Slope brings a lot of things together. It ties up a burgeoning understanding of expressions and satisfying equations, rate intuitions and on top of all of that it’s also a geometric measure.
This lesson, via Fawn Nguyen, has students build their geometric intuitions of slope by developing a metric for measuring steepness. Her post is also a rich collection of links to related problems:
I began the lesson by showing a bunch of images, like these, and asked, “What do you see?”:
Kids’ responses: going down, going up, all using their legs, exercise, at an angle.
Related is this post by Dan Meyer:
Tonight’s homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.