grade 7 mathematics unit 4 circles and area · unit 4: circles and area grade 7 math curriculum...

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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

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Page 1: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 2: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

Grade 7 Mathematics Curriculum Outcomes 130

Outcomes with Achievement Indicators

Unit 4

Page 3: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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)

Page 4: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 5: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 6: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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= .

Page 7: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 8: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 9: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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)

Page 10: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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)

Page 11: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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)

Page 12: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 13: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 14: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 15: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 16: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 17: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 18: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 19: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 20: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 21: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 22: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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%.

Page 23: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 24: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 25: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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%

Page 26: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 27: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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

Page 28: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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.

Page 29: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

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)

Page 30: Grade 7 Mathematics Unit 4 Circles and Area · Unit 4: Circles and Area Grade 7 Math Curriculum Guide 131 Unit 4 Overview Introduction Students will focus on several major ideas related

Strand: Statistics and Probability (Data Analysis)

Grade 7 Mathematics Curriculum Outcomes 158

Outcomes with Achievement Indicators

Unit 4