Next Lesson: Polynomial Inequalities and End Behavior

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.

How do you have students take and use notes?

This is a huge problem at our school! Our students have never learned how to effectively take notes in class, and then how to actually use them for completing homework and studying.

When I was in high school (and even college), all I did was write down what was on the board and important things said by the teacher, and it was clear to me how to use these notes to study from. I don't know how or when I learned to do this - and, since I don't remember learning how to make sense of this, I don't have a good idea of how to teach it.

The "just write it down and study it" method does not work at all for most of my students. I can get them to write things down pretty well, but I have had no success with getting them to actually keep and use these notes effectively. My students end up having binders stuffed with notes and handouts, but whenever they need to find something, they start at the beginning and flip randomly through it until they either find it (rare) or give up (frequent).

This coming year, a colleague of mine and I have decided to have students organize their binders conceptually instead of chronologically. He will do this for Geometry and I will do this for Algebra 2. Our idea is that, at the beginning of the year, we will come up with a conceptual structure for the entire year, and students will be required to keep their binders structured that way. For example, Geometry may have the following categories: Points, Lines, Planes, and Angles; Triangles; Quadrilaterals; General Polygons; Circles and Spheres; Logic and Proof; Synthesis. These will be the sections in their binders. Additionally, there will be a summary sheet at the beginning divided into these same categories. At the end of each lesson, students will be asked to file their notes in the appropriate section (this will be scaffolded away as the year progresses), and to make an entry on their summary sheets. Homework and other handouts will also be filed in the appropriate section, next to any relevant notes.

We think that this will help students access their notes much more effectively - which will encourage them to actually use them for studying! For example, if they see a problem with a diagram of a right triangle and a missing side length, they may not know what information they need, or when that information was taught (do we even remember when we taught what?) but they will know to look in the "triangle" section. Then, if their notes are complete, it should be a lot easier to find something useful.

Any thoughts on this? Has anyone tried this? How do you get students to effectively keep and use their notes?