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# Got zeros? Polynomials do! Multiplicity of Zeros (Day 1 of 3)

Lesson 5 of 15

## Objective: SWBAT• Determine the degree of polynomial functions and understand the relationship between degree and end behavior of polynomial functions. • Write possible equations for a polynomial function, given information about its zeros. • Write polynomial equations in factored form, given the graphs of three functions.

## Big Idea: Using Nspire Calculators, students investigate the relationships of polynomial functions, their degree, end behaviors, zeros and x-intercepts.

*50 minutes*

**Preparation: **If you plan on having students use the .TNS file from Texas Instruments, you will want to have that ready to load onto the students’ calculators. I recommend doing a test run on on at least one calculator. One time I found that I needed to do a complete update of the operating system on all of my calculators! I was so glad that I had tried before the lesson or it would have been a disaster!

**Narrative: **To introduce today’s activity, I have students review what it means to find the zeros of a polynomial function. I plan to begin by having students complete page 2 of Multiplicity of Zeros Day 1 (p.1-2). If students do not recall zeros of a function, you can remind them. Here is a straight forward hint: find the values for x that make f(x) = 0. Push the students to come up with an algebraic solution, but do not restrict them to this approach. I think it is important in math that we allow students to solve problems in ways that make sense to them. I am curious, however, if some students may use a calculator and find the zeros from a table or graph.

While students are working, teams of students can come up and transfer the calculator file.

Note: At first, I was very nervous/intimated about loading TI programs during class. The first few times I used my TI calculators I would have all the files pre-loaded before students came in. But, it really does go quickly and save scarce time for other tasks.

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Multiplicity of Zeros, video narrative, Calculator Investigation is a detailed presentation of my plan to approach this lesson. In this video I discuss the following:

- Why I like to use Texas Instruments Lessons
- How to adapt the lesson for non-Nspire calculators, see
**Teacher Resource - Equations for Non-Nspire Calculators**for more information too - Importance of giving students time to 'play'
- Using page 3-5 of the flipchart to show students how to find the degree
- How this tasks builds on students' prior knowledge and is aligned to the CCSS
- How this task introduces end behaviors
- Mathematical practice 5 and 7 and how they relate to this activity

Below in my video reflection: **Got zeros day 1, video reflection, Review Vocabulary**, I highlight some of the obstacles that my students encountered as they completed today's exploration: Student Handout - Multiplicity_of_Zeros_of_Functions.

I would expect my students to probably only complete questions 1 and 2 today. Also, you may want to note for your students that on page 1.2 of this activity there is a slider to click at the top of the graph to change the function. There are six functions (#0-5). They should be on #0 for the first question.

**Resources Use in this Lesson:**

Multiplicity of Zeros of Functions. (accessed September 3, 2013 on Education.TI.com)

##### Resources (6)

#### Resources

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With about 5 minutes remaining in class today, I am going to have students do a Stand up, hand up, pair up** **(with someone not already at their table!).** **I want students to then compare their answers in the table for question 2. This will allow each team of three to check their answer with possibly three other teams. Then when we come back tomorrow, individual students can bring back their findings to their team and teams can discuss revisions to their answers or how they wrote their answers.

#### Resources

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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