Generalize patterns for the expansion of binomials and explain the connection to the entries of Pascal's triangle.
Expand binomial expressions using the Binomial Theorem.
Quick Lesson Plan
In this final lesson of Unit 2, students continue practicing producing equivalent forms of polynomial functions, in this case by expanding binomials. The activity has students expand the binomial (x+y) to the second, third, and fourth power, using an area model. It is important that students combine all like terms and write each expression in standard form before moving on to the next expansion.
The area model truly illuminates many of the patterns that emerge in a binomial expansion. Students see how the coefficients of each term are related to both the previous terms and as an artifact of combining like terms. Before getting too explicit about any patterns, we want students to have the opportunity to notice on their own. In our experience students generate the vast majority of the patterns without any kind of prompting. If students are stuck, start by having them look at the exponents of x and y before moving on to the coefficients of the terms, as that can be more difficult to articulate.
How does the area model demonstrate the distributive property?
What do you notice about the coefficients?
What do you notice about the sum of the exponents on x and y in each term?
(Question 3) Which products in the expansion will give an x^3y term? Why?
How are the coefficients of this expansion related to the previous one? Why does this happen?
The debrief should consist of the teacher facilitating a discussion around students’ noticings. Be prepared that the patterns students see will be hard to articulate. As much as possible, use color and annotations to represent a student’s observations on the board. Ask other students to rephrase and summarize what they heard their peers say. If needed, restate the student’s contribution to make it more accessible to the class but avoid cleaning up the response to your “preferred” solution.
As students move to the Check Your Understanding, they may struggle with problems where one of the terms in the binomial expression is a number instead of a variable. We suggest going over question 1 as a class so students understand how to expand and simplify binomials of this form.
While this lesson is rich with patterns and helps students review fundamental concepts around the distributive property and how to raise numbers to a power, there is a temptation to increase difficulty by doing long expansions like (3x+4y)^9. We don’t find these exercises particularly useful mainly because it is rare that students would be expanding binomials like this by hand. If the goal is to assess patterns, consider asking for only a part of the expansion or even giving students the expansion and asking a question about why a certain term is what it is.