MED 607/I2T2 Project—5 day lesson plan on geometric probability ...

Day 1—use GSP to make a large target, embed the areas and hide them.Learner outcomes- the students will use technology to develop materials needed for alater <strong>lesson</strong>. They will demonstrate an understanding of symmetry and relative area inpreparation for a relative frequency definition of probability.MST Standard #5—Students will use computers as tools for design, and for generatingideas.Materials: Access to Geometer’s SketchpadAnticipatory set—The students have an entrance pass, two questions handed to them asthey enter the classroom. These are worth points and are collected in journals.1. Assume snowfall is spread evenly over Buffalo. What does this mean aboutthe probability of a snowflake falling inside a 100’ x 100’ square as compared to a 10’ x10’ square?2. Assume you are competing in the high dive at the Olympics. Does it changeyour chances of hitting the water below if the pool is round or square or triangular? Whydo you care?Instruction: Direct teaching format as we explore the answers generated as part of theanticipatory set.1. If snow is falling evenly, a smaller ‘target’ collects a smaller quantity of snow.However, it does so at a smaller rate than the larger target, so that it takes the sameamount of time to cover both areas. Does your experience confirm this? Does it take thesame time to cover the roof of a car as to cover the roof of a house? Hopefully, yes.Look at a single snowflake. Isn’t it reasonable to assume that it is more likely to strike alarge target as opposed to a smaller? If I’m throwing darts, isn’t it more likely to hit theboard itself than to score in the center ring?2. Since we agree that the probability of striking targets increases with target size,and that we really care about missing the pool below, I claim that if I’m a diver, I want apool that is a circle centered at the point I aim for in my dive. If I’m missing a little tomy left, I still have pool. If I’m overshooting a little, I still have pool.On the other hand, if I’m a snowflake falling from a great height, I’m not really ‘aiming’for a spot. I could fall anywhere, so a square is a good target.

2. Same game, but the opening to the container is twice as large.Instruction: Direct teaching format as we explore the answers generated as part of theanticipatory set.1. Answers will vary, but the point here is that the probability of winning isproportional to the size of the opening of the container.2. Since all that has changed is the size of the opening, they should be willing toaccept a prize that is half as large, so that the total expected winnings are the same.3. Play the game attached with the targets generated on GSP. Feel free to modify therules as needed. For example, I specified three roles for team members, but I wasconcerned about keeping all three members of my 6 th grade team engaged. The purposeof stepping back until misses begin to occur is to introduce errors, and errors that areprobabilistic. The ratio of the number of hits in two regions should be proportional to theareas of the regions. Score doesn’t matter; just the number of times a region is struck.Closure and assessment: For each target, have printout copies with areas displayed. Handeach group such a copy of their target. Compare the ratio of hits in a pair of regions tothe ratio of their areas on one target. Assign the following as Homework: Find at leastthree ratios of hits and corresponding areas. Compare these numbers and answer thequestion—How good is this technique in estimating areas? How would you improve it?----------------------------------------------------------------------------------------------------------Day 3—find pi by simulation on TI-83, circle inscribed in square.Learner Outcomes: the student will use technology to better assess the value of usingsimulation techniques to evaluate probabilities, and will be able to predict the number oftrials to get within a certain margin of error in such a probability.MST Standard #5—Students will use computers as tools for design, and for generatingideas.New York State Mathematics Standard #6D—Students will use simulation techniques toestimate probabilities.Materials: TI-83 calculator and “circleinsquare.GSP”, attached.

Day 4—Venn diagrams and geometric probability, using websiteLearner outcomes- the students will use access to the Internet to develop ideas leading toa geometric interpretation of probability.MST Standard #5—Students will use computers as tools for design, and for generatingideas.Materials: Access to the Internet, or to Geometer’s Sketchpad.Anticipatory set—The students have an entrance pass, two questions handed to them asthey enter the classroom. These are worth points and are collected in journals.1. My wife and I are outrunning errands. We try to meet up at Starbuck’s forcoffee. If we say we’ll be there between 2:00 and 3:00, and we’ll wait for 30 minutes,and leave if the other person doesn’t show, what are the chances we get to have coffeetogether?2. Assume 2 numbers A & B, with 0

I went into GSP to make Venn GSP. It took me ten minutes to make somethingcomparable to the website, in that now I can change the area of the intersection bydragging one of the circles around, and I can change the areas of the circles by draggingthe endpoint of the radius around. I had to define intersection as arc segments, since arcsectors include other parts of the circle. So, it looks like if I can’t get to the web, I canmake a nice substitute with GSP. I can’t do the part where they click on A c and that partof S gets highlighted, but I can make that an extra credit project for somebody. I havesome sketches, Venn2.gsp & Venn3.gsp.I wanted to give the students a chance to do some free exploration on this website, so Iwould just ask some questions about how to define the conditional probability of A givenB, and what happens when the sizes associated with these events change.Day 5—Bayes theorem. Conditional probability will be part of their prior knowledgenow, it’s covered extensively elsewhere, and I want to use this as an example ofinvertibility, an extension I’ve never seen anywhere, and I think it’s very interesting.Inversion is a very powerful concept, and I’ve never seen it done in probability. We do itin algebra all the time, in calculus and geometry, why not probability?Learner Outcomes: The student will be able to define and give an example of Bayes’Theorem.Anticipatory set—The students have an entrance pass, two questions handed to them asthey enter the classroom. These are worth points and are collected in journals.1. If I know P(A), P(B), P(A & B), how do I find P(A given B)?2. Given the above, how do I find P(B given A) in terms of P(A given B)?Instruction: Direct teaching format as we explore the answers generated as part of theanticipatory set.1. P(A given B) = P(A & B)/ P(B). This is a standard definition. In words, weknow both A and B have to happen, and we’re interested in the relative size of (A & B) toB; B has become our new sample space, and we’re defining a probability as a relativearea.2. If P(A given B) = P(A & B)/ P(B), we know P(A & B) = P(A given B) *P(B). We can employ a symmetry argument to derive P(A & B) = P(B given A) * P(A).This gives us P(A given B) * P(B) = P(A & B) = P(B given A) * P(A), or P(B given A)= P(A given B) * P(B)/P(A).