This is beautiful, right? Put enough straight lines in the right places and your eyes see a curve.
How many linear equations did the student use to create it? You might start counting lines and assume it required dozens. For some students, you’d be right. They typed 40 linear equations and corrected a handful of typos along the way.
But other students created it using only four linear equations and many fewer errors!
The seventh mathematical practice in the Common Core State Standards asks students to “look for and make use of structure.” The second half of that standard is a heavier lift than the first by several hundred pounds.
Because it’s easy enough for me to ask students, “What structures do you notice?” It’s much more difficult for me to put them in a situation where noticing a mathematical structure is more useful than not noticing that structure.
First, we ask students to write the linear equations for a couple of parallel lines.
Then four lines. Then nine lines.
It’s getting boring, but also easy, which are perfect conditions for this particular work. A boring, easy task gives students lots of mental room to notice structure.
Next we ask students, “If you could write them all at once – as one equation, in a form you made up – what would that look like?” Check out their mathematical invention!
Next we show students how Desmos uses lists to write those equations all at once, and then students put those lists to work, creating patterns much faster and with many fewer errors than they did before. With lists, you can create nine lines just as fast as ninety lines.
What are the four equations that created this graph? Personally, I find it almost impossible to discern by just looking at the graph. I have to write the equation of one of the lines. Then another. Then another. Then another, until that task becomes boring and easy. Only then am I able to notice and make use of the structure.