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Discovering π (adapted from Mary Laycock’s Hands-on Math for Secondary Teachers) Materials circular or cylindrical objects (plates, jar lids, cans, film canisters, etc.) measuring tape (or meter stick and string) basic function calculator (optional) To Do and Notice Recall that diameter is the distance across a circle and circumference is the distance around the circle. Let’s go hunting for circles: Use centimeters as your units to measure the diameter and circumference of each circle you find. (Remember that the base of a cylinder is also a circle!) Then record your data on the chart below. To complete the chart, use your circumference and diameter measurements from each circle and find the sum, difference, product, and quotient of each set of data. Make sure you include your units. What do you notice about the data? (Hint: check out the last column.) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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| What’s Going On? You probably noticed that one of the columns gives you almost the same value for every circle. Your neighbor’s chart will be the same. This number is usually a little more than 3—very close to the constant ratio pi. The more carefully you make your measurements, the closer this value comes to pi. Pi is the ratio of any circle’s circumference to its diameter. Pi is an irrational number, which means that it cannot be written as a ratio of two integers and that its decimal expansion goes on forever and is non-repeating. If we stop the decimal expansion of pi at a certain place, we get only an approximation of the number pi; the more decimal places we keep, the better the approximation we get. Pi = 3.141592653589793..., and a very common approximation is pi 3.14. Notice that the linear centimeter units remain as linear measurements when you add or subtract the circumference and diameter. The product of two linear measurements gives you square units. The quotient has no units, as centimeters/centimeters “cancel out.” Pi is unitless. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||