Here Are Ten New Desmos Activities

The Desmos quarter just ended and this was a huge one for the team of teachers I support.

First, we made substantial upgrades to our entire activity pool. Second, we released ten new activities in the same amount of time it took us to release one activity two years ago. This is all due to major improvements to our technology and our pedagogy.

Technologically, our engineers created a powerful scripting language that hums beneath our activities, enabling us to set up more meaningful interactions between teachers, students, and mathematics.

Pedagogically, my teaching team has spent the last year refining our digital mathematics pedagogy through daily conversations, lesson pitches, lesson critiques, summary blog posts, occasional lunch chats with guests like the Khan Academy research team, and frequent consultation with our Desmos Fellows.

The result: we cut an activity pool that once comprised 300 pretty good activities down to 127 great ones, and we gave each one of those 127 a serious upgrade, making sure they took advantage of our best technology and pedagogy. Then we added ten more.

I don’t think I’ve learned as much or worked as hard in a three-month span since grad school, and I owe a debt of gratitude to my team – Shelley Carranza, Christopher Danielson, and Michael Fenton – for committing the same energy. Also, it goes without saying that none of our activity ideas would have been possible without support from our engineers and designers.

In future posts, I’ll excerpt those lessons to illustrate our digital pedagogy. For today, I’ll just introduce the activities themselves.

Picture Perfect

Hang loads of pictures precisely and quickly using arithmetic sequences.

Game, Set, Flat

Your shipment of tennis balls has been contaminated. Use exponential functions to find the bad ones.

Graphing Stories

Graphing Stories comes to Desmos just in time for its tenth birthday.

Pool Border Problem

One of the oldest and best problems for exploring algebraic equivalence. We wouldn’t have touched it if we didn’t think we had something to add.

Laser Challenge

Use your intuition for angle measure to bounce lasers off mirrors and through targets.

Lawnmower Math

Use Algebra and the properties of circles to help you mow ten lawns automatically and quickly.

Land the Plane

Use linear equations to land airplanes safely and precisely.

Circle Patterns

Practice circle equations by completing artistic patterns.

Constructing Polynomials

Develop your understanding of the behavior of polynomial graphs by creating them piece by piece, factor by factor.

What’s My Transformation?

This is my favorite introduction to the concept of a transformation. “Actually, there’s really only one parabola in the world – we just move it around to make new ones.”

We are still testing these activities. They are complete, but not complete complete, if you know what I mean. You won’t find all of them in our search index yet. We welcome your feedback.

[Pseudocontext Saturdays] Tornado!

This Week’s Installment


What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #10

  • Calculating probabilities of independent events (69%, 238 Votes)
  • Interpreting bar graphs (20%, 70 Votes)
  • Calculating area of parallelograms (11%, 38 Votes)

Total Voters: 346

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(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Team Me: 5
Team Commenters: 4

Pseudocontext Submissions

William Carey has offered two additional genres of pseudocontext that are worth your attention:


One motif in pseudocontextual questions seems to be treating as a variable things that, you know, don’t vary.


The car question follows a fascinating pattern that shows up in lots of physicsy work: it begs the question. Physicists like to measure things. Sometimes measuring something directly is tricky (or impossible), so we measure other things, and then calculate the thing we actually want.

Questions like that have as their givens the thing we can’t measure and ask us to calculate the thing that we can measure. It’s absolutely backwards.


Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)


The commenters bit down hard on the lure this time, folks. The correct answer – “calculating area of parallelograms” – was selected least.

Delicious pseudocontext, right? The judges all suffered massive strokes when they saw this problem so I couldn’t get their official ruling, but I don’t think it matters. This context fails the “Come on, really?” test for pseudocontext.

“This unpredictable force of nature is threatening a precisely-bounded parallelogram? Come on, really?”

How could we neutralize the pseudocontext? I would be thrilled to see a task that invited students to select and approximate important regions with various quadrilaterals, but let’s not lie about where our tools are useful.

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.

My Winter Break in Recreational Mathematics

Chase Orton asked all of us, “What is your professional New Year’s resolution?

I said that I wanted to stay skilled as a math teacher. As much as I’d like to pretend I’ve still got it in spite of my years outside of the classroom, I know there aren’t any shortcuts here: I need to do more math and I need to do more teaching. I have plans for both halves of that goal.

In support of “doing more math,” I’ll periodically post about my recreational mathematics. Please a) critique my work, and b) shoot me any interesting mathematics you’re working on.

When Should You Bet Your Coffee?

Ken Templeton sent me an image from his local coffee shop.

Should you bet your free coffee or not? Under what circumstances?

This question offers such a ticklish application of the Intermediate Value Theorem:

If the bowl only has one other “Free Coffee” card in it, you’d want to bet your own card on the possibility of a year of free coffee. But if the bowl had one million cards in it, you’d want to hold onto your card. So somewhere in between one and one million, there is a number of cards where your decision switches. How do you figure out that number? (PS. I realize the IVT doesn’t hold for discrete functions like this one. Definitions offer us a lot of insight when we stretch them, though.)

I asked some of my fellow New Year’s Eve partygoers this question and one person offered a concise and intuitive explanation for her number, a number I personally had to calculate using algebraic manipulation. Someone else then did his best to translate some logistical and psychological considerations into mathematics. (eg. “Even if I win it all, I won’t likely go get a drink every day. Plus I’m risk averse.”) It was such an interesting conversation. Plus what great friends right?

Here’s my work and the 3-Act Task for download.

How Many Bottles of Coca-Cola Are in That Pool?

When I watched this video, I had to wonder, “How many bottles of Coca Cola did they have to buy to fill that pool?”

I tweeted the video’s creator and asked him for the dimensions of the pool.

12 feet across by 30 inches high,” he responded.

Even though the frame of the pool is a dodecagon, the pool lining itself seems roughly cylindrical. So I calculated the volume of the pool and performed some unit conversions to figure out an estimate of the number of 2-liter bottles of Coca Cola he and his collaborators would have to buy.

Here’s my work and the 3-Act Task for download.

How Do You Solve Zukei Puzzles?

Many thanks to Sarah Carter who collected all of these Japanese logic puzzles into one handout.

Carter describes the puzzles as useful for vocabulary practice, but I found myself doing a lot of other interesting work too. For instance, justification. The rhombus was a challenging puzzle for me, and this answer was tempting.

So it’s important for me to know the definition of a rhombus – every side congruent – but also to be able to argue from that definition.

And the challenge that tickled my brain most was pushing myself away from an unsystematic visual search towards a systematic process, and then to write that process down in a way a computer might understand.

For instance, with squares, I’d say:

Computer: pick one of the points. Then pick any other point. Take the distance between those two points and check if you find another point when you venture that distance out on a perpendicular line. If so, see if you can complete the square that matches those three points. If you can’t, move to the next pair.


  • What are your professional resolutions for 2017?
  • What recreational mathematics are you working on lately?
  • If any of you enterprising programmers want to make a Zukei puzzle solver, I’d love to see it.

2017 Jan 2. Ask and ye shall receive! I have Zukei solvers from Matthew Fahrenbacher, Jed, and Dan Anderson. They’re all rather different, each with its own set of strengths and weaknesses.

2017 Jan 2. Shaun Carter is another contender.

What’s Wrong with This Experiment?

If you’re the sort of person who helps students learn to design controlled experiments, you might offer them W. Stephen Wilson’s experiment in The Atlantic and ask for their critique.

First, Wilson’s hypothesis:

Wilson fears that students who depend on technology [calculators, specifically –dm] will fail to understand the importance of mathematical algorithms.

Next, Wilson’s experiment:

Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didn’t. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.

I trust my readers will supply the answer key in the comments.

BTW. I’m not saying there isn’t evidence that calculator use will inhibit a student’s understanding of mathematical algorithms, or that no such evidence will ever be found. I’m just saying this study isn’t that evidence.

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Scott Farrand:

The most clarifying thing that I can recall being told about testing in mathematics came from a friend in that business: you’ll find a positive correlation between student performance on almost any two math tests. So don’t get too excited when it happens, and beware of using evidence of correlation on two tests as evidence for much.