Slope. (AKA If at first you don't succeed, try, try again.)

I'm teaching a two week, three hour a day, summer enrichment for students who will be in Algebra with me this year. After we did a lot of "fun math" last week (WODB, Estimation 180, Would You Rather?, etc), I asked them what they'd like to work on this week. Slope was a nearly unanimous answer.

We began yesterday with Which is Steepest? They could easily discern which was the steepest but struggled to explain how they knew. Eventually we had these rectangles drawn around the line segments (normally I'd use a slope triangle, but this was what they came up).

Our slopes were 6 and 4 (represented as 6/1 and 8/2, respectively). We tried all sorts of things like making both denominators (although they weren't written as fractions at the time) 2 and comparing the other number. Eventually, one of my girls did this fascinating math in the photo. She said she'd find the LCM of 6 and 8 (24). Then she'd figure out the numerator for the equivalent fractions. Whichever was smaller had to be steeper since it was going up the same amount in a narrower space. WHAT? I was blown away by this incredible thought process. I absolutely would never have thought about doing something like that.

Despite all of this rich discussion, there was almost a gasp of shock when I told them that steepness is basically slope. They had no idea that we'd been talking about slope the entire time. We had clearly made progress in their understanding but had a long way to go. 

Today we returned to Desmos doing both Put the Point on the Line and Investigating Rates of Change. They are still struggling with interpreting steepness in a quantitative fashion. We've talked about it as a relationship between the horizontal and vertical changes at length, but something still doesn't seem to be clicking. When I ask them to reflect on how their understanding has changed, they respond positively. They don't hate slope anymore (which is a huge win in and off itself), but they shared that they don't totally get it yet. 

We've established that:

-slope is independent of length (a longer line is not necessarily steeper)

-it's easier to figure out steepness if you have the line on a grid rather than a blank background

-we need to count the vertical and horizontal distances 

I'm not sure where to go with them next. It's clear that we need to spend another day (maybe two) on this topic, but I can't think of anything we could do that would be different than what we've already done. I have a staircases activity that I've used in class before that I'm thinking about starting with tomorrow, but I'm sort of at a loss for how else to approach slope with them.

I've really enjoyed teaching this summer enrichment because we have the time to delve into rich tasks, but it's also become apparent quickly where students have misconceptions (and we have time to address those). I'll admit that I am somehow shocked that we've done all of these things and they still have questions. I wouldn't even have done this much in a regular classroom structure. I love how Desmos allows students to make clear what they do and don't understand so much more than a practice problem ever could.