

meters h = 4 1 _ 2

meters

EXAMPLE 1

V = ℓwh

= 3 ∙ 2 1 _ 4

∙ 4 1 _ 2

= 3 ∙ 9 _ 4

∙ 9 _ 2

= 243 ___

8

= 30 3 _ 8

cubic meters

Reflect1. Several of the small cubes in the Explore Activity are arranged into

a medium-sized cube as shown.

Show two different ways to find the volume of this cube.

EXPLORE ACTIVITY (cont’d)

Can you also use the formula V = Bh to find the

volume? Does it matter which face you choose as

the base?


6.G.2COMMONCORE

Volume of a Rectangular Prism

V = ℓwh, or V = Bh

(where B represents the area of

the prism’s base; B = ℓw)

Write each mixed number as an improper fraction.

Multiply.

Write as a mixed number in simplest form.

h

wl

2) Each small cube has edge length 1 _ 3

, and there are 2 small

cubes along each edge. The edge length of the medium-

sized cube is 2 · 1 _ 3

= 2 _ 3

, and its volume is 2 _ 3

· 2 _ 3

· 2 _ 3

= 8 __ 27

.

1) 8 small cubes each have volume 1 __ 27

; 8 · 1 __ 27

= 8 __ 27

Yes; No; the calculation will still give the same result.

Unit 6426

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EXPLORE ACTIVITY

1 unit

ESSENTIAL QUESTIONHow do you find the volume of a rectangular prism?

L E S S O N

15.2Volume of Rectangular Prisms

Using Fractional Edge LengthsA cube with edge length 1 unit and volume 1 cubic unit is filled with

smaller cubes as shown.

How many small cubes are there?

How does the combined volume of the small cubes

compare to the volume of the large cube?

∙ ? =

What is the volume of one small cube? cubic unit(s)

Each edge of the large cube contains small cubes.

∙ ? =

What is the edge length of one small cube? unit(s)

Complete:

Each small cube has edge length unit(s) and

volume cubic unit(s).

The formula for volume of a cube with edge length ℓ is V = ℓ · ℓ · ℓ,

or V = ℓ 3 . Find the volume of one small cube using this formula.

V = =

A

B

C

D

Number of small cubes

∙Volume of one

small cube=

Volume of large cube

Number of small cubes per edge

∙Edge length of one small cube

=Edge length of

large cube

6.G.2

Find the volume of a right rectangular prism with fractional edge lengths....

COMMONCORE

6.G.2COMMONCORE

27

1 __ 27

1 __ 27

1 __ 27

1 _ 3

∙ 1 _ 3

∙ 1 _ 3

3

1 _ 3

1 _ 3

The volumes are equal.

27

3

1

1

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PROFESSIONAL DEVELOPMENT

Math BackgroundWhen students first learned about the exponent

2, they were told that an expression like 5 2 is read

“five squared.” That is related to the fact that a

square with sides of length 5 units has an area of

5 2 , or 25 square units.

Similarly, an expression like 6 3 can be related to a

cube whose sides have a length of 6 units. By the

formula for the volume of a rectangular prism,

this cube has a volume of 6 × 6 × 6, or 6 3 cubic

units. Just as the model of a square illustrates

why we refer to 5 2 as “five squared,” the model of

a cube illustrates why we refer to 6 3 as “six cubed.”

CC Integrate Mathematical Practices MP.4

This lesson provides an opportunity to address

the Mathematical Practices standard that calls for

students to model with mathematics.

Throughout this lesson, students use tools such

as diagrams and formulas to solve problems

arising in everyday life and society.

Volume of Rectangular Prisms 426

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ADDITIONAL EXAMPLE 2A deck of playing cards is shaped like a

rectangular prism. The prism is 2 inches

long, 1 __ 2 inch wide, and 3 1

__ 2 inches high.

What is the volume of the deck of

cards? 3 1 __ 2 cubic inches

EXAMPLE 2Questioning Strategies Mathematical Practices • Could another face be used as the base of the prism? Give an example. Yes; for example,

if the base is the 13 1 __ 2 ×16 inch face, then B = 216 and h would be 25 1

__ 2 ; 216 × 25 1

__ 2 = 5,508.

• If you were to double the length, width, or height of the terrarium, how would the volume

be affected? If you double one dimension of the terrarium, the volume doubles as well.

Engage with the WhiteboardHave a student find the volume using the formula V = lwh on the whiteboard. Then

explain to students that the two formulas are equivalent by showing that B = lw.

YOUR TURNAvoid Common ErrorsSome students may forget to write the units or may write square units to represent volume.

Remind students that cubic units, not square units, are used to measure volume.

Elaborate

Talk About ItSummarize the Lesson

Ask: Describe two ways you can find the volume of a rectangular prism. Find the

area of one base and multiply by the height, or use the formula V = lwh.


For Exercises 3-4, have students fill in the blanks and calculate the volumes of each

prism. Ask students to find the volume using both formulas as a check.

Avoid Common ErrorsExercises 6–7 Some students may multiply the fraction parts and whole number parts

separately when multiplying. Remind them that this method results in an incorrect product,

and emphasize that mixed numbers must first be written as improper fractions before

multiplying.

CC

3 in.12

in.12 2 in.

427 Lesson 15.2

10 m

3.2 m

5 m

4 m

8 m

7 m14

4 ft

7.5 ft

9.25 ft

Guided Practice

A large cube is made up of smaller unit cubes as shown on the right.

Each small cube has an edge length of 1 _ 2

unit. (Explore Activity)

1. Each edge of the large cube is units.

2. The volume of the large cube is cubic units.

Find the volume of each prism. (Example 1)

3.

V = × ×

= cubic meters

4.

B = × = m 2

V = cubic meters

5.

V = cubic feet

6.

4 in.12

6 in.12

18 in.

V = cubic inches

8. Which two formulas can you use to find the volume of a rectangular

prism? Why are these two formulas equivalent?

ESSENTIAL QUESTION CHECK-IN??

7. A cereal box is 8 1 _ 2 inches long, 3 1 _

2 inches wide, and 12 inches high.

What is the volume of the box? (Example 2)

V = lwh and V = Bh; The area of the base, B, is really

lw (length times width), and since both B and lw are

multiplied by h, the two are equivalent.

357 cubic inches

277.5

526 1 _ 2

2

8

10

7 1 _ 4

160

232

3.2

4

5

29

Unit 6428

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Math Trainer


Personal

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Math Trainer


Personal

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7 in.12

2 in.12

4 in.

Solving Volume ProblemsWhen you solve a real-world problem involving the volume of a prism, you can

choose to use either of the volume formulas you know.

A terrarium is shaped like a rectangular prism. The prism is 25 1 _ 2

inches long,

13 1 _ 2

inches wide, and 16 inches deep. What is the volume of the terrarium?

Choose one side to be the base, and find its area.

Find the volume.

V = Bh

= 1,377

____ 4

× 16

= 1,377

_____ 4

× 16 __

1

= 5,508

4

1

The volume of the terrarium is 5,508 cubic inches.

EXAMPLEXAMPLE 2

STEP 1

B = 25 1 _ 2

× 13 1 _ 2

= 51 __

2 × 27

__ 2

= 1,377

____ 4

STEP 2

4. A rectangular swimming pool is 15 meters long, 10 1 _ 2 meters wide,

and 2 1 _ 2 meters deep. What is its volume?

YOUR TURN

Find the volume of each rectangular prism.

2.

3. length = 5 1 _ 4 inches

width = 3 1 _ 2 inches

height = 3 inches

YOUR TURN

6.G.2COMMONCORE

Use the 25 1 __ 2

-inch by 13 1 __ 2

- inch face as the base.

Substitute 1,377

_____ 4

forB and 16 for h.

Simplify before multiplying.

The area of the base is 1,377

_____ 4

square inches. You need to perform another operation, so you don’t need to write this value as a mixed number.

393 3 _ 4 cubic meters

75 cubic inches 55 1 _ 8

cubic inches

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ManipulativesHave students bring a box, such as a cereal box

or cracker box, from home. Have them find the

volume of the box. Make a table of the

dimensions and volumes on the board. Then ask

students if two rectangular prisms can have

different heights but the same volume. Ask them

to explain their reasoning. Sample answer:

Yes, it is possible for two rectangular prisms to

have different heights but the same volume.

For example, a prism with a 2 × 3 unit base

and a height of 2 units has the same volume

(12 units 3 ) as a prism with a 6 × 2 unit base

and a height of 1 unit.

Critical ThinkingHave students build a variety of prisms using

centimeter cubes. Ask students to count the

cubes to find the height, area of the base, and

volume of each prism and to record the data in a

table as shown.

Ask students to find a relationship between the

area of the base, the height, and the volume.

Sample answer: The volume is the product of

the area of the base and the height, or V = Bh.

Additional ResourcesDifferentiated Instruction includes:

• Reading Strategies

• Success for English Learners ELL

• Reteach

• Challenge PRE-AP

Area of base Height Volume



and Intervention

Personal

Online homework assignment available

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15.2 LESSON QUIZ

1. A rectangular prism is 20 inches long, 10 inches wide, and 15 inches high. What is the volume of the prism?

2. What is the volume of a rectangular prism with a length of 2 1 __ 2 inches, a width of 4 1 __ 4 inches, and a height of 3 inches?

3. Find the volume of a rectangular prism with a base that has an area of 22 cm 2 and a height of 8 cm.

4. A classroom is 32 1 __ 2 feet long, 30 feet wide, and 7 1 __ 2 feet high. What is the volume of the classroom?

EvaluateGUIDED AND INDEPENDENT PRACTICE

Concepts and Skills Practice

Explore ActivityUsing Fractional Edge Lengths

Exercises 1–2

Example 1Finding Volume

Exercises 3–6

Example 2Solving Volume Problems

Exercises 7, 9–10, 12–15

Additional ResourcesDifferentiated Instruction includes: • Leveled Practice Worksheets

6.G.2 COMMONCORE

COMMONCORE 6.G.2



11 3 Strategic Thinking MP.2 Reasoning



15 2 Skills/Concepts MP.4 Modeling

16–17 3 Strategic Thinking MP.2 Reasoning

18 3 Strategic Thinking MP.6 Precision


COMMONCORE

ft32

30 ft

12

ft7 12

Answers1. 3,000 cubic inches

2. 31 7 __ 8 cubic inches

3. 176 cm 3

4. 7,312 1 _ 2 ft3

429 Lesson 15.2

DO NOT EDIT--Changes must be made through "File info"CorrectionKey=D




36 in.

72 in.12

24 in.12

4 cm14

2 cm34

5 cm

Name Class Date

Independent Practice15.2

9. A block of wood measures 4.5 inches by 3.5 inches by 7 inches. What is

the volume of the block of wood?

10. A restaurant buys a freezer in the shape of a rectangular prism. The

dimensions of the freezer are shown. What is the volume of the freezer?

11. Rectangular prism A measures 6 inches by 4 inches by 5 inches.

Rectangular prism B’s dimensions are twice those of prism A. Find

the volume of each prism. How many times as great is prism B’s volume as

prism A’s volume?

12. Leticia has a small paper weight in the shape of a rectangular prism.

The dimensions of the paper weight are shown. What is the volume of

the paper weight?

13. A company is designing a juice box. The box is in the shape of a

rectangular prism. The base of the box is 6 1 _ 2 inches by 2 1 _

2 inches, and

the box is 4 inches high. If juice fills 90% of the box’s volume, find the

volume of juice in the box.

14. Science Density is the amount of mass in a certain volume of an object.

To find the density in grams per cubic centimeter of a substance you

can use this relationship:

Density = mass in grams

_____________________ volume in cubic centimeters

A gold bar that is 16 centimeters by 2.5 centimeters by 5 centimeters has

a density of 19.3 grams per cubic centimeter. What is the mass of the

gold bar?

15. A suitcase is a rectangular prism whose dimensions are 1 1 _ 4 feet by

1 3 _ 4 feet by 1 1 _

4 feet. Find the volume of the suitcase.

6.G.2COMMONCORE

110.25 cubic inches

63,945 cubic inches

A: 120 in 3 ; B: 960 in 3 ; 8 times as great

58 7 __ 16

cubic centimeters

58.5 cubic inches

2 47 __

64 cubic feet

3,860 grams

429Lesson 15.2

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Work Area

16. The Smith family is moving and needs to decide on the size of

the moving truck they should rent.

a. A moving van rents for $94.50 per day, and a small truck rents

for $162 per day. Based on the amount of space inside the

van or truck, which is the better deal? Explain your answer.

b. How much greater is the volume of the large truck than the

volume of the small truck?

c. The family estimates that they need about 1,100 cubic feet to

move their belongings. What should they rent?

17. Persevere in Problem Solving A cube has a volume of 1 ___

512 cubic meter.

What is the length of each side of the cube? Explain your thinking.

18. Communicate Mathematical Ideas Think about two rectangular prisms,

one labeled prism P and one labeled prism Q.

a. Suppose the bases of the prisms have the same area, but the height of

prism Q is twice the height of prism P. How do the volumes compare?

b. Suppose the area of the base of prism Q is twice the area of the base

of prism P. How do the volumes compare?

19. Critical Thinking The dimensions of a rectangular prism are 3 1 _ 4 feet by 2 1 _

2

feet by 5 feet. Lee found the volume by multiplying 12 1 _ 2 by 3 1 _

4 . Lola found

the volume by multiplying 16 1 _ 4 by 2 1 _

2 . Who is correct? Explain.


Inside Dimensions of Trucks

Type Length (ft)

Width (ft)

Height (ft)

Van 10 1 _ 2

6 6

Small Truck

12 8 6 3 _ 4

Large Truck

20 8 3 _ 4

8 1 _ 2

They cost the same per unit volume; the van’s volume

is 378 ft 3 , so the cost is $0.25 per ft 3 and the small

truck volume is 648 ft 3 , so the cost is $0.25 per ft 3 .

839.5 cubic feet

1 _ 8

m; find a fraction that, when multiplied by itself 3

times, equals 1 ___

512 .

the large truck

The volume of Q is twice the volume of P.

The volume of Q is twice the volume of P.

Both are correct. Lee used the 2 1 _ 2

by 5-foot face as the base and

3 1 _ 4

as the height, then multiplied the base area, 12 1 _ 2

, by the

height. Lola used the 3 1 _ 4

by 5-foot face as the base and 2 1 _ 2

as

the height, then multiplied the base area, 16 1 _ 4

, by the height.

Unit 6430

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Activity A 5-inch section of a hollow brick

measures 12 inches tall and 8 inches wide on the

outside. The brick is 1 inch thick. Find the volume of

the brick, not the hollow interior.

180 in 3

8 in.

12 in.

5 in.

1 in.

1 in.


15.3L E S S O N

Solving Volume Equations


How do you write equations to solve problems involving volume of rectangular prisms? Sample answer: Use V = lwh to write an equation. Substitute known values and solve.

Motivate the LessonAsk: Have you ever wondered about the volume of a swimming pool, or how much water it takes to fill an aquarium? Begin Example 1 to find out.

ExploreEngage with the Whiteboard

Make a model of a rectangular prism using cubes (4 × 2 × 3) on the whiteboard. Ask students how many cubes make up this prism. Tell them they have just

found the volume of the prism. Then tell them that they can also use V = lwh to find the volume.

ExplainEXAMPLE 1

Focus on Math Connections Mathematical PracticesPoint out to students that finding a missing dimension in a problem involving volume is similar to finding an unknown value using an area formula.

Questioning Strategies • How do you know what operation to use? The formula for volume is a multiplication equation, so you will need to use division to find any one of the missing dimensions.

YOUR TURNAvoid Common ErrorsRemind students that dividing by a number is the same as multiplying by its reciprocal.

EXAMPLE 2Focus on Critical Thinking Mathematical PracticesPoint out to students that just as with problems involving area, you can use the formula for volume to find a missing or unknown dimension. The process is the same; you just have an additional variable to work with.

Questioning Strategies Mathematical Practices • In Step 1, why is 30 gallons divided by 7.5 gallons per cubic foot? To find the length of the aquarium in cubic feet, you need to express 30 gallons as a measure in cubic feet. To do so, you need to divide 30 gallons by the conversion factor, 7.5 gallons per cubic foot.

YOUR TURNAvoid Common ErrorsRemind students that they must first convert the amount of water the aquarium holds to cubic feet before they can find the width of the aquarium.

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ADDITIONAL EXAMPLE 1Abby has a shoebox with a volume of 609 cubic inches. The length of the shoebox is 14.5 inches, and the width is 7 inches. What is the height of the shoebox? 6 in.

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ADDITIONAL EXAMPLE 2One cubic foot of oil equals approximately 0.2 barrel and weighs approximately 55 pounds.

A rectangular oil storage tank holds 210 barrels of oil. It is 7 feet wide and has a height of 10 feet. What is the length of the tank? 15 ft

CC

CC

CC


Expressions and Equations—6.EE.7

Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all nonnegative rational numbers.

Geometry—6.G.2



MP.6 Precision

COMMONCORE

COMMONCORE

COMMONCORE

431 Lesson 15.3




and Intervention

Personal

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16 cm

18 cm

V = 6,336 cm3

? cm

Solving Multistep ProblemsOne cubic foot of water equals approximately 7.5 gallons and weighs approximately 62.43 pounds.

The classroom aquarium holds 30 gallons of water. It is 0.8 feet wide and has a height of 2 feet. Find the length of the aquarium.

Find the volume of the classroom aquarium in cubic feet.

30 gallons

__________________ 7.5 gallons per cubic foot

= 4 cubic feet

Find the length of the aquarium.

The length of the aquarium is 2.5 feet.

EXAMPLE 2

STEP 1

STEP 2

4 = ℓ · 0.8 · 2

4 = ℓ(1.6)

4 ___ 1.6 = ℓ(1.6) ____ 1.6

2.5 = ℓ

How much does the water in the classroom aquarium

weigh? Explain.

3. An aquarium holds 33.75 gallons of water. It has a length of 2 feet and a

height of 1.5 feet. What is the width of the aquarium?

YOUR TURN

1. Use an equation to find the width of the rectangular prism. (Example 1)

2. One clay brick weighs 5.76 pounds. The brick is 8 inches long and 2 1 _ 4 inches wide. If the clay weighs 0.08 pounds per cubic inch, what is the volume of the brick? Find the height of the brick. (Example 2)

Guided Practice


6.G.2, 6.EE.7COMMONCORE

Divide both sides of the equation by 1.6.

Use the formula V = ℓwh to write an equation.

Multiply.

Divide the total number of gallons by the unit rate to find the number of cubic feet.

249.72 pounds; one cubic foot weighs 62.43 pounds. 62.43 × 4 = 249.72

1.5 feet

6,336 = 16(18)w; w = 22 centimeters

V = 72 in 3 ; 4 inches

Unit 6432

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6_MFLESE056695_U6M15L3 432 23/02/13 7:01 AM

How do you write equations to solve problems involving volume of rectangular prisms?

?



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ft34

ft12

ESSENTIAL QUESTION

L E S S O N

15.3Solving Volume Equations

Writing Equations Using the Volume of a Rectangular PrismYou can use the formula for the volume of a rectangular prism to write an equation. Then solve the equation to find missing measurements for a prism.

Samuel has an ant farm with a volume of 375 cubic inches. The width of the ant farm is 2.5 inches and the length is 15 inches. What is the height of Samuel’s ant farm?

V = ℓwh

375 = 15 · 2.5 · h

375 = 37.5h

375 ____ 37.5 = 37.5h ____ 37.5

10 = h

The height of the ant farm is 10 inches.

Reflect1. Communicate Mathematical Ideas Explain how you would find the

solution to Example 1 using the formula V = Bh.

EXAMPLEXAMPLE 1

2. Find the height of this rectangular prism, which has a volume of 15 __ 16 cubic feet.

YOUR TURN

6.G.2

. . . Apply the formulas V = ℓwh and V = bh . . . in the context of solving real-world and mathematical problems. Also 6.EE.7

COMMONCORE

6.G.2, 6.EE.7COMMONCORE

Multiply.

Write the formula.

Use the formula to write an equation.

Divide both sides of the equation by 37.5.

Multiply 15 × 2.5 to find B, 37.5. Then divide the

volume, 375, by 37.5.

The height is 5 _ 2 ft or 2 1 _ 2 ft.

431Lesson 15.3

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Home ConnectionStudents may be unfamiliar with the volumes of rectangular prisms commonly found in their homes. Point out, for example, that they can calculate the volumes of items such as a juice box, a cereal box, a container of milk, a jewelry box, or even a rectangular cake. Ask students to select an item they can find at home, and have them estimate the volume of that item. Then have them share their work with the class. Make a list of all the items students choose.

Critical ThinkingAsk: How does increasing the area of a base of a rectangular prism affect its volume if the prism’s height remains the same? Explain and give an example. Sample answer: If the area of the base increases, the volume also increases because V = Bh = lwh. For example, doubling the area doubles the volume. Increasing the length of a side of the base also increases the area, so the volume becomes greater as well (e.g., If a prism is 5 in. × 4 in. × 3 in., V = 60 in3, but If the width (4 in.) becomes 5 in., then V = 75 in3.).

Additional ResourcesDifferentiated Instruction includes: • Reading Strategies • Success for English Learners ELL

• Reteach • Challenge PRE-AP


Solving Volume Equations 432



and Intervention

Personal

Online homework assignment available

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ElaborateTalk About ItSummarize the Lesson

Ask: How can you use equations to solve problems involving the volume of right rectangular prisms? Use A = lwh, the formula for the volume of a rectangular prism,

to write an equation. Then substitute known values for variables into the equation and solve for the unknown dimension.


For Exercise 2, have students circle or highlight the important information in the problem on the whiteboard. Ask them to sketch the figure, labeling known

dimensions. Then have them find the height of the brick and explain their reasoning.

Avoid Common ErrorsExercise 2 Remind students that they must first convert the volume of the brick from pounds to cubic inches before they can find the height of the brick.

EvaluateGUIDED AND INDEPENDENT PRACTICE

Concepts & Skills Practice

Example 1Writing Equations Using the Volume of a Rectangular Prism

Exercises 1, 3–5

Example 2Solving Multistep Problems

Exercises 2, 6–10

Additional ResourcesDifferentiated Instruction includes: • Leveled Practice Worksheets


15.3 LESSON QUIZ

1. The volume of a prism is 48 cubic centimeters. If the length is 2 centimeters and the width is 4 centimeters, what is the height?

2. Jeremy has a wooden box that has a volume of 160,000 cubic centimeters. The length of the box is 80 cm, and the height is 40 cm. What is the width of the box?

3. One cubic foot of water equals approximately 7.5 gallons. A rectangular restaurant sink holds 50.4 gallons of water. It has a length of 3 ft and a height of 1.4 ft. What is the volume of the sink? What is the width of the sink?

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6.EE.7, 6.G.2COMMONCORE

COMMONCORE 6.EE.7, 6.G.2



7 2 Skills/Concepts MP.3 Logic


10 2 Skills/Concepts MP.3 Logic

11 3 Strategic Thinking MP.4 Modeling

12 3 Strategic Thinking MP.7 Using Structure


14 3 Strategic Thinking MP.3 Logic

COMMONCORE

80 cm

40 cm

Answers1. 6 cm

2. 50 cm

3. 6.72 ft3, 1.6 ft

433 Lesson 15.3





24.25 in.

V = 3,758.75cubic inches 12.5 in.

?

3 ft

1 ft

ft

12

312

8 in.

4 in.

4 in.

Name Class Date

3. Jala has an aquarium in the shape of a rectangular prism

with the dimensions shown. What is the height of the

aquarium?

Height =

4. The area of the base of a rectangular juice box is 4 1 _ 2 square

inches. If the volume of the box is 18 cubic inches, how tall

is the box?

Height =

5. A box of cereal is shaped like a rectangular prism. The box is

20 centimeters long and 30 centimeters high. Its volume is

3,600 cubic centimeters. Find the width of the box.

Width =

6. About 7.5 gallons of water fill up 1 cubic foot of space. How

many gallons of water will fill a goldfish pool shaped like the

prism shown?

7. Physical Science A small bar of gold measures 40 mm by

25 mm by 2 mm. One cubic millimeter of gold weighs about

0.0005 ounce. Find the volume in cubic millimeters and the

weight in ounces of this small bar of gold.

8. History A typical stone on the lowest level of the Great Pyramid in

Egypt was a rectangular prism 5 feet long by 5 feet high by 6 feet deep

and weighed 15 tons. What was the volume of the average stone? How

much did one cubic foot of this stone weigh?

9. Hank has cards that are 8 inches by 4 inches. A stack of these cards

fits inside the box shown and uses up 32 cubic inches of volume.

How tall is the stack of cards? What percent of the box’s volume is

taken up by the cards?

Independent Practice15.36.EE.7, 6.G.2COMMON

CORE

118 1 _ 8

gallons

The volume of the bar is 2,000 cubic millimeters,

and the weight of the bar is 1 ounce.

1 inch; 25%

150 cubic feet; 0.1 ton

12.4 inches

4 inches

6 centimeters

433Lesson 15.3

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Work Area

10. A freshwater fish is healthiest when there is at least 1 gallon of water for

every inch of its body length. Roshel wants to put a goldfish that is about

2 1 _ 2 inches long in her tank. Roshel’s tank is 7 inches long, 5 inches wide, and

7 inches high. The volume of 1 gallon of water is about 231 cubic inches.

a. How many gallons of water would Roshel need for the fish?

b. What is the volume of Roshel’s tank?

c. Is her fish tank large enough for the fish? Explain.

11. Multistep Larry has a clay brick that is 7 inches long, 3.5 inches wide,

and 1.75 inches thick, the same size as the gold stored in Fort Knox in the

form of gold bars. Find the volume of this brick. If the weight of the clay

in the brick is 0.1 pound per cubic inch and the weight of the gold is

0.7 pound per cubic inch, find the weight of the brick and the gold bar.

Round all answers the nearest tenth.

Volume of the brick or bar = cubic inches

Weight of the brick = pounds

Weight of the gold bar = pounds

12. Represent Real-World Problems Luisa’s toaster oven, which is in the

shape of a rectangular prism, has a base that is 55 cm long by 40 cm

wide. It is 30 cm high. Luisa wants to buy a different oven with the same

volume but a smaller length, so it will fit better on her kitchen counter.

What is a possible set of dimensions for this different oven?

13. Multiple Representations Use the formula V = Bh to write a different

version of this formula that you could use to find the area of the base B of

a rectangular prism if you know the height h and the volume V. Explain

what you did to find this equation.

14. Communicate Mathematical Ideas The volume of a cube is 27 cubic

inches. What is the length of an edge? Explain.


42.9

4.3

30.0

Sample answer: If the base area B and the height h

are the same, the volume will also be the same. The

length could be 50 cm, the width 44 cm, and the

height 30 cm.

Begin with V = Bh and divide both sides of the equation

3 inches; 3 · 3 · 3 = 27

by h to get B = V __ h

.

245 cubic inches

No, the fish

2.5 gallons

would need 577.5 cubic inches of water.

Unit 6434

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Activity Jeff's new tent is shaped like a triangular prism. Each triangular face of the

prism has a base length of 4.5 feet and a height of 4 feet. The height of the prism is

6.5 feet. What is the volume of the tent? Explain.

58.5 ft3; use the formula, V = Bh. For a triangular prism, B = area of the

triangular base = 1 __ 2 bh = 1

__ 2 (4.5 · 4 ft) = 9 ft2. The height of the prism is 6.5 ft.

So, V = 9 · 6.5 = 58.5 ft3.

The volume of Jeff's old tent was 46.8 cubic feet. How many times as great as the

volume of the old tent is the volume of the new one?

1.25 times as great

Solving Volume Equations 434

Math Trainer


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12 cm

12 cm8 cm

6 m

8 m38

8 m14

5 ft

6 ft12

2 ft14

12 cm12 cm

8 cm

MODULE QUIZ

15.1 Nets and Surface Area

A square pyramid is shown sitting on its base.

1. Draw the net of the pyramid.

2. The surface area of the pyramid

is square centimeters.

15.2 Volume of Rectangular Prisms


3.

V = cubic meters

4.

V = cubic feet

15.3 Solving Volume Equations


5. The volume inside a rectangular storage room is 2,025 cubic

feet. The room is 9 feet high. Find the area of the floor.

6. An aquarium holds 11.25 cubic feet of water, and is

2.5 feet long and 1.5 feet wide. What is its depth?

7. How can a model help you to solve surface area and volume problems?

ESSENTIAL QUESTION

414 9 __ 16

225 ft 2

3 ft

73 1 _ 8

336

A net shows faces and helps you find surface area; a

drawing helps you choose a base and height when

finding volume.

435Module 15

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2

3Response toIntervention

Intervention Enrichment

Access Are You Ready? assessment online, and receive

instant scoring, feedback, and customized intervention or

enrichment.

Online and Print Resources


• Reteach worksheets

• Reading Strategies ELL

• Success for English

Learners ELL


• Challenge worksheets

PRE-AP

Extend the Math PRE-AP

Lesson Activities in TE

Additional ResourcesAssessment Resources includes:

• Leveled Module Quizzes

Ready to Go On?

Assess MasteryUse the assessment on this page to determine if students have

mastered the concepts and standards covered in this module.


Lesson Exercises Common Core Standards

15.1 1–2 6.G.4

15.2 3–4 6.G.2

15.3 5–6 6.EE.7, 6.G.2

435 Unit 6 Module 15

8 in.

6 in.

3 in.




MODULE 15 MIXED REVIEW

Selected Response

1. Indira is wrapping the box below. How

much wrapping paper does she need?

A 34 in. 2 C 144 in. 2

B 90 in. 2 D 180 in. 2

2. Colin has an ice cube tray with 12 identical

compartments. Each compartment is

a prism that is 4 centimeters long,

3 centimeters wide, and 3 centimeters

high. Given that 1 cubic centimeter holds

1 milliliter of water, how many milliliters of

water can the tray hold?

A 36 mL C 432 mL

B 66 mL D 792 mL

3. A store manager set up a cardboard display

to advertise a new brand of perfume. The

display is a square pyramid whose base is

18 inches on each side. The height of each

triangular face of the pyramid is 12 inches.

How much cardboard was used to make

the display?

A 516 in 2 C 756 in 2

B 612 in 2 D 1,080 in 2

4. Which expression is equivalent to 24 + 32?

A 8 × (3 + 4)

B 8 × (3 + 32)

C 6 × (4 + 32)

D 6 × (4 + 6)

5. A bathtub in the shape of a rectangular

prism is 5 feet long, 3 1 _ 2 feet wide, and 4 1 _

4 feet

high. How much water could the tub hold?

A 14 7 _ 8

ft 3 C 74 3 _ 8

ft 3

B 25 1 _ 2

ft 3 D 87 1 _ 2 ft 3

6. The point (-1.5, 2) is reflected across the

y-axis, What are the coordinates of the

point after the reflection?

A (-1.5, -2) C (2, -1.5)

B (1.5, 2) D (2, 1.5)

Mini-Task

7. An cardboard box is open at one end

and is shaped like a square prism missing

one of its square bases. The volume of the

prism is 810 cubic inches, and its height is

10 inches.

a. What is the length of each side of

the base?

b. Draw a net of the box.

c. How much cardboard is used for the

box?

Assessment Readiness

DD

CC

CC

AA

CC

BB

441 square feet

9 inches

10 in.

10 in.

9 in. 10 in.10 in.

436 Unit 6

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Assessment Readiness Tip Students can draw a net of the figures to

help solve for surface area.

Item 1 If students have difficulty getting started, encourage them

to draw a net of the figure to help them see the faces of the prism.

If they do they will see that there are six faces for this figure.


to draw a net of the figure to help them see the faces of the prism.

If they do they will see that there are five faces for this figure.

Avoid Common ErrorsItem 2 Some students may solve for the volume of one

compartment and quickly select choice A. Remind students to

read the problem carefully to be sure they are answering the

correct question. Point out that there are 12 compartments in the

tray, and they need to find the volume of all 12 compartments.


to draw a diagram first. Then they should be able to choose the

correct formula to find the surface area of the rectangular prism.

Additional Resources

Assessment Readiness


Items Grade 6 Standards Mathematical Practices

1 6.G.4, 6.EE.7 MP.4

2 6.G.2, 6.EE.7 MP.4

3 6.G.4, 6.EE.7 MP.4

4* 6.EE.3 MP.7

5 6.G.2, 6.EE.7 MP.4

6* 6.NS.8 MP.2

7 6.G.2, 6.G.4 MP.4

* Item integrates mixed review concepts from previous modules or a previous course.


surface area and volume of solids 15 - my.hrw.com · surface area and volume of solids 416 do not...

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