##
* *Reflection: Complex Tasks
A Gallery of Cubic Functions, Day 1 of 2 - Section 3: Solving Problems Collaboratively

Since this problem set is so complex, you can expect quite a few errors along the way. Watch carefully for them, but don't rush to correct every one of them immediately. I often find that my students learn more when then are given time and space to recognize their errors for themselves. Once they realize that something doesn't make sense, they will be motivated to look for a solution and recognize its significance.

It's common for students to mismatch a few graphs. This may happen because they're being too hasty, or it may be due to some deeper misconception. Typically, they notice this mistake when they try using the roots visible on the graph to find the factors of the equation. In this case, I'll ask how they went about matching graphs to equations. From here, I can typically help them to see that this particular graph doesn't match this equation.

I also find students sometimes spend a long time trying to factor a quadratic equation that does not have real solutions (I'm thinking of the second & third equations). In this case, I want to encourage them to use the graph! If they've found the correct graph for these equations, they can see that this function only has one real root. If they've already found one linear factor, they can be pretty confident there aren't any more.

Finally, I've found that some of my students seem to forget that they already know how to solve quadratic equations. For some reason, when they find a quadratic factor of a cubic equation, they are hesitant to treat it like other quadratics. If I suggest making use of the discriminant of the quadratic formula, for instance, they may ask, "Does that still work in this case?"

*Errors to Watch For*

*Complex Tasks: Errors to Watch For*

# A Gallery of Cubic Functions, Day 1 of 2

Lesson 5 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.

*45 minutes*

#### Setting the Stage

*5 min*

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:

- Factor the equation completely.
- Identify the root(s) of the equation.
- Match the equation to its graph. (They'll need scissors & tape to do this.)

At this point, it's important to emphasize to the class that they may do these things *in whatever order is most convenient*.

#### Resources

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#### Making Sense of the Problems

*10 min*

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**)

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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.

#### Resources

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#### Wrapping Up

*5 min*

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.

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- UNIT 1: Modeling with Algebra
- UNIT 2: The Complex Number System
- UNIT 3: Cubic Functions
- UNIT 4: Higher-Degree Polynomials
- UNIT 5: Quarter 1 Review & Exam
- UNIT 6: Exponents & Logarithms
- UNIT 7: Rational Functions
- UNIT 8: Radical Functions - It's a sideways Parabola!
- UNIT 9: Trigonometric Functions
- UNIT 10: End of the Year

- LESSON 1: Intro to Power Functions
- LESSON 2: Applications of Power Functions
- LESSON 3: The Biggest Box
- LESSON 4: Factoring Cubic Equations
- LESSON 5: A Gallery of Cubic Functions, Day 1 of 2
- LESSON 6: A Gallery of Cubic Functions, Day 2 of 2
- LESSON 7: The Dynamic Cubic Function
- LESSON 8: The Factor Theorem & Synthetic Substitution
- LESSON 9: Graphs of Cubic Functions
- LESSON 10: Graphs of Cubic Functions, Day 2
- LESSON 11: The Lumber Model Problem
- LESSON 12: Cubic Equations Practice
- LESSON 13: Cubic Equations Quiz