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. Will paper lessons plans ever be offered for any of these acts? (It would really help with my planning).

  2. Assuming it isn’t paper itself that’d help with your planning, rather something on the paper, what would that be? What would help you out here?

  3. Something similar to Mathalicious lessons. A lesson plan and a student guide broken down by acts. Given your emphasis on discovery, the student guide would presumably be a sparse outline, whereas the teacher’s guide would offer details.

  4. Do his work for him dan! Hehe! Question about this video for some reason I really want to see what the dollar looks like every time you pull it out of the copier *:)

  5. Pepe: I think the general idea is that students should learn how to structure their thoughts themselves. The goal is to get students to approach these problems like we adults would. We get out a sheet of paper, we sketch some diagrams, we write questions, we perform calculations.

    Getting a kid to do that is pretty hard, of course.

    So, to that end I think you could give a common handout that has structures that should work for any of these three act problems. Remember that the concept of the acts are not exactly for students: they are ways for us as teachers to break down what we are saying and giving to the students.

    A structure that I may give to a class unused to this type of activity may just be dividing the paper up into 4 sections: “your questions” “data/info” “diagrams” “calculations”. I may just do this by telling the students to fold their paper into four parts.

    Another key element to this type of project is to have the students revisit the problem — twice. After they have “solved” it, make them write a 2nd draft solution. They should write out their complete solution: clean out all the unneeded stuff, adjust their diagrams, clean up the flow of their calculations, and add in some actual sentences. Just because we’re in math class doesn’t mean we can’t write.

    You can offer comments on their 2nd drafts and how to improve their reasoning. Then, finally, have them make a 3rd draft solution. They tune up their 2nd draft and fix mistakes, adjust their commentary, and make it neat.

    Its hard. Even high-skilled math students have a lot of trouble explaining their work. They often spend time explaining their arithmetic operations instead of the concept. A made up example: “We multiplied 5*6 and then divided by 8 and that’s how we got the answer” vs. “The area of the rectangle is 30, thus each of the 8 people get over three […]” And the low skilled math students often struggle with writing in general.

    But this is why they should have practice. A computer can solve a quadratic equation. A human can interpret and communicate the solution. We want our kids to be human, yes?

  6. Certainly I’d like to support teachers, but “which supports?” is a huge, outstanding question and I’m not sure handouts and lesson plans are the answer.

  7. apologies for combining problems here, but when i saw this video my head went to your Guggenheim thing – i really wanted to know how many *more* of these shrunken dollars it would take to create that artwork.

  8. This looks very cool Dan. I’m going to try it with some under performing Juniors and Seniors I work with. Smart enough kids, they have just chosen to/been allowed to dumb down over the years.

    I’ll let you know how it goes.

    Thanks again.

  9. For what it’s worth, I think Dan’s stuff is valuable in large part because it doesn’t include the type of explicit scaffold that Mathalicious lessons do. The approach is predicated on students’ coming up with their own questions, and Dan’s gift is in crafting opportunities for them to do this (while still providing some structure, albeit subtly).

    In the end, I think WCYDWT lessons and Mathalicious lessons are similar in spirit–they’re both about exploring the world–but fundamentally different at the same time. The types of topics that many Mathalicious lessons address require a more explicit scaffold; whereas students might be able to come up with “Is Wheel of Fortune rigged?” on their own, it’s unlikely that they’d transition from NikeiD.com to a discussion of paralysis by analysis without the narrative afforded by the handout.

    I don’t think either one is better or worse. They’re just very different, and therefore require a different approach.

  10. I attempted this 3acts today in my 8th grade algebra class. I gotta say it was a bit disappointing. Potential problems: End of the day; 35 kids in the class.
    Of the nine groups in the class, I think 3 were actually engaged in the problem. After they struggled with it for about five minutes, I stopped the class and we discussed how you could get started on the problem. But then I struggle with not leading them down the path of how to do it. At what point do you throw in the white flag and move on.

    On a slightly positive note, one kid right off the bat, eyeballed 75% and sketched the drawings on the sheet of paper, and was within two millimeters of the answer (we projected his image over final cut from act 3)

  11. Hey Josh, thanks for the feedback. It’s a lot easier to dish out positive results than the negative and I’m obliged you’d make the effort.

    Presumably you didn’t blindly implement the problem. You saw it yourself and thought it’d be interesting. Do you have any sense why the problem didn’t land for your students? Any sense how I might scaffold it differently? Do you think your students would have experienced the problem any differently had I shown them the last dollar and asked, “How many times was it copied?”

  12. I am going to give the problem to my geometry classes as well. I’ll give you more complete thoughts tomorrow after they see it also.

  13. Before giving my thoughts on why it bombed, let me say that the geometry kids ate it up in about one minute. Aside from the questioning of who this guy is who is able to copy money, they dug right in and groups finished the original problem and most finished the followup questions. Many came up to ask me how small it would have to get to be invisible. “Not sure,” I would respond. “how small do you think it should be?” which got about two of them actually thinking about the size of .00001 mm.

    The algebra classes struggled more because they don’t underhand percents very well. I felt the initial hurdle was too high. I handed out the dollar outline to each student (I think one sheet to each group would have been better). I don’t want to go into too much detail, but some went right to perimeter or area, which brings up other questions.

    I do think your idea of having the original and the ninth copy moght have helped as a launch point. I also think showing them either the first copy and blanking out the percent would have also been a good problem. the question could then have been, “what setting was entered?” The followup could then have been how big is the ninth copy.

    Gonna try the dominos next week.

  14. I used this with a grade 11 precalculus class for an introduction to geometric series. They had not been given any formulas or explanations. We did an example of geometric growth previously by looking at the distances in a spider web. For the most part the students really enjoyed the challenge especially when I refused to say when or if they had the “answer”. We then took the class to the copier and did the reductions and measured to verify which had the right and wrong solutions and why. Went very well and sets up our discussion of geometric sequence formula work tomorrow. We had trouble with when the bill disappeared because we couldn’t recognize the limit of the copier but tomorrow we will look at how we can determine the limits of the copier by being able to predict the shrinking bill size at each stage. Thanks for the work. I really appreciate you posting the ideas.