Story Time in Mathland

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.

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  1. If it leads to them wanting to combine functions in operations, this is great! The idea of a story is a neat way to get them to notice features of a graph, and inherently invites switching representations from visual to verbal and back again. Good stuff!

  2. To the story I would add the discussion of how many possible turns could the graph have. Love the rest, especially how the a value stretches the graph but it doesn't change the zeros.

    1. Great point. Yes we did talk about n-1 maximum or minimum extrema, failed to mentiion it. I have an goose bump story to share of one student's work around to avoid factoring at all costs!

  3. I love pushing them to realize that a positive definition of a polynomial is anything that can be broken down into a series of linear multiplications. f(x) = (2x-5)(3x-sqrt(2))(x+5)(x-2i)(x+2i) etc. Once they grasp this, then they realize what the last year and a half of algebra is for, and why linear functions are so vitally important.

    1. Never thought to mention that, only the single variable part. Like. So where do we give them the practice factoring 9x^2-2 as a valid difference of squares. This one throws them for a loop looking for REAL linear factors. Love having students factor over the rationals, reals, AND linear factors, finally they think deeply about what those numbers mean.
      Thanks for reading and giving me yummy brain food.

  4. Hello! This post was recommended for MTBoS 2015: a collection of people's favorite blog posts of the year. We would like to publish an edited volume of the posts and use the money raised toward a scholarship for TMC. Please let us know by responding via email to whether or not you grant us permission to include your post. Thank you, Tina and Lani.


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