Sketch a smaller square inside the given square so that the area of the smaller square is half the area of the larger square. Write an explanation to convince another that your new, smaller square is truly half the area of the original square.

I’m sure that I didn’t write this question, but I don’t remember from where it came.

What would your students get right?

What misconceptions might they have?

I was surprised at some of my students’ misconceptions (and miscalculations) and explanations.

Measuring errors.

Calculation errors. Lots of “simple” calculation errors: Half of 36 is 16. Half of 25 is 13.25. Half of 5 is 3.5.

More measuring errors.

Square vs. rectangle errors.

And someone getting at the immeasurability of irrational numbers.

The test was summative. But even though the test was summative in my gradebook, I can still use the responses to inform the learning experiences that I give my students in the future. And so I will, as the journey continues …