A few weeks ago, Kate Nowak posted on her blog that she was frustrated at the lack of online resources to support the teaching of the binomial expansion. She threw out a challenge asking people to create a better way to teach the binomial expansion.
Here’s her challenge:
“Objective : Present the binomial expansion in a way that makes sense. Bonus points if students are able to as a result completely expand a power of a binomial and find a specific term in an expansion.”
Our revised curriculum in Alberta includes a pedagogical shift that asks teachers to create opportunities for students to establish their own meaning through exploration, problem solving, investigation and developing personal strategies. One of the common concerns I hear from teachers is that we can’t teach higher level mathematics this way, because the material is so hard that students won’t be able to construct their own meaning.
Kate’s challenge gave me an opportunity to try to teach in this manner in one of our academic courses. In Alberta, binomial expansion shows up in a pre-calculus course called Math 30 Pure. Students go from that course into Math 31, which is a calculus course covering limits, derivatives, application of derivatives, and basic integration. Binomial expansion, and the binomial theorem are two pretty dry topics. I took the challenge and attempted to create a lesson that was both engaging, and allowed students to construct their own meaning through looking at patterns. A colleague was nice enough to let me try the lesson in his class.
My lesson went over fairly well with the students, and I think I met all of Kate’s objectives. The introduction/hook was more compelling than anything I had done in the past on that topic. Students were able develop the patterns in the binomial expansion on their own, and were able to apply those patterns to expand more complicated binomials. What surprised me was that they were able to come up with the formula to find any term in an expansion all on their own.
I’d love to hear some feedback on the lesson.
An 11 minute compressed version of the lesson can be watched here: Binomial Expansion Lesson
If you’d prefer to just read the lesson plan, here it is.
Introduction: I showed them a mathemagician video from TED.
Hook: I claimed that I was a mathemagician, too. I wrote (a + b)^2 on the board, and asked them to expand it. Then I wrote (a + b)^3 on the board and had them expand it. Then I wrote (a + b)^4 and (a + b)^5 and challenged them to a race. They could use friends, pencil, paper, calculators or whatever they wanted, and I would just use my brain. I pretended to struggle, and then wrote down the answers as quickly as I could. I had to hurry because one kid was darn quick, and was getting at (a + b)^4 by multiplying the answer to (a + b)^3 by (a + b) rather than expanding the whole thing as I had expected them to do. If you watch the video, we appear to tie on that one, but he already started while I was blabbering.
Students look for patterns: I explained that I am not really that smart, and that I was cheating using a pattern. I wrote “In the expansion of (a + b)^n, ” on the board, and asked them to spend a few minutes together coming up with ways to complete that statement. They got that each term had the same degree as the exponent on the binomial, and that there were n+1 terms in the expansion. It took a little longer and some direction from me for them to notice that the a’s started at a^n and decreased to a^0, while the b’s did the same thing in the opposite direction. One girl put Pascal’s triangle on the board when she noticed the pattern of coefficients.
Expand a binomial: I asked them to use all that they had learned to expand (x + 2y)^5. They worked together and there was much discussion about how to handle the 2y part. They managed to figure it out.
Develop a formula for a general term: I gave them one like (x + 2y)^12 and asked them if they could figure out the 8th term without writing out the first 7. I gave them three blanks to fill in: coefficient, “a”, and “b”. They got the a and b part just fine, but struggled with whether the coefficient should be 12 choose 7 or 12 choose 8. They figured it out by counting. I told them that they had just figured out the formula. I wrote the formula tk+1 = nCk x^(n-k) y^k on the board.
(At this point I fell back into my old bad lecture style, and did two examples with them using the formula, rather than making them do it themselves.)
Example: Find the term containing y^7 in the expansion of (x – 3y)^7.
Example: Find the constant term in the expansion of (3x – 2/x^5)^12.