##
* *Reflection: Developing a Conceptual Understanding
Inequalities: The Next Generation - Section 3: Extend

A really important conversation came up when we were discussing the sign diagram for the rational inequality; students were wondering why we put the* x*-intercepts and undefined points on the sign diagram. Why wouldn't a *y*-intercept go on the sign diagram?

This was important to consider and I had students take a short period of time to think. A few students understood right away that in order for a graph to switch from above the *x*-axis to below there can only be an *x*-intercept or an asymptote - nothing else. Afterwards I realized that this idea was essential to their understanding of the sign diagram!

*Why Do We Set Up the Sign Diagram Like That?*

*Developing a Conceptual Understanding: Why Do We Set Up the Sign Diagram Like That?*

# Inequalities: The Next Generation

Lesson 9 of 12

## Objective: SWBAT solve nonlinear inequalities.

*55 minutes*

#### Launch and Explore

*15 min*

This lesson starts off with a problem that can be stated very simply - **solve a quadratic inequality**. I have a quick discussion with my students about the fact that today we are going to be using inequality symbols instead of equal signs. I ask my students what the implications of that are. Most will say that there are going to be an infinite number of solutions, instead of a finite number when we solve equations.

In this task I have asked students to try to solve this problem in more than one way so that are not fixated solely on the algebraic or graphical representation.

Ideally, students should be working in groups of 2 or 3 to work on this first problem. Inequalities that are not linear may be a new concept for students and they can be tricky. It is important to push students to think of multiple ways to think about the problem, because the graphical approach can be the key to clear up the misconceptions that students hold while solving the inequality algebraically. So if a student found their answer algebraically, ask them how they could use a graph to solve. The good news is that students have a lot of experience with quadratic functions and they can easily factor and graph, and those strategies will help them make sense of this problem.

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#### Share

*20 min*

I plan to followup the Launch task with a class discussion of the misconception that I am positive that will show up in my class! Usually about 95% of my class will overgeneralize the application of the **Zero Product Property** and when they solve this problem. My plan is to choose a student who solved the problem in this way and have him/her explain his/her solution to the class.

Here is a video that describes the mistake that I see in students use of the Zero Product Property (I suspect that this mistake creeps into your classroom as well):

From the discussion of this student's solutions to the problem, we will discuss a couple of approaches to help students unpack the mathematics in this solution.

**One approach:** Use their solution of x> -5 or x >3 and test different values. The numbers 4 and 0 satisfy both inequalities but only the four works in the original inequality. Students may then say that we should change the word "or" to "and." We could do that and then try to plug in -4. That works in the original inequality but does not satisfy our answer inequalities.

**A different approach: **Solve the equation x^2 + 2x - 15 = 0 using the zero product property. Remind students about what the zero product property actually says (if two things multiply to equal zero, one of them has to be zero). Now go back to the inequality. If two things multiply together to be greater than zero, do both things have to be greater than zero? Students will likely say no, since two negative can multiply to get a positive, and will see the fault in using the zero product property in this instance.

At this point, I expect that much of the class realizes the misconception. I will now ask another student to try to find the root of the thinking that led to the misconception. Students will definitely realize that -5 and 3 are the important points in this inequality; in fact, they are critical. So I plan to add the two critical points to a number line and create a sign diagram. Then, I will select a student volunteer who can explain how to use the sign diagram to solve the inequality.

After the algebraic solution is completed, I will ask a student who solved using a graph to present his/her solution. If no one used that method, I will take us in that direction by asking students how they could use a graph to solve the inequality. (I will give students a few minutes to work on it before having a class discussion).

To wrap up the segment of the class, I use this image to show the connection between the sign diagram and the graph of the function. It is crucial to helping students develop a clear understanding of this topic.

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#### Extend

*20 min*

After students comfortably understand the first example, I will have them try the additional inequalities on this worksheet.

Questions #1a) is pretty similar to the first problem that was given. After students understand how to do the initial problem, they can use the same method of factoring and creating a sign diagram to test the intervals.

Question #1b) can be solved by using a number line, but there needs to be additional conversation about what points should be on the number line. Since there is a fraction, the critical points will be where the numerator is equal to zero and where the denominator is equal to zero. All of those points will be added to the number line and the correct intervals can be selected. The function can be graphed using graphing software and the inequality can be analyzed in that way.

#### Resources

*expand content*

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Quadratic Function Jigsaw
- LESSON 2: Sketching Graphs of Polynomial Functions
- LESSON 3: Roots of Polynomial Functions - Day 1 of 2
- LESSON 4: Roots of Polynomial Functions - Day 2 of 2
- LESSON 5: Polynomial Function Workshop
- LESSON 6: Ultramarathon Pacing and Rational Functions
- LESSON 7: Homecoming and the Five Pound Gummy Bear
- LESSON 8: Graphing Rational Functions
- LESSON 9: Inequalities: The Next Generation
- LESSON 10: Rational Functions and Inequalities Formative Assessment
- LESSON 11: Unit Review Game: Pictionary
- LESSON 12: Polynomial and Rational Functions: Unit Assessment