In the past few days, my 9th graders have worked through a chapter on polynomials: multiplying, factoring, solving, simplifying. There’s a lot of process here, and often my fear is that students attempt to memorize short-cuts (such as the old stand-by…FOIL) without fully understanding the reasons WHY procedures are valid. It’s an easy “out” to tell students they will need procedures for their next class – I drink from this well sometimes – but I need something more my students. I want them to be able to clearly articulate and verify, using precise vocabulary, the rationale for all steps they take in math class.
In today’s opener students were presented with two problems on the board I had “solved”, and were asked to comment on my procedures:
The problem on the left is one we had completed yesterday in class, and a number of students noticed that one of the solutions, zero, was missing. I asked students to identify reasons why my solution gave a different solution set:
Because you don’t get the right answer.
Because zero is supposed to be an answer.
We’re not quite getting to the heart of the matter….I asked students to look over my solution to the problem on the right and comment.
You have to subtract the 5 over first.
You need to set it equal to zero.
In each case, students were fixated more on what I should have done, rather than what was presented in front of them as a solution. Time to re-direct the conversation – I asked students to think about each step I had done in the problems, and tell me specifically which step was in error. This is a much more uncomfortable experience. For each problem, the steps “feel” right. In both of my classes, the breakthrough eventually came, with some coaxing:
Students: You set both factors equal to 5. You need to set them equal to zero.
Me: Why can’t I set them equal to 5? The equation equals five.
Students (eventually): Because if two things multiply to make zero, one of them must equal zero.
Now we are getting someplace. The zero-product property is often taken for granted in this unit, but it is a powerful little engine. Name two numbers which multiply to make a product of 5….is it guaranteed that one of the two numbers MUST be 5? Nope. Zero is the hero. Hoepfully, some new conenctions were made regarding the nature of zeroes here.
The problem on the left was a much tougher nut to crack. The conversation eventually focused on the “other” solution – zero – and the perils of dividing by zero. Definitely look for more “devil’s advocate” moments as we explore rational expressions further.