GEOMETRIC PROBABILITIES: FROM FRACTIONS TO DEFINITE INTEGRALS USING TECHNOLOGY Sergei Abramovich State University of New York, Potsdam, USA

National Council of Teachers of Mathematics (PSSM: 2000) Probability is a curriculum strand that spans across grades 1 – 12. Calculation of probability is a quantification of the likelihood of an event. Measurement includes assigning a number to a characteristic of a situation. Geometric representations can help students make sense of area and fractions.

Combining Probability, Geometry, and Measurement while using Technology Geometric approach to the measurement of likelihood involved construction of regions in the sample space and finding measures of those regions. The approach lends itself to the use of technology: “…it influences the mathematics that is taught” (Technology Principle, p.24)

Instructional context Elementary teacher candidates (Content and Methods courses) Secondary mathematics teacher candidates (The capstone course)

STEM education “Engineering might not be a discipline that people expect to find in elementary or even secondary school but it is a powerful way of teaching, learning, and extending educational methods.” Rogers & Portsmore, Bringing Engineering to Elementary School, Journal of STEM Education, 2004, 5(3/4), p. 25.

Technology used --Spreadsheets --The Graphing Calculator 3.5 (Pacific Tech), alternatively NuCalc --The Geometer’s Sketchpad (Key Curriculum Press) --Maple

History Buffon – a French naturalist, mathematician, and cosmologist of the 18 th century. Worked on the development of fair games by introducing calculus into probability. The Needle Problem (the first geometric probability problem): determining the probability of a needle landing on one of a set of parallel lines when dropped at random on a plane.

Formal definition of geometric probability Consider a trial consisting in a random choice of a point over the region D. The outcomes of this trial are uniformly distributed over D. Let E be an event that a point chosen randomly from D belongs to its part d. Then P(E)=measure(d)/measure(D)

The didactic of geometrization “To find a lucid geometric representation for your nongeometrical problem could be an important step toward the solution.” G. Polya, How to Solve It, 1945, p. 108

Early elementary curriculum “There are three chances out of four” to pick up an even number from the set {1, 2, 4, 6}. How can one measure the likelihood (or chances) for such an event to realize? Recourse to geometry and Application of fractions What fraction of the square is filled with even numbers? (A dart game).

Spreadsheet as a geometric medium (elementary classroom) A machine changed a 25-cent coin (a quarter) into dimes, nickels, and pennies. Assuming that there is equally likely to get any combination of the coins, find the probability that there are no pennies in the change. What fraction of the (staircase-like) chart is shaded? P(E)=3/12=1/4

Comparing probabilities through geometrization Billy wants to eat red M&Ms only. There are two bags of M&Ms available, plain and peanut. If there are 3 red plain out of 5 total and 4 red peanut out of 7 total, for which bag does Billy have higher probability to get a red M&M? X’s --- 3/5; O’s ---4/7 21 X’s; 20 O’s. Higher probability for the bag with plain candies. This is not the end of the story!

Fractions may be challenging not only for children! Billy and Mary want to eat red M&Ms only. If there are 3 red plain out of 5 total and 4 red peanut out of 7 total, how many red M&Ms of each kind does Billy have to eat in order for Mary to have higher probability to get a red peanut M&M? How can we replicate such a situation?

Probabilistic approach to exploring systems depending on parameters “…systems may have unexpected effects that can not be predicted from the behavior of individual subsystems.” Committee on K-12 Engineering Education (2009).

Behavior of linear functions with parameters A point M (a, b) is chosen at random over the rectangle with the vertices (-2, 0), (-2, 5), (4, 5), (4, 0). What is the probability that the function f(x) = ax + b at the points x = 2, x = -1, and x = -0.5 assumes values such that f(2) > -2, f(-1) > 1, f(-0.5) < 3? Triangular region of parameters responsible for the behavior of f(x)

Multiple ways of finding the area Traditional integration (pencil-and-paper) Integration using math software (e.g., Maple) The use of The Geometer’s Sketchpad Pick’s formula (Area = 0.5B + I – 1: B = 9, I = 4, A = 7.5, P = 7.5/30 = 1/4)

Using a spreadsheet Geometric probability vs. experimental probability

Pick’s formula extended: Constructing points using two-variable inequalities

Construction technique made possible by the Graphing Calculator 3.5 The point (a 0, b 0 ) can be defined as the set where ε is the “thickness” of the point. In the plane of parameters (a, b) this set can be defined through the inequality

Geometric probabilities defined by curvilinear regions in the plane of parameters Parameters a and b are chosen at random over a large square with side 2R centered at the origin. What is the probability that the systems of simultaneous equations and both have two real solutions?

The case R < 4 (what fraction of the square is dark?) P = R/24 + 1/4  5/12 as R  4 Technology used: The Graphing Calculator 3.5, Maple

The case R > 4 (what fraction of the square is dark?) P = f(R) 5/12 < f(R) < 1

The case R = 4 (what fraction of the square is dark?) P = 5/12

Experimental Probability (R = 2) = Geometric Probability (R = 2) = 0.333….

Experimental Probability (R = 30) = Geometric Probability (R = 30) = 0.758…

Conclusion Geometric approach to probability: Gives new meaning to traditional concepts; Provides new context for technology use; Connects theory and experiment; Offers new avenues for problem solving, Enhances the “E” component of STEM education.