interpreting graphs 4 · blm 4–2 degree circle 4.1 choosing a graph 180–240 min (tr page 159)...

4General OutcomeDevelop statistical reasoning.

Specifi c OutcomeS1 Solve problems that involve creating and interpreting graphs, including:

bar graphshistogramsline graphscircle graphs.

General OutcomeDevelop number sense and critical thinking skills.

Specifi c OutcomeN1 Analyze puzzles and games that involve numerical reasoning, using

problem-solving strategies.

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Interpreting Graphs

978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 149

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

Section Understanding Concepts, Skills, and Processes

4.1 create graphs�

choose possible graphs to represent a given data set�

explain the advantages and disadvantages of each type of graph�

4.2 describe trends in a graph�

interpolate and extrapolate values from a graph�

determine if predictions and estimates are reasonable�

4.3 determine if a graph accurately represents data�

explain how the same graph can show more than one conclusion�

explain how a graph can be misrepresented to emphasize a point of view�

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

Chapter 4 Planning Chart

Section/Suggested Timing Prerequisite Skills Materials/Technology

Teacher’s ResourceBlackline Masters

Chapter Opener15–20 min

(TR page 155)•

Students should be familiar withtypes of graphs (bar, line, circle)reading values from a graph

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BLM 4–1 Chapter 4 Self-Assessment

Get Ready60–90 min

(TR page 157)•

Students should be familiar withcalculating and estimating percentsconverting fractions to and from decimalsreading values from a graphcreating types of graphs (bar, line, circle) with and without technology

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calculatorgrid paper or graphing technology

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Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid Paper

Master 4 1 _ 4 Inch Grid Paper

BLM 4–2 Degree Circle

4.1 Choosing a Graph

180–240 min(TR page 159)•

Students should be familiar withcreating types of graphs (bar, line, circle)reading values from a graph

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calculatorgrid paper or graphing technologyruler

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BLM 4–2 Degree CircleBLM 4–3 Chapter 4 Warm-UpBLM 4–4 How to Make a Histogram

in Microsoft ExcelBLM 4–5 Section 4.1 Extra Practice

4.2 Interpolating and Extrapolating Values

180–240 min(TR page 170)•

Students should be familiar withcreating types of graphs (bar, line, circle, histogram)reading values from a graph

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BLM 4–3 Chapter 4 Warm-UpBLM 4–6 Section 4.2 Extra Practice

4.3 Graphic Representations

180–240 min(TR page 178)•

Students should be familiar withcreating types of graphs (bar, line, circle, histogram)reading values from a graph

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BLM 4–3 Chapter 4 Warm-UpBLM 4–7 Section 4.3 Extra Practice

Chapter 4 Skill Check

50–60 min(TR page 185)•


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BLM 4–1 Chapter 4 Self-AssessmentBLM 4–5 Section 4.1 Extra PracticeBLM 4–6 Section 4.2 Extra PracticeBLM 4–7 Section 4.3 Extra Practice

Chapter 4 Test Yourself

45–60 min(TR page 186)•


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BLM 4–1 Chapter 4 Self-AssessmentBLM 4–8 Chapter 4 Test


Exercise Guide Extra Support

Assessment

Assessment as Learning

Assessment for Learning

Assessment of Learning

Math at Work 11 Online Learning Centre

TR page 154 TR page 154

Adapted: #1, #3, #4, #6, #8, #10Typical: #1–#10


TR page 158

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


TR pages 162, 169 TR pages 163, 165–167

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


TR pages 171, 177 TR pages 173–176

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


TR pages 180, 184 TR pages 181–183

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

TR page 185

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: #1–#5, #8

TR page 187 TR page 187BLM 4–8

Chapter 4 Test


Section/Suggested Timing Prerequisite Skills Materials/Technology

Teacher’s ResourceBlackline Masters

Chapter 4 Project60–80 min

(TR page 188)•

grid paper or graphing technologyrulercoloured pencils or other art materialscomputer with word processing or design software

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


BLM 4–9 Chapter 4 Project Checklist

Chapter 4 Games and Puzzles

30–40 min(TR page 190)•

3 colours of pen or 2 colours of pen and a pencil

• Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid Paper


BLM 4–10 Chapter 4 BLM Answers


Exercise Guide Extra Support

Assessment



Assessment of Learning

TR page 189Master 1 Project

Rubric

TR page 190


Assessment Supporting Learning


Use the Before column of BLM 4–1 Chapter 4 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 152–153 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 150–151 in Math at Work 11 to activate students’ prior 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 graphing and what they know about jobs, careers, or hobbies that involve creating and interpreting diff erent types of graphs.

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 4, 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 4–3 Chapter 4 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 from previous chapters, 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 4.1, students learn about choosing the best type of graph to represent a data set. Th ey learn to create a histogram with and without technology. Students compare two graphical representations of a data set and practise creating graphs from given data.

In section 4.2, students examine the trends in a graph and interpret the trend’s meaning. Th ey learn to interpolate and extrapolate values from a graph, and consider the reasonableness of their estimates and predictions.

In section 4.3, students see how graphs can be used to represent data in a way that may be misleading, or as a way to emphasize a particular point of view.

Planning NotesUse the cartoon in the opener to start a discussion on the diff erences and similarities of the two graphs. To help students answer the questions posed in the opener, ask:

What is represented on the horizontal axis? Is it the same for both graphs?What is represented on the vertical axis? Is it the same for both graphs?What was the wage in 2000? in 2006? Do both graphs give the same value?What is the scale for wages on each graph?What is the minimum value for wages represented on each graph?

Th ese questions will help determine why the graphs look diff erent. To help students answer the last two questions, ask:

If you were working for a union trying to raise the salaries of construction workers, which graph would you use? Why?If you were an employer saying that construction workers do not need a raise, which graph would you use? Why?

As a class, discuss the Key Words. Which words do students already know? Which words will need discussion throughout the chapter?

Discuss the photographs and information given in the Career Link. Ask:How is math used in each job?Why must all workers, not just the union representative, know how to interpret graphs?

Meeting Student NeedsIt is important for students to focus on which graph is a more accurate representation of the data presented. Ask students for the criteria by which they would judge one representation to be more accurate than the other.Divide students into three groups—two “companies” and an evaluation group. Have the two companies discuss and then present their representation of the data to the evaluation group. Th e evaluation group can discuss the criteria they used to decide which group was more convincing in their arguments.Ask students how they could fi nd out the average amount that the workers’ wage increased by over the ten-year period. Th e idea of slope may come up here even though it will not be addressed in detail until Chapter 6.

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Math at Work 11, pages 150–151

Suggested Timing15–20 min

Blackline MastersBLM 4–1 Chapter 4

Self-Assessment

Key Wordsdiscrete

histogram

continuous

trend

interpolate

extrapolate


ELLTake the time to review the terms that will come up in this chapter and start to create a word wall. Discuss with students the typical visual representations that they have encountered. Th is will allow you to review terminology connected to these representations. For a line graph, you may discuss the axes, labels, variables (dependent or independent), data (discrete or continuous), and so on.

Gifted and EnrichmentStudents may be able to talk about how much the average increase in wages was in dollars per year, but encourage them to address the question of how much of a percent increase this was. Ask students how they arrived at a value. Some may be clear on what the change in salary was over the ten years, but students are oft en unclear on which number to divide by in order to get the percent increase.

Career Link

Trade union representatives are members of a trade union who represent their fellow workers in dealings with an employer. The main part of the job is negotiating collective agreements, which involves securing fair wages and working conditions. They must also be able to interpret and apply a contract. To get more information about a career as a trade union representative, go to www.mcgrawhill.ca/school/learningcentres and follow the links.

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4Get ReadyCategory Question Numbers

Adapted (minimum questions to cover the outcomes) #1, #3, #4, #6, #8, #10

Typical #1–#10

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: Have students complete the Get Ready exercises as the need arises. For example, do Get Ready questions #4, #6, and #8 before doing the section 4.1 Explore.Method 3: Discuss the questions in the Get Ready and have students brainstorm what they know about these topics already. Organize the information that they share in a graphic organizer or in their notebook.

Meeting Student NeedsBe sure to show students multiple strategies for solving the questions in #2. Some may prefer to set up proportions, whereas others might be more fl exible in their thinking.Help students fi nd convenient numbers to estimate with for #2. For example, to fi nd 35% of 200, fi nd 10% fi rst and then add that value three times.Review the use of protractors and compasses to enhance students’ understanding of how the representation in #5 will be created. You may also wish to provide students with BLM 4–2 Degree Circle to assist them in creating circle graphs.For the Puzzler, have students verbalize the operations needed to complete each set of equations before attempting to complete the tables. Remind them that each column in each table follows the same pattern of operations.

ELLFor #2, discuss that the word of oft en means multiplication. Are there other words that hint at multiplication as the operation (such as and)? What words imply the other operations?

Gifted and EnrichmentGive students multiple scenarios to create discussion about what kind of representation would fi t the scenarios best, given a choice of line graphs, bar graphs, and circle graphs.

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Materialscalculatorgrid paper or graphing technology

Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5 Centimetre

Grid PaperMaster 4 1 _ 4 Inch Grid Paper

BLM 4–2 Degree Circle

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 make rounding errors.

Rx Give students the following numbers to round, reminding them of the rule of rounding up if the place value is 5 or larger:

8.235 (round to the nearest hundredth)0.96 (tenth)1.0281 (thousandth)1.0281(hundredth)40.45 (one)

Students sometimes confuse the terms ascending and descending.Rx Use these defi nitions to help students visualize the meanings:

Ascending: going up in an airplane (or stairs), thus low to high.Descending: going down in an airplane (or stairs), thus high to low.

Students may not scale the axis of a graph uniformly.Rx Have students compare their axes to a ruler. Note the uniform scaling on the

ruler. Th e scale on an axis must be similar.



Get ReadyHave students complete the Get Ready exercise on pages 152–153 in Math at Work 11.

Have students 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.Have students keep a journal of when they personally see examples of graphs in newspapers and magazines, online, and in other media.

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4.1Choosing a GraphCategory Question Numbers

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

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

Planning NotesHave students complete the section 4.1 warm-up questions on BLM 4–3 Chapter 4 Warm-Up to reinforce prerequisite skills needed for this section.As a class, discuss the photograph and the opening text. Do an informal survey, by raising hands, of the methods students use to get to school. Lead a discussion by asking:

How do the methods of transportation aff ect the principal’s decision on the operating hours of the school?How do the methods of transportation aff ect how many buses should be purchased or hired?How would the town or city use this transportation information?Would the principal gather the information like we did with a show of hands? If not, what method may be used?How could this information be organized and displayed?

Explore Graphs for Specifi c SituationsIn this exploration, students collect data, examine diff erent methods of displaying the data, and discuss which method is preferable.

You made need to adjust the intervals in the table depending on the distance travelled by your students, or to explain how a tally and a frequency are related. Point out the diff erence between a bar graph and a histogram. Students may have seen histograms but not be familiar with their use or how to create them. Histograms will be explored in greater detail in On the Job 2.

Students can create their graphs by hand or by using technology. Use BLM 4–2 Degree Circle and one of Master 2 Centimetre Grid Paper, Master 3 0.5 Centimetre GridPaper, or Master 4 1 _ 4 Inch Grid Paper for graphs created by hand. BLM 4–4 Howto Make a Histogram in Microsoft Excel can be used to provide support for creating a histogram using graphing technology.

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Materialscalculatorgrid paper or graphing technologyruler


Grid Paper


BLM 4–2 Degree CircleBLM 4–3 Chapter 4 Warm-UpBLM 4–4 How to Make a

Histogram in Microsoft ExcelBLM 4–5 Section 4.1 Extra

Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)


bar graphshistogramsline graphscircle graphs

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Aft er students have completed their fi rst graph, display the graphs around the classroom according to type of graph. Th is will allow students to complete step 4. Be aware that some students may not accurately represent the data in their graphs. Th is may be why the data appear to be diff erent on some of the graphs.

Step 5 could also be completed by groups looking at and discussing the diff erent type of graphs.

Meeting Student NeedsDiscussion of at least three ways to present the data shown in the table in step 2 is a must. Have students defend the representation they have chosen.Explore reducing the data table in step 2 into only two distance categories, 0–6 km and 6.1–12 km. Ask students what the purpose of having only two categories may be in a real-life scenario. For example, a school division may only give bus passes to students who live more than 6 km away from the school. In this case, more divisions are not required to obtain the necessary data.Create a sample bimodal data set for the table in step 2; that is, one in which there are many students in the fi rst two categories and the last two categories, but very few in the middle ones. Ask students what this might tell them about the school population.

Gifted and EnrichmentHave students try to create questions for which their representation is the best choice. For example,

Th e school district knows that they can only aff ord to bus 30% of the students to the school. What representation would best help them decide who can take the bus to school?Th e school district wants to know the average distance that their students travel to school. Which representation best depicts this?

Common ErrorsStudents may represent the data with a combination of bar graph and histogram components.

Rx Show students an example of a bar graph and an example of a histogram. Discuss the diff erences and similarities between the two graphs by asking:

How are the graphs the same?How are the graphs diff erent?Why are the bars joined in the histogram?Why are the bars not joined in the bar graph?

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Answers

Explore Graphs for Specifi c Situations1. and 2. Example:

Distance Travelled (km) Tally Frequency

0.0–2.0 |||| |||| |||| | 16

2.1–4.0 |||| ||| 8

4.1–6.0 ||| 3

6.1–8.0 ||| 3

8.1–10.0 | 1

10.1–12.0 | 1

3. Examples:

a) Bar graphb) Example: A bar graph shows the frequency of each range accurately, and it shows that each

range is independent from the other ranges.4. Examples:

a) line graphs or circle graphsb) A circle graph may have been used to show the percent of students who must travel each

distance range.c) Although the data should all be identical, a line graph can give the impression of time passing.

5. Example:

Type of Graph Advantages Disadvantages

Bar graph • Frequency is easy to identify accurately• Easy to compare frequencies of diff erent

distance ranges

• Data are not shown as a percent of the whole• Graph can imply trends that may not exist (i.e., the

farther the distance to school, the fewer students)

Circle graph • Shows percent of total for each distance range (easy to see which range has the most students)

• Visually interesting

• The precise frequency is not easy to identify• Some distance ranges that have the same

frequency may appear to have diff erent frequencies

6. Examples:a)

b) Th e exact number of students in each distance category is not shown, but it is apparent that half of the class travels 2 km or less to get to school.

Freq

uenc

y

1816141210

86420

Distance Travelled (km)0.0–2.0 2.1–4.0 4.1–6.0 6.1–8.0 8.1–10.010.1–12.0

How Far Do You TravelFrom Home to School?

Freq

uenc

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1816141210

86420

Distance Travelled (km)0.0–2.0 2.1–4.0 4.1–6.0 6.1–8.0 8.1–10.010.1–12.0

How Far Do You TravelFrom Home to School?

How Many Kilometres Do YouTravel From Home to School?

0.0–2.050%

2.1–4.025%

4.1–6.010%

6.1–8.09%

8.1–10.03%

10.1–12.03%

How Many Kilometres Do YouTravel From Home to School?

0.0–2.050%

2.1–4.025%

4.1–6.010%

6.1–8.09%

8.1–10.03%

10.1–12.03%




Refl ectPost the graphs students have drawn under categories, such as circle, bar, line, and histogram, throughout the classroom.

Have students share their list of advantages and disadvantages after they have had time to think of a list independently.Use the posted graphs to discuss the advantages and disadvantages of each as a group.

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Extend Your UnderstandingSome students may have chosen the most appropriate type of graph for their fi rst graph. Suggest they try another type of graph so they can see the disadvantages of the second type.

Have students state which type of graph they think is the most appropriate and why.

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On the Job 1Open the class with a discussion about charities and how they receive funding. Some students may have personal experiences working for charities, or know someone who has.

Encourage students to examine the features of the circle graph and the bar graph. You may wish to have them practise creating the circle graph by hand and the bar graph using graphing technology. Supply students with BLM 4–2 Degree Circle for assistance in creating a circle graph. Ask students if other types of graphs, such as those encountered in the Get Ready and the Explore, would be suitable for Martha to use. Ensure they justify their choices.

For part b), ask:How can you defi ne discrete in your own words?If the data are not discrete, what are they called?Can you think of other advantages or disadvantages for the types of graphs shown in part a)?Using the list of advantages and disadvantages, which type of graph best represents the data? Explain.

For the Your Turn, remind students to scale the axes of their graphs uniformly, if they are drawing a bar or line graph. When students are fi nished drawing their graph, do a survey to see the diff erent types of graphs drawn and which type was drawn most frequently. Ask students to support their choice for the best type of graph to use by listing its advantages, and then listing the disadvantages of the other types of graphs.

Meeting Student NeedsTh is section focuses on comparing a bar graph to a circle graph. Th e strength of a bar graph is that it communicates the totals in a situation. Th e strength of a circle graph is that it communicates the relative comparison as a percent of the whole.Remind students that they may consider the time and diffi culty in producing each type of graph when deciding which type of graph to draw.Have visual learners cut out photocopied circle graphs in order to compare the relative sizes of the things being compared. By folding the sectors they may be able to discover whether one sector is double another and so on.

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Help guide students through the process of creating a circle graph for Martha’s presentation by changing the donation values into percents prior to creating the circle graph. Doing this example on an interactive whiteboard is an excellent demonstration for students. Involve them in the process.

ELLYou may need to explain the term sector to students completing a circle graph by hand. A visual example, such as a slice of pie or pizza, may be helpful.

Answers

On the Job 1: Your TurnExamples:a) A bar graph is best.b) Diff erent values can be easily compared.c)



On the Job 1Have the students do the Your Turn. Check that

If students graphed the data as a line or bar graph, they included a title, labelled and scaled the axes uniformly, and plotted the data accurately.If students graphed the data as a circle graph, they included a table with conversions to percent, a legend (or labelled the sections of the circle), and a title, and that they drew the sections of the circle accurately.

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Students may benefi t by working with a partner.Provide a labelled and scaled grid if students are drawing a bar or line graph.Make graphing technology available and provide instruction, if necessary, on how to use it.

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Check Your UnderstandingTry ItFor #1, have students recall the discussion about how to create accurate graphs by hand or with technology.

For #2, have students recall the advantages and disadvantages listed in On the Job 1. Th ey can use this information to help justify their choice for this question.

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ber o

f Wor

kers

12 00010 000

8 0006 0004 0002 000

0

Type of Job

Number of People Employedin Various Types of Jobs in PEI

Constru

ction

Educati

on

Food an

d hosp

itality

Health

care

Manuf

acturing

Public a

dminis

tration

Trades

Num

ber o

f Wor

kers

12 00010 000

8 0006 0004 0002 000

0

Type of Job

Number of People Employedin Various Types of Jobs in PEI

Constru

ction

Educati

on

Food an

d hosp

itality

Health

care

Manuf

acturing

Public a

dminis

tration

Trades


For #3, ask students if any of the types of graphs that they have been using would allow for exact representation of the data. Suggest that students round the data to the nearest hundred or thousand. Since students are choosing the best type of graph for the data, they may have trouble selecting the most appropriate type of graph. Refer students to the advantages and disadvantages listed in part b) of On the Job 1.

Apply ItTh ese questions require students to apply their knowledge of choosing the most appropriate graph. Consider having students work in pairs. Have each student graph and answer the questions, and then exchange their knowledge about the problem and its graph with their partner.

For #4, remind students what types of graph serve what purposes:Circle graphs: discrete data, showing parts of a wholeBar graphs: discrete data, showing data in categoriesLine graphs: continuous data, showing changes in data over time

For part c), ask how the number of people who voted for each category could be determined when the total number of people who voted is known, as well as the percent for each category.

Meeting Student NeedsFor students who have diffi culty recording their thoughts, pair them with another student so they can describe their methods orally.Provide students with BLM 4–2 Degree Circle.You may want to provide students with rounded values.Provide students with labelled and scaled axes on grid paper.

Gifted and EnrichmentIt can be benefi cial for students to work backward from a representation to the question itself. Th is opens up the question to critical thinking pathways and can help students connect to the diff erent types of representations and their strengths and weaknesses.

Common ErrorsStudents may make rounding errors.

Rx Give the students the following numbers to round, reminding them of the rule of rounding up if the place value is 5 or larger:

8.235 (round to the nearest hundredth)0.96 (tenth)1.0281 (thousandth)1.0281(hundredth)40.45 (one)

Students may not scale the axis of a graph uniformly.Rx Have students compare their axes to a ruler. Have them note the uniform scaling

on the ruler. Th e scale on an axis must be similar.

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

Encourage students to use graphing technology.Students may need to work through questions using more than one type of graph before they can select the best type for the data.

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On the Job 2Discuss the photograph and ask:

When will the next Canada Winter Games be held?Where will the games be held?

Lead students through a discussion of the data in the table. Ask:What is the age range of the players?What is the height range of the players?Who might want to know this information? Why?

Have students consider which types of data would and would not be appropriate to display in a graph on the Canada Winter Games web site.

For part b), you may wish to have students explore creating a histogram in Microsoft Excel, and then compare the two processes. Provide students with BLM 4–4 How to Make a Histogram in Microsoft Excel.

For the Your Turn, if students choose to create a histogram, they will need to graph the frequency as a range of values. For example,

Fish Length (cm) Frequency

19.5–29.5

29.5–39.5

39.5–49.5

49.5–59.5

59.5–69.5

69.5–79.5

Students may need guidance with setting up the ranges for the frequency table. Ensure they consider the range of values and the most frequent values.

Meeting Student NeedsUse a template of the table with the heights of the hockey players to start the discussion of this section. Ask students to represent the data with any type of graph they have studied, provided it is suitable to answer a specifi c question they have in mind.Students may need extra time to work with a spreadsheet in order to create a table and a histogram. Th is is an excellent opportunity to get students comfortable with building a table and using the chart wizard as opposed to talking them through it.For the Your Turn, ask questions to prompt students to think about the best ways to represent the data. For example,

To fi nd the percent of cod under the size limit, what type of representation would be best?What is the most common size of cod to come out of Trinity Bay?

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ELLTh e use of the words discrete and continuous with respect to data is an extremely diffi cult concept for many students. Take the time to discuss and give examples of situations that involve discrete and continuous data. Some students may say that since there are no data in our table between the heights, the points should not be connected. Take plenty of time to discuss the defi nitions of discrete and continuous given in the student resource, and place as many examples as possible around the classroom.

Answers

On the Job 2: Your Turna) Example: a histogram could be appropriate to show frequency of diff erent sizes of fi sh.b) Example:

c) 7d) Example: Only about 58% of the cod that they caught were large enough to keep.



On the Job 2Have students do the Your Turn. Check that students include the tables they used to draw the graphs, and that the graphs include a title, appropriately scaled and labelled axes, and accurately plotted data.

Provide a range for the data.Provide a labelled and scaled grid if students are drawing a bar or line graph.Make graphing technology available and provide instruction, if necessary, on how to use it.

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Check Your UnderstandingTry ItFor #1a), remind students of the purpose of the diff erent types of graphs:

Circle graph: discrete data, showing parts of a wholeBar graph: discrete data, showing data in categoriesLine graph: continuous data, showing changes in data over timeHistogram: frequency of a range of data, usually continuous data

For #4, ask students:Do you own a car?How much insurance do you pay on the car?What is the purpose of the insurance?If you do not own a car, do your parents have to pay insurance so you can drive the family car?

For #4a), you may need to discuss Morgan’s purpose for drawing graphs.

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Freq

uenc

y

6543210

Length ofFish Caught

Cod Length (cm)

54.5or less

54.5–59.5

59.5–64.5

64.5–or more

Freq

uenc

y

6543210

Length ofFish Caught

Cod Length (cm)

54.5or less

54.5–59.5

59.5–64.5

64.5–or more


Apply ItTh ese questions require students to apply their knowledge of choosing the most appropriate graph. Consider having students work in pairs. Have each student graph and answer the questions, and then exchange their knowledge about the problem and its graph with their partner.

For #5c), some students may fi nd the diff erence of the two percent dropout rates, while others may fi nd how many times larger the rate was in 1990–1991 than in 2009–2010.

Meeting Student NeedsSome students may need help with rounding data. Review the rules for rounding.Review what types of graphs serve what purpose. See the notes for #1 above.Prepare the data sets and graphs that you will be discussing with students in a presentation mode using overhead transparencies, an interactive whiteboard, or a computer graphing program that can be projected on a large screen.

Gifted and EnrichmentStudents could create a circle graph using Microsoft Excel and data they have researched on the Internet.

Common ErrorsStudents may use discrete data to draw a continuous graph.

Rx To help students decide whether the data are continuous, ask:Can there be values between the known values?Can there be a fractional number of people/cars/animals/etc.?Can a value between two known values be a decimal or a fraction?

If the answer to any of these questions is no, then the data cannot be continuous.



Try ItStudents should be able to answer #1, #2, and #5.

For #1, encourage students to sketch the graph for each scenario.Allow students to work in pairs.Provide students with labelled and scaled axes on grid paper.Encourage students to describe the data before selecting an appropriate type of graph.

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Work With ItStudents have now completed On the Job 1, 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 On the Job 1 and On the Job 2 in practical situations.

For #1, ask students to add the percents in the table. Th en, ask:What is the total of the “% of Volunteers” column?Why does the total not equal 100%?Even though the data are represented in percents, is the data set suitable for a circle graph? Why or why not?

Give students a number other than 440, and ask them to predict the number of volunteers in each category assuming the same percent distribution.

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For #2a) and b), ask:Could either graph be used to answer the question? If yes, explain how.How many times was each sport viewed? How did you determine the answer?

For #3a), remind students of how to determine whether data are continuous. Th en, ask them to recall the types of graph suitable for each type of data.

Discuss ItTh ese questions give students an opportunity to explain their understanding of choosing an appropriate graph for given data. Look for reasonableness and justifi cation of answers. Some students may benefi t from a class or group discussion prior to recording their own answers.

For #4, it may be benefi cial to discuss the responsibilities of a movie theatre manager. Th ese could include

scheduling staff and movie timesmarketing and advertisingsecurityemployee interviews and trainingordering suppliespaperwork related to accounting, bank deposits, and evaluating performance

Students use their imagination to develop a scenario to fi t the data in #5. Remind students to consider the following:

Is the data set represented as discrete or continuous?What labels will you provide for the vertical axis?What is the title of the graph?Does the scenario include a completed data set?

Meeting Student NeedsAs a class, create a checklist for students to use when deciding which type of graph to use to represent a data set.

Question Answer Type of Graph

Is the data set discrete? Yes 1: Bar2: Circle

Is the data set continuous? Yes 1: Line2: Histogram

Do you want to show part of a whole? Yes 1: Circle2: Bar

Do you want to show a change over time? Yes 1: Line2: Bar

Do you want to show data in categories? Yes 1: Bar2: Circle

Do you want to show the frequency of a range of data? Yes 1: Histogram

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, 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.

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Prepare data sets and graphs that you will be discussing with students in a presentation format, using overhead transparencies, an interactive whiteboard, or a graphing technology program that can be projected on a large screen.Provide BLM 4–5 Section 4.1 Extra Practice to students who would benefi t from more practice.

Gifted and EnrichmentEncourage students to conduct a survey of their classmates with a question similar to those encountered in this section. For example, “How much money do you spend on entertainment each month?”Have students record the results in a table similar to the one below, depending on the range of answers given.

Amount Spent on Entertainment Tally Frequency

0.50–20.50

20.50–40.50

40.50–60.50

60.50–80.50



Discuss ItThese questions provide students with an opportunity to explain their thinking verbally or by producing graphs. Have all students complete #1, #2, and #4.

In #1, students may need an explanation for why a circle graph may not be appropriate even though the data set is given as percents.In #2, students may not notice that both graphs could be used to answer the questions. Discuss the possibility that more than one type of graph can represent the data appropriately.In #4, discussion of the responsibilities of a theatre manager may help students determine which type of graph to choose.

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4.2 Interpolating and Extrapolating ValuesCategory Question Numbers

Adapted (minimum questions to cover the outcomes) Explore #1–#4On the Job 1 #1, #2, #5On the Job 2 #1–#3Work With It #1, #2

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

Planning NotesHave students complete the section 4.2 warm-up questions on BLM 4–3 Chapter 4 Warm-Up to reinforce prerequisite skills needed for this section.

As a class, discuss the photograph, graphs, and opening text.Discuss the defi nition of trend, and how trends apply to mathematics.Discuss the diff erence between a positive and negative trend.

Students may have a preferred method of creating graphs. Give those who prefer to graph by hand Master 2 Centimetre Grid Paper, Master 3 0.5 Centimetre GridPaper, or Master 4 1 _ 4 Inch Grid Paper.

Explore the Trends in a Set of DataTh e purpose of the Explore is for students to examine data in a table and a graph to

determine a trend, if one existsdetermine the direction of a trend (increasing or decreasing)estimate values from the graph between known values (interpolate)predict values from the graph beyond known values (extrapolate)

Put students in pairs or small groups. Have each group examine the data from one year to the next year. Ask them to describe the diff erence between the CPI for their two years as increased, decreased, or not changed. Th en, have the entire class discuss their fi ndings to come up with a general trend for the data.

As you discuss step 2a), you may want to introduce the term extrapolate, which is defi ned in On the Job 1.

As you discuss step 2b), you may want to introduce the term interpolate, which is also defi ned in On the Job 1.

For step 5, some students may mistakenly assume that data must be in a straight line for a trend to exist. Remind students that a trend is a general pattern in the data.

For step 6a), have students share their fi ndings or do the research with students and display it on the interactive whiteboard or projected computer screen. For part b), students should compare their predications with the actual numbers. An informal survey could be done to see how close students were to the actual numbers.

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Grid Paper


BLM 4–3 Chapter 4 Warm-UpBLM 4–6 Section 4.2 Extra

Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)



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Meeting Student NeedsEnsure that the introduction to the exploration involves an opportunity for students to use the web link provided to research the nature of the CPI and how it is calculated.Project the graph and have students come to the front and use meter sticks to demonstrate how to predict a value of the CPI given a year in which it has not been measured.

ELLTrend is a diffi cult word for students, since it implies direction more than a precise pattern. Try to connect students to the societal notion of trends that increase and decrease.

Gifted and EnrichmentAsk students:

What would be the change to the trend if we disregarded the information for 1996 and 2010?How would this change our prediction for the years aft er 2010 or prior to 1996?

Have one group of students make projections using all data except 1996 and 2010. Have another group make projections using the original data.

Common ErrorsStudents may confuse negative and positive trends.

Rx Have students think about a staircase and relate it to the trend. Going up from one step to the next causes one to rise above the fl oor. Going down from one step to the next causes one to come closer to the fl oor.

Answers

Explore the Trends in a Set of Data

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1. Th e Canadian CPI is increasing.2. a) Example: 2012: 120; 2016: 130

b) Example: 753. Example: 102.54. a) Example: 112

b) Example: 905. Example: Th e CPI does not necessarily

increase by the same amount each year.

6. a) 1997: 90.37; 2003: 102.75; 2007: 111.45b) Example: Th e predictions were fairly

close to the actual values.c) Example: Th e trend seems to be

increasing less quickly for the past two years, so the CPI may not increase as much in the next few years.



Refl ectListen to how students approach the Refl ect question. How much detail is needed to inform their opinion of the trend in the data? Prompt students to look generally at its direction.

Discuss why a graph with a visible trend does not always follow a straight line.

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Extend Your UnderstandingListen as students discuss their fi ndings from the Explore. Encourage students to generalize and reach a conclusion about their fi ndings.

Rather than asking students to research the CPI for various years independently, have student groups research and present their fi ndings to the class.If students’ predictions were not very close to the actual values, prompt discussion of why this happened:

What unit was the time axis scaled by?Did the actual CPI value follow the trend?

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On the Job 1Ask students to examine the data. Ask:

Was the height information recorded at the same time intervals, that is, every second day or every third day?Can you tell the rate of growth by examining the data in the table? Why or why not?

Aft er discussing the meaning of interpolate, ask students where they interpolated in the Explore. Do the same with the word extrapolate.

Prepare a graph for display on an overhead, interactive white board, or projection screen, so you can refer to it as the questions and solutions are discussed. As you discuss the solution with students, ask:

What was the scale used for the horizontal axis?Was there a height associated with each of the scaled dates on the horizontal axis? Why or why not?

On the graph, point out the steps used to interpolate to fi nd the possible height of the plant on November 28 and December 5. Remind students that interpolated values are estimates.

On the graph, point out the steps used to extrapolate to fi nd the possible heights on December 13 and 25. Emphasize that these values are predictions, made by assuming the trend shown for the known values continues. Discuss why the extrapolated values may not be accurate:

Could the plant have reached its maximum height on or before December 13? If so, what would the trend look like?If maximum height was reached on or before December 13, what would be the height on December 25?

For the Your Turn, suggest that studentsPut speed on the horizontal axis and distance on the vertical axis.Scale and label the horizontal axis by showing a break in the graph, and starting at 68 km/h, using an interval of 2 km/h. For the vertical axis, do the same and start the distance at 35 m with an interval of 5 m.Since the question involves extrapolating, remind students not to end the ranges of their axes at the smallest or largest data value given. Leave room on the graph for extensions.

Meeting Student NeedsStress that the process of interpolation or extrapolation can only be done based on a clear visualization of the data. Students must make a decision about what type of model will work best based on the trends in the data points.Some students will gravitate toward using a visual approach to decide on the interpolated points between two known values; others may be comfortable using a more algebraic method.Have students attempt the interpolations and extrapolations at the board to let them model their technique to other students.

ELLTh e terms interpolate and extrapolate may be diffi cult for some students. Make connections with the prefi xes, such as inter for inside the points and extra for outside the points.

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Gifted and EnrichmentHave students debate whether they agree with the extrapolation of the amaryllis data beyond December 7. Some students may have noticed that there is a plateau occurring in the data between November 25 and December 5 and thus would predict a similar occurrence somewhere in the middle of December. Th is would suggest that the student resource’s extrapolation of 140 cm may be an overestimate.

Answers

On the Job 1: Your Turn

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a) b) Example: 60 mc) Example: 45 md) Example: the upper limit of the

distance that Cory can throw the ball, or the upper limit of the speed at which Cory can throw the ball.

Dis

tanc

e (m

)

100908070605040302010

0

Speed (km/h)

Speed and Distance of Cory’s Throws

70 72 75 80 83 85

Dis

tanc

e (m

)

100908070605040302010

0

Speed (km/h)

Speed and Distance of Cory’s Throws

70 72 75 80 83 85



On the Job 1Have students do the Your Turn. Check that

the axes are labelled and scaled uniformlythe values are plotted accuratelythe interpolated and extrapolated values are reasonable

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You may wish to have students work in pairs.Encourage students to verbalize their thinking, discuss, and compare their answers with those of their partner.Suggest students study the On the Job 1 questions and solution before seeking assistance.Using two rulers, one for the horizontal axis and one for the vertical axis, may help students fi nd unknown values.

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Check Your UnderstandingTry ItFor #1, students will only use interpolation. Suggest students fi nd the values by using the same methods shown in the examples in On the Job 1.

Apply ItTh ese questions allow students to apply their knowledge of interpolation and extrapolation of graphs to solve problems. Consider having students work in pairs, so they can share and compare methods and knowledge.

Discuss stopping distance in #4d). Ask:What aff ects reaction time?What aff ects the length of time it takes the brakes to engage and bring the vehicle to a stop?

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Meeting Student NeedsHave students attempt #4 using a technology tool to create their graph.Suggest that students use two rulers, one for the horizontal axis and one for the vertical axis, to help them fi nd unknown values.

Common ErrorsStudents may make mistakes reading values from a graph.

Rx Suggest that students do the following when reading values:Use a ruler.Ask themselves: Is the value that I am reading reasonable? Does it follow the trend of the data?



Try ItDo #1, #2, and #4.

Encourage students to verbalize their thinking, discuss, and compare their answers with those of their partner.Suggest students study the On the Job 1 questions and solutions before seeking assistance.Encourage students to pay attention to the values on the axes and what they represent.

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On the Job 2Discuss with students the picture of the data analyst. You may wish to supply more information about the career of a data analyst. For more information, go to www.mcgrawhill.ca/school/learningcentres and follow the links.

Ask the following questions about the data:What age can a male who was 65 in 2000 expect to live to?What age can a male who was 65 in 2008 expect to live to?Why do you think life expectancy varies by gender?Why do you think life expectancy is aff ected by the country you live in?

Students will need to identify whether the data are continuous or discrete. Discuss why the data are continuous by asking:

Is the remaining life expectancy limited to whole numbers?Is it possible to use the graph to interpolate what the life expectancy was in mid-2002? What was it?

For the Your Turn, suggest the following strategy:Round the values to the nearest hundred.Determine whether the data are continuous or discrete to help decide what type of graph to use.

To answer part d), refer students to part c) of On the Job 2.

Meeting Student NeedsDiscuss with students whether they think that Chelsea’s displayed graph is somewhat misleading. Is there a way to draw the graph so the increase in life expectancy is not quite so dramatic? Students could suggest starting the vertical-axis data at the origin or increasing the scale.Th e Your Turn questions may lend themselves to a research type of assessment, in which students have the opportunity to not only interpolate and extrapolate but also to analyse a trend and its regularities and irregularities.

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ELLTh e idea of a life expectancy aft er a certain time frame may be confusing to some students. Ask: If a person turned 65 in the year 2000, what age would you expect them to live to? What about a person who turned 65 in 2008?At fi rst, students may think these data refer to the aging of a particular person. Help them to understand that the data are collected province-wide.

Answers


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a)

Num

ber o

f Fem

ale

Ath

lete

s 5000

4000

3000

2000

1000

0

Year

Number of Female Athletes in theSummer Olympic Games

1980 1984 1988 1992 1996 2000 2004 2008

Num

ber o

f Fem

ale

Ath

lete

s 5000

4000

3000

2000

1000

0

Year

Number of Female Athletes in theSummer Olympic Games

1980 1984 1988 1992 1996 2000 2004 2008

b) Th ere is an upward, or positive, trend.

c) Example: More sports are open to female competitors

d) Example: Eventually, the number of female athletes will level out because there is a limit on the number of competitors at the Olympic Games.



On the Job 2Have students do the Your Turn. Check that

the axes are labelled and scaled uniformlythe values are plotted accuratelythe answers given are reasonable

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You may wish to have students work in pairs.Encourage students to verbalize their thinking, discuss, and compare their answers with those of their partner.Suggest students study the On the Job 2 questions and solution before seeking assistance.Change the values in the question to make them easier to graph.Allow students to predict their answer for part b) before creating the graph.Students may need to create more than one graph to help them identify the trend in part b).

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Check Your UnderstandingTry ItFor #1, suggest students use the terms increasing, decreasing, levelling off , small change, large change, or steady change when describing and comparing the trends.

For #2, suggest students use the same terms as in #1 to describe the change in temperature.

For #3, when discussing an appropriate graph to represent the data, ask students:Do the percents add up to 100? Why or why not?Since the data involve percents, is a circle graph suitable? Why or why not?Do the data represent a change over time? Why or why not?What type(s) of graph would be suitable to represent the data?

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Apply ItTh ese questions allow students to apply their knowledge of identifying trends in graphs to solve problems. Consider letting students work with a partner, so they can share and compare methods and knowledge.

For #4c), encourage students to think of reasons there may or may not be a need to increase funding other than the number of registrants increasing. For example,

Can the price of the registration fees be increased?Does more equipment need to be purchased?Do more instructors need to be hired?

Meeting Student NeedsFor #2, some students may have diffi culty drawing the bar graph by hand since it involves negative quantities. You may choose to prepare a labelled and scaled grid for the data, or show students a bar graph that was prepared using graphing technology.For #3 and #4, to help students decide what type of graph would be appropriate, ask:

Are the data discrete or continuous?Do you want to show a part of a whole?Do you want to show change over time?Do you want to look at data in categories?Do you want to show frequency of data?

Gifted and EnrichmentAsk students to fi nd examples of graphs in a media source such as a magazine, a newspaper, or the Internet that show the following:

a positive trenda negative trendno trend

Students could present the graphs to the class with explanations of the purpose of the graphs and the trends shown.



Try ItStudents should be able to correctly answer #1, #3, and #4 before moving on to the rest of the questions.

Encourage students to verbalize their thinking, discuss, and compare their answers with those of a classmate.For #1, students may need to cover up part of the graph so they can focus on a particular section.For #4c), encourage students to consider the state of the swimming program both if it receives increased funding and if it does not before determining their answer.

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Work With ItFor #1, discuss:

Why were no Olympic Games held in 1944?What time period do the data cover? Suggest that when data cover such a long time period, there may be instances where there are slight deviations to the trend.What would be a trustworthy online location to fi nd the answer to part e)?

For #2, ask the following question, which leads in to the topic in section 4.3: How could the graph be redrawn, accurately, to show a stronger trend?

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Suggest students do research to fi nd the oil production in 2010. Discuss why the trend may not continue indefi nitely. Example:

reserves run drythe growing use of alternative fuelpolitical reasons, such as war or the decision to not export oilthe cost of recovery may not justify the production

Discuss ItMeeting Student Needs

For #3, some students may assume that all trends are linear. Depending on students’ knowledge of exponential graphing, they may see a defi nite positive trend, but not recognize it as increasing so rapidly. You may wish to show students other examples of exponential graphs, such as the graphs of the squares or cubes of numbers.Provide BLM 4–6 Section 4.2 Extra Practice to students who would benefi t from more practice.

Gifted and EnrichmentFor #1, suggest students research another Olympic event, record the data in a table, graph it, and prepare questions that involve trends, interpolation, and extrapolation.For #2, suggest that students fi nd oil production data for two other countries and compare them to Canadian production through graphs.

Common ErrorsStudents may become confused about which variable they are examining when looking for a trend in the data.

Rx Remind students to use the axes to determine what variable they are examining.



Discuss ItThese questions give students an opportunity to explain their thinking verbally or by graphing. Have all students complete #1 and #2.

Encourage students to use class discussions as a source of ideas when preparing their answers.Consider having students work in pairs to share methods and ideas.

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4.3 Graphic RepresentationsCategory Question Numbers

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

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

Planning NotesHave students complete the section 4.3 warm-up questions on BLM 4–3 Chapter 4 Warm-Up to reinforce prerequisite skills needed for this section.

Discuss the opening picture and text. Lead the discussion with the following questions:What is recycling?How is it good for the environment?What items are recycled?What incentives are off ered to people and companies who recycle?What information would Anna and Yuri collect in their research?How could they organize this information?

Explore Using Graphs to Accurately Represent DataIn this exploration, students examine two graphs of the same data. One of the graphs misrepresents the data either by the scaling of the axis or by the width of the bars in a graph. Students determine how the graphs are similar and how they are diff erent. Students are then asked how a graph could be drawn to infl uence the interpretation of the data.

Step 3 may lead to some discussion about accuracy. Ask:Are the data represented accurately in both graphs? Check a few dates and amounts to be sure.What is the trend in each graph?Does one graph show a stronger trend than the other? If so, which one? Explain why.

Aft er students make their prediction in step 5, graph the data with the new scale as a class, or ask students to do it individually. Th ey can then check their prediction. Display the two graphs given and the graph drawn for step 5 to help students complete step 6. 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 when they are graphing by hand.

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Grid Paper


BLM 4–3 Chapter 4 Warm-UpBLM 4–7 Section 4.3 Extra

Practice


Connections (CN)



Reasoning (R)

Technology (T)

Visualization (V)



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Meeting Student NeedsAt the beginning of this section of the chapter, consider having students collect samples of misleading graphic displays from newspapers or magazines.It is important to make connections to the two most common ways of changing the look of data in line graphs. Students must be aware of where a scale starts and the relative size of the scaling on the axes.It is important to make the connection to a common way of changing the look of data in bar graphs. Students must be aware of the fact that the width of the bar does not aff ect its value.

ELLMany students have not had much exposure to measurements on axes that are adjusted to make the actual numbers appear more reasonable (in this case 100s of tonnes). It is important to highlight this to students and see whether they can correctly interpret this axis.

Gifted and EnrichmentStudents could research the following regarding recycling in their school or community:

How many recycling locations are there?What type of recycling is done at these locations?

Have students conduct a survey of family and friends regarding recycling. Graph the results of the survey using a method that represents the data accurately. Th en, graph the results of the survey using a method that misrepresents the data.

Answers

Explore Using Graphs to Accurately Represent Data

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1. Example: Th e x-axis values, title of the graph, and axes labels are all the same, as well as the points that are plotted on the graph.

2. Example: Th e steepness of the lines and the y-axis values are diff erent.

3. Example: Th e fi rst graph gives a more accurate representation of the trend because the second graph gives the viewer the impression that very little was recycled in 2000, compared to the amount recycled now.

4. a) Example: Th e graphs have the same axis labels and values, and the same title. Th e “bottles” bar in the fi rst graph is much wider than the “cans” bar.

b) Example: Th e “bottles” bar appears to take up about the same amount of space on the graph as the “cans” bar, but the “bottles” value is actually much less than the “cans” value.

5. Example: Th e value of the “bottles” bar will be much higher because bottles weigh more than cans.

6. Example: Values that are very close may appear very diff erent if the scale does not start at zero.

7. a) Anna’s fi rst graph supports this statement.b) Yuri’s fi rst graph supports this statement.c) Yuri’s second graph supports this

statement.8. Example: Businesses that would like to see

more recycling facilities built could use Yuri’s fi rst graph to promote their views.




Refl ectNote the type of discussions students have on how the scale on a graph aff ects the analysis.

Ask students to prepare the graph with the suggested scaling in step 5. From this graph and the two line graphs provided with the Explore, students can check whether their prediction was correct.

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Extend Your UnderstandingListen to students’ discussions to ensure that they have understood the purpose of the Explore, which is to show that data can be misrepresented in diff erent ways.

Provide students with other examples of misrepresented data. These examples could be displayed in the classroom.

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On the Job 1Using the illustration and introductory paragraph, ask:

How many text messages do you send a day?What charge do you pay per month for sending text messages?Compare the number of messages you send per day to the daily average given in the table. Are you above, below, or very close to the averages?

Before students examine the solution to part a), ask them to state their own observations. Do the same for part b). Aft er this, they can compare their answers to the solutions given.

Th e answer to part c) could vary depending on the observations that students make in part a). Explain that the solution given is not the only possible solution.

Suggest that students complete the Your Turn using the solution to On the Job 1 as a guide. Compare and discuss the diff erent graphs that students produce for Your Turn part c). Ask each student:

How does your graph diff er from the one given?Does your graph support the title of the graph?

Meeting Student NeedsAsk students to consider the following when making their observations in part a):

Is there really enough information here to be confi dent that boys and girls in PEI send the fewest messages?Are the axes scaled appropriately?What would happen if the double bars were combined to create only one bar for each province that represented the texts of all the people surveyed?

Use the Your Turn to help improve students’ ability to recognize misleading graphs. Ask students to count the number of squares coloured in the advertising bar and the newsstand bar (18 squares vs. 4.5 squares).

ELLWatch for students who misinterpret the meaning of “average daily text message rates.”Some students may think that the vertical axis shows how many people a day send text messages. Ask students to interpret the graph verbally in their own words.

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Answers

On the Job 1: Your Turna) Example: It looks like the newspaper

gets almost twice as much money from advertising as it gets from subscriptions.

b) Advertising: 45%, Newsstand sales: 23%, Subscriptions: 32%

c) Example: No. Th e graph makes it look like the newsstand sales and subscriptions total less than the money earned from advertising.

d) Example: circle graph Where We Get Our Money

Advertising45%

Newstandsales23%

Subscriptions32%

Where We Get Our Money

Advertising45%

Newstandsales23%

Subscriptions32%



On the Job 1Have students do the Your Turn. Make sure students present reasonable conclusions and create accurate graphs.

Encourage students to talk about their conclusions to parts a) and b) of the Your Turn before putting the conclusions to paper.Provide students with a scale to use for their graph in part c).

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Check Your UnderstandingTry ItFor #1a), students may incorrectly agree that store B is spending more money on wages than store A because the percent spent for store B is greater than the percent spent for store A. Ask:

Are the total expenditures for each store given?If the total expenditures for store A were $100 000, what amount would be spent on wages?If the total expenditures for store B were $50 000, what amount would be spent on wages?Can a conclusion about the amount spent on wages be made without knowing the total expenditures?

For #2b), ask students:Can you read the number of tablet PCs sold from the line graph? Why or why not?Can you read the number of laptop PCs sold from the line graph? Why or why not?Which graph allows you to read the number of each product sold?

For #3, ask students to refer to their answers to #1 before answering part b).

Apply ItIn #4, students apply their knowledge of the diff erent methods of representing data and of how some of these methods can lead to misinterpretation. You may suggest students work with a partner so a discussion and comparison of the methods can occur. Ask students to refer to the answers to On the Job 1 when they answer parts a) and b).

For #5, remind students of the diff erent methods used to represent data by referring back to On the Job 1 and the Your Turn.

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Meeting Student NeedsFor #1c), students may need suggestions to determine how the graph could be drawn:

If comparing wages only, how many pieces of data will be represented in the graph?Could a line graph give a suitable representation of the data? Why or why not?Could a circle graph give a suitable representation of the data? Why or why not?Which graph would be suitable? Explain.

For #2, some students may say that both graphs represent the data equally well because the same scale and minimum value are used.



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

You may wish to have students work with a partner.Suggest students study the On the Job 1 questions and solutions before seeking assistance.Students may struggle to answer #1a) realizing that they do not have enough information to accurately answer the question. You may want to have students start with part b) and omit part a).For #2b), encourage students to describe an advantage of each type of graph.

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On the Job 2Using the illustration and opening text, ask:

Do you know anyone who is a boilermaker?What do you know about the trade?

Before students examine the solution to part a), ask them to state their own observations. Do the same for part b). Th ey can then compare their answers to the solutions given. Th e answer to part c) could vary depending on the observations students made in part a). Explain that the solution given is not the only possible solution.

For the Your Turn, suggest that studentsread from the graph(s) the number of sales for each car dealership per monthdetermine the total sales for each car dealershipcompare the total car sales

Meeting Student NeedsEncourage students to recall which type of graph is better to use for diff erent situations, such as when considering the total amount and when comparing a part to the whole. You may wish to post diff erent types of graphs and summaries of their relative strengths and weaknesses in the classroom.

ELLIt can be helpful to students to use a consistent and methodical approach to solve the critical thinking questions in this section. Try to start by discussing the similarities and diff erences when a choice of alternative graphing scenarios is presented.

Answers


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a) Example: Th e graph makes it seem that All-Star Cars has much higher sales than its competitor.

b) 0c) Th eir total sales were the same.




On the Job 2Have students do the Your Turn.Check that students

include a reason for their answer to part a)give reasonable answers for parts b) and c)

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Encourage students to talk about their conclusions to parts a), b), and c) of the Your Turn before putting the conclusions to paper.Ask students whether either display really shows which company sells the most cars. Using a cumulative graph would be the best option here.Have students work in pairs. Each person can determine the total sales for one of the car companies. Students can then use both sums to calculate the diff erence in total sales. Make sure students record their answers individually.

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Check Your UnderstandingTry ItFor #1, ask:

From the size of the bars, how many times larger does the price of gasoline on weekends appear to be than the price on a weekday?What makes the price of gas on weekends appear twice as large as the price of gas on a weekday?

For #2c), ask students to consider what representation each company is trying to make based on the number of wet days per year.

Apply ItFor #4b), ask:

What must be computed in order to determine each percent?How do you fi nd a percent given an amount and the total?

For #5, have students consider what would have to change in each graph so that it emphasizes a diff erent point of view.

Meeting Student NeedsFor #1c), ask:

Does the type of graph need to change?What needs to change to make the graph represent the data more accurately?What would be an appropriate scale to use?

For #2, refer students back to the Explore in section 4.3.For #4, some students may need help estimating percents from a circle graph. It may be helpful to present the data in a 2-D circle graph. Students could then compare the sections of the circle to known percents on a circle, such as 50% and 25%.



Try ItOn the Job 2: #1, #2, #4, and #6

You may wish to have students work with a partner.Encourage students to verbalize their thinking, discuss, and compare their answers with those of their partner.Encourage students to work backward. For example, in #1, ask students to describe the diff erence in appearance between the given graph and a graph that starts at $0/L on the vertical axis. Students can use a similar strategy when working through #3.

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Work With ItFor #1, ask students:

What could be changed on the new graph? Scale? Minimum value? Or is a diff erent type of graph needed?How would using technology help you create the graphs?

For #2, you may need to discuss what each person would want to emphasize. Th at way, students will have a better understanding of which graphical representation to choose.

Discuss ItFor #3, you may wish to have students share their fi ndings with the class.

Meeting Student NeedsFor #3, you may wish to display the graphs that students found by categories in a table similar to the one below. Th e table would need to be large enough to hold the cut-out or printed graphs.

Example PurposeHow is the graph

misleading?How could the graph be

redrawn accurately?

Provide BLM 4–7 Section 4.3 Extra Practice to students who would benefi t from more practice.

Gifted and EnrichmentAsk students to select an issue that is important to them and that they can support with data. Have them collect data, organize it in a table, and represent it with two graphs. Examples:

Reality television shows take up too much weekly programming time.Th e government’s trend to cut back on social programs is too drastic.Th e declining population of wolves may lead to their extinction.

Encourage students to create at least one graph that may be misleading to support their point of view.



Discuss ItThese questions give students an opportunity to recognize when data have been misrepresented in graph form and how graphs can be used to emphasize a point of view.Have all students complete #1, #3, and #5.

Consider having students share their ideas for #3 in small groups.Distinguish inaccurate data and misleading data before students work on #4.For #4, suggest that students graph the data before they determine examples of how the data could be misrepresented.Encourage students to discuss their ideas orally before recording their answers.

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4Skill CheckPlanning NotesHave students use the “What You Need to Know” in the Skill Check section to help them determine what skills they understand. From this list, they can make a list of skills that they have mastered and a list of skills that need more work.

Have students use BLM 4–1 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.

You may wish to give students Master 2 Centimetre Grid Paper, Master 30.5 Centimetre Grid Paper, or Master 4 1 _ 4 Inch Grid Paper.

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.

Students should complete all questions to meet the related curriculum outcomes.

Meeting Student NeedsStudents who require more practice on a particular topic may refer to BLM 4–5 Section 4.1 Extra Practice, BLM 4–6 Section 4.2 Extra Practice, and BLM 4–7 Section 4.3 Extra Practice.



Chapter 4 Skill CheckThe Chapter 4 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 revisit any section that they are having diffi culty with prior to working on the Test Yourself.Review with the class by referring to a classroom display or students’ notes that show the diff erent types of graphs and the purpose of each.

Circle graph: discrete data, showing parts of a wholeBar graph: discrete data, showing data in categoriesLine graph: continuous data, showing changes in data over timeHistogram: the frequency of a range of data, usually continuous

Review with students how graphs can be misrepresented:using a minimum value other than zero when labelling the vertical axischanging the scalechanging the size of barsusing 3-D rather than 2-D circle graphs

Review by referring to a classroom display or student notes that show diff erent trends, and examples of interpolation and extrapolation.

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Grid Paper


BLM 4–1 Chapter 4 Self-Assessment

BLM 4–5 Section 4.1 Extra Practice



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4 Test YourselfPlanning 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 t

interpreting graphs 4 · blm 4–2 degree circle 4.1 choosing a graph 180–240 min (tr page 159)...

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