Review of Geometric Solids: Part 1 - Everyday Math · ... Prisms, Pyramids, Cylinders, Cones, and...

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856 Unit 11 Volume

Advance PreparationFor Part 1, make and display five labels around the room: Prisms, Pyramids, Cylinders, Cones, and Spheres. Each group of students will

need the following: at least one prism, pyramid, cylinder, cone, and sphere (such as boxes, cans, party hats, balls, and so on); a set of

polyhedral dice that includes one tetrahedral die (4-sided), one octahedral die (8-sided), one decahedral die (10-sided), one dodecahedral

die (12-sided), and one icosahedral die (20-sided); and a set of Math Masters, pages 323–326. Make one additional set to construct models.

For the optional Extra Practice activity in Part 3, consider copying Math Masters, pages 329 and 330 on different-colored construction paper.

Teacher’s Reference Manual, Grades 4–6 pp. 186 –192

Key Concepts and Skills• Compare and classify geometric solids.

[Geometry Goal 2]

• Describe and classify polyhedrons

according to their faces. [Geometry Goal 2]

Key ActivitiesStudents sort geometric solids into groups:

prisms, pyramids, cylinders, cones, and

spheres. They build models from paper

patterns and use the models and solid-

geometry vocabulary to describe and

compare solids.

Ongoing Assessment: Informing Instruction See page 858.

Ongoing Assessment: Recognizing Student Achievement Use journal page 369. [Geometry Goal 2]

Key Vocabularygeometric solid � surfaces � faces � edges �

vertex � vertices (vertexes) � prisms �

pyramids � cylinders � cones � spheres �

polyhedrons � regular polyhedrons

MaterialsMath Journal 2, pp. 369 and 370; p. 428

(optional)

Student Reference Book, pp. 147–149 and 152

Math Masters, pp. 323 –326

slate � 5 index cards or sheets of construction

paper � per group: scissors, tape or glue,

geometric solids, polyhedral dice

Volume of a Rectangular PrismMath Journal 2, pp. 371A and 371B

Student Reference Book, pp. 196

and 197

Students find the volumes of

rectangular prisms.

Math Boxes 11�1Math Journal 2, p. 371

Geometry Template

Students practice and maintain skills

through Math Box problems.

Study Link 11�1Math Masters, p. 327

Students practice and maintain skills

through Study Link activities.

ENRICHMENTComparing the Faces, Vertices, and Edges of PolyhedronsMath Masters, p. 328

Class Data Pad � decahedral die

Students compare the number of faces,

vertices, and edges of polyhedrons.

EXTRA PRACTICE

Building Models for Geometric SolidsMath Masters, pp. 329 and 330

scissors � tape or glue

Students use patterns to construct a

rectangular prism and an octahedron.

ELL SUPPORT

Describing Geometric Solidschart paper

Students describe the properties of prisms,

pyramids, cylinders, cones, and spheres.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�

Review of GeometricSolids: Part 1

Objective To review the names and properties

of geometric solids.o

Common Core State Standards

856_EMCS_T_TLG2_G5_U11_L01_576914.indd 856856_EMCS_T_TLG2_G5_U11_L01_576914.indd 856 3/28/11 1:53 PM3/28/11 1:53 PM

Geometric SolidsLESSON

11�1

Date Time

Each member of your group should cut out one of the patterns from Math Masters,pages 323–326. Fold the pattern, and glue or tape it together. Then add this model to

your group’s collection of geometric solids.

1. Examine your models of geometric solids.

a. Which solids have all flat surfaces?

b. Which have no flat surfaces?

c. Which have both flat and curved surfaces?

d. If you cut the label of a cylindrical can in a straight

line perpendicular to the bottom and then unroll and

flatten the label, what is the shape of the label?

2. Examine your models of polyhedrons.

a. Which polyhedrons have more faces than vertices?

b. Which polyhedrons have the same number of faces and vertices?

c. Which polyhedrons have fewer faces than vertices?

3. Examine your model of a cube.

a. Does the cube have more edges than vertices, the same

number of edges as vertices, or fewer edges than vertices?

Is this true for all polyhedrons? Explain.

b. How many edges of the cube meet at each vertex?

Is this true for all polyhedrons? Explain.

cut

line

�

Pyramids and prisms— including cubes

spheres

Cylinders and cones

rectangle

none

Cubes, triangular prisms, and rectangular prisms

More edgesthan vertices

3 edgesMore than 3 edges

could meet at the vertex of a pyramid.

At least 3 edges areneeded to form 1 vertex.

Triangularpyramids and square pyramids

Yes

No

Math Journal 2, p. 369

Student Page

Lesson 11�1 857

Getting Started

3

_ 4 0.75, 75%

1

_ 5 0.2, 20%

4

_ 5 0.8, 80%

Math MessageLook at your group’s collection of objects and name the geometric solid that each item represents. Use pages 147–149 of the Student Reference Book as a resource.

Mental Math and Reflexes Have students mentally convert between fractions, decimals, and percents (or refer to the Probability Meter in the reference section of their journals, as needed). Suggestions:

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Student Reference Book, pp. 147–149)

Discuss Student Reference Book, pages 147 and 148. Emphasize that each geometric solid is identified by its surfaces (flat and curved), faces, edges, and vertex or vertices (or vertexes).

Display the five prepared labels for types of geometric solids around the room. Name one type of geometric solid from the five labeled groups (prisms, pyramids, cylinders, cones, and spheres). Ask volunteers to show an example of the solid from their group’s collection and describe the object in terms of its surfaces, faces, edges, and vertices. Ask students to discuss how it is similar to and different from the other solids. For example, cones and pyramids have one base, but the cone has a curved surface and the pyramid does not have a curved surface. Have a student from each group place the object in the area with the appropriate label. Repeat until all five geometric solids have been identified and sorted and a list of properties has been generated.

Read Student Reference Book, page 149 as a class. Ask: Are all geometric solids also polyhedrons? no Which solids in the collection are not polyhedrons? Cylinders, cones, and spheres Why aren’t they polyhedrons? Because at least one of their surfaces is curved

▶ Exploring Characteristics

SMALL-GROUP ACTIVITY

of Geometric Solids(Math Journal 2, p. 369; Math Masters, pp. 323–326)

Give each group of four a rectangular prism, a cylinder, a cone, and a sphere from the class collection. They will also need one set of Math Masters, pages 323 –326, scissors, and glue or tape.

NOTE Math Masters, pages 323 –326 provide patterns (nets) for a cube, a

triangular prism, a triangular pyramid, and a square pyramid.

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12 1

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87 1

_ 2 %

7

_ 8 , 0.875

0.6 ⎯ 6 2

_ 3 , 66

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0.375 3

_ 8 , 37

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Links to the Future

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

Students will take a closer look at prisms

and pyramids in Lesson 11-2.

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Math Boxes LESSON

11�1

Date Time

4. Solve.

a. 4 _ 5 of 25 = 20

b. 5

_ 7 of 35 = 25

c. 3

_ 12 of 16 = 4

d. 6

_ 8 of 20 = 15

e. 1 _ 2 of 1 _ 4 =

5. Write the prime factorization for 180.

22 ∗ 32 ∗ 5, or

2 ∗ 2 ∗ 3 ∗ 3 ∗ 5

1. Subtract.

a. 10 - (-2) = 12

b. 5 - 8 = -3

c. 15 - (-5) = 20

d. -15 - (-5) = -10

e. -4 - 7 = -11

3. The students in Ms. Divan’s class took a

survey of their favorite colors. Complete the

table. Then make a circle graph of the data.

2. Which triangle is not congruent to the other

three triangles? Circle the best answer.

A. B.

C. D.

Favorite Colors

24%red 12%

purple

8% yellow

40%blue

16%orange

(title)

Favorite Number of Percent Color Students of Class

Red 6

Blue 10

Orange 4

Yellow 2

Purple 3

Total

24%

40%

16%

8%

12%

100%25

92–94 155

47 8990 126

12 73 1 _ 8

369-392_EMCS_S_MJ2_U11_576434.indd 371 3/4/11 7:04 PM


Student Page

Polyhedral Dice and Regular PolyhedronsLESSON

11�1

Date Time

A set of polyhedral dice includes the following polyhedrons:

Examine the set of polyhedral dice that you have. Answer the following questions.

1. Which of the dice is not a regular polyhedron? Why?

2. Which regular polyhedron is missing from the set of polyhedral dice?

3. a. How many faces does an octahedron have? faces

b. What shape are the faces?

4. a. How many faces does a dodecahedron have? faces


5. a. How many faces does an icosahedron have? faces


6. Explain how the names of polyhedrons help you to know the number of their faces.

the faces are not regular polygons.

Decahedral die;

The first part of the name tells me the number of faces:

octa- means 8; dodeca- means 12; icosa- means 20.

Equilateral triangles

Regular pentagons

Equilateral triangles

20

12

8

cube

Tetrahedral

dieOctahedral

die

Dodecahedral

die

Icosahedral

dieDecahedral

die

369-392_EMCS_S_MJ2_G5_U11_576434.indd 370 3/7/11 3:41 PM


Student Page

858 Unit 11 Volume

Each student uses a pattern on one of the masters to construct the model of a geometric solid. Each group will then have one constructed model for each of the four solids.

Ongoing Assessment: Informing Instruction

Watch for students who have difficulty folding the patterns. Suggest that they

score the fold lines with a ruler and pen before folding. To support English

language learners, model the meaning of score in this context.

If students have difficulty adhering the flaps of the patterns, suggest that they

tape the flaps to the outside of the models.

Have students use the objects they selected from the labeled groups and the geometric models they constructed to answer the questions on journal page 369. When all groups have completed the journal page, bring the class together to discuss the answers.

Ongoing Assessment: Journal

69 �Recognizing Student Achievement

Use journal page 369 to assess students’ knowledge of the properties of

geometric solids. Students are making adequate progress if they correctly

complete Problem 1.

[Geometry Goal 2]

Have students display the objects and constructed models with the appropriate labels. The constructed models will be used in Lesson 11-2.

▶ Investigating Regular

SMALL-GROUP ACTIVITY

Polyhedrons(Math Journal 2, p. 370; Student Reference Book, p. 152)

Distribute one set of polyhedral dice to each group of four and assign students to complete journal page 370. Students examine the dice and use the information on Student Reference Book, page 152 to complete the journal page.

When most students have finished, ask volunteers to describe a regular polyhedron. Sample answers: Regular polyhedrons are geometric solids with faces that are all the same size and shape. Each face is formed by a regular polygon. Every vertex looks exactly the same as every other vertex. The five kinds of regular polyhedrons are tetrahedrons, cubes, octahedrons, dodecahedrons, and icosahedrons.

ELL


371A

Date Time

3.

a. The solid holds 50 cubes.

b. Volume: 50 cubic units

2.

3 cubes

5 cu

bes

4 cubes

a. The solid holds 60 cubes.

b. Volume: 60 cubic units

Volume of a Rectangular PrismLESSON

11�1

1. a. The volume of this cube is 1 cm3.

b. Write a number model

for the volume of the cube. 1 ∗ 1 ∗ 1 = 1 cm3

c. This cube is called a cube.

Find how many unit cubes can be packed in each container. Then give the volume

of the container. Some cubes have already been packed in the container.

unit

1 cm 1 cm

1 cm

4. Circle the formulas below that give the volume of a rectangular prism.

(B is the area of the base; l, w, and h are the measures of length, width, and height.)

V = B ∗ h V = B ∗ w ∗ h V = l ∗ w V = l ∗ w ∗ h

5. This box can be filled with unit cubes, each 1 inch long on a side.

a. To fill the box, you need 240 unit cubes.

b. Write a number model for the volume of the box.

48 ∗ 5 = 240 or 6 ∗ 8 ∗ 5 = 240 in3

c. Explain why either of these two expressions can be used to find the

volume of the box: (6 ∗ 8) ∗ 5 or 6 ∗ (8 ∗ 5).

1 in.1 in. 1 in.

Sample answer: Both produce the same product of the

length, width, and height. According to the Associative

Property of Multiplication, (6 ∗ 8) ∗ 5 = 6 ∗ (8 ∗ 5).

369-392_EMCS_S_MJ2_G5_U11_576434.indd 371A 4/7/11 3:58 PM

Math Journal 2, p. 371A

Student Page

371B

Date Time

Volume of a Rectangular Prism continuedLESSON

11�1

Find the volume of each rectangular prism. Use either of these formulas:

V = B ∗ h or V = l ∗ w ∗ h.

6. 5 ft

15

ft

10 ft

7. 12 cm 9 cm

8 c

m

V = 750 ft3 V = 864 cm3

(unit) (unit)

Solve each problem.

8. What is the volume of a cube that

has sides that are 15 yards long?

V = 3,375 yd3

10. A classroom is 30 feet long, 20 feet wide, and 10 feet high.

a. What is the area of the classroom floor? b. What is the volume of the classroom?

A = 600 ft2 V = 6,000 ft3 (unit) (unit)

11. Each small cube in the drawing measures 1 foot on each side.

The large cube represents 1 cubic yard. How many cubic feet

are in 1 cubic yard?

27 cubic feet

12. A rectangular ditch has a base with dimensions 30 ft by 90 ft.

It has a depth of 10 ft.

a. What is the volume of the ditch? 27,000 ft3

Try This

b. How many cubic yards of dirt must be dug out to make the ditch? 1,000 yd3

1 yd (3 ft)

1 yd (3 ft)

1 yd (3 ft)

1 cubic yard

9. The inside measurements of a

refrigerator are about 36 in. wide, 69 in.

high, and 28 in. deep. What is its volume?

V = 69,552 in3

369-392_EMCS_S_MJ2_G5_U11_576434.indd 371B 4/7/11 3:58 PM

Math Journal 2, p. 371B

Student Page

Lesson 11�1 859

Ask students to think about the Everyday Mathematics games that are played using six-sided dice. Explain that six-sided dice are considered fair because there is an equal chance of any one of the six sides landing on top when the dice are thrown. Discuss whether or not each of the polyhedral dice is fair. Yes, because all of the faces on polyhedral dice have the same size and shape.

2 Ongoing Learning & Practice

▶ Volume of a Rectangular Prism

INDEPENDENT ACTIVITY

(Math Journal 2, pp. 371A and 371B; Student Reference

Book, pp. 196 and 197)

Students find the volumes of rectangular prisms. Remind students that they can do so by counting unit cubes or applying one of the formulas: V = B ∗ h or V = l ∗ w ∗ h.

▶ Math Boxes 11�1 INDEPENDENT

ACTIVITY (Math Journal 2, p. 371)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 11-3. The skill in Problem 5 previews Unit 12 content.

▶ Study Link 11�1


(Math Masters, p. 327)

Home Connection Students are given four patterns, three of which can be folded into a cube. Students select which one of the four cannot be folded into a cube and

check their selection by cutting out and folding that pattern.

3 Differentiation Options

ENRICHMENT


▶ Comparing the Faces, Vertices, 5–15 Min

and Edges of Polyhedrons(Math Masters, p. 328)

Social Studies Link The Swiss mathematician and physicist Leonard Euler is said to have discovered that the

sum of the faces and vertices of any polyhedron is 2 more than the number of edges. Euler’s Theorem can be expressed by the formula F + V - E = 2, where F stands for the number of faces, V for the number of vertices, and E for the number of edges.

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LESSON

11�1

Name Date Time

Exploring Faces, Vertices, and Edges

� A flat surface of a geometric solid is called a face.

� A corner of a geometric solid is called a vertex. The plural of vertex is vertices.

� An edge of a geometric solid is a line segment or curve where two surfaces meet.

face

face

face edges

edges

edges

edges

vertices

vertices

Polyhedron Faces Vertices Faces + Vertices Edges

Cube 6 8 6 + 8 = 14 12

Tetrahedron 4 4 4 + 4 = 8 6 Octahedron 8 6 8 + 6 = 14 12 Dodecahedron 12 20 12 + 20 = 32 30 Icosahedron 20 12 20 + 12 = 32 30

1. Complete the table.

2. Compare the values in the Faces + Vertices column with the Edges column. What

do you notice?

The sum of the number of faces and vertices is always 2 more than the number of edges.

3. Two of the patterns below can be folded to make a tetrahedron. Cross out the

patterns that will not make a tetrahedron. Circle the patterns that will make a

tetrahedron. Explain your solution strategy.

I looked for patterns that would make ashape with 4 faces, 4 vertices, and 6 edges.

323-347_EMCS_B_MM_G5_U11_576973.indd 328 3/9/11 8:44 AM

Math Masters, p. 328

Teaching Master

STUDY LINK

11�1 Cube Patterns

Name Date Time

There are four patterns below. Three of the patterns can be folded to form a cube.

1. Guess which one of the patterns below cannot be folded into a cube.

My guess: Pattern (A, B, C, or D) cannot be folded into a cube.

2. Cut on the solid lines, and fold the pattern on the dashed lines to check your

guess. Did you make the correct guess? If not, try other patterns until you find

the one that does not form a cube.

My answer: Pattern D (A, B, C, or D) cannot be folded into a cube.

A

B

D

C

Answers vary.

147–149

323-347_EMCS_B_MM_G5_U11_576973.indd 327 3/9/11 8:44 AM

Math Masters, p. 327

Study Link Master

860 Unit 11 Volume

To apply students’ understanding of the properties of geometric solids, have students explore relationships between the faces, the vertices, and the edges of polyhedrons. Students list the number of faces, vertices, and edges for the five regular polyhedrons and find the sum of the number of faces and the number of vertices. Then they analyze patterns for a tetrahedron.

When students have finished, discuss their responses to Problem 2. Write Euler’s Theorem and the formula on the Class Data Pad. Have students use a decahedral die to verify Euler’s Theorem by counting the faces, vertices, and edges and by substituting the values in the formula. 10 + 12 - 20 = 2

Discuss students’ solution strategies for Problem 3. Emphasize how faces, vertices, and edges are used to recognize when a pattern will make a tetrahedron.

EXTRA PRACTICE


▶ Building Models for 15–30 Min

Geometric Solids(Math Masters, pp. 329 and 330)

Students build models for a rectangular prism and an octahedron from patterns. Display the constructed models with their names and property descriptions.

ELL SUPPORT


▶ Describing Geometric Solids 15–30 Min

To provide language support for geometric solids, have students describe the properties of prisms, pyramids, cylinders, cones, and spheres. They record their observations on chart paper for display as a reference during this and future lessons.

Planning Ahead

Collect open cans, preferably with the labels removed, or other cylindrical and watertight containers of different sizes. In Lesson 11-3, students will need at least 1 can per partnership.


Review of Geometric Solids: Part 1 - Everyday Math · ... Prisms, Pyramids, Cylinders, Cones, and...

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