## 3 Cup System - Providing Help Where it is Most Needed - Section 2: Sketching and Interpreting Graphs of Polynomials

*3 Cup System - Providing Help Where it is Most Needed*

# Performance Task - Representing Polynomials

Lesson 13 of 15

## Objective: SWBAT demonstrate the ability to translate between algebraic and graphical representations of polynomials and graph transformations of polynomial functions.

## Big Idea: The graphs of polynomial functions can be transformed by altering the algebraic form in specific ways. These transformations have the same effect on all the function types studied in Algebra 2 and Precalculus.

*90 minutes*

#### Pretest

*20 min*

Representing Polynomials is a Mars Assessment Task designed to asses how well students can translate between graphs and algebraic representations of polynomials.

To be successful in the task, students must be able to:

- Determine the end behavior of a polynomial presented algebraically
- Recognize the connection between the zeros of a polynomial and its linear factors
- Understand the connection between transformations of the graphs of polynomials and changes to the algebraic form of the function, for example replacing f(x) with f(x + k), f(x) + k, -f(x), or f(-x).

To warm up for the days work and assess what they already know, my students complete the Representing Polynomials Pretest.

#### Resources

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I place students in groups of three for the next part of this lesson. These groups are selected so that they are of mixed ability and students who do not normally work together have a chance to collaborate.

Representing Polynomials Matching Activity asks students to match graphical and algebraic representations of polynomials [MP2, MP7]. Students have done work like this before earlier in the unit, but todays activity is more challenging for two reasons.

- Some of the linear factors have a multiplicity other than one, so students must decide whether the graph touches or crosses at each root.
- In some cases, no graph matches the given algebraic form and the students have to draw in their own function that matches.

The first time this activity is used, the cards should be printed out on colored card stock and laminated. This way, students can use a wet-erase marker to draw in their own graphs and the cards can be saved for future use.

As groups work to complete the activity, I circulate around the room, using the 3 Cup System to determine which groups are struggling. Although I do not specifically tells students the correct pairings, I will give small hints to point them in the right direction when appropriate.

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#### Structured Discussion

*25 min*

When groups have finished matching up the cards and drawing in their own graphs where appropriate, the class comes together for a discussion of the activity. The goal of the discussion is for students to understand each other's matching strategies and make sense of the mathematical logic of each strategy [MP3]. To foster this type of discussion, I ask the following types of questions:

- Who used a strategy different from the one shown?
- Who can build on what ___ has said?
- Can anyone find two pairings that are similar in some way?
- Can anyone find two graphs that are reflections of one another?
- Can anyone find two graphs that are translations of one another?

Through this discussion I hope that my students will solidify their understanding of translations of polynomial graphs and the relationship between the verbal and algebraic representations of polynomial functions [MP2, MP7].

#### Resources

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#### Review Responses to Pre-Test

*20 min*

After we have thoroughly discussed the activity and my students have had the opportunity to ask any questions, I return the Representing Polynomials Pretest., which they completed at the beginning of the lesson. I ask them to use a different color to correct any responses that they feel differently about. Students turn this in to me for grading when it is complete.

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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