Today in my Algebra 1 class we were practicing moving between different representations of linear relationships. One of my students was talking with me about her thinking about writing an equation for this table:
My student came to me with the following:
y = x + 4 – x
I think she included the first x for a reason something like (and this is my extrapolation) “you have to start with some x.” At the same time, she could see that the patten was decreasing by 1 each time—and so as x got bigger and bigger by 1, you’d want to take x away.
Her formula, as we soon worked out, was not giving the correct outputs. In fact, it gave 4 each time, as you can see from the example we worked out on the board.
Taking away x once only got us back to where we started, so she had the idea that we could take away x twice:
y = x + 4 – 2x.
Makes sense, and a very fresh-eyed way of looking at things. After that, we simplified the expression to 4-x, and this seemed satisfying to her. She saw how this expression captured her original feeling about starting at 4 and decreasing as x increased.
It was a roundabout way to get to the answer, but illuminating, too—about equivalent expressions, about rates of change, and about my student’s thinking.