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Please be sure to state your assumptions as you solve each problem. Answers to the 46(2) Hexa Challenge Problems will be featured in the 47(1) issue of Virginia Mathematics Teacher.

Contest Alert!

Virginia Mathematics Teacher is conducting a contest for educators and students who can solve the great est number of problems correctly by 12/01/2020. The winner will receive a prize and will be featured in the next issue of the VMT. Send your solu

tions to vmt@radford.edu with the email subject line: Hexa Challenge

Challenge 6: In Bill ’ s trigonometry class one morning, he finally understood polar coordinates when he correctly plotted the graph of the cardioid r = 2(1 - sin Ɵ ). Later that afternoon in his statistics class, he was fascinated by how his teacher approximated π. She selected N points randomly from the unit box by selecting x i and y i uniformly from the interval [0,1]. She let p N count the number of these points that were also inside the unit circle—that is, x i 2 + y i 2 < 1. She says (and he observes) that 4 p N / N is approximately π due to the ratio of the areas of the quarter circle and the unit box being π /4 . Bill realizes that his cardioid is contained within the circle of radius 4 centered at the origin, and so he decides to estimate its area by selecting r i and Ɵ i uniformly from the intervals [0,4] and [0,2 π] and then letting P N count the number of points in polar coordinates that satisfy r i < 2(1 - sinƟ i ). Using N = 1000, his repeatedly finds that 16 π P N / N ≈ 25 and assumes this is a good approximation of the area of the cardioid. Unfortunately, the actual area is 6 π ≈ 18.85 . Why is Bill so far off?

NASA Resources: Building Leaders for Advancing Science & Technology For Elementary School Teachers & Students

For Teachers

NASA ’ s modern Hidden Figures toolkit. Teachers can bring Katherine Johnson ’ s story to their class room. Suitable for all grades because it is broken down into grade bands; K - 4, 5 – 8, 9 – 12: https://www.nasa.gov/stem - ed - resources/nasa - modern - figures - toolkit.html Street Math Space Shuttle Style This includes activities from second grade through high school: https://www.jpl.nasa.gov/edu/teach/activity/street - math - space - shuttle - style/

Virginia Mathematics Teacher vol. 46, no. 2

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