##
* *Reflection: High Expectations
Graphs of Cubic Functions, Day 2 - Section 1: Picking Up Where We Left Off

I expect some students to struggle with the second problem because although the three roots are given, the y-intercept is not. They will need to evaluate create an equation based on the three roots, evaluate it at *x* = -1 (a great opportunity to put synthetic substitution to use), and then determine the necessary constant factor.

I expect almost *everyone* to struggle with the third problem since, in this case, only two of the roots are given! Encourage them to think about the structure of the equation, and to create two unknowns for the missing values. You can see the kind of thought process I'm looking for in this student's work. She created let *r* be the missing root and *a* the missing constant factor. Then, with some help from classmates along the way, she was able to solve for both of them. Conveniently, in this case *a* = 1!

I believe these kinds of challenges are important for students. They push them to think more deeply about the subject and to avoid reducing it merely to a set of procedures for solving routine problems. It's also important to bear in mind that my ultimate goal for all students is the ability to solve problems like the first one on their own. After working through these more challenging examples, the first problem begins to look like a cake walk!

# Graphs of Cubic Functions, Day 2

Lesson 10 of 13

## Objective: SWBAT write an equation for a cubic function from a graph with given roots. SWBAT explain how the factor theorem helps them write a cubic equation.

## Big Idea: How many points does it take to determine a cubic equation? Four - and it helps if three of them are roots!

*45 minutes*

#### Picking Up Where We Left Off

*20 min*

This lesson begins very simply with the instruction to "pick up wherever you left off" in the assignment. Students will quickly begin forming small groups to collaborate or they may choose to work independently.

I will take a quick walk around the room to get a sense of how far everyone still has to go. I'll also be on the lookout for student who are either significantly ahead or significantly behind. I might pair some of these students up so that the more advanced students get a chance to explain their thinking to someone, and those who are behind get some assistance in catching up.

For the remainder of the time, I'll continue to circulate, answering questions and checking for understanding.

I expect some students to struggle with the second problem because although the three roots are given, the y-intercept is not. They will need to evaluate create an equation based on the three roots, evaluate it at *x* = -1 (a great opportunity to put synthetic substitution to use), and then determine the necessary constant factor.

I expect almost *everyone* to struggle with the third problem since, in this case, only two of the roots are given! Encourage them to think about the structure of the equation, and to create two unknowns for the missing values. (**MP 1**)

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#### Student Solutions

*20 min*

At some point during the previous section, I will have sketched the graphs for problems two and three on the board with some empty space nearby for students to present their solutions. (**MP 3**)

Now, it's time to call a student to the board to explain the solution to problem 2. Typically, I will ask for a volunteer, but I may also call on someone I identified earlier in the lesson. This student will explain how she identified the three linear factors and how she made use of the fourth given point to determine the correct constant factor. After a few minutes for the class to ask questions, she'll take her seat.

Another volunteer will be called to explain the solution to problem 3. This time I'll be very careful only to call on someone with whom I've already discussed the solution during the previous section of the lesson. This student will need to explain how he made use of the two given roots to form a quadratic equation, and how he used the other two points to identify the missing linear factor and the missing constant factor. The focus should first be on the creation of the equation with two unknowns and then on how he "plugged in" the other two given points to create a system of linear equations. (You can see my solutions for an idea of what I'm expecting.)

Finally, I'll return to the board to summarize the solution one more time and to answer any remaining questions.

#### Resources

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#### Wrapping Up

*5 min*

As class ends, I think it's important to praise your students not only for their ability to solve these problems, but for the clarity of their explanations. They're thinking through some very sophisticated arguments, and solving some very challenging problems!

At this point, some students may be feeling a bit overwhelmed. Sure, they think, we solved these problems *together*, but I'm going to have to do it alone on the quiz! It's important to reassure them that on the quiz they will be given all three roots and the y-intercept. At this point, everyone should be able to confidently solve a problem like that - especially with a little more practice.

See the video narrative: Wrapping Up.

#### Resources

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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