Multiplying Polynomials Investigation

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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.

Warm Up

15 minutes

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. 

Investigation

20 minutes

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:

  1. Number of terms in the first expression
  2. Number of terms in the second expression
  3. 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.

Closure

5 minutes

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.