Teil2

Let's celebrate Pi Day together!

PROJECT’S TEAM invites all our students to take part in the contest of creative works (poems, drawings, installations or cookery) dedicated to the number Pi and all students from 8th to 12th grade to take part in the Game

The game and the Competition summarizing will be held on March 13 at the time of the 6 lesson.

Plan of Pi Day Activity  The preparatory stage Of course, this stage each school can spend on its own. In our school, we will declare a contest of creative works (poems, drawings, installations or cookery) dedicated to the number Pi. The students of 5th-12th grades will prepare reports and make presentations about the number Pi - an important concept in mathematics and science, its history.  Festive event for students from 8th - 11th grades. Grade 12 students organize this event.  In the hall where the event will take place, an exhibition of creative works of students will be made.  Students from 8th to 11th grade are divided into teams of 4 -5 students from different classes in advance.  Each team takes place at a separate table.  The members of each team come up with a mathematical name of their team and write the name on the emblems, which are prepared in advance.  The song sounds. “The day they discovered Pi…” by Ken Ferrier and Antoni Chan via YouTubehttp://www.piday.org/2010/mathematical-pi-song/  Grade 12pupils congratulate everyone present on the occasion of Pi Day, give Pi Day cards to all participants (Appendix 1) and explain therules of the game"What is Pi? Where is Pi? When is Pi?"  Game "What is Pi? Where is Pi? When is Pi?" First stage – individual game Each participant receives an answer sheet (Appendix 2). At my school too many participants and the event will be held in the large hall. But you can use Kahoot!.

While demonstrating slides with questions (each question - 20sec.) students mark their answers on the answer sheet. Questions also shall be read aloud.

Answer sheets are collected and are given to the jury for summarizing. For each correct answer the team gets 1 point (Appendix 3).

When all participants have given their answers to the jury, the right answers are demonstrated on the screen.

Second stage - team game 1. Who is who?

Teams receive answer sheets (Appendix 4).

Portraits of famous mathematicians are demonstrated on the screen.

Task for teams - to match the names of the mathematicians with the description of their achievements.

Answer sheets are collected and are given to the jury for summarizing. Each correct answer 1 point (Appendix 5).

When all participants have given their answers to the jury, the right answers are demonstrated on the screen.

2. Questions for 1 minute The question is demonstrated on the screen and read out loud.

Within 1 minute, teams should discuss the solution, write the answers on a piece of paper and give it to the jury.

When all the teams have given their answers to the jury, the right answers are demonstrated on the screen.

Rating scoring. For Example: If the game is played by 5 teams and only one team gives the correct answer, it gets 5points. If two teams give the correct answer, they receive 4 pointseach team. If three teams give the correct answer, they get 3 points each team. If 4 teams give the correct answer, they get 2 points each team. If al l5 teams give the correct answer, they get1 point each team.

3. Challenge tasks The question is demonstrated on the screen and read out loud.

Within 5 or more minutes, teams should discuss the solution, write the answers on a piece of paper and give it to the jury.

Rating scoring When all the teams have given their answers to the jury, the right solution is demonstrated on the screen.

4. Make a Pi Chain Each team must have paper strips of ten different colors and a stick of glue.

At the time of this competition lively music may sound (5 min).

Use 10 different colors of construction paper. Choose a different color to represent each digit, 0 to 9. For example: 1=red, 2=blue, 3=green, 4=yellow‌ In that way, 3.141 would have rings of green, then red, then yellow, then red. Continue creating the chain for 3.14159‌ How many decimeters is the length of the chain, made by team, the same number of points is given to the team. The jury monitors that the color sequence of the chain corresponds to digits of Pi. Summarizing and rewarding the winners In our school prizes are usually pizzas with different diameters.

We hope that the event will be interesting and fun. Dear colleagues, we would like to know your impressions.

Ď€ = 3.14159265358979323846264...

Nikita Opletins 12 B 13.03.2015.

The Definition of Pi • Pi, not Pie. • The 16th letter of the Greek alphabet is P or p, corresponding to the roman p.

• A number, represented by said letter, expressing the ratio of the circumference of a perfect circle to its diameter. The value of pi has been calculated to many millions of decimal places, to no readily apparent pattern.

Beginning of Pi • It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives π = 3. Not a very accurate value of course and not even very accurate in its day.

Beginning of Pi • The earliest values of π including the 'Biblical' value of 3, were found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC.

Archimedes (Known for his work with Pi)

• 287 BC – 211 BC • Spent most of his life in Syracuse, Sicily.

• Studied in Alexandria, Egypt under the followers of Euclid.

Archimedes

• Generally regarded as the greatest mathematician and scientist of antiquity.

The “father of integral calculus.”

Archimedes Archimedes calculated pi to a very close degree, how?

He found an approximation of pi by determining the length of the perimeter of a polygon inscribed within a circle and dividing it by the diameter of the polygon.

How Many Numbers?

• 1699 – Only 71 digits were correctly discovered.

• 1719 – 112 digits were found to be correct in France.

How Many Numbers?

1841 – 440 were calculated in England.

1946 – 620 digits were correctly ordered.

Pi Facts

• So far, the largest count of Pi was done by a supercomputer at the University of Tokyo in Sept of 2002. 1.2411 trillion decimal digits were calculated.

That looks like 1,241,100,000,000 numbers

Pi Facts • There are no repeating parts in Pi. This means that at no part during the 1.2411 trillion counts of Pi did the pattern begin to repeat itself. It is the only number like this in the world. • Albert Einstein was born on Pi Day in 1879

Pi Facts • Pi Day begins at precisely 1:59 p.m. on March 14. It reads 3/14/1:59 which corresponds to 3.14159‌

Pi Equals?

• π = circumference/diameter ≈ 3.14 • π ≈ 22/7 or 3 and 1/7

Use Pi to Find: • The area of a circle = πr2 • Circumference of a Circle = πd or 2πr • Surface Area of a Sphere, Volume of a Cone, Cylinder and Sphere all are formulas that involve π.

Use Pi to Find: Right Cylinder – V = πr2h SA = 2πr2 + 2πrh

Right Cone – V = 1/3 πr2h SA = πr (l + r)

Sphere – V = 4/3 πr3

SA = 4πr2

Pi in trigonometry • Radian to degree conversion: Two pi radians is a full circle, since pi radians = 180 degrees, two pi converts to 360 degrees.

One-half pi radian is equal to 90 degrees

What is "pi"?

• Mathematician: Pi is the ratio of the circumference of a circle to its diameter.

• Engineer: Pi is about 22/7.

• Physicist: Pi is 3.14159 plus or minus 0.000005 • Computer Programmer: Pi is 3.141592653589 in double precision.

Sources

http://www.pen.k12.va.us/Div/Winchester/jhhs/math/humor/pijokes.html http://www.fourmilab.ch/gravitation/orbits/ http://www.math.nyu.edu/~crorres/Archimedes/contents.html http://www.ifoce.com/records.php http://www.joyofpi.com/pifacts.html http://www.math.com/tables/constants/pi.htm http://www.mathforum.com/library/drmath/view/57543.html http://www.sixflags.com http://www.worldslargestthings.com

WHAT IS PI? WHERE IS PI? WHEN IS PI?

Dear Project ÂŤMMM: More Meaningful MathÂť participants!

We salute you on the eve of the twenty seventh annual Mathematical feast - Pi Day! Let us celebrate this never-ending number (3.14159 . . .) and Einstein's birthday together!

Founded at the Exploratorium by physicist Larry Shaw, Pi Day has become an international holiday, celebrated live and online all around the world.

Latvian team

ď ?Choose the correct answer!

Individual game

1. What is the formal definition of Pi? • a) The surface area of a sphere of diameter 22/7.

• b) The radius of a unit circle.

• c) The ratio of a circle's circumference to its diameter.

• d) 3.14159.

2. What is the relationship between the symbol "�" and the word "Pi"?

• a) A "�" was chosen for "perimeter" of circles.

• b) This is the first letter of the name Pythagoras.

• c) This is the first letter of the word "perpendicular".

• d) This is the first letter of the word "plane".

3. For how many years has Pi Day been celebrated?

• • • •

a) 25; b) 26; c) 27; d) 28.

4. March 14 is also whose birthday?

• • • •

a) Archimed; b) William Jones; c) Euler; d) Einstein.

5. Clock with a bell ring 6 times within 5 hours. For how many hours the clock will ring 12 times?

• • • •

a) 10; b) 11; c) 12; d) 13.

6. The clock shows 4 o'clock .

What angle is between the clock rates?

• • • •

a) 60°; b) 90°; c) 120°; d) 150°.

7. The distance around a bicycle wheel is 21.98 feet.

What is its diameter?

• • • •

a) 3,5 feet; b) ≈ �, � �eet; c) 7 feet; d) 21 feet.

Use � ≈ 3.14 to calculate your answers

8. The area of a DVD is 28,26 square centimeters.

What is its diameter?

• • • •

a) ≈10 cm; b) 6 cm; c) 3 cm; d) 9,42 cm.

Use � ≈ 3.14 to calculate your answers

9. Why are manhole covers make round and not square?

• a) In order to save metal.

• b) For the beauty.

• c) In order to cover not failed into the hatchway.

• d) It's just a tradition.

10. On the table one coin 2 euro lies still while the other 2 euros coin rolls around the first touching it. How many times did it turn around its center, before returning to the starting position.

•

a) 1; • b) 2; • c) 3; • d) 3,14.

ď ? Who is who?

Team game

Rene Descartes (1596 – 1650)

Pythagoras (569 – 475 BC)

Carl Gauss (1777 – 1855)

Euklid (325 - 265 BC)

Al – Horezmi (780 – 850 BC)

Pierre de Fermat (1601 – 1665)

Blaise Pascal (1623 -1662)

Leonhard Euler (1707 – 1783)

Sofia Kovalevskaya (1850 – 1891)

Erastophen (276 – 194 BC)

Nikolai Lobachevsky (1792 – 1856)

ď ?Questions for 1 minute

Team game

Question 1 A lawn sprinkler sprays water 5 m in every direction as it rotates. What is the area of the sprinkled lawn?

Use ďż˝ ≈ 3.14 to calculate your answers

Answer: 78,5 m2

Question 2 What is the circumference of a 12-inch pizza?

Use � ≈ 3.14 to calculate your answers

Answer: 37,68 in

Question 3 A dog is tied to a wooden stake in a backyard. His leash is 3 meters long and he runs around in circles pulling the leash as far as it can go. How much area does the dog have to run around in?

Use ďż˝ ≈ 3.14 to calculate your answers

Answer: 28,26 m2

Question 4 An asteroid hit the earth and created a huge round crater. Scientists measured the distance around the crater as 78.5 miles.

What is the diameter of the crater?

Use ďż˝ ≈ 3.14 to calculate your answers

Answer: 25 mi

Question 5 A storm is expected to hit 7 miles in every direction from a small town. What is the area that the storm will affect?

Use ďż˝ ≈ 3.14 to calculate your answers

Answer: 153,86 mi2

Question 6 If Juan can consume in one sitting one medium (12 cm diameter) deep dish (1 cm thick) pizza, what is the minimum volume of his stomach?

Use ďż˝ ≈ 3.14 to calculate your answers

Answer: 113,04 cm3

ď ? Challenge tasks

Team game

Problem 1 If a circular track is 5 meters wide and it takes a horse, traveling its fastest, Pi more seconds to travel the outer edge than the inner edge, what is the horse's speed?

Solution

�=

�

Answer: 10 m/s

� �

� �= =

�

�

+� +�

+� +�

+� = � � = �

� ∙� �= = �

+�

�/

Problem 2 Earth globe pulled hoop around the equator, and in the same way "around the " equator " pulled hoop orange. Imagine that the length of each hoop has increased by 1 meter. In this case, between the surfaces of bodies and hoops a gap is formed. In which case this gap will be larger the globe or in orange?

Solution

Answer: ∆� = ∆

Make a Pi Chain Use 10 different colors of construction paper.

Choose a different color to represent each digit, 0 to 9. For example: 1=red, 2=blue, 3=green, =yello ‌ In that ay, .1 1 ould ha e rings of green, then red, then yellow, then red.

Continue creating the chain for .1 1 9‌

Congratulations to all the participants of the project on the holiday – Pi Day!

http://ed.ted.com/lessons/the-infinite-life-of-pi-reynaldo-lopes http://www.piday.org/2010/mathematical-pi-song/

PI cartoons and comics

picture on: https://www.pinterest.com/pin/564287028283062951/

picture on : https://www.pinterest.com/pin/407083253791281711/

http://www.pi314.net/imagespi/Pi_day/2011_i-hate-pi-day.jpg

http://www.pi314.net/imagespi/Pi_day/2008_PiDayCartoon_Grand.jpg

http://www.pi314.net/imagespi/Pi_day/2007_pi_day_dinosaurs_mini.jpg

https://www.pinterest.com/pin/107523509823449860/

picture

http://imgarcade.com/1/cartoon-pi-symbol/

https://gjismyp.wordpress.com/2013/03/01/math-cartoon-the-wife-of-pi/

https://www.pinterest.com/pin/556898310144273023/

Congratulations to all the participants of the project on the holiday – Pi Day!

.1 1 ‌

Individual Game answer sheet NAME:_________________ Question Answer

a b c d

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Individual Game answers NAME:_________________ Question Answer

a b c d

1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Who is who? - Worksheet 1. René Descartes (1596 – 1650)

A. At the age of 3 he corrected arithmetic mistakes made by his father.He proved that any natural number can be represented as a multiple result of prime numbers.

2. Pythagoras (569 – 475 y.B.c.)

B. He has written 2 books in mathematic, which helped to popularize the decimal system in the world. The word ‘algebra’ is in title of one of his books.

C. At the age of 19 he created his own adding machine. In fact, he is a founder of the theory of probability.

3. Carl Gauss (1777 – 1855) 4. Euklid (325 – 265 y.B.c.)

5. Al-Horezmi (780 – 850 y. B.c.)

6.Pierre de Fermat (1601 – 1665) 7. Blaise Pascal (1623 – 1662)

D. "I think, therefore I exist" - this scientist said . He was the first who introduced the letter symbols to designate variables.

E. He is a mathematician, an author of the largest number of works (more than 800), a founder of the graph theory.

Without this theory, it would be impossible to make microchips.

F. The author of the first non-Euclidean geometry. Not to shock the audience with his geometry he called it "imaginary geometry" (but was still laughed at). Only after his death he received international recognition and fame.

G. He proved that there are infinitely many primes and that a square root of 2 is an irrational number, author of the famous textbook of geometry.

8. Leonhard Euler (1707 – 1783)

H. His most meaningful achievement was a calculation of the circumference of the Earth when people did not know that the Earth was round.

9. Sofia Kovalevskaya (1850 – 1891)

J. He taught that the basis of things was a number. Ancient chroniclers have named him as the author of the most famous theorem in mathematics: in a right angled triangle the square of the long side is equal to the sum of the squares of the other two sides.

K. The first woman in the world - a mathematician. A professor at Stockholm University. One of the three women in Europe, who has received the degree of Doctor of Science.

L. He has made a great contribution to the theory of numbers. The Author of the famous theorem. It took more than 300 years, to prove the validity of this theorem.

10. Erastophen (276 – 194 г y.B.c.)

11.Nikolai Lobachevsky (1792 – 1856)

Who is who? - Answers 1. – D 2. – J 3. – A 4. – G 5. – B 6. – L 7. – C 8. – E 9. – K 10. – H 11. – F

Ideas to celebrate PI-day by Dorian T. (germany)

Ideas to celebrate PI-day

by Dorian T. (germany)

1. Simons says 

it´s an instant memory challenge



A volunteer must start with the first digit of pi



The next person has to add with the next digit and it goes on like that...



Try to get the most digits of pi



http://teachpi.org/activities/fun-with-digits/

2. Stand up for pi 



Go out with the complete school

Split the digits, so every grade have another digit



Get the most digits as you can



http://teachpi.org/activities/fun-with-digits/

3. Book of recipes 

Collect recipes for a pi cake, cookies and more



One teacher or student make a book of recipes



Bake some of the cakes and bring them to class



http://teachpi.org/activities/eats/

4. Albert Einstein and Pi

4. Here you see the use of Pi

Pi Rap 



https://www.youtube.com/watch?v=qsjrjPquqiA

This is a pi rap with all necessary informations about pi and the story about pi

The pi song 





https://www.youtube.com/watch?v=eDiSYp_51i Y Here you find the most famous song about pi with about one million views The song tell you much digits of pi

The first 1000 digits of pi

What Pi sounds like 



https://www.youtube.com/watch?v=wK7tq7L0N 8E The person in the video try to play pi with instruments

5. That´s how you calculate Pi

6. Domino 





https://www.youtube.com/watch?v=Vp9zLbIE8z o In this video a girl build with dominos a spiral with pi in the middle The spiral can be calculated

7. Students make videos For extra credit, encourage small groups of students to make a short film about pi.

Suggest both music videos and live-action skits.

http://teachpi.org/activities/projects/

8. Compose songs or music Ask students to compose an original song, poem, or piece of art about Pi Day or the number pi. Hold presentations/exhibits, and present any artistic awards, on Pi Day.

» How about a “Pi-ku” poetry contest? (Think Haiku.) For example: Unending digits…

Why not keep it simple, like Twenty-two sevenths?

http://teachpi.org/activities/projects/

9. PI competition in class PGU did it last year in grade 8.

Students have built groups of 3-4 students and prepared presentations. Unfortunatly all presenta-tions were not recorded because there had been a problem with the recorder.

Students had the following ideas: The story of Mrs. PI, a self-made video to introduce students to PI, a video how to to approximate PI, a computer animation with PI, a PI cake, explanations around PI with a wooden self-made PI symbol, a PI song, a video of funny measurements of round objects, ‌

C I N D Y D. K R O O N

Teacher to Teacher

Playing around with

“Mono-pi-ly” A

CCORDING TO THE GEOMETRY STAN-

dard in Principles and Standards for School Mathematics, “In grades 6–8, all students should precisely describe, classify, and understand relationships among types of two- and three-dimensional objects” (NCTM 2000, p. 232).

The Measurement Standard goes on to state, “In grades 6–8, all students should develop and use formulas to determine the circumference of circles” (NCTM 2000, p. 240). In addition, South Dakota’s Measurement Standard for Grade 7 delineates what mathematics students should know, such as “Given the formulas, find the circumference, perimeter, and area of circles” (South Dakota Department of Education 2004).

This article describes how to play “mono-pi-ly,” a mathematical game for two to five players that furthers the intent of both NCTM’s and South Dakota’s Standards. It was created as part of a Pi Day activity for use with a mixed-level geometry class. While playing, students review their understanding of mathematics vocabulary involving circles and practice area and circumference calculations. The playing time is approximately forty to fifty minutes.

The game generated a great deal of student enthusiasm and was also a good review of circle properties. Students were by turns competitive and sup-

CINDY KROON, cindy.kroon@k12.sd.us, teaches grades 9–12 mathematics at Montrose High School in Montrose, SD 57048. She is interested in problem solving and connecting mathematics and science and enjoys presenting professional development workshops.

294

portive as they played the game. All students were engaged, whether completing calculations as part of their own turns or checking the accuracy of answers calculated by others.

Although created for Pi Day, this game would serve as a good class activity whenever circle concepts are taught or reviewed. Making the game’s questions either easier or more difficult can vary the level of difficulty. The game board is shown in figure 1, and the questions are shown in figure 2.

These materials are needed for play: • Game board (see fig. 1) • Questions (see fig. 2) (Note: Question cards will be more durable and easier to use if printed on heavy paper or card stock.)

• Two dice • Playing pieces (buttons, coins, and so on; one per player) • Calculator • Scratch paper The following information is important to note before playing: • The official rules are in figure 3.

• Use 3.14 as an approximation for pi in calculations.

• The student whose birthday is closest to Pi Day (3/14) plays first. After that, play passes to the left.

• When a student lands on a “?” symbol, he or she will answer a question for a bonus roll. Someone else should read the question, since the answer appears at the bottom of the card.

MATHEMATICS TEACHING IN THE MIDDLE SCHOOL Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

Fig. 1 The “mono-pi-ly” game board VO L . 1 1 , N O . 6 . FEBRUARY 2006

295

Find the area of a circle with a radius of 8 meters.

The radius of a circle is 13 inches. What is its circumference?

The diameter of a circle is 10 cm. What is its area?

200.96 m2

81.64 inches

78.5 cm2

True or false: A circle is a polygon.

What is the formula for circumference?

The radius of a circle is 10 m.

What is its area?

False

C = 2pr, or pd

314 m2

The diameter of a circle is 8 cm. What is its circumference?

What is the decimal approximation for p to 5 decimal places?

What do you get when you divide the circumference of a pumpkin by its diameter?

3.14159

Pumpkin pi!

The area of a circle is 78.5 ft.2. Find its circumference.

Circle A has a diameter of 24 mm. What is its radius?

How many degrees are in a semicircle?

31.4 ft.

12 mm

180

The area of a circle is 100 cm2.

What is the radius of the circle?

How many degrees are in a circle?

5.64 cm

360°

Have the person to your left make up the radius of a circle. Calculate its area.

What is the area of a circle with a diameter of 12 cm?

What does circumference divided by diameter equal?

Find the area of a circle with a diameter of 26 cm.

113.04 cm2

p

530.66 cm2

Find the area of a circle with a circumference of 18.84 cm.

Have the person to your left make up the radius of a circle.

Calculate its circumference.

25.12 cm

28.26 cm2

The area of a circle is 28.26 m2. The circumference of a circle is 6.28 m. What is its diameter?

What is the circumference?

2m 18.84 m Define radius.

The distance from the center of a circle to any point on the circle.

Have the person to your left make up the diameter of a circle. Compute its area.

What is the circumference of What is the equation for finda circle with a radius of 9 cm? ing the area of a circle?

56.52 cm A = pr2

Fig. 2 The game’s questions, divided into individual cards 296

MATHEMATICS TEACHING IN THE MIDDLE SCHOOL

Find the area of the 68 degree sector of a circle with a radius of 4 cm.

9.49 cm2

The radius of a circle is 2.5 in. What is its diameter?

5 in.

What is the area of a circle with a diameter of 12 cm?

113.04 cm2 What is the circumference of a circle with 24 m as the radius?

150.72 m

How to Play The student whose birthday is closest to Pi Day (3/14) plays first. After that, play passes to the left.

1. Roll two dice. The sum of these two numbers is the circumference of a circle. Calculate the diameter of the circle. (Use 3.14 as an approximation for pi.)

2. If you answer correctly, round to the nearest unit and move that many spaces. If the answer given is incorrect, you lose your turn.

3. Follow the directions on the space that you land on. More than one player can be on the same space at one time.

4. If you land on a space with a ?, you must answer a question card. If the answer is correct, roll one die and advance that many spaces.

Your turn ends. If the answer is incorrect, stay on the ? space. Your turn is over.

5. To win the game, you must be the first player to go around the game board twice and correctly answer a question at the finish line.

Fig. 3 The rules of “mono-pi-ly”

• Other spaces on the game board—decorated with circles and letters representing circumference, area, radius, diameter, and so on—are decoration only and do not affect game play.

• Cards should be shuffled and placed facedown at the start of the game.

As a follow-up writing assignment, students critiqued the game and made suggestions for improvement (see fig. 4). Each student also submitted three problems with solutions, suitable for inclusion in a subsequent question set.

It is hoped that others will echo Pat, who wrote this comment after playing the game, “It makes it very easy to remember all the formulas involved with circles.”

References National Council of Teachers of Mathematics (NCTM).

Principles and Standards for School Mathematics.

Reston, VA: NCTM, 2000.

South Dakota Department of Education. “Measurement Standard 6–8.” Office of Curriculum, Technology, and Assessment. doe.sd.gov/contentstandards/math/ new/standards. 2004.

Fig. 4 Student questions and opinions about the game VO L . 1 1 , N O . 6 . FEBRUARY 2006

297

π - day Activity 1: Circle, Circles everywhere Like many interesting shapes, circles are all around us every day. But how often do you notice them? Circles have fascinated people throughout the ages, so let's explore some of the most famous and mysterious circles in history.

In Ancient Greek culture the circle was thought of as the perfect shape. Can you think why?

How many lines of symmetry does a circle have, for instance? To the Greeks the circle was a symbol of the divine symmetry and balance in nature. Greek mathematicians were fascinated by the geometry of circles and explored their properties for centuries.

Circles are still symbolically important today -they are often used to symbolize harmony and unity. For instance, take a look at the Olympic symbol. It has five interlocking rings of different colours, which represent the five major continents of the world united together in a spirit of healthy competition.

π is the symbol used for a special number which we call pi (pronounced "pie"). π comes from working with circles.

That is why we will first explore circles and their properties.  Write down the definition of a circle.

1

 In the space below draw a circle using a compass.

 Working in groups, take the tools provided: the stones, the meter ruler and the bottle of water. Each person should take a stone and place it one meter away from the bottle of water. What shape is being formed? What is the significance of the stones, the meter ruler and the bottle of water in this formation?

 Would you now like to redefine a circle?

 What do we call the diameter of a circle?

 What do we call the circumference of a circle?

2

π - day Activity 1: Circle, Circles everywhere Like many interesting shapes, circles are all around us every day. But how often do you notice them? Circles have fascinated people throughout the ages, so let's explore some of the most famous and mysterious circles in history.

In Ancient Greek culture the circle was thought of as the perfect shape. Can you think why?

How many lines of symmetry does a circle have, for instance? To the Greeks the circle was a symbol of the divine symmetry and balance in nature. Greek mathematicians were fascinated by the geometry of circles and explored their properties for centuries.

Circles are still symbolically important today -they are often used to symbolize harmony and unity. For instance, take a look at the Olympic symbol. It has five interlocking rings of different colours, which represent the five major continents of the world united together in a spirit of healthy competition.

is the symbol used for a special number which we call pi (pronounced "pie"). working with circles.

comes from

That is why we will first explore circles and their properties.  Write down the definition of a circle. Ε ώ ο ς αφή ο

α ράψο

ό ι θέ ο . Και άθος α ί αι

ο ιορθώ ο

.

1

 In the space below draw a circle using a compass.

 Working in groups, take the tools provided: the stones, the meter ruler and the bottle of water. Each person should take a stone and place it one meter away from the bottle of water. What shape is being formed? What is the significance of the stones, the meter ruler and the bottle of water in this formation?

A circle is being formed Stones (set of points) Meter ruler (radius of circle) Bottle of water (Centre of the circle)  Would you now like to redefine a circle?

A set of points equidistant from a given point, called the center.

 What do we call the diameter of a circle?

In geometry, the diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

The word "diameter" is derived from Greek ιά τ ος (diametros), "diameter of a circle", from ια- (dia-), "across, through" + έτ ο (metron), "measure".[

 What do we call the circumference of a circle?

The circumference of a circle is the distance around it You can use the rope to measure it

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Ď€ - day Activity 2: Slope, Pi, and Lines In this activity, you will measure the circumference and diameter of several circles, and then graph the relationship between circumference and diameter.

Your group should have a roll of masking tape, a pair of scissors, and at least three circular objects of various sizes.

A. Wrap masking tape around one of the circles, overlapping the edges. Select a place to cut through the masking tape, peel it off the object, and label it circu fere ce.

Measure its length and write the measurement on the tape.

B. Stretch a length of tape from one side of the object to the other that passes through the centre of the circle. Trim the tape at the edges of the circle and label this strip as dia eter. Measure its length and write the measurement on the tape.

C. On the graph, place the tape of the diameter under the x-axis. Then place the circumference tape vertically, so that it is parallel to the y-axis. Plot a point at the top of the circumference strip and label its coordinates. (See diagram below.)

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1.

Repeat Steps A–C for the other circles. Record a summary of the results in the table below.

Object

Diameter

Circumference

2.

Draw a straight line that approximates the points on your graph.

3.

What number represents the slope of the line?

4.

What is the equation of the line?

5.

Write down a formula relating the circumference to the diameter.

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π - day Activity 3: Area of circles As a warm-up, give students an opportunity to estimate the area of the circular objects that they have brought to class. Working in groups students should individually complete the first two columns:  

Description of the object Their estimate for the area of the object

In this activity, you will explore ways to estimate the area of circles.

First, use the circular objects you have in front of you to estimate their area. Think of different methods you can use to achieve this. Record your results in the following table (fill in the first two columns only).

DESCRIPTION OF THE OBJECT

YOUR ESTIMATE OF THE AREA (IN SQUARE CENTIMETERS)

RADIUS OF THE OBJECT

ACTUAL AREA

Table 1

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Students may use any method they like to estimate the area of their objects. Some possible methods include:  



Students can trace the shape of their object on a piece of centimeter grid paper and count how many square centimeters make up the total area of the circle.

Students can divide the circle into wedges by drawing various radii. They can approximate the area of each wedge using the triangle formula. (This method is similar to a method used by Archimedes, and it is the method that will be used later in this lesson. For a connection to mathematical history, you may want to include a brief overview of Archimedes and his method for calculating the area of a circle.)

Students can inscribe the circle in a square, hexagon, or some other polygon. Then, the same shape could be inscribed within the circle. Students could determine the area of the inscribed and circumscribed shapes to get lower and upper estimates, respectively. (You may need to provide a sample drawing of this method, like the one shown below.)

After students have estimated the area of several objects, allow them to physically discover the area formula of a circle.

1. Now take the coloured paper circle, fold it into fourths, and then unfold it. Cut along the creases created dividing your circle into four wedges. Arrange the shapes so that the points of the wedges alternatively point up and down. When arranged in this way, do the pieces look like any shape you know?

Students will likely suggest that the shape is unfamiliar.

2. Now divide each wedge into two thinner wedges so that there are eight wedges total and arrange the shapes alternately up and down. Does this arrangement look like a shape you know?

This time, students will be more likely to suggest that the arrangement looks a little like a parallelogram.

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3. Finally, divide each wedge again into two thinner wedges so that there are sixteen wedges in total and arrange the wedges so that they alternately point up and down.

When the circle is divided into wedges and arranged like this, does it look like another shape you know? What do you think would happen if we kept dividing the wedges and arranging them like this?

Lead the discussion so students realize the shape currently resembles a parallelogram, but as it is continually divided, it will more closely resemble a rectangle.

4. Do you have any suggestions which can improve this shape into making it look more like a rectangle instead of a parallelogram?

Cut the last section in two and arrange these two smaller pieces on either side of the parallelogram.

5. What are the dimensions of the rectangle that is formed?

The length of the rectangle is equal to half the circumference of the circle, or πr.

Additionally, it should be obvious that the height of this rectangle is equal to the radius of the circle, r. Co se ue tly, the a ea of this e ta gle is πr × r = πr2. Because this rectangle is equal in area to the original circle, this activity gives the area formula for a circle:

6. So the area of the circle is given by the formula: _________________ A = πr2

7. Now return to the objects for which you estimated the area at the beginning of the activity. Measure the radius of each object and record it in the third column of table 1.

Then, use the formula that you have just discovered for the area of the circle, to calculate the actual area of each object, and record the area in the fourth column.

Once all groups have completed the measurements and calculations, a whole-class discussion and presentation should follow. On the board, the presenter should record the areas for the objects for each group. The students should compare the results of each group and discuss the accuracy of the areas found.

The class should also compare their original estimates with the actual measurements.

On their recording sheets, have them highlight the objects for which their estimates were very close to their actual. Using a few sentences, have the students explain (on the recording sheet) why some estimates were closer than others.

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During the class discussion, the following are some key points to highlight: 



Emphasize that 3.14 is only one approximation for π. Refe to the Circumference lesson, and discuss the various estimates that were found for π a d hat aused these variations. Also explain that there are other approximations, but typically 3.14 is used because it is accurate enough for most situations and it is easy to remember. If students are curious, other approximations for π a e gi e o the Pi Approximation sheet.

The total area is almost always an approximation. Because the value of π a o ly be approximated, any time the area of a circle is stated without the π sy ol, it must be an approximation. For instance, a circle with radius of 5 cm has an exact a ea of 25π cm2 and an approximate area of 78.54 cm2.

1.

What number represents the slope of the line?

2.

What is the equation of the line?

3.

Write down a formula relating the circumference to the diameter.

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Ď€ - day Activity 3: Area of circles In this activity, you will explore ways to estimate the area of circles.

First, use the circular objects you have in front of you to estimate their area. Think of different methods you can use to achieve this. Record your results in the following table (fill in the first two columns only).

DESCRIPTION OF THE OBJECT

YOUR ESTIMATE OF THE AREA (IN SQUARE CENTIMETERS)

RADIUS OF THE OBJECT

ACTUAL AREA

Table 1

1. Now take the coloured paper circle, fold it into fourths, and then unfold it. Cut along the creases created dividing your circle into four wedges. Arrange the shapes so that the points of the wedges alternatively point up and down. When arranged in this way, do the pieces look like any shape you know?

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2. Now divide each wedge into two thinner wedges so that there are eight wedges total and arrange the shapes alternately up and down. Does this arrangement look like a shape you know?

3. Finally, divide each wedge again into two thinner wedges so that there are sixteen wedges in total and arrange the wedges so that they alternately point up and down.

When the circle is divided into wedges and arranged like this, does it look like another shape you know? What do you think would happen if we kept dividing the wedges and arranging them like this?

4. Do you have any suggestions which can improve this shape into making it look more like a rectangle instead of a parallelogram?

5. What are the dimensions of the rectangle that is formed?

6. So the area of the circle is given by the formula: _________________

7. Now return to the objects for which you estimated the area at the beginning of the activity. Measure the radius of each object and record it in the third column of table 1.

Then, use the formula that you have just discovered for the area of the circle, to calculate the actual area of each object, and record the area in the fourth column.

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Pi Filling, Archimedes Style STEPS 1) Clear the screen

RESULT  Go To GeoGebra  First we will create regular polygons circumscribed around a circle.  Go to Graphics window, right click on the empty space and select Axes to remove them

 Your screen should look like this:

2)

Circumscribed Regular Polygon

 Go to I put bar o the botto

of the page a d type

=3 .

 In the Algebra window right click on n=3 and select Object Properties





Kj

Pi memorization contest π is the symbol used for a special number which we call pi (pronounced "pie"). π is the ratio of the circumference of a circle to its diameter. π is needed to find the area of a circle using the formula πr2 What is the value of Pi? In the Old Testament of the Bible (I Kings 7:23) it is suggested that π is equal to 3. The Babylonians, in about 2000 BC, use 3 or 3 1/8. In the ancient Rhind papyrus, the Egyptian scribe Ahmes said that π was equal to 16/9 squared. This calculates to . ….

Archimedes worked on the problem of finding π by calculating the area of regular polygons, with up to 96 sides. He decided that π was somewhere between 310/71 and 310/70. In decimals this would be . … and 3.14285 … (remember the decimal places keep going on and on).

Over the centuries, many mathematicians, such as Ptolemy (the ancient Greek astronomer), Tsu Ch'ung-Chi (of China) and Ludolph van Ceulen (of Germany) kept trying to find more accurate values for pi using a variety of different methods. During the last few centuries people have been trying to find as many decimal places as possible so they can look for patterns in the long string of digits. In 1949 a computer was used to calculate pi to 2037 places. In 1967 in France 500 000 digits were found. In 1983 a Japanese team found 16 777 216 decimal places for pi. The current record is about 51 billion decimal places.

Here are some of the decimal places that have been found: π= .

…

To help remember these digits, people like to make up sentences or rhymes, called mnemonics.

For example, "May I have a large container of coffee?" is quite a famous one for the first eight digits. You work out the numbers by counting the letters in each word.

Here's one for the first 31 digits: "Now I will a rhyme construct, By chosen words the young instruct.

Cunningly devised endeavour, Con it and remember ever.

Widths in circle here you see, Sketched out in strange obscurity."

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And one more: Pie I wish I could recollect pi.

"Eureka!," cried the great inventor.

Christmas pudding, Christmas pie, is the problem's very centre!

Try making up your own mnemonic for the digits of pi.

For this challenge you have to work in groups.

Choose a member of your team whom you believe is most capable of memorizing as many digits of π as possible.

Once you have decided about who is going to represent your team, find ways to help him/her memorize as many digits of π as possible. Below you can see the first 200 or so digits of pi.

Good Luck!!!

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π is the symbol used for a special number which we call pi (pronounced "pie"). π is the ratio of the circumference of a circle to its diameter. π is needed to find the area of a circle using the formula πr2

What is the value of Pi? In the Old Testament of the Bible (I Kings 7:23) it is suggested that π is equal to 3. The Babylonians, in about 2000 BC, use 3 or 3 1/8. In the ancient Rhind papyrus, the Egyptian scribe Ahmes said that π was equal to 16/9 squared. This calculates to 3.16 ….

Archimedes worked on the problem of finding π by calculating the area of regular polygons, with up to 96 sides. He decided that π was somewhere between 310/71 and 310/70. In decimals this would be . … and . … (remember the decimal places keep going on and on).

Over the centuries, many mathematicians, such as Ptolemy (the ancient Greek astronomer), Tsu Ch'ung-Chi (of China) and Ludolph van Ceulen (of Germany) kept trying to find more accurate values for pi using a variety of different methods. During the last few centuries people have been trying to find as many decimal places as possible so they can look for patterns in the long string of digits. In 1949 a computer was used to calculate pi to 2037 places. In 1967 in France 500 000 digits were found. In 1983 a Japanese team found 16 777 216 decimal places for pi. The current record is about 51 billion decimal places.

Here are some of the decimal places that have been found: π= .

…

To help remember these digits, people like to make up sentences or rhymes, called mnemonics.

For example, "May I have a large container of coffee?" is quite a famous one for the first eight digits. You work out the numbers by counting the letters in each word.

Here's one for the first 31 digits:

"Now I will a rhyme construct, By chosen words the young instruct.

Cunningly devised endeavour, Con it and remember ever.

Widths in circle here you see, Sketched out in strange obscurity."

Try making up your own mnemonic for the digits of pi.