About Math is a Piece of This module is dedicated to and inspired by the Math Group, a group of elementary children working in enrichment topics in mathematics in the Ames School District. The Math Group persistently requested a lesson on during the fall semester and these activities were put together partly in response to that interest. There are 5 activities associated with this module. All require a calculator. 1. Measuring by circumference/diameter--measure and calculate. 2. Measuring by area/ radius2 (green circle on grid) Measure and calculate. 3. Probably : a probability result -- each visitor can contribute to this experiment. 4. Measuring area (by measuring water depth) of an 8x8 square pan and a 9" round pan to see that the area is the same. Calculating the ancient Egyptian value for . 5. Making a polygon on a circular geoboard to see that it looks pretty close to a circle. Calculate a decimal value for Archimedes' fractional approximation for . The math behind the activities: 1. C = D = 2R. If the kids measure more than one they should discover that is a constant: the ratio of circumference to diameter is independent of the size of the circle. 2. Area of circle = R2 3. The probability derivation requires calculus. The result is fun, though. 4. Area of square = 64. Area of circle = x (4.5)2. The calculation is solving for , step by step. Should produce a value for of 3.16. 5. Actually calculating the perimeter of a polygon with more than 6 sides involves math beyond the elementary level. Some students might be able to figure out the perimeter of a hexagon. The point here is to see that a regular polygon with many sides looks a lot like a circle. Equipment needed for Math is a Piece of Photo of equipment. 1. Some round objects to measure (playdough pies). Another possibility is to use plastic lids of various sizes as your round objects. You could put pictures of pies inside them. 2. String and a couple of rulers for measuring. 6. At least one calculator. 2. 2 7/8" long toothpicks. 4. Data sheet and grid of lines for Buffon's needle (in Printable Material). I recommend printing two copies of the grid lines and taping them together to make a larger area for the dropped toothpicks. 8. Scratch paper. 7. A square 8x8 cake pan and a round 9" cake pan. 9. Two pyrex measuring cups and a funnel. 10. Circular geoboard and rubber bands. (Some geoboards are two-sided: one side is the ordinary square grid and the other side has pegs evenly spaced around a circle. The two-sided geoboard is what is needed here). Notes on the activities. 1. Remember that this is a measurement and is subject to measurement error. When I tried this out the value for tended to run low. Expect values from 3.05 to 3.25 for very careful measurements. You might want to try this yourself before kids arrive. 2. I count 78 squares inside the circle. The radius is 5 units long. This gives a value of =3.12. Again, there's measurement error in estimating the number of squares shaded. The precision could be improved by making a finer grid but then counting squares would be more tedious. 3. Good results for require some 200 drops. It would be a good idea to do a few runs of this yourself to get it started. Fill in four or so rows of the data sheet before you have customers or in between visits. A good probability question for your customers might be "If you drop a few more toothpicks, is your value for p guaranteed to be closer to the actual v alue for p?" Probability says that a larger data set will get you a more accurate value for p, but doesn't guarantee that four or five more drops will produce a more accurate value. 4. You will need to take care of pouring the water from the pans back into the pyrex measuring cups. That's what the funnel is for. 5. You'll probably have to help kids understand what's meant by a regular polygon in the circle. Here's a picture which shows an eight-sided rubber band polygon on a circular geoboard: Reference material on Children with any inclination to like math seem to be fascinated by . The Joy of by David Blatner contains a lot of background on the history of and may appeal to middle school and high school readers, and perhaps some elementary school readers. It is presented in an appealing format. The Joy of Cooking by Irma Rombauer contains an oblique reference to ancient Egyptian : the section on Cakes, Cupcakes, Torten, and Filled Cakes has a table of comparative pan sizes under the heading About Cake Pans. Substitution of an 8x8 cake pan for a 9" round cake pan is recommended if you don't have the square pan. Janice Van Cleave's Geometry for Every Kid has a activity which is not in this list: finding the area of a circle by cutting into sectors and reassembling them to look almost like a rectangle. It makes the transition from the circumference definition of to finding out that the area of a circle is R2. There are many websites. Here are a couple of good ones: http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Buffon.html http://mathforum.org/dr.math/tocs/pi.middle.html http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html#126 http://www.mste.uiuc.edu/reese/buffon/buffon.html Written 12/17/01 by Janet A. Dixon Revised 1/10/06