## Trying to find math inside everything else

### Steepest Stairs and Wacky Measurements

After reading Dan Meyer’s post mentioning a Steepest/Shallowest stairs contest, I decided to go for it. But Dan had them do it for homework after they knew what slope was. I decided that I thought steepness of stairs would be a great way to introduce the concept, and then we can have the contest after. So Ms. Barnett (my co-teacher) and I went around the area and took lots of pictures of stairs we can find. Then I put them up as a warm-up and asked them which were steepest and which were shallowest.

In every class, there was near-universal agreement on which stairs were the shallowest (the top-right), but lots of different votes for the steepest. So then I asked them, “How can you know? What does it mean to be steep?” I got a lot of good, intuitive answers from that (My favorite was that something is steeper when it is closer to being vertical). I asked them what they needed to know to find out which was steeper, and they said we should measure it.

But what exactly should we measure? That took a little cajoling and probing, until we eventually decided on the height of the step and how deep it was. So I gave it to them:

Alright, now we have these measurements, what can we do with them? I lead them on a discussion on how best to use these numbers (a ratio), and we looked at another example. This is a pretty clear example (1/2), but not all of them are. So we used our estimating skills.

And my personal favorite…

(They really asked if I had 11 cell phones. I guess my Photoshop skills are better than I thought.)

The best part of these pictures is that they so naturally prompted them to question the units of measurement. “That one is cell phones, but the other one is hands. How can we compare them?” And so it’s natural to talk about slope as a ratio with no units. I didn’t have to artificially insert it. I even had a picture of a curved slide at the end, so we could theorize about the steepness of that.

Finally at the end I mentioned the contest. Unfortunately, I’m afraid Ms. Barnett and I did too well finding stairs. I’ve had students say they’ve been looking, and some say they found some (but don’t have pictures yet, though they have one more week). I hope someone can knock us off our thrones:

## SLIDES:

Steep Stairs (PDF)

Steep Stairs (PPT)

### No Better Feeling

At the beginning of yesterday’s lesson, I threw up this monster of a problem:

I told my students that, by the end of the lesson, they would be able to solve it. They flipped and freaked out. “No way, Mr. Cleveland, not going to happen.”

In all 4 sections, 1 hour later, every student correctly solved the problem. And they were all so proud of themselves for doing so. There’s no better feeling than that.