##
* *Reflection: Student Ownership
Quiz and Intro to Graphs of Polynomials - Section 3: Group Time to Generate Questions

In my own enthusiasm for the content, it can sometimes be difficult to stop myself from simply dictating what is interesting or intriguing to me. It can be equally difficult to trust that, if left to their own devices, students can and will recognize their own patterns and ask their own questions. By the time students reach high school it is sadly common that their enthusiasm for inquiry has been stifled. It takes time to get it back, but is well worth the effort.

In this exercise, I ask students to look at some mathematics and ask their own questions: first individually, then in collaboration with a small group of peers, and finally as a large group analysis of the ‘best questions.’ Hesitant students might need an initial probe from me. I might ask, “If you were a scientist and wanted to sort these graphs, how would you do that? How about the equations? Do your sorted groups help you see any patterns?”

It would be a lot faster and easier if I simply wrote my question on the board at the start: “Is there a relationship between the degree of a polynomial and the shape of its graph?” However, when the students inevitably ask this question themselves, along with several other probing questions, they own this lesson and are more motivated to find the answers. There is hardly a question they can ask that will not take us in a direction that is rich, useful, and even standards aligned.

*Directing Student Inquiry*

*Student Ownership: Directing Student Inquiry*

# Quiz and Intro to Graphs of Polynomials

Lesson 8 of 15

## Objective: SWBAT demonstrate understanding of polynomial theorems and ask good questions about the graphs of polynomial functions.

## Big Idea: The graph of a polynomial can look a variety of ways depending on the degree, lead coefficient, and linear factors.

*90 minutes*

#### Quiz on Polynomial Theorems

*45 min*

To be successful on Quiz Polynomial Theorems, students must know the Remainder Theorem and the Fundamental Theorem of Algebra and understand the implications of these theorems for finding roots of polynomials. Although I do not require my students to memorize a specific wording of these theorems, I do ask them to state them in their own words using polynomial vocabulary terms correctly[MP6].

The questions on this quiz probe for understanding of the idea that a consequence of these theorems is the relationship between linear factors and zeros of polynomials. I hope they can explain that any polynomial, P(x) ca be written as (x-a)*Q(x) + P(a), and that if P(a)=0 then (x-a) is a factor and a is a root of P(x) [MP2].

#### Resources

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#### Post-Quiz Warm-up

*15 min*

As students turn in their quiz, I hand them Polynomial Graph Questions. The goal of this exercise is for students to graph some polynomials using a graphing calculator and then try to develop a set of questions about why the graphs look the way they do. They first work independently to generate questions.

I anticipate that students will ask questions like

- Why do polunomials sometimes open up and sometimes open down?
- Why do the functions sometime cross the x-axis and sometimes not cross it?
- Why do some polynomials seem to "wiggle" more than others?
- What is the relationship between the degree of a polynomial and the graph of the polynomial?

#### Resources

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After students have had some time to generate questions individually, they join together with 3 or 4 other students to assemble a group set of questions.

Each group is given a sheet of post-it paper and markers. The goal is to find commonalities among their questions and come up with their favorite 5 questions. These posters will be put on the wall at the end of the group work time.

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#### Structured Discussion

*15 min*

I ask each group to put their poster on the wall. Then students circulate around the room to read the questions developed by each group. We have a brief discussion about commonalities between the sets of questions generated by the groups and begin a discussion about how we might go about answering the questions generated. The posters will remain on the wall for the next class period, when we will attempt to answer the questions [MP3].

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- LESSON 1: Seeing Structure in Expressions - Factoring Higher Order Polynomials
- LESSON 2: Proving Polynomial Identities
- LESSON 3: Polynomial Long Division and Solving Polynomial Equations
- LESSON 4: The Remainder Theorem
- LESSON 5: The Fundamental Theorem of Algebra and Imaginary Solutions
- LESSON 6: Arithmetic with Complex Numbers
- LESSON 7: Review of Polynomial Roots and Complex Numbers
- LESSON 8: Quiz and Intro to Graphs of Polynomials
- LESSON 9: Graphing Polynomials - End Behavior
- LESSON 10: Graphing Polynomials - Roots and the Fundamental Theorem of Algebra
- LESSON 11: Analyzing Polynomial Functions
- LESSON 12: Quiz on Graphing Polynomials and Intro to Modeling with Polynomials
- LESSON 13: Performance Task - Representing Polynomials
- LESSON 14: Review of Polynomial Theorems and Graphs
- LESSON 15: Unit Assessment: Polynomial Theorems and Graphs