##
* *Reflection:
Detailed Descriptions of our Puzzling Polynomials - Section 2: Describing Polynomials: Increasing and Decreasing

As students were working on part a, I took some time to observe their responses. I found that a lot of my students skipped indicating the regions where the graph is either increasing or decreasing. A few brave students attempted to respond and I really liked this creative answer: Student Work - Interesting Answer.** ** During their struggles through part A students would ask for help, I would tell them there is no right or wrong answer just write what you think. I would then prompt students to show me where the graph is increasing or decreasing. They had no problem doing this. Yet they still didn’t write anything! So clearly, they just didn’t know HOW to write it.

I predicted that students would struggle over describing the regions, but I didn’t expect them to just flat out skip it. However, I still like that students had this mini-struggle before they learned interval notation for describing the regions. I think it helps students to have a greater appreciation for interval notation or at least understand WHY we need it.

One of my students also came up with a great response that created an opportunity for a Favorite No.** **discussion: Student Work - Favorite Wrong Answer**. ** It gave us a chance to discuss that common misconception. Just because a region in the graph is going off to the right it does not mean it is increasing. I asked students if anyone could come up with a counter-example. They did and drew it on the board.

*Student Work: "Favorite No" & Most Interesting*

# Detailed Descriptions of our Puzzling Polynomials

Lesson 4 of 15

## Objective: SWBAT describe polynomial functions using accurate terminology, identify the number of turning points and state whether the functions are even or odd.

## Big Idea: Students build on their intuitive understanding of increasing and decreasing functions and explore even/odd functions using a calculator.

*50 minutes*

#### Warm-up

*4 min*

Have students answer the built in clicker questions on pages 2-4 of the Flipchart - Describing Polynomials to assess students intuitive understandings about increasing, decreasing, and constant functions. I am predicting that students already have an intuitive understanding about anything increasing or decreasing should make these go smoothly. Students may need to be reminded to “read” the graphs from left to right just like we read a book.

*expand content*

Students will first work through just part 1 of the Student Notes - describing polynomials to have a little more practice identifying increasing and decreasing parts of a graph. I am going to allow calculator use to sketch this function as my goal in this part of the activity is not whether or not they can make an accurate graph by hand, but whether they can identify increasing and decreasing parts of a graph.

In part a, the question does not specifically address how students should ‘indicate the regions.’ And I think this is the great part about this question. I am curious to see how students go about showing these regions. Any method at this point would be fine. And this really is the point, which leads us into the next section.

*expand content*

Next present page 6 of the flipchart to students and have them add the formal definition of *increasing functions* to their** **Personal Dictionaries (or maybe just write a statement about what increasing or decreasing means to them… that might be even better!).

Since this mathematical definition is pretty complex for students (especially in comparison to their intuitive understanding of increasing) it is a great opportunity to have them break apart what the math words and symbols are communicating. On page 7 of the Flipchart - Describing Polynomilas (p.6-11) there are some discussion questions to prompt the class to think about what this formal definition means. I don’t intend on even showing pg. 7 to my classes, but instead just using these as prompts to break down the definition while I am displaying the definition to the class. So I will print this page ahead of time and use the questions/prompts to get the students talking about the definition.

While students are communicating about this definition it is a great time for students to hone in on their skills at **Mathematical Practice 3: Construct viable arguments and critique reasoning of others. **Present pages 8 and 9 and have students add these definitions to their dictionaries.

Model the example problem for students on page 10. During this time, I plan to introduce interval notation as the uniform way we will notate the intervals on which a function is increasing and decreasing. Now students should complete part 2 on the describing polynomials notes. Toward the end of this allotted time, be sure to take questions from students over these practice problems.

*expand content*

#### Closure

*5 min*

To assess students understanding of even and odd functions and to help students summarize their learning today, have students answer the 5 questions on pages 14-18 of Flipchart - Describing Polynomilas (p.14-18).

*expand content*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Puzzling Polynomials and Quizzical Quadratics
- LESSON 2: Building Connections: Building Polynomials (Day 1 of 2)
- LESSON 3: Building Connections: Building Polynomials (Day 2 of 2)
- LESSON 4: Detailed Descriptions of our Puzzling Polynomials
- LESSON 5: Got zeros? Polynomials do! Multiplicity of Zeros (Day 1 of 3)
- LESSON 6: Got zeros? Polynomials do! Multiplicity of Zeros (Day 2 of 3)
- LESSON 7: Got zeros? Polynomials do! Multiplicity of Zeros (Day 3 of 3)
- LESSON 8: Long, Synthetic, and Square! Oh my! Polynomial Division
- LESSON 9: Rational Roots and Remainders: Important Theorems of Polynomials
- LESSON 10: Polynomials with Complexes… Complex Zeros that is!
- LESSON 11: Putting the Pieces of Polynomials Together (Day 1 of 2)
- LESSON 12: Putting the Pieces of Polynomials Together (Day 2 of 2)
- LESSON 13: Roller Coaster Polynomials
- LESSON 14: Test Review
- LESSON 15: Polynomials Unit Test