Category: lessons

Total 94 Posts

Pomegraphit & How Desmos Designs Activities

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.

It’s yours to use.

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.

Plates Without States

Hey history teacher-friends!

Lately, I’ve been interested in the math teaching opportunities that arise when we delete and then progressively reveal details of a task. Digital media offers us that luxury while paper denies it.

I saw an opportunity to apply the same approach in history and geography. I took the license plates from all fifty United States and removed explicit references to the state or its outline. Then Evan Weinberg turned it into an online quiz.

Feel free to send your students to that quiz, or to use the images themselves [full, deleted, animated] in any way you want. If you’re feeling obliging, stop by and let us know how it went in the comments.

What Do You Do on the First Day of School?

I know. I know. Too early, right? But Ali Grace is a go-getter.

My contributions:

Help the rest of us out in the comments. What do you do on the first day of school?

2016 Jul 27. A Collection of First Week Activities.

2017 Aug 21. Sarah Hagan has 21 ideas for the first week of school.

2017 Aug 21. YouCubed has its Week of Inspirational Math.

2017 Aug 21. Sara Van Der Werf’s 100 Numbers to Get Students Talking.

The Money Animal Marketplace Was The Most Fun I Had Doing Math This Summer

In my modeling workshops this summer, we first modeled the money duck, asking ourselves, what would be a fair price for some money buried inside a soap shaped like a duck? We learned how to use the probability distribution model and define its expected value. We developed the question of expected value before answering it.

Then the blogosphere’s intrepid Clayton Edwards extracted an answer from the manufacturers of the duck, which gave us all some resolution. For every lot of 300 ducks, the Virginia Candle Company includes one $50, one $20, one $10, one $5, and the rest are all $1. That’s an expected value of $1.27, netting them a neat $9.72 profit per duck on average.

That’s a pretty favorable distribution:


They’re only able to get away with that distribution because competition in the animal-shaped cash-containing soap marketplace is pretty thin.

So after developing the question and answering the question, we then extended the question. I had every group decide on a) an animal, b) a distribution of cash, c) a price, and put all that on the front wall of the classroom – our marketplace. They submitted all of that information into a Google form also, along with their rationale for their distribution.


Then I told everybody they could buy any three animals they wanted. Or they could buy the same animal three times. (They couldn’t buy their own animals, though.) They wrote their names on each sheet to signal their purchase. Then they added that information to another Google form.

Given enough time, customers could presumably calculate the expected values of every product in the marketplace and make really informed decisions. But I only allowed a few minutes for the purchasing phase. This forced everyone to judge the distribution against price on the level of intuition only.

During the production and marketing phase, people were practicing with a purpose. Groups tweaked their probability distributions and recalculated expected value over and over again. The creativity of some groups blew my hair back. This one sticks out:


Look at the price! Look at the distribution! You’ll walk away a winner over half the time, a fact that their marketing department makes sure you don’t miss. And yet their expected profit is positive. Over time, they’ll bleed you dry. Sneaky Panda!

I took both spreadsheets and carved them up. Here is a graph of the number of customers a store had against how much they marked up their animal.


Look at that downward trend! Even though customers didn’t have enough time to calculate markup exactly, their intuition guided them fairly well. Question here: which point would you most like to be? (Realization here: a store’s profit is the area of the rectangle formed around the diagonal that runs from the origin to the store’s point. Sick.)

So in the mathematical world, because all the businesses had given themselves positive expected profit, the customers could all expect negative profit. The best purchase was no purchase. Javier won by losing the least. He was down only $1.17 all told.

But in the real world, chance plays its hand also. I asked Twitter to help me rig up a simulator (thanks, Ben Hicks) and we found the actual profit. Deborah walked away with $8.52 because she hit an outside chance just right.


Profit Penguin was the winning store for both expected and actual profit.

Their rationale:

Keep the concept simple and make winning $10s and $20s fairly regular to entice buyers. All bills – coins are for babies!

So there.

We’ve talked already about developing the question and answering the question. Daniel Willingham writes that we spend too little time on the former and too much time rushing to the latter. I illustrated those two phases previously. We could reasonably call this post: extending the question.

To extend a question, I find it generally helpful to a) flip a question around, swapping the knowns and unknowns, and b) ask students to create a question. I just hadn’t expected the combination of the two approaches to bear so much fruit.

I’ve probably left a lot of territory unexplored here. If you teach stats, you should double-team this one with the economics teacher and let me know how it goes.

This is a series about “developing the question” in math class.

Personality Coordinates Icebreaker

With school starting up, I thought I’d share the most interesting icebreaker I found last year. Copy, cut, and pass out these half-sheets [pdf].


Each person in a group picks a dot and writes her name next to it.

Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.

That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.

Previously. This Who I Am sheet, which I adapted from my first student-teaching placement, has been popular.

[I got this particular idea from a workshop I led with Jo Boaler, Kathy Sun, Jennifer Ruef. It may be a Complex Instruction staple for all I know. I’m not claiming ownership, just passing along the fun.]

2013 Aug 7. I am informed by Marty Joyce this icebreaker is from the College Preparatory Mathematics series.

2013 Aug 9. Rachel Rosales used the Math Forum’s Noticing / Wondering framework on her first day.

Featured Comments

Laura Hawkins:

I learned this from Carlos Cabana, at the Creating Balance conference at Mission HS in SF. I’ve used it with much success for a couple years now.

Be aware, however, that it can surface issues you might not be prepared to deal with. I teach at a private school, and there is huge income inequality between my kids’ families. This year, a group labeled one axis “number of bathrooms in my house” with the two quantities being 4 and “more than 4″. In my surprise, I don’t think I handled it well to support the kids in the class who might have just 1 or 2 bathrooms in their house and suddenly had their lack of wealth put in their face in math class.

Cathy Yenca:

I’ve used your “Who I Am” sheet for several years – after reviewing them, I file them away, then surprise students on the last day of school by returning them. They giggle at how much they’ve changed in 9 short months, often barely remembering filling out the page in the first place. The “self-portraits” are always a favorite to revisit.

Nathan Amrine:


Desire to read
# of hours spent swimming
# of summer vacations
Love of Ice Cream
# of pets
Distance from school

2013 August 13. First day activities around the web from:

Featured Email

Joey Warren, with a nice modification:

I thought I would share with you an activity that I stole from you and adjusted a bit which worked out very well for my first day freshmen Alg students. I read your Personality Coordinates Ice breaker activity and made a little twist. I put the x and y axis on the ground and used the classroom for students to move around. I gave them different labels for the x-axis and y-axis and let them move around the room and meet each other and see who was similar to them and who was different. I used labels like desire to read, distance traveled this summer, and # of hours you sleep each night. It lead into some good movement and good discussions of where students traveled over the summer, what they read, etc. The best graph was “love for math” vs “desire to get better at math.” After they moved around I categorized students into ones that were going to be my project to get them to like math, ones that would be easy to teach, and the rest I was going to push further away from the origin to get them to love math more.