Using an Area Model to Multiply Binomials
Lesson 6 of 18
Objective: SWBAT use symbols to express the areas of rectangles created by altering squares with sides of length x.
I begin class with a reminder about what students have learned so far about quadratics and where they are headed. I might use some generic text on the SmartBoard like:
Where are we?
I will ask students to share out what we've learned about quadratics so far, being sure to elicit the importance of the vertex and how vertex form makes it easy for us to find this maximum or minimum.
I let students know that like the Fireworks problem, many quadratics are not yet in vertex form. We will be using algebra to change this equation into vertex form.
Next, I tell students that we will prepare for this work by studying ways to multiply, square, and factor algebraic expressions. Along the way, they will see how to transform a quadratic from from vertex form into standard form.
I inform to my students that we will be taking a little side trip to learn how to multiply binomials, today. I emphasize that this process is helpful in transforming quadratics from vertex form into standard form.
Next, I hand out Factor Fixin' and we read the introductory paragraph together. I draw a square on the board with a length and width of x. I make sure students understand that this would be the size of the original quilt square. Then, I ask students how they can add to or change this diagram in order to represent the new dimensions outlined in Question 1. I emphasize with students that they will be creating a diagram for each of the questions 1 through 7. They will also be writing two expressions for the area of the new quilt as a product of the new length and width (one with parentheses and one without).
Once I have gone through all the steps in Question 1, I let students get to work in small groups.
Things to watch for while students work:
- Students may struggle with Question 3 where only one dimension is changed. Let them fiddle around with their diagrams and see if they can represent this problem without getting the “answer” somewhere else.
- Make sure students are writing the area in two ways. Students may be tempted to only write the area as the sum of all the rectangles in the diagram. Make sure they write the area as a product of the two binomials as well.
- Students may struggle with numbers 6 and 7 as they involve subtracting area, rather than adding it. Some students will use the original model anyway, which will work, while others will try to make sense of it in a new way. I try to provide students with the opportunity in the discussion part of class later to share their representations of these problems.
Discussion + Closing
I leave plenty of time for the discussion of the Factor Fixin' activity. Key points to I stress in the Discussion area:
- In Question 3, I emphasize the equivalence of both expressions.
- In Question 2, I have students choose a number to substitute for the original size of the square. Then I have them substitute this number into both expressions and make sure they come out the same. Next, I have students enter both expressions into the graphing calculator and check to make sure their graphs are the same. I repeatedly emphasize that the two binomials in parenthesis are equivalent to the expressions they got without parenthesis when they added up the areas of all of the rectangles.
- I give students the opportunity to share their interpretations of Questions 5 and 6. I pay special attention to alternative representations they may have come up with to take into account subtracting length. I make sure to assure students who followed the initial model that this works even though it doesn't make much sense to add a rectangle when they are taking away length.
This activity is a good opportunity for students to work on SMP 7: Look for and Make Use of Structure. They are developing an understanding of how the distributive property works by using an area model. They are also laying a foundation for learning how to factor a quadratic later in the unit. To see more about how SMP7 and area models connect, watch this video! area_model.MP4
The second part of this activity (Questions 8 through 17) take students in the opposite direction, working from quadratics in standard form to factored form. For my class, this is not the point of today's activity and will be included in the next lesson, so I might offer it as an extension for students who are ready for more.
If students need more practice multiplying binomials, this is a good opportunity for them to do some practice at home. I might use a Kuta Software worksheet or something similar to keep this idea fresh in their minds.
Factor Fixin' is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.