Polynomial Application Problems

We have finished our unit on Polynomials in Algebra II, and two of the last activities we do in class are the set of Application Problems and set of Rational Root Theorem graphing problems.

The five application problems that I use are these five colorful slips. They’re not really thin-slicing problems, since they don’t get progressively more difficult or introduce new skills as they build. BUT they’re five great modeling problems that require students to really think and extend what they’ve learned to create their own polynomial equations.

Most students only did the following three problems. They worked in random pairs at the vertical boards and had one and a half periods to complete them. Most got through them all, but some got hung up on small algebra errors, like combining like terms incorrectly or applying exponents incorrectly.

Ideally, the orange slip is replaced with a whole box-building activity that spans two days where students work in pairs to build boxes (each pair creates a different sized box) to help them visualize the box with the largest volume. I love that activity but we didn’t have time for it this year.

My favorite slip that they did was the blue one, copied below. I made this the first slip so that I could guarantee every pair would fully finish it in the time given.

I love this question because it requires students to slow down and carefully consider units. They like to rush through them as quickly as they can, and it’s not possible to rush through this one since they have to carefully label their axes and think about zeros as they interpret slope properly. Part (c) is also great because it gets them to think about profit and how to visualize it. Some end up sketching a graph of the profit function and find zeros. Some look at the graphs of revenue and cost and look at the intersection points. Others don’t use a graph but consider test points in between and outside of the zeros to check the sign.

Whenever I noticed one person taking charge of part (c) and doing too much work, I made sure to ask the other student to explain to me their thinking before checking them off on my record sheet and having them move on to the next slip. Typically they cannot explain fully so I ask them to discuss more while I do another loop of the room.

Before the test, they go through the following Rational Root Theorem slips in different random pairs to help them practice the entire polynomial graphing process. Students fully factor and solve each and then graph them with an additional point using the Remainder Theorem.

These are more similar to thin-slicing problems, since the first one is easier and they gradually add on multiplicity, irrational zeros, imaginary zeros, and more difficult p/q testing. I changed the order so that the purplish one goes fourth and the aqua one goes last. Not everyone got to the aqua problem in the two class periods they had. This slip had two fractional zeros, so the p/q testing takes the longest.

Now it’s on to all things related to Exponents!