-
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.
••••
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�
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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
••
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
•
•
••
calculatorgrid paper or graphing technology
••
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
•
•
calculatorgrid paper or graphing technologyruler
••
•
Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid
Paper
Master 4 1 _ 4 Inch 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
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
•
•
calculatorgrid paper or graphing technologyruler
••
•
Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid
Paper
Master 4 1 _ 4 Inch Grid Paper
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
•
•
calculatorgrid paper or graphing technologyruler
••
•
Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid
Paper
Master 4 1 _ 4 Inch Grid Paper
BLM 4–3 Chapter 4 Warm-UpBLM 4–7 Section 4.3 Extra Practice
Chapter 4 Skill Check
50–60 min(TR page 185)•
calculatorgrid paper or graphing technologyruler
••
•
Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid
Paper
Master 4 1 _ 4 Inch Grid Paper
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)•
calculatorgrid paper or graphing technologyruler
••
•
Master 2 Centimetre Grid PaperMaster 3 0.5 Centimetre Grid
Paper
Master 4 1 _ 4 Inch Grid Paper
BLM 4–1 Chapter 4 Self-AssessmentBLM 4–8 Chapter 4 Test
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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 151
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
Math at Work 11 Online Learning Centre
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
Math at Work 11 Online Learning Centre
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
Math at Work 11 Online Learning Centre
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
Math at Work 11 Online Learning Centre
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
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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
•
••
•
Master 1 Project RubricMaster 2 Centimetre Grid PaperMaster 3
0.5 Centimetre Grid Paper
Master 4 1 _ 4 Inch 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
Master 4 1 _ 4 Inch Grid Paper
BLM 4–10 Chapter 4 BLM Answers
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Exercise Guide Extra Support
Assessment
Assessment as Learning
Assessment for Learning
Assessment of Learning
TR page 189Master 1 Project
Rubric
TR page 190
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Assessment Supporting Learning
Assessment as 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.
•
Assessment for Learning
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.
•
Assessment as Learning
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.
•
•
Assessment for Learning
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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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 155
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.
•••••
•
•
••
•
•
•
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
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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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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 157
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.
•
•
•
•
•
•
Math at Work 11, pages 152–153
Suggested Timing60–90 min
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.
Assessment Supporting Learning
Assessment for Learning
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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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 159
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.
•
•
••
•
Math at Work 11, pages 154–167
Suggested Timing180–240 min
Materialscalculatorgrid paper or graphing technologyruler
Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5
Centimetre
Grid Paper
Master 4 1 _ 4 Inch 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
Mathematical Processes Communication (C)
Connections (CN)
Mental Math and Estimation (ME)
Problem Solving (PS)
Reasoning (R)
Technology (T)
Visualization (V)
Specifi c OutcomeS1 Solve problems that involve creating and
interpreting graphs, including:
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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•
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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 161
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
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?
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%
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Assessment Supporting Learning
Assessment as Learning
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.
•
•
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.
•
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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978-1-25-901239-6 Chapter 4 Interpreting Graphs • MHR 163
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)
Assessment Supporting Learning
Assessment for Learning
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.
•
•
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.
••
•
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.
•
•
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
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
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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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Assessment Supporting Learning
Assessment for Learning
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.
Assessment Supporting Learning
Assessment for Learning
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
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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.
Assessment Supporting Learning
Assessment for Learning
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
Assessment Supporting Learning
Assessment as Learning
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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Math at Work 11, pages 168–181
Suggested Timing180–240 min
Materialscalculatorgrid paper or graphing technologyruler
Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5
Centimetre
Grid Paper
Master 4 1 _ 4 Inch Grid Paper
BLM 4–3 Chapter 4 Warm-UpBLM 4–6 Section 4.2 Extra
Practice
Mathematical Processes Communication (C)
Connections (CN)
Mental Math and Estimation (ME)
Problem Solving (PS)
Reasoning (R)
Technology (T)
Visualization (V)
Specifi c OutcomeS1 Solve problems that involve creating and
interpreting graphs, including:
bar graphshistogramsline graphscircle graphs
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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.
Assessment Supporting Learning
Assessment as Learning
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.
•
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
•
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
Assessment Supporting Learning
Assessment for Learning
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?
Assessment Supporting Learning
Assessment for Learning
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
On the Job 2: Your Turn
•
•
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.
Assessment Supporting Learning
Assessment for Learning
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.
Assessment Supporting Learning
Assessment for Learning
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.
Assessment Supporting Learning
Assessment as Learning
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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Math at Work 11, pages 182–195
Suggested Timing180–240 min
Materialscalculatorgrid paper or graphing technologyruler
Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5
Centimetre
Grid Paper
Master 4 1 _ 4 Inch Grid Paper
BLM 4–3 Chapter 4 Warm-UpBLM 4–7 Section 4.3 Extra
Practice
Mathematical Processes Communication (C)
Connections (CN)
Mental Math and Estimation (ME)
Problem Solving (PS)
Reasoning (R)
Technology (T)
Visualization (V)
Specifi c OutcomeS1 Solve problems that involve creating and
interpreting graphs, including:
bar graphshistogramsline graphscircle graphs
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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
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Assessment Supporting Learning
Assessment as Learning
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%
Assessment Supporting Learning
Assessment for Learning
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.
Assessment Supporting Learning
Assessment for Learning
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
On the Job 2: Your Turn
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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.
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Assessment Supporting Learning
Assessment for Learning
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%.
Assessment Supporting Learning
Assessment for Learning
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.
Assessment Supporting Learning
Assessment as Learning
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.
Assessment Supporting Learning
Assessment for Learning
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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Math at Work 11, pages 196–197
Suggested Timing50–60 min
Materialscalculatorgrid paper or graphing technologyruler
Blackline MastersMaster 2 Centimetre Grid PaperMaster 3 0.5
Centimetre
Grid Paper
Master 4 1 _ 4 Inch Grid Paper
BLM 4–1 Chapter 4 Self-Assessment
BLM 4–5 Section 4.1 Extra Practice
BLM 4–6 Section 4.2 Extra Practice
BLM 4–7 Section 4.3 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