Replacing Synthetic with Real, long division…and other shifts to CCSS

I am thinking most teachers of upper level high school math have at least started to wrap their minds around the profound shift we toward which we are being led by the CCSS and the standardized tests that are also moving in that direction.  I am hoping colleges are, likewise, recognizing and expecting those shifts in our students.  But I sense the specifics are often fuzzy as evidenced by the continued Twitter discussions of which procedures are faster and easier to execute. The progressions docs (page 7) should have clued us in immediately that it was time to say  farewell to synthetic division ; but for some reason the word hasn’t gotten around to everyone yet.

I have replaced memorized formulas for factoring sums and differences of cubes with long division, but there are at least two connections of long division that are left to discuss.

  1.  We can connect long division of polynomials with long division with integers
  2. We can connect long division of polynomials with area models used for factoring (the reciprocal operation).

It is possible to connect polynomial long division with both long division of integers and with area models, but I struggle with doing that well.  Most of my students arrive in my classroom with a very limited understanding of area, perimeter, and area models.  Much of their understanding of math is memorized steps with no understanding of why many of those steps are logical.   Last year I attempted to build area models from the ground up.  It started off well but when we got to factoring, many students reverted back to, “Where do I put this number?  Does it go in there or out here?”   We finished fine but I had many sleepless nights worrying about whether trying to build math sense with area models was too difficult.

My conclusion was that area models are perfect for kids who relate to them from kindergarten arrays all the way through as there is a solid progression of understanding, and the consistent flow should reap phenomenal retention.  However, with poor vertical alignment of area models through lower grades, starting from scratch was too challenging for my students given the time that we had together.

What has your experience been?  How are you progressing with the shifts to connected, extensible math that makes sense in the upper high school levels?  I know the main issue is finding textbooks that will walk us through. is helping with that, but it takes awhile to filter down into the classrooms.  I would love to hear from upper level mathematics who have made intentional moves toward meeting the modeling standards and connecting topics in the upper levels of high school math.


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