Last class we introduced the binomial theorem. My approach was to have them expand \sum_{k=0}^{n} n_C_k x^n y^{n-k} without them knowing that. I explained it to them this way: your job is to identify n (the power to which the binomial is being raise). Example: (3a -2b)^4. Start by seeing that n=4 and start writing 4_C_0 (3a)^4(-2b)^0 + 4_C_1 (3a)^3(-b^1) + ... This was prefaced by a discussion about ascending and descending terms so they would know what it meant for the 3a terms to descend and for the -2b terms to descend. That was prefaced by expanding (a+b)^n for n=0,1,2,3 to see that ascending and descending terms naturally occur in binomial expansions.
The next day I assign two problems for a warm-up:
(1) (2a - 3b)^4
(2) (5x + y^2)^5
Most students make the error of expanding (1) without the correct parentheses around the 2a and -3b terms which I correct when they come to show me their work. They go on to successfully expand (2).
Several of my better students do (1) correctly but complain that I had never "taught" them how to do (2).
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