Full Circle Reflection

It’s almost the end of the trimester already which made today my last official “teaching” day with my Algebra class.  I’ve used a lot of the Math Forum’s Problems of the Week in this class.  Since this is a college algebra class, I use the POWS more as problems of the day rather than the week.  As a member, I have access to the library of problems, which I scour quite frequently to find just the right problem to fit the topic at hand.

Today’s adventure:  Rational Functions

I used a POW in which the first four terms of a patterned sequence of A’s and B’s are shown.  The students are asked to create an expression to represent the number of B’s in the nth term and then create an expression to represent the ratio of B’s to the total number of letters in the nth term.  What I like about this task in particular is that it isn’t a completely obvious fraction-ladened, asymptote-wielding, makes-a-student-want-to-cry rational function.  The students are able to work through most of the problem forgetting that this is in fact THAT type of function.  In fact, since they weren’t immediately scared off with a 1/x or the like, it seemed easier for them to make connections from their solutions to the graph and equation of the function.

What was particularly fantastic about this problem was that the growth of these students in the problem solving process was so evident.  It was clear as I circulated the room that over the course of a trimester, these students’ goals as mathematicians were evolving:  from “fast and correct” to “patient and curious.”

For example, when asked to find the term that results in 35% B’s, I had many students make a table with # of A’s, B’s, Total letters, and ratio of B’s to Total letters.  At the beginning of the trimester, these kids would accept their correct answer, but then reject their method of arriving at the answer because it was not as quick as those able to recall an equation or procedural method.  Now, after 13 weeks, these same kids are able to look at their table and appreciate the extra questions they can now address about this pattern scenario.  Additionally, some students were willing to attempt multiple methods in arriving at the answer.    It was a pretty profound moment for them as problem solvers and me as an algebra teacher.  I don’t know who was more proud, them or me.

Here are some samples of their work:


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  1. Y’know, I’ve solved that problem multiple times and I never even thought of it as a rational expressions problem. I’ve always focused on tables, never even thought to graph it… How cool to see the different ways students approached it, and the ways they’ve grown as perseverers and sense makers!!!

  2. Pingback: What We Talk about When We Talk about Thinking . . . Online | NCTM Regional Conferences

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