This Warm-up follows from the previous lesson. Before students experiment with different polynomials, I ask them think about the question for 30 seconds. Then, I use a non-verbal cue to determine which students think the answer to the question is yes or no. Then, I let them make up some polynomials with their partner and add them together. This exploration encourages students to practice reasoning abstractly and quantitatively (MP2).
I expect students to come up with two polynomials that sum to zero. After they identify a few examples, I open the conversation to let the class reflect on whether the sum is still a polynomial.
Instructor's Note: If no students come up with an example like that, ask the class yourself to get students thinking about this case. Because even zero is a polynomial (it is a numeric term), polynomials are "closed" under addition and subtraction. Meaning if you add or subtract two polynomials you will always get another polynomial (A.APR.1).
To end the warm-up, I explain to students that this lesson will be about showing that polynomials are also closed under multiplication.
Slide 3: This slide is meant to remind students how polynomials are multiplied together. I have students look at the two polynomials being multiplied and notice how their powers are related to the power of the answer (added together).
Teaching Point: If needed, show a few more examples of this type. I like to use two colors as shown on the slide so students can see how the two parts come together.
Slide 4: This slide allows students to see the multiplication of two polynomials using an area model. The area model can serve as a valuable tool for students when mulitplying polynomials (MP5). Because this is only a model, all of the squares (areas) are the same size. I like to explain to students that we are not saying that all of the segments in this rectangle are actually the same. The area of each section is put in the box. Let students work with their partners to see why this model is a demonstration of finding the product of the polynomial given (in other words, where does each of the six products "come from?". Have students do a think-pair-share on this slide. When sharing out whole group, let each student build on the ideas of the previous student before giving their own idea.
Slide 5: In this slide, we want students to try to find the product using the distributive property. Depending on the ability of my class, first, I will will draw a basic representation of the distributive property (example: 3(x+2)). Then, I show students how the distributive property has been applied to write the expression the second way and then let the students find the final product and show that it matches what they obtained in the previous slide.
I give students an initial round of practice with the area model and with using the distributive property to write equivalent expression. I structure the practice as follows:
I think that this structure allows students to critique each other's reasoning and construct viable mathematical arguments (MP3).
The multiply_polynomials_closure Ticket Out requires students to think abstractly MP2. I give them a little time to think about the problem. I do not ask them to verify until they have identified the degree of the product. I have given the expression x^5+3 as the degree 5 expression to show students how simple an expression can be. They will make up their own degree 2 expression and verify that the product will be of degree 7.