Author: Huck Stewart
1.Discovery of the number Pi
2.Objective: Students will use their existing knowledge on how to find the perimeter of a regular polygon to approximate the number Pi using the relationship between an increasing n-sided regular polygon as n approaches infinity. This relationship will correspond with the common core standards found on page 77 of http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf where it is stated under “Circles” to understand and apply theorems about circles. Students will be using knowledge of measurements of polygons supported by the common core standards found at the bottom of page 25 of the same link document.
3.Experiment: The students will use the Pre-Made existing Geogebra applet to be able to visualize how an n-sided regular polygon’s perimeter will begin to look like the circumference of a circle as n gets large. They will be asked to record their measurements, using their knowledge of the formulas of areas and perimeters of each n-sided regular polygon, and their algebra skills for a few numbers of their choice. They will follow the task sheet provided.
Reflect and Explain: We will discuss as a class the results from their task sheet and note some of the students’ results on the board and see if we can come up with a good estimation of the number Pi.
Verify and Refine: The students will come up as a class a reason as to why they believe the constant Pi kept appearing in their experiment. They will write this down on their paper. This lesson can be taken further and one could find a paragraph or two for your students to read from “A History of Pi” by Petr Beckmann or finding a great article to sum up some of the historical discoveries of pi: http://www.pcworld.com/article/191389/a_brief_history_of_pi.html (but be cautious about the complete accurateness of such articles).
How to construct this Applet on your own:
Step 1: Select the regular polygon button and construct a regular polygon. Note that the distance you pick your first side AB will ultimately decide the rest of your sides and that the longer AB is the larger your 100 sided polygon will be because those sides (in applet’s case) remain constant.
Step 2: Click on two points to create your first side AB of your regular polygon. Note that you may want to leave the x and y axis and put your side AB on the number line so that measuring the perimeter of your n-sided regular polygon will be easier to calculate. Otherwise you might choose to simply display the Geogebra distance measurement approximation tool to show the length of side AB so that the students can calculate the perimeter and area. I chose the unit value length of 2 because it was more interesting that just 1 but not hard to calculate with.
Step 3: When the regular polygon prompt appears change the number of sides to 3 and hit ok.
Step 4: Select the “Slider” button from the tool bar and click in your work space to make the prompt appear.
Step 5: In the slider bar properties change the minimum value to positive 3 and the maximum value to 100. (You may choose to go larger or smaller but 100 seems sufficient since Archimedes used a 98 sided polygon to approximate Pi). Finally change the increment to 1. Then click apply.
Step 6: Click on the “Move” button then, under your Algebra bar, right click on “poly1” (named this way be default or whatever you named your regular polygon) and select “object properties”. Under the “Basic” tab in the Definition box you will see it reads like: Polygon[A, B, 3] 3 is the value that indicates the number of sides. Delete 3 and change that to “d” which is the name of your slider bar. This now means that the slider bar value indicates the number of sides of your regular polygon. Close the properties box. Right click on the slider bar “d” and click on the “Basic” tab and change the name to “Sides”. Close this window as well and slide the “Sides” slider bar back and forth to make sure your polygon is changing as well.
Step 7: Change the number of sides back to 3 (so you see a triangle again) then select the “Circle Through Three Points” button and click on the points A, then B, then C to form a circle. The triangle is now inscribed in this circle as are all the n-sided regular polygons. Slide the bar back and forth to test out the circle.
Step 8: Select the “Midpoint or Center” button and click anywhere on the circumference of the circle. This will create the center of your ever-changing circle. This center “D” will also be the center of any n-sided regular polygon inscribed in the circle. Now, using the same “midpoint” button click on point A then B to find the midpoint of that side.
Step 9: Create the radius of the circle by selecting the “segment between two points” and click on D then A. Now, using the same “segment” tool make a segment from D to E (click on D then E). This creates the altitude of all regular polygons inscribed in this circle.
Step 10: We are nearly done! Click on the “Distance or Length” button and have Geogebra display the distance of both the altitude and the radius. (with the button selected click A then D for the radius then click on point E then D for the altitude).
