Here is how your unit on linear equations might look:

- Writing linear equations.
- Solving linear equations.
- Applying linear equations.
- Graphing linear equations.
- Special linear equations.
- Systems of linear equations.
- Etc.

On the one hand, this looks *totally normal*. The study of the linear functions unit *should* be all about linear functions.

But a few recent posts have reminded me that the linear functions unit needs also to teach *not linear functions*, that good instruction in *[x]* means helping students differentiate *[x]* from *not [x]*.

Ben Orlin offers a useful analogy here:

If I were trying to teach you about animals, I might start with cats and dogs. Theyâ€™re simple, furry, familiar, and lots of people have them lying around the house. But Iâ€™d have to show you some other animals first. Otherwise, the first time you meet an alligator, youâ€™re gonna be like, â€œThat wet green dog is so ugly I want to hate it.â€

Michael Pershan then offers some fantastic prompts for helping students disentangle rules, machines, formulas, and functions, all of which seem totally interchangeable if you blur your eyes even a little.

Not all rules that we commonly talk about are functions; not all functions are rules; not all formulas have rules; not all rules have machines. Pick two: not all of one is like the other. A major goal of my functions unit to help kids separate these ideas. So the very first thing I do is poke at it.

And then I was grateful to Suzanne von Oy for tweeting the question, “Is this a line?” a question that is both rare to see in a linear functions unit (where *everything* is a line!) and important. Looking at *not lines* helps students understand *lines*.

So I took von Oy’s question and made this Desmos activity where students see three graphs that look linear-*ish*. The point here is that not everything that glitters is gold and not everything that looks straight is linear. Students first make their predictions.

Then they see the graphs again with two points that display their coordinates. Now we have a reason to check slopes to see if they’re the same on different intervals.

Finally, we zoom out to check a larger interval on the graph.

I’m sure I will need this reminder tomorrow and the next day and the next: teach the controversy.

**BTW**. In addition to being good for learning, controversy is also good for curiosity.

**Bonus**. Last week’s conversation about calculators eventually cumulated in the question:

“Calculators can perform rote calculations therefore rote calculations have no place on tests.” Yay or nay?

I’ve summarized some of the best responses â€“ both yay and nay â€“Â at this page. (I’m a strong “nay,” FWIW.)