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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”.
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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
At the end of the lesson, the students must be able to:
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
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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.)
B. Presentation of the Lesson
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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.)
B. Presentation of the Lesson
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
At the end of the lesson, the students must be able to:
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).
B. Presentation of the Lesson
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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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
The Egyptian Octagon MethodThe Egyptian Octagon Method
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
The Egyptian Octagon MethodThe Egyptian Octagon Method
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
The Egyptian Octagon MethodThe Egyptian Octagon Method
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
The Egyptian Octagon MethodThe Egyptian Octagon Method
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
The Egyptian Octagon MethodThe Egyptian Octagon Method
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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