Day 7 – “Slope Functions”

After figuring out yesterday that curvy things in fact DO have slopes that are just different at every point, the question today was “Well, how do we calculate those slopes then?” I introduced students to the hard way in order to motivate the need for a pattern – draw a tangent line to a point, estimate the slope and write it down. Then we made a table as a class and tried to see a pattern. I chose x^2 as the function so that the pattern would be obvious. Even though some slope estimates were way off, both classes came to the conclusion that for x^2 the slope was always twice the x coordinate – so the “slope function” is 2x!
I liked this lesson because to the students who were really following along, it felt like they were discovering the math themselves. I did have trouble keeping everyone with me though. Every year I have done this, I get lots of students who can’t draw tangent lines, have huge difficulties estimating the slope, and generally get swamped in little little details and miss the big ideas. I’m not sure how I can rearrange things in order to help them out with that. I guess they just need more time for these big ideas to sink in too.