I usually only share the good stuff and I think this qualifies.
I have an interesting group 3rd period. It's a geometry/exploratory math class and the kids in there are very curious. About everything. Today they were arguing about the world's tallest mountain and what considerations are used to determine height.
One student went to Wikipedia and pulled up this page and the highlighted graphic got us talking about relative maximum and minimum.
So I sketch a run-of-the-mill cubic function on the board and I ask what the maximum and minimum are.
"It goes on forever in both directions, Mr. Cox."
"Is it increasing or decreasing?"
"It goes up, then down, them up again."
"Ok, so what do these two points have in common?"
*pointing to relative max and min*
We talked about why the points were considered relative in terms of being a maximum or minimum and discussed how right before the relative max, the function is increasing. Right after the max it's decreasing.
They quickly connected their understanding of slope to the increasing or decreasing nature of the function. Then I pointed to the relative maximum and minimum and asked:
"So what's the slope here?"
"And, so what do they have in common?"
"Zero! The slope is zero at both points."
"How do you think we can figure the slope at different points on the graph?"
This time I sketched y = x2 and we talked about how we need two points to determine a slope.
"Would we be more accurate if the points are far apart or close together?"
"How close is close enough?"
It wasn't long before we had it.
4th quarter projects should be interesting.
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