Students took to the number line models pretty well in the last lesson. I think it can be really eye-opening to see how lines, parabolas, cubics, and so on are all related in a very simple and elegant way. Some of the students started recognizing the alternating positive/negative pattern that occurs when all the x-intercepts are real and there are no repeated roots. In this next lesson, we will look at polynomial functions (still in factored form) that have repeated roots, such as f(x)=(x-2)^2(x+1).

The students will then take notes on what end-behavior means, and what it looks like for a polynomial function. We will use quasi-limit notation like: "as x→+∞, f(x)→-∞". There will also be direct instruction on how to solve factored-form polynomial inequalities. But I think this will go relatively quickly, as they have already practiced solving quadratic and absolute value inequalities graphically (and writing the solutions in interval notation), and because they are understanding the number line models so well. Scaffolding = Success. (This equation reminds me of the team name of the two smart but disruptive boys I put together in one of my other classes - "Sexy + Math = Johnny + Ben" [names changed to protect the dorky].)

After the lecture portion, students will work individually or in groups on the practice work. The problems ask them to make number line models of polynomials to sketch a reasonable graph, solve inequalities, and describe end-behavior. The last page is a mini-exploration on how to find the end-behavior of a polynomial in standard form. I'm not sure if there will be time to adequately understand that, but we'll give it a go. It's one of those concepts that seems really easy (i.e. just plug in a big positive or negative number into the leading term, and see if your result will be positive or negative), but last year, most students took a long time figuring out how to do those problems.

The homework will be more similar problems to review for a quiz on Friday. I have also recommitted to assessing students on their note-taking and organization, as they tend not to do things unless they are assessed (long-term goals are still not immediately accessible to many sophomores). The once-per-unit binder checks I did last semester were not very effective: organized students didn't need me to check on them, and they just got free points; disorganized students would try to get it all together the day before the test (when I checked binders), and they never could do it. This semester, I am going to check the notes and table of contents for the week each Friday as they take a weekly quiz. These short-term objectives should help some of the more disorganized students stay on top of things.

A beautiful combinatorics argument

19 hours ago

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