I recently tried Kate’s lesson on the discriminant. Like most of Kate’s stuff, it worked great, and so I tried to do something similar with graphing polynomials. Instead of just telling students how to find the end behavior and what the multiplicity tells us about the graph. Instead, I tried to have students figure something out own their own. I did this lesson over 2 days. The first day we looked at the end behavior. I wanted to develop an informal way to talk about the ends of the graph. So I showed my students this slide:

After gaining so street cred, I had students try a few examples on their own.

Next, I had students pair up and find 2 different polynomials, of different degrees, using Desmos that satisfied the different Drake end behaviors.

After about 15 minutes, we compiled our different polynomials on the board and I threw the “What do you notice/wonder?” question at them. Both classes had great discussions. One class got all the important information. The other classes got a lot of the important information. Now, I don’t know if this will make students remember how to find the end behavior of a polynomial better than if I just told them. But just the skill of figuring something out on your is worth spending time prating. And worse come to worse, if students don’t figure anything out, I can still tell them at the end of the lesson and all I would have lost is 15-20 minutes.

We then filled this slide out .

Then did some whiteboard practice.

Day 2 was on to multiplicity. Here was the opener I used.

In pairs, students used Desmos to complete the following handout:

We then TPS over this slide:

Again, some classes figured out more than others. Some students figure out more than others. But at least they figured something out first, which is way better then me just telling them.

Here are the handouts and Keynote/PowerPoint.

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Certainly one of the reasons I love these is

that it makes college students think about what they should know

in order to choose the higher possibility https://math-problem-solver.com/ .

Also papers on variational (in-)equalities, optimization and probability and statistics are welcomed.