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Chapter 11Measurement: Perimeter, Area, and Volume

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Chapter 11Measurement: Perimeter, Area, and Volume

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http://ca.gr5math.com/

Lesson 11-1 Perimeter

Lesson 11-2 Area of Parallelograms

Lesson 11-3 Problem-Solving Strategy: Make a Model

Lesson 11-4 Area of Triangles

Lesson 11-5 Problem-Solving Investigation: Choose the Best Strategy

Lesson 11-6 Volume of Rectangular Prisms

Lesson 11-7 Surface Area of Rectangular Prisms

1111Measurement: Perimeter, Area, and Volume



Five-Minute Check (over Chapter 10)

Main Idea and Vocabulary

California Standards

Key Concept: Perimeter of a Square

Key Concept: Perimeter of a Rectangle

Example 1

Example 2

11-111-1 Perimeter


11-111-1 Perimeter

• I will find the perimeters of squares and rectangles.

• perimeter


11-111-1 Perimeter

Standard 5MG1.4 Differentiate between, and use appropriate units of measures for two- and three-dimensional objects (i.e., find the perimeter, area, volume).


11-111-1 Perimeter


11-111-1 Perimeter


The base of the Eiffel Tower is shaped like a square with each side measuring 125 meters. What is the perimeter of the base?

P = 4s Perimeter of a square

P = 4(125) Replace s with 125.

11-111-1 Perimeter

P = 500 Multiply.

Answer: The perimeter of the base of the Eiffel Tower is 500 meters.


The park is shaped like a square with each side measuring 100 yards. What is the perimeter of the park?

A. 400 feet

B. 400 yards

C. 200 yards

D. 100 yards

11-111-1 Perimeter


Find the perimeter of the rectangle.

Answer: The perimeter is 22 meters.

11-111-1 Perimeter

7 m

4 m

Write the formula.

P = 2(7) + 2(4)

P = 14 + 8 Multiply.

P = 22

Add.

P = 2 + 2w

Replace with 7 and w with 4.


Find the perimeter of the rectangle.

A. 7 cm

B. 6 cm

C. 8 cm

D. 10 cm

11-111-1 Perimeter

3 cm

1 cm



Five-Minute Check (over Lesson 11-1)

Main Idea


Key Concept: Area of a Parallelogram

Example 1

Example 2

Example 3

11-211-2 Area of Parallelograms



• I will find the areas of parallelograms.

• base

• height



Standard 5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle.

Standard 5MG1.4 Differentiate between, and use appropriate units of measures for two- and three-dimensional objects.





Find the area of the parallelogram.

Answer: The area is 30 square units or 30 units2.

A = bh Area of parallelogram

A = 3 • 10 Replace b with 3 and h with 10.

A = 30 Multiply.

The base is 3 and the height is 10.




A. 35 units2

B. 28 units2

C. 49 units2

D. 64 units2



Answer: The area is 36.9 square centimeters or 36.9 cm2.



A = 8.2 • 4.5 Replace b with 8.2 and h with 4.5.

A = 36.9 Multiply.



A. 68 mm2

B. 70 mm2

C. 68.64 mm2

D. 70.42 mm2




A particular area rug is shaped like a parallelogram. Estimate the area of the floor it will cover.

Answer: The area of the rug is about 66 ft2.


A = 11 • 6 Replace b with 11 and h with 6.

A = 66 Multiply.

Estimate 6 is about 6 and 10 is about 11.14

12


A parking lot is shaped like a parallelogram. Estimate the area of ground it will cover.

A. 7,000 sq. yds.

B. 7,200 sq. yds.

C. 7,140 sq. yds.

D. 7,080 sq. yds.





Main Idea


Example 1: Problem-Solving Strategy

11-311-3 Problem-Solving Strategy: Make a Model



• I will solve problems by making a model.



Standard 5MR2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.




While volunteering at the local farm market, Julia was asked to make a display for the oranges. She needs to stack the oranges in the shape of a square pyramid. The base should have 100oranges and one orange needs to be on top. There are 400 oranges total. Are 400 oranges enough to make a square pyramid with a base of 100 oranges?


Understand

What facts do you know?

• The oranges need to be in the shape of a square pyramid with 100 oranges in the base and 1 orange on top.

• There are 400 oranges altogether.

What do you need to find?

• Are 400 oranges enough to make a square pyramid with a base of 100 oranges?



Plan

Make a model using pennies to find the number of oranges needed.



SolveBegin with 100 pennies. For each consecutive layer, place 1 penny where 4 meet.


bottom layer

100

second layer

81

third layer

64

fourth layer

49


SolveAnswer: By continuing this pattern, 100 + 81 + 64 + 49

+ 36 + 25 + 16 + 9 + 4 + 1 or 385 oranges will be needed. Since 385 < 400, 400 oranges are enough to make a square pyramid.



Check


Look back at the problem. 400 – 100 – 81 – 64 – 49 – 36 – 25 – 16 – 9 – 4 – 1 leaves 15 oranges.




Main Idea


Key Concept: Area of a Triangle

Example 1

Example 2

Example 3

11-411-4 Area of Triangles

Area of Triangles



• I will find the areas of triangles.



Standard 5MG1.1 Derive and use the formula for the area of a triangle and of a parallelogram by comparing it with the formula for the area of a rectangle.






Find the area of the triangle.

Area of a triangle

Replace b with 5 and h with 8.

By counting, you find that the measure of the base of the triangle is 5 and the height is 8.

A = bh 12

A = (5) • (8) 12



Multiply.

Multiply.

A = (40) 12

A = 20

Answer: The area of the triangle is 20 square units.



A. 25 sq. units


B. 16 sq. units

D. 20 sq. units

C. 12 sq. units

12




Area of a triangle

Replace b with 16.4 and h with 7.9.

A = bh 12

A = (16.4) • (7.9) 12



Multiply.

Divide. 129.56 ÷ 2 = 64.78

A = (129.56)

12

A = 64.78

Answer: The area of the triangle is 64.78 square meters.


Clio cut out a banner in the shape of a triangle. What is the area of the banner?


B. 99 cm

A. 99 sq. cm

C. 49 cm12

1249 sq. cmD.




Area of a triangle

Replace b with 12 and h with 6.

A = bh 12

A = (12) • (6) 12



Multiply.

Divide. 72 ÷ 2 = 36.5

A = (72) 12

A = 36.5

Answer: The area of the triangle is 36.5 square inches.


Kira drew a triangle on the sidewalk with chalk. What is the area of her triangle?

A. 21 sq. ft

B. 10.5 sq. ft

C. 20 sq. ft

D. 11 sq. ft





Main Idea


Example 1: Problem-Solving Investigation

11-511-5 Problem-Solving Investigation: Choose the Best Strategy



• I will choose the best strategy to solve a problem.



Standard 5MR2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.



ROSS: I want people to find out about a party I’m having, so I will tell Jamie and Cara and have each of them tell two friends, and so on. I wonder how many people would be invited to the party in three minutes if two friends tell another two friends each minute?

YOUR MISSION: Find the number of people who would be invited to the party in three minutes.



Understand

What facts do you know?

• You know that Ross tells Jamie and Cara about the party, and then each friend tells two other friends each minute.

What do you need to find?

• You need to find the number of people who would be invited to the party in three minutes.



Plan

Draw a diagram to show the number of people who would be invited to the party.



Solve


Answer: So, 14 people would be invited to the party.


Check


Look back at the problem to see if the diagram meets all of the requirements. Since the diagram is correct, the answer is correct.






Key Concept: Volume of a Rectangular Prism

Example 1

Example 2

11-611-6 Volume of Rectangular Prisms



• I will find the volume of rectangular prisms.

• rectangular prism

• volume

• cubic units



Standard 5MG1.3 Understand the concept of volume and use the appropriate units in common measuring systems to compute the volume of rectangular solids.

Standard 5MG1.4 Differentiate between, and use appropriate units of measures for two- and three-dimensional objects (i.e., find the perimeter, area, volume).





Find the volume of the rectangular prism.

Estimate V ≈ 10 m × 10 m × 5 m or 500 m3

In the figure, the length is 10 meters, the width is 8 meters, and the height is 5 meters.

Use V = wh.



Volume of rectangular prism

V = 10 • 8 • 5

V = 400 Multiply.

Answer: The volume is 400 cubic meters.

Replace with 10, w with 8, and h with 5.

V = wh



Since we overestimated, the answer should be less than the estimate. 400 < 500.

Check for Reasonableness


Find the volume of a rectangular prism with a length of 14 mm, a width of 6 mm, and a height of 3 mm.

A. 250 mm3

B. 300 mm3

C. 252 mm3

D. 254 mm3




A closet is 6.2 feet long, 2.8 feet wide, and 8.1 feet high. Find the amount of space contained within the closet for storage. Round to the nearest foot.

Estimate V ≈ 6 ft × 3 ft × 8 ft or 144 ft3



Volume of rectangular prism

V = 6.2 • 2.8 • 8.1

V = 140.686 Multiply, then round to the nearest foot.

Answer: The storage space in the closet is about 141 cubic feet.

Replace with 6.2, w with 2.8, and h with 8.1.

V = wh



Compare to the estimate. 141 ≈ 144.

Check for Reasonableness


A tissue box is 12 inches long, 5 inches wide and 5 inches high. Find the amount of space contained within the box for tissues.

A. 300 cubic inches

B. 250 cubic inches

C. 125 cubic inches

D. 315 cubic inches







Key Concept: Surface Area of a Rectangular Prism

Example 1

Example 2

11-711-7 Surface Area of Rectangular Prisms

Using a Net to Build a Cube



• I will find the surface areas of rectangular prisms.

• surface area



Standard 5MG1.2 Construct a cube and rectangular box from two-dimensional patterns and use these patterns to compute the surface area for these objects.





Find the surface area of the rectangular prism.

Find the area of each face.

top and bottom:

2( w) = 2(8 × 4) or 74



front and back:

two sides:

2(wh) = 2(4 × 3) or 24

Add to find the surface area.

Answer: The surface area is 74 + 48 + 24 or 136 square centimeters.

2( h) = 2(8 × 3) or 48


Find the area of a rectangular prism that is 12 in long, 6 in wide, and 5 in high.

A. 320 square inches

B. 300 square inches

C. 324 square inches

D. 342 square inches



A box measures 13 inches long, 7 inches wide, and 4 inches deep. What is the surface area of the box?


Surface area of a prism

S = 2(13 × 7) + 2(13 × 4) + 2(7 × 4)

S = 2(91) + 2(52) + 2(28) Simplify within parentheses.

S = 182 + 104 + 56 Multiply.

S = 342 Add.

Answer: The box has a surface area of 342 square inches.

S = 2 w + 2 h + 2wh

= 13, w = 7, h = 4


A flat screen television measure 48 inches long, 2 inches wide, and 25 inches high. What is the surface area of the TV?

A. 2,692 square inches

B. 2,700 square inches

C. 1,874 square inches

D. 2,962 square inches





Five-Minute Checks

Area of Triangles

Using a Net to Build a Cube


Lesson 11-1 (over Chapter 10)

Lesson 11-2 (over Lesson 11-1)







(over Chapter 10)

Draw the three-dimensional figure whose top, side, and front views are shown. Use isometric dot paper.

B. A.

top side front


(over Chapter 10)


D. C.

top side front


(over Chapter 10)


B.


(over Chapter 10)


B. A.

top side front


(over Chapter 10)


C. D.

top side front


(over Chapter 10)


C.


(over Lesson 11-1)

A. 11 in

B. 28 in

C. 3 in

D. 22 in

Find the perimeter of each square or rectangle.

length 7 in, width 4 in


(over Lesson 11-1)

A. 52 cm

B. 25 in

C. 36 cm

D. 62 cm


sides 13 cm


(over Lesson 11-1)

A. 24 ft

B. 48 ft

C. 144 ft

D. 140 in


length 14 ft, width 10 ft


(over Lesson 11-1)

A. 100 yd

B. 25 yd

C. 255 ft

D. 125 ft


sides 25 yd


(over Lesson 11-2)

A. 128 yd2

B. 28 yd2

C. 248 yd2

D. 112 yd2

Find the area of each parallelogram.

base 14 yd, height 8 yd


(over Lesson 11-2)

A. 143 ft2

B. 141 ft2

C. 144 ft2

D. 144 yd2


base 13 ft, height 11 ft


(over Lesson 11-2)

A. 48 in2

B. 44 in2

C. 52 in2

D. 48 ft


base 8 in, height 6 in12


(over Lesson 11-2)

A. 49.92 cm2

B. 45.12 cm2

C. 14.8 cm2

D. 24.8 cm2


base 9.6 cm, height 5.2 cm


(over Lesson 11-3)

A. 48 cans

B. 52 cans

C. 44 cans

D. 70 cans

Solve. Use the make a model strategy. Cans of tuna are stacked into a 4-layer pyramid-shaped display. The bottom layer is 8-cans long and 4-cans wide. There is 1 less can in the length and width of each layer above it. How many cans are on display?


(over Lesson 11-4)

A. 25 ft2

B. 25 ft

C. 12.5 ft2

D. 12.5 ft

Find the area of each triangle.

base 5 ft, height 5 ft


(over Lesson 11-4)

A. 557 cm2

B. 71 cm2

C. 554 cm2

D. 600 cm2


base 48 cm, height 23 cm


(over Lesson 11-4)

A. 2.32 cm2

B. 8.32 cm2

C. 15.23 cm2

D. 18.32 cm2


base 5.2 cm, height 3.2 cm


(over Lesson 11-4)

D. 18 in34


base 5 in, height 7 in12

C. 2 in

A. 12 in12

B. 30 in12


(over Lesson 11-5)

A. $22

B. $24

C. $19

D. $34

Tell what strategy you used. Desta saved $1 the first week. After that she saved $2 more each week than she had the week before. How much money did she save in the tenth week?


(over Lesson 11-6)

A. 168 yd3

B. 18 yd3

C. 59 yd3

D. 88 yd3

Find the volume of each prism.

length 8 yd, width 7 yd, height 3 yd


(over Lesson 11-6)

A. 240 cm3

B. 480 cm3

C. 144 in

D. 25 in


length 12 cm, width 8 cm, height 5 cm


(over Lesson 11-6)

B. 141 ft334

A. 144 ft3


C. 3,444 in

D. 1,333 cm

length 5 ft, width 3 ft, height 9 ft14


(over Lesson 11-6)

A. 156.222 cm3

B. 147.744 cm3

C. 24 cm3

D. 444 in


length 2.4 cm, width 17.1 cm, height 3.6 cm

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