Surface Area 1 - Morell Regional High School Technology · 1.3 use formulas to determine the surface area of rectangular and triangular prisms, pyramids, and cylinders 1.4 use formulas

1General OutcomeDevelop spatial sense through direct and indirect measurement.

Specifi c OutcomeM1 Solve problems that involve SI and imperial units in surface area

measurements and verify the solutions.

General OutcomeDevelop algebraic reasoning.

Specifi c OutcomesA1 Solve problems that require the manipulation and application of formulas

related to:surface area

A3 Solve problems by applying proportional reasoning and unit analysis.

General OutcomeDevelop number sense and critical thinking skills.

Specifi c OutcomesN1 Analyze puzzles and games that involve numerical reasoning, using

problem-solving strategies.

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Surface Area

978-1-25-901239-6 Chapter 1 Surface Area • MHR 1

By the end of this chapter, students will be able to

Section Understanding Concepts, Skills, and Processes

1.1 draw nets of 3-D objects�

determine the area of the 2-D shapes that make up a 3-D object�

determine the surface area of 3-D objects�

1.2 use length references to estimate the dimensions and the surface area of an object�

use area references to estimate the surface area of an object�

1.3 use formulas to determine the surface area of rectangular and triangular prisms, pyramids, and cylinders

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1.4 use formulas to determine the surface area of cones and spheres�

2 MHR • Math at Work 11 Teacher’s Resource 978-1-25-901239-6

Chapter 1 Planning Chart

Section/Suggested Timing Prerequisite Skills Materials/Technology

Teacher’s ResourceBlackline Masters

Chapter Opener30–45 min

(TR pages 7–8)•

Students should be familiar with2-D shapescalculating the area of shapesfi nding missing side lengths of right trianglescircumference of a circlerelationship between radius and diametermultiplying fractions by whole numberscalculating area of rectangles

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BLM 1–1 Chapter 1 Self-Assessment

Get Ready45–60 min

(TR pages 9–10)•

Students should be familiar withdiff erence between SI and imperial systems of measurementlinear conversionsolving basic proportionswriting mixed numbers as improper fractionsmultiplying fractions by whole numberscalculating the area of various 2-D shapesequivalent fractionsPythagorean relationshiporder of operations

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calculator•

1.1 Nets and Surface Area of 3-D Objects

120–140 min(TR pages 11–19)•

Students should be familiar with2-D shapescalculating the area of shapesfi nding missing side lengths of right trianglescircumference of a circlerelationship between radius and diameter

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calculatorcardboard boxrulerscissorsgrid papertape

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Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid Paper

Master 4 1 _ 4 Inch Grid Paper

BLM 1–2 Chapter 1 Warm-UpBLM 1–3 Nets of 3-D FiguresBLM 1–4 3-D FiguresBLM 1–5 Section 1.1 Extra Practice

1.2 Estimating Surface Area

120–140 min(TR pages 20–29)•

Students should be familiar withmultiplying fractions by whole numberscalculating area of rectangles

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ruler or measuring tape• BLM 1–2 Chapter 1 Warm-UpBLM 1–6 Section 1.2 Extra Practice

1.3 Using Formulas for Surface Area of 3-D Objects

160–180 min(TR pages 30–43)•

Students should be familiar withorder of operationsPythagorean relationshiparea of 2-D shapescircumference of a circlerelationship between radius and diameter of a circle

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boxrulergrid paper

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1.4 Surface Area of Cones and Spheres

90–120 min(TR pages 44–52)•

Students should be familiar witharea of circlePythagorean relationshiprelationship between diameter and radiuspercent of a number

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cone-shaped paper cupsrulergrid paperscissorstennis ball or other small ball

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Exercise Guide Extra Support

Assessment

Assessment as Learning

Assessment for Learning

Assessment of Learning

Math at Work 11 Online Learning Centre

TR page 6 TR page 6

Adapted: #3, #4, #7, #9Typical: #1–#4, #7, #9

TR page 10

Adapted: Explore #1–#4; On the Job 1 #5, #6; On the Job 2 #4; Work With It #1, #2, #4, #7Typical: Explore #1–#5; On the Job 1 #6, #7; On the Job 2 #5; Work With It #1, #3, #5, #7, #8


TR pages 13, 19 TR pages 14–16, 18

Adapted: Explore #1–#3; On the Job 1 #5, #7; On the Job 2 #8; Work With It #1, #3, #6Typical: Explore #1–#4; On the Job 1 #5, #7, #8; On the Job 2 #8, #9; Work With It #1–#3, #5, #6


TR pages 22, 29 TR pages 23–25, 28

Adapted: On the Job 1 #3; On the Job 2 #3; On the Job 3 #3; On the Job 4 #3; Work With It #1, #3, #4, #6, #8Typical: On the Job 1 #3, #5; On the Job 2 #3, #5; On the Job 3 #3, #5; On the Job 4 #3, #4; Work With It #1–#4, #6, #8, #10, #11


TR pages 32, 43 TR pages 33, 34, 36–41

Adapted: On the Job 1 #6; On the Job 2 #5; Work With It #3, #6Typical: On the Job 1 #6; On the Job 2 #2, #5; Work With It #3–#6


TR pages 47, 52 TR pages 49–51


Section/Suggested Timing Prerequisite Skills Materials/Technology

Teacher’s ResourceBlackline Masters

Chapter 1 Skill Check

45–60 min(TR page 53)•

rulergrid papercalculator

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BLM 1–1 Chapter 1 Self-Assessment BLM 1–3 Nets of 3-D FiguresBLM 1–5 Section 1.1 Extra PracticeBLM 1–6 Section 1.2 Extra PracticeBLM 1–7 Section 1.3 Extra PracticeBLM 1–8 Section 1.4 Extra Practice

Chapter 1 Test Yourself

45–60 min(TR pages 54–55)•

rulergrid papercalculator

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BLM 1–1 Chapter 1 Self-AssessmentBLM 1–3 Nets of 3-D FiguresBLM 1–9 Chapter 1 Test

Chapter 1 Project120–150 min

(TR pages 56–57)•

cardboardcoloured paperscissorsglue and/or tapegrid paperruler

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Master 1 Project RubricMaster 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid Paper


BLM 1–3 Nets of 3-D FiguresBLM 1–10 Chapter 1 Project

Checklist

Chapter 1 Games and Puzzles

60–90 min(TR page 58)•

thick red paper and blue paperrulerscissorsgluemarkerdicecalculator

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BLM 1–11 Chapter 1 BLM Answers


Exercise Guide Extra Support

Assessment




Have students do at least one question related to any concept, skill, or process that has been giving them trouble.

TR page 53

Provide students with the number of questions they can comfortably do in one class. Choose at least one question for each concept, skill, or process.Minimum: #3, #4, #7, #8

TR page 55 TR page 55BLM 1–9

Chapter 1 Test

TR page 57Master 1 Project

Rubric

TR page 58


Assessment Supporting Learning


Use the Before column of BLM 1–1 Chapter 1 Self-Assessment to provide students with the big picture for this chapter and help them identify what they already know, understand, and can do. You may wish to have students keep this master in their math portfolio and refer back to it during the chapter.

During work on the chapter, have students keep track of what they need to work on. They can check off each item as they develop the skill or process at an appropriate level.

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Method 1: Use the Get Ready on pages 4–5 in Math at Work 11 to activate students’ prior knowledge about the skills and processes that will be covered in this chapter.Method 2: Use the visuals and introduction on pages 2–3 in Math at Work 11 to activate student knowledge about the skills and processes that will be covered in this chapter.Method 3: Have students develop a journal entry to explain what they personally know about area and what they know about jobs, careers, or hobbies that involve estimating and calculating area using SI and imperial measurement.

Have students use their list of what they need to work on to keep track of the skills and processes that need attention. They can check off each item as they develop the skill or process at an appropriate level.

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As students work on each section in Chapter 1, have them keep track of any problems they are having.

As students complete each section, have them review the list of items they need to work on and check off any that have been handled.Encourage students to write defi nitions for the Key Words in their own words, including reminders and tips that may be helpful for review throughout the chapter.

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BLM 1–2 Chapter 1 Warm-UpThis reproducible master includes a warm-up to be used at the beginning of each section. Each warm-up provides a review of prerequisite skills needed for the section.

As students complete questions, note which skills they are retaining and which ones may need additional reinforcement.Use the warm-up to provide additional opportunities for students to demonstrate their understanding of the chapter material.Have students share their strategies for completing math calculations.

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What’s AheadIn section 1.1, students learn how to calculate the surface area of prisms and cylinders. Students use two-dimensional (2-D) nets to help them fi nd the area of each face. Th ey discover that the total area of all of the faces is the surface area of the three-dimensional (3-D) fi gure.

In section 1.2, students learn how to estimate surface area. Th ey develop references to help them with their estimation skills. References for linear and area measurements help students estimate the area of various real-world objects.

In section 1.3, students develop formulas to determine the surface area of a rectangular prism, triangular prism, square-based pyramid, and cylinder. Students then use their formulas to determine the surface area of those fi gures. Once students become comfortable with the formulas, they answer questions in which they must make adjustments to the formulas based on missing sides or surfaces that do not need to be covered.

In section 1.4, students explore the formulas for calculating the surface area of a cone and a sphere. Th ey learn that they can calculate the total surface area of a cone by fi nding the sum of the area of the curved surface and the area of the base. Students use the formulas to determine the surface area of various cones and spheres in real-world contexts.

As you complete each section with your class, it might be helpful to summarize the formulas and procedures covered in the section.

Planning NotesStart by discussing the diff erence between 2-D and 3-D fi gures. Students should see the connection that all 3-D shapes are comprised of 2-D shapes.

Students may fi nd it challenging to visualize that the curved surface of a cylinder is actually a rectangle. You may wish to have paper towel rolls or some similar objects on hand to cut and show students that the middle circular section rolls out to become a rectangle.

As a class, go through the Key Words. Encourage students to defi ne them in their own words and to provide an example of each one. Going through this list as a class will allow you to gauge students’ prerequisite knowledge. Create cards with these Key Words and examples, and place them around the room for reinforcement throughout the chapter.

Go through each of the photos and discuss how knowing how to calculate surface area would be an asset for each job. For example, to tile a bathtub you would need to know how many tiles to buy to cover the wall; to build a greenhouse, you would need to determine how much glass to buy.

Move the discussion to the types of work that involve surface area calculations. Examine the Career Link and discuss how surface area would be used when painting an oil tank.

Meeting Student NeedsFor the introductory activity, it may be helpful for students to build their own model of a cylinder using construction paper. In this way, students can more easily visualize the 2-D shapes that make up this 3-D fi gure. Have students work in pairs to speed up the process and allow for dialogue with a partner.

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Math at Work 11, pages 2–3

Suggested Timing30–45 min

Blackline MastersBLM 1–1 Chapter 1

Self-Assessment

Key Wordsrectangular prism

net

surface area

triangular prism

cylinder

diameter

radius

square-based pyramid

slant height

cone

sphere


Due to the emphasis on measurement in this chapter, it may be worthwhile to have students exposed to some measuring tools from the workplace, such as vernier callipers and micrometers.Since calculating area is crucial to determining surface area, consider having students make cue cards outlining how to calculate the area of basic 2-D shapes. Th is will be helpful to those who are struggling with this concept or with remembering the area formulas.For students who fi nd it challenging to determine surface area, provide pre-made nets for rectangular and triangular prisms (see BLM 1–3 Nets of 3-D Figures). Assist students in labelling all of the dimensions on the net. Th en, have them calculate the area of each shape right on the net. Some students may fi nd it easier to work from a 3-D model. Have available pre-made rectangular and triangular prisms. Help students label all of the dimensions on each 3-D object, and then have them do their calculations directly on the object.

ELLEncourage students to defi ne all the Key Words in a special section of their notebook. Defi nitions should be in simple language and contain an example or diagram illustrating the meaning of the word. Have students refer to these terms throughout the chapter. Have them add new words as the chapter progresses.Remind students of the various ways of representing units of measure: cm/centimetre, foot/ft /′, and so on.Remind students that square units are used to represent area. Remind them of the various ways of representing square units, such as square inch/sq. in./in.2 and square metre/m2.

Gifted and EnrichmentChallenge students to develop a formula to calculate the surface area of

a rectangular prism with length l, width w, and height han isosceles triangular prism with triangle base b, height h, slant height s, and length of the rectangular section, l

Career Link

There are many careers in the oil fi eld. Training is available online and through distance learning, conferences, and courses. To get information about a career in the oil fi eld and the training needed, go to www.mcgrawhill.ca/school/learningcentres and follow the links.

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Web LinkTo learn more about exterior cleaning of industrial storage tanks, go to www.mcgrawhill.ca/school/learning centres and follow the links.

Tools of the Trade

A micrometer is used to make precise measurements of small distances in a number of mechanical trades. A vernier calliper is used to measure the distance between two opposite sides of an object, and is used in many trades, such as metalworking and woodworking.


1Get Ready

Category Question Numbers

Adapted (minimum questions to cover the outcomes) #3, #4, #7, #9

Typical #1–#4, #7, #9

Planning NotesMethod 1: You may wish to assign all of the questions in the Get Ready as a means of preparing for the chapter. Correct and review the questions with the class, ensuring that everyone is comfortable with the skills and knowledge needed to complete the questions.

Method 2: Based on the class discussion of the Key Words, you may decide that your students need to complete only part of the Get Ready. Select questions that specifi cally meet the needs of the class and their understanding of the prior material.

Method 3: Look through the prerequisite skills for the Get Ready listed in the Planning Chart on page 2 of this Teacher’s Resource. Have students work in pairs to create a study sheet outlining how to perform each skill. If students are able to explain each concept and include an example, they do not need to complete the Get Ready to show their understanding of these concepts.

Meeting Student NeedsIn #1b), where the thickness of a coin is mentioned, you may wish to introduce the use of a micrometer and a vernier calliper.In #6, discuss the equivalence of various proportional statements. Have studentsexplore if they would get the same answers for 3 _ 8 = x _ 56 and 3 _ x = 8 _ 56 .

ELLStudents may be less familiar with imperial units and their origins. Take the time to explain the origins and history of this system, and discuss why the SI system is currently used by the majority of countries.Discuss with students that the prime symbol (′) represents feet and the double prime symbol (�) represents inches.

Gifted and EnrichmentDiscuss the answer to #4b). Students will express their answer as either 3.5 feet or3 1 _ 2 feet. Ask them to write their answer in a way diff erent from the form they used.

Challenge students who are attempting #7c) to determine at least two diff erent strategies for calculating the area of the shape.

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Materialscalculator

Mathematical Processes Communication (C)

Connections (CN)

Mental Math and Estimation (ME)

Problem Solving (PS)

Reasoning (R)

Technology (T)

Visualization (V)

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Common ErrorsStudents may have forgotten the diff erence between imperial and SI systems of measurement.

Rx Post examples around the classroom for student reference. Remind students that the SI system is used in Canada. Give them examples of SI units of distance: centimetres, metres, and kilometres. Tell them that imperial units are used in the U.S. Give them examples of imperial units of distance: inches, feet, yards, miles.

Students may not remember how to calculate the area of various 2-D shapes.Rx Have groups of students make cue cards for some of the basic shapes. Each

group is responsible for completing the cue card for one shape. Th ey must include a picture of the shape, the area formula for the shape, and an example of how to calculate the area. Each group can then share its card with the class.

Some students may have forgotten how to use and apply the Pythagorean relationship.

Rx Remind students of the relationship among the sides of a right triangle: the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

a2 + b2 = c2



Get ReadyHave students complete the Get Ready exercise on pages 4–5 in Math at Work 11.

Have students use their math journal to keep track of the skills and processes that need attention. As they work on the chapter, they can check off each item as they develop the skill at an appropriate level.Encourage students to record on cue cards how to determine the area of various 2-D shapes. Suggest that they keep the cue cards clipped inside the cover of their notebook. They can add notes to them as they work through the chapter.

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b2

a2

c2


1.1Nets and Surface Area of 3-D Objects


Adapted (minimum questions to cover the outcomes) Explore #1–#4On the Job 1 #5, #6On the Job 2 #4Work With It #1, #2, #4, #7

Typical Explore #1–#5On the Job 1 #6, #7On the Job 2 #5Work With It #1, #3, #5, #7, #8

Planning NotesHave students complete the section 1.1 warm-up questions on BLM 1–2 Chapter 1 Warm-Up to reinforce prerequisite skills needed for this section.

Prior to introducing section 1.1, ask students to bring in a product to show its packaging. Encourage students to bring in shapes other than rectangular prisms. Discuss the various 3-D shapes and see which are most commonly used.

You might want to have several examples of nets on hand. You may wish to provide BLM 1–3 Nets of 3-D Figures. Students could cut out the nets and fold them together to form 3-D fi gures. Hang the 3-D fi gures around the classroom with their name labelled on them. Th is could open up a discussion about how 3-D objects are classifi ed by diff erent names. For example, discuss the diff erence between prisms and pyramids or cones and cylinders.

Explore the Net and Surface Area of a Rectangular PrismIn this exploration, students explore how to calculate the surface area of a rectangular prism. You may wish to have students label the dimensions on the box using a marker. Th is will enable them to clearly see which pairs of faces have identical measurements. Consider providing students with Master 2 CentimetreGrid Paper, Master 3 0.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper to draw their nets.Once students have identifi ed the pairs of faces that are identical in step 1, ask them if this is always the case for rectangular prisms. How many pairs of faces does a rectangular prism have?For step 2, ask students whether they will measure the dimensions in SI or imperial units. Which measurement units would make the most sense? Ask them to justify their choice.In step 3, by cutting the box open, they will see the relationship between the 3-D fi gure and the 2-D net. Have students label the dimensions and calculate the area of the faces directly on the net.For step 4, have a class discussion about students’ results in case their calculation in step 2b) does not match their calculation in step 3c).



Materialscalculatorcardboard boxrulerscissorsgrid papertape

Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5 Centimetre

Grid Paper


BLM 1–2 Chapter 1 Warm-UpBLM 1–3 Nets of 3-D FiguresBLM 1–4 3-D FiguresBLM 1–5 Section 1.1 Extra

Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)

Specifi c OutcomesM1 Solve problems that involve SI and imperial units in surface area measurements and verify the solutions.

A1 Solve problems that require the manipulation and application of formulas related to:

surface area


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Have a class discussion about students’ results in step 5c) in case their measurements do not match the given dimensions. Discuss why this may have happened.

Meeting Student NeedsEncourage students to try using imperial units in this exploration to give them an opportunity to become more familiar with this system of measurement.Divide the class into groups and have them measure the same box with diff erent SI units and imperial units in order to look for connections within and between the systems of measurement.

ELLSince the word net may be confusing, emphasize the defi nition in the margin and give students multiple chances to work with and build their own nets. Ask students if the net in the margin is the only possible net for a cube.

Gifted and EnrichmentChallenge students to create as many nets as possible for one rectangular prism.Give students a rectangular prism, or the net of a rectangular prism, that has the side lengths labelled with variables. Challenge them to develop the formula for the surface area of any rectangular prism using these variables.

Common ErrorsStudents may forget to use square units for area.

Rx Remind students that linear measurements, such as length and width, are one-dimensional (1-D). Linear measurements can be measured with a ruler. Area is a 2-D measurement made up of length and width. So, area is measured in square units.

Answers

Explore the Net and Surface Area of a Rectangular Prism

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1. b) Opposite sides will be identical. Example: 1 and 6, 2 and 5, 3 and 4

2. a) Example:

First Area# of

Identical FacesTotal Area of

Identical Faces

72 cm2 2 144 cm2

48 cm2 2 96 cm2

96 cm2 2 192 cm2

b) Example: 432 cm2

3. b) Example: large rectangle: 12 cm × 28 cm = 336 cm2; top rectangle: 6 cm × 8 cm = 48 cm2; bottom rectangle: 6 cm × 8 cm = 48 cm2

c) 432 cm2

4. Th ey are the same.5. a)

c) Th e measurements should be the same.d) Example: staples




Refl ectStudents should see the connection between the sum of the area of the faces on the 3-D fi gure and the area of the 2-D net. You might want to ask students if they can derive a formula for the surface area of a rectangular prism.

Rather than students fi nding the area of all six faces, they should recognize that there are three pairs of faces. If students prefer, they can fi nd the total area of all six faces, but encourage them to double the total area of the three diff erent faces.You may wish to have students make the 3-D fi gure again once each 2-D shape has been labelled with its area. This might help students see the pairs of sides that are equal: top and bottom, front and back, and the two sides.

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Extend Your UnderstandingOnce students understand the relationship between the 3-D object and its 2-D net, have them create a 2-D net to build a 3-D shape.

Remind students that the net in step 3 consisted of three pairs of faces. When they are drawing the net, encourage students to label the pairs of identical faces: front and back, top and bottom, and side 1 and side 2. Have a 3-D fi gure on hand to show students so that they can see how the sides relate to one another.

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On the Job 1Ask students the diff erence between a triangular prism and a rectangular prism. Students should recognize that there is a diff erence in the number of faces and their shapes. Tell them that prisms are named by the shape of their base. Students might need help visualizing that a triangular prism has its name because the base is triangular.

Encourage students to imagine that the base of the prism (side 3 on the diagram in the student resource) is fastened to the ground and that the tent has collapsed around the base. Make sure students label all of the dimensions needed to fi nd the area. Ask students:

Are there any pairs of sides that are the same shape and size?Would 102 ft 2 of material be suffi cient to make the tent? What about overlap or seams?

Meeting Student NeedsStudents may need to be reminded of the formula for calculating the area of a triangle. Start with a simple right triangle on grid paper and help them compare it to a rectangle with the same length and width. See if they can determine the formula by recognizing that a triangle has half the area of a rectangle with the same dimensions. Th en, show students a triangle on grid paper with an obtuse angle. Place them in groups to discuss whether the same formula for the area of a triangle applies.Some students have diffi culty creating the nets of 3-D objects to calculate the surface area. Provide them with pre-made nets for the various fi gures. For example, students could take the template of a net for a triangular prism, label the dimensions on the net, and then calculate the area of each face. Provide students with BLM 1–3 Nets of 3-D Figures.Some students may prefer to make a large 3-D model, label the dimensions, and then calculate the area of each face. Provide them with BLM 1–4 3-D Figures.

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Gifted and EnrichmentPose the following problem as a challenge to students: A tent in the shape of a triangular prism needs to hold two sleeping bags that are 2 ft by 6 ft . Th e tent will have a height of 6 ft . What is the least amount of material that a tent can have that will fi t the two sleeping bags? (If the base of the triangle is 6 ft and the length of the triangular prism is 4 ft , the surface area is approximately 114 ft 2, which is the least amount of material the tent can have.)Encourage students to try to develop a formula to calculate the surface area of a triangular prism.

Answers

On the Job 1: Your Turna)

b) 74 cm2



On the Job 1Have students do the Your Turn by drawing a net and then calculating the area of each face, or by calculating the area of each face using the given 3-D model. Check that students calculate the area of three rectangular faces and two triangular faces. Ensure that students use the correct units in their answer.

Encourage students to draw the net of the triangular prism and label all the dimensions. Ask students which sides have equal dimensions and, therefore, the same area.Encourage strong students to use the least number of calculations necessary to determine the total area.

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Check Your UnderstandingTry ItFor #2, ask students to research the most common units used to measure these items. Ask them to look at fl yers and see the units used to advertise the size of computer screens. Do maps and atlases use the same units to express the surface area of lakes? Encourage them to compare answers.

Aft er completing #3, you might also want to work through examples with students of using the Pythagorean relationship to fi nd the slant height of a triangle, which will be useful later in the chapter. You could also provide an example of a triangle with the slant height given, and work through how to use the Pythagorean relationship to determine the height. Th is will be helpful for students when they work on #8.

For #5, provide students with grid paper to use when drawing the net. You may wish to hand out Master 2 Centimetre Grid Paper or Master 3 0.5 Centimetre Grid Paper.

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3.5 cm5 cm

4 cm

4 cm

3.5 cm5 cm

4 cm

4 cm


Apply ItFor #6, make sure students understand that the tent has a fl oor.

For #8, you may need to assist students in using the Pythagorean relationship to determine the height of the triangular face. Divide students into groups to discuss what shape of packaging would minimize the amount of material needed to package the three golf balls. Ensure students realize that in part d) they need to calculate the total area of cardboard needed for a rectangular prism that would hold three golf balls.

Meeting Student NeedsELL

Ensure students know what each of the objects in #2 and #4 are. Make available a photo of each item for student reference.

Common ErrorsSome students may not understand the diff erence between height and slant height.

Rx Draw a triangle on the board and ask students to label the height, base, and slant height. Remind students that the height of the triangle is the perpendicular distance from the highest point of the triangle to its opposite side. Th e slant height is the distance from the highest point of the triangle to the edge of its base.

Some students may think that triangular prisms have six faces.Rx Draw the triangular prism resting on the triangular base. Students will

then see that the top and bottom are triangles (two faces) and that there are three rectangular faces connecting each side of the triangular bottom to the triangular top (2 + 3 = 5 faces).



Try ItStudents should be able to correctly answer #3 before moving on to #5, #6, and #7.

For #8, students need to apply the Pythagorean relationship to fi nd the length of the leg of a right triangle given its hypotenuse and other leg.You may wish to have students practise fi nding the slant height of a triangle when given the triangle’s base and height.

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On the Job 2Students will likely understand that fi nding the surface area of a cylinder includes fi nding the area of the top and bottom circles. Th ese circles are the same, so fi nding the area of one and multiplying by 2 gives the area of the two circles combined. Some students may fi nd it challenging to determine the area of the curved surface. To help in their understanding, bring in some models of cylinders so that students can see that the curved surface of a cylinder is a rectangle. You can also demonstrate by using a piece of paper rolled into a cylindrical shape. Emphasize that the length of the rectangle is the circumference of the circle and the width of the rectangle is the height of the cylinder. Ensure students recall the defi nition of circumference and the formula for the circumference of a circle.

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Meeting Student NeedsFor students have diffi culty creating the nets of 3-D objects, provide them with a pre-made net of a cylinder. For example, students could take the template of a net for a cylinder, label the dimensions on the net, and then calculate the area of each face. Provide them with BLM 1–3 Nets of 3-D Figures.

Gifted and EnrichmentEncourage students to try to develop a formula to calculate the surface area of a cylinder using the variables r and h.

Answers

On the Job 2: Your Turna)

b) ≈ 240 cm2



On the Job 2Have students do the Your Turn by drawing a net and calculating the area of each face, or by calculating the area of each face using the given 3-D model. Check that students are calculating the area of the rectangle and the two circular faces. Ensure that students use the correct units in their answer.

Encourage students to draw the net of the cylinder and label all of the dimensions: the height of the cylinder, the radius of the circular bases, and then the circumference of the circles. Ask students which sides have equal dimensions and, therefore, the same area.Encourage strong students to use the least number of calculations necessary to determine the total area.

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Check Your UnderstandingTry ItFor #1, ask students what to do if they are given the diameter of the base rather than the radius. Which of the calculations will this aff ect?

For #3, ask students what happens if you are given the radius and not the diameter. Can the formula be revised? How?

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4 cm

9 cm

9 cm

4 cm

9 cm

9 cm


Apply ItFor #4, ask students if they need the net of a cylinder to calculate the answer to this question. If they are using the net, which faces do they not need to calculate the area of?

For #5, ask student what part of the net the label is. Ask students to predict how the thickness of the plastic might aff ect the area needed to make one container.

For #6, ask students if they think it is best to convert square inches to square feet or convert the measurements from inches to feet and then fi nd the area. You may wish to have them try both methods to see if they arrive at the same answer. Th en, ask them which method they prefer.

Meeting Student NeedsMake sure students know the diff erence between the diameter and radius of a circle. Ask students when they need each dimension. Ask students if it is possible to calculate the surface area of a cylinder using only the radius or only the diameter.Students may struggle with shapes that are not linear in nature. Start with a tennis ball canister and ask them which is longer, the distance around the top of the can or the height of the can. Have a student come up and measure with a string.For students who are having diffi culty visualizing the net of a cylinder, use a cylinder made of construction paper to show students the three pieces.Give students an opportunity to recall and practise using the formulas for the circumference and area of a circle.When drawing a net of a cylinder, ask students whether it matters where you draw the two circles above and below the rectangular area. Again, this may improve their confi dence with visualization.

Gifted and EnrichmentIf students devised a formula for calculating the surface area of a cylinder, have them test their formula on questions for which they already found the area using a net. Which method do they prefer? What are the pros and cons of each method?Ask students to predict what the net of a pentagonal prism, hexagonal prism, and octagonal prism would look like.Discuss that the formulas for area and circumference are usually written in terms of the radius. Challenge students to write these formulas in terms of the diameter instead. Ask: Do you prefer diff erent formulas depending on whether you are determining circumference or area?

Common ErrorsWhen converting area from one unit to another, students may misinterpret based on the linear model, such as 1 ft 2 = 12 in.2.

Rx To help in their understanding, draw diagrams, such as a square that is 1 ft by 1 ft and made up of 144 squares. You may also wish to show students the correct conversion by working with linear dimensions. Example:1 ft = 12 in.1 ft 2 = 12 in. × 12 in. = 144 in.2

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Students may have diffi culty calculating the area of a circle correctly when given the diameter instead of the radius.

Rx Explain that if the question gives the diameter, students need to divide the diameter by 2 to fi nd the radius. Th en, they can use the radius in the formula.

Students may have diffi culty calculating the circumference of a circle correctly when given the radius.

Rx Students may be familiar with only one formula for determining the circumference of a circle: C = πd. Explain that since the radius multiplied by 2 equals the diameter, students can use C = 2πr when given the radius.



Try ItStudents should be able to correctly answer #1, #2, and #3 before moving on to #4 and #5.

Students need to know when to use the formula for the area of a circle and when to use a formula for the circumference of the circle. Remind them that area is the amount of 2-D space an object covers. This is why we need to calculate the area of the circles at each end of the cylinder. Circumference is the distance around the outside of the circle. We need to calculate the circumference in order to know the length of the rectangle.

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Work With ItStudents have now completed On the Job 1 and On the Job 2 and the related Check Your Understanding questions. In the Work With It section, students have an opportunity to use the skills from both On the Job 1 and On the Job 2 in practical situations.

For #1, ask student by how much the surface area is increased if it is not doubled. Ask:What is the surface area if all of the dimensions are doubled?In this case, does the surface area double?

Students may want to fi gure out the above question using an example with actual dimensions.

For #2, students could calculate the area, multiply by 48, and then round. Th ey could compare this result to the one obtained by rounding the area and then multiplying by 48. You might wish to use this opportunity to discuss waste.

For #3, have students work in pairs. Th is will allow them to discuss how to calculate the area of the casing that wraps around the circles at each end.

For #4, if students seem to have a thorough understanding of the relationship between a triangular prism and its net, you may choose for them not to complete parts c) and d).

Discuss ItTh ese questions provide students with an opportunity to explain their understanding of surface area. Look for reasonableness and justifi cation of answers. Some students may benefi t from discussing the answers as a class before recording their own answers. Others may prefer to complete them individually in a journal.

In #5, students have an opportunity to demonstrate their understanding of the connection between the total area of the individual faces and the surface area of a 3-D shape.

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In #6, refer students back to On the Job 1 #7. Ask:What other shapes of packaging might a manufacturer use?Which shape would have the least amount of material waste?Which shape would result in the greatest amount of waste?

For #8, ask students:What is a cube? Is a cube a rectangular prism?Is a rectangular prism a cube?Can you fi gure out a way to calculate the surface area of a cube without using a net?How can knowing the area of one face of a cube help determine the total surface area?

Meeting Student NeedsProvide BLM 1–5 Section 1.1 Extra Practice to students who would benefi t from more practice.For students who are struggling, provide pre-made nets for rectangular prisms, triangular prisms, and cylinders (see BLM 1–3 Nets of 3-D Figures). Students can then label the dimensions and calculate the area of each face.Continue to encourage students to follow a logical sequence for solving word problems. Aft er they read and understand a problem, they should sketch a diagram (if applicable), estimate the answer, calculate the answer, and then check the reasonableness of the answer. Reinforce the importance of using estimation to help determine if a solution makes sense.

ELLTh e concept of the battery casing in #3 may be confusing and will likely have to be demonstrated to students. Discuss other types of situations in which an overlap of a surface area may be required, such as tabs for gluing together a box or seams on a tent.

Gifted and EnrichmentHave students sketch the net of the box in Work With It #1 with the top opening along the centre as in the visual.Challenge students to devise a formula to fi nd the surface area of each 3-D shape.Challenge students to calculate surface area without using a net.



Discuss ItThe purpose of these questions is for students to connect the ideas learned in this section.

Encourage students to explain their answers to a partner before recording them.Encourage students to use concrete examples as part of their explanations.Have students trade their work with a partner to receive constructive feedback.

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Estimating Surface Area


Adapted (minimum questions to cover the outcomes) Explore #1–#3 On the Job 1 #5, #7On the Job 2 #8Work With It #1, #3, #6

Typical Explore #1–#4 On the Job 1 #5, #7On the Job 2 #8, #9Work With It #1–#3, #5, #6

Planning NotesHave students discuss with a partner jobs that require exact measurements. In what types of jobs would accuracy be essential? For example, it is important to have exact measurements when building houses or road structures. Being inaccurate in these measurements could be a safety concern. Students could also discuss and brainstorm types of jobs in which estimation is used. For example, estimation skills are used in interior design and landscaping. Estimates are a good way to get started in determining the amount of materials and supplies needed. Consider using this as an opportunity to talk about underestimation and overestimation. Discuss situations in which each of these might be useful.

Have students complete the section 1.2 warm-up questions on BLM 1–2 Chapter 1 Warm-Up to reinforce prerequisite skills needed for this section.

Explore Length and Area ReferencesIn this exploration, students use various frames of reference to estimate the dimensions of many common items. Developing these estimation skills is useful for determining the reasonableness of measurements in the real world. Using your body as a frame of reference is a convenient way to estimate the height and width of everyday items.

For step 1, ask students: What are some common references on your body or in your surroundings that might help you estimate the dimensions of other objects? Have students work in pairs to measure various parts of their body: arm span, height, fi nger span, length of foot, width of hand, and so on. Talk to them about where they might use these measurements to estimate the height or width of items. Ask:

Which units will you use for these common body measurements: SI or imperial?Which units are easier for you, personally, to estimate?

For step 2, students can come up with everyday items as a frame of reference for common measurements in SI or imperial units. Have students try to develop the table individually and then compare with a partner. Note the diff erent frames of reference they come up with. You might want to create a class list of the references that work best for students and post it in the classroom so that they can refer to it when they are estimating measurements (e.g., 1 cm = width of pinkie fi nger, 1 ft = distance from shoulders to top of a person’s head).

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1.2Math at Work 11, pages 15–24


Materialsruler or measuring tape

Blackline MastersBLM 1–2 Chapter 1 Warm-UpBLM 1–6 Section 1.2 Extra

Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)

Specifi c OutcomeM1 Solve problems that involve SI and imperial units in surface area measurements and verify the solutions.

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Meeting Student NeedsFor students who are struggling with estimation, have them measure an item with a measuring tape. Th en, keeping that length on the measuring tape, walk around the classroom and fi nd other items that are close to that measurement. Students could also create their own measuring tape with their own frames of reference marked on it. For example, they could mark the 1-ft point on their tape and label it “foot-long sub,” and for the measurement of 165 cm, students could mark on the tape “my height.” Students could have a lot of fun creating this tape with their own meaningful measurements.Have students measure the length of a page using their pinkie fi nger, and compare measurements across the class to see the variation. Discuss the problem of using this technique to measure the distance across the classroom (measurement error and time).

ELLEncourage students to look at home for household items that would be useful references. Th is exercise will help promote a stronger vocabulary (“the length of an iron is about a foot”). Place these references on the class list as well.

Gifted and EnrichmentTh e more challenging references will be those for area. Encourage students to come up with references for 1 cm2 (e.g., the face of a die) or 1 sq. mi (e.g., the area of a city district).

Common ErrorsStudents sometimes confuse linear measurements with area measurements.

Rx Explain to students that linear measurements are used to measure the height/depth, length, or width of objects. You can use a ruler to make these measurements. Area measurements are used to measure the amount of 2-D space an object takes up. Th ese measurements cannot be made with a ruler.

Answers

Explore Length and Area References

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•Web LinkYou may wish to show students a web site that provides information on the SI system and SI conversion. The site includes a quick metric converter, common errors and how to avoid them, and even a cooking converter. Go to www.mcgrawhill.ca/school/learningcentres and follow the links.

1. Examples:a) Th e width of my pinkie fi nger is about

1 cm.b)

Reference SI Length Imperial Length

My hand width 10 cm 4 in.

My arm span 1.8 m 2 yd

My desk height 72 cm 2 ft, 4 in.

My foot length 25 cm 10 in.

2. Examples:a) A new pencil is about 6 in. long.b)

SI Length Reference

Imperial Length Reference

1 cm Length of keyboard key

1 in. Width of my watchband

50 cm Width of two textbooks

1 ft Floor tile

1 m Half the height of a door

1 yd Width of a single bed

3. Examples:a) length of keyboard key; area of

keyboard keyb) 1 ft 2: a fl oor tile; 1 m2: area of half a door

4. Examples: area of a parking space: 160 ft 2; area of a football fi eld: 7150 yd2; area of a residential lot: 7500 ft 2




Refl ectListen to how students approach the Refl ect question. How do they translate their references to1 cm to references for 1 cm2? To help students visualize, ask: If you have a square with side length 1 cm, what is the area of the square? (1 cm2)

You may need to help students who are having diffi culty with this question. Students need to take an object that measures about 1 cm and then imagine that this measurement is the side lengths of a square. Explain that the space that the resulting square would take up would represent 1 cm2. For example, a pinkie nail is about 1 cm wide and 1 cm long. Students could think of their entire pinkie nail as a reference for 1 cm2.

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Extend Your UnderstandingObserve whether students estimate the area of the parking space by using their reference for 1 ft2 or 1 m2, or whether they estimate the area by using their reference for 1 ft or 1 m and then calculating the area.

Some students struggle with estimation and visualization. For these students, suggest that they pace the parking space and use their steps to estimate the length and width. They could draw a sketch of the space, label the dimensions on it, and then calculate the area that way. Sometimes it seems overwhelming to estimate a large area. Breaking it down into simpler steps or parts can be helpful.

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On the Job 1Ask students how they might estimate the area of wall to be painted in the closet.

What reference would you use to estimate the dimensions?Would you want to overestimate or underestimate the area?What would happen if you overestimated the area to be painted?What would happen if you underestimated the area to be painted?

Work through Dylan’s solution with students. In particular, ask them how to calculate5 1 _ 2 × 8. Some students might suggest changing the mixed number to an improperfraction and the whole number to an improper fraction, and then multiplying the numerators and denominators together. Others might suggest changing the fraction to a decimal number and then multiplying. Work through both methods, as well as the method in the text, and discuss the pros and cons of each.

For the Your Turn question, you may wish to have students estimate for a room other than the classroom, since they will be estimating the classroom in the Check Your Understanding. Have students work in pairs. Th ey should use their own reference to estimate the dimensions of the room. Note whether students are using references for the dimensions and then calculating the area or using references for area. Since students will have diff erent ways to approach this question, it might be useful to share these methods as a class. Have students share their estimates also. How accurate are their estimates? If the estimates are not accurate, discuss possible reasons.

For the Puzzler, you may wish to provide students with three sizes of blocks to help them work on the solution. Aft er students have a chance to work on the Puzzler, have them share their solutions and strategies. Th en, discuss the solution as a class:Put one face of the 6-cm cube in contact with a face of the 8-cm cube. Th at will cover two areas that are the size of one face of the 6-cm cube. Put two faces of the 2-cm cube in contact with a face of the 6-cm cube and a face of the 8-cm cube.

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SA = surface area of 8-cm cube + surface area of 6-cm cube + surface area of 2-cm cube - (2 × area of face of 6-cm cube) - (4 × area of face of 2-cm cube)SA = 6(82) + 6(62) + 6(22) - (2 × 62) - (4 × 22)SA = 536 cm2

Th e least surface area is 536 cm2.

Meeting Student NeedsYou can help students estimate the area of the four walls by measuring in “feet.” Students can walk toe to heel along the four walls, counting how many steps it takes. Th is will give them the approximate length of the four walls in feet. Next, students could estimate the height of the walls by taking off their shoe and seeing how many times their shoe fi ts to the ceiling. Th is would give them the approximate height in feet. Th ey then have one simple calculation to fi nd the approximate area of all four walls.For students who fi nd it challenging to estimate area, provide them with hands-on references. For example, cut out a cardboard square that measures 1 ft by 1 ft . Th is would show students what 1 ft 2 looks like. Th ey could hold the cardboard square to the wall and imagine how many pieces it would take to fi ll the entire wall. Alternately, they could fl ip it along the fl oor and see how many cardboard squares fi t along the length of the wall and similarly up the height of the wall, and then they could estimate the area by multiplying the two dimensions. Since there are about 3 ft in 1 m, 1 m2 would measure about 3 ft by 3 ft . Th is means there would be about 9 ft 2 in 1 m2. Students could divide their estimate for the area of the wall in square feet by 9 to determine the area of the wall in square metres.

Answers

On the Job 1: Your TurnExample: 1000 ft 2



On the Job 1Have students complete the Your Turn question. Encourage them to use the measurement references they listed in the Explore.

Students should work with a partner and then share their strategies with the class. Allow students to estimate in whatever way they prefer, and then discuss the pros and cons of the methods they chose. It would be benefi cial to have an actual area measurement of the four walls of the room so students could see how close their estimates were to the actual measurement.

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Check Your UnderstandingTry ItFor #1 and #2, if student estimates are not close to their calculations, have them redo the Explore length and area references to improve their ability to estimate area. It is sometimes hard to estimate area, and improving their abilities now is important for when they need to use these skills in the future.

For #3, ask students how they are estimating. Ask them: Is it easier to estimate the area or estimate the dimensions and then calculate the estimated area?

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Apply ItFor #4, encourage students to estimate the area using a diff erent room and diff erent method from the Your Turn question. Ask them:

Do you prefer to estimate in square feet or square metres?Why does this unit of measure appeal to you?Which unit of measure would most likely result in an estimate further from the actual measurement?

When it comes time to measure the classroom walls using a measuring tape, have students work in pairs to increase the accuracy of their measurement.For #5, students may need assistance in understanding that they can use the length of 20 ft to help them estimate the width and height of the container.For #6, ask students what parts of the picnic table need to be covered in sealer. Have them create a net or a list of shapes and dimensions that need to be covered. Ask them to determine which shapes or parts of the table have the same dimensions and area. Th ey may need assistance in understanding that they can use the known dimension to help them estimate the width of the table and the dimensions of the legs. For example, “If the length of the table is 6 ft , the width looks to be about half of that, so it’s about 3 ft .”For #7, encourage students to draw a net of the prism. Ask them:

Which faces are the same?Do you have all of the dimensions you need to calculate the area of each side?How do you calculate the height of the triangle?Which unit of measurement will you use?

Note whether students gravitate toward SI or imperial units. Discuss why they prefer to work with their chosen unit of measurement.

Meeting Student NeedsAs a lead-up to #4, have students estimate the area of the four walls in your classroom. Have multiple groups use diff erent approaches. For example, have one group use a smaller SI reference. Have another group use a larger SI reference. Do the same with imperial references.Discuss which references work best for a task and why. Talk about whether it makes more sense visually to use a height reference (such as the height of a door) rather than a length reference (such as a fl oor tile) to measure a vertical distance. You may also have a debate as to whether using an area reference is superior to determining dimensions using a length reference and then calculating the approximate area.

Gifted and EnrichmentMore creative students have the opportunity to use a variety of references to answer questions such as #6 and #7. Have students share the diff erent references they use, which will result in a discussion benefi cial to all students.



Try ItStudents should be able to correctly answer #1 and #2 before moving on to the remainder of the questions.

Students need to have an established set of reference measurements to be able to estimate surface area. They could choose two common units of measurement for SI (such as centimetres and metres) and for imperial (such as inches and feet) and make sure they have reliable references for them. Give students plenty of opportunities to estimate dimensions and areas, since this is an important skill.

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On the Job 2Work through this question with students and have them evaluate Ria’s method. Initiate a class discussion by asking:

Do you like the way Ria estimated the area needed for fl ooring?What did you like about it?What, if anything, would you change about her method?Why did she get 24 cases? What is the benefi t of doing that?How does rounding up to 24 cases change Ria’s estimate?How much would you add on to account for errors in measuring or cutting?Would you add to the area of the fl oor, to the number of fl oor tiles, or to the number of boxes of fl oor tiles?

For the Your Turn question, have students work in pairs. Observe the various methods they use to calculate the area. Do students use the ceiling as a means to estimate? Do they estimate the dimensions of the room using their linear references and then calculate the area? Do they estimate the area of the room using their area references? Have students make their estimates and then share their methods and estimates with the class.

Have students measure the actual area of the fl oor with a partner. See how close students’ estimates were to the actual measurement. Which method resulted in the most accurate estimate? Discuss as a class why this method yielded the most accurate results. Brainstorm as a class how to improve the other methods.

Meeting Student NeedsSome students may fi nd it challenging to estimate area. Provide them with a hands-on reference, such as a square piece of cardboard that is 1 ft by 1 ft . Students could use it to measure the area of the wall or the dimensions of the wall.

Gifted and EnrichmentHave students think about various types of materials that could be used for a new classroom fl oor, such as carpet, laminate, linoleum, hardwood, and tile. Discuss the pros and cons of each type of fl ooring for a classroom. Students could then research the cost of each type of fl ooring and make a presentation with all the pros and cons and costs of each type. Have students recommend a fl ooring type based on all of these factors.

Answers

On the Job 2: Your TurnExample: 600 ft 2



On the Job 2Have students complete the Your Turn question. Encourage them to use the references for linear measurements or for area that they listed in the Explore.

Have students work with a partner and then share their strategies with the class. Allow students to estimate in whatever way they prefer and then discuss the pros and cons of the method they chose. Have on hand an actual area measurement of the classroom fl oor so students can see how close their estimates were to the actual measurement.

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Tools of the Trade

Many contractors buy 10% more fl oor tiles than they need for the estimated area. This is in case of errors in measuring or cutting.


Check Your UnderstandingTry ItFor #1, ask students what a square inch looks like. Consider having students cut out a model showing 1 in.2. Students might fi nd it easier to estimate each object if they use a hands-on model. Provide the actual area measurements for students to see how accurate their estimates are. Have students compare their estimates with a partner’s. Who was more accurate for each measure? How can they improve their estimates?

For #2, give visual and hands-on learners a sheet of 8 1 _ 2 � by 11� paper so they canplace it along the perimeter of each surface to estimate the dimensions and calculate the estimated area.

For #3, students may need help converting square inches to square feet. Since there are 12 in. in 1 ft , students may think there are 12 in.2 in 1 ft 2. Suggest they draw a model to help them in their understanding. Some students may need help determininga fraction of a whole number. Ask them how to determine 1 _ 2 of a number. Ask: How can you do that by multiplying? by dividing?

For #4, teach struggling students a trick to converting linear measurements. One way to remember the proper order of prefi xes is to make a sentence using the fi rst letter of each prefi x; for example,

King Henry Danced, Uncle Didn’t Care Much

Kilo Hecto Deca Unit Deci Centi Milli

To convert between measurements, identify the “start point” on the linear scale, where the given amount is, and the “endpoint,” where the amount you need to convert to is. Count the number of spaces and direction you need to move from start to end. Th en, on the given amount, move the decimal point that number of spaces in that direction. Place zeros in any empty spaces. Th e result is the converted amount.

Example: Change 4.5 m to millimetres.

Kilo Hecto Deca Unit (m)START

Deci Centi MilliEND

3 spaces right

4.5 m = 4500 mm

You can also use this idea to convert measurements of area. Since area has square units, move each decimal point double the number of spaces.

Example: Change 3.2 cm2 to square millimetres.

Kilo Hecto Deca Unit (m2) Deci CentiSTART

MilliEND

2 spaces right

3.2 cm2 = 320 mm2


For #7a), some students might fi nd it easier to answer this question using the same units. For example, they would determine the fraction of 1 ft 2 aft er converting the dimensions of the tile to feet, or they would determine the fraction aft er converting 1 ft 2 to square inches. Suggest they pick one unit or the other and draw a picture of the tile and the area. Th ey may wish to do the same for part b).

Apply ItStudents might need help with #8 since it is the fi rst time they have been asked to estimate the surface area of a cylinder. Th is would be a good question for them to work on in pairs or small groups. Suggest they use their model for square inch since the dimension is in inches. How many square inches fi t around the curved surface? Ask students if they have to calculate the area of the top and the bottom or if they see a shortcut.

Meeting Student NeedsReturn to #4 of Check Your Understanding for On the Job 1. Ask students to estimate the surface area of the four walls using an area reference.You may have a debate about whether an area reference is superior over a length reference for making surface area estimates. Encourage the class to suggest strengths and weaknesses of the two methods.

ELLShow students the multiple ways that surface area may be expressed, such as square foot, sq. ft , and ft 2. Reinforce that the exponent 2 refers to a square measure. Check that students understand that ′ represents foot and � represents inch.Students might not know what all the objects in Try It #1 look like. Have these items on hand for students to work with.

Gifted and EnrichmentChallenge students in Try It #2 to calculate the area of an 8 1 _ 2 � by 11� sheet of paper in square feet. Is the estimate of 1 1 _ 2 sheets being about 1 ft 2 a reasonable one?

Common ErrorsStudents may have diffi culty converting square inches to square feet.

Rx Show students a piece of paper that measures 12 in. by 12 in. Calculate the area of the paper (144 in.2). Th en, measure the dimensions of the sheet in feet. Students will see that the sheet measures 1 ft by 1 ft . Calculate the area (1 ft 2). Students will visually see that although the units have changed, the size of the paper has not changed. Th erefore, 1 ft 2 is equivalent to 144 in.2.

Students may fi nd it challenging to determine a fraction of a whole number.Rx Show students that 1 _ 2 of a number means dividing the number by 2, 1 _ 3 of a number means dividing the number by 3, and so on. Have them fi nd the pattern: 1 _ n of a number means dividing the number by n.

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Web LinkRefer students to a web site for some interesting math facts related to hockey pucks. Go to www.mcgrawhill.ca/school/learningcentres and follow the links.


In #7b), students may correctly convert the area of the mirror to 625 cm2 but then mistakenly believe that this is equivalent to 6.25 m2.

Rx Th ere are a number of ways to assist students in their thinking:Model a square with an area of 1 m2 on grid paper. Discuss that 1 m is equivalent to 100 cm, so the square is 100 cm by 100 cm, or 10 000 cm2. Th en, show what fraction of this is a square with an area of 625 cm2. It will become clear to students that 625 cm2 could not be equivalent to 6.25 m2 because the 625-cm2 square is much smaller than 1 m2.Have students convert 25 cm to metres fi rst, and then multiply to determine the area of the mirror. Th ey will see that the answer is 0.0625 m2, not 6.25 m2.Remind students how to use “King Henry Danced, Uncle Didn’t Care Much” to perform conversions involving square units.



Try ItStudents should be able to correctly answer #1 before moving on to answer #2, #5, and #6.

Provide students with a variety of models and visualization techniques to estimate area, since students will likely fi nd estimating area more diffi cult than estimating length and width.

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Work With ItFor #1, have students work in pairs. Make sure they use a room that has not been used for any other question in this section. Challenge some pairs to estimate the area of the four walls using linear references. Challenge other pairs to estimate using area references. Compare the results of the two diff erent methods. Discuss which method they think would be the most accurate.

For #2, students may need to be reminded that the bubble wrap covers both sides.

For #4, discuss with students the need to add on fabric to their estimation for the seam. Talk about what size seam they think is common for upholsterers to use. You might then have them research the actual size of seams.

Discuss ItFor #6, make sure students know that a tube is an open-ended cylinder. Discuss with them the diff erence in the units. Ask how they can convert to work with the same units. Would it make sense to work in feet or inches?

For #7, refer students back to On the Job 2. You might wish to have students contact a hardware store to fi nd out what their policy is on returning fl ooring. Have students contact a few stores to see how the policies vary. Ask students how the various store policies would aff ect their decision on where to buy fl ooring.

Meeting Student NeedsProvide BLM 1–6 Section 1.2 Extra Practice to students who would benefi t from more practice.For students who have diffi culty recording their thoughts, pair them with another student and ask them to describe their methods orally.

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ELLProvide an example of the items students are writing about. It is sometimes easier for students to write about an object if they have seen it.

Gifted and EnrichmentFor #2, have students estimate the amount of overlap needed to wrap the plate in bubble wrap.For #5, students could make a net, drawn to scale, for the amount of plastic wrap needed to wrap a DVD. Ask them to cut out this net and use it to wrap an actual DVD. Seeing how much material is needed for overlap will help strengthen students’ estimation skills.




Encourage students to explain their answers to a partner before recording them.Encourage students to use concrete examples as part of their explanations.Have students exchange their work with a partner to receive constructive feedback.

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1.3 Using Formulas for Surface Area of 3-D Objects


Adapted (minimum questions to cover the outcomes) On the Job 1 #3On the Job 2 #3On the Job 3 #3On the Job 4 #3Work With It #1, #3, #4, #6, #8

Typical On the Job 1 #3, #5On the Job 2 #3, #5On the Job 3 #3, #5On the Job 4 #3, #4Work With It #1–#4, #6, #8, #10, #11


To facilitate this discussion, bring in four or fi ve examples of rectangular prisms (tissue box, shoe box, cereal box, etc.). Ask students the process for determining the surface area of each box. Regardless of the size of the box or the dimensions, students should be able to convey that they need to determine the area of each face and then add to calculate the total area of all faces. Ask them if this is also the case for a triangular prism and for a cylinder. Students should see that while the shape of each face may diff er from one 3-D fi gure to another, the process for fi nding surface area remains the same.

Explore Formulas for the Surface Area of Rectangular PrismsTh e purpose of this Explore is for students to derive the formula for calculating the surface area of a rectangular prism. It may be benefi cial for students to have masking tape and a marker to label the dimensions of the box they are working with.

For step 1, ask students to list the pairs of faces that are the same. To make it easier, they can label the faces from 1 to 6. Ask them to identify the pairs of faces using numbers fi rst. Th en, you may wish to ask them to label the faces front, back, top, bottom, and side (both sides). It might be a good idea for students to become comfortable thinking of a rectangular prism as a set of three pairs of identical faces: front/back, top/bottom, and side/side.

For step 3, ask students what shape the top/bottom of the prism is. Ask them how to fi nd the area of this shape, so that they come to understand that the area of one face is length × width. When they move on to the front/back faces in step 4, make sure they refer to them using their names. Again, ask them the shape of those faces and the formula for the area of that shape. Area of a rectangle is length × width, but the



Materialsboxrulergrid paper


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surface area


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dimensions of the front/back could be thought of as length and height; thus, the area of the front/back could be written as length × height. Students may struggle with this and need to revisit it a couple of times before it sinks in. Repeat this process for the side faces, which could be thought of as having an area of width × height.

In step 6a), students are asked to come up with a formula for the surface area of a cube. In order to do this, they need to think of all sides as having a length of s instead of each side being unique (length, width, and height). So, the areas of each face will be s2 and the formula becomes SA = 6s2. Th is may be a leap in thinking for some students, so emphasize that the formula SA = 2lw + 2lh + 2wh still works for a cube (since a cube is a rectangular prism) but that the formula SA = 6s2 is more effi cient. For step 6c), you may wish to provide students with Master 2 Centimetre Grid Paper or Master 3 0.5 Centimetre Grid Paper for drawing the net of the cube.

Allow students time to work on the Puzzler, either individually or in pairs. Th en, discuss the solution (25 cubes). Challenge students to determine the surface area of the fi gure if the cubes have an edge length of 1 cm.

Meeting Student NeedsAssist students in writing a formula for the surface area of a rectangular prism. Provide students with a rectangular prism (e.g., a tissue box). Have them label the length 10 in., width 4 in., and height 6 in. (does not have to be to scale). Ask students to describe how to calculate the area of each face and to identify faces with identical areas. Students can oft en calculate surface area with numbers quite easily. Get them to describe the process to you and write down what they are saying. Th en, ask them to identify the length, width, and height of the prism. Label the length (l), width (w), and height (h) on the box beside each measurement. Review the process they described to you using the numbers. Th en, using the same process, replace the numbers with the variables. It might help students to see this transition from numbers to variables.Aft er students have written their formulas, discuss the fl exibility in the formulas. Ask students if the terms in the formula can be written in diff erent orders. Ask if there is something common to all of the terms in the formula.

ELLEnsure that students understand the diff erence between height, width, and length of a 3-D object. Get them to add these defi nitions and a visual example of them to their personal dictionary.Students may fi nd step 6 challenging due to the need to change from working with l = length, w = width, and h = height to working with s = side. Allow for some fl exibility here and demonstrate that the original formula is still valid.

Gifted and EnrichmentChallenge students to fi nd a general formula for the volume of a rectangular prism.

Common ErrorsStudents may neglect to double all the areas: SA = lw + wh + lh.

Rx Remind students that in a rectangular prism, each face has an identical face opposite it; there are six faces or three pairs of faces: SA = lw + lw + wh + wh + lh + lh or SA = 2lw + 2wh + 2lh.

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Some students may try to calculate area by multiplying all of the dimensions: SA = lwh.

Rx Remind students that surface area is the sum of the area of all faces. Discuss that multiplying the dimensions results in volume. Tell students that they will explore volume in Chapter 3, including examining the diff erences between surface area and volume.

Answers

Explore Formulas for the Surface Area of Rectangular Prisms

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1. a) 6 sides b) 3 pairs2. Example:

3. Examples:a) Multiply the length and width to determine

the area of the top (the white face).b) the bottom (the opposite side)

4. Example: Multiply the width and height to determine the area of each of the two sides (the light grey face). Multiply the length and height to determine the area of the front (the dark grey face) and of the back.

5. Examples: Surface Area = 2 × (length × width) + 2 × (length × height) + 2 × (width × height) or SA = 2[(length × width) + (length × height) + (width × height)] or SA = 2lw + 2wh + 2lh

6. a) Examples: Surface Area = 6 × (side × side) or SA = 6(side)2 or SA = 6s2

b) 54 cm2

c) Example:

d) 54 cm2

e) Example: Yes

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Refl ectListen as students fi gure out how to use the areas of the individual faces to compose a formula for surface area. Encourage them to write the most condensed formula possible.

To write the formula in step 5, students need to pull together the information they gathered in the previous steps. Ask them how to use the fact that there are three pairs of identical faces. Have them describe, in their own words, how to fi nd the surface area of a prism before they try to put variables into the mix. For example, students may say that the surface area of a prism is the total of the areas of the front/back, top/bottom, and side/side.

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Extend Your UnderstandingListen as students try to make the connection between a rectangular prism and a cube.

Students need to see that a cube is a special type of rectangular prism, in which all the edges are equal in length. Ask them to use one dimension to describe the cube.

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On the Job 1Lead a class discussion focusing on why it might be important to be more exact on big paint jobs than on smaller ones. Ask:

Can you think of other examples where accuracy is really important?What would happen if you overestimated the amount of paint needed?What would happen if you underestimated the amount of paint needed?Which consequences would be greater?

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For the Your Turn question, ask students if they fi nd it easier to apply the formula for the surface area of a rectangular prism if they have a diagram of the prism. Does it help them to draw the net of the prism? Initiate a class discussion on how students approach these questions so that students are exposed to multiple strategies and can determine which ones work best for them. For students who prefer to use a net, you may wish to provide them with Master 2 Centimetre Grid Paper, Master 3 0.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper, or provide them with the pre-made nets on BLM 1–3 Nets of 3-D Figures.

Meeting Student NeedsHave students state the formula for the surface area of a rectangular prism and then identify the values to be substituted into the formula: l = 45, w = 9.5, and h = 8.Discuss why the formula SA = 2lw + 2lh + 2wh is simplifi ed to SA = 2(lw + lh + wh). Encourage students to work with the version that feels most comfortable to them.

Gifted and EnrichmentChallenge students to calculate the amount of paint needed to paint the container, not including the bottom. Since the bottom of the container is always on the ground, it would save time and money not to paint it. Ask them how they would adjust the formula for surface area of a rectangular prism to account for this.Have students explore the eff ects of changing some of the dimensions in the equation. What if the length is doubled? What if both the length and width are doubled? What eff ect would this have on the amount of paint needed to cover the container’s surface area?

Answers

On the Job 1: Your Turn267.2 m2



On the Job 1Have students do the Your Turn question. Check that their calculations are correct, that all work is shown, and that they answer with the appropriate units.

Have student pairs share their answer and strategies with another pair. Suggest they draw a picture of the rectangular prism with the dimensions labelled. You may wish to provide them with BLM 1–4 3-D Figures. Do they prefer to use the formula to calculate the surface area or do they prefer to calculate the surface area using the net?

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Check Your UnderstandingTry ItFor #2, observe whether students use the formula for the surface area of a rectangular prism or the formula for the surface area of a cube that they developed in the Explore. Ask students which formula they fi nd easier to work with for the surface area of a cube. Students could work in pairs, with each student using a diff erent method to fi nd the surface area, and then they could compare answers. Ask them:

Does the surface area double when the side length doubles?If not, by how much does it increase?

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Apply ItIf students are struggling with #3b), ask them what face will not be painted. Th en, review the formula and ask students which part of the formula represents the top and bottom piece. Since they do not need to calculate the area of the bottom, how does the formula change? If students are having diffi culty working with the formula for this question, ask them to draw the net of the cabinet. Ask students to draw an X through the face that represents the bottom of the cabinet and fi nd the area of the remaining faces. Also, discuss the strategy of using the formula and then subtracting the area of the bottom since some students may be more comfortable with this method.

For #4, if students are struggling to fi nd the error, encourage them to cover up Jake’s work and do the question themselves. Th ey might be able to fi nd Jake’s error by comparing their work to his.

For #5, suggest that students sketch a net of the lid if that helps them. Ask students which fi ve faces of the rectangular prism make up the parts being covered with fabric.

Meeting Student NeedsEncourage students to draw a net of the rectangular prism in each question. Provide them with the template of a net for a rectangular prism for them to label, such as on BLM 1–3 Nets of 3-D Figures. Students could label the prism with the dimensions and cross out any faces not needed for the question. Encourage students to determine the surface area of the prism in as few steps as possible.

Gifted and EnrichmentFor #2d), challenge students to come up with an algebraic explanation for the increase to the surface area of a cube if the side lengths are doubled.For #3, challenge students to determine the surface area that needs to be painted if Kevin does not paint the bottom of the cabinet but he does paint the inside of the cabinet.




Students should try to use the formula for the surface area of a rectangular prism. If they fi nd this challenging, suggest that they use a net to help them determine the surface area. Encourage them to see a rectangular prism as three pairs of identical opposite faces.

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On the Job 2It may be helpful for students to draw a diagram of the greenhouse before working through this example. Ensure students understand that they are calculating the exterior surface area of the triangular prism, not including the base.

Ask students how they think they would calculate the surface area.How is a triangular prism diff erent from a rectangular prism?What side is missing?Are there any pairs of sides equal in length?What is the minimum number of calculations needed to fi nd the surface area of a triangular prism?

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Having this discussion before showing students the formula may promote greater understanding of the formula. Note that this student resource includes only triangular prisms with triangular faces that are equilateral or isosceles triangles; it does not include scalene triangles.

Discuss the defi nition of slant height. Ensure students understand the diff erence between slant height and height.

Some students might prefer to leave the total area of the two triangular ends as2 × 1 _ 2 × w × h. Encourage them to use the method they prefer. Allow students tofocus on applying the formula rather than memorizing it.

Ask students what they would do if they did not know one of the dimensions of the triangle.

What if you did not know the height of the triangle?How might you calculate it?What if you did not know the slant height of the triangle?How might you calculate it?

For the Your Turn question, discuss with students that if they leave the triangular prism positioned as it is in the diagram, they may wish to adjust the formula. Th e surface area would be the area of the top/bottom piece, the two slanted sides, and the rectangular end. Th ey may wish to draw the pizza box so that it is in the same position as the triangular prism in On the Job 2. Have them label the dimensions. Th en, they can use the formula shown in On the Job 2. Encourage students to sketch a net of the triangular prism if that helps them.

Meeting Student NeedsSome students struggle with plugging numbers into the variables of a formula because the formula has little meaning to them. For these students, suggest they use the formula SA = area of the base + area of the 2 triangular ends + area of the 2 slanted walls. Th en, they will have a better understanding of exactly what they are calculating.Discuss how each of the surfaces is represented in the formula. Some students will be able to think from the 3-D model and others may require the net. When discussing the formula, it is important to have both visuals present.Students need to determine which side, the length or the width, forms the base of the triangular prism. Note that length multiplied by width always represents the area of the base, but the area of the triangles (and the area of the lateral sides) varies depending on whether the length or width acts as the base of the triangle.

ELLStudents may need clarifi cation on the diff erence between the slant height and the height. Th ey need to understand which is used to calculate the area of the triangles versus which is used to calculate the area of the lateral sides. Encourage them to use visuals before writing the formula for the question.

Answers

On the Job 2: Your Turn465.2 cm2

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On the Job 2Have students complete the Your Turn question. Check that their calculations are correct and that they answered with the correct units.

Students may fi nd it necessary to make adjustments to the Your Turn question. The triangular prism has an orientation diff erent from the one in the example. Students will either have to adjust the formula or redraw the image so that the rectangular end becomes the base. Some students may have strong skills in visualization and 3-D perspective and not need to redraw the prism.

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Check Your UnderstandingTry ItFor #1, observe whether students gravitate toward using the formula or the net. In the case of the triangular prism, either method is acceptable. Encourage students who are using the net to devise their own formula. Th ey will need to calculate the area of each face on the net. Encourage them to identify pairs of faces in order to simplify the equation.

For #2, ask students to predict the change to surface area when each of the dimensions is doubled. Have them calculate the surface area with the original dimensions and then with each dimension doubled. How does the surface area compare? Ask them to predict the eff ect of doubling the slant height. How could they check if their prediction is accurate?

Apply ItFor #5, students might benefi t from drawing the net of the triangular prism. Ask them:

If the prism is open, what side is not needed when calculating the surface area?How does this translate to the formula?What part of the formula is not needed?

Meeting Student NeedsFor #2b), challenge students to make generalizations. Ask them what the eff ect on surface area is of doubling the length (changing l to 2l), what the eff ect is of doubling the slant height (changing s to 2s), and what the eff ect is of doubling the height (changing h to 2h).Highlight #5 as showing how it may be necessary to manipulate the formula under special conditions (no top). Ask students to brainstorm situations in which this may be necessary or advantageous in real-world applications (e.g., if a lid is not necessary for a container, the cost of material will be reduced).It may be benefi cial for some students to draw a net of the triangular prism for each question. You may wish to provide them with a template for a net of a triangular prism, such as on BLM 1–3 Nets of 3-D Figures. Students could label the prism with the dimensions and cross out any faces not needed in the question.

Gifted and EnrichmentFor #3, ask students what they notice about the slant height and base length of the triangular faces (they are the same). Ask them what this means about the area of the slanted faces and the area of the base (they are also the same). Discuss what type of triangle the triangular face is (equilateral). Have students write a revised formula for the surface area of equilateral triangular prisms. (For example: SA = wh + 3le, where l = length, w = width, h = height, e = edge length of triangle.)

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Common ErrorsStudents may use the height of the triangle as a dimension for the rectangular sections.

Rx Show students on a 3-D model that they need to use the slanted part (slant height) of the triangle for the rectangular section. If they use the height of the triangle for the rectangular sections, the calculation of the surface area will be too small.



Try ItStudents should be able to correctly answer #1 before moving on to answer #3 and #5.

Students should try to use the formula for surface area of a triangular prism. If they fi nd this challenging, suggest they use a net and develop their own formula for calculating the surface area. Encourage them to visualize a triangular prism as a set of equal opposite faces.

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On the Job 3Discuss with students that pyramids are named by their base. Since this is a square-based pyramid, we know the base is a square. How does knowing this help us with the other faces of the pyramid? We know the base has equal side lengths; therefore, the triangles are equal as well. Ask students to explain how they know that all the triangles are equal. Based on this, ask them how many calculations are needed to determine the surface area of the square-based pyramid. In the formula shown, students may struggle with why it is 2lh when there are four triangles. Discuss thisas a class. If students prefer, let them use 4 ( 1 _ 2 lh) to show the area of the four triangles.

Meeting Student NeedsEncourage students to create the net of a square-based pyramid. Since they have not yet worked with pyramids in this chapter, they might have diffi culty jumping right to the formula. Ask them how to calculate the area of the square base. Ask them how to calculate the area of one of the triangles. Is the triangle equal in area to any of the other triangles? How do they know? If all the triangles are equal, what can you do to the area of one to determine the combined area of all of them? Aft er going through the question in this manner, show students the formula again and ask if they can explain it to you given what they now know.Discuss what each of the surfaces will become in the formula. Some students will be able to think from the 3-D model and others may require the net. When discussing the formula, it is important to have both visuals present.It is helpful to compare the use of s for the slant height in the pyramid to the triangular prism. Ask students to describe the similarities and diff erences between the slant heights in these two 3-D fi gures.

Gifted and EnrichmentGive students the dimensions of the square base and the height from the centre to the highest point to see if they can determine the surface area. Also, give them an example in which they know the area of the base and the distance from the highest point to a corner of the pyramid’s base. Both require good visualization and an understanding of the Pythagorean relationship.

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Answers

On the Job 3: Your Turn2100 m2



On the Job 3Have students do the Your Turn question. Check that their calculations are correct and that they answered with the correct units.

Check to see whether students are using the formula or a net to fi nd the surface area. If students are using the formula, do

they understand the simplifi ed 2lh or are they using 4 ( 1 _ 2 lh) ?

If they are using the formula, it is important that they understand why the formula works. If students are involved in developing the formula, they have a deeper understanding of it.

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Check Your UnderstandingTry ItFor #1, students can calculate the surface area using a formula or a net. Have a class discussion to see which method most students are gravitating toward. Discuss the pros and cons of each method, especially since students have not worked with pyramids previously in this chapter. While using the formula is quicker, make sure students choosing this method understand it, why they multiply those particular dimensions, and why they need to add as they do.

Apply ItFor #3, have students work in pairs to brainstorm ideas for the perfume. Get pairs to pick their favourite name and then share the name with the class. You could allow the class to decide on one name and then challenge students to design the box with this name. Working in pairs, students could construct the net of the box, cut it out, and put it together to form the 3-D model of the box. Have them put the name of the perfume on the box and decorate it in a way that fi ts with its name. Allow students to use these dimensions to design packaging diff erent from perfume packaging if they wish.

For #4, have students predict what doubling the height the triangular face does to the surface area of the box. Will it double as well? Get them to explain their reasoning. What if the manufacturer doubled the length of the base instead? What eff ect would that have on the surface area of the box?

For #5, ask students:If you use a net to calculate the surface area of the square-based pyramid, which face is not needed for this question?Based on the faces of the net you have left , can you develop your own formula to fi nd the surface area of what remains?If you use the formula to calculate the surface area of the square-based pyramid, which part is not needed for this question?What would the new formula become?

Allow students time to work with a partner to attempt the Puzzler. Have students share their solutions and strategies. Th en, work through the solution as a class:Cut each cube into eight smaller cubes. Th e surface of each small cube is 1 _ 4 thatof the large cube, so the total surface area is doubled.

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Meeting Student NeedsDiscuss how each of the faces is represented in the formula. Some students will be able to work from the 3-D model and others may require the net. When discussing the formula, it is important to have both visuals present.Compare the slant height of a pyramid to the slant height of a triangular prism. Ask students to describe the similarities and diff erences between the two slant heights.

ELLStudents may not know what a pyramid is. Have several models on hand so that students can see that a pyramid comes to a point. Show them pyramids and prisms, and ask them to describe how they are similar and how they are diff erent. Let students know that pyramids are named according to the shape of their base, such as square-based pyramid and triangular pyramid.

Gifted and EnrichmentAsk students to create a formula for calculating the surface area of a rectangular pyramid. Give them a picture of a rectangular pyramid and label the base dimensions as l and w and the height of each triangle as h. For an even greater challenge, ask them to draw a triangular pyramid and label dimensions with a variable of their choice. Can they determine a formula for the surface area of a triangular pyramid? Check to ensure they do not assume that the triangle is equilateral or right.Give students the dimensions of the square base and the height from the centre to the apex and see if they can determine the surface area of the square-based pyramid.Give students the area of the square base and the distance from the apex to a corner of the pyramid’s base and see if they can determine the surface area.




Students should try to use the formula for the surface area of a square-based pyramid. If they are struggling with this, suggest they use a net and develop their own formula for fi nding the surface area. Encourage them to see a square-based pyramid as a square base and a set of four equal triangular faces.

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On the Job 4Developing the formula for a cylinder is perhaps the most challenging of the 3-D shapes. Before working through this question, you may want to present several examples of cylinders of various shapes and sizes and, as a class, try to derive the formula. Cover examples in which the radius of the cylinder is known and examples where the diameter is known. Students should come to see a cylinder as a set of two equal and opposite circular bases and a rectangular middle section that has been wrapped around the circular bases.

You may need to review how to calculate the area of the curved surface. Use a diagram of the net of a cylinder to show students that the curved surface is a rectangle with a width equal to the height of the cylinder and a length equal to the circumference of the circular base. Remind students that to determine the area of the circle, they need the radius of the circle. Ask them how to fi nd the radius given the diameter.

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For the Your Turn question, give students the option of using a net or using the formula. Students can work in pairs and compare their answer and their method with those of another pair. Ask students to discuss with their partner how they would approach a question for surface area of a cylinder if they knew only the height of the cylinder and the radius.

Meeting Student NeedsFor some students, it might be benefi cial to use a hands-on approach. Take a sheet of paper and ask students what shape it is. Once they agree that the shape is rectangular, label the length and width of the paper. Ask students how to fi nd the area of a rectangle and label it on the sheet of paper as well. Th en, take the sheet of paper and roll it into a cylinder shape. Ask them if the width of the rectangle has changed. So, the width of the rectangle is also the height of the cylinder. Ask them if the length of the rectangle has changed. Th e length of the rectangle now forms the shape of a circle. What is the distance around the outside of a circle called? Th e length of the rectangle is also the circumference of the circle.

Answers

On the Job 4: Your Turn≈ 1558 in.2



On the Job 4Have students do the Your Turn. Check that their calculations are correct and that they answered with the correct units.

Check to see whether students are using the formula or a net to fi nd the surface area. If students are using the formula, do they understand how to determine the area of the curved surface? If they are using the formula, it is important that they understand why the formula works. If students are involved in developing the formula, they have a deeper understanding of it.

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Check Your UnderstandingTry ItFor #1b), students are calculating the surface area of a cylinder with the radius, not the diameter, given. Ask them what eff ect, if any, this has on fi nding the surface area. For students fi nding surface area using a formula, have them state and defi ne all of the parameters fi rst. For example,

SA = 2πr2 + πdl π = r = d = l =

For students using a net, have them label the net with all the necessary dimensions.

In #2, students are fi nding the surface area of an open-ended tube. You may wish to draw a diagram to ensure they understand what a tube looks like. If they are using the formula for the surface area of a cylinder, what part of the cylinder are they trying to fi nd the surface area of? What part of the formula do they need? Why do they not need the other part of the formula? Make sure students recognize that the dimensions are in two diff erent units and that they need to convert fi rst.

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Apply ItFor #3, ask students:

What part of the cylinder does the mesh cover?To calculate the area of the curved surface, what information do you need?What part of the formula is needed to fi nd the area of the mesh?When drawing a net, what part of the diagram do you not need to include?How can you fi nd the area of just the curved surface?

Remind them that height or length can be used to describe the width of the rectangular part of the cylinder, depending on its orientation.For #5, again, make sure students recognize that the dimensions are in two diff erent units and that they need to convert fi rst.



Try ItStudents should be able to correctly answer #1 before moving on to #2 and #3.

Students should try to use the formula for surface area of a cylinder. If they are fi nding this challenging, suggest they use a net and develop their own formula for determining the surface area. Encourage them to see a cylinder as a rectangular middle section and a set of two equal circular bases.

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Meeting Student Needs

Gifted and EnrichmentChallenge students to write the formula for surface area of a cylinder using only the radius (r) and the height (h). Th en, ask students to write the formula for the surface area of a cylinder using only the diameter (d) and the height (h). Students can check the accuracy of their formulas by using them to solve the problem in On the Job 4.Have students explore the change in the surface area if you double the height of the cylinder versus doubling the radius.

Common ErrorsStudents may square the circumference instead of the radius.

Rx Since the formula contains a string of variables and constants as well as an exponent (SA = 2πr 2 + 2πrh), students may make errors in the order of operations. Review their skills with the order of operations.

Work With ItFor #1, if students are using a formula, they need to determine what 3-D shape they are working with. Once they have identifi ed the formula they need to use, ask them if all sides of the rectangular prism will be included in the calculation. Since the ceiling or “top” of the rectangular prism is not being sprayed, what changes need to be made to the formula? If students are working with a net, have them draw and label the net with all of its dimensions. Ask them what sides, if any, will not be used in the fi nal calculation.

For #2, students might want to approach this question in two parts: fi nd the surface area of the upper part of the greenhouse and then fi nd the surface area of the lower part. Th e upper part is a triangular prism. Ask students to identify the faces that need glass and the faces that do not. Th e base of the greenhouse is a rectangular prism. Again, ask students to identify the faces that need glass and the faces that do not. Now, students need to decide if they are using a formula or a net. Whichever method they choose, they will have to account for sides that do not need glass.

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Tools of the Trade

Builders use water-resistant cardboard tubes to make concrete posts. The tubes can also be used as moulds for simple columns. For more information on using cardboard tubes for concrete, go to www.mcgrawhill.ca/school/learning centres and follow the links.


Encourage students to draw a diagram for #5. Ask them their plan to approach this question: Will they fi nd the surface area that will be painted for one cabinet and multiply it by eight? Ensure students do not think they can fi nd the surface area of one big cabinet (eight cabinets long). If students do not understand why they cannot use this second method, have half the class use the fi rst method and the other half use the second method and then compare answers. Th ey will see that the results are diff erent. Using diagrams, discuss why the second method is incorrect.

For #7, have students draw a net for the amount of metal needed to make the can. You may wish to provide them with BLM 1–3 Nets of 3-D Figures.

What adjustments would they make to their net for the label of the can?What part(s) do they not need in their calculation?How would the 1-cm overlap aff ect their net?Which dimension would change?When the can doubles in height, which dimensions will be aff ected?Will the amount of metal needed to make the can double as well?Will it aff ect the size of the label?

Th is would be a good question to take up and discuss as a class.

Discuss ItFor #8, some students may benefi t from a class discussion and then the opportunity to refl ect before answering using their own ideas and words. Others may benefi t from recording their thoughts individually fi rst and then engaging in a class discussion. Choose which method works best for your class. Encourage students to off er a lot of detail in their response. Perhaps they could include a diagram or an example in their explanation. Get them to trade their answer with a partner to see how their answers compare and to provide feedback to each other. Ask them to revise their own answers, as necessary, based on the feedback they receive.

For #9, students should recognize that surface area is the amount of 2-D space it takes to cover the surface of an object. Ask students if they know the term for the amount a box can hold. You might make the analogy that surface area is the amount of wrapping paper needed to wrap a present, and volume is what is inside the present. Tell students that they will be studying volume in Chapter 3.

For #10, help students understand the diff erence between linear, area, and volume measurements by off ering the following example/explanation:

Th e length of 10 cm can be measured with a ruler. It is a 1-D measurement of length (1 dimension: unit with an exponent of 1).Th e area of 10 cm2 is the amount of space a 2-D shape occupies. It is a 2-D measurement made up of width and length (2 dimensions: unit with an exponent of 2).Th e volume of 10 cm3 is the amount of space a 3-D fi gure occupies. It is a 3-D measurement made up of width, length, and height (3 dimensions: unit with an exponent of 3).

For #11, make sure students recognize that the triangular faces of the prism are equilateral triangles.

Meeting Student NeedsProvide BLM 1–7 Section 1.3 Extra Practice to students who would benefi t from more practice.

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Gifted and EnrichmentChallenge students to make a single net for the greenhouse in #2. Instruct them not to think of it as a triangular prism on top of a rectangular prism but as a fi gure with six sides (they should not include the fl oor). Th ey can then use their net to fi nd the surface area. As an additional challenge, students could research the cost of glass needed to build the greenhouse.Challenge students to develop a formula to calculate the volume of the 3-D fi gures covered in this section. Discuss that volume is the amount of space a 3-D fi gure occupies. Get students started by telling them that the volume of any object can be found by multiplying the area of the base by the height of the object. Students could then work on the specifi c formulas for each fi gure.





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1.4 Surface Area of Cones and Spheres


Adapted (minimum questions to cover the outcomes) On the Job 1 #6On the Job 2 #5Work With It #3, #6

Typical On the Job 1 #6On the Job 2 #2, #5Work With It #3–#6


Elicit from students other examples of cones that they have encountered in their everyday life. Ask them what shapes they think the net of a cone would be made up of. Ask them how they think cones and cylinders are diff erent and how they are similar. In this way, you can check what students know about cones. Ask students if they know of any other 3-D fi gures that have curved faces. Perhaps students already know about spheres, which could also be a point of discussion.

Explore the Surface Area of Cones and SpheresIn the fi rst part of this Explore, students conduct a hands-on experiment to fi nd the surface area of a cone. Students estimate the dimensions of the cone and use those estimates to estimate the area. Ask students to identify the parts of the cone. How do these parts compare to the parts of a cylinder?

For step 1a), it might be diffi cult for students to come up with an accurate measurement for diameter of the cone without knowing where the centre of the circle is. Encourage students to measure the diameter a couple of times to get as accurate a result as possible.

For step 1b), ask students if it would be better to measure the radius using a ruler or to use their measurement for diameter.

For step 2a), you may wish to provide Master 2 Centimetre Grid Paper, Master 30.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper for students to use when drawing the circle.

For step 3c), students may need assistance in understanding that they can label the circumference of the circle on their diagram of the fl attened curved surface. Before they cut in part a), you may wish to have them colour around the circumference of the circular base of the cup with a marker. Th en, when they cut the cup in step 3a), they will see which part of the fl attened shape represents the circumference of the circle. Th is will help them to identify the circumference in step 3c).

For step 4, ask students:Can you draw a cone in which the area of the curved surface is greater than the area of the base of the cone?

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Materialscone-shaped paper cuprulergrid paperscissorstennis ball or other small ball


Grid Paper



Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)



surface area

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Can you draw a cone in which the area of the curved surface is smaller than the area of the base of the cone?What is the main diff erence between the two diagrams?Which parameter changes between each diagram?

For Part B of the Explore, you may wish to model the steps with the class. For step 6a), it might be diffi cult for students to come up with an accurate estimate for the diameter of the sphere. Encourage students to use a ruler to “measure” the diameter as best they can. Have them perform the measurement a couple of times to get as accurate an estimate as possible.

Some students may need assistance in simplifying their formula in step 11b):SA = d × 2πrSA = 2r × 2πrSA = 4πr 2

An alternate approach to Part B of the exploration is to have students peel an orange and determine that the peel has the equivalent area to four circles with the same radius as the orange. Th en, students use the formula for the area of a circle to determine the total area of the four circles, and generalize this to a formula for the surface area of a sphere.

Meeting Student NeedsFor step 2, it may help some students to estimate the area of the circular base by counting the number of squares within the traced circle. You could also provide students with a piece of string to measure the circumference. Students could then measure the length of the piece of string.As part of step 2, discuss which methods for estimating would be most accurate and would minimize error.It may benefi t some students to add radius, diameter, circumference, and area to their notebook and write a rule and/or formula for each one.

ELLHave students draw four circles in their notebook. On the fi rst circle, have them draw and label the radius. On the second circle, have them draw and label the diameter. Ask them to explain the relationship between the radius and the diameter. On the third circle, have them outline the circumference with a marker and write the formula to determine the circumference of a circle. Have them record two formulas for the circumference: one for when the radius is given and one for when the diameter is given. Have them shade in the entire fourth circle to illustrate area. On the inside of the circle, have them write down the formula for area of a circle. Ask students how they would calculate the area of the circle if only the diameter is given.

Gifted and EnrichmentChallenge students to fi nd a cone in which the area of the curved surface is equal to the area of the base. Have students determine what kind of cone comes closest to doing this. Would it be tall and narrow or short and wide?Ask students to identify conditions for which the curved surface of the cone would be greater in area than the base of the cone, and conditions for which the curved surface would be smaller in area than the base of the cone.Ask students if they can fi nd the area of the curved surface given the angle measure of the curved surface. Challenge students to fi nd the area of the whole circle. Th en, have them determine what percent of the circle the curved surface is and use this information as an alternative way of determining the area of the curved surface.

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Challenge students to determine if a rectangle with any width and length could have an area that is equivalent to a sphere with a corresponding diameter and circumference. Why or not why? Have them include examples with their explanation.

Common ErrorsStudents may use the height of the cone to fi nd the area of the curved surface rather than the slant height.

Rx Remind students they use the height of the cylinder to fi nd the surface area of the curved surface because the circular ends are perpendicular to the curved surface. In a cone, the curved surface runs at a slant from the circular base to the highest point. Th is is why we use the slant height, not the height.

Answers

Explore the Surface Area of Cones and Spheres

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1. Examples: a) diameter: 6 cm; height: 9 cmb) 3 cm c) ≈ 9.5 cm d) 95 mm

2. Examples:a)

b) 18 cm c) ≈ 18.8 cm d) 27 cm2

e) ≈ 28.3 cm2

3. b) and c) Example:

4. Examples:a) Th e curved surface appears to have a

greater surface area.b) Yes, the curved surface will always have

a greater surface area than the base.c) 90 cm2

5. Examples:a) ≈ 89.4 cm2

b) My estimate was greater than the calculated surface area.

c) SA = πrs + πr2 or SA = π(rs + r 2) 6. a) Example: Th e diameter of a tennis ball is about 2 1 _ 2 in.

7. b) Example: Th e strip is about 8 1 _ 4 in. long. 8. and 9. Examples:

Column 1 Column 2 Column 3

Rectangle width

= 2 1 _ 2 in.

length

= 8 1 _ 4 in.

Area

= 20 5 _ 8 in.2

Sphere diameter

= 2 1 _ 2 in.

circumference

= 8 1 _ 4 in.

Surface Area

= 20 5 _ 8 in.2

10. a) Area = width × lengthb) Surface Area = diameter × circumference

11. a) C = πdb) SA = πd2, or SA = 4πr 2

circumference

slant height

circumference

slant height




Refl ectBy adjusting the radius and the slant height of a cone, students should be able to see which section of the cone has the greater area. If the radius of the cone equals the slant height, the areas are the same. If the radius is greater than the slant height, the area of the base is greater. If the radius is smaller than the slant height, the area of the base is smaller.Observe whether students can extend their understanding of the formula for the area of a rectangle to the formula for the surface area of a sphere.

If students are having diffi culty with Refl ect in Part A, provide them with the drawings of three cones:

Cone 1: The radius has the same measurement as the slant height.Cone 2: The radius has a greater measurement than the slant height.Cone 3: The radius has a smaller measurement than the slant height.

Students can then fi nd the dimensions of the radius and slant height and identify which is bigger or the same. Then, when they calculate the area of each part, they can see the connection between the two measurements.You may wish to have students follow the steps in Part 2 for a sphere of a diff erent size to help them generalize what they have learned.

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Extend Your UnderstandingIf students understood how to fi nd the area of the curved section of the cone, they can use their understanding to determine the area of the circular base. They can then add the two amounts together, since there are only two parts to the net of the cone.Observe whether students are able to substitute the formula for the circumference of a circle into the formula SA = diameter × circumference.

Remind students that the net consisted of two parts: the curved section and the circular base. Since they have just learned the formula for the area of the curved section, they merely need to add the area of the circular base.Allow students to work with diameter or radius when simplifying their formula for the surface area of a sphere. Students who work with diameter should answer step 11b) with SA = πd 2. Students who work with radius should answer with SA = 4πr 2. To help students as they move forward in this section, have them share and discuss each variation of the simplifi ed formula.

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On the Job 1Ask students again what the diff erence is between the height of the cone and the slant height of the cone. Which measurement is needed to fi nd the surface area of the cone? What do they think the height of the cone could be used for? Ask students how to fi nd the slant height of the cone given the diameter and the height of the cone. Students may need to be reminded that the radius, not diameter, is needed to fi nd the slant height. Once students have calculated the slant height, have them review the formula for surface area. What is the area of the curved surface? What is the area of the base?

For the Your Turn question, have students draw a diagram of the cone before they draw the net. You may wish to provide them with BLM 1–4 3-D Figures. Discuss that this question is diff erent from the On the Job 1 question. Ask them whether it is easier to calculate the surface area of a cone when given the radius or when given the diameter.


Meeting Student NeedsSome students will benefi t from drawing a net of the cone and labelling the relevant parts. Provide them with BLM 1–3 Nets of 3-D Figures.Students need to understand how the Pythagorean relationship can be used to determine the slant height, given the height (and diameter or radius). Compare and contrast with the triangular prism and the pyramid. What are the similarities and diff erences?Ask students questions about the spatial relationships within the cone: Would doubling the radius have the same eff ect on the total surface area of the cone as doubling the slant height? What about the eff ect on just the area of the curved surface?

ELLCheck that students are profi cient in identifying both the slant height and the height. Try inverting and rotating visuals of the cone to ensure that students are confi dent with these measures.

Gifted and EnrichmentChallenge students by giving them the surface area and radius of the cone and asking them to solve for the slant height.Extend On the Job 1 by asking students to account for the seams of the tent. What dimensions would be aff ected if a seam allowance is needed? Challenge them to calculate the amount of fabric needed to construct the tent if a 3-cm seam is needed.Have students rewrite the equation for the surface area of a cone in terms of the diameter instead of the radius.

Common ErrorsStudents may fi nd it challenging to set up a correct Pythagorean relationship using the height and radius to determine the slant height.

Rx Remind students that the slant height is the hypotenuse.

Answers

On the Job 1: Your Turna) b) ≈ 101 ft 2 c) ≈ 50 ft 2 d) ≈ 151 ft 2

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On the Job 1For the Your Turn question, give students the option of using the formula, or drawing a labelled net of the cone.

Whether students draw a net or use the formula, encourage them to state all the information given in the question that they need in order to fi nd the surface area. Remind them to show all their calculations and provide a detailed solution.

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Check Your UnderstandingTry ItStudents struggling with questions #1 to #5 will need to review the material in the Explore.

Apply ItFor #6, students should see that they need to calculate only the area of the curved surface. If students are solving this question using a net, have them sketch the net of a cone and then ask them what part of the net they do not need to calculate the area of. You may wish to provide them with BLM 1–3 Nets of 3-D Figures. If students are solving this question using the formula, ask them to explain the parts of the formula to you. What part(s) do they not need to use?

For #7, ask students why they think they are calculating the minimum amount of felt needed. Might they need more fabric? Is it possible to calculate the maximum amount of felt needed?


Gifted and EnrichmentOnce students have calculated the area of the curved surface of the cone, ask them to determine the fraction of the whole circle that the curved surface represents.

Common ErrorsStudents may use the diameter instead of the radius to determine the slant height.

Rx Create a 3-D drawing of a cone (or use BLM 1–4 3-D Figures) and ask students to identify the highest point of the cone. Ask them to identify the centre of the circular base. Draw a line segment perpendicular from the centre to the highest point. Explain that to make a triangle, you connect the centre of the circle to any point on the circle’s circumference. Th e triangle that is formed has sides of radius (not diameter), height, and slant height.



Try ItStudents should be able to correctly answer #1 to #5 before moving on to #6 to #8.

Students need to be able to apply the Pythagorean relationship to fi nd the slant height of a right triangle given its legs (radius and height).

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On the Job 2Consider asking students what they think the net of a sphere would look like. Have students work in small groups to try to devise a net of a sphere. Students should quickly see that it is impossible to make a perfect sphere from a net. Why is that? Discuss that you cannot bend paper in two directions at the same time. Th e net of a sphere is an approximation. Th is would be a good introduction to the formula and why we can use only a formula to determine the surface area of a sphere.

Meeting Student NeedsSee if students can develop their own formula for the surface area of a sphere using the diameter as their variable. Discuss which formula they prefer and why.Introducing the surface area of a hemisphere may be helpful if you wish to work with composite objects. Give students the formula SA = 3πr 2 and see if they can fi gure out why it is not simply half the formula of a full sphere.

Gifted and EnrichmentChallenge gift ed students to rewrite the formula for the surface area of a sphere using the diameter, not the radius. Students may need a hint to get them started. Encourage them to write the formula by breaking it down into its prime factorization (i.e., SA = 2 × 2 × π × r × r).Have students attempt surface area problems involving composite fi gures (perhaps a grain silo that is cylindrical with a hemi-spherical roof).Ask questions that compare surface areas. For example: A pyramid and a cone have the same height but the pyramid has a side length of 4 cm and the cone has a diameter of 4 cm. Which has the larger surface area? Why?

Common ErrorsStudents may misinterpret what should be squared in the formula.

Rx Have students place brackets around the quantity being squared: SA = 4π(r 2).

Answers

On the Job 2: Your Turn5026.5 mm2



On the Job 2Have students use the formula for the Your Turn question.

Check students’ calculations to make sure they are squaring the radius rather than multiplying by twice the radius.Also check that students answer with the correct units.

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Check Your UnderstandingTry ItFor #2, ask students to predict whether the surface area will double if the radius doubles before they make their calculations and comparisons. Once they have a rule, ask them to try their rule on another sphere to test its accuracy.

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Apply ItFor #3b), some students may need to have their heights measured.

In #4, students might not recognize that they need to make two calculations. Ask them how to calculate the surface area of Earth given its diameter. Th en, ask them how they can fi gure out the surface area of the water on Earth if 71% of Earth’s surface is covered with water.


Gifted and EnrichmentHave students algebraically determine what happens to the surface area of a sphere if the radius doubles. Similarly, they can algebraically determine the eff ect on the surface area if the diameter doubles.

Common ErrorsStudents may think that r 2 = 2 × r.

Rx Ask students the meaning of the exponent 2. Where would the 2 be if it was intended to multiply r by 2?



Try ItStudents should be able to correctly answer #1 before moving on to the rest of the questions.

In #1, #3, and #5, students need to use the formula to determine the surface area of a sphere. Encourage students who are struggling to use brackets or fi ll-in-the-blanks to solve these questions. Mistakes in these questions will likely be caused by simple calculation errors.

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Work With ItFor #3, ask students what shape the entire badminton birdie is similar to (a cone). What measurements of the cone are needed to calculate the surface area of the entire birdie? Discuss that students need to determine the surface area of only the blue plastic part. How might they do this? What do they need to subtract from the surface area of the entire birdie? Discuss why the resulting answer is only an approximation.

Discuss ItEncourage students to answer #4 without making any calculations. Just using what they know about surface area, cones, and cylinders, have them determine which will have the greater surface area. For students struggling with this question, provide them with drawings of a cone and cylinder with the dimensions labelled. You may wish to use BLM 1–4 3-D Figures. Th ey could then calculate to determine the answer.

For #5, ask students what the height of a sphere is. Th ey should recognize that the diameter of a sphere is both its height and width. Encourage students to answer this question without making any calculations but just using what they know about surface area, spheres, and cylinders. For students struggling with this question, provide them with drawings of a sphere and a cylinder with the dimensions labelled. You may wish to use BLM 1–4 3-D Figures. Th ey could then calculate to determine the answer.

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Meeting Student NeedsProvide BLM 1–8 Section 1.4 Extra Practice to students who would benefi t from more practice.





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1Skill Check

Planning NotesTh ese review questions will help students check their progress on topics covered throughout the chapter. Aft er completing the Skill Check, students should know which skills they have mastered and which skills they need further work on. For any skills needing further work, refer them back to the appropriate On the Job section for further review.

For students who need to draw nets to assist them in answering questions, you may wish to provide Master 2 Centimetre Grid Paper, Master 3 0.5 Centimetre GridPaper, or Master 4 1 _ 4 Inch Grid Paper, or provide BLM 1–3 Nets of 3-D Figures.

Have students use BLM 1–1 Chapter 1 Self-Assessment to assess their current progress. Encourage them to review the appropriate section or sections of this chapter that deal with areas where they are having diffi culty.

Have students who are not confi dent discuss strategies with you or a classmate. Encourage them to refer to their notes, On the Jobs, and previously completed questions in the related sections of the student resource.

Have students make a list of questions that they need no help with, a little help with, and a lot of help with. Th ey can use this list to help them prepare for the Test Yourself.

Th ese are the minimum questions that will meet the related curriculum outcomes: #5, #7–#10.

Meeting Student NeedsFor students who fi nd drawing nets challenging, you may wish to provide BLM 1–3 Nets of 3-D Figures.Students who require more practice on a particular topic may refer to BLM 1–5 Section 1.1 Extra Practice, BLM 1–6 Section 1.2 Extra Practice, BLM 1–7 Section 1.3 Extra Practice, and BLM 1–8 Section 1.4 Extra Practice.



Chapter 1 Skill CheckThe Chapter 1 Skill Check is an opportunity for students to assess themselves by completing selected questions in each section and checking their answers against answers in the student resource.

Have students check the contents of the What I Need to Work On section of their math journal and do at least one question related to each listed item.Have students revisit any section that they are having diffi culty with prior to working on the Test Yourself.

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Materialsrulergrid papercalculator


Grid Paper



BLM 1–3 Nets of 3-D FiguresBLM 1–5 Section 1.1 Extra

PracticeBLM 1–6 Section 1.2 Extra



Practice

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1Math at Work 11, pages 52–53


Materialsrulergrid papercalculator


Grid Paper



BLM 1–3 Nets of 3-D FiguresBLM 1–9 Chapter 1 Test

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Test Yourself

Planning NotesHave students start the Test Yourself by writing the question numbers in their notebook. Have them indicate which questions they need a lot of help with, a little help with, or no help with.

Have students fi rst complete the questions they know they can do.Th en have students work on the questions they know something about. Encourage them to get peer coaching for any diffi culties.Finally, encourage students to deal with the questions they fi nd diffi cult. Review these questions with students. Depending on the question, refer them to the specifi c On the Jobs and Explores listed in the study guide that follows. You may also wish to review specifi c sample questions they have already handled. Once they have reviewed this material, help them to think through the diffi culty they are having.

It is important for students to know how to do the questions in this Test Yourself, since the chapter test will be modelled along the same lines.

Th is Test Yourself is a practice test that can be assigned as an in-class or take-home assignment. Provide students with the number of questions they can comfortably do in one class. Th ese are the minimum questions that will meet the related curriculum outcomes: #3, #4, #7, #8.

For students who need to draw nets to assist them in answering questions, you may wish to provide Master 2 Centimetre Grid Paper, Master 3 0.5 Centimetre GridPaper, or Master 4 1 _ 4 Inch Grid Paper, or provide BLM 1–3 Nets of 3-D Figures.For students who fi nd drawing nets challenging, you may wish to provide BLM 1–3 Nets of 3-D Figures.

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Study Guide

Question(s) Section(s) Refer to The student can …

#1 1.2 On the Job 2 #8, #9

estimate the surface area of a cylinder�

#2 1.2 On the Job 2 #8, #9

estimate the surface area of a cylinder�

#3 1.1 ExploreOn the Job 1On the Job 2

determine the 3-D fi gure from the net�

#4 1.3 On the Job 3 use a formula to determine the surface area of a pyramid

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#5 1.3 On the Job 1 On the Job 1

#2

use a formula to determine the surface area of a rectangular prismdetermine the eff ect of dimensional changes on surface area

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#6 1.3 On the Job 1On the Job 2Work With It

#2

solve a problem that involves the surface area of a rectangular prism and a triangular prism

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#7 1.4 Work With It#3

solve a problem that involves the surface area of a cone, and that requires the manipulation of a formula

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#8 1.4 On the Job 2 use a formula to determine the surface area of a spheresolve a problem within measurement systems

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Chapter 1 Self-AssessmentHave students review their math journal notes and earlier responses on BLM 1–1 Chapter 1 Self-Assessment.

Have students use their responses on the practice test and work they completed earlier in the chapter to identify areas in which they may need to reinforce their understanding of skills or concepts. Before the chapter test, coach students in the areas in which they are having diffi culty.

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Chapter 1 TestAfter students complete the Test Yourself, you might wish to use BLM 1–9 Chapter 1 Test as a summative assessment.

Before the test, coach students in areas in which they are having diffi culty.Consider allowing students to use their chapter summary notes to complete the practice test.

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1 Chapter Project

Planning NotesTh e Chapter 1 Project allows students to apply their skills to a fun context. Students create their own board game, including the board and playing pieces. Th e game is based on answering questions related to the material learned in this chapter. Have a few game boards on hand to help students in creating their own design. Once students have created their board design, they create question cards. Encourage students to go through the chapter and make a list of topics covered. Students could then create two or three questions per topic for players to answer as they play the game.

Ask students guiding questions to help them in creating their game:How many questions do you want in your game?How many spaces will you have on your game board?Will you have categories of questions that will allow the player to move a diff erent number of spaces if he or she answers correctly?Will the players be given paper, pencil, and calculator in your game?What is the ideal number of players for your game?

Set aside a day in which students have the opportunity to play each other’s games.

For drawing the nets, you may wish to provide Master 2 Centimetre Grid Paper,Master 3 0.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper.

Students can use the BLM 1–10 Chapter 1 Project Checklist to assist them in developing a board game with all of the required elements.

Review the Master 1 Project Rubric with students so that they know what is expected.

Meeting Student NeedsYou could do this project on a smaller scale and leave out Part C. Students could save their board games and use them to create a game for the next strand of math you will be teaching.For students who fi nd drawing nets challenging, you may wish to provide BLM 1–3 Nets of 3-D Figures.

ELLHave an example of a board game on hand so that students can see what an actual board game comprises. Th ey may follow the format of the board game you show them or create their own.

Gifted and EnrichmentEncourage gift ed students to create a challenging board with high-level questions. Gift ed students could then play each other’s games and give feedback to one another based on the level of diffi culty. Once their games are perfected, they could challenge you to play a round with them.

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Math at Work 11, page 54


Materialscardboardcoloured paperscissorsglue and/or tapegrid paperruler

Blackline MastersMaster 1 Project RubricMaster 2 Centimetre Grid PaperMaster 3 0.5 Centimetre

Grid Paper


BLM 1–3 Nets of 3-D FiguresBLM 1–10 Chapter 1 Project

Checklist


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)



surface area

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Chapter 1 ProjectThis chapter project gives students an opportunity to apply and display their knowledge of surface area. It should be evident that students can draw a net to represent a 3-D object, determine the area of a net, and calculate the surface area of a 3-D object.Master 1 Project Rubric provides a holistic descriptor to help in assessing student work on this project. See the notes that follow about how to use this rubric.

BLM 1–10 Chapter 1 Project Checklist can be used as a self-assessment prior to handing in the project. If students have not used this master to help them organize their project, have them use it to try to improve what they have done.You might wish to use Master 1 Project Rubric, or photocopy and distribute the following chart, leaving the Specifi c Level Notes blank. Work with students to develop what should go into those notes.

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Th e chart below shows Master 1 Project Rubric for the Chapter 1 Project and provides notes that suggest possible ways to score answers for this project.

Score/Level Holistic Descriptor Specifi c Level Notes

5(Standard ofExcellence)

Applies/develops thorough strategies and mathematical processes, making signifi cant comparisons/connections that demonstrate a comprehensive understanding of how to develop a complete solutionProcedures are effi cient and eff ective and may contain a minor mathematical error that does not aff ect understandingUses signifi cant mathematical language to explain their understanding and provides in-depth support for their conclusion

Demonstrates a comprehensive understanding of drawing nets of 3-D objects, determining the area of the 2-D shapes that make up a 3-D object, and determining the surface area of a 3D objectTransfers knowledge to given situations and builds on understanding to enhance project designSupports detailed drawings mathematicallyCommunicates clearly and logically

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4(Above

Acceptable)

Applies/develops thorough strategies and mathematical processes, making reasonable comparisons/connections that demonstrate a clear understandingProcedures are reasonable and may contain a minor mathematical error that may hinder the understanding in one part of a complete solutionUses appropriate mathematical language to explain their understanding and provides clear support for their conclusion

Demonstrates a very good understanding of drawing nets of 3-D objects, determining the area of the 2-D shapes that make up a 3-D object, and determining the surface area of a 3-D objectTransfers knowledge to given situations and builds on understanding to enhance project designSupports detailed drawings mathematically, but may lack precisionCommunicates clearly and logically

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3(Meets

Acceptable)

Applies/develops relevant strategies and mathematical processes, making some comparisons/connections that demonstrate a basic understandingProcedures are basic and may contain a major error or omissionUses common language to explain their understanding and provides minimal support for their conclusion

Demonstrates a good understanding of drawing nets of 3-D objects, determining the area of the 2-D shapes that make up a 3-D object, and determining the surface area of a 3-D objectTransfers knowledge to given situations and builds on understanding to enhance project designSupports drawings mathematically; the playing surface overlaps and/or the nets and construction of the playing piece may not be precise; calculations may have numerical errorsCommunicates in an understandable manner, but may miss some important ideas

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2(Below

Acceptable)

Applies/develops some relevant mathematical processes, making minimal comparisons/connections that lead to a partial solutionProcedures are basic and may contain several major mathematical errorsCommunication is weak

Demonstrates an acceptable understanding of drawing nets of 3-D objects, determining the area of the 2-D shapes that make up a 3-D object, and determining the surface area of a 3-D objectThe playing surface overlaps and/or the nets and construction of the playing pieces are not correctly drawn, or their area calculations are not accurate.Communicates in an understandable manner the basics of nets, area, and surface area, but may have signifi cant numerical errors

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1(Beginning)

Applies/develops an initial start that may be partially correct or could have led to a correct solutionCommunication is weak or absent

The following are not suffi ciently demonstrated: the drawing of nets of 3-D objects, determining the area of the 2-D shapes that make up a 3-D object, and determining the surface area of a 3-D object.The information needed for assessment is either missing or incorrect.

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1 Games and Puzzles

Planning Notes

High RollerFor this game, each time they roll the dice, students must quickly make six calculations using the four operations. Ask them for which operations the order of the two numbers does not matter so they do not need to repeat those operations. You might want to provide a template for the six calculations to assist students. For drawing the nets, you may wish to give students Master 2 Centimetre Grid Paper,Master 3 0.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper.

Surface Area Dice ChallengeFor this game, students roll three dice and then choose to use two of the numbers as the dimensions of a cylinder or cone or three of them as the dimensions of a rectangular prism. Allow students to use calculators to perform the calculations. You might want to have students play more rounds for other 3-D fi gures. You can also do some activities related to the surface area of the dice. For example, have students estimate the surface area of a partner’s dice, and then measure and calculate to see how accurate their estimates were.

Meeting Student NeedsFor High Roller, you may wish to help students remember which cube is positive and which is negative this way:red = hot = positiveblue = cold = negativeFor Surface Area Dice Challenge, it may help students with the calculations if they draw the fi gure (or net) with the dimensions that they rolled.



High RollerMaking the dice helps students see the connection between a 2-D net of an object and the 3-D object itself. It tests students’ number sense and ability to fi nd the greatest/least answer to a number sentence.

Allow students to use a calculator.Encourage students to determine the sign of the answer themselves and to use the calculator only to fi nd the numeric value of the answer.

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Surface Area ChallengeThis game requires students to calculate the surface area of cones, cylinders, and/or rectangular prisms.

Allow students to use a calculator.Have students write out the formulas before beginning the game. Then, they only need to substitute each value into the formula and calculate the surface area.

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Math at Work 11, page 55


Materialsthick red paper and blue paperrulerscissorsgluemarkerdicecalculator


Grid Paper


Specifi c OutcomesA1 Solve problems that require the manipulation and application of formulas related to:

surface area

M1 Solve problems that involve SI and imperial units in surface area measurements and verify the solutions.

N1 Analyze puzzles and games that involve numerical reasoning, using problem-solving strategies.

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Documents

Surface Area 1 - Morell Regional High School Technology · 1.3 use formulas to determine the surface area of rectangular and triangular prisms, pyramids, and cylinders 1.4 use formulas