My husband teaches Multi-Media studies at the Santa Rosa Junior College. He started as a K-8 music teacher, including band, chorus, classroom music, and drama. He did musicals with all the grades.
What does this have to do with Math, and in particular, graphing? Well my husband can tie his careers together too, because in the end, he is really a storyteller. This is his mantra. Does your photo tell a story? Does your website tell a story? Does your logo, animation, digital portfolio, tell a story? If not, go back and make it tell one.
So when I was about to teach Polynomial Functions, it dawned on me, equations are little stories. They tell us about their shape, their domain, their range, their limits, their relative extrema, their x-intercepts, their y-intercepts, etc... They also tell us what family they belong to and how they are different from their parents. Each is unique and this became really clear when the students got to building a polynomial function.
For this lesson I wanted the students to tell the story of the equation through its graph.
The students had to understand:
→What role the degree played in the graph’s end behavior.
→What role the multiplicity of the zeros played.
→What was the effect of the leading coefficient have on the graph.
First we used Amy’s Polynomial card sort (thanks Amy) to make sense of these equations. I loved Amy’s script and pretty much followed it.
After we got used to the shape of the graphs, we explored methods of finding the roots. We used all the typical methods: factoring, rational root test, quadratic formula, long division and synthetic division. Along the way, we found out that some of the roots were imaginary, that they came in these conjugate pairs (we giggled, bc someone said, “oh like “Orange is the New Black.” Now I can’t think of these pairs of numbers without immediately translating it into “conjugal.”)
It was fun to watch them discuss and grapple with zeros vs. intercepts, (more on that later) and found out that their factoring skills stink. (How do we involve CCSS with the mundane task of factoring, I want my Pre-Calc kids to be able to have TOOLS to factor nearly anything.) Then they got this gift:
Find the equation of a third degree polynomial the following roots such that f(1) = -60.
2, 3 + 4i
Most students got this far: f(x)=x^3-8x^2+37x-50 but couldn’t figure out what to do with the
f(1) = -60.
So we talked about what story does f(x)=x^2-x-6 have? How many different graphs have zeros at 3 and -2? What does f(1) = when a is 2, 3, -5, 6, -1, 1, etc…
The students understood from this exploration that there are infinitely many equations with roots 2 and 3 + 4i. Next time I will use the slider function in Desmos to help the students find “a” before we do it by hand. (Why I didn’t think of it this year is beyond me.)
I wish I had done this first: (I am a tad intimidated by sliders)
In the end, the students did get that the leading coefficient makes f(x) have a unique story.
It was then way more fun to move on to rational expressions. The students were now curious to see what an analysis of f(x)=N(x)D(x) would produce.