We just finished our course material in our AP Stats class on Friday. We are using the delightful text by Starnes, Yates, and Moore – The Practice of Statistics, 4th Ed. and the last two sections of the course focus on linear regression and transformation of data. This course, for many of my students, has been a relatively algebra-free zone. In this last chapter when we are talking about transforming data, there is NO way around trying to remember some algebra and precalculus ideas. When we looked at some example scatter plots and talked about what shape it looked like to them, there was a bit of a gulf in terms of confidence and comfort in my students. Some of this is fatigue at the end of the year but some of it is an indication of the fact that many students are willing to let some of these facts just kind of disappear. I know that my students have worked through graphing functions of the form y = 1/x , y = 1/x^2, and y = ln x. All of these functions were referenced in class the last two days. On Thursday we spent 40 minutes on one data set that ended up being a very close fit to y = k / x and we had some real transformation work to do to find the missing k value. I was really pleased with the patience and attention of each of my two sections. On Friday, I had these notes prepared and big hopes to make it through the problems on the note sheet. The setting of the problem (tossing M & M’s on to a table and eating only those with the M showing) made it pretty clear that some sort of exponential function was at play. In fact, in the discussion of the data set we touched on the idea that the proportion of remaining candies at each turn should be about 0.5 of the previous value. The number of M & M’s remaining after each round of this set up was 30, then 13, then 10, then 3, then 2, then 1, then 0. I was pleased that in each class a student immediately asked whether we could find the original amount in the bag. They’re thinking like statisticians! However, they are not completely comfortable thinking like precalculus students although they all have been already. Quick conversation led to thinking about a half-life formula but I really wanted to push them in the direction of trying to find a linear model for the data somehow. Playing with the data entered in the TI was slower than I wanted it to be but resulted in some great conversations. We wanted to think about logs since we were thinking about exponents. We debated whether to take the log of the # of candies or to raise e to the # of tosses involved. We tried the exponent first and did not like the looks of the scatter plot much. We tried to take the natural log of the # of candies and got a dire warning about domains. I thought that this might trip people up but in each class I was quickly reminded that 0 is a bad input for the log function. One student even answered about WHY that’s a problem, not just THAT it’s a problem. So, we tossed out the data point with the zero output. We looked at scatter plots of both and decided we liked it better when we took the log. Some kids seemed suspicious of the log idea but were convinced that the natural log was okay after seeing the scatter plot on the TI. Each class asked for a linear regression on this new scatter and they were impressed when the correlation coefficient was -0.99 and the linear regression equation was y = 4. 059 – 0.681x Here is where each class got interesting and why I think this was worth blogging about tonight. I anticipated that someone in each class would tell me to use this to figure out an estimate of the original amount. I was prepared to remind them that the y here is really ln y and we can solve for the ‘real’ y. What happened instead in each class is that someone recognized that the slope was familiar. Now, I’ve been teaching longer than my students have been alive. I recognize and remember certain helpful numbers and I knew that the slope needed to be related to the natural log of 2 since this is a half-life problem. What surprised me was that each class contained a student who knew this. I excitedly congratulated the student in each case for recognizing that and talked my class through why this was so. But as I pounced on this recall with my complimentary response, I noticed that certain students looked dispirited. I made a point of backing up now. I reminded everyone that it was a great thing to be able to recall this kind of number and I tried to impress upon them the power of noticing these connections. But I tried to make sure that they understood that I would never set up a problem where they needed to make this sort of jump. It was interesting to think about this. I want to reward (with enthusiasm) cleverness and the ability to make connections. I want to celebrate this kind of create and thoughtful analysis. However, I do NOT want to create a stressful environment where the majority of my students are wondering whether this is what is expected of them. I tried to patiently point out that I was thrilled and surprised by this recognition. I am happy that I have made it to the point where I am able to feel that stress in the class, where I can see the almost visible sighs of some of my students as they recognize that some of their peers can do things that they don’t del that they are capable of doing. I want to think that I am creating an environment where students feel that they are safe in making guesses publicly and eel safe in not being able to understand where those guesses come from sometimes. When I am the one making this kind of guess it is easy for my students to raise their hand and press me about why I made the connection I made. When one of their peers is the one making a creative connection, I think it is a little more intimidating. I hope that I reassured the vast majority of my students that knowing the fact that ln 2 is approximately 0.69 is a nice thing to know but not a crucial thing to know.