## Working Backwards - Section 2: Investigation: Building Polynomial Functions

# Building Connections: Building Polynomials (Day 1 of 2)

Lesson 2 of 15

## Objective: SWBAT explain the relationship between linear factors of a polynomial function and the graph of the function.

## Big Idea: Love, love, love(!) this lesson from NCTM where students build a polynomial function based on linear factors.

*51 minutes*

On page 2 of the flipchart students will be asked to write the equation of the line. Students have already had experience with writing the equation of lines in this class (Arthur’s spyglass problem is one example I can think of off the top of my head) so hopefully the warm-up with go smoothly. It is important that students are comfortable writing the equation of a line to be successful in today’s activity.

**Differentiation: **To help struggling students write the equation of the line, I would first ask them what method they are using: slope-intercept, point-slope, using a table to find a rule. And then try to encourage them to ask me a question to help clarify what they don’t understand. It is important for students to be able to identify what they don’t understand. Not just “I don’t get this.” Like they know nothing… I don’t think so. If students are able to articulate a question, then great! I will answer it straight forward. However, if they don’t even know where they are stuck I would encourage them to make a table of values. Being sure to list where x is zero. Then try to find the function rule (I know this is kind of a longer way, but I think it has more meaning to students). If they are still stuck, I would try to guide them into finding the slope by giving the hint that it’s the ‘change in y divided by the change in x.’ I think this change is easier for students to see in a table. I will then let them know that this slope is part of the step of getting to the next number. We multiply by it. But what else must we do? Now realistically, all of this probably isn’t going to happen in that 5 minute window along with everything else teachers need to do in those first few minutes. So I would encourage students to ask teammates for help first and then let students know I will be around for more help on how to write the equation of a line during the upcoming student investigation time.

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For the next two class periods, I want students to start making connections to how a polynomial function (in particular quadratic functions today) relate to the zeros of its linear factors. The NCTM Illuminations lesson Buidling Connections is very student centered and is very clear in helping students to make this connection.

Today I expect students to only have enough time to work though the Building Polynomials worksheet. But if you have more time in your period or your students work though this quickly just have them continue on to the next worksheet: Working Backwards.

This lesson does a great job at having students work toward becoming better mathematicians. While working through this activity many of the mathematical practice standards will be addressed. I am going to encourage my students to be aware of how they are practicing the following two standards:

**Mathematical Practice 2: Reason abstractly and quantitatively –** Students will need to be able to reason about their answers to the questions presented in order to make conclusions on how linear factors relate to the quadratic function. For example, in question 8 students will need to be able to reason quantitatively about the actual numerical values of b/m and how this relates to the graph and then students will be asked to make conclusions about these linear factors in an abstract form (x-c).

**Mathematical Practice 7: Look for and make use of structure - **In particular, students will be examining the structure of a quadratic function in factored form and making connections to how this relates to the graph. A great example where students will practice this habit of mind is in problem 12. Students will be asked to make a connection of the relationship between the linear factors’ x-intercepts and y-intercepts and the graph of the parabola.

I definitely recommend working through the worksheet yourself before having students complete it. Check out the video in the section below (which is a bit long, sorry… you may be better just working through the activity!) for an overview of the activity and some tips on differentiation.

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To wrap-up today’s learning and to be sure all students made the connections that they were supposed to, have students do a Think-Pair-Share on the prompting question on page 3 of today’s flipchart. If you have time in the period share some student responses with the rest of the class. This is a good opportunity for struggling students to hear their peers put today’s learning into perspective.

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I really enjoyed using this lesson. Â I did the activity cold and wrote down some of the struggles I went through in order to ask my students questions. The graphs in Working Backwards (#11) had y-intercepts as fractions which were challenging for the students to determine. As an afterthought did you give the students the fraction value at where the x and y intercepts occurred Â or did you lead them to use the vertical transformation of a factor of 5/9 to calculate the y -int to be (0, 20/9)?

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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