As I was presenting the remainder theorem yesterday, I stopped the lesson before the last slide where I show that that if you plug a from divisor (x-a) into the function, the value of the function is the remainder when divided by (x-a)…saying that backwards 3 times just for effect (not really, but…). What am I doing?!? My kids feel “ahhhhhh” when they see the relationship between “zero remainder” and the fact that the divisor is a factor of the dividend. Also the quotient is a factor of the dividend. That is interesting, but who the heck would go through long division to discover what f(a) is? In previous years, my students were kind to not have called me out on that.
One of the faults of Common Core math, in my opinion, is they never tell us what they left out. So I dove into the progressions documents for APR.2 and found this:
It is important to regard the Remainder Theorem as a theorem, not a technique. A consequence of the Remainder Theorem is to establish the equivalence between linear factors and zeros that is the basis of much work with polynomials in high school.
The rest of the paragraph explains further connections that can be built in very precise mathematical notation, but important thing is for students to have a conceptional understanding of what a factor is. For example, my students know that 27/5 = 5 with a remainder of 2. Now they see that p(x) = (x-a)q(x) + r. In My point is that if we are going to build relevance, we need to stop while the students are saying “Ahhhhhhhhhh.” Not only does this save time to explore modeling problems as intended by the standards, but it also helps maintain credibility for relevance in the eyes of our students.
This time I felt really good about the lesson. I could see my students found the information fairly interesting and felt it was reasonable to explore the idea further in their homework problems.