SWBAT match function rules to graphs of cubic functions and to describe the connections between the graphs and the rules for these functions.

What connections can you make between the graphs of cubic functions and their equations? Students complete a matching activity to help them develop a more abstract understanding of these functions.

40 minutes

This lesson’s warm-up continues spiraling the more abstract problems of the unit so far. The Level A Warm-Up version focuses on two types of problems—creating the data tables using a verbal description and find rules to fit data tables. The General Version of the Warm-up shows a full range of problems, from data tables to graphs. There can be a lot of student choice with this warm-up, as long as students understand the big ideas.

This is the general coaching I give students, both as a whole class and individually:

- Identify your strengths and weaknesses so far. Make sure you have a big picture understanding of what you are expected to know—based on looking over the entire warm-up.

- Choose problems that will help you practice—problems that are not so difficult you have no clue how to start but not problems that are so easy you can do them without thinking.

I find that presenting such ideas to the class can be a good starting point, but nothing really happens until I have individualized conversations with students about how to use each strategy. I try to have these conversations in a relevant context—when I se a student choosing problems that don’t seem quite right, or when I see a student who has done a great job choosing problems—I ask students to talk to me about their process of choosing problems and how they are deciding which notes and resources to use.

30 minutes

This is a simpler task than the previous two days. It can be condensed to 15 or 20 minutes. This seemed to work well because the investigations from the previous two days left a lot more for students to figure out—after the warm-up, students can spend time finishing their work on these two tasks, or working on more challenging problems on the same tasks.

The Matching Cubic Graphs to Equations Task is trying to get students to make the connection between the functions and the graphs—we have really been focused on data tables, and the next representation we will focus on is graphs.

It is really important that computers *not* be available when students first tackle this because it is way too easy for them to just graph the equations given and then look at the graphs on the computer. Computers can be made available to students who have tried the match-ups already, and written some rough justifications, so that they can check their own work. This is sort of a trust situation—and I try to make this clear to students. I say something like, “I am trusting that you try the match-ups before checking your answers on the computer—show evidence of this by writing some notes about how you made your match-ups.”

I post some key words on the board and ask students to use these words in their justifications:

- Roots
- Ractors
- X-intercepts
- Inputs/outputs
- Coefficient

Each part of the investigation focuses on a different aspect of the graph—and the idea is for students to use data tables or key points to start identifying the key information that is relevant to each part. Part 1 focuses on the connection between the factors and the *x*-intercepts of cubic functions, Part 2 focuses on the idea that two functions can have the same roots and factors but not be the same graph. Part 3 focuses on the affect that the coefficient has on the graph.

5 minutes