A Gallery of Cubic Functions, Day 2 of 2
Lesson 6 of 13
Objective: SWBAT match graphs to equations for a variety of cubic functions. SWBAT convert a cubic equation from expanded form to factored form when roots are given.
I like to begin class by quickly helping students check their work on the first two pages. First, I'll call on a student to name the roots he or she found for a particular equation. Then, if everyone agrees with this answer (and if it's correct), I'll call on someone else for the roots of the next one. We'll quickly check the roots for the first two pages. Next, I'll ask students to get in pairs (preferably not someone they worked with yesterday) to check their graphs. As they're doing this, I'll take a quick walk around to settle any disputes and see that everyone completed the homework.
Now is also the time to ask whether any student has a strategy they'd like to share with the class. Since the problems on the final page are harder, any advice would be helpful! In particular, I'm looking for someone to explain that the y-intercept helps them identify the graph and that the graph then helps them identify the factors/roots. (MP 7)
Once this is progress check is finished, it's time for everyone to get back to work with the aim of wrapping up the final page.
For the next 20 minutes, students will work individually or in small groups to finish the remaining problems. I will spend that time moving around the room to answer questions and keep an eye on things. I will make a point of asking every student at some point or another how they're doing. I can usually tell just by looking and listening, but it's important to ask. Some students are doing just fine, and I'll move on. Others are struggling and I will stop to work with them individually.
At this time, I'm checking to see that everyone is taking care to factor every equation completely, that they're writing down the roots, and that they're correctly matching the graphs. I can't catch every mistake now, but I'll try to get as many as I can. See an example of some typical student work.
Finally, during the final 15 minutes of class, I will begin a discussion of the results. These conversations always begin somewhat differently, but I like to open with the broad question, "So, what did you learn from this exercise?" or perhaps, "What do you think you were supposed to learn from this?" (MP 8)
These questions are so open ended that they are bound to elicit a variety of responses. We'll all laugh when someone says, "I learned that factoring is hard," but then we'll get more serious. If I need to, I might follow up with a more pointed question, such as, "How does the graph relate to the two different forms of the cubic equation?"
The points I will be sure to touch on are the following:
- the y-intercept can be "seen" in the expanded form
- the roots can be "seen" in the factored form
- the factors can be identified from the roots of the graph
- the coefficient on the cubic term determines the end behavior of the graph
- the number of distinct linear factors determines the number of distinct roots
- when a linear factor is doubled, the graph merely touches the axis
- a cubic function may have 1, 2, or 3 real roots
Finally, after a conversation like this, something like the 321 Exit Ticket is a good strategy for formative assessment. I'll use one today.