## What is this?

## What did the course look like?

- What do the dots represent?
- How can we quantify this?
- Does speed matter?
- Did the bike have gears?
- What do we mean when we say, "what does the course look like?

So, the next day they came back and and I had them map the course for these two:

It's fun to watch the students construct an understanding of antiderivatives. The three-day span reminded me of a few things:

- the importance of listening to how students are thinking about and making sense of problems

- our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
- allowing them the time to sort things out together

I feel like there are a couple ways to go with this trajectory next:

- Stay with resistance, but use a continuous curve
- Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
- Stay with discrete and resistance, but use a more complicated readout

I'm leaning towards #1. What do you think would be best for the students?