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.

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Love this… Sometimes I’m tempted to “correct” understandings like this but your student came up with a meaningful and lasting understanding her way.

I think one of the most valuable lessons in teaching that I’ve learned—and continue to learn—is patience. If I can create a safe and supportive space for kids, they so often figure things out on their own.

Thanks for reading, Bowman!