## Trying to find math inside everything else

### Estimation180 and Absolute Value Graphs

So as I was getting ready to teach absolute value graphs a little while ago, I came across this post from Kate Nowak about a lesson she did with it. I liked the idea but…I didn’t like the idea of having to “get my butt into overdrive” to collect data from staff and students about such a thing. I wanted a lesson for the next day, so I didn’t really have time for that.

But then I thought, well, my students have been doing Estimation180 all year long. Maybe there’s a way I can use that? I even tweeted Kate about it, but was left to my own devices. (Though I suppose this is finally the write up I promised.)

I thought about what was different between what we’ve done with Estimation180 and Kate’s task, and then it hit me – Kate’s lesson is all about one guessing event, but we have loads of different ones. At that point we have done ~30 estimations. What if we could do some comparisons?

My premise was this – Mr. Stadel, who runs the Estimation180 site, wants to implement a ranking system where all the estimations are listed as “easy” “medium” “hard” etc. But how can he tell when one is hard or not? He knows all the answers, so he can’t used himself to judge. So I told Mr. Stadel that we have lots of data from our class and we could probably use it to come up with a system.

[Aside – this was the 3rd or 4th day of the new semester, and to complete the task I asked students to use the estimation sheets from the previous semester. They revolted, because they claimed I had told them they could throw those out! Which I vowed I didn’t…though, to be honest, it’s possible I did, since I hadn’t thought of this lesson yet. Luckily enough students had not thrown them out so that it could still work.]

So I reviewed what the estimations we did were and I told each group that they have to pick one estimation that they wanted to evaluate. Then they had to collect data from their classmates (and from the binders of other classes, through me) – the estimate each person made and what their error was. Once they have collected enough data, they have to make an Error vs Estimate graph and see what happens. Then I had them make some analysis on whether this counted as a difficult task or not. I didn’t have them compare graphs at the time, but I totally should have.

I think it worked pretty well and many of the students understood why it should be a V-shaped graph. They were at first surprised about where the vertex was, but then it made sense, especially comparing many different error graphs.

Estimation Difficulty Rating (Word format)

### The Carnival Guesser

Have you ever been to a carnival or amusement park and seen one of those people who will try to guess your weight, height, or age? If they get within a certain range of your weight, they win and keep your money. If they are wrong, you win and get a prize. I’ve occasionally wondered how they determine what their range is. This clip from Steve Martin’s The Jerk makes me wonder, instead, how they determine which prizes they can give away.

That’s the lead-in I give the students. Steve Martin can only give away that small section of prizes because he is a terrible guesser, so he often loses. If you worked for the carnival as a guesser, what can you give away?

I have the students go around and guess the weight and height of any 10 willing participants in the class. (Any student can turn down being guessed, so students had to ask first. Also, I think my students were always worried about insulting someone, because they almost always under guessed. Also, many of your students may not know how much they weigh, so doing this lesson near when they have a fitness test and get weighed in gym is a good idea.) They record their guess, the actual amount, and the difference between the two.

Then, in groups, they try to determine a metric for figuring out who is the best guesser. We talk about how being 10 pounds off for a super thin person, child, or baby is much worse than being 10 pounds off for a very large person. We also talk about how being over or under doesn’t really change how good the guess is. After I push back on their metrics, some students pick up on the proportionality of the guess to the real amount, and lead us into relative error.

I somewhat drop the conceit at that point, mostly because I’m not sure of the best way to finish it off. But I like the start, and it’s a very natural intro to relative error, and the relative size of numbers in general.