WHAT’S IN A VOLUME OR SURFACE AREA? 2

TEACHER EDITION

List of Activities for this Unit:

ACTIVITY / STRAND / DESCRIPTION
1 – Hand me the Net / ME / Nets and surface area of rectangular solids.
2 – What’s my Area / ME / Surface area, chg in dimensions
3 – Hide in a Sphere They Can’t Corner You There! / ME / Spheres, formulas, chg of dimensions table
4 – Change my Size…Cubes / ME / Cube Table
5 – Change my Size…Rectangular Prisms / ME / Rectangular Solid Table
6 – Sphere Fun / ME / Sphere Table
7 – The Box-It Company / ME / Surface area and volume apps
8 – Name that Volume / ME / Determine Volume w/various dimensions
9 – Backyard Greenhouse / ME / Surface Area and Volumes of Cylinders
10 – Changes to the Greenhouse / ME / Find the effect on volume when changing dimensions
11 – The Sand Box / ME / Determine Volume
12 – Practice Problems- MC / ME / (Prob #46-49) What’s in a Volume Multiple Choice

Vocabulary: Mathematics and ELL

aquarium / excavation / prism
bases / faces / radius
cylindrical / foundation / rectangular prism
determine / greenhouse / similar
diagrams / justify / solid
diameter / linear / sphere
dimensions / manufacturer / surface area
doubles / net / triples
edge / perimeter / vertices
effect / polyhedron
COE Connections / Installing a Pool
Zachary's Pyramid
Spherical Marbles
MATERIALS / Calculators
Warm-Ups
(in Segmented Extras Folder) / Paper Posers
Painters Problem
Ice Cream Dilemma
A Fishy Story

Essential Questions:

  • What is a net?
  • How does the change in one linear dimension affect the volume?
  • What is meant by the terms solid, prism, edge, vertex, face and polyhedron?
  • What is a rectangular prism?
  • What is a cube?
  • What is a sphere?
  • What are the appropriate labels when determining area or volume?
  • What is the difference between a 2-dimensional and a 3-dimensional figure?
  • How does one calculate the surface area and volume of a cube, rectangular prism and sphere?
  • How can a conclusion be supported using mathematical information and calculations?
  • How can one of the dimensions of a solid be determined when the volume is known?
  • What would happen if one linear dimension of a cube, rectangular prism or sphere is changed?
  • How is perimeter of a semi-circle determined?
  • How is the area of a semi-circle determined?
  • What is meant by linear foot?
  • How is volume of a cylinder determined?

Lesson Overview:

  • Before allowing the students the opportunity to start the activity: access their prior knowledge regarding how to do an investigation. What do you already know regarding determining the volume of a solid and what will affect the volume of a solid. Also what they already know regarding terminology involving solids and determining surface area.
  • Before allowing the students the opportunity to start the activity: access their prior knowledge with regards the students’ experiences with buying products by the linear foot, building greenhouses, the value of having a greenhouse vs. trying to grow plants without one, and experiences with shopping in a hardware store.
  • A good warm-up for this activity is Paper Posers, Painters Problem, Ice Cream Dilemma or a Fishy Story.
  • How is a problem situation decoded so that a person understands what is being asked?
  • What mathematical information should be used to support a particular conclusion?
  • 1 gallon 0.13 ft3
  • A warm-up involving calculating volume and area of a circle could be used.
  • Focus on ratios for sides, perimeter, area and volume.
  • Numbers 28 and 29 should be done in groups and then discussed as a class after the students have had time to work on solving the problems.
  • How will the students make their thinking visible?
  • Use resources from your building.

Performance Expectations:

4.5.CIdentify missing information that is needed to solve a problem.

6.2.DApply the commutative, associative, and distributive properties, and use the order of operations to evaluate mathematical expressions.

6.4.ADetermine the circumference and area of circles.

6.4.CSolve single- and multi-step word problems involving the relationships among radius, diameter, circumference, and area of circles, and verify the solutions.

6.4.EDetermine the surface area and volume of rectangular prisms using appropriate formulas and explain why the formulas work.

7.3.ADetermine the surface area and volume of cylinders using the appropriate formulas and explain why the formulas work.

7.3.CDescribe the effect that a change in scale factor on one attribute of a two- or three-dimensional figure has on other attributes of the figure, such as the side or edge length, perimeter, area, surface area, or volume of a geometric figure.

7.3.DSolve single- and multi-step word problems involving surface area or volume and verify the solutions.

8.4.CEvaluate numerical expressions involving non-negative integer exponents using the laws of exponents and the order of operations.

G.6.DPredict and verify the effect that changing one, two, or three linear dimensions has on perimeter, area, volume, or surface area of two- and three-dimensional figures.

Performance Expectations and Aligned Problems

Chapter 16
“What’s in a Volume or Surface Area?”Subsections: / 1-
Hand me the Net / 2-
What’s my Area / 3-
Hide in a Sphere… / 4-
Ch-ange my Size…Cubes / 5-
Ch-ange my Size…Rect-angu-lar / 6-
Sphere Fun / 7-
The Box-it Com-pany / 8-
Name that Vol-ume / 9-
Back-yard Green-house / 10-
Ch-anges to the Green-house / 11-
The Sand box / 12-
MC
Prob-lems
Problems Supporting:
PE 4.5.C / 4, 10, 11 / 1, 2 / 2 / 3, 4 / 2 - 4
Problems Supporting:
PE 6.2.D / 10-13 / 6, 7 / 1, 2 / 1 / 1 / 1 / 1 – 5 / 1, 2 / 1 – 3 / 1 – 4 / 1 – 4 / 3, 4
Chapter 16
“What’s in a Volume or Surface Area?”Subsections: / 1-
Hand me the Net / 2-
What’s my Area / 3-
Hide in a Sphere… / 4-
Ch-ange my Size…Cubes / 5-
Ch-ange my Size…Rect-angu-lar / 6-
Sphere Fun / 7-
The Box-it Com-pany / 8-
Name that Vol-ume / 9-
Back-yard Green-house / 10-
Ch-anges to the Green-house / 11-
The Sand box / 12-
MC
Prob-lems
Problems Supporting:
PE 6.4.A ≈ 6.4.C / 1 / 1 – 4 (partially) / 1 – 4 (partially) / 4
Problems Supporting:
PE 6.4.E ≈ 7.3.D / 1 – 14 / 1 – 7 / 1, 2 / 1 / 1 / 1 / 1 – 5 / 1 / 1 – 4 / 1 – 4 / 3, 4 / 3
Problems Supporting:
PE 7.3.A / 1 – 4 / 1 – 4 / 4
Problems Supporting:
PE 7.3.C / 1 / 1 / 1 / 3 – 5 / 1, 3, 4 / 1, 3, 4
Problems Supporting:
PE 8.4.C / 1 – 5 / 1, 2 / 1 / 1 / 1 – 4 / 1, 3, 4 / 1, 4 / 4
Problems Supporting:
PE G.6.D / 1 / 1 / 1 / 3 – 5 / 1, 4 / 3, 4

Assessment: Use the multiple choice and short answer items from Measurement and Geometric Sense that are included in the CD. They can be used as formative and/or summative assessments attached to this lesson or later when the students are being given an overall summative assessment.

1-HAND ME THE NET

A solid is a three dimensional figure. A solid has length, width, and height. If a solid has any flat surfaces they are called faces. The line segment where two faces meet is called an edge. Verticesare the points where the edges meet. (The singular term for vertices is vertex.) If the faces of a solid are all polygons, it is called a polyhedron. Some of the solid figures that you may have studied before are cones, pyramids, rectangular solids, cubes, spheres and cylinders. A rectangular solid is a figure made up of rectangles and/or squares. It has 6 faces. A prism is a special type of polyhedron in which two of the faces are parallel and congruent. The parallel faces are called bases. The surface areaof a prism is the sum of the areas of all the faces of the prism. Surface area is important when figuring the amount of material needed to cover packages, furniture, and other solid figures.

When you find the surface area of a solid, you are finding the area of the entire surface. Sometimes it helps to cut the figure apart and lay it flat. This cutting a prism and flattening it out has a shape called a net. Shown below is one possible net for the rectangular prism. If you find the area of each of the six rectangles formed and add them up, you find the surface area of the prism.

Rectangular Prism (solid)

Vertex

Edge Net for the rectangular prism

Face

1. What is the length and width of rectangle A? The length is 3cm and the width is 2cm.

2. What is the area of rectangle A? The area is 3cm • 2cm = 6cm2.

Show your work using words, numbers and/or diagrams.

3. What other rectangular face has the same area as A? F

4. What is the length and width of rectangle B? The length is 4 cm and the width is 2 cm.

5. What is the area of rectangle B? The area is 4cm • 2cm = 8cm2.

Show your work using words, numbers and/or diagrams.

6. What other rectangular face has the same area as rectangle B? E

7. What is the length and width of rectangle C? The length is 4cm and the width is 3cm.

8. What is the area of rectangle C? The area is 4cm • 3cm = 12cm2.

Show your work using words, numbers and/or diagrams.

9. What other rectangular face has the same area as rectangle C? D

10. What is the surface area of this rectangular prism (solid)? 52cm2

Show your work using words, numbers and/or diagrams.

2(6cm2) + 2(8cm2) + 2(12cm2) = 12cm2 + 16cm2 + 24cm2 = 52cm2

11. A storage box measures 8 inches by 4 inches by 5 inches. The box is sitting in Myra’s front

room. She wants to cover the box with paper.

What is the surface area of the box, including the top and bottom? 184in2

Be sure to label your answer.

Show your work using words, numbers and/or diagrams.

2(4in•5in) + 2(5in•8in) + 2(4in•8in) = 40in2 + 80in2 + 64in2 = 184in2

12. When displaying books at the book fair, Renee likes to cover the display boxes with pretty fabric.

To determine how much fabric she needs, she needs to know the surface area of the display box.

What is the surface area of the display box? 413.34cm2

Be sure to label your answer.

Show your work using words, numbers and/or diagrams.

2(8.3cm8.3cm) + 2(8.3cm8.3cm) + 2(8.3cm8.3cm) = 6(8.3cm8.3cm) = 6(68.89cm2) = 413.34cm2

13. John said that, to find the surface area of the rectangular solid below, he added the following areas: 48 sq. cm + 42 sq. cm + 56 sq. cm + 48 sq. cm + 42 sq. cm + 56 sq. cm. Leroy said that the way that he found the surface area was to add the areas, 48 sq. cm + 42 sq. cm + 56 sq. cm, and multiply this sum by 2. Who was correct? Both are correct; their answers are equivalent.

Justify (support) your answer.

48cm2 + 42cm2 + 56cm2 + 48cm2 + 42cm2 + 56cm2

48cm2 + 48cm2 + 42cm2 + 42cm2 + 56cm2 + 56cm2

2(48cm2) + 2(42cm2) + 2(56cm2)

2(48cm2 + 42cm2 + 56cm2) = 292cm2

14. A cube is a special kind of rectangular solid. All of the faces have the same area. Here is a cube.

a. If you look at the top of a cube what do you see? A square.

b. If you look at a cube from the side, what 2-dimensional figure do you see? A square.

c. If you look at a cube from the front, what 2-dimensional figure do you see? A square.

d. What is true about all of the 2-dimensional figures that make up the faces of a cube?

They are all squares…6 of them.

2-What’s My Area

1. The surface area of a cube is easier to determine(find) than the surface area of other rectangular solids.

Why? All the faces are all squares; 6(area of one face).

Given a cube, what does it mean to find the surface area? Find the area of one face and multiply by 6.

2. The surface area refers to the sum of the areas of the faces of a solid. What is meant by the

volume of a cube solid? The amount of space a solid occupies or how many cubic units the solid can contain.

3. Given a cube with edge length of 9 inches, find the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 6(9in)(9in) = 6(81in2) = 486in2 Volume (9in)(9in)(9in) = (9in)3 = 729in3

Show your work using words, numbers and/or diagrams (pictures).

4. Given a cube with edge lengthe, find the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 6(e)(e) = 6(e2) Volume (e)(e)(e) = e3

Show your work using words, numbers and/or diagrams.

5. A cube has a volume of 125 cubic cm. What is the edge length? (5cm)3 = 125cm3 5cm is the edge length OR = 5cm

What is the surface area? 6(5cm)(5cm) = 6(25cm2) = 150cm2

Show your work using words, numbers and/or diagrams.

6. Given a rectangular prism with length of 5 cm, width of 8 cm, and height of 11 cm. Determine the surface area and the volume. Be sure to label all drawings and your answer.

Surface area 366cm2 Volume 440cm3

Show your work using words, numbers and/or diagrams.

SA = 2({5cm•8cm} + {5cm•11cm} + {8cm•11cm}) = 2(40cm2 + 55cm2 + 88cm2) = 2(183cm2) = 366cm2

V = (5cm)(8cm)(11cm) = 440cm3

7. A rectangular prism has a volume of 120 cubic feet. Give three different combinations of dimensions (measurements) for the prism. Determine the Surface Area using your dimensions.Be sure to label.

a. 1ft x 1ft x 120ft Surface Area: 2({1ft•1ft} + {1ft•120ft} + {1ft•120ft}) = 482ft2

b. 3ft x 4ft x 10ftSurface Area: 2({3ft•4ft} + {3ft•10ft} + {4ft•10ft}) = 164ft2

c. 4ft x 5ft x 6ftSurface Area: 2({4ft•5ft} + {4ft•6ft} + {5ft•6ft}) = 148ft2

3-Hide in a Sphere… They Can’t Corner You There!

1. Given a sphere (a ball shaped solid)of radius r, the formulas to determine the surface area and volume are:

Surface Area = 2 Volume = 3

a. A sphere has a radius of 15 cm. Determine the surface area and volume. Be sure to label all your answers.

a. Surface area SA = 4(15cm)2 = 900π cm2 ≈ 2827.43 cm2

b. Volume V = 3 = 4500π cm3 ≈ 14,137.17 cm3

Show your work using words, numbers and/or diagrams.

2. A sphere has a volume of 288π cm3. Determine the radius, the diameter, and the surface area. Be sure to label all your answers.

The radius of a circle is the distance from the center of a circle to the edge of the circle.

The diameter of a circle is the distance from one edge of the circle to the opposite edge of the circle that passes straight through the center of the circle.

a. Radius 6cm

b. Diameter 12cm

c. Surface Area SA = 4r2 = 4(6cm)2= 144 ≈ 452.39cm2

Show your work using words, numbers and/or diagrams.

Volume

288 = 4/3  r3

R3 = 216

r = 6 cm

4-Change My Size…Cubes

  1. The cubeon the right mayhelp you to complete the table.

Length of edge / Surface Area / Volume / Change in each edge length / New
Surface
Area / Describe surf area change / New Volume / Describe
Volume change
2 cm. / 24 cm.2 / 8cm3 / Doubles
(times 2)
length / 96 cm 2 / 4 times larger / 64cm3 / 23 = 8 times larger
3 cm. / 54cm2 / 27cm3 / Triples
(times 3) length / 486cm2 / 32 = 9 times larger / 729cm3 / 33 = 27 times larger
5in / 150 in 2 / 125in3 / Doubles length / 600in2 / 22 = 4 times larger / 1000in3 / 23 = 8 times larger
4 m / 96m2 / 64m3 / Multiplies length by 4 / 1536m2 / 42 = 16 times larger / 4096m3 / 43 = 64 times larger
2.5 m / 37.5m2 / 15.625m3 / Multiplies length by 10 / 3750m2 / 100 times larger
(102 = 100) / 15,625m3 / 103 = 1000 times larger
K units / 6K2units2 / K3units3 / Multiplies
length by m / m2•6K2units2 / m2 times larger / m3•K3units3 / m3times larger

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 1 of 28

5-Change My Size…Rectangular Prisms

1. The rectangular prism on the right may help youto complete the table.

A / B / C / Surface
Area / Volume / Change in edges / New Surface Area / Describe surf area change / New Volume / Describe volume change
A / B / C / Surface
Area / Volume / Change in edges / New Surface Area / Describe surf area change / New Volume / Describe volume change
4 cm / 5 cm / 9 cm / 202 cm² / 180 cm³ / Double all sides / 808 cm² / 4 times as big (2²) / 1440 cm³ / 8 times as big (2³)
2” / 6” / 6" / 120 in² / 72 in3 / Multiply all edges by 3 / 1080 in² / 9 times as big (3²) / 1944 in³ / 27 times as big (3³)
4 m / 7 m / 8 m / 232 in² / 224 m3 / Multiply A & B by 4 / 1600 in² / 6.89655 times as big / 3584 in³ / 16 times as big (4²)
6 in / 5 in / 4 in / 148 in2 / 120 in³ / Multiply all edges by 3 / 1332 in² / 9 times as big (3²) / 3240 in³ / 27 times as big
2 ft / 10 ft / 7 ft / 208 ft2 / 140 ft³ / Multiply all edges by 4 / 3328 ft² / 16 times as big (4²) / 8960 ft³ / 64 times as big (43)

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 1 of 28

6-Sphere Fun

1. Use the sphere on the right to help complete the table. (SA = ; V = )

Radius / Surface Area / Volume / Change in radius / Change in diameter / Change in circumference / Change in area / Change in volume
4 in. / 64 in² ≈ 201.06in2 / 85 in3 ≈ 268.08in3 / Double radius
(2 times larger) / 2 times larger / 2 times larger / 22 = 4 times larger / 23 = 8 times larger
5.2 in / 108.16 in² ≈ 339.79in2 / 187.48 in³ ≈ 588.98in3 / 3 times larger / Triple diameter
(3 times larger) / 3 times larger / 32 = 9 times larger / 33 = 27 times larger
3.76 cm. / 56.5504cm² ≈ 177.66cm2 / 70.88 cm³ ≈ 222.67cm3 / 2 times larger / 2 times larger / Double
Circumference (2 times larger) / 22 = 4 times larger / 23 = 8 times larger
3 in / 36 in² ≈ 113.1 in2 / 36π in3 ≈ 113.1in3 / 3 times larger / 3 times larger / 3 times larger / 9 times larger / x by 2197
3 m. / 36 m² ≈ 113.1 m2 / 36 m³ ≈ 113.1m3 / 13 times larger / 13 times larger / 13 times larger / 132 = 169 times larger / 133 = 2197 times larger
K units / 4k²units2 / k3units3 / 20 times larger / 20 times larger / 20 times larger / 202 = 400 times larger / 203 = 8000 times larger

Teacher Ch 16 What’s in a Volume or Surface Area? 7/1/08 Page 1 of 28

7-The Box-it Company

1. The Box-It Company is producing a new design for one of its rectangular cardboard boxes. The area of the top of the box is 192 inches, the area of the front of the box is 192 inches, and the area of the side of the box is 144 inches.

What is the volume of the box? Be sure to label correctly. 2304in3

Show your work using words, numbers and/or diagrams.

Factors to get 144: 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16, 12x12

Factors to get 192: 1x192, 2x96, 3x64, 4x48, 6x32, 8x24, 12x16

Look for factors that you can find throughout all 3 side areas: the sides are 12x12, 12x16, and 12x16

Volume: 12in x 12in x 16in = 2304in3

2. The Box-It Company is producing another new design for one of its rectangular cardboard boxes. The area of the top of the box is 192 inches, the area of the front of the box is 108 inches2, and the area of the side of the box is 144 inches.

What is the volume of the box? Be sure to label correctly. 1728in3

Show your work using words, numbers and/or diagrams.

Factors to get 108- 1x108, 2x54, 3x36, 4x27, 6x18, 9x12

Factors to get 144- 1x144, 2x72, 3x48, 4x36, 6x24, 8x18, 9x16, 12x12

Factors to get 192- 1x192, 2x96, 3x64, 4x48, 6x32, 8x24, 12x16

Look for factors that you can find throughout all 3 side areas: the sides are 9x12, 9x16, and12x16

V = (9in)(12in)(16in) = 1728in3

3. The company decides to double all of the dimensions of the box from problem 2.

a. What happens to the surface area of the box? The SA will be 22 = 4 times larger. So

the new surface area is 4• 888in2 = 3552in2.

Show your work using words, numbers and/or diagrams.

Original SA =2({l • w} + {l • h} + {w • h}) = 2 lw rectangles + 2 lh rectangles + 2wh rectangles

= 6 faces

Doublingallthe dimensions:

New SA = 2(2l •2w +2l • 2h + 2w • 2h) = 2(4{l •w} + 4{l •h} + 4{w •h}) = 4 • 2({l • w} + {l • h} + {w • h}) = 22• Original SA = 4• Original SA

b. What happens to the volume of that box? The volume will be 23 = 8 times larger. So

the new volume is 13,824in3

Show your work using words, numbers and/or diagrams.

Original V = lwh

Doubling the dimensions:

New V = 2l2w2h = 23(lwh) = 8(lwh) = 8• Original V 8• 1728in3 = 13,824in3

4. A firework explodes and is spreading from its center in a spherical pattern. The sphere currently covers a surface area of 240 sq. meters. The diameter of the firework is expected to double in the next 3 seconds. What will be the new surface area of the sphere? 22 = 4 times larger  4(240m2) = 960m2.

Show your work using words, numbers and/or diagrams.

5. A coffee shop is having rectangular doughnut boxes made. The small box is 11” by 17” while the larger one is twice as long and twice as wide, but the same depth. The box-making company charges by the cubic inch for boxes; the cost of the smaller box is $0.45 per box.