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Grade 7 Mathematics Curriculum Outcomes 129
Outcomes with Achievement Indicators
Unit 4
Grade 7 Mathematics
Unit 4
Circles and Area
Estimated Time: 20 Hours
[C] Communication [PS] Problem Solving
[CN] Connections [R] Reasoning
[ME] Mental Mathematics [T] Technology
and Estimation [V] Visualization
Grade 7 Mathematics Curriculum Outcomes 130
Outcomes with Achievement Indicators
Unit 4
Unit 4: Circles and Area
Grade 7 Math Curriculum Guide 131
Unit 4 Overview
Introduction
Students will focus on several major ideas related to circles and their areas. The development of the area
formulas in this unit will be built using formulas learned about geometric shapes introduced in lower
grades. Exploration activities will be useful for strengthening understanding of these new formulas. The
big ideas in this unit are:
• The introduction of the number π and the fact that it represents the ratio of the circumference of
any circle compared to the diameter of that same circle.
• The conservation of area; an object can be separated into an infinite number of smaller objects
which can be then rearranged. The combined area of those smaller objects stays equal to the area
of the original object.
• Objects can be constructed using a wide variety of techniques.
• Area (which is two-dimensional) is found by multiplying two one-dimensional quantities together.
• A circle graph is one method of organizing and displaying data. It is used to compare parts of a whole to the whole. Two circle graphs together can be used to compare parts of two separate
wholes to each other; i.e. percent of blue cars in Nova Scotia compared to percent of blue cars in
Newfoundland and Labrador.
Context The students will learn various geometric definitions and how to construct a variety of geometric objects.
These constructions will utilize an assortment of tools and techniques. This unit puts emphasis on
exploration and hands-on creation.
The students will develop the formulas for the areas of triangles, parallelograms, and circles. They will
not be given these formulas directly. Exploratory activities will be used so that students can learn and
understand conservation of area. These explorations will allow students to generalize formulas for the areas of a triangle, parallelogram, and a circle.
Students will collect data, organize it and then use the data to create circle graphs. They will use the circle
graphs to solve problems.
Why are these concepts important?
Developing a good understanding of Circles and Area will permit students to:
• Recognize that circles are found everywhere; both naturally occurring and man-made.
• Be prepared to work with cylinders and spheres in higher grades.
• Understand the concept of dimensions in that length and width are one-dimensional, area is two-
dimensional and volume is three-dimensional. Students will be better able to understand the
concepts of surface area and volume, as well as the differences between them.
• Understand information as they encounter it in media and entertainment.
“Do not disturb my circles!”
Archimedes (c. 287 BC – c. 212 BC)
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 132
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
Elaborations: Suggested Learning and Teaching Strategies
Students have been introduced to the concept of circles, area,
and perimeter in previous grades. It is assumed that students
can:
• recognize circles, triangles, and parallelograms.
• calculate the area of a rectangle.
• measure perimeter in linear units, and measure area in
square units.
A circle consists of all the points in a plane that are a given
distance from a given point called the centre.
The radius is the distance
from the centre of a circle
to any point on the circle.
The diameter is a line
segment passing through
the centre of the circle
with both endpoints on
the circle. The diameter is
twice the length of the
radius.
The circumference of a circle is the distance around, or the
perimeter, of a circle.
Pi, π , is defined as the ratio of circumference to diameter.
(Note: This relationship should be discovered through investigation.)
Pi, π is a non-repeating, non-terminating decimal that cannot
be expressed as a fraction (i.e. irrational).
Pi, π = 3.1415926535897932384626433832795 ... The value
of π is often approximated as 3.14 although most calculators
have a π button. However, for estimates, students may use 3
as an approximate value forπ .
Diameter
nceCircumfere=π
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 133
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Resources/Notes
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 134
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Note:
Circumference, radius, and diameter are measured in linear
units such as mm, cm, m, km, etc.
Angles are measured using degrees where one full revolution
equals 360o.
It is important to explore the relation between the diameter and
the radius in both directions. That is, not only rd 2= but
rd
=
2as well.
7SS1.1 Illustrate and
explain that the diameter
is twice the radius in a
given circle.
OA ,OB , OC are radii.
AC is the diameter.
AC is two times the length of OA , OB , andOC .
The diameter is always twice the radius.
Or rd 2= .
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 135
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Journal
1. Make a list of sports in which circles play an important role.
Estimate the radius of each circle you describe.
2. In your own words answer these questions.
A. Do you need to use hands-on measurement for every one
of the three measures (radius, circumference, diameter)
requested?
B. If you know the radius, what can you do to get the
diameter?
C. If you know the diameter, what can you do to get the
radius?
Resources/Notes
Math Makes Sense 7
Lesson 4.1
Unit 4: Circles and Area
TR: ProGuide, pp. 4–6
Master 4.10, 4.15, 4.24
PM 19
CD-ROM Unit 4 Masters
ST: pp. 130–132
Practice and HW Book
pp. 80–81
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 136
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Although there are many ways to draw a perfect circle, some
work better than others in certain situations. Here are a few
techniques;
Method 1 (without a compass using diameter)
Find a perfectly round object that is the desired size. The
outermost edge should be smooth and without bumps.
Put the above object on your paper and hold it down firmly
with one hand while you trace it with your other.
Method 2 (without a compass using radius)
Tie a piece of string near the bottom of a pencil. Hold the
string the length of the radius away from the pencil with your
finger.
Hold the string down against the paper where you want the
centre of the circle to be. Draw around the centre while
keeping the string tight and the pencil upright.
7SS1.2 Draw a circle with
a given radius or diameter,
with and without a
compass.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 137
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Pencil and Paper
Practice drawing circles with your compass. Draw a circle with a 10
cm radius, with 5 cm radius, and with 3 cm radius.
Technology/Web Resources
1. Also, investigate the Circle Song at this link:
http://www.teachertube.com/view_video.php?viewkey=2fca331
343d8eade9ec2
This website was found at: www.teachertube.com
2. Also, investigate the conversation about circles at this link:
http://www.teachertube.com/view_video.php?viewkey=6ef15c2
72415206e1028
This website was found at: www.teachertube.com
3. Also, investigate crop circle examples at this link:
http://youtube.com/watch?v=9bmrN9rIdro
This website was found at: www.youtube.com
Resources/Notes
Math Makes Sense 7
Lesson 4.1
(continued)
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 138
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Method 3 (with a compass using radius)
Secure a sharp
pencil in the clamp
of a compass so the
point of the
compass and the
point of the pencil
are level when the
compass is closed.
Adjust the angle
of the arms so
that they span
the full desired
radius. Ensure
the hinge is
tightened so the
radius does not
adjust while the
circle is being
made. Put the
sharp end of a
compass down
firmly wherever
you want the
middle of your
circle to be. Put
the pencil point
gently down on
the paper. Keep the compass upright and hold the compass at
the top. Turn the compass so that the pencil draws a circle.
7SS1.2 Draw a circle with
a given radius or diameter,
with and without a
compass. (continued)
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 139
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Resources/Notes
Math Makes Sense 7
Lesson 4.1
(continued)
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 140
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
The following achievement indicators are addressed together.
Circumference: The circumference of a circle is the
distance around the edge, or the perimeter, of a circle.
Recall: For a circumference of C units and a diameter of d
units or a radius of r units, dC π= or rC π2= .
The concept of pi having a value very close to 3 can be very
well explored with the activity below. This excellent activity is
found on page 133 of the text but it does not contain a column
for calculating the ratio of circumference to diameter; the table
below does have this column.
Other than this adaptation, the activity in the text should be
followed as it is written.
Object Circumference Radius Diameter d
C
Can
Plate Frisbee
Provide these objects, or other round items, to the students and
have them perform the necessary measurements to complete
the table. They may use their knowledge from the previous
achievement indicator and calculate some of these measures
from one hands-on measure.
Ultimately, we want the students to realize that the ratio is a
constant value that is close to 3. The ratio 3.141592... is π .
Calculator use is encouraged in determining the value of the
ratio in the activity and any exercises that follow.
7SS1.5 Solve a given
contextual problem
involving circles.
7SS1.3 Illustrate and
explain that the
circumference is
approximately three times
the diameter in a given
circle.
7SS1.4 Explain that, for
all circles, pi is the ratio of
the circumference to the
diameter ( )C
d and its
value is approximately
3.14.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 141
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Group Activity
Students can play the game called Circle Mania(Master 4.6).
Graphic Organizer (Foldable)
Teachers may choose to use a 3-tab foldable as a method of helping
students keep organized notes. (Refer to Appendix 4–A for the
instructions on how to create this foldable.)
Journal/Interview
Ask the students these questions.
A. What is the best estimate for the circumference of a circle
with a diameter of 12 cm? Justify your choice.
(i) 6 cm (ii) 18 cm (iii) 36 cm
B. What is the best estimate for the circumference of a circle
with a radius of 10 cm? Justify your choice.
(i) 30 cm (ii) 60 cm (iii) 90 cm
Paper and Pencil
1. Jackie is constructing a round dining room table that will seat
12 people. She wants each person to have 60 cm of table space
along the circumference. Determine the diameter of the dining
room table.
2. A manufacturing company is producing dinner plates with a
diameter of 30 cm. They plan to put a gold edge around each
plate. Determine how much gold edging they need for an eight
plate setting. If gold edging costs $4 per cm, what would it cost
to trim all of the plates? (limiting example)
Resources/Notes
Math Makes Sense 7
Lesson 4.1
(continued)
Math Makes Sense 7
Lesson 4.2
Unit 4: Circles and Area
TR: ProGuide, pp. 7–11
Master 4.16, 4.25
PM 20
CD-ROM Unit 4 Masters
ST: pp. 133–137
Practice and HW Book
pp. 82–83
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 142
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS2. Develop and apply a
formula for determining
the area of:
• triangles
• parallelograms
• circles.
[CN, PS, R, V]
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Area can be defined as a measure of the space inside a region
or how many square “units” it takes to cover a region.
Having students understand Conservation of Area is critical.
That is, an object retains its size when the orientation is
changed or when it is broken into smaller parts and the parts
are rearranged.
The following achievement indicators are addressed together.
Area of a Parallelogram
Students should recognize that the area of a parallelogram is
the same as the area of a related rectangle (one with the same
base and height). Students should be able to determine the base
or height, given the area and the other dimension, and
recognize that a variety of parallelograms can have the same
area.
The diagram displayed above represents an activity found on
page 139 of the textbook. The activity develops the formula
for the area of a parallelogram, and builds awareness of
conservation of area.
Note:
Students already know that, for a rectangle,
( )( )heightbaseArea = . Students may refer to it as length x
width.
This activity will indicate that the area of the related
parallelogram is also given by ( )( )heightbaseArea = .
7SS2.1 Illustrate and
explain how the area of a
rectangle can be used to
determine the area of a
parallelogram.
7SS2.2 Generalize a rule
to create a formula for
determining the area of
parallelograms.
base base
height height
7SS2.3 Solve a given
problem involving the
area of triangles,
parallelograms and/or
circles.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 143
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Paper and Pencil
1. Ask the student to draw a parallelogram, on grid paper, with an
area of 24 cm2. Then ask him/her to create three other
parallelograms with a different base length but the same area.
Technology/Web Resources
Also, investigate the Area of a Parallelogram at this link:
A. http://illuminations.nctm.org/ActivityDetail.aspx?ID=108
B. http://illuminations.nctm.org/ActivityDetail.aspx?id=47
This website was found at: www.illuminations.nctm.org
Resources/Notes
Math Makes Sense 7
Lesson 4.3
Unit 4: Circles and Area
TR: ProGuide, pp. 13–16
Master 4.17, 4.26
PM 23
CD-ROM Unit 4 Masters
ST: pp. 139–142
Practice and HW Book
pp. 84–86
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 144
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS2. Develop and apply a
formula for determining
the area of:
• triangles
• parallelograms
• circles.
[CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
The following achievement indicators are addressed together.
Students should see that the area of a triangle is just one-half
of the area of its related parallelogram. They should also be
able to connect this idea to the relationship between the
formulas of a parallelogram and rectangle. Students can use
this relationship to find areas of triangles. Students should
understand that, as long as the base and height are the same,
the areas of visually-different triangles are the same.
Note: On page 143 of the text there is an Explore activity that
develops the concept of a triangle having half the area of its
related parallelogram. An alternate activity developing the
same concept utilizing rectangles is found below.
Explore: Finding the area of a triangle.
1. On grid paper, draw a rectangle that has a base of 8 units
and a height of 5 units.
2. Using scissors cut out the rectangle.
3. Count the number of squares in the rectangle and have the
students record the number of squares as the area of the
rectangle. This reinforces the idea of square units for area.
4. Draw a diagonal line from one corner of the rectangle to the
opposite corner. Inform students that this line is actually called
a diagonal. Cutting along the diagonal separate the rectangle
into two sections. What shapes have been created?
5. Place these two shapes on top of each other. How do they
compare?
6. How does the area of one of the triangle compare to the area
of the original rectangle?
7. Have students suggest a formula for the area of a triangle
recalling that the area of a rectangle is bhA = .
8. Have students report their formulas and discuss as a group
any similarities and/or differences in their formulas.
7SS2.4 Illustrate and
explain how the area of a
rectangle or a
parallelogram can be used
to determine the area of a
triangle.
7SS2.5 Generalize a rule
to create a formula for
determining the area of
triangles.
2
bhA =
or
7SS2.3 Solve a given
problem involving the
area of triangles,
parallelograms and/or
circles.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 145
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Paper and Pencil
1. Daniel just bought a used sailboat
with two sails that need replacing.
How much sail fabric will Daniel
need if he replaces sail A?
2. How much sail fabric will Daniel
need if he replaces sail B?
Performance
An activity, entitled Area Diagram, which explores areas of
rectangles and triangles. (Refer to Appendix 4–B.)
Technology/Web Resources
1. A Triangle Explorer for Area can be found at:
http://www.shodor.org/interactivate/activities/TriangleExplorer
This website was found at: www.shodor.org
2. A link that explores triangles with constant base and height but
different shapes is:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=106
This website was found at: www.illuminations.nctm.org
Resources/Notes
Math Makes Sense 7
Lesson 4.4
Unit 4: Circles and Area
TR: ProGuide, pp. 17–21
Master 4.18, 4.27
PM 23, 25
CD-ROM Unit 4 Masters
ST: pp. 143–147
Practice and HW Book
pp. 87–89
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 146
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS2. Develop and apply a
formula for determining
the area of:
• triangles
• parallelograms
• circles.
[CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Many students tend to memorize formulas that we use in
geometry or other mathematics areas without much
understanding. This activity introduces estimating the area of a
circle without using a formula. Development of a formula will
follow. This activity is found on page 151 of the text.
However, it should be addressed well before developing the
formula for the area of the circle. As it appears in the text it is
an Assessment Focus and occurs after the formula is
developed.
Activity
1. Each student should have a piece of grid
paper.
2. Using a compass, each student draws a
circle on grid paper.
3. Each student counts squares inside the
circle and estimates the area.
4. Each student draws a square outside the
circle and calculates the area of the square.
5. Each student draws a square inside the
circle and calculates the area of the square.
6. Estimate the area of the circle by relating
it to areas of the outer and inner squares. (Hint: average the
areas of the two squares.)
7. Discuss the advantages and disadvantages of the above
method for measuring the area of a circle.
7SS2.6 Illustrate and
explain how to estimate
the area of a circle without
the use of a formula.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 147
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Group Activity
Download Estimating Areas of Circles worksheet. (Refer to
Appendix 4–C.)
Developing the formula for the area of a circle is done by cutting a
circle into many equal “triangular” sections and arranging them
into the approximate shape of a parallelogram. The more sections
you use the more accurate the formula becomes because more and
more space inside the parallelogram gets “filled up”.
Mathematically this is known as the Method of Exhaustion. This
method can also be used to estimate the area of a circle.
Have students estimate the area of the circle using
the octagon as a benchmark. Notice that the
octagon fills more of the circle than a square
would.
You will need a bag of beans for this next part.
Have the students fill in as much of the circle as
possible with beans. Because of their curved
shape the beans should fill more space inside the
circle than the octagon.
Move the beans they used onto the
squares. These squares are the four
squares from the diagram. They can
be called r-squares since their sides
are the same as the radius of the circle. Now, count the number of
smaller squares that are covered by the beans to get an estimate of
the circle’s area.
Students should have covered a little more than 3 of the larger
squares. So using this method they get an estimate that is
approximately 3 r-squares.
This is similar to the known formula for the area of a circle and
provides a reasonable estimate.
Resources/Notes
Math Makes Sense 7
Lesson 4.5
Unit 4: Circles and Area
TR: ProGuide, pp. 22–26
Master 4.19, 4.28
PM 22
CD-ROM Unit 4 Masters
ST: pp. 148–152
Practice and HW Book
pp. 90–92
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 148
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Specific Outcome
It is expected that students will:
7SS2. Develop and apply a
formula for determining the
area of:
• triangles
• parallelograms
• circles.
[CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Students should develop the formula for the area of a circle
through investigations that connect a circle, cut into equal
sectors, to a parallelogram. The exploratory work done by
students in estimating the areas of circles also provides a
foundation for developing the formula for the area of a circle.
The Explore activity found on page 148 of the textbook is the
model found in many textbooks and on numerous websites.
Ultimately, the Explore arrives at the formula 2rA π= .
7SS2.7 Apply a formula
for determining the area of
a given circle.
ATTENTION: Students have NOT been exposed to
powers or exponents. When we develop the formula 2
rA π= , introduce it as rrA ××= π . The textbook
does use 2
rA π= and teachers are reminded to use
caution when using this notation. Students would have
seen this notation when working with area units.
It is not a Specific Outcome.
7SS2.3 Solve a given
problem involving the
area of triangles,
parallelograms and/or
circles.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 149
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems
Suggested Assessment Strategies
Paper and Pencil
1. Mr. McGowan made an apple pie with diameter of 25 cm. He
cut the pie into 6 equal slices. Find the approximate area of each
slice.
2. The outer ring on the Canadian Toonie
has an outside radius of 14 mm, and an
inside radius of 8 mm. What is the area
of the outer ring? (Limiting Example)
Journal
Jackie’s mom was decorating Jackie’s bedroom and placed a round
mat on the floor near the bed. Jackie had just learned about circles
in math class and wondered about the area of the mat. The tag on
the mat said that it was 60 cm wide. She performed the following
calculations;
Are Jackie’s calculations
reasonable?
Explain why or why not.
Graphic Organizer (Foldable)
For the foldable called Tri-fold Foldable, refer to Appendix 4–D.
Students can keep notes about the three shapes discussed in this
unit; parallelograms, triangles, and circles. Topics like estimation,
definitions and area formulas could be included for fast and easy
access.
Resources/Notes
Math Makes Sense 7
Lesson 4.5
(continued)
2
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 150
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Specific Outcome
It is expected that students will:
7SP3. Construct, label and
interpret circle graphs to
solve problems.
[C, CN, PS, R, T, V]
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Circle graphs are similar to bar graphs in that they provide
information arranged in categories. Circle graphs are different
from bar graphs in that they display the categories as parts, or
percents, of a whole. Recall that bar graphs display how many
items are actually in the category, not what part of the whole
that the category represents. In a circle graph, categories are
represented by sectors, and in a bar graph, categories are
represented by bars.
You can compare two wholes, by comparing two circle graphs.
For example, one circle graph may display the percentage of
people in each age group for a city and the other may show the
same information for the province. Since circle graphs display
ratios rather than quantities, the small set of data can be
compared to the large set of data. That could not be done with
bar graphs (Van de Walle and Lovin, 2006, p. 324).
The title, legend and labels (illustrated clearly on page 157 in
the text) are crucial to interpreting circle graphs. Use real data
if at all possible when interpreting or drawing circle graphs.
When constructing circle graphs, data would typically be given
as percents or as raw data to be converted to percents.
The sum of the percents of all the parts will always be 100%.
Likewise, the sum of the central angles will always be 360˚.
Note: This curriculum guide covers the following:
Explain, using an illustration, that the sum of the central angles
of a circle is 360o.
This is presented on shortly. It is discussed in Lesson 4.7 of the
student text.
7SP3.1 Find and compare
circle graphs in a variety
of print and electronic
media, such as
newspapers, magazines
and the Internet.
7SP3.2 Identify common
attributes of circle graphs,
such as:
• title, label or legend
• the sum of the
central angles is
360o
the data is reported as a
percent of the total, and
the sum of the percents is
equal to 100%.
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 151
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Suggested Assessment Strategies
Portfolio
On page 160 of the text there is a Reflect section. This would be an
appropriate activity for the purpose of find and compare circle
graphs in a variety of print and electronic media, such as
newspapers, magazines and the Internet.
Paper & Pencil
For the activity entitled Parts of a Circle Graph, refer to Appendix
4–E.
Parts of a Circle Graph
Mike is a student in grade 7 and is learning about circle graphs. Mike has to study
regularly in order to keep his grades up. He decided how he should use his study
time and recorded it in the table below.
Using the data in the table, label the circle graph correctly. Match the correct
percentages with the correct sectors, and create an appropriate title for the graph.
Complete the legend and shade the circle graph to match.
Math 30 %
Social Studies 15 %
Lang. Arts 25 %
Science 20 %
French 10 %
Legend
Resources/Notes
Van de Walle and Lovin,
2006, p. 324
Math Makes Sense 7
Lesson 4.6
Unit 4: Circles and Area
TR: ProGuide, pp. 30–34
Master 4.20, 4.29
CD-ROM Unit 4 Masters
ST: pp. 156–160
Practice and HW Book
pp. 93–95
Math Makes Sense 7
Lesson 4.7
Unit 4: Circles and Area
TR: ProGuide, pp. 35–38
Master 4.12, 4.21, 4.30
CD-ROM Unit 4 Masters
ST: pp. 161–164
Practice and HW Book
pp. 96–99
Title
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 152
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Specific Outcome
It is expected that students will:
7SP3. Construct, label and
interpret circle graphs to
solve problems.
[C, CN, PS, R, T, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
In Unit 3 of the text, students learned how to find the percent
of a number. They will use this skill when solving problems
involving circle graphs.
Students should be able to interpret a circle graph. They can
then use a percentage to determine what portion of the total
group corresponds to that percentage.
Example:
If there are 435 nuclear reactors in operation, how many are in
the United States? (Hint: Find 24% of 435.)
24% is equivalent to 0.24
Therefore, (0.24)(435) ≈ 104.
So, there are approximately 104 nuclear reactors in the United
States.
7SP3.3 Translate
percentages displayed in a
circle graph into quantities
to solve a given problem.
7SP3.4 Interpret a given
circle graph to answer
questions. Nuclear Reactors in Operation 2007
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 153
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Suggested Assessment Strategies
Paper and Pencil
For the activity entitled Analyzing Circle Graphs refer to Appendix
4–F.
Jan wants to show that the sales of chocolate milk are higher at the
end of the week, so that more chocolate milk can be ordered for that
time. He creates the circle graph above. Analyze the graph and
answer these questions.
1. What percentage of the milk is sold on Wednesday?
2. Identify a group of days that accounts for about half of the total
sales. (there is more than one possible answer)
3. A. If Friday is a holiday, discuss how that would affect ordering
chocolate milk for that week.
B. In a regular week 500 cartons of chocolate milk are sold.
How many cartons should be ordered if Friday was a holiday?
4. If weekly sales of chocolate milk are $200, how much of that is
made on Monday?
5. Why do you think chocolate milk sales increased steadily as the
week progressed?
Resources/Notes
Math Makes Sense 7
Lesson 4.6
Lesson 4.7
(continued)
Tech Activity:
pp. 165–166
Amount of Chocolate Milk Sold in a Week
Monday
10%
Tuesday
16%
Wednesday
18%Thursday
26%
Friday
30%
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 154
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems.
Specific Outcome
It is expected that students will:
7SS1. Demonstrate an
understanding of circles by:
• describing the
relationships among radius,
diameter and circumference
• relating
circumference to pi
• determining the sum
of the central angles
• constructing circles
with a given radius or
diameter
• solving problems
involving the radii,
diameters and
circumferences of circles.
[C, CN, PS, R, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Why does a circle have 360o?
A line of ancient peoples who lived in Mesopotamia (now
southern Iraq) invented writing, observed the skies, and
invented a 360-degree circle to describe their findings. About
3000 BC, the Sumerians invented writing. They also had a
calendar, dating from 2400 BC, that divided the year into 12
months of 30 days each, that is, 360 days.
They noticed the circular track of the Sun's annual path across
the sky and knew that it took about 360 days to complete one
year's circuit. So, they divided the circular path into 360
degrees to track each day's passage of the Sun's whole journey.
That's how we got a 360 degree circle.
But the year has 365 days, not 360. We seem to be five
degrees short. Why? Standards of scientific measurement in
ancient days were not as precise as they are today. Three
hundred sixty was also readily divisible into thirds, fourths,
fifths, sixths, etc. This gave mathematicians a great advantage
when it came to doing calculations.
7SS1.6 Explain, using an
illustration, that the sum
of the central angles of a
circle is 360o.
Diagram is
not to scale.
Note that this is a geocentric, not heliocentric, illustration which is
inaccurate. However, this is how the majority of the ancient
astronomers understood the solar system; with the sun orbiting the
earth.
Strand: Shape and Space (Measurement)
Grade 7 Mathematics Curriculum Outcomes 155
Outcomes with Achievement Indicators
Unit 4
General Outcome: Use direct or indirect measurement to
solve problems.
Suggested Assessment Strategies
Paper and Pencil
1. Using a protractor and a ruler create, and classify these angles.
A. 30o
B. 107o
C. 180o
D. 220o
2. Draw a circle using a method of your choice and locate the
centre. Divide your circle into 5 sections with the following
angle measures; 35o, 80
o, 60
o, 135
o, 50
o.
3. Several circles are divided into 3 sections. The central angles of
two of the sections are given below. In each case, determine the
measure of the third central angle.
A. 55o, 117
o
B. 91o, 74
o
C. 1o, 163
o
Resources/Notes
Math Makes Sense 7
Lesson 4.7
Unit 4: Circles and Area
TR: ProGuide, pp. 35–38
Master 4.12, 4.21, 4.30
CD-ROM Unit 4 Masters
ST: pp. 161–164
Practice and HW Book
pp. 96–99
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 156
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Specific Outcome
It is expected that students will:
7SP3. Construct, label and
interpret circle graphs to
solve problems.
[C, CN, PS, R, T, V]
(Cont’d)
Achievement Indicators
Elaborations: Suggested Learning and Teaching Strategies
Note: It is very important that constructing and interpreting
data are not addressed independently of each other. When
students take the time to construct data displays, these displays
should also be used for interpretation.
The text introduces this idea with the use of a percent circle. A
percent circle is a circle divided into tenths, where each tenth
is further sub-divided to create hundredths.
The ability to find the percent of a number and the ability to
use a protractor are very useful skills for the formal
development of circle graphs when they are constructed from
raw data.
The construction of a circle graph using pencil and paper can
be a time consuming activity and should always be done with
at least the aid of a basic calculator to perform the tedious
calculations associated with percentages and conversions to
degree measures.
Once students have generated one or two circle graphs by
hand, the focus should be on when a circle graph is the most
appropriate data display and how to use technology to
construct one. Technology options include, but are not limited
to, Microsoft Excel, websites, and graphing calculators.
A frequency distribution table is useful for organizing data
when constructing circle graphs.
Category
(Types of
Pie)
Frequency Fraction of
Total
Fraction as
a percent
% as an
angle
( )o360% ×
Apple 12 12/50 24% 86o
Lemon 10
Cherry 15
Coconut 13
7SP3.5 Create and label a
circle graph, with and
without technology, to
display a given set of data.
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 157
Outcomes with Achievement Indicators
Unit 4
General Outcome: Collect, display, and analyze data to solve
problems.
Suggested Assessment Strategies
Group Activity
1. Keeping with the philosophy of making the data real, or
pertinent, to the students here is a list of possible surveys that
can be conducted very quickly in the classroom. Each of these
will lead to a useful circle graph that will promote conversation
in the class.
• How many children are in your family?
• What kind of pet do you have?
• In what month were you born?
• What color are your eyes?
• Is your family vehicle a car, truck, van, or S.U.V.?
• What is your favourite hockey team?
2. Make a ‘human circle graph’. Have students choose their
favorite of four hockey teams and line them up so that students
favouring the same team are together. Have students form a
circle. Tape the ends of four long strings in the middle and
stretch them out to show the divisions (Van de Walle and
Lovin, 2006, p.324).
3. Have students make bar graphs. When completed, cut out the
bars and tape them end to end. Tape the two ends together to
form a circle. Estimate where the centre of the circle is, draw
lines to the points where the different bars meet, and trace
around the full loop. You can now estimate the percentages
(Van de Walle and Lovin, 2006, p.324).
Technology/Web Resources
1. A useful website to find statistics is at the Stats Canada website:
www.statcan.ca
2. For a pie chart maker, refer to the National Library of Virtual
Manipulatives:
http://nlvm.usu.edu/en/NAV/frames_asid_183_g_2_t_1.html
This website was found at: www.nlvm.usu.edu
Resources/Notes
Math Makes Sense 7
Lesson 4.7
(continued)
Strand: Statistics and Probability (Data Analysis)
Grade 7 Mathematics Curriculum Outcomes 158
Outcomes with Achievement Indicators
Unit 4