Waterline & Taking Textbooks Out Of Airplane Mode

tl;dr – This is about a new digital lesson I made with Christopher Danielson and our friends at Desmos. It’s called Waterline and its best feature is that it shares data from student to student rather than just from student to teacher. I’ll show you what I mean while simultaneously badgering publishers of digital textbooks. (As I do.)

Think about the stretches of time when your smartphone or tablet is in airplane mode.

Without any connection to the Internet, you can read articles you’ve saved but you can’t visit any links inside those articles. You can’t text your friends. You can’t share photos of cats wearing mittens or tweet your funny thoughts to anybody.

In airplane mode, your phone is worth less. You paid for the wireless antenna in your tablet. Perhaps you’re paying for an extra data plan. Airplane mode shuts both of them down and dials the return on those investments down to zero.

Airplane mode sucks.

Most digital textbooks are in airplane mode:

  • Textbooks authored in Apple’s iBooks Author don’t send data from the student’s iPad anywhere else. Not to her teacher and not to other students.
  • HMH Fuse includes some basic student response functionality, sending data from the student to the teacher, but not between students.
  • In the Los Angeles Unified iPad rollout, administrators were surprised to find that “300 students at three high schools almost immediately removed security filters so they could freely browse the Internet.” Well of course they did. Airplane mode sucks.

The prize I’m chasing is curriculum where students share with other students, where I see your thoughts and you see mine and we both become smarter and life becomes more interesting because of that interaction. That’s how the rest of the Internet works because the Internet is out of airplane mode.

Here’s one example. In Waterline we ask students first to draw the height of the water in a glass against time. We echo their graph back to them in the same way we did in Function Carnival.


But then we ask the students to create their own glass.


Once they successfully draw the graph of their own glass, they get to put it in the class cupboard.


Now they see their glass in a cupboard right alongside glasses invented by their friends. They can click on those new glasses and graph them. The teacher sees all of this from her dashboard. Everyone can see which glasses are harder to graph and which are easier, setting up a useful conversation later about why.

We piloted this lesson in a local school and asked them what their favorite part of the lesson was. This creating and sharing feature was the consensus winner.

A selection:

  • Making my own because it was my own.
  • Trying to create your own glass because you can make it into any size you want.
  • Designing my own glass because I was able to experiment and see how different shapes of the glass affects how fast the glass filled up.
  • My favorite part of the activity was making my own glass and making my other peers and try and estimate my glass.
  • My favorite part of the activity was solving other people’s glasses because some were weird shapes and I wanted to challenge myself.

Jere Confrey claimed in her NCSM session that “students are our most underutilized resource in schools.” I’d like to know exactly what she meant by that very tweetable quotation, but I think I see it in the student who said, “I also liked trying out other’s glasses because we could see other’s glasses and see how other people solved the problem.”

I know we aren’t suffering from too many interactions like that in our digital curricula. They’re hard to create and they’re hard to find. I also know we won’t get more of them until teachers and administrators like you ask publishers more often to take their textbooks out of airplane mode.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Dying here – this is fantastic!

    My next question is… how can teachers whose names aren’t Dan Meyer or Christopher Danielson begin creating more lessons like this?

    So far, my tool of choice has been Nearpod for creating my own goodies. While sharing instant work samples through Nearpod checks the boxes of teacher student sharing and student student sharing, activities like Waterline add a level of student buy-in that existing digital tools don’t yet provide.

    Also, using graphs to model changes over time works well in Waterline and Function Carnival, but how can we begin creating such digital learning experiences with content besides graphs over time? I’d love to see something like this happening at least once per unit, versus concentrating all the cool stuff for interpreting graphs.

    Thanks for creating, and sharing!

  2. Awesome!!! love it. The best part is the creating your own. I can see students competing with each other…..”I bet you can’t match me graph” Doing this!

  3. What a great way to connect functions to real life! Did you create the animation/program yourself? I have done this w/kid using paper/pencil (have them draw a the graph of a function and write its corresponding “story” to explain what is happening over time).

  4. This bottle-graphics activity is older. You can see here [http://www.xtec.cat/~aaubanel/Fitxes/F22.pdf] (in catalan) the non-virtual version of the activity.

    Anton Aubanell is a passionate of mathematics, teacher of mathematics also. He has many physical resources for teaching old things: pi, hipsòmetre, etc. See that [http://www.xtec.cat/~aaubanel/] (also in catalan).

    Sergi del Moral is a disciple [http://www.sergidelmoral.net/]

    Just for info.

    At the end, all the knowledge is reused ;-)

  5. @Xavier, one challenge for us is to recognize that those who came before have done a lot of valuable work that we should learn from. The challenge for those who came before is to recognize that new technology offers mathematical experiences to students that weren’t possible earlier.

    @Zach, for a constant flow rate, aren’t those the same question? We decided that height vs. time is less abstract to beginning algebra students than volume vs. height.

  6. Awesome how you’ve taken the idea behind graphingstories, added a new tool and collaboratively, and BAM! you get function carnival and now this. Will definitely be using it.

  7. Unfortunately, part of the issue is fear of lawsuits when parents sue the school for something bad a kid did on the Internet while he was at school. Schools are so restricted by needing to monitor everything kids do at school that it is easier to limit accessibility.

  8. You changed the rate of the water flow so the glasses always take the same amount of time to fill. To simplify? It’s nice to get an average slope of 1, I imagine, to keep the students from having an awkwardly steep graph, but did you see any confusion from students comparing answers for a thin straight glass and a thick straight glass?

    It might be more interesting to them that the slope is the same, the way you did it, as a kind of “forget units, we’re just talking about relative rates of change here” way.

    Do you think it will be a confusing surprise later when they realize that the water is pouring out at different rates?

  9. Oh, also, this is really, really interesting. Thanks for doing this work and sharing it! I have been aiming a little simpler and more general but your (Dan, Christopher, & Desmos’) work is inspiring.

  10. Once again, thank you for your creativity. I can see lots of applications for a program like this, from beginning algebra to calculus. Of course, there are similar activities out there. But what I love about what you’ve done here is the sharing of student designs. I can definitely see some students trying to create very challenging designs.

    If you have this piece of student sharing figured out, it would be great if the critique on the graphs could also come from within the class. This would allow students to give real feedback to each other, who then might want an opportunity to go back and revise their graphs.

    The teacher dashboard of information is nice to have around after class, but difficult to monitor while students are working and problem solving. So having students give each other some peer critiques will help with that monitoring – tapping into student expertise, not only to create more examples, but also to help with understanding concepts.

  11. Awesome. Dan, I remember you discussing this idea a year ago – very glad it’s seen fruition. The simplicity of the glass drawing screen is genius.

    Also, I like that you guys chose not to show a score or error calculation. I can just see how close my line is to the waterline, and judge myself whether the graph needs improvement or is good enough.

  12. Awesome!

    I look forward to using this work, the other desmos projects you have been working on and graphing stories in my curriculum. I can’t think of a better source for time – distance/etc graphs.

    The only thing I might add into the mix is logger pro and their motion detectors to add physical motion to the experience.


  13. I’d like to know what percentage of high school students will end up drawing cups that looks like bongs or whatever. Looking forward to that blog post.

    OK, so obviously I love this. The make-your-own-cup stuff is amazing and I can’t wait to put it in front of kids.

    The one weak spot that I’ve had in teaching your Desmos lessons is the “Help ____ fix his graph” questions. I think that I need to do a better job of sharing interesting explanations or suggestions. I can do that without technology, but is there a way to easily share the kids’ explanations or suggestions in the teacher’s page? Something like a board where I can just select and view kids explanations or graphs on an otherwise blank page?

  14. This is a great way to get students to interact with one another and get them interested in what they are learning.

  15. This is one of the most beautiful and practical integrations of mathematics and technology that I have ever seen. What makes this most amazing for me is the real time feedback students get to help them adjust their ability to interpret functions. I have done similar activities with paper and pencil, and even with graphingstories.com and this is a whole new level of incredible that has not previously existed.

    This is a great time to be in mathematics education.

  16. I find this lesson fascinating especially because my county is providing one-on-one device to every student by 2018. I was having a difficult time envisioning how lessons should be when that change is implemented, and this lesson just provided a perfect example!

  17. @Dan: Perhaps the “old-fanshioned” style is also useful. In some cases, paperboard, scissors and matching are better than digital ones ;-)

    Perhaps people are obsessed by technology. You could take advantage of both things: new-technological and old-technological