For a few blog posts now, I have been singing the praises of Visual Patterns and how they have helped my classes to make concrete connections between an abstract formula and a visible pattern.

Friday was our test on Quadratics.  I put pattern #86 on the test the students had 2 tasks:

1.  Find the number of circles in step #6.  [I wanted a question that most students would do correctly, to keep the barrier to entry low.  Step #5 resulted in the number that math teachers hate the most, so I went with step 6 instead.]

2.  Write a simplified formula for the nth step.

I took multiple pictures of student work to compare their different methods of arriving at the simplified answer of n^2 +8n + 4.  My favorite though (and the one that most represents the growth I’ve been hoping for) is the student that abandoned the algebraic method of creating a system of equations from the table for a more conceptual representation derived from generalizing based on the picture of the pattern.

Here is that student’s work:

I love how the student got most of the way through the algebra and still opted for the other approach.  The student even figured out the “c” value of the quadratic, which is the value that students opting for the algebraic approach struggle with the most.  These students are realizing, after 8 weeks of struggle, that they cannot easily recall since they were usually memorized rather than grounded in conceptual understanding.

Here is another student’s work:

This student came in before school to take the test because she was going to miss the class period due to a field trip.  (which is why her test was printed in color).  Anyway, it took her a while to finish this problem.  I was so proud that she was able to arrive at the correct answer with lots of persistence and without relying on an algebraic algorithm.  At the beginning of the trimester, this is one of the students that would stop when she reached difficulty in a problem and then would wait for someone else to put up the correct answer so she could memorize the method.  I’ve seen her move from that lack of confidence to a place where she is willing to make mistakes so she can formulate her own methods for solving problems.  I’m very impressed with this girls (and this class’s) efforts.