Eight years ago, this XKCD comic crossed my desk, then into my classes, onto my blog, and through my professional development workshops.

That single comic has put thousands of students in a position to encounter the power and delight of the coordinate plane. But I’ve never been happier with those experiences than I was when my team at Desmos partnered with the team at CPM to create a lesson we call Pomegraphit.

Here is how Pomegraphit reflects some of the core design principles of the teaching team at Desmos.

**Ask for informal analysis before formal analysis.**

Flip open your textbook to the chapter that introduces the coordinate plane. I’ll wager $5 that the *first* coordinate plane students see includes a grid. Here’s the top Google result for “coordinate plane explanation” for example.

A *gridded* plane is the formal sibling of the *gridless* plane. The gridded plane allows for more power and precision, but a student’s earliest experience plotting two dimensions simultaneously shouldn’t involve precision or even numerical measurement. That can come later. Students should first ask themselves what it means when a point moves up, down, left, right, and, especially, diagonally.

So there isn’t a single numerical coordinate or gridline in Pomegraphit.

**Delay feedback for reflection, especially during concept development activities.**

It seemed impossible for us to offer students any automatic feedback here. After a student graphs her fruit, we have no way of telling her, “Your understanding of the coordinate plane is incomplete.” This is because there is no *right* way to place a fruit. Every answer could be correct. Maybe this student *really* thinks grapes are gross and difficult to eat. We can’t assume here.

So watch this! We *first* asked students to signal tastiness and difficulty using *checkboxes*, a more familiar representation.

*Now* we know the quadrants where we should find each student’s fruit. So when the student then *graphs* her fruit, on the next screen we don’t say, “Your opinions are *wrong*.” We say, “Your graph and your checkboxes *disagree*.”

Then it’s up to *students* to bring those two representations into alignment, bringing their understanding of both representations up to the same level.

**Create objects that promote mathematical conversations between teachers and students.**

Until now, it’s been impossible for me to have one particular conversation about the tasty-easy graph. It’s been impossible for me to ask one particular question about everyone’s graphs, because the answer has been scattered in pieces across everyone’s papers. But when all of your students are using networked devices using some of the best math edtech available, we can collect all of those answers and ask the question I’ve wanted to ask for years:

“What’s the most controversial fruit in the room? How can we find out?”

Is it the lemon?

Or is it the strawberry?

What will it be in your classes? Find out and let us know.

**2017 Jun 16**. Ben Orlin adds several different graphs of his own. Delete his objects and ask your students to choose and graph their own. Then show Ben’s.