Module in Teaching Circle

Introduction

This teaching module is written with an aim of integrating history of

Mathematics in teaching circle to grade six elementary students. Included in

the discussions are the parts, circumference and area of a circle. The

lessons employ strategies such as exploration, discovery and discussion to

deepen students understanding and to make Math learning meaningful to

the students. A CD-ROM disk is provided. It includes videos and PowerPoint

Presentations which can be used in teaching the lessons.

Time Frame: 1 session (1 hour)

1

Objectives

At the end of the lesson, the students must be able to:

1. Define a circle

2. Identify the different parts of a circle

3. Determine the relationship between the radius and diameter of a circle

4. Solve problems involving basic ideas related to circles

Subject Matter

Topic: Parts of a Circle

Materials: “Circle” Worksheet (Quantity depends on the Number of Students)

Scissors

Colored Pencils

Yarn

Rulers

“Circle Parts” Worksheets

“Reflection” Worksheets

PowerPoint Presentation on “Parts of a Circle”

Values: Appreciation of the importance of circles in our daily life

Lesson Proper

(Note: Use the PowerPoint Presentation on Parts of a Circle provided in the cd.)

A. Motivation:

Present the video “Circle Around Us” (This is included in the PowerPoint.) to

the students. After watching the video, ask the students to site other examples of

things that can be seen inside the classroom that are circular in shape. Then, tell

the students that the study of the circle goes back beyond the recorded history. The

invention of the wheel is a fundamental discovery of properties of a circle (show

pictures of Sumerians and their wooden wheel included in the PowerPoint). Ask the

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students what they think would happen if the wheels of the vehicles are oblong or

square.

B. Presentation of the Lesson

1. Tie a string around a piece of chalk and use it to draw a circle on the chalkboard.

Hold the string down with your thumb or finger and pull the string tight with the

chalk. Now use the chalk to draw a circle. Emphasize the idea that to be able to

construct a circle there must be a fixed point in the center and the distance from

the center (length of the string) is also fixed. Define a circle. (The definition is on

the PowerPoint slide 4)

2. Pass out copies of the “Circle” worksheet, and have students cut out the circle.

“Circle” worksheet

3. Tell the students to fold the circle in half and then unfold it. Ask them to use a

colored pencil and a ruler to trace the line segment formed by the crease they

just made, and ask them to label this colored line “diameter” (see “Labeled

Circle: Front” sample).

4. Have students fold the circle in half again, but not along the same fold as before.

Again tell them to unfold, and ask whether they notice anything special about

where these two lines intersect. (It is the center of the circle.) Have them draw a

dot there and label it “center”.

3

Circle•a set of points that have the same distance from a fixed point.

5 687

5. Repeat step four twice more for a total of four diameters.

6. Have the students use a different colored pencil and a ruler to trace the line

segment along any diameter from the center to the edge of the circle. Tell them

to label this line segment “radius”.

7. Have students make a final crease by making a “flap” (making any fold that

does not go through the center). Have them use a different colored pencil and a

ruler to trace and label this line “chord”.

8. Now, have the students turn the circle over and use a different colored pencil to

trace around the periphery (boundary) of the circle. Have them label this

“circumference”.

9. Ask the students to use their labeled circle to verbally define chord, diameter,

radius and circumference. Give guidance as necessary. Point out that the

diameter is a special chord because it goes through the center. Use the

PowerPoint Presentation (slides 5, 6, 7 & 8) to present the definitions formally.

10.To see if the students know how to identify the parts of a circle, use the figure on

the PowerPoint presentation slide 9. The answers are on slide 10.

4

Given the circle with center at P, name the following:

• Chord (s)

• Radius/Radii

• Diameter (s)P

C

F

E

B

D

G

Answers:

Chords

DE, CF, BE, BG

Radii

PB, PF, PC, PE

Diameters

BE, CF

P

C

F

E

B

D

G

11.Have students use the ruler and yarn to measure each circle part. Compare

answers and discuss. Through discussion, be sure students notice that all

diameters have the same length, that the radius is half the diameter and,

conversely, that the diameter is twice the radius, and that the chords can have

many different lengths. The teacher may use slide 11 of the PowerPoint

presentation to present these relationships formally.

12. Give problems involving the diameter and radius of a circle. Use the

examples on slides 12 and 13.

Evaluation

Have students complete the “Circle Parts” Worksheet.

5

All the radii of a circle are congruent.

All the diameters are congruent.

The measure of the diameter, d, is twice the measure of the radius r. Formulas that relate these measures are:

d = 2r and r = ½d

121. Find the radius of a circle with

diameter of 10 cm.

Answer: 5 cm

13

2.Find the diameter of a circle with radius 2 in.

Answer: 4 in

Assignment

Have students complete the “Reflection” Worksheet.

Time Frame: 2 sessions (1 hour per session)

Objectives


1. Recall the definition of circumference

2. Calculate the ratio of circumference to diameter

3. Compare the different approximations of the value of pi across history

4. Use calculator to calculate for the different approximations of the value of pi

6

5. Discover the formula for the circumference of a circle

6. Find the circumference of a circle

Subject Matter

Topic: Pi and Circle Circumference

Materials: pieces of string, approximately 48” long

Cans of different sizes

PowerPoint Presentation of “Pi and Circle Circumference”

ruler

calculator

Value: Cooperation and Unity

Day 1

Lesson Proper

A. Motivation

As a warm-up, ask students to measure the length and width of their

desktops. Ask them to decide which type of unit should be used. Then, have

students measure or calculate the distance around the outside of their desktops.

With the class, discuss the following:

1. What unit did you use to measure your desks? Why? (The students must

agree that due to the size of the desks, the most appropriate units are

probably inches or centimeters.)

2. Why did some of your classmates get different measurements for the

dimensions of their desks? (The students must realize that measurements will

obviously differ because of the units. Moreover, the level of precision may

give different results.)

3. What do you call the distance around the outside of an object? (The students

must answer that the distance around the outside of a polygon is known as

the perimeter. The distance around the outside of a circle is known as the

circumference.)


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1. Inform the class that they will be measuring the circumference of the bases of

the cans during today’s lessons.

2. Divide the class into groups of four students. Each student in the group will be

given a different job.

Task Leader: Ensures all students are participating; lets the teacher

know if the group needs help or has a question.

Recorder: Keeps group copy of measurements and calculations from

activity.

Measurer: Measures items. However all students should check

measurements to ensure accuracy.

Presenter: Presents the group’s findings and ideas to the class.

In grouping let the students give their insight on the effect of cooperation and unity

in solving real life problems.

3. Pass out a copy of “What’s in a Circle?” Worksheet to each group. Let the

students do the activity in the worksheet. Remind the class that they have 15-20

minutes to finish the activity.

4. When all groups have completed the measurement and calculations, let each

group present their findings and ideas to the class.

5. Explain to the students that they have just discovered the value of pi which is

use for many calculations having to do with circles. Discuss what pi is and its

approximations. Use the PowerPoint Presentation slides 2 and 3.

6. Present the symbol for pi ( ) and the numerical value of pi. Tell the class that it

was William Jones, a self taught English Mathematician born in Wales, who

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• Pi is the ratio of the circumference to the diameter.

• It is an irrational number . That means that it can not be written as the ratio of two integer numbers. It takes an infinite number of digits to give its exact value (You can never get to the end of it!!!)

• One popular approximation for the value of pi is 22/7 which equals about 3.14…

PI IS NOT :

• A whole pie

• A slice of pie

The First 500 Digits of3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198

selected the Greek letter for the ratio of a circle’s circumference to its diameter

in 1706. (Use the PowerPoint presentation slide 6 )

7. Discuss some uses of pi. (Use slide 6)

8. Point out that groups within class may have obtained slightly different

approximations for . Explain that determining the exact value of is very hard to

calculate, so approximations are often used. Let students discover various

approximations of throughout the history. Pass out copies of “Solve A Round-

The-World Puzzle” Activity Sheets. Let the students to work in pairs. After 10

minutes, call for volunteers to discuss each item on the activity.

Day 2

Lesson Proper

A. Motivation/ Review

Recall the relationship between circumference, diameter and from the

previous activity. Students must recall the relationship below:

or

Tell the class that it was Euclid of Alexandria (325-265 BC) who proved that

the ratio of C over d is always the same, regardless of the size of the circle. He did it

by inscribing regular polygons inside circles of different sizes. He was able to show

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• The symbol for pi is π.

• It was William Jones, a self taught English Mathematician, who selected the Greek letter π for the ratio of a circle’s circumference to its diameter in 1706.

• Pi was chosen as the letter to represent 3.141592…because the letter π in Greek, pronounced like our “p”, stands for perimeter.

Pi

William Jones

(1675-1749)

Pi has many uses:

Engineering

Signals: Radio, TV, radar, telephones

Navigation: Global paths, global positioning

Other problems involving circle

that the perimeter of the polygon was proportional to the radius (which is half of the

diameter), regardless of its size. He then increased the number of sides of the

polygon, realizing that as he increased them, the perimeter of the polygon got

closer and closer to that of the circle. Therefore, he was able to prove that the

perimeter of the circle (circumference) is proportional to the radius and also to the

diameter. Emphasized that this method was similar to the one used by Archimedes

of Syracuse, Sicily (287-212 BC) who did the first theoretical calculation of . (Use

the PowerPoint Presentation to show pictures of Euclid and Archimedes. The teacher

may also share some information on how Archimedes approximated pi.)


1. Ask the students to come up with a formula that would allow them to calculate

the circumference of a circle if they knew only the diameter of the circle and the

value of pi.

2. The students must agree that the formula for Circumference is . Ask the

students what will be the formula for the C if the given is the radius. They must

come up with the formula .

3. Students should practice solving problems involving the circumference of a circle.

Emphasize that circumference of a circle is just an approximation if they use 3.14 as

the value of . This is because 3.14 is only one approximation of . Use the

examples provided on the PowerPoint Presentation.

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1. The world’s largest Ferris wheel in Yokohama City, Japan, has a diameter of 328 ft. Find its circumference.

Solution:

C = πd

≈(3.14)(328)

≈1029.92 ft.

2. A costume designer is making costumes for an Elizabethan play. An actress’s neck has a 14 in. circumference. The ruff (collar) will be made from a circle with a 6.2 in. radius. How much lace is needed to accent the circumference of the ruff?

Solution:

C = 2πr

≈2 (3.14)(6.2 in.)

≈ 38.936 in. or 39 in.

Queen Elizabeth I of England (1533-1603)

3. The earth’s circumference is approximately 25,000 mi. Find the approximate diameter of the earth.

Solution:

C = πd, thus d = C/π

d ≈ (25, 000)/ (3.14)

d ≈ 7 961 mi

4. The side of the square is 18 cm long. Find the circumference of its inscribed circle.

Solution:Since the side of the square is equal to the

diameter of the circle, therefore the diameter of the circle is also equal to 18 cm.

C = πd≈ (3.14)(18 cm)≈ 56.52 cm

18 cm

4. To sum up the lesson on Pi and Circumference , teach the song “Circle of

Friends”. (Use the PowerPoint to play the song.)

Evaluation:

Ask the students to answer the following problems:

Solve the following problems:

1. According to Guinness, the world’s largest rice cake measured 5.83 feet in

diameter. What is the circumference of this cake? (Answer: C ≈ 18.31 ft)

2. The tallest tree in the world is believed to be the Mendicino Tree, a redwood

near Ukiah, California, that is 112 meters tall. Near the ground, the

circumference of this tree is about 9.85 meters. The age of a redwood can be

estimated by comparing its diameter to trees with similar diameters. What is

the diameter of the Medicino tree? (Answer: d ≈ 3.14 m)

Assignment

Have the students complete the “Reflection” worksheet.

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Circle of FriendsCall me Ra(y) – Dius

You know my friends and meWe move in proper circles of plane

geometry.Lovely Lady Di Ameter

She’s twice as great, its true.And our good friend Sir Circumference

He’s always with us too.Our relationships are constant

You can change them if you try.Circumference divided by diameter is Pi.

Or should I day 3.14159 or soThe truth is nobody seems to know.

Time Frame: 1 session (1 hour)

Objectives


1. Discover the formula for the area of a circle

2. Find the area of a circle

3. Estimate the area of circles using methods used throughout history

Subject Matter

Topic: Area of a Circle

Materials: “Fraction Circles” activity sheet

scissors

PowerPoint Presentation of “Area of a Circle”

Value: Be optimistic in dealing with problems

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Lesson Proper

A. Review

Ask students to define area. (Area is the number of square units it takes to

cover a 2-dimensional figure.) Recall how to find the area of a rectangle or square

(Area = length x width).


1. Pass out copies of the “Fraction Circles” activity sheet.

2. Ask the students to cut the circle from the sheet and divide it into four wedges.

(The students will cut along the solid black lines.) Ask the students to arrange the

shapes as shown below: (Use the figure on slide 2 of the PowerPoint Presentation

on Area of a Circle to demonstrate the shape.)

The points of the wedges alternately point up and down.

3. Ask the students this question: “When arranged in this way, do pieces look like

any shape you know?” Students would likely suggest that the shape is unfamiliar.

4. Have students divide each wedge into thinner wedges so that there are eight

wedges in total. (Students will cut along the thicker dashed lines.) Ask the students

to arrange the wedges alternately up and down. (Use the PowerPoint to guide the

students.)

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http://illuminations.nctm.org/Lessons/ApplePi/ApplePi-AS-FractionCircles.pdf

5. Ask the students if this arrangement look like a shape they know. This time,

students will be more likely to suggest that the arrangement looks like a

parallelogram.

6. Have students divide each wedge into thinner wedges so that there are sixteen

wedges total. (Students cut along all of the dashed lines.) Ask them to arrange the

wedges alternately point up and down, as shown below: (Use the PowerPoint

Presentation to guide the students.)

7. Ask, “When the circle is divided into wedges and arrange like this, does it look

like another shape you know? What do you think will happen if we kept dividing the

wedges and arranging them like this?” The students must realize that the shape

resembles a parallelogram, but if it is continually divided, it will more closely

resemble a rectangle. (If time permits, the class may continue this activity by

dividing the wedges even further.)

8. Ask the students, “What are the dimensions of the rectangle that is formed? (Use

slide 6 of the PowerPoint.) From the lesson on Circumference, students should

realize that the length of the rectangle is equal to half of the circumference of the

circle which is r. The students must also realize that the width of the rectangle is

equal to the radius of the circle which is r. Recall that to find the area of a rectangle,

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8 wedges

16 wedges

multiply the length and the width. Consequently, the area of the rectangle formed

by the wedges of the circle is xr2= r2. This activity gives the formula for the

area of a circle A = r2.

9. Give examples of problems involving the area of a circle. Use the problems on

the PowerPoint presentation. Again, emphasize that the total area is just an

approximation if we used 3.14 as the value of .

In solving difficult problems remind the students to always be positive in

dealing with them. No matter how circuitous and difficult they might be, there’s

always a way to solve them.

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As the number of wedges , the transformed shape becomes more and more like a rectangle.

What are the dimensions of this rectangle?

½C

r

What is its area in terms of the radius?

RememberC = 2πr

A = πr x r = πr2

= πr

2.The size of a jaguar’s territory depends on how much food is available. In a situation where there is plenty of food, such as in a forest, the c ircular territory of the jaguar may as small as 3 mi in diameter. Find the area of the region.

Solution:

= π(1.5)2 r = ½d; r =1.5 mi≈ (3.14)(2.25)≈ 7.065 mi2

2rA

3. The diameter of the Aztec calendar stone shown at the right is 12 feet. This stone, which weighs over 24 tons, may have enabled the Aztecs to calculate the motions of the planets. Find the area and circumference of the face of the calendar stones.

Solution:A = πr2 C = πd≈(3.14)(6)2 ≈(3.14)(12)≈113.04 ft2 ≈ 37.68 ft

4. If the area of a c irc le is 49π ft2, what is its diameter?

Solution:If A = πr2 = 49π

r2 = 49r = 7 ftd = 14 ft

1. The length of a radius of a c irc le is 12 cm. Find the area.

Solution:

= π(12)2

≈ (3.14)(144)≈ 452.16 cm2

12cm

2rA

10. Show how Ancient Egyptians solve the area of a circle. (Use slides 11-19 of the

PowerPoint Presentation.)

11. To summarize the lessons on circle play the video on slide 20.

Evaluation:

Ask the students to answer the following problems:

1. A round clock has a diameter of 15 in. Find the area. (Answer: A ≈ 176.625 in2)

2. The radar screens used by air traffic controllers are circular. If the radius of the

circle is 12 centimeters, what is the total area of the screen? (Answer: A ≈

452.16 cm2)

Assignment

16

11

2 feet

How would you calculate the area of this circle ?

...probably using the formula A = r2

Since the diameter is 2 feet,

The constant , called “pi”, is about 3.14

so A = r2

3.14 * 1 * 13.14 square feet

means “about equal to”

?r

1 foot

“r”, the radius, is 1 foot.

12

2 feet

?

LETS explore how people figured out circle areas before all this

business ?

The ancient Egyptians had a fascinating

method that produces answers remarkably close to the formula

using pi.

13

2 feet

?

The Egyptian Octagon MethodThe Egyptian Octagon Method

Draw a square around the circle just touching

it at four points.

What is the AREA of this square ?2

feet

Well.... it measures 2 by 2, so the

area = 4 square feet.

14

2 feet


2 fe

et

Now we divide the square into nine equal smaller squares.

Sort of like a tic-tac-toe game !

Notice that each small square is 1/9 the area of the large one --we’ll use that fact later !

15

2 feet


2 fe

et

Finally... we draw lines to divide the small squares in the corners in half, cutting them on their diagonals.

Notice the 8-sided shape, an octagon, we have created !

Notice, also, that its area looks pretty close to that

of our circle !

16

2 feet


2 fe

et

The EGYPTIANS were very handy at finding the area of this Octagon

19

After all, THIS little square has an area 1/9th of the big one...

19

19

19

19

And so do these four others...

And each corner piece is 1/2 of 1/9 or 1/18th of the

big one

1. 18

1. 18

1. 18

1. 18

17

2 feet


2 fe

et

...and ALTOGETHER we’ve got...

1. 18

1. 18

1. 18

1. 18

4 pieces that are 1/18th

or 4/18ths which is 2/9ths19

19

19

19

19

Plus 5 more 1/9ths

For a total area that is 7/9ths of our original big

square

18

2 feet


2 fe

et

FINALLY... Yep, we’re almost done !

The original square had an area of 4 square feet.

So the OCTAGON’s area must be 7/9 x 4 or 28/9

or 3 and 1/9

or about 3.11 square feet

We have an OCTAGON with an area = 7/9 of the original square.

79

19AMAZINGLY CLOSEAMAZINGLY CLOSE

to the pi-based “modern” calculation for the circle !

3.11 square feet 3.14 square feet

only about 0.03 off... about a 1% error !!about a 1% error !!

OS

As an extension to the discussion, students will use the Internet to research various methods for approximating the area of circles throughout history. In pairs, students would try the various methods and determine the accuracy of their results as compared to the formula that they found. They have to answer the following questions:

What cultures used good methods that produced accurate results? Did anything surprise you about these methods or the results?

Each pair of students would report back to the class using a poster, overhead transparencies, or PowerPoint presentation. (A rubric for group project/output will be use to grade the students.)

Circle (Chapter Assessment)

Score:__________

Name:________________________________Grade & Section________Date:_____________

I. Use the circle at the right for each of the following:

1. Name the circle.__________________________

2. Name the radii of the circle._____________________

3. Name a diameter of the circle.__________________

4. If the radius of the circle is 3 cm, how long is its diameter?________

5. If the diameter of the circle is 16.4 in., how long is its radius?______

II. Solve the following problems:

1. Find the circumference of the circle.

2. A circular man-made lake has a radius of 30m. What is its area?

3. A circle has a circumference of 19.45 m. Give its radius, diameter and area.

17

RT

U

References:

Charles, R., Dossey, J., Leinwand, S., Seeley, C., and Embse C. (1999). Middle School

Math (pp.456-460). USA: Adison Wesley Longman, Inc.

Oronce, O., and Mendoza, M. (2007). E-Math III (pp.492-500).Manila: Rex Book Store

Inc.

Kenda, M., and Williams, P. (1995). Math Wizardry for Kids (pp.44-45).New York City:

Scholastic Inc.

History of Pi. (1997). The Math Forum. Retrieved November 23, 2008, from http://mathforum.org/dr.math/faq/faq.pi.html

Interesting Facts about Pi. Retrieved November 10, 2008, from http://www.middleweb.com/INCASEpi.html

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Module in Teaching Circle

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