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Friday, August 28, 2009

WCYDWT: Maxine

I don't really know if I can use this. But this pooch Maxine lives a few doors down from my parents and does this routine for every passerby. It reminds me of that problem where the fly is flying between the two trains. (Would need a better quality video for class, natch. Unfortunately I don't have a rolling-tripod thing, and the walking is kind of critical. Maybe someone stationary could tape both the walker and the dog.)

The obvious question is "How far does Maxine run?" but a more critical thinking question would be "What minimum information do you need to determine how far Maxine runs?" What do you all think?


  1. Oh boy, does she have an invisible fence!

  2. The average speed the dog runs at plus how long it takes you to traverse the lawn.

    Since the dog runs on average this speed for the duration you cross the lawn the product of the two numbers will tell you how far the dog runs.

    Great video though. This is the kind of stuff that makes these problems fun.

  3. Joseph, if the dog runs at anything but a constant speed you've got it exactly backwards. Because what is the "average speed" but the total distance covered divided by the elapsed time. Saying that the total distance is the product of the average speed and the elapsed time is content-free.

  4. Since knowing the dog's average speed would require knowing the distance she covers in advance, could we do it with a different piece of information? That it would be feasible to measure?

  5. Never mind...I was thinking the walker's data could be useful, but in the shower I realized No. That's what I get for BWU (Blogging While Undercaffeinated.)

  6. The average speed would be feasible to measure. You don't need to know anything about distance:

    You hold up a speed detector, like they use to measure how fast baseball's move, to record its speed at every second, plot it (as a function of time :) ) with respect to time your clock is giving you to measure how long it takes to traverse the lawn, then use standard calculus.

    However, I was confused by what is meant by "minimal". I will try to think of a more clever thing to do that will be considered "feasible". (But I really think you really could do the above.)

  7. Okay, I just realized you have to extract your speed from the speed on the detector since they will give an inaccurate reading for the dog.

    But you still don't need to know distance if your friend, who is standing still, records your speed while you are recording the dog's.

    So still no distances required. But you do need this extra friend I admit so I will try to think of something more minimal.

  8. Sure, I will just whip out my pocket speed detector... :-)

    We actually do have such a gizmo at school that hooks up to the graphing calculator, but I don't think it has a long enough range to see the dog for the whole width of the yard.

  9. First of all, thank you for doing this post because it actually makes me think and problem solve. I really appreciate this.

    Okay, now that I know we need solutions that are practical for the average person, I will go back to the drawing board.

  10. I saw this earlier today and kept thinking about it. I think if you actually wanted to find the answer you could just drop a penny every time the dog met you on your walk. Then go back and measure each penny's distance from the beginning of the lawn and label these distances x1, x2, etc. Then the length of the dog's path (where y=the length of the lawn) is just y + 2(y - x1) + 2(y - x2) + ...

    This isn't the optimal solution, but hey, it's practical. To actually calculate the points at which the dog meets the walker seems to me to require knowing the dog's speed.

  11. This might be one where the word problem is superior to the video. Be still my nerdy heart.

  12. What is the word problem here, exactly?

  13. Two trains 150 miles apart are headed toward each other on the same track. One train goes 60 mph, the other 90 mph. A fly buzzes from the first train to the second train, turns around immediately and flies back to the first train, turns around again. It goes on flying back and forth between the two trains until they collide. If the fly's speed is 120 mph, how far will it travel?

    120 mph sounds unrealistic for a fly, but this is the version I found on Ask Dr. Math.

  14. The video might be a fun mini-break while students are thinking about a problem like that. Or when someone says that sort of thing never happens.

  15. Kate's version posits a constant value for the fly's speed, which doesn't seem to hold for the dog. That's why the fly problem has a shortcut, related to Joseph's original suggestion.

    What you need is the list of positions (along the length of the yard) where the dog turns.

  16. But of course you can't have a constant speed if you're reversing direction. That reminds me of a book my brother used for a grad course in mathematical modeling (for non-math-majors), Consider a Spherical Cow.

    To model real-life events with math, you often have to simplify - a lot!

    So I'm thinking we could pretend the dog's speed is constant.

  17. I still maintain constant speed is irrelevant since you can determine the average speed if you know the speed as a function of time and use calculus to find the average. In some sense it would be good to teach high school students how to do this.

    And if they are not calculus students. Give them the speed in one second intervals, then see if they can figure out how to sum up the absolute values of the speeds at which point you are done. (Assuming one second intervals.)

  18. I've got another idea... How many passerby's would it take for Maxine to dig so far into the ground you couldn't see her anymore? You weren't kidding that she does it every time someone passes by. It reminds me of those cartoons where a character paces so much they dig a hole. Okay, sorry about the tangent, good post!

    So, taking the penny idea but making the students do a little more math, what if you gave them the total distance of the lawn, and put a timer on the screen where they could see the times that the dog stopped? That should be enough information for them to figure it out. (They could find the speed of the walker, figure out the distance that person traveled at each time the dog stopped and add up the remaining distances)

  19. hey. i've known about the *fly* since the sixties
    (though i'll only have understood the series solution
    since... well, what does one mean by "understand"
    here again, exactly?).

    but this video? coolest doggone
    illustration of the mad-back-and-forth
    (that this problem type depends on)
    that i can remember seeing, um, ever.
    almost makes me wanna go out and
    learn about some of these hightech tools
    everybody always seems to be so keen on.

    anyhow. now. how can we recover info
    about distances and walking speeds
    and so on from the video itself
    (with tools at hand)?

  20. Good one, Kate. This one's fun. I'm also glad I checked out the comments. No way I'd catch vlorbik's succumbing to (even brief) technophilia at my place. Made my evening.

  21. I think I could work with Sue's idea of "cute break from thinking about the fly problem" and Mr Sweeney (I just typed McSweeney twice and had to delete) 's "timer plus distance" - thanks for the ideas. And all the comments in general. Nice discussion for a spontaneous clip i took w/my phone.

    @vlorbik the basic flip cam is like $60 and tiny and unbelievably simple to use. It's basically one button that both starts and stops recording. And it doesn't even have any cords, the little usb plug pops right out.

  22. Oh and for the how-to, I suppose I should figure out one of these days how to superimpose a timer on a video. It can't be that hard, right? Measuring or providing kids with a good enough estimate of the distance of the initial leg of her run wouldn't be a big deal...

    If I got serious about it, I'd get my mom to stand with the Flip while I walk by, to capture both the dog and walker. Though, I wonder if someone else's nearby presence would alter Maxine's behavior. She doesn't do the same thing if you walk the opposing direction down the street, for example. Crazy dog.

  23. Okay, so I really dig the kind of WCYDWT problems that end with some kind of definitive answer we can contrast against our own. Sort of like finding out that the ball only looks like it's going to land in the trash can but really lands to the side.

    This one seems like such a finicky situation, though, unless Maxine freaks like that every time she sees you. If it were me, I'd chalk this one up as a win and fabricate reasonable inputs after the students tell you which inputs matter.

    Adding a timecode is, unfortunately, non-trivial. It's ridiculously easy within Final Cut Pro but FCP is a ridiculously expensive solution for adding a timecode track. I haven't given it a shot yet but the best shareware option seems to be QTSync, if you're on a Mac.

  24. I recently added a timer to a few videos trying to make some of my own "Graphing stories" videos like Dan's, but used a PC. After installing and trying a hojillion other programs (all of which pretty much failed) I settled on Corel VideoStudio 12. It's full featured and free for 30 days, but is ~$80 if you want to continue after the trial. Adding a timer using it was pretty simple.

  25. Reminds me of the hummingbird question.

    Incidentally, there's some great easy puzzles from that site that may be appropriate for the classroom.

  26. I LOVE that video! I taught middle school math, and am tutoring college students. My sister is a HS algebra/geom teacher and several friends teach math and physics, also. I can already 'see' the problem, along with the video, using the computers/presentation type things, maybe? Fun to figure out how to use it - and more fun to develop it!


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