This chapter is more uses for the derivatives, and how we can graph without being dependent on the graphing calculator. It seems to be an obsolete topic nowadays, but I think it offers a lot of thinking opportunities for students. They have to organize a lot of information for themselves, what each derivative tells them about the graph, how to apply the intervals and turn them into graphs. Even with the graphing calculators and apps and computer programs, it is definitely necessary. The other part of this unit I find important is the use of the theorems. These theorems really force the students to think about the hypotheses and whether they’re applicable to each problem. I think it is one of the first times since geometry that students really have to consider the hypotheses and what they are telling them. By calculus, students are definitely more mature math students, so it is nice to come back to this. Special shout out to Math=Love for the vocabulary sheets! They have been working out awesomely this semester, usually as an introduction to the chapter. I have to find a way to get my students to do this more independently next year.
This chapter did not turn out as well as it could have…we had state testing in the middle of it, and the distractions and days off from class did not help a lot of my students. It’s nothing I can control, so it is what it is. I changed my organization of the chapter in my notebook a little bit, and tried my best to adjust for all of the interruptions.
I start this chapter with a factoring review since this chapter expects the students to be able to factor any factorable polynomial. I found a great flow chart to include in my notebook on a precalculus blog. I used a few more of his ideas later in the chapter! I love finding resources out there, so hopefully someone else will be able to benefit from my posts as well. I also gave students practice problems to complete after their last test.
The chapter starts with quadratics, a quick review. I’m finding that no matter how much time is spent on these functions, students struggle. It’s a lot to sort out, when to set it equal to zero, when to look for the vertex, etc.
Next, we talked about division and the theorems that utilize the operation. And then some vocabulary terms.
This part of the chapter, students are creating polynomials given some “zeros”. I showed them how to more easily multiply polynomials, or to organize their work a little more using charts. These are honors students, but they appreciated not having to distribute as usual and to try to add up like terms that someone go all over their paper. It’s good that they have as much trouble reading their own work as I do sometimes!
We then go through a few tests that help students to make a challenging polynomial more factorable, and then graph these polynomials. I tried to teach end behaviors using limit notation, to introduce the students to this topic. We will work more with this notation later in the class, it is just meant to be an introduction. During this time, I have students explore the shape of the graphs around the zeros on this WS_Exploring_Polynomial_Functions_with_repeated_linear_fac.
At the end of the chapter, I gave students a worksheet I borrowed from Sam Shah. It gets them thinking about polynomials in reverse, writing equations from the graph. They have to pay attention to the end behaviors, zeros, even y-intercepts. equations from graphs ws
I made a few changes to my first honors precalculus chapter this semester. Having gone through a whole course with my INB once, I decided I had a lot more room and didn’t need to cram a lot of information on one page. The unit starts out discussing functions and all that is related, domain, range, notation… I got the vocabulary sheet from Math=Love blog, just retyped it myself to have 2 per page rather than 6.
The inside of this notesheet is modified a bit from last term…I think students made the transition from inequality notation to interval notation a lot easier by starting with that, then discussing domain and range from the graphs. I’m trying to emphasize the importance of color a lot more this term, and students seem to be taking to that better than last semester as well. Some were even using color on the short answer portion of their test at the end of the chapter.
This even/odd functions page is new…Students weren’t sure about taking notes without a notesheet in their INBs, but mine were planned out enough that I could tell them to divide up the page into three sections to be sure they had enough room.
So is the brochure-like page about Analyzing Functions. I wanted it to make more sense to me, hopefully then making sense to the students a little better. The Left Hand Side activities and problems are also causing me to assign less homework problems, because there is time set aside in class to do some practice that most students seem to get done because they know they will get immediate feedback on it.
My last INB did not have a Difference Quotient/Rate of Change page…I changed that here.
DQs are still a work-in-progress. I have to remember to keep assigning one of these here and there. Next, graphing topics and operations on functions.
I think my students appreciated the little graph squares on the piecewise functions pages. It made it easier to budget space as well as to keep the graphs neat. I think as far as the composition of functions go, I need to put more emphasis on determining functions that were composed together (breaking down the composition) because that is how they will be used the most in Calculus. This unit, I did a few group activities too. One is a translations assignment: Translations_and_the_Coordinate_a_b_GW1. The others are saved at school, and I will try to remember to post them later!
In a previous post (found here), I talked about using some things from other bloggers relating the derivatives and their graphs. In that post, I forgot to mention the work we did dissecting the first derivative’s graph. At first, my students weren’t so sure, but by the end, it seemed to help them put all the theorems we learned throughout the chapter in order for them, and helped them make more sense, for some 😉
given f’ determine f (PDF of the worksheet above)
Here are a couple of more worksheets my students worked on. Most of these problems come from old calc books, but I thought they were great problems for my students to better show understanding: