Lesson 15 of 18
Objective: SWBAT expand a binomial that is raised to an integer power.
Launch and Explore
The focus of much of this unit has been recognizing patterns and describing them recursively and explicitly. Similarly, expanding a binomial raised to a power into the sum of its terms can be thought of as another type of sequence.
Today, we are going to expand binomials in the form (a + b)n and look at the pattern of the terms. I find that my students know a few shortcuts with this topic. Most know about Pascal’s Triangle and most know about the notation nCr, but they usually do not know the connection between the two. If your students do not know about Pascal’s Triangle or combinations, you may have to add in some more information to this lesson.
I give students this worksheet with little explanation and have them work in their table groups for about 15 minutes. As they are working I will circulate to make sure that the focus is on finding shortcuts. I know that they could find (a + b)4 the long way, but I want them to notice patterns. My questioning will reflect this intent of the lesson.
Once a majority of the class has gotten through the worksheet, we will regroup and go through the questions. Usually we can get through #1 and #2 very quickly; students don’t usually have many problems on these. For #3, I ask some students how they can find the sixth term of the expansion. The most common response that I usually get is to use Pascal’s Triangle until we get to the appropriate row. My students will usually know that these are the coefficients of the expansion. However, they usually need a reminder that we want the tenth row since the first row is the coefficient for (a + b)0.
Then I ask students if they know any other ways to find the coefficients of the expansion. If there is a student who knows the connection to combinations (nCr), it is usually only one or two. My students have learned combinations in Algebra 2, so I usually have to refresh their memories about the notation and what it means exactly. I talk more about this in this video.
Using combinations is helpful because it gives us an explicit way to find the coefficients of an expansion, rather than the recursive way with Pascal’s Triangle.
Finally, students will make the connection to a main focus of this unit - the sum of a sequence. We can write an expansion in summation notation just like we do with an arithmetic or geometric series. My students did well writing the exponents in summation notation, but they could not write the coefficients since they were using a recursive pattern. Now that we have talked about combinations I see if they can revise their formula.
We also have a discussion about using a summation that starts with 0 versus a summation that starts with 1. It certainly simplifies the formula if we start with 0, so we will go over both approaches.
To summarize this lesson, I will ask a few questions that hit on the main points of expanding a binomial. Here are some suggestions to ask your students to get them thinking about the big ideas of the lesson.
- If (a + b) is raised to the nth power, how many terms will the expansion have?
- What happens to the exponents of a from the first term to the last in the expansion? The exponents of b?
- Why can we use summation notation to represent a binomial expansion?
- Why are the coefficients of expansion symmetrical?
After this wrap-up, I will give a homework assignment from the textbook to give students some practice with this section.