Ancient Mathematicians Ancient Mathematicians for A Day:for A Day:

The calculation of piThe calculation of pi

Georgia CTAE Resource Georgia CTAE Resource NetworkNetwork

Instructional Resources OfficeInstructional Resources OfficeWritten By: Philip Ledford and Dr. Frank FlandersWritten By: Philip Ledford and Dr. Frank Flanders

July 2009July 2009

ObjectivesObjectives

► Learn the origin of pi.Learn the origin of pi.

► Examine the process in which pi was developed.Examine the process in which pi was developed.

► Test the process in which pi was developed.Test the process in which pi was developed.

► Create graphs from discovered data.Create graphs from discovered data.

► Evaluate and compare discovered data.Evaluate and compare discovered data.

► Solve basic mathematical problems with discovered data.Solve basic mathematical problems with discovered data.

Introduction to the ActivityIntroduction to the Activity

► Around 4000 years ago the ancient Egyptians and Around 4000 years ago the ancient Egyptians and Babylonians discovered the mathematical mystery Babylonians discovered the mathematical mystery of pi. They did this by drawing different sized circles of pi. They did this by drawing different sized circles and measuring the circumference and the diameter and measuring the circumference and the diameter of the circle. They would determine how many of the circle. They would determine how many diameter lengths it took to equal the length of the diameter lengths it took to equal the length of the circumference. This process gave a number, a ratio, circumference. This process gave a number, a ratio, that was consistent for every circle. This number is that was consistent for every circle. This number is now known as pi.now known as pi.

► This activity will simulate the work of ancient This activity will simulate the work of ancient mathematicians and help in understanding pi.mathematicians and help in understanding pi.

Ancient Mathematicians for A Ancient Mathematicians for A DayDay

►Ever wondered where some of the Ever wondered where some of the math problems you do come from?math problems you do come from?

► A lot of the math we deal with today was developed and used A lot of the math we deal with today was developed and used over 4000 years ago by ancient civilizations.over 4000 years ago by ancient civilizations.

► One of the biggest mysteries of math is “pi”.One of the biggest mysteries of math is “pi”.

► Pi was discovered by the ancient Egyptians and Babylonians.Pi was discovered by the ancient Egyptians and Babylonians.

PiPi

►Pi is a mathematical constant whose Pi is a mathematical constant whose value is the ratio of any circle's value is the ratio of any circle's circumference to its diameter circumference to its diameter

► In this lesson we will participate in the In this lesson we will participate in the same procedures the ancient Egyptian same procedures the ancient Egyptian and Babylonian mathematicians did to and Babylonian mathematicians did to discover pi.discover pi.

Materials You Will NeedMaterials You Will Need

►MaterialsMaterials► PaperPaper► Various sizes of circles to outline Various sizes of circles to outline

(Use anything that is circular like (Use anything that is circular like cups, cans, etc.)cups, cans, etc.)

► StringString► RulerRuler► TapeTape► ScissorsScissors

►Work in groups of Work in groups of two or more.two or more.

Instructions Instructions

► Step One: Tracing…Step One: Tracing…► Trace different size circular objects. Trace different size circular objects.

• • When Drawing….When Drawing….RememberRemember to trace the best circle you can, to trace the best circle you can, the more accurate your drawings are the the more accurate your drawings are the more accurate your calculations will be.more accurate your calculations will be.

Instructions (con’t)Instructions (con’t)

► Step Two: Taping….Step Two: Taping….► Tape the string down over the Tape the string down over the

outline of the circles. Remember to outline of the circles. Remember to try and cover the outline precisely try and cover the outline precisely for accuracy.for accuracy.

► Cut the string to form a perfect Cut the string to form a perfect circle. circle.

Instructions (con’t)Instructions (con’t)

► Step Three: Measuring Circumference…Step Three: Measuring Circumference…► After taping the string around the circle outline take it off and After taping the string around the circle outline take it off and

measure it. measure it. ► Record this data as your circumferenceRecord this data as your circumference

Instructions (con’t)Instructions (con’t)

► Step Four: Cutting and Folding…Step Four: Cutting and Folding…► Take the string off the circle. Cut out the circle on the paper.Take the string off the circle. Cut out the circle on the paper.

Now fold the paper circle in half.

Instructions (con’t)Instructions (con’t)

► Step Five: Measuring Diameter….Step Five: Measuring Diameter….► The circle is folded straight through the middle to determine the diameter. The circle is folded straight through the middle to determine the diameter.

► Take a piece of string and lay it out across the folded line. Now cut the string Take a piece of string and lay it out across the folded line. Now cut the string and and measure itmeasure it. This will give you the length of your diameter.. This will give you the length of your diameter.

Instructions (con’t)Instructions (con’t)

► Step Six: Data….Step Six: Data….► Every circle should have a diameter and a circumference measured, Every circle should have a diameter and a circumference measured,

and recorded.and recorded.

Instructions (con’t)Instructions (con’t)

► Step Seven: The Fun Part….Step Seven: The Fun Part….► This step is where the ancient mathematicians developed pi.This step is where the ancient mathematicians developed pi.

► If the diameter string of a circle is laid on the outline of its circumference it If the diameter string of a circle is laid on the outline of its circumference it should take the diameter string a little more than should take the diameter string a little more than three three times to equal the times to equal the circumference. This number is the value of pi and should work with every size circumference. This number is the value of pi and should work with every size circle.circle.

► Repeat the previous steps on different sizes of circles. It is recommended to do at least 4 Repeat the previous steps on different sizes of circles. It is recommended to do at least 4 different sizes.different sizes.

RecordsRecords► Examine the data that has been recorded. 4 different diameters and 4 Examine the data that has been recorded. 4 different diameters and 4

different circumference should be written down.different circumference should be written down.

► Label a piece of paper in the same manner as the picture below.Label a piece of paper in the same manner as the picture below.

► Put the diameter under the X-AXIS and circumference under Y-AXIS.Put the diameter under the X-AXIS and circumference under Y-AXIS.

► Make sure the data being recorded is for the correct circle. Both diameter and Make sure the data being recorded is for the correct circle. Both diameter and circumference for the first circle traced and measured is recorded on the first circumference for the first circle traced and measured is recorded on the first circle line.circle line.

Graphing DataGraphing Data► Once diameter and circumference from all circles is gathered, draw a graph Once diameter and circumference from all circles is gathered, draw a graph

similar to the one below.similar to the one below.

► Now it is time to plot points on this graph.Now it is time to plot points on this graph.

Graphing Data (con’t)Graphing Data (con’t)► Diameter measurements go on the X-AXIS and circumference Diameter measurements go on the X-AXIS and circumference

measurements go on the Y-AXIS. measurements go on the Y-AXIS. It is important to plotting these It is important to plotting these points correctly or your end result will come out wrong.points correctly or your end result will come out wrong.

► You should have a graph similar to the one below. Notice that your You should have a graph similar to the one below. Notice that your graph graph will notwill not look exactly like the on below. look exactly like the on below.

► When connecting the points on your graph it should make a straight When connecting the points on your graph it should make a straight line.line.

Testing the DataTesting the Data

► To test the data recorded we are going to use a simple formula for To test the data recorded we are going to use a simple formula for slope.slope.

► Plug in any two points that have been graphed into this formula.Plug in any two points that have been graphed into this formula.

Testing the DataTesting the Data

► Example: Point 1 ->(5.8 , 18.2) and Point 2 -> (2.16 , 6.8) are my Example: Point 1 ->(5.8 , 18.2) and Point 2 -> (2.16 , 6.8) are my points.points.

► I plug the points into the formulaI plug the points into the formula 6.8 – 18.2 -11.46.8 – 18.2 -11.4

_________________ = _____ = 3.13186 ~ 3.14_________________ = _____ = 3.13186 ~ 3.14

2.16 – 5.8 2.16 – 5.8 -3.64-3.64Plug the “y” values in first and subtract.

Plug the “x” values in next and subtract.

You should come out with a fraction like the one above.

Divided the numerator from the denominator.

This number will equal the slope and it should be close to around 3.14 (pi)

Testing the DataTesting the Data

► Any point that has been plotted on the graph can be used to find Any point that has been plotted on the graph can be used to find slope. It does not matter which two are chosen.slope. It does not matter which two are chosen.

► If the slope does not come out to be close to 3.14 (pi) look back at If the slope does not come out to be close to 3.14 (pi) look back at the formula make sure the y’s are the numerator and the x’s are the the formula make sure the y’s are the numerator and the x’s are the denominators.denominators.

SummarySummary

►What have we learned?What have we learned?► Pi was developed over 4000 years ago by ancient Egyptians and Pi was developed over 4000 years ago by ancient Egyptians and

Babylonians.Babylonians.► We experienced the measurement of pi the way the ancient We experienced the measurement of pi the way the ancient

Egyptians and Babylonians did.Egyptians and Babylonians did.► We learned how to collect data by measuring and recording the We learned how to collect data by measuring and recording the

circumference and diameter of different circles.circumference and diameter of different circles.► We learned ways to calculate pi.We learned ways to calculate pi.► We learned how to graph the data we collected.We learned how to graph the data we collected.