I’m moving into graphing polynomials in one section of Pre-Calc. Most students have done this before. Most also don’t remember polynomials very well. We just wrapped up some time working with function transformations, composition, inverses, and touched on even and odd functions, as well as plenty of time on quadratics. Here are five tasks that I have for students to work through. I’m not sure how to order them, and what instruction to give before, during, and after the tasks. Curious for ideas, tips, or feedback.
Graph 6 polynomials written in factored form, with linear and square factors, using technology if students find it helpful. Link here.
Characteristics in a Table
Students open a Desmos graph and record characteristics of 8 different polynomial graphs written in standard form. Link here.
10 statements about polynomials. Students label each as always true, sometimes true, or never true, and justify their answer. Link here.
24 cards. Each card is either an equation, a graph, or a few statements about the function. Equations are written in a variety of forms — some standard, some factored. Link here.
Desmos Graph Challenge
In Desmos Activity Builder, students graph a number of functions that meet specific criteria. Link here.
As a side note, I’m unconvinced that polynomials need to come strictly before rationals. It seems like two big ideas at the heart here, which will echo through calculus, are that factored terms in the numerator are zeros, factored terms in the denominator are vertical asymptotes, and factored terms that appear in both the numerator and the denominator are removable discontinuities. I think I may introduce those big ideas to start, and then move deeper and deeper, reasoning through end behavior and other interesting properties, linking polynomial and rational functions together whenever possible.
I don’t know that I have the grasp on this conception of polynomials and rationals to teach it well, but I really like it. That’s what I get teaching topics for the first time.