Last year I made a lesson about determining the steepest stairs, using pictures my co-teacher and I took and based on an idea from Dan Meyer. It took about a period, and was mostly teacher-led. But after arguments and deep thinking about slope, I wanted to go into the lesson deeper, so I turned it into a lab.
I started the same way, throwing up the (new and improved) opening slide and asking which they thought was the steepest and which was the shallowest.
I really like this new improved one because I took a picture of the toy staircase from the board game 13 Dead End Drive (middle left). Last time, there were overall agreement on the shallowest (the Holiday Market) while there was disagreement on steepest. This time, because the toy was tiny (if not shallow), we had some disagreement there, which really let us tease out some definitions of “steeper” and “shallower.”
Once we had definitions of steeper (which usually came out to something like “closer to vertical” or “at a bigger angle”), I handed out the pictures on a sheet of paper and asked them to develop a method for determine which was steeper, or the steepest. I mentioned coming up with some sort of “steepness grade” (because I thought it would be amusing to throw the word “grade” in there).
So I let them struggle, and come up with what information they had to ask me for, which I would then provide. If I had to do it again, I would also have pictures of the width of each stair, as a distracter, because some kids asked for it. Interestingly, some also asked for the angle, because of our prior experience in the year with the clinometers. I told them I didn’t have the clinometer with me at the time. One kid called me on it, because she knew I had a clinometer app on my iPad. So I told her (truthfully) some of the pictures were taken last year, before I had it.
So I had them come up with their own measures. If they tried to base it off of only height or only depth, I deflected with examples of really tall, really shallow stairs, or really short, really steep stairs.
By the end of the classes, students usually came up with one of three different measures: slope, the inverse of slope (depth over height), and grade (that is, slope as a percentage).
So they had to then reason as to why they might prefer height/depth to depth/over. (Their logic: it seems more natural to have bigger numbers be steeper stairs, rather than the other way around.) And so it was that point that I told them this “steepness” grade that they developed was often called “slope” by mathematicians.
At which point, I got a big “Ohhhhhhhhhhh.” Which always makes it worthwhile.