854 Unit 11 3-D Shapes, Weight, Volume, and Capacity

Objective To review the properties of common geometric solids.

Additional InformationAdvance Preparation For Part 1, construct a cube with 16 twist-ties and 12 straws, all the same length. In four boxes, place enough twist-ties and full-size, �

12�-size, and �

34�-size straws

that each pair of students can have 16 twist-ties and 8 straws of each length.

Teaching the Lesson materials

Key ActivitiesStudents review common geometric solids—including prisms, pyramids, cylinders, cones, and spheres—and investigate their properties.

Students construct rectangular prisms using straws and twist-ties.

Key Concepts and Skills• Identify parallel and intersecting line segments and parallel planes. [Geometry Goal 1]• Describe, compare, and classify plane and solid figures. [Geometry Goal 2]• Identify congruent faces. [Geometry Goal 2]• Construct a rectangular prism. [Geometry Goal 2]

Key Vocabularygeometric solid • rectangular prism • cylinder • triangular prism • cone • sphere • square pyramid • face • congruent • curved surface • edge • vertex (vertices) • cube • flat surface

Ongoing Assessment: Recognizing Student Achievement Use journal page 290.[Geometry Goal 2]

Ongoing Learning & Practice materialsStudents determine how many of each U.S. coin are needed to make a 1-ounce weight.

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students explore the relationships among thenumber of vertices, faces,and edges of polyhedrons.

Students use a Venndiagram to compare geometric solids.

Students create a Word Wallof geometry vocabulary.

� Teaching Aid Masters (MathMasters, pp. 389 and 390)

� models of geometric solids� computer with Internet access

ELL SUPPORTEXTRA PRACTICEENRICHMENT

3

� Math Journal 2, pp. 291 and 292� Study Link Master (Math Masters,

p. 328) � balance or scale capable of

measuring 1 ounce� pennies, nickels, dimes, and

quarters

2

� Math Journal 2, pp. 289 and 290� Student Reference Book, p. 101 � Study Link 11�1� models of geometric solids (See

Planning Ahead in Lesson 11�1.)� straws and twist-ties� Class Data Pad (optional)

See Advance Preparation

1

Technology Assessment Management System

Journal page 290, Problem 8See the iTLG.

See the Web site on page 859.

� Math Message Follow-Up(Math Journal 2, p. 289)

Display models of the six geometric solids—rectangularprism, cylinder, triangular prism, cone, sphere, and square pyramid—shown on journal page 289.

Begin with the rectangular prism. To support English languagelearners, attach a tag and label it. Hold it up and ask the class toshare examples of rectangular prisms in the classroom. You maywish to keep a list on the board or the Class Data Pad. Repeat thisprocedure for the remaining solids.

When all of the solids have been discussed, ask students thefollowing questions:● Which solids were easy to find? Probably the rectangular prism,

cylinder, and sphere● Which were hard to find? Probably the pyramid, triangular

prism, and cone● Why do you think some solids are more common than others?

Probably because they are easier to make or are more useful forstoring things

Rectangular Prism

Cylinder Triangular Prism

Cone Sphere Square Pyramid

WHOLE-CLASS

ACTIVITY

1 Teaching the Lesson

289

Geometric SolidsLESSON

11� 2

Date Time

101 102

Geometric shapes like these 3-dimensional ones are also called geometric solids.

Look around the classroom. Try to find examples of the geometric solids picturedabove. Draw a picture of each. Then write its name (for example: book).

Example of rectangular prism:

Name of object:

Example of cylinder:

Name of object:

Example of triangular prism:

Name of object:

Example of cone:

Name of object:

Example of sphere:

Name of object:

Example of square pyramid:

Name of object:

Rectangular Prism

Cylinder Triangular Prism

Cone Sphere Square Pyramid

Answers vary.

Math Journal 2, p. 289

Student Page

Lesson 11�2 855

Getting Started

Math MessageComplete journal page 289.

Study Link 11�1 Follow-UpAsk small groups of students to compareanswers and to pose and solve the problemsthey created.

Mental Math and ReflexesWrite large numbers on the board, and have volunteers readthem aloud. Suggestions:

7,540,3122,560,37116,436,280

43,290,517831,247,906372,815,206

1,206,598,3462,165,307,498172,039,598,563

Ask questions like the following:• What is the value of the digit x?• Which digit is in the millions place?

Geometric solids

856 Unit 11 3-D Shapes, Weight, Volume, and Capacity

Links to the Future

� Reviewing Vocabulary forGeometric Solids(Math Journal 2, p. 289; Student Reference Book, p. 101)

Use the display models of the six geometric solids and StudentReference Book, page 101 to review vocabulary associated withgeometric solids. To support English language learners, discussthe meaning of each term. Pose questions like the following:● Which of these geometric solids has 6 faces? Rectangular

prism● Which solids have congruent faces? Rectangular prism,

cylinder, triangular prism, square pyramid● Which solids have a curved surface? Sphere, cone, and

cylinder● Which has the most edges? Rectangular prism● Which two have the fewest vertices (corners)? Cylinder and

sphere What is the singular form of the word vertices? vertex● Which has two faces and one curved surface? cylinder

Have students look around the classroom and point out the faces,edges, and vertices of objects that have shapes similar to those inthe display.

Encourage students to use the geometry vocabulary, but do not expect them tobe precise at this time.

� Modeling Geometric Solids(Math Journal 2, p. 290)

Show the class the cube you constructed out of straws. (SeeAdvance Preparation.) Point out that it shows only the edges ofthe faces. It is a “frame” for the geometric solid; the flat surfacesof the cube must be imagined.

PARTNER

ACTIVITY

WHOLE-CLASS

DISCUSSION

Check Your UnderstandingCheck Your Understanding

1. a. How are cylinders and cones alike? b. How do they differ?2. a. How are spheres and cones alike? b. How do they differ?

Check your answers on page 343.

Geometry and Constructions

Geometric Solids

Polygons and circles are flat, 2-dimensional figures. Thesurfaces they enclose take up a certain amount of area, butthey do not have any thickness and do not take up any volume.Three-dimensional shapes have length, width, and thickness.They take up volume. Boxes, chairs, and balls are all examples.

A geometric solid is the surface or surfaces that surround a3-dimensional shape. The surfaces of a geometric solid may beflat or curved or both. A flat surface of a solid is called a face.A curved surface of a solid does not have any special name.

A cube has 6 square faces that are the same size. Three of thecube’s faces cannot be seen in the figure at the right.

A cylinder has 3 surfaces. The flat top and flat bottom arefaces that are formed by circles. A curved surface connects thetop and bottom faces. A food can is a good model of a cylinder.

A cone has 2 surfaces. The flat bottom is a face that is formedby a circle. A curved surface is connected to the bottom face andcomes to a point. An ice cream cone is a good model of a cone.However, keep in mind that a cone is closed; it has a “lid.”

The edges of a geometric solid are the line segments or curveswhere surfaces meet. A corner of a geometric solid is called avertex (plural vertices). A vertex is usually a point at whichedges meet, but the vertex of a cone is an isolated corner. It iscompletely separated from the edge of the cone.

A sphere has one curved surface but no edges and no vertices.A basketball or globe is a good model of a sphere.

cylinder

face

curved surface

face

cone

sphereno edges, no vertices

curved surface

face

vertex

edge

verticesedgesvertices

edges

edge

edge

no vertices

cube

face

faceface

Student Reference Book, p. 101

Student Page

290

Modeling a Rectangular PrismLESSON

11� 2

Date Time

101 102

After you construct a rectangular prism with straws and twist-ties, answer the questions below.

1. How many faces does your rectangular prism have? face(s)

2. How many of these faces are formed by rectangles? face(s)

3. How many of these faces are formed by squares? face(s)

4. Pick one of the faces. How many other faces are parallel to it? face(s)

5. How many edges does your rectangular prism have? edge(s)

6. Pick an edge. How many other edges are parallel to it? edge(s)

7. How many vertices does your rectangular prism have? vertices

8. Write T (true) or F (false) for each of the following statements about the rectangular prism you made. Then write one true statement and one false statement of your own.

a. It has no curved surfaces.

b. All of the edges are parallel.

c. All of the faces are polygons.

d. All of the faces are congruent.

e. True

f. False

FTFT

83

121

66

vertices

faces

edges

�

Answers vary.Answers vary.

0,or 2

Math Journal 2, p. 290

Student Page

Ask students what geometric solid this construction represents.Cube, or rectangular prism Demonstrate how the vertices are put together.

Lesson 11�2 857

Cube made out of straws and twist-ties

Distribute straws and twist-ties. (See Advance Preparation.) Havepartners work together to make a rectangular prism. One way isto start with a rectangle and build up. Have the straw cube, as wellas other models of rectangular prisms, available for inspection.When their rectangular prism is finished, partners shouldcomplete journal page 290.

NOTE Problem 3 on journal page 290 asks students to identify the number ofsquare faces in the rectangular prisms they have made. Depending on the strawsizes used, the prisms will have either 0 or 2 square faces. Students cannot construct cubes (which have 6 square faces) because they have only 8 straws of each length per partnership.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 290, Problem 8 to assess students’ ability to describe arectangular prism. Students are making adequate progress if they are able tocorrectly identify the given statements as true or false and write their own trueand false statements. Some students may write statements that involvecomparisons among geometric solids.

[Geometry Goal 2]

� Making a 1-Ounce Weight(Math Journal 2, p. 291)

Students use a balance or scale to determine how many of eachavailable type of U.S. coin are needed to make a 1-ounce weight.Students can display the results of their experiments in the Gram& Ounce Museum.

SMALL-GROUP

ACTIVITY

2 Ongoing Learning & Practice

Journal

page 290 �Problem 8

Try This

291

Making a 1-Ounce WeightLESSON

11� 2

Date Time

140

1. Estimate how many of each coin you think it will take to make a 1-ounce weight. Then use abalance or scale to determine exactly how many of each coin are needed.

2. Describe how you estimated how many of each coin it might take to make a 1-ounce weight.

Sample answer: I know there are about 28 g in 1 ounce, and a nickel weighs 5 g. Therefore, 6 nickels should weigh 30 g (5 � 6 � 30), so 6 nickels equal about 1 ounce. I estimate that pennies weigh �

12� as much as nickels, so it should take twice as

many pennies, which is 12. Dimes are smaller than pennies, so I guessed 15 dimes. Quarters are heavier than nickels, so I guessed 4.

3. About what fraction of an ounce does each coin weigh?

1 penny � oz 1 nickel � oz 1 dime � oz 1 quarter � oz

Explain how you found your answers.

Sample answer: I used the number of coins that equal 1 oz as the denominator. For example, because 12 pennies equal 1 oz, 1 penny equals �1

12� oz.

�15��1

13��

16��1

12�

Coin Estimated Number Actual Number of Coins of Coins

penny 11 or 12nickel 5 or 6dime 12 or 13

quarter 5

Answers vary.

Sample answers:

Math Journal 2, p. 291

Student Page

858 Unit 11 3-D Shapes, Weight, Volume, and Capacity

292

Math Boxes LESSON

11� 2

Date Time

4. Insert �, �, or � to make a true numbersentence.

a. �12 �19

b. �44 26

c. �64 �0.43

d. ��12� ��

48�

e. �0.28 �0.37��

�

��

6. A cinnamon raisin bagel has about 230 calories. How many calories are in one dozen bagels?

About calories2,760

1. The object below has the shape of ageometric solid. What is the name of thesolid? Circle the best answer.

A. rectangular prism

B. cone

C. cylinder

D. square pyramid

2. Draw the figure after it is rotated clockwise �

14�-turn.

3. Write a number model to estimate theanswer. Then correctly place the decimal point.

a. 0.97 � 4 � 3.8 8

Number model:

b. 1 8.7 � 74.8 � 4

Number model:

5. Round each number to the nearest tenth.

a. 2.34

b. 0.68

c. 14.35

d. 1.62

e. 5.99 6.01.6

14.40.72.3

VOLUME0 2 4 6 8 10

101 102

6 60

47182 183

106 107

1 � 4 � 4

80 � 4 � 20

Math Journal 2, p. 292

Student Page

� Math Boxes 11�2(Math Journal 2, p. 292)

Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 11-4 and 11-6. The skills in Problems 5 and 6 preview Unit 12 content.

Writing/Reasoning Have students write a response to thefollowing: For Problem 6, how would you determine thenumber of calories in 3�

12� bagels? Sample answer: Multiply

the number of calories in 1 bagel by 3; 230 * 3 � 690. Then dividethe number of calories in 1 bagel by 2; 230 / 2 � 115. Add the quotient to the number of calories in 3 bagels; 115 � 690 � 805.

� Study Link 11�2(Math Masters, p. 328)

Home Connection Students identify geometric solids represented by various objects. They also identify the vertices and the number of edges in two geometric solids.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

STUDY LINK

11� 2 Solids

101 102

Name Date Time

1. The pictures below show objects that are shaped approximately like geometric solids. Identify each object as one of the following: cylinder, cone, sphere, triangular prism, square pyramid, or rectangular prism.

a.

Type:

d.

Type:

b.

Type:

e.

Type:

c.

Type:

f.

Type:

square cone spherepyramid

cylinder triangularprismprism

2. Mark Xs on the vertices of the 3. How many edges does therectangular prism. tetrahedron have? edges6

4. Circle the numbers that are multiples of 7. 132 7,000 63 560 834 91

5. Circle the numbers that are multiples of 12. 24 120 38 600 100 75

Practice

rectangular

Math Masters, p. 328

Study Link Master

� Exploring Euler’sPolyhedral Formula(Math Masters, p. 389)

Technology Link To apply students’ ability to describe solidfigures, have them explore the relationships among vertices,

edges, and faces of polyhedrons at http://nlvm.usu.edu/en/nav/frames_asid_128_g_2_t_3.html?open�instructions.

On an Exit Slip, ask students to record their observations and drawa conclusion. Sample answer: The sum of the numbers of faces andvertices is 2 more than the number of edges: E � 2 � F � V.

Euler’s polyhedral formula states that the number of verticesminus the number of edges plus the number of faces is alwaysequal to 2.V � E � F � 2

NOTE The Web site is part of the National Library of Virtual Manipulatives for Interactive Mathematics developed by Utah State University. Seehttp://nlvm.usu.edu.

� Comparing Geometric Solids(Math Masters, p. 390)

To practice comparing the attributes of solid figures, havestudents choose two geometric solids and use them to

complete the Venn diagram on Math Masters, page 390.

� Creating a Word WallTo provide language support for geometry vocabulary, havestudents illustrate definitions of key geometric terms for a classroom display.

NOTE It might appear that there is an overwhelming number of geometry terms to know and understand. Each of these terms should be discussed in thecontext of solving problems and should relate to students’ experiences. Teachingthese terms in isolation or reducing them to a vocabulary list that needs to be memorized will not produce successful results in most cases. Students shouldhave opportunities to work with each of these terms, build models of them, writethem, and discuss them.

30+ Min

SMALL-GROUP

ACTIVITYELL SUPPORT

5–15 Min

INDEPENDENT

ACTIVITYEXTRA PRACTICE

15–30 Min

INDEPENDENT

ACTIVITYENRICHMENT

3 Differentiation Options

Lesson 11�2 859

Venn Diagram

Name Date Time

Math Masters, p. 390

Teaching Aid Master