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. In the Robotics club I’m coaching, I had the students try putting on different propellers (actually fan blades for computer fans) onto the bilge-pump motor and feeling how much thrust they generated running forward and backward. The best of the blades had a very different forward and backward thrust.

    Later (by next week?) they’ll build a rig so that they can actually measure the thrust and the current into the motor to choose which propeller to use. I got a different propeller in the mail yesterday (one designed to be used in water, not air), so I’m hopeful that they can get more thrust from that design.

  2. In the last week, maybe nothing. But a few weeks ago, the Escape from the Textbook conference got me thinking mathematical thoughts all day. And a conversation on the Math Future group (aka Math 2.0, a google group), along with a post by Dave Richeson, got me writing down all of my thinking on the problem I posted about here.

    Waldorf teachers are encouraged to continue growing, and a good friend of mine decided that meant learning the cello. I love her spirit of inquiry. I’ve seen myself blossom in my mathematical inquiry in the past 3 years because of my connections online.

  3. Are you trying to trick me into giving away all my blog post ideas? Well, it won’t work.

    Ok – it works. I made a video of my daughter doing a kip (gymnastics).

  4. My classes saw a play about the bombing of the the 16th Street Baptist Church in 1962 Birmingham. It was a powerful show on stage told by African-American and white kids. Powerful.

    Before the show, we watched clips of videos and read newspaper articles. After the show, a student asked about the church today. I thought, great question. Then I found an NPR clip from 2003 that told the present day history. I played the four minute clip and it got a round of applause from my 6th graders!

  5. I kept the times for the cars at my son’s pinewood derby this past weekend. I managed to get the weight of every car as well as all the times. I’m finally going to get to play with the data tonight.

  6. Logarithms.
    I met a great teacher last weekend who challenged me to think about the ways I engage students with mathematical tasks.
    So I looked at my textbook today for contexts to play with having to do with logarithms. It turns out that pH is a logarithmic scale so I’m trying to get my hands on some pH strips to see what they’ll tell me about the strength of various orange juice mixtures.

  7. I want a video of a tall thin container of water being siphoned into a shorter container with a larger radius & perhaps some valuable electronics scattered around the container being filled up to add a little drama.

    I would also like to get a video of a cars mpg & speed as it changes speeds. No luck on youtube & I don’t have a mpg meter on my car. I have googled data for this in the past – it is roughly quadratic.

  8. I am brand new to teaching this year and am at a school where the textbook is king. As such, I love this blog and the memes that live here because they keep me excited about how I might adjust the curriculum for next year. Lately, I have seen lots of migrating birds overhead, flocks of thousands. As I watch the birds I consider how they fly (the physics of the situation), how they navigate, their number (as an estimate and over time), their paths in 3-space as a function of time, their amazing view of my beautiful city, and the chances they’ll poop on me. I am not very practiced at turning such inquiry into valid and useable lessons …yet. But it is exciting to know that other teachers can and do and that the students of those teachers are being encouraged to think, explore, and ask questions.

  9. I heard a news story on NPR about Tunisians collecting supplies for Libyan refugees. They listed a few things being collected, but diapers caught my ear. So then I started thinking, how many diapers does a baby wear every day? How many refugees are there? When they said there could be tens of thousands, I started making a mental image of all the trucks they would need just to carry the diapers!

    I teach elementary school, and a question like this would be right at the edge of my students’ capabilities. The numbers are really big, but I might use it as an extension for our next unit on multiplication. I’m so tired of multiplying a 3×4 array of marbles.

  10. Phil: I am not very practiced at turning such inquiry into valid and useable lessons …yet.

    Hey Phil, welcome to the blog. I just wanted to point out that, going into the cheese block experiment, I was 90% sure a usable classroom lesson wasn’t going to come of it. (I’m still not sure I made one.) But when we chase down these lines of inquiry to their terminal point, we’re sharpening certain senses that pay off dividends both inside and outside of the classroom. No need to force a lesson out of your fascination with the birds. One may present itself, but curiosity is an end of its own.

  11. During a unit on slope and measurement we were tasked to build stairs for a woman’s basement across the street from the school. In one lessons homework, I wanted them to measure and calculate the slope of stairs at their homes. One learner asked how she was supposed to calculate the slope of her spiral staircase. I admitted to not knowing and encouraged her to take a picture and think about it.

    While walking down a spiral staircase to enter the subway, the answer to the question became evident. I snapped a pic and that became an interesting opener for the next day.

    I’d like to think that they saw me not know the answer to a question and then see how my view of the world tends to elicit the solution.

  12. In the kitchen…

    Would a can of beans fit into a [rectangular] Tupperware?
    How many pepperonis would fit on a large pizza?
    Speaking of which, what percent is crust?
    Per square inch, which size (S, M, L) is the cheapest?
    Does 4x(5-Hour Energy) = 20 hours?

  13. Oh, I almost forgot:

    A jar of peanut butter could cover how many Ritz (circle), Club (rectangle), and Saltine (square) crackers? Also, could a jar of peanut butter cover an entire desk?

    Also, could Mathalicious get sued if we urge teachers to bring peanut butter to class?

  14. Failure distributions of popcorn.

    I’ve video recorded microwave popcorn. Next step is to extract the time for each pop. The resulting distribution should uncover information about the way the popcorn kernels “fail” to produce popcorn.

    The learning here would be how to interpret the data and to discuss ideas for representing the distributions. What kind of distribution best fits the data? Gaussian? Weibull? That kind of stuff.

  15. Adam Poetzel

    March 2, 2011 - 8:51 pm -

    Bought an elliptical exercise machine….Do the pedals actually travel in a true ellipse?

    Was looking at my car’s speedometer…imagined a video of 3 cars lined up for a long straight race of a set distance…then just show the speedometers…each car’s speedometer tells a different “speed story”… Can you tell what order the cars finished the race?

    Was standing in an elevator and saw the sign “16 passenger maximum or 2400 lbs”… Was wondering if all elevators have a similar “avg weight per passenger”…If we looked at the floor space of different elevators, could we rank them by how efficient they are?

  16. Recently my students just finished an inquiry into what it means to be human, inspired by the song “Human” (Grant/Sparks). As a result of that unit they decided they wanted to do something just like the “unexpected heroes” (their term) that we read about. They’re making plans to build a garden and use the produce/proceeds to benefit “hungry people.” How great is that?! What’s even more amazing was one comment from one of my second graders: “Guys, we can’t just give stuff to people…we need to teach them, too. That’s way better.” So I’m sure we’ll also put together a garden demonstration or video to educate others.

    Natalie Grant also sings a song called “Beauty Mark,” and while I listened to it the other day, it inspired me to create a unit of inquiry around “true beauty.” I’m going to use fairytale picture books as the vehicle to drive this question forward, as well as supplemental materials like the Dove Campaign for Real Beauty stuff online (they have an awesome video). In short, I want to nudge students to challenge the cultural definition of beauty and create a definition that centers on what truly matters.

  17. @Adam, the speedometers are a great idea. I actually had to record my dashboard for a lesson on speeding tickets recently, and have a few dozen recordings of both the speedometer and the view from the windshield. (Unfortunately for your lesson, my speeds are all constant). Let me know if you’re interested.

    @Laura, if you teach math, you could structure the beauty lesson around two different concepts, depending on the standard you want to hit. One could be the golden ratio, which many believe is the standard of mathematical beauty. Another is how symmetrical a face is, where the math standard could be reflections, and then figuring out a way to calculate some “symmetry factor” (which would involve controlling for the size of the face). I haven’t written the reflection lesson yet for Mathalicious, but may have an old GSP sketch for symmetry that I’ll be happy to send you.

  18. @Laura, oh, my bad. I just saw that you teach 2nd grade. The golden ratio may be a big high-brow (pardon the pun!), but the symmetry one could still work, even if you didn’t get into the math nitty-gritty of it.

  19. I see Laura’s inquiry being of a type unfamiliar to many math folks-qualitative inquiry. She wants her students to cast a critical eye on what different cultures consider “beautiful” and to ask “why?”

    We math folks always want to quantify stuff. (e.g. golden ratio and symmetry measures)

    But the inquiry process is the same in the social sciences. Let’s not accept what we are told to pay attention to; let’s ask our own questions. It’s lovely and I hope Laura will report back on her second graders’ ideas about beauty.

  20. I’ve been thinking about division. If we can model it as partitive and quotitive, how do these models continue into high school mathematics? Is there a place for quotitive division in high school? And if we teach children that division is about fair shares (partitive), why do we teach them a standard algorithm that is quotitive (repeated subtraction)? Well, I don’t teach that, but it’s out there!
    So, if exponentiation can be represented by multiplication (I know that’s not all it does, Dr. Devlin), can logarithms be represented by repeated division? Does that make sense?

  21. Lately I’ve been teaching power functions and radical equations, and then exponential functions, so I’ve been doing some informal experiments at my house with pendulums (pendula?), since the period of a pendulum is almost exactly a radical function of its length, and the motion of a pendulum decays almost exactly exponentially.

    The other night I heated a tupperware container full of water to 99 degrees, stuck a digital thermometer in it, sealed it, and recorded some data on how it cooled over the next several hours, to simulate (somewhat) the cooling body of a murder victim.

    As a side note to #22’s side note to Dr. Devlin, I think his position on multiplication vs. repeated addition, and his apparently similar position on exponentiation vs. repeated multiplication, to be deeply asinine, and possibly indicative of a harmful approach to mathematics teaching.

  22. I videotaped my students while they were competing in World Maths Day. I had to laugh at them because they were so enthusiastic about doing math problems and not realizing that THEY WERE DOING MATH PROBLEMS. Dan, you’ve done such a wonderful job of showing us that if we change the dynamics of a classroom or a math problem, that students can not only enjoy doing math but be enthusiastic about it.

  23. This is admittedly abstruse, but honestly what’s had my notebook out all week is trying to figure out what the Riemann surface of y^3-xy-x looks like. I think I’ve figured out how the sheets permute at the branch points, but I don’t think I’ll really be satisfied until I’ve in some way drawn a picture of the mapping of the y-plane to the x-plane, and I don’t see that happening without computational help. I’ve played around with WolframAlpha but it’s not really giving me what I need. Time to learn how Sage‘s graphics features work?

    @ Rhett – was that a Princess Bride reference? (“It HAS worked! You’ve given everything away!!”)

  24. I am teaching conics right now. A student pointed out that if he used a table to graph a hyperbola he sometimes got imaginary numbers. I told him to figure out what it looks like in the complex plane. I still can’t stop thinking about it…

    I’m fairly certain that hyperbolas on the Cartesian coordinate system are circles/ellipses when graphed in the complex plan and visa versa, but I’m not 100% sure.

    The funniest part was when the student got an answer and asked me if he was right. When I told him that I had no idea he told me to stop pretending to not know. It really shows the misconception students have about math teachers knowing everything about mathematics.

    I sometimes write about the mathematics that keeps me up at night.

  25. These responses are so much fun to read. Here’s mine: a standard introductory optimization problem is “what’s the largest rectangular area you can enclose with a fence of total length P.” The standard approach starts by setting A = LW and P = 2L+2W, then getting A in one variable and setting A’ to zero. My puzzle is: it seems to me that since A=LW and P=2L+2W is symmetric (in the sense that the labels L & W can be swapped without changing anything), that there can’t really be any other answer than L=W. But if I said that to my students and they asked me to explain I’d be stumped. So I’m trying to write a proof that the max area is L=W, that, say, my students could understand, based on symmetry and not on the derivative. Maybe its screamingly obvious to one of you, but don’t post the explanation here — I want to work this out for myself!

  26. @RWW I’m getting ready to go over square root equations next week. I’m interested in how the “body cooling” water activity worked out for you. Id like to introduce that topic to my class in a different manner than just notes then homework. Anyone else have suggestions?

  27. I brought in a selection of different size drink containers.
    E.g. Carton, Coke can, Red Bull can, Different sized water bottles etc.
    (Emptied and washed before class!) Left the labels on!

    Then asked then students to order them from largest to smallest. Much debate in the class how to do it, particularly the tall skinny Red Bull compared to the shorter squat Coke.

    Finally someone suggests looking at the labels.

    Promoted a discussion about mL and L. Also then led nicely in fractions, as all the containers were a proportion of a liter, 1/5, 1/4, 1/3*, 1/2.

    * The coke can is marked as 330mL, which is not quite a third of a liter. Though can mention about rounding and accuracy here.

    Main problem was only having one set of bottles! So now going to collect more sets over time.

  28. Current puzzle: Is there a way I can track my food intake and changes in my body measurements (weight, but also size) WITHOUT negatively impacting my self-image.

    Backstory: I sew. Not that often in recent years, but enough that I actually have records of my measurements. So when my friend asked if I was dieting (because she noticed my weight loss before I did) I was able to compare my measurements to 6 months prior. I’d lost an 1-1.5 inches on all of them. That explains why my clothes aren’t fitting as well. Here I was blaming my laundry decisions.

    So I was curious and have decided to track my measurements once every 3 weeks or so. (Note, not too often, because I don’t want to become obsessed by this. It’s a curiosity not a body image issue, but verges enough that I can imagine crossing the line.)

    Current story: Two nights ago I measured myself. The next day had multiple “special meals” so I ate more than usual. At the end of the night I felt like my stomach had expanded. Kinda gross, kinda cool. I got home and compared my waist measurement to 24 hours before. Had increased by a whole inch.

    So now I’m curious about weight fluctuations. But I’m also positive that if I track it as much as I want to, it will veer to the point of diet and changing my habits in a way that I don’t really want to.

  29. Roots of polynomials. Cubic ones in particular. They are always on math contest problems in some form or another and elicit such cool patterns. Why can’t that be in my curriculum? Why is it “reserved” for the mathletes? That and a tangent line question that had me going polar coord to rectangular and writing equations of lines, and equations of lines perpendicular to other lines, and I found a solution that was different (ever so slightly) from the published solution. I felt smart in that math geeky sort of way.

  30. #32: For a long time now, I’ve been wanting to keep track of my exact dietary intakes (broken down into carbs, fat, etc.) and daily or at least weekly weight changes, to see what kind of correlations there might be. Heck, it would be interesting just to see if there are any regular patterns in the intakes themselves.

  31. Glad I’m not the only one curious. I just know that sort of tracking is one of the favorite suggestions of diet magazine articles. And I don’t really want to change my habits right now. Not quite yet.

  32. Didn’t really know where to leave this comment but I “discovered” Dan about a week ago, and have been hooked ever since. I started doing Concept quizzes last week with our fraction unit. I wanted to know if anyone would be willing to share ideas like the ones on this site for 5th grade math class, decimals, percents, anything. Thanks

  33. # 32, 34, 35: If you’re not trying to change your eating habits, and body-image issues can be a problem for you, I wouldn’t really advise tracking your weight…

    However, there’s a book “The Hacker’s Diet”:
    …and some of his ideas might perhaps help you track yourself without the negative body-image impact.

    He points out that someone with a stable weight will still see a lot of fluctuation in their daily weight measurements, depending on how much water and “solids” are in your system at the moment you happen to weigh yourself that day.

    So he basically suggests taking a running average of your daily weight, so that you can see the trend over time instead of taking the daily fluctuations too seriously. The particular way he takes the average is a bit complicated, but even if after each measurement you just take a simple average of the past few weight-ins, it’s probably a decent indication of the trend.

    Since you AREN’T trying to change your weight or diet, doing regular measurements of weight/waist and tracking them against a moving average could give you a sense of the natural variation in your measurements. If there’s a particular weight you don’t want to go over, you don’t need to worry if a single day’s weight happens to be higher — only once the moving average gets over that weight limit.