content standards 6.g.2, 6.g.4 cc investigation 4: measurement · cc investigation 4: measurement...

Measurement 35

CC Investigation 4: Measurement

Teaching Notes

Mathematical Goals DOMAIN: Geometry

• Find the volume of a right rectangular prism with fractional edge lengthsby packing it with unit cubes, and show that the volume is the same asfound by multiplying the prism’s edge lengths.

• Apply the formulas V = lwh and V = bh to solve real-world problemsinvolving the volumes of right rectangular prisms with fractional side lengths.

• Represent three-dimensional figures using nets made of rectangle and triangles.

• Use nets of three-dimensional figures to solve real-world problemsinvolving surface area.

Vocabulary

• net

• prism

• surface area

• volume

At a Glance

Show students examples of rectangular prisms and pyramids. To represent athree-dimensional shape on a flat page, such as in a book, you must draw atwo-dimensional representation of that shape. Read the introduction withstudents and review the dimensions of three-dimensional figures; length,width, and height. Point out that three-dimensional objects consist of suchparts as faces, edges, and vertices, and that by turning the object it can havethe appearance of different two-dimensional shapes.

To begin the instruction on surface area, show students a box. Rotate thebox to several positions. Review the shape on each side of the box. Askstudents to explain how to find the area of one side. Review the formula forarea of a rectangle with the class. Show the class a sheet of wrapping paper.Ask the class if they believe there is enough paper to cover the box. Let thestudents explain how they can find this information. Then develop adefinition for surface area using the faces of the box to show the total areathat must be covered by the wrapping paper.

Use unit cubes in an exercise to give the students a hands-on experiencewith cubic measurement and volume. Show that the cubes can berearranged into numerous configurations without the volume changing.

Note that the solid shapes used in this investigation are all right shapes:the bases of the rectangular and triangular prisms are aligned vertically; theapexes of the pyramids are directly above the centers of the bases.

PACING 3 days

Content Standards 6.G.2, 6.G.4

06_CMP2_TRG_Inv4_035-038.qxd 1/26/11 12:40 PM Page 35

Problem 4.1

Before Problem 4.1, present various models of prisms and pyramids tostudents and review the faces that make up the figures. Ask:

• What two-dimensional shapes make up the faces of a rectangular prism?(6 rectangles)

• What two-dimensional shapes make up the faces of a square pyramid?(1 square, 4 triangles)

During Problem 4.1 A, ask:

• What two-dimensional shapes make up Ashley's net? (6 rectangles)

• What are the dimensions of each rectangle in the net? (14 in. � 4 in.;14 in. � 8 in.; 14 in. � 4 in.; 14 in. � 8 in.; 8 in. � 4 in.; 8 in. � 4 in.)

• What do you notice about how some of the rectangles' sizes are related?(There are 3 pairs of equal rectangles.)

During Problem 4.1 B, ask:

• What two-dimensional shapes make up the faces of the figure?(2 triangles and 3 rectangles)

• What do you notice about the two triangular faces? (They are the samesize and shape.)

• What are the base and height lengths of the triangles? (b = 6 in.;h = 8 in.)

Problem 4.2

During Problem 4.2 A, ask:

• How many blocks long is the base of Ashley’s model? (10)

• How many blocks wide is the base? (6)

• How can you find the number of blocks in the base layer?(Multiply 10 � 6.)

During Problem 4.2 B, ask:

• How can you find the volume, in cubic inches, of one of Brandon’s

blocks? (Multiply .)

• Could Brandon have used Ashley’s blocks to make his model? (No; he

couldn’t model in. using only -in. blocks.)

Summarize

To summarize the lesson, ask:

• What shapes are the faces of a cube? (6 squares)

• What faces does a square pyramid have? (1 square, 4 triangles)

• How can you find the surface area of a prism? (Find the area of eachside of the prism and add the areas.)

• How can you find the volume of a rectangular prism? (Multiply its basetimes its height, or its length times its width times its height.)

• How is finding the volume of prisms with edge lengths that are fractionsor mixed numbers like finding the volume of a prism with integer sidelengths? (Use the same formulas to find the volume.)

12

634

14

�14

�14

�1

64

36 Common Core Teacher’s Guide


Assignment Guide forInvestigation 4Problem 4.1, Exercises 1–15Problem 4.2, Exercises 16–25

Answers to Investigation 4

Problem 4.1A. 1. rectangular prism

2. Find the area of each rectangular piece ofthe net and then add the areas to find thetotal area of the net.

3. A = (14 � 4) + (14 � 8) + (14 � 4) + (14 � 8) + (8 � 4) + (8 � 4) = 56 + 112 + 56 + 112 + 32 + 32 = 400 in.2

4. The area of the net is equal to the surfacearea of the three-dimensional figure thatcan be made with the net.

B. 1.

2. Find the area of each piece of the net andthen add the areas to find the total area ofthe net.

3. A = (14 � 8) + (14 � 10) + (14 � 6) +

+ = 112 + 140 +

84 + 24 + 24 = 384 in.2

Problem 4.2A. 1. 60 blocks

2. 600 blocks

3. 5 in. � 3 in. � 5 in.

4. V = lwh = in.3

5. V = lwh = 5 � 3 � 5 = 75 in.3

6. The volume of a block that measures 1 in.on each side is 1 in.3, so one of Ashley’s

blocks has a volume that is that volume.

7. The volume of the model is the product ofthe number of blocks and the volume of

each block: .

B. 1. blocks

wide; blocks long

2. A = lw = in.2

3. blocks

4. 10 � 12 � 27 = 3,240 blocks

5. V = Bh =

in.3

6. V = Bh =

= in.3

7. The volume of a block that measures 1 in.on each side is 1 in.3, so one of Brandon’s

blocks has a volume that is that volume.

8. The volume of the model is the product ofthe number of blocks and the volume of

each block: .5058

� 3,240 �1

64

164

5058

712

� 634

�152

�274

�15 � 272 � 4

�1

64

( 14

�14

) �14

�1

4 � 4 � 4

634

�14

�274

�14

�274

�41

� 27

212

� 3 �52

� 3 �152

� 712

3 �14

� 12

212

�14

�52

�14

�52

�41

� 10

75 � 600 �18

18

12

�12

�12

�1

2 � 2 � 2�

18

12

(6) (8)12

(6) (8)

Measurement 37

6 in.6 in.

6 in.8 in.

8 in.8 in.

14 in.

10 in.

06_CMP2_TRG_Inv4_035-038.qxd 1/26/11 12:40 PM 7


Exercises1. Sample (not to scale):

2. 6 rectangles

3. 2 triangles and 3 rectangles

4. 1 square and 4 triangles

5. Sample (not to scale):

6. Sample (not to scale):

7.

8.

9. 96 in.2

10. 184 m2

11. 243 cm2

12. 144 ft2

13. a. 4 ft � 6 ft, 4 ft � 6 ft, 12 ft � 6 ft,12 ft � 6 ft, 12 ft � 4 ft, 12 ft � 4 ft

b. 24 ft2, 24 ft2, 72 ft2, 72 ft2, 48 ft2, 48 ft2

c. 288 ft2

14. No; Surface Area = 15 � 12 + 15 � 12 + 12 �8 + 12 � 8 + 15 � 8 + 15 � 8 = 792 in.2;792 > 750.

15. a. square

b. No, you also need the height of a triangular face.

16. 256 m3

17. 1,728 cm3

18. 294 in.3

19. ft3

20. in.3

21. in.3

22. in.3

23. a. -inch blocks

b. 280 blocks

c. 35 in.3

24. A

25. a. 8 blocks

b.

c. in.3

d. Possible answers:

14

in. � 9 in.

12

in. � 412

in.;34

in. � 3 in.;1 in. � 214

in.;

112

in. � 112

in.;

338

34

in. � 214

in. � 2 in.

12

1318

558

178

23518


Name Date Class

Additional Practice Investigation 4

Measurement

1. Seth needs to cover the box below with fabric so that it can be used as a propin the school play. Seth has 375 in.2 of fabric.

a. Draw a net of the box. Name the shapes that make up the faces of the box.

b. Does Seth have enough fabric to cover this box? Explain how you know.

c. What is the longest rectangular box with a square base that measures 5 in.on each side that Seth could wrap with his fabric? Explain how you foundyour answer.

2. A tea company sells a flavored tea in the rectangular boxes shown below.

a. Without calculating, which box do you think holds the most tea? Explainyour answer.

b. What is the volume of each box? How does this compare to your prediction?

c. Which box of tea would you expect to cost the least? Explain your reasoning.

© Pe

arson

Educ

ation

, Inc

., pu

blish

ing as

Pears

on Pr

entic

e Hall

. All r

ights

reserv

ed.

Measurement 39

18 in.

5 in.

5 in.

6 cm6 cm

12 cm

8 cm

8 cm

7.5 cm

10.2 cm

8 cm

5 cm

06_CMP2_TRG_Inv4_ANC#96D51B.qxd 2/8/11 10:24 AM Page 39

Name Date Class

Skill: Find the Surface Area Investigation 4

Measurement

Find the surface area.

1. 2.

3. 4.

Skill: Find the VolumeFind the volume.

7. 8.

9. 10.

© Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.


3 cm

5 cm

4 cm

14 cm8 in.

3 in.6 in.

2.5 m2.5 m

2.5 m

16 cm

20 cm

12 cm

5 cm

9 in.

35 in.9 in.

in.12

in.12

in.12

2 in.12

4 in.14

2 in.

3 yd34

7 yd12

1 yd34

06_CMP2_TRG_Inv4_ANC#96F1AC.qxd 2/11/11 4:19 PM Page 40

Name Date Class

Check-Up Investigation 4

Measurement

1. Jimmy needs to wrap this rectangular box for a birthday present.

a. Draw a net of the box and label the dimensions.

b. Jimmy has 600 in.2 of wrapping paper. Is this enough for him to wrap thebox? Explain how you found your answer.

2. Jimmy has more presents he needs to wrap. The other presents each come inits own box with the shape shown below.

a. Draw a net of the box and label the dimensions.

b. How many of these boxes could Jimmy wrap with 600 in.2 of wrappingpaper? Explain how you found your answer.

© Pe

arson

Educ

ation

, Inc

., pu

blish

ing as

Pears

on Pr

entic

e Hall

. All r

ights

reserv

ed.

Measurement 41

15 in.

3 in. 14 in.

6 in. 6 in.

8 in.10 in.

06_CMP2_TRG_Inv4_ANC#96D594.qxd 2/11/11 4:06 PM Page 41

Name Date Class

Check-Up (continued) Investigation 4

Measurement

3. Latoya is going to build a rectangular storage shed for her backyard. The shedwill have a volume of 432 ft3. Latoya sketched this net of the shed.

a. The base of the shed is the face labeled C. What is the height of the shed?Explain how you found your answer.

b. Latoya will paint the inside walls and ceiling of the shed, but not the floor.She finds cans of paint that will cover 48 ft2 and cost $5.29 each. She alsofinds a gallon of paint that will cover 350 ft2 and costs $29.99. Whichshould Latoya buy? Explain your answer.

4. Latoya finds plans for another rectangular storage shed that has dimensionsof 10 ft � 5 ft � 7.5 ft. If she wants the shed with more storage space, whichshed should she build? Explain your choice.

© Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.


9 ft

A

C

E

F

D 6 ft

6 ft

B

06_CMP2_TRG_Inv4_ANC#96D594.qxd 2/11/11 4:06 PM Page 42

CC Investigation 4: MeasurementDOMAIN: Geometry

A net is a two-dimensional model that can be folded into a three-dimensional figure. Prisms are three-dimensional figures that have twocongruent and parallel faces that are polygons, such as rectangles ortriangles. The rest of a prism’s faces are parallelograms. You can use nets of rectangular and triangular prisms to find their surface areas.

Ashley cuts nets from poster board and folds them to make three-dimensional models of buildings.

A. Ashley first cuts out the net shown at the right.

1. What three-dimensional figure can shefold from this net?

2. Ashley knows that she can use theformula A = lw to find the area of arectangle, where l represents therectangle’s length and w represents itswidth. Explain how to find the totalarea of the net.

3. What is the area of the net?

4. How is the area of the net related tothe surface area of the prism?

B. Ashley wants to model the roof of a building.The triangular faces of the figure are parallelright triangles.

1. Draw a net of the figure. Label the lengthsof the sides.

2. Explain how you can use the net to find thefigure’s surface area.

3. What is the surface area? Remember that the formula is

used to find the area of a triangle with a base b and a height h.

A 5 12bh

Measurement 41

Problem 4.1

14 in.

4 in.

4 in.

8 in. 8 in.

8 in.

8 in.

8 in.

10 in.

6 in.

14 in.

Notes

STUDENT PAGE

Measurement (41)42

ST

UD

EN

T P

AG

E

06_CMP2_TRG_Inv4_SE_41-46_42.qxp 2/15/11 1:39 PM Page 41

Remember that you can find the volume of a rectangular prism bymultiplying the area of its base by its height. Use one of these formulas:

V = Bh, where B is the area of the base, and h is the height, or

V = lwh, where l and w are the length and width of the base, and h isthe height.

Ashley and Brandon use various cubic blocks to model some of thebuildings near their school.

A. Ashley’s blocks are inch on each side.

She starts her model by making a 5-inch by 3-inch rectangular base with the blocks.

1. How many blocks does she use?

2. Ashley adds 9 more layers to the base layer. How many blocks doesshe use in all?

3. What are the length (l), width (w), and height (h) of the model,in inches?

4. Use the formula V = lwh to find the volume, in cubic inches, of oneof Ashley’s blocks. Show your work.

5. Use the formula V = lwh to find the volume, in cubic inches, ofAshley’s model. Show your work.

6. How is the volume of 1 block related to the volume of a block thatmeasures 1 in. on each side?

7. Look at the volume of each block, the number of blocks used, andthe volume of the model. How are these numbers related?

12

42 Common Core Additional Investigations

Problem 4.2

5 in.

3 in.

ST

UD

EN

T P

AG

E

Notes

(42)42 Common Core Teacher’s Guide

STUDENT PAGE


B. Brandon measures his blocks to be inch on each side. He models

a building in the shape of a rectangular prism that is in. wide,

3 in. long, and in. high.

1. How many blocks wide and long is the base of Brandon’s model?

2. What is the area, in square inches, of the model’s base?

3. How many blocks tall is the model?

4. How many blocks did Brandon use for the model?

5. Use the formula V = Bh to find the volume, in cubic inches, of oneof Brandon’s blocks. Show your work.

6. Use the formula V = Bh to find the volume, in cubic inches, ofBrandon’s model. Show your work.

7. How is the volume of 1 block related to the volume of a block thatmeasures 1 in. on each side?

8. Look at the volume of each block, the number of blocks used, andthe volume of the model. How are these numbers related?

Exercises1. Draw the three-dimensional figure modeled by this net.

Name the two-dimensional shapes used to make the faces of each object.Tell how many there are of each shape.

2. a rectangular prism, such as a shoebox

3. a triangular prism, like a tent

4. a pyramid, like the structures built by the Egyptians

634

212

14

Measurement 43

Notes

STUDENT PAGE

Measurement (43)42

ST

UD

EN

T P

AG

E


Draw a net for each figure.

5. 6.

7. 8.

Find the surface area of each figure.

9. 10.

11. 12.

13. A container has two rectangular ends that measure 4 ft by 6 ft, andanother side that has a length of 12 ft.

a. What are the measurements of each of the faces of the container?

b. What are the areas of all of the faces of the container?

c. What is the total surface area of the container?

14. Keira has 750 square inches of wrapping paper. Her package is shapedlike a right rectangular prism that is 15 inches long, 12 inches wide,and 8 inches high. Does she have enough paper to cover her package? Explain.

6 ft

3 ft

6 ft

9 cm9 cm

9 cm

6 m

10 m

5 m

4 m

4 in.

4 in.

4 in.


ST

UD

EN

T P

AG

E

Notes


STUDENT PAGE


15. The pyramid at the right has four faces that are congruent triangles.

a. What shape is the base of the pyramid?

b. If you know just the length of a side of the base, doyou have enough information to find the surfacearea of the pyramid? Explain.

Find the volume of each rectangular prism.

16. 17.

18. 19.

For Exercises 20–22, each rectangular prism is built with -in. blocks.Find the length, width, height, and volume of the prism.

20. 21. 22.

12

6 ft

34

4— ft

14

8— ft

12 in.

12

3— in.7 in.

12 cm

12 cm

12 cm4 m4 m

16 m

Measurement 45

Notes

STUDENT PAGE

Measurement (45)42

ST

UD

EN

T P

AG

E


23. a. What size cubic blocks would you use to make a rectangular prism

that is in. by in. by 4 in.? Explain your choice.

b. How many blocks would you need?

c. Give the volume of the model in cubic inches.

24. Multiple Choice Which set of dimensions describes the rectangularprism with the greatest volume?

A. in. by 2 in. by 5 in.

B. 3 in. by 3 in. by in.

C. 4 in. by 2 in. by 4 in.

D. in. by 2 in. by 7 in.

25. Megan uses 216 cubic blocks to make a rectangular prism 9 blocks long

and 3 blocks tall. Each block measures inch on each side.

a. How many blocks wide is the prism?

b. What are the prism’s dimensions in inches?

c. What is the prism’s volume, in cubic inches?

d. Megan uses all of the bricks to make a new prism that is 6 brickstall. Give 3 possible sets of dimensions of the base of the prismMegan could have made.

14

214

312

312

312212


ST

UD

EN

T P

AG

E

Notes


STUDENT PAGE


content standards 6.g.2, 6.g.4 cc investigation 4: measurement · cc investigation 4: measurement...

Documents