Surface Area - Everyday Math · triangular prism, square pyramid Plotting and Analyzing Insect Data ... Add the areas of the faces to find the total surface area. Area of 2 triangular

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CurriculumFocal Points

890 Unit 11 Volume

Advance PreparationFor Part 1, organize workstations for groups of 4 students (2 partnerships each). Each group will need 2 cans

(see Lesson 11-3) and 1 each of the triangular prism and square pyramid models (Math Masters, pages 324

and 326) constructed in Lesson 11-1. Place a cardboard box with the top taped shut near the Math Message.

Teacher’s Reference Manual, Grades 4–6 pp. 220–222

Key Concepts and Skills• Measure the dimensions of a cylinder

in inches and centimeters.

[Measurement and Reference Frames Goal 1]

• Use rectangle and triangle area formulas

to find the surface area of prisms and

cylinders. Apply a formula to calculate

the area of a circle.

[Measurement and Reference Frames Goal 2]

• Identify and use the properties of prisms,

pyramids, and cylinders in calculations.

[Geometry Goal 2]

Key ActivitiesStudents find the surface areas of prisms,

cylinders, and pyramids by calculating the

area of each surface of a solid and then

finding the sum.

Ongoing Assessment: Informing Instruction See page 892.

Ongoing Assessment: Recognizing Student Achievement Use journal pages 389 and 390. [Measurement and Reference Frames

Goals 1 and 2]

Key Vocabularysurface area

MaterialsMath Journal 2, pp. 389 and 390

Study Link 11�6

slate � calculator � cardboard box �

per workstation: 2 cans, tape measure, ruler,

triangular prism, square pyramid

Plotting and Analyzing Insect DataMath Journal 2, pp. 390A and 390B

Students plot insect lengths in

fractional units on a line plot. They use

the line plot to analyze the data.

Math Boxes 11�7Math Journal 2, p. 391

Students practice and maintain skills

through Math Box problems.

Study Link 11�7Math Masters, p. 341

Students practice and maintain skills

through Study Link activities.

ENRICHMENTFinding the Smallest Surface Area for a Given VolumeMath Masters, p. 342

per partnership: 24 cm cubes

Students find the rectangular prism with the

smallest surface area for a given volume.

EXTRA PRACTICE

Finding Area, Surface Area, and VolumeMath Masters, p. 343

Student Reference Book, pp. 196 and 197

Students practice calculating area, surface

area, and volume.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

�

Surface AreaObjective To introduce finding the surface area of prisms,

cylinders, and pyramids.c

Common Core State Standards

890_EMCS_T_TLG2_G5_U11_L07_576914.indd 890890_EMCS_T_TLG2_G5_U11_L07_576914.indd 890 3/28/11 1:54 PM3/28/11 1:54 PM

Surface AreaLESSON

11�7

Date Time

The surface area of a box is the sum of the areas of all 6 sides (faces) of the box.

1. Your class will find the dimensions of a cardboard box.

a. Fill in the dimensions on the figure below.

b. Find the area of each side of the box. Then find the total surface area.

Area of front � in2

Area of back � in2

Area of right side � in2

Area of left side � in2

Area of top � in2

Area of bottom � in2

Total surface area � in2

2. Think: How would you find the area of the metal used to manufacture a can?

a. How would you find the area of the top or bottom of the can?

b. How would you find the area of the curved surface between the top and bottom of the can?

c. Choose a can. Find the total area of the metal used to manufacture the can.

Remember to include a unit for each area.

Area of top �

Area of bottom �

Area of curved side surface �

Total surface area �

38.48 cm2

878

187

187

99

99

153

153

Sample answers:

Sample answers for a canwith a 7 cm diameter and13 cm height:

First measure the diameter of the can, and divide by 2 to

find the radius. Then calculate the area (A � π º r 2).

Calculate the circumference of the base (c � π º d ), and

multiply that by the height of the can.

38.48 cm2

285.88 cm2

362.84 cm2

top

front

right

side

in.

in.

in.9

17

11

�

Math Journal 2, p. 389

Student Page

Lesson 11�7 891

Getting Started

Math MessageIf you were to wrap this box as a gift, how would you calculate the least amount of wrapping paper needed?

Study Link 11�6 Follow-UpBriefly review the answers. Ask: How would you explain the difference between capacity and volume? Volume is the measure of how much space a solid object occupies. Capacity is the measure of how much a container can hold.

Mental Math and Reflexes Have students write numbers from dictation and mark the indicated digits. Suggestions:

5,024,039 Circle the digit in the thousands place, and put an X through the digit in the hundred-thousands place.

28,794,052 Circle the digit in the ten-thousands place, and put an X through the digit in the hundreds place.

31.005 Circle the digit in the thousandths place, and put an X through the digit in the hundredths place.

150,247.653 Circle the digit in the thousandths place, and put an X through the digit in the thousands place.

587.624 Circle the digit in the tens place, and put an X through the digit in the hundredths place.

94,679,424 Circle each digit in the millions period, and put an X through each digit in the ones period.

1 Teaching the Lesson

▶ Math Message Follow-Up

WHOLE-CLASS ACTIVITY

(Math Journal 2, p. 389)

Ask volunteers to use geometry vocabulary to describe the box. The box is a rectangular prism that has six rectangular faces. Then have students explain their solution strategies. If wrapping paper covers each face of the box exactly, the least amount of wrapping paper needed is the sum of the area of each of the six faces. Tell students that the sum of the areas of the faces or the curved surface of a geometric solid is called surface area. If wrapping paper covers each face of the box exactly, the area of paper required is the surface area of the box. If the area of the available wrapping paper is less than the surface area of the box, the paper will not completely cover the box.

Ask a pair of students to measure the length, width, and height of the box to the nearest inch. Have students record these dimensions on the figure in Problem 1 on journal page 389, find and record the areas of the six sides of the box, and add these to find the total surface area. Remind students that opposite sides (top and bottom, left and right, front and back) of the box have the same area. This means that students need to calculate only three different areas.

▶ Finding the Surface Area of a Can PARTNER ACTIVITY


Algebraic Thinking Distribute one can to each partnership. Ask students to imagine that the top lids of their cans have not been removed. Ask: How would you find the surface area of your can?

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BBBBBBBBBBBBBBBBBBBB EELELEMMMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBBOOROROROORORORORORORORORO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGLLLLLLLLLLLVINVINVINVINVINNNNVINVINVINVINVINVINVINGGGGGGGGGGGOLOLOOLOLOLOLOOLOO VINVINVLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLLOO VVVLLLLLLLLLLVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOSOOSOOOOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

891-894_EMCS_T_TLG2_G5_U11_L07_576914.indd 891891-894_EMCS_T_TLG2_G5_U11_L07_576914.indd 891 2/24/11 7:27 AM2/24/11 7:27 AM

Analyzing Fraction DataLESSON

11�7

Date Time

Kerry measured life-size photos of 15 common insects to the nearest 1

_

8 of an inch.

His measurements are given in the table below.

InsectLength

(nearest 1

_ 8 in.) Insect

Length (nearest

1

_ 8 in.)

American Cockroach 1 3

_ 8 Heelfly

3

_

8

Ant 1

_ 4 Honeybee

3

_ 4

Aphid 1

_ 8 House Centipede 1

3

_ 8

Bumblebee 1 1

_ 2 Housefly

1

_ 4

Cabbage Butterfly 1 1

_ 4 June Bug 1

Cutworm 1 1

_ 2 Ladybug

5

_ 8

Deerfly 3

_ 8 Silverfish

3

_ 8

Field Cricket 7

_ 8

1. Make a line plot of the insect lengths on the number line below. Be sure to label the

axes and provide a title for the graph.

X X X X X XX

XXX

XX

XX

X

18

14

38

12

58

34

78

0 1 181 1

41 381 1

21

Insect Lengths

Length (inches)

Num

ber o

f Ins

ects

369-392_EMCS_S_MJ2_U11_576434.indd 390A 3/3/11 9:23 AM

Math Journal 2, p. 390A

Student Page

3. Use your model of a triangular prism.

a. Find the dimensions of the triangular and rectangular faces. Then find the areas

of these faces. Measure lengths to the nearest 1 _ 4 inch.

base = in. length = in.

height = in. width = in.

Area = in2 Area = in2

b. Add the areas of the faces to find the total surface area.

Area of 2 triangular bases = in2

Area of 3 rectangular sides = in2

Total surface area = in2

4. Use your model of a square pyramid.

a. Find the dimensions of the square and triangular faces. Then find the areas

of these faces. Measure lengths to the nearest tenth of a centimeter.

length = cm base = cm

width = cm height = cm

Area = cm2 Area = cm2

b. Add the areas of the faces to find the total surface area.

Area of square base = cm2

Area of 4 triangular sides = cm2

Total surface area = cm2

Surface Area continuedLESSON

11�7

Date Time

Formula for the Area of a Triangle

A = 1 _ 2 º b º h

where A is the area of the triangle, b is the length of its base, and h is its height.

2

1 3 _

4

1 3 _

4

4 1 _

2

3 1 _

2

27

2

9

108.99

30 1 _

2

6.36.3

39.69

39.6969.3

6.35.5

17.325

�

369-392_EMCS_S_MJ2_U11_576434.indd 390 3/4/11 7:04 PM


Student Page

892 Unit 11 Volume

Remind students that in this problem they are finding the area of the surface of their can, not the volume. Give students plenty of time to explore solution strategies among themselves, without your help. Add the areas of the top, bottom, and curved surface to find the total surface area. Ask a volunteer to explain how to find the area of the circular top and bottom. Use the formula A = π ∗ r2. Ask another volunteer to explain how to find the area of the curved surface between the top and bottom of the can. Calculate the circumference of the base using the formula c = π ∗ d, and then multiply the circumference by the height of the can.

If students are unable to suggest an approach for finding this area, remind or show them that the label on a food can is shaped like a rectangle if it is cut off and unrolled.

If you cut a can label in a straight line

perpendicular to the bottom, it will have

a rectangular shape when it is unrolled

and flattened.

Draw a rectangle on the board:

● How would you measure the can to find the length of the base of the rectangle? Guide students to recognize the following two possibilities:

1. Measure the circumference of the base of the can with a tape measure.

2. Measure the diameter of the base of the can. Calculate the circumference of the base, using the formula c = π ∗ d.

● How would you find the other dimension (the width) of the rectangle? Measure the height of the can.

Have partners measure the dimensions of their cans and complete Part 2 on journal page 389. Circulate and assist.

Ongoing Assessment: Informing Instruction

Watch for students who seem to have difficulty with visualizing the cutting and

unrolling of the can label. Have them use the following procedure to cut a sheet

of paper to use as a label for the side of the can:

1. Mark the top and bottom rims of the can at the can’s seam.

2. Lay the can on the sheet of paper with the marks touching the paper. Mark

these points on the paper.

3. Roll the can one complete revolution, until the marks touch the paper again.

Mark these points on the paper.

4. Connect the four marks on the paper to form a rectangle. Cut out

the rectangle.


Analyzing Fraction Data continuedLESSON

11�7

Date Time

Use the data and the line plot you made on journal page 390A to solve these problems. Give all answers in simplest form.

2. What is the maximum length of the insects measured? 1 1 _ 2 in.

3. What is the minimum length? 1 _ 8 in.

4. What is the range of insect lengths? 1 3 _ 8 in.

5. What is (are) the mode length(s) of the insects? 3 _ 8 in.

6. What is the median length of the insects? 3 _ 4 in.

7. How much longer is the American cockroach that Kerry measured than the field cricket?

1 _ 2 in.

8. Suppose 5 ladybugs were placed end to end. How long would they be? 3 1 _ 8 in.

9. How many times as long is the bumblebee than the housefly? 6 times as long 10. a. Suppose all of the insects less than 3 _ 4 inch long were placed end-to-end.

How long would they be?

2 3 _ 8 in.

b. Suppose all of the insects greater than 5 _ 8 inch long were placed end-to-end. How long would they be?

9 5 _ 8 in.

c. What is the total length of all 15 insects placed end to end? 12 in.

11. What is the mean length of the 15 insects that were measured? 4 _ 5 in.

369-392_EMCS_S_MJ2_G5_U11_576434.indd 390B 4/7/11 3:58 PM

Math Journal 2, p. 390B

Student Page

3. Find the volume of the rectangular prism.

Volume of rectangular prism

V = B * h

V = 12 cm3

Math Boxes LESSON

11�7

Date Time

5. Solve the pan-balance problems below.

a.

One weighs as much as 3 Xs. One weighs as much as 6 Xs.

b.

One weighs as much as 5 One weighs as much as 5

paper clips. paper clips.

1. Write each number in simplest form.

a. 80

_ 5 = 16

b. 53

_ 6 =

c. 8 24

_ 48 =

d. 11 38

_ 54 =

4. Find the area and perimeter of

the rectangle.

Area = 24.5 cm2

(unit)

Perimeter = 21 cm

(unit)

2. How are cylinders and cones alike?

Both have a circularbase and onecurved surface.

8 5 _

6

8 1 _

2

11 19

_

27

4 cm3

cm

1 cm

X XX X X X X

X X X X X X

7 cm

1

23 cm

62 63 147 148

197 186 189

228 229

369-392_EMCS_S_MJ2_U11_576434.indd 391 3/4/11 7:04 PM


Student Page

Lesson 11�7 893

▶ Finding the Surface Area PARTNER ACTIVITY

of a Prism and a Pyramid(Math Journal 2, p. 390)

Algebraic Thinking Have partners measure each solid constructed in Lesson 11-1 from Math Masters, pages 324 and 326 and record the dimensions on journal page 390. Then have them calculate and record the face areas and total surface area for each solid. Circulate and assist.

Ongoing Assessment: Journal pages

389 and 390 �Recognizing Student Achievement

Use journal pages 389 and 390 to assess students’ ability to measure to the nearest 1 _ 4 inch and 0.1 cm and to find the area of circles, triangles, and rectangles. Students are making adequate progress if they correctly answer Problems 2 and 3 and use rectangle, circle, and triangle area formulas to find the surface area of prisms and cylinders. [Measurement and Reference Frames Goals 1 and 2]

2 Ongoing Learning & Practice

▶ Plotting and Analyzing

INDEPENDENT ACTIVITY

Insect Data(Math Journal 2, pp. 390A and 390B)

Students plot insect lengths in fractional units on a line plot. They use the line plot to analyze data and solve problems involving addition, subtraction, multiplication, and division with fractions. As needed, review how to make a line plot based on fractional units and how to perform operations with fractions.

▶ Math Boxes 11�7



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

▶ Study Link 11�7


(Math Masters, p. 341)

Home Connection Students practice using formulas to calculate volume and surface area. If students use calculators, remind them to enter 3.14 for π if necessary.


LESSON

11�7

Name Date Time

A Surface-Area Investigation

In each problem below, the volume of a rectangular prism is given. Your task is to find

the dimensions of the rectangular prism (with the given volume) that has the smallest

surface area. To help you, use centimeter cubes to build as many different prisms as

possible having the given volume.

Record the dimensions and surface area of each prism you build in the table. Do not

record different prisms with the same surface area. Put a star next to the prism with

the smallest surface area.

1. Dimensions (cm) Surface Area (cm2) Volume (cm3)

2 × 6 × 1 40 12

3 × 4 × 1 38 12

3 × 2 × 2 32� 12

12 × 1 × 1 50 12

Dimensions (cm) Surface Area (cm2) Volume (cm3)

1 × 2 × 12 76 24

1 × 3 × 8 70 24

1 × 6 × 4 68 24

2 × 3 × 4 52� 24

2 × 2 × 6 56 24

24 × 1 × 1 98 24

2.

3. If the volume of a prism is 36 cm3, predict the dimensions that will result in the

smallest surface area. Explain.

4. Describe a general rule for finding the surface area of a rectangular prism in words or

with a number sentence.

find the sum of these areas.Find the area of each face. Then

that would result in the given volume.3 º 3 º 4; I used the smallest 3 dimensions

323-347_EMCS_B_MM_G5_U11_576973.indd 342 3/9/11 8:44 AM

Math Masters, p. 342

Teaching MasterLESSON

11�7

Name Date Time

Area, Surface Area, and Volume

Record the dimensions, and find the volume and surface area for each figure below. Round results to the nearest hundredth.

Area of rectangle: A = l ∗ w

Volume of rectangular prism: V = B ∗ hor V = l ∗ w ∗ h

Circumference of circle: c = π ∗ d

Area of circle: A = π ∗ r 2

Volume of cylinder: V = π ∗ r 2 ∗ h

1. Record the dimensions and find the area.

Length = 5 1 _ 2 ft Width = 2 5 _ 8 ft Area = 14 7

_ 16 ft2

How many square tiles, each 1 ft long on a side, would be needed to fill the rectangle?

14 7

_ 16

2. Record the dimensions and find the volume.

Length of base = 5 cm

Width of base = 4 cm

Area of base = 20 cm2

Height of prism = 2 cm

Number of 1-centimeter unit cubes needed to fill the box: 40

Volume = 40 cm3

3. Rectangular prism

Length of base = 8 cm Width of base = 5 cm Height of prism = 7 cm Volume = 280 cm3

Surface area = 262 cm2

4. Cylinder

Diameter = 7 ft Height = 9 ft Volume = 346.36 ft3

Surface area = 274.89 ft2

125 ft

582 ft

5 cm 2 cm4 cm

8 cm 5 cm

7 cm

7 ft

9 ft



Teaching Master

STUDY LINK

11�7 Volume and Surface Area

Name Date Time

1. Kesia wants to give her best friend a box of chocolates.

Figure out the least number of square inches of wrapping

paper Kesia needs to wrap the box. (To simplify the

problem, assume that she will cover the box completely

with no overlaps.)

Amount of paper needed:

Explain how you found the answer.

Sample answer: I found the area of each of the 6 sides and then added them together.

2. Could Kesia use the same amount of wrapping paper to

cover a box with a larger volume than the box in Problem 1? Explain.

A 4 in. × 4 in. × 3 1

_ 2 in. box has a volume of

56 in3 and a surface area of 88 in2.

Find the volume and the surface area of the two figures in Problems 3 and 4.

3. Volume: 4. Volume:

Surface area: Surface area:

Yes

216 in2

216 in3

351.7 cm2

502.4 cm3

88 in2

6 in.

2 in.4 in.

10 c

m

8 cm

6 in.

cube

197 198200 201

Area of rectangle: A = l º w

Volume of rectangular prism: V = l º w º h

Circumference of circle:c = π º d

Area of circle: A = π º r 2

Volume of cylinder:V = π º r 2 º h



Study Link Master

894 Unit 11 Volume

3 Differentiation Options

ENRICHMENT PARTNER ACTIVITY

▶ Finding the Smallest Surface 15–30 Min

Area for a Given Volume(Math Masters, p. 342)

Algebraic Thinking To apply students’ understanding of volume and surface area, have them use given volumes to find the dimensions of rectangular prisms. Partners use the patterns in the dimensions to identify the dimensions of a prism that would have the smallest surface area.

When students have finished, discuss any difficulties or curiosities they encountered.

EXTRA PRACTICE


▶ Finding Area, Surface Area, 5–15 Min

and Volume(Math Masters, p. 343; Student Reference Book, pp. 196 and 197)

Students practice calculating area, surface area, and volume of various geometric figures.


LESSON

11�7

Name Date Time

Area, Surface Area, and Volume

343

Cop

yrig

ht ©

Wrig

ht G

roup

/McG

raw

-Hill Record the dimensions, and find the volume and surface area for each

figure below. Round results to the nearest hundredth.

Area of rectangle: A = l ∗ w

Volume of rectangular prism: V = B ∗ hor V = l ∗ w ∗ h

Circumference of circle: c = π ∗ d

Area of circle: A = π ∗ r 2

Volume of cylinder: V = π ∗ r 2 ∗ h

1. Record the dimensions and find the area.

Length =

Width =

Area =

How many square tiles, each 1 ft long on a side, would be needed to fill the rectangle?

2. Record the dimensions and find the volume.

Length of base =

Width of base =

Area of base =

Height of prism =

Number of 1-centimeter unit cubes needed to fill the box:

Volume =

3. Rectangular prism

Length of base =

Width of base =

Height of prism =

Volume =

Surface area =

4. Cylinder

Diameter =

Height =

Volume =

Surface area =

125 ft

582 ft

5 cm 2 cm4 cm

8 cm 5 cm

7 cm

7 ft

9 ft

323-347_EMCS_B_MM_G5_U11_576973.indd 343323-347_EMCS_B_MM_G5_U11_576973.indd 343 4/7/11 9:43 AM4/7/11 9:43 AM

Documents

Surface Area - Everyday Math · triangular prism, square pyramid Plotting and Analyzing Insect Data ... Add the areas of the faces to find the total surface area. Area of 2 triangular