Tag Archives: geometry

Puzzling Problems Preferred

Yesterday morning I determined that I would find something from my day to blog about, and hey! I did! :)

My geometry kids are learning about quadrilaterals. The prior day’s lesson was about properties of rectangles, rhombi, and squares. Here’s the warm-up problem I gave them:

In rhombus QRST, the diagonals QS and RT meet at point U. If QS=12 and RT=16, find the perimeter of the rhombus.

I think it’s a strong problem, because they have to recognize all of these things:

  1. that since the rhombus is a parallelogram, the diagonals bisect one another;
  2. that the diagonals are perpendicular;
  3. that the four small triangles are right triangles;
  4. that the Pythagorean theorem will allow them to find the length of one side of the rhombus; and
  5. that all four sides of the rhombus are congruent.

Almost all of my students struggled with the problem. I gave some hints – “What do you know about the diagonals of a rhombus? Okay, now think about triangles.” One student asked me if it was a “trick question;” I told her that it’s not a trick, just a puzzle.

But once they solved the problem, whether they needed hints or not, they felt proud of their work. That was awesome to me. Some of them told me that they really liked the problem. It was a challenge to them, so they had a sense of accomplishment once they had solved it. One student asked if I could give them more problems like that for their homework; the book’s problems are pretty simplistic and generally don’t require them to make connections across multiple concepts. I really need to get better about giving them more challenging problems for practice.

In other news, I am ordering sample texts for almost every mathematics course we offer for grades 6-12. The math department (all 3 of us) will be meeting at the end of March for a day of articulating who we are, and then using those decisions about curriculum. I’m really excited about this. :)

As part of that process, I’m reviewing the calculus materials available from The Worldwide Center of Math, and so far I am loving their model for the modern textbook. Check it out.

“Could we have an oral quiz?”

What would you do (or have you done) if your students asked you that?

I said sure, and tried to figure out a way to make it happen.

This was my 1st period geometry class, but I decided to use the same method for 5th period as well. The unit was on angles, and to be honest, I was feeling very very bored with the whole chapter. I understand the necessity of having students grasp the basic vocabulary and concepts, but it’s so tedious to have to go through it.

So for their quiz, I had the students do an angle scavenger hunt. They worked with a partner, and each pair had a list of terms (same for everyone) and a picture of a quilt block (varied by group).* They had to label points on their pictures and then identify on the list what examples they’d found & labeled for each item. The items on the list were things like “pair of adjacent angles,” “segment that bisects an angle,” etc.

When they’d finished labeling these things with their partners, I called them up individually and asked them questions. “Okay, you wrote that angle ABC and angle CBD are complementary. What do you mean by that? How do you know they’re complementary? You drew them so that they share a side – do they have to share a side in order to be complementary?” And so on. I had my list of Learning Targets beside me and was able to mark their scores as they showed me their examples and explained them to me.

The Good:

  1. I was able to see and correct errors in understanding much more immediately than when students take completely written assessments.
  2. I was able to see some really awesome things about my students’ thinking – one student’s spatial reasoning is really strong, and I know he wouldn’t have written out all the words he said to me about how “if you flip the angle over like this and then slide it over here, it will fit exactly on top of the other one.” Another student tends to rush through written work but was taking his time and thinking carefully so he could communicate using extremely precise language.
  3. I had worried that the working together part might result in a poor measure of individual students’ understandings, but it didn’t – the kids that had different levels of understanding still demonstrated that when they spoke with me one on one, even if they had the exact same things written on their papers.
  4. The “scavenger hunt” aspect made it a little more interesting, at least in my opinion. Don’t know if the kids agree or not – I need to ask them.

The Bad:

  1. This is a classroom management nightmare. Once the kids were done with their own written work, they had to sit around and wait for their turn to talk to me (or wait after they talked to me). Not a productive use of their time, and I need to come up with something else for them to do while they wait if I’m going to use this idea again. It was poor planning.
  2. It took a long time. A traditional assessment would have taken one class period. This took two, and in 1st period where I have kids who take longer to think through things, I still need to talk to a couple more kids. Combined with the previous note especially, I need to work on this. At the same time, though, I don’t want to skip any questions, because I need to make sure they truly have understanding on all the Learning Targets.

I’m sure there were more drawbacks, but those were the really big ones. The kids felt nervous about talking to me individually and getting a grade for it, but hey, I’m defending my thesis on Wednesday and am right there with them on the nervousness thing. Doing this more often would help them feel more confident in their ability to communicate their understanding verbally, I think.

Anyway, just reviewing and reflecting on how this little experiment went. I’m not going to do something like this for every unit, but I may well try something like it again later in the year and/or with other classes.

* The pictures I used were this, this, and this, all of which came from this quilt. Hey, if you have a hobby with mathematical tie-ins, use it, right?