I start this lesson with the following expressions on my front board:(x+y)^2 (x+y)^3 (x+y)^4
I let my students discuss strategies and then ask them to individually expand these expressions. (MP1) Most students can do this fairly easily although the fourth degree expansion may take a bit more time for some and is more likely to result in arithmetic and algebraic errors. I find that some students really don't have any tools for multiplying polynomials other than FOIL, so this is a good opportunity to help them add to their mathematical toolbox as they see the variety of strategies their classmates use. As they finish, I ask for volunteers write out their solutions on the board, including all the intermediate steps they used. This gives me a chance to reinforce the fact that the distributive property works for polynomials. When all the solutions are posted I ask my students to review them for accuracy. I then put up two more difficult polynomial expansions (x+y)^9 (x+y)^12 and ask if anyone wants to work these out on the board for us. If I get any volunteers, I let that individual work through the ninth degree expansion, commenting all the while about how long it's taking and how much work it is. (Not as a criticism of the student, but rather a complaint about the problem - I make this clear to the class while I'm griping.) When he/she is done I thank him/her and ask if anyone would like to know an easier way. All this work and my complaining has hopefully set my students up to be ready for an easier way even if it means learning something new.
You will want a copy of Pascal's triangle to show the class for this section of the lesson. This Educreations video shows students how to make their own Pascal's Triangle and also discusses some cool patterns in the Triangle. My Two for One video explains why I have them work through all the expansions first. I've already posted two more difficult polynomial expansions on the board and worked through at least one of them. My students are now ready to see a new approach for these so I tell them about Blaise Pascal who published his first mathematical piece at 16. I explain that Pascal's father had quite a bit of arithmetic to do as a tax collector so Blaise invented a mechanical calculator to make adding and subtracting easier. I go on to say that he also created a table to make working with polynomial expansions easier. I show Pascal's Triangle and ask my students to compare the numbers in the third, fourth and fifth rows to their work. I expect at least a few students to notice the coefficients match and to share that with the class. About this time someone will also compare the numbers in row ten with the ninth degree polynomial and observe that those also match up. Generally there is great appreciation for this triangle as well as a bit of frustration with the effort put in earlier to find the same coefficients. (MP7) To those who are frustrated I reply that they wouldn't have necessarily believed me if I'd simply said that Pascal's triangle gave the coefficients for any polynomial expansion. I go on to demonstrate to my students how they can generate their own triangle any time they need it to as many rows as they need.
I then tell my students that they will get to work individually to apply Pascal's Triangle to more polynomials expansions. (MP1) I ask them to write the original polynomial and then put the expansion equal to it, so I can see which ones are which. The polynomials I use are:
(x+y)^15 (x+y)^7 (x+y)^22 (x+y)^14
but you can change these, add more or reduce the number depending on your class.
To close this lesson I ask my students to write a rule generalizing the results of their work today in the form (x+y)^n = ... This provides an opportunity to expand their thinking beyond just individual problems to the overall patterns of what binomial expansions are. (MP8)
If this seems to open-ended for your students you might ask them to write a short paragraph describing how they would use Pascal's Triangle and the Binomial Theorem to justify these statements: (x + y)1 = x + y AND (x + y)0 = 1