At the beginning of this lesson, I will begin with a brief summary of what we've learned about cubic equations so far. We seen that cubic equations can often (always?) be written as the product of one linear and one quadratic factor, and that the quadratic factor can sometimes be divided further into two more linear factors.
We've also seen one situation in which a cubic function arose naturally, and we've examined its graph - which was kind of weird!
Today, we're going to make the connection between the equation and graph a little tighter. As I pass out the Gallery of Cubics Functions, I explain what students are going to do. They'll see that they are given a bunch of cubic equations written in expanded form. I'd like them to do three things with each equation:
At this point, it's important to emphasize to the class that they may do these things in whatever order is most convenient.
Initially, I will ask my students to work individually so that they have some time to consider different ways to approach the problems. It's during this time that they'll also quickly cut out all the graphs they were given. I also make paperclips available to keep these little graphs together so they don't get lost.
I expect that some students will begin by trying to match all the graphs to the equations first. This is the "fun" part of the assignment for many of them, and I'll encourage them to do it. Others will begin by factoring the first three functions because we've recently practiced factoring sums and differences of cubes. Whatever approach they choose is just fine at this point, but as they go deeper into the assignment I'll start encouraging them to look for the most efficient approach. (MP 1, 5)
As students work individually or in small groups, I spend my time moving among them listening to their conversations, observing their work, asking questions, and offering suggestions. I am especially interested to see how students are using the equation to understand the graph and also the graph to understand the equation. (MP 7 & 8)
The first "Aha!" moment that I expect is the recognition that the y-intercept (i.e. the final constant in the given equation) can be very helpful in identifying the appropriate graph.
The next discovery should be that the roots indicated on the graph can be used to help factor the equation. Most students won't notice this connection until they've factored a few equations by hand. At that point, they'll not only have some examples to work with but they'll also be strongly motivated to find an easier way to factor. This "discovery" paves the way for the Factor Theorem in coming lessons.
More subtle patterns will also be noticed. For instance, when the graph just touches the x-axis without crossing it, the equation will have two of the same factors (a "double root"). Also, when the initial coefficient is negative, the end behavior of the function changes. Finally, all cubic polynomials must have at least one linear factor.
My answer key is full of notes on these things.
As class ends, I remind students to clean up any scraps of paper around their desks and then assign homework. I don't expect anyone to be completely finished yet, so I'll assure them that we'll have some time tomorrow to finish and to discuss what we've seen.
For homework: Finish correctly matching all graphs to equations and completely finish first two pages (i.e. factor and identify roots). NOTE: There will be time in the next lesson to complete all the problems.