##
* *Reflection:
Rational Roots and Remainders: Important Theorems of Polynomials - Section 2: Investigation: Remainder Theorem

All my students completed the tables perfectly. But most failed to make the full connection and master the learning target in the observation question. Here are some sample student responses for the observation section of this investigation:

- Some students simply missed the mark and
**did not demonstrate mastery**:

* “The opposite of the divisor equals the remainder.” - *very common answer!

* *“Almost the same remainder."

* *

- Many students got that the remainder was the same as the function being evaluated at that same value of k. But I don’t think they understand what k was or how k was related to the problem in first place. So most students actually only showed a
**partial mastery:**

* “The remainder of the first box and the answer of the second box are the same.”*

* *“I observe that the remainders and f(k) are the same exact answers.”

* *“The remainders are the same.”

* *“The remainder is related to f(k).”

* *“If you substitute the k values in the function you get the remainder.”

* *“It’s just backwards. Remainders are the same as f(k).”

**

Many students were not precise in documenting their observations. A majority of my students fell into the 3^{rd} category above, so it make me think that the “what did you observe?” question should be more direct. Or maybe I should add a question about “what is k? Or how can we find k?” I would really like students to make this connection on their own without me walking them through it though. In the future, I will monitor student work more closely on this and ask students to be more precise in their answers by asking questions like “What do the values in the table that you found represent? What is their relationship? This is true when? What is k? How can we find it?”

*Student Work – More practice needed on Mathematical Practice 6!*

# Rational Roots and Remainders: Important Theorems of Polynomials

Lesson 9 of 15

## Objective: SWBAT identify the connections between dividing polynomials and evaluating polynomials and determine the possible rational zeros of a polynomial using the Rational Root Test.

## Big Idea: Using polynomial division students discover the Remainder Theorem & then learn about the Rational Root Test.

*51 minutes*

Students should begin working on the two polynomial division problems on page 2 of the Flipchart on a scratch piece of paper (they will record their findings later). After about 5 minutes, I plan to go over the first problem using whichever method (long division, synthetic division, or division with generic squares) that students request, maybe even all 3.

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Today was the last day of new learning and I definitely plan on sharing this with students. So for the next week students are just going to be presented with opportunities to put all the pieces together and build on their current knowledge. Since we are getting close to the end of the unit, I wanted to get a sense for how students feel that they are progressing. At the end of today’s lesson I am going to have my students complete a **3-2-1 Assessment** to accomplish this.** **The prompts for students are on the last slide of the power point. We will be reviewing students’ responses at the start of class tomorrow and may use some of their feedback to drive the instruction in the upcoming lessons.

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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