WCYDWT: The Slow Runner

A black rectangle and the pause button will take you 70% of the way.

Where do you put the black rectangle? What question does that provoke?

BTW: I would put a black rectangle above Rich Eisen’s speed and ask the students to guess at his speed. Then they’re converting from yards per second to miles per hour and you have a nice conversation about good time intervals for that calculation. (If you time him over the entire forty-yard dash, your estimate will be well below his top speed.)

Where do you press pause? What question does that provoke?

BTW: I would press pause on the clip where Eisen gets a head start on Jacoby and ask them how long it will take Jacoby to catch up to Eisen. Which is a nice enough way to introduce systems of equations.

2011 Feb 16: Rich Eisen calls me out on Twitter.

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. I would put the black rectangle over the time during the first run through. I would hope the students would ask how fast the man in the suit ran the 40. We would have to know how fast Jacoby runs the 40. After some time with google we would find this: http://tinyurl.com/m6zxf6. Hopefully some math magic would ensue.

  2. Zachary Brown

    July 29, 2010 - 12:49 pm -

    On first glance, I’d put the black rectangle over the number that displays the speed (leaving the text: “mph” visible) to provoke the question “what is the speed?”

    yards. miles. seconds. hours. some good stuff here.

    I’d pause it around 6 seconds, when the suit hits the 40 yard line… hopefully to provoke the question, “How’d he do?”

    What tool would you recommend for adding a black rectangle?

  3. Black rectangle over the speed, as mentioned above.

    I’d hit pause at the 10sec mark, just when the suit is paired up against Jacoby.

    “Does the guy in the suit win or lose?”
    followed immediately by:
    “How badly does the suit lose? How far ahead do you think Jacoby will be when he crosses the line?”

  4. After all the calculating from the previous questions, you could also pause it on the head start race right before Jacoby takes off. Will Rich Eisen win or lose this race? By how much? How much of a head start will Rich need in order to at least tie Jacoby? The same could be done with the other runners.

  5. @Karim: Not sure if this is what you’re looking for, but..

    My sweet spot: I like having/stating clear objectives before we start an activity. Something involving a race, I would probably ask the kids simply to figure out who wins, and use names of kids in the class in the story to get them competitive. If I don’t have hi-fi media, I might make the problem into multiple representations. One person’s info is given in a table, another’s in a graph, another’s as an equation, yet another as a story. (This is not an innovative idea, obviously… I’m sure most middle-school teachers have used a lesson like this.)

    The kids would figure out/decide many things for themselves, such as the fact that they might need to know the speed and/or where each person starts, and have a way of comparing the different modes of representation (no written scaffolding there provided, but maybe orally scaffolded using questioning when I circulate… I definitely would not ask, “how fast is each person going?” to set them up for finding the winner), but most likely, that conversation will take place in small groups of 3 or 4 instead of as a class. (I find that it maximizes learning when all kids are talking amongst themselves.)

    Towards the end of the assignment, I will have some concrete skills portions where they demonstrate something specific, like where is each person at t = 15? When does one person take over the lead?

    If the meta-goal is to have the kids process the process as much as possible, then maybe at the end I would ask the kids to answer (ie. write out explanations for) “what would happen if […]?” situations. I think this is important, but since we’re always running out of time, it’s not always achievable, unfortunately.

    I would love to think that all of this, plus excellent hi-fi media, is achievable in 50 minutes daily and be massively successful consistently. (I can dream, can’t I?) But, what I find typically is that even if you shoot for the moon and fail, the lesson tends to turn out OK in the end, as long as you manage to get some of those desired elements in there and change up what you incorporate from day to day.

  6. @Karim

    I teach kids physics and physical science (second year, still pretty green), so this fits perfectly as a visual demonstration/jumping off point for my speed/velocity/acceleration unit.

    I could use this to draw distance over time graphs, velocity over time graphs, etc.

    I’d love to try visualizing the information myself, a la Dan’s Graphing Stories (https://blog.mrmeyer.com/?p=213), and have the students experiment with representing the “story” in a graph.


    I would concur with you on the group work for this. The challenge will be to make sure every group get enough, individualized support. Also, explicit and clearly modeled group roles will be really important. (http://larkolicio.us/blog/?p=533)

    As far as where the “sweet spot” is in making lessons like this available to teachers in a non-stifling way: I think Dan’s blog and comments section is a great model to start off from. But I would take it a step further. I see a wiki where users can not only comment but upload their materials and reflections on the lesson.

    I checked out your website (http://www.mathalicious.com/?p=1789). I dig the lesson a lot, but I’d like to see you “be less helpful.” Now, this doesn’t mean I want any less content (videos, slides, etc.) in your lesson plans. That’s great stuff. I’d like to see more encouragement for teachers to take this in different directions and share their experiences.

  7. I think this is fantastic, and I want to use it, but I’m having a hard time figuring out a logical place in my curriculum for it. I teach Geometry and Algebra II with Trig. At first glance, I thought of using it to introduce unit conversions – exploring the different ways to express speed – but it seems tangential to the point of the video. Perhaps I could use it as graphing practice, dabbling in some distance v time or velocity v time graphs as Eric mentioned, since most of my geometry kids will also be taking physics.

    Ideally, I would hope for a unit in which we’re using math we’ve learned to help solve the problem, and not the problem to help learn more math. Seems like the former would be far more gratifying.

    If anyone has any thoughts as to what other topics I could use this with, I’d appreciate it. It’s possible that’s there’s an obvious connection and summer has succeeded in frying my brain.

  8. Okay, textbook

    Topic – linear equations? Fractions? graphing functions? triangles? quadratic? whatever.

    1st lesson: video, other visually/kinetically enhanced problem without criteria, for students to converse, deliberate on what is needed to solve this problem
    (do something to introduce each new topic thoughout the text).
    (similar to water tank and the cd/ticket roll)

    After the students determine these needs, and, aided, solve the problem, do it again with a like topic. They should be quicker.

    Do it again – it should be easy for everyone. (This is all group or small group/report back discussion – the last one, or maybe the second, can be journaling or small group discussion and reporting back to class.

    If there is a way to tweak it, if the book tweaks it in other problems, do a visual/kinesthetic of the tweak one or 2 times.

    Next, take it to the pure math equation. go over these 2 or 3 times as a class/small group. Relate it back to the visual.

    Then, take it to pure mathematics. Find/tell a way in which this is used in an actual career (construction, engineering, medicine, science, computing, surveying, whatever.)

    Play-act being mathematicians – whether in the career, or as research at a university, PhD, whatever. Solve the problems in the mathematician mode as a group of researchers working together, with a fellow group leader guiding (take turns with everyone).

    Break out in centers (maybe with various career motif?), where each student individually completes several pages of the problem type and hands it in. Teacher helps when needed.

    Do this a couple of days, with testing timeline for them to individually take the tests at their desk or at a testing center.

    How does that sound?

  9. @ Karim –

    Love the setup. I think the followup imposes too much structure. Not mathematically, but in terms of the question of what the right measurement is that will make the comparison fair. In each comparison, you’re taking the raw accomplishment (the distance jumped; the weight benched; the time in the 100m dash) and highlighting a measurement of the person / animal against which to scale the accomplishment (body length; weight; height). To me making this choice of what to highlight removes from the students the most mathematically fertile thing for them to think about. It also sends the message that there is an obvious best answer to this question, but actually the question is far more subtle.

    For example, just to explicate the subtlety, the lesson as presented is highlighting that “how much can you lift?” as a piece of raw data favors larger people/animals, and seems to imply that the “fair” comparison is the ratio (quantity lifted) / (body weight). This ratio, however, is also unfair, in favor of small people/animals. The reason for this has to be one of the most awesome things I ever learned in school:

    Muscle strength is roughly proportional to cross-sectional area, which varies with the square of linear size. Weight varies with the cube of linear size. So if I were scaled down by a factor of 10 (from 2 meters to 20cm, say), I would be 1/100th as strong but weigh only 1/1000th as much. In proportion to my own weight I would therefore be able to lift 10 times as much. This is why ants can carry hundreds of times their own weight. It’s also why all land animals with exoskeletons (bugs, crabs, etc.) are small – an exoskeleton weighs a lot and as you get bigger this weight is going up with the cube of your length but your muscle strength only with the square of your length, so there’s a threshold of size beyond which a bug wouldn’t be able to lift itself. And why somewhat larger animals with exoskeleta can be found in the water e.g. lobsters, since it’s easier to lift your body in the water than on land. And why large animals have to have thicker legs in proportion to their bodies than smaller animals. (Compare an ant or a mouse to an elephant.) How awesome is all of this??? Okay let me stop now.

    To the extent that the muscle-strength/cross-sectional-area thing is true, this allows a sort of “more fair” comparison: (quantity lifted) / (body weight)^(2/3). What makes this “more fair” is the assumption of an “ideal” relationship where strength varies with length^2, weight with length^3. Then this ratio measures how much stronger you are than what would be predicted by this model. The extent of the truth/strength of the model is a science question and a statistics question.

    Things are much murkier when it comes to height and the 100m dash, because the relationship between height and speed isn’t clear at all. Usain Bolt is both taller and faster than everybody else, but in the other races for which you gave data it isn’t true that the tallest person is the fastest, and since the lesson repeatedly broadens out to the animal world, it’s worth noting that cheetahs are faster in a sprint than horses which are faster than elephants. Is being taller even an advantage in a sprint? If not, then the idea that a shorter person should run a shorter distance in order to run an “equivalent” race to a taller person seems contrived.

    What I’m getting at is that “what’s the fair way to measure strength / speed, that allows you to compare people / animals of different sizes?” is a really rich question that has the potential to go very deep, so I think it’s important not to impose a particular measurement or ratio and act like that’s clearly the fair one. In fact, depending on the intended level of challenge, this question could actually be the lesson’s central question.

    Still, I think opening by shaking up people’s notions of strongest/fastest with the examples of smaller folks (ants, frogs, aditya dev…) is a great way to get people thinking about this question. So it’s not that the ratios (weight lifted)/(body weight) and (distance run)/(height) shouldn’t be part of the lesson, I just don’t think they should be treated as the end goal.

  10. @Eric: Since you’re taking about “Being Less Helpful,” you might be interested in my post “Win? Fail? Physics!”


    And the wiki that goes with it:


    Currently, I have wiki comments turned off. Based on what Eric is saying, I could turn comments on and individual teachers leave ideas and example lessons.

    What do you (all) think? I want this to be as helpful (without being too helpful) to everyone.

  11. I made something similar to this with tricycle races between my colleagues and myself…
    http://bit.ly/9Aq8D3 (.mov file)
    http://bit.ly/aSAIrA (.avi file)

    Can’t wait to see what the kids will do with it!

    Would love to see what other people have been able to put together in a similar vain…

  12. Quick note that I suck, basically, and can’t keep up with anything in my life at all, much less the interesting comments flying back and forth here. I did add my own response to the prompt in the post itself, which basically aligned with josh g.’s.

    bree: I thought of using it to introduce unit conversions — exploring the different ways to express speed — but it seems tangential to the point of the video.

    Agreed. It’s tangential to the video until you cover up the suit’s speed and then it’s central to the video.

    Zachary: What tool would you recommend for adding a black rectangle?

    I’d import the video into your slide software of choice. Then I’d create a black rectangle in that software and position it just-so. Really simple, no expensive video editing software required.

    Cool video, also, Scott. What question do you anticipate your students asking about it?

  13. @Dan, thanks. Just trying to explore the media. As for questions, trying to decide what’s best…
    -speed and unit conversion as mentioned above
    -if everyone started at the same time, how far apart were they when they started? (X-intercepts)
    -what time will each person be passed by the fastest racer? (Solving systems)
    -How long is the race? -if the race lasted for 400 meters, how far apart would everyone be?

    Appreciate any input from the community… Thanks.

  14. My two cents.

    Speed and unit conversion is nice. Converting to miles per hour is a useful way to talk about speed. (Here’s hoping you measured the distance between fence poles.)

    Looking over your other hooks, I’m inclined to ask this: what is the predominant question when anyone watches a race?

    The perplexing question is “who is going to win?”

    Depending on what information you provide or obscure, you can then push the conversation towards different skills, but the most engaging way to start this problem is with that question. Other questions might lead to a better standard or skill but you’re working against what is naturally perplexing about your video.

    I find this kind of curriculum design very tricky and I’m grateful for the chance to bounce some ideas around.

  15. I am adapting this lesson for Monday and struggling with the concept of “half speed” as mentioned in the video. If Julio Jones travels 40 yards in 4.39 seconds, my feeling is that half speed would mean it takes him twice as long to travel the distance (8.78 seconds). But the video works out to be about 6.585 seconds or 1.5 times his original speed. Shouldn’t we say then that Jones is running at 2/3 of his original speed? What am I missing here?

  16. Hm. At first my thought was that acceleration accounts for the missing time. But those differences should be uniform if the entire video is slowed down.

    If it were me, I’d just tell your students that the announcer was wrong. That it’s 2/3 the speed.

  17. thanks for the quick reply. I am going to try an adapted version of the lesson and mention the 2/3

    My students will follow up with a project, titled “Me and Julio”. We will set them up in the gym and sprint so that they can compare their times to Julio Jones. I’ll send samples or post (if I ever get a chance)