## Thursday, April 7, 2011

### Impossible Trapezoid In my science teaching seminar, we just read a paper on differentiated instruction, and I assigned for homework a writing assignment to discuss how differentiated instruction connects to what we have been learning about formative assessment (something we have been learning about all semester).

For class I was hoping to give to the learning assistants (LAs) a few different teaching scenarios. One scenario was going to involve some fake data from a "quiz" where math students had to do two different but related problems. In the scenario, 1/3 of your students bomb one question on the quiz, 1/3 of your students bomb the other question, and 1/3 of your students ace both of them.

I was going to have to the LAs look over the quizzes to decide what skills which students were struggling with, and have the come up with a plan for their class the following day that would further all students along in their understanding (hence differentiating the instruction based on formative assessment).

For one of the quiz problems, I was going to use this problem, taken from Dan Meyer's sample geometry tests. That is until I started to try to solve the problem my self. Can you figure out what's wrong with this problem?

1. Last semester I would assign four problems every class period but instead of collecting them I would randomly select one and have them do that one in 10 minutes at the beginning of class without any notes. I started getting students who would simply memorize all four solutions (apparently the solution manual for our text was floating around) so I began to twist the problems around (given this, find these where the "this" and the "these" would be swapped). I found it interesting when I would start to ask students what the minimum number of "this" things were so that you could find all of the "these" things. We'd have a discussion about that for a while and then I'd randomly select students to shout out random numbers for all of the "this" variables. The reason I'm writing all of this here is that I'd always have to remember to ask whether the number combinations were possible before letting them begin. I would tell them that sometimes you can't just pick any numbers for parameters. I found those conversations with the class to be very fun and educational. Some of them agreed on my student evaluations while others said it was a waste of time.

2. There is a graduate student here who speaks highly of this kind of system you speak of.

I think the interesting thing with this problem is that you can get 2 or 3 for different answers, depending on how you approach it. But I think the assessment hand in mind that students would just use A = 1/2 (B1 +B2) * H. It makes it an unfortunate assessment item.

However, I learned a lot about trapezoids in solving the problem 3 different ways, and understanding how they are all related in some deeper way than I never learned before.

3. If you drop another altitude of length 6 on the right hand side, the base of the triangle formed has length 8 (it's a 6-8-10 right triangle). This means the rest of the base of the trapezoid has a length of 5...which needs to be greater than the length of the top. Oops.

This is why I like having students create problems of their own. At first they just throw random numbers at each side, which leads to some interesting geometry when they realize that you can't always do this.

4. Nice concise explanation Avery!

It makes me wonder how many students would just use the formula A = 1/2 (B1+B2)*h, and get "an" answer; And thus with the assessment, maybe get full credit for just plugging and chugging.

Then, on the other hand, I imagine you could have students who work the problem conceptually by adding (or subtracting) the areas of rectangles and triangles, and they wouldn't be able to arrive at an answer (because the problem makes no sense).