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## Class Opener – Day 73 – You Can’t…Because You Can’t

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.

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## Class Opener – Day 72 – Fermi Questions

Today’s class opener caused a student to string together a sequence of words I don’t think I’ve ever heard in this history of sentences:

Can we please just factor now?

Today I exposed my freshmen classes to Fermi Questions, a series of unusual estimation experiences, like the one we started with below:

The Fermi site here provides a slider where you can change the power of 10 to integer values. Some students had trouble wrapping their heads around the expectation, until some students summarized the ideas quite nicely:

• It has to be in the thousdands.
• I don’t think it could be in the millions.
• Think about how many it would take to go along the side and multiply.

Students really got into the estimates, and I enjoyed listening to them argue their position with neighbors while attempting to estimate unknown quantities. I facilitated the group-think by moving the slider based on loud “higher” or “lower” from the group, until it seemed we were satisfied.  The site then gives you a result and a score based on how close you were.  There are a few thousand questions on the site, and we got through about a dozen today before settling into class.

Some of my favorite questions are those which demand a negative exponent, such as this one:

Determine the diameter of a 22 caliber bullet divided by the length of the Nile river.

Do we think it is one-tenth the length? One hundredth? One millionth? This was a fun way to re-visit laws of exponents, especially negative exponents.

While most of the class was engaged in the discussion, a few shyed away, which led to the quote at the start of this post. Are this questions really so threatening to students that they would RATHER factor? It also plants the seeds for some potential stats data collection, down the road.

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## Class Opener – Day 71 – Factoring Drills

What’s the first rule of factoring?