Polynomial Puzzles 3: Multiplying and Factoring Polynomials

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SWBAT multiply and factor binomial and trinomial expressions.

Big Idea

This lesson will help students to see that multiplying and factoring trinomials are inverse operations


15 minutes

This lesson is a variation of Polynomial Puzzles 1: Adding and Subtracting. The difference is that in today's lesson the puzzle tables are based on multiplication and not addition. I begin class today by leading students through the polynomials_multiply_factor_launch presentation.

Slide 1

I start by asking students investigate the first slide individually.  Since we have done these puzzles a week or so ago with addition, I expect students to see that multiplying the left two entries in each row gives the product in the right entry of each row. Similarly, multiplying the top two entries in each column gives the product in the bottom entry in each column.  

Slide 2 and 3

Now, I let students work with their partner to determine the missing entries on Slide 2. I encourage students to "think aloud" so that their partner can gain some insight into their strategy (MP3).  Notice that the bottom right cell serves as a check.  Both the cells to the left of that cell and the cells above it should have the same product.

Before letting students solve the puzzle on Slide 3, pause to have students do a think-pair-share to determine a strategy for finding the missing entries.  I plan to call on one or two students to share their ideas with the class before giving students the okay to start solving the puzzle.

Slide 4 and 5

For Slides 4 and 5 I let students work the puzzles with their partner.  

When we go over the answers I will call on multiple students to fill in the remainder of the puzzle.  With the original table projected on the board, I call on a student to explain which cell they tried to fill in first.  Then, I call on another student for the second, etc.  Going over this puzzle out loud will help all members of the class to see reasoning that others are using to find a solution. This opportunity to reflect helps students to develop their own strategy (MP2).

Since Slides 4 and 5 contain monomials and polynomials, I will ask students to discuss how they factored or multiplied to find each missing entry.


20 minutes

The work of solving the puzzles in polynomial puzzles 2 practice will help students to become more fluent with multiplying and factoring polynomials.  Rather than only factoring or only multiplying, the puzzles give students the opportunity to employ both operations.  In my classes, students typically move back and forth between factoring and multiplying as they fill in missing entries.

I let students work with a partner. I encourage students to discuss their ideas and strategies (MP1, MP3).  If a pair of students gets stuck, I encourage them to speak with another group for help. Students can learn by seeing the steps that another group used find the missing entries. I continually remind students to use the bottom-right cell as a check to ensure that all products moving down and across the columns/rows are correct.

If time permits, I plan to ask students share their thinking about how to solve each puzzle. I use a similar process as in the launch sharing. My goal is to get as many students involved as possible in sharing their ideas with the class.  I think it is worth managing the time during he lesson so that sharing can take place. Sharing is especially powerful when students took different approaches. Throughout this activity I observe students work to identify ideas that can be compared as the class ends.


5 minutes

Today's closing will serve as an application for the abstract work that was done on the polynomial puzzles.  I expect my students to observe on the fact that the given polynomial should have three factors.  These can be seen as the length, width and height of the rectangular solid.  Have students turn and talk with a partner about the second question dealing with the square base.  Listen for students who are having some insight as to why x^2+4x+4 is a "square".  Have students share their ideas with the class.

Often, students do not see (x+2)(x+2) or x^2 +4x+4 as a square polynomial (MP2).  If students can begin to see this structure as today's lesson closes, it will help in a future lesson when they are asked to complete the square for a quadratic function.


This lesson was adapted from Polynomial Puzzler on the NCTM Illuminations website.

Source URL: http://illuminations.nctm.org/Lesson.aspx?id=2938