##
* *Reflection: Developing a Conceptual Understanding
Multiplying Polynomials Investigation - Section 2: Investigation

This lesson is certainly a departure from the "typical" approach to teaching arithmetic with polynomials. That said, it is focused on a topic that all math teachers mostly take for granted. Sometimes I find it useful to make these implicit traits really explicit for students. The more they can understand why polynomials "work" the way they do the deeper their understanding will be. This investigation did a good job of making the relationship between the number of terms in the factors and the number of terms in the product of two polynomials really explicit. It also gave students the tools to anticipate what an answer will look like before beginning their calculations. This can be a really powerful habit of mind.

*Look for the structure!*

*Developing a Conceptual Understanding: Look for the structure!*

# Multiplying Polynomials Investigation

Lesson 8 of 18

## Objective: SWBAT multiply polynomials with a various number of terms and degree.

## Big Idea: Students will discover that the number of terms in the product of two polynomials is connected to the number of terms in the expressions being multiplied.

*40 minutes*

#### Warm Up

*15 min*

Students should work in pairs on this WarmUp activity. The purpose is to allow students to practice using vocabulary associated with polynomials. Students should be able to complete the chart in pairs. Then, I allow as many pairs of students as possible share out all that they learned about each polynomial using the vocabulary. For example, students might say that the first polynomial (row 1) had degree 2, it is a quadratic polynomial and it is a trinomial.

**Resource Note**: Page 2 of multiply_polynomials_investigation_warm_up is an answer key. Page 3 could be used as a matching activity if time allows in your particular class.

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#### Investigation

*20 min*

I designed this Investigation to help students notice the structure associated with multiplying polynomials (MP7).. The lesson is sequenced after students have learned how to multiply various polynomials. The emphasis today is not on the arithmetic of polynomial operations but rather on the structure of the product.

First, I go through and have students individually find the product of each set of polynomials. This task will take approximately 7-10 minutes. Then I have students pair up and compare their answers with their partner. During this time, I encourage students to critique each other's thinking. While most students will be fairly proficient at multiplying, they will still make some procedural errors that can be corrected in small groups.

Once students are confident with their products, I have them fill out the boxes for:

- Number of terms in the first expression
- Number of terms in the second expression
- Number of terms in the unsimplified product

Before students start to fill in these boxes, I ask them to be looking for a pattern as they work.

Once students have completed the boxes, I have them generate a conjecture about the relationship between the number of terms in the factors and the number of terms in the unsimplified product. I let several students share their thinking. I guide them towards understanding that the relationship is multiplicative. As we discuss student work, I refer to earlier lessons which have used an area model to demonstrate multiplication of polynomials to make this relationship clear (e.g. a 2-term polynomial times a 2-term polynomial will result in a 4-term polynomial before simplifying).

Finally, I ask my students to determine the simplified answer for each product. (Note: only the last two products produce like terms that can be simplified). This brings up an important point: even a trinomial could have started as two binomials. Here I am laying the foundation for factoring in future lessons.

#### Resources

*expand content*

#### Closure

*5 min*

On a half sheet of paper, I will have students answer the questions on the lesson close individually. If time allows, I let students compare their answer with a partner to determine if they agree on the solution. If there is disagreement, I encourage students to come to consensus on which solution is correct.

*expand content*

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- LESSON 1: Adding and Subtracting Monomials
- LESSON 2: Adding and Subtracting Polynomials
- LESSON 3: More with Adding and Subtracting Polynomials
- LESSON 4: Polynomial Puzzles 1: Adding and Subtracting Polynomials
- LESSON 5: Multiply and Divide Monomials-Jigsaw day 1 of 2
- LESSON 6: Multiply and Divide Monomials-Jigsaw Day 2 of 2
- LESSON 7: Multiplying Higher Degree Polynomials
- LESSON 8: Multiplying Polynomials Investigation
- LESSON 9: Polynomial Vocabulary
- LESSON 10: Polynomial Puzzles 2: Distributive Property
- LESSON 11: Factoring Using a Common Factor
- LESSON 12: What if There is No Common Factor?
- LESSON 13: Factoring Trinomials
- LESSON 14: More with Factoring Trinomials
- LESSON 15: Polynomial Puzzles 3: Multiplying and Factoring Polynomials
- LESSON 16: Seeing Structure in Factoring the Difference of Squares
- LESSON 17: Factoring Completely
- LESSON 18: More with Factoring Completely