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.
Factored Form
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.
Always/Sometimes/Never
10 statements about polynomials. Students label each as always true, sometimes true, or never true, and justify their answer. Link here.
Card Sort
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.
Another interpretation:
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.
Five Twelve Thirteen 
My first thoughts are to have a constructional approach. Take the graph of any function y=f(x), where f is described by the plot alone, and let them figure out what happens to the plot if
a) you add 1 to the function value at all points (so you now have y=f(x)+1), and then some other number
b) you multiply the values by a number
c) you multiply by x
d) then by x+2
and so on.
Then say, Ok what if we started with f(x)=x, or 2x
Soon we get a parabola, y=2x^2
Now multiply this by x, or by x-1
Some things start to stick out, like “the graph crosses the x axis at x=1
Go towards cubics and 4th degree in the same way.
If you want a REAL application have a look at cubic spline curves.
I can give you some pointers on this.
Have fun!
One interesting feature that this approach brings out is the x-intercepts of linear functions. This is not a feature typically explored in beginning Algebra classes, but understanding those properties seems to me a central piece of building the type of understanding you are talking about. It’s something to think about, for sure.
I did some googling on cubic spline curves. Really fascinating — although I wonder if they fit better in a calculus course? Given my position teaching material students have, for the most part, seen before, I’m not sure how well they work — they add value in that they are unfamiliar for all students, but at the same time I don’t want to get bogged down forever in polynomials.
The really nice thing about the development of cubic splines, which I figured out a long time ago, is that by looking at the simplest representation, on the interval 0 to 1, where we have two points, (0,3) and (1,5) for example, the straight line joining them has the equation y=3(1-x)+5x. This is a weighted average of the two y values, using x and 1-x as the weights. If we had a third point, at x=2, say (2,8) and the line from the second to the third being y=5(2-x)+8(x-1), check it out! then the equation of the quadratic through all three points is a weighted average of the two line equations, and so on, for two quadratics generating the cubic through 4 points. See if you can work this out !!!
What would concern me deeply is the statement, “Most also don’t remember polynomials very well.” Speaks volumes about their previous mathematics education. Every teacher they had previously should share responsibility for that, because regardless of whether any given teacher was responsible for “covering” polynomials (and that would comprise at least two of them, in the majority of cases), all of them failed to instill an attitude towards and habits of mind in studying mathematics that would make it close impossible for any of their students to go through a major topic that appears repeatedly in mathematics down the road to be in the position you describe. It’s not a matter of drilling, but of getting at fundamental meaning. If students aren’t making sense of mathematics, they aren’t learning mathematics.
Thanks, Michael. I’m pretty reluctant to lay blame anywhere — in my experience, the curriculum is as much to blame as any given teacher. If polynomials are given short shrift in a curriculum, and emphasis is mostly on FOILing and synthetic division, it seems unlikely students will bring anything significant to the table.
You mention making sense of mathematics. I’m still not sure of what the big ideas about polynomials are that I want students to make sense of. I have some ideas, Glenn offered another approach, and Howard a third take above. What are the concrete, transferable understandings students need about polynomials?
1: Two points determine a line, 3 points a quadratic, 4 points a cubic etcetera, so for any number of points there is a unique lowest degree polynomial passing through those points.
2. Every factor is either linear or irreducible quadratic (no real roots)
3. All the simple ones, y=x^2, y=5x^4 … touch the x axis at the origin, and this can be used to determine the slope at any point (call it post-pre-calculus, no explicit limits needed, only gentle hand waving)
Yes, I know that these are all “mathematical” reasons.
I always favor teaching as much about the connections between the functions/equations and their graphs, so I agree with Glenn on that. Looking at how behaviors of odd & even polynomials, respectively, cohere as “families.” Perhaps one of the biggest ideas is that polynomials are continuous everywhere and defined over all real numbers. Most function families I can think of are only defined over part of the real numbers or have other restrictions (some trig functions are continuous and defined over all x, but not all trig functions can make that claim): our friendly polynomials are continuous and defined everywhere. And that has implications for calculus that students might benefit from hearing about in non-technical, non-analytic language (the notion that the slope of ANY polynomial is always both defined and calculable for any real number in the domain is pretty spiffy, and while it may require calculus to compute, it doesn’t require it to have some intuitive understanding. Comparing that with functions from other classes of functions should prove worth a look.
The main thing is for the teacher to have a collection of meaningful mathematical residues s/he wants students to take away from a given unit or lesson, and a game plan for helping students encounter as many of those as s/he sees fit. It’s not necessary to get to everything, but it’s vital that some things are presented so that students will in fact take away some key notions: things that connect to previously learned mathematics, to key future ideas, to some practical & meaningful applications, etc. If any student who has given my classes a half-way fair shake walks away saying, “I don’t really understand polynomials or know much about them,” I’ve failed that student.
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