Follow Up: Will It Hit The Corner?

I’m very impressed by the commentary in the kick-off post. The default WCYDWT stance has the eager math teacher stroking his chin and musing that “we should really tie this into gas prices somehow … ” while studiously avoiding the essential, practical details of constructing a framework for that learning. Instead, at freaking last, our commenters are starting to attack those logistics with a certain thrilling mania, developing full-bodied worksheets, manipulatives, and Geogebra appletsHats off to: Steve Phelps, Jack Bishop, Josh Giesbrecht, Dan Schellenberg, Nick Hershman, and Justin Lanier. Great work..

I’m not exactly sure of the best route through this problem. In fact, the one that interests me most is one I don’t know how to solve. I hope you can help me with that. I only know one thing:

We can’t learn much from an obscure background element of a video clip unless we drag it into the foreground. We need our own copy of that bouncing DVD screensaver. So I made one in AfterEffects. [download clip]

Your goal with these intro clips should be to infect your students with as much of PB&J’s anticipation as you can:

Take bets: will it hit a corner with five minutes? Ten minutes? Put a few students on record.

Now ask your students, “what matters here?” There are nearly ten variables you can define together. Ask, “what are good ways to measure what matters?” Pixels, angles, speed, time, etc.

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Play this clip. It features information that should, ideally, surprise no one. Your students have abstracted all this information already. You’re just taking their hard wor and pressing play. [download clip]

From there, take your pick. You could give them something fairly explicit like this [download image]:

Or you could just give them this grid, 720 by 480 with ten-pixel increments, go frame-by-frame through the movie, and pick out some data points together. [download image]

The awesome observation they should make, regardless of what route they take, is that, once that icon starts moving, the rest of its natural life is foretold. It’s totally predictable in this frictionless environment.

By my count, we’re still missing a clip.

We need video of the solution. It’s one thing for you to consult your answer key (the full measure of your authority) and confirm a student’s answer. (A: lower-left corner at 1:34.) It’s another thing entirely to say, “It doesn’t matter what I think. Let’s check the tape.”

So here are five minutes of the DVD screensaver.

Now will someone teach me how to solve this algebraically?

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.

21 Comments

1. Mr. K.

February 12, 2010 - 2:25 am -

>> Now will someone teach me how to solve this algebraically?

Srsly?

My immediate take on it is to run it next to the Donald duck pool video, and ask why one is different from the other.

Proper scaling will turn this into a diophantine problem, no?

2. How to solve it? Let S be the number of squares the box has to travel before it hits the corner.

The box hits the top after 19 squares, and every 38 squares thereafter. So S = 19 + 38a = 19(1 + 2a).

It hits the side after 31 squares, and every 62 squares thereafter. So S = 31 + 62b = 31(1 + 2b).

We need whole-number solutions to 19(1 + 2a) = 31(1 + 2b). Both are prime, so 1 + 2a = 31 and 1 + 2b = 19. Then a = 15, b = 9, and S = 19 x 31 = 589.

Answer: the box hits the corner after travelling 589 squares, or 5890 pixels. On the way it bounces off the horizontal edges 31 times, and the vertical edges 19 times.

If the box is travelling at 57 squares every 10 seconds (by inspection), then it will hit the corner after 1:43.33333…. seconds.

3. Note: actually an easier problem that some kids could then answer is “when will it hit the corner again?”

4. Barry Lewis

February 12, 2010 - 4:53 am -

I’m preparing to teach math and I don’t mind checking my understanding (or naÃ¯vetÃ©) publicly, in front of this legion of esteemed commenters; I want to know that I’m seeing the physics correctly. The logo will never hit the corner for the same reason that it never has, right? Once hitting the corner means forever hitting the corner. Is getting to that aha-point one of the goals of the lesson being considered here?

5. Dan Meyer

February 12, 2010 - 5:59 am -

@Alex, that’s great. And I’m pretty sure I see how that translates to different screen/icon sizes. One correction:

On the way it bounces off the horizontal edges 15 times, and the vertical edges 9 times.

@Barry, I’m not entirely clear. It does hit the corner â€” 1m43s into the video. I presume it will hit that corner again, and only at that point will the loop close and repeat over and over again.

6. Barry Lewis

February 12, 2010 - 6:08 am -

Yes, of course. Only if the rectangle is a square would a logo that hits the diagonally opposite corner on its maiden voyage be eternally committed to that one reflective path. I should check the shape of the box that I’m trying to think outside of!

7. Hey Dan,

So I just worked this problem for a period with two kids (crazy testing schedule meant that everyone else was testing…). We used the handout I linked to, and found that if you start a (1×1) logo in a corner it’s positively sloped diagonal lines (picture the line of grid squares hit as the logo rises from bottom left to upper right) cover half of the squares. It’s negatively sloped diagonals will cover half the remaining squares (so 3/4) and if you were to choose an untouched square and set the logo running it would pass through all the rest. Basically, there are two possible paths the logo can take, and we found that either way it will hit two corners and miss the other two.

Proof forthcoming, and it would be so awesome to have a python app to model the grid and the box and to be able to resize and test theories on. That may be a weekend project, though I’m not sure I’ll be able to make it happen. If anyone does, I’d love to hear about it.

Suggestion When I do this in a larger class with more reluctant students, I might suggest that students start by working the problem on a 4-by-5 grid, then an 8-by-10… working up to the 64-by-80. I don’t like the idea of having the worksheet build up to that (seems a little forced), but I might suggest it if folks are stuck.

8. Update I made the python program and I feel glorious. The parameters are simple provide a width and height and starting width and height and the program runs until it hits a corner. Then a grid pops up of your specified width and height and shows all the squares that were covered with a color coding (green is the initial square, orange are passing squares and red is the final square).

You can download it here.

It’s written on python 2.5.x and might not work on python 3.x.
The four variables to set are at the top of the file (lines 6 – 11)

9. Gilbert Bernstein

February 13, 2010 - 1:32 am -

I used to amuse myself with a slightly more discrete version of this phenomenon while using the bathroom, or sitting around anywhere else with a tile floor for an extended period of time. Imagine firing a little particle or laser beam at a 45 degree angle from one corner of a rectangle made up of… say 5×3 square bathroom floor tiles. I’m bad at this question approach to things, but here’s a go:

What’s the shape/image traced out by the particle/laser beam?
How many corners (of the rectangle) does the (let’s just say) laser beam hit?
Which corners does the laser beam hit?
how long is the complete path traced by the laser beam?
How many corners of the square tiles are hit by the laser beam?
Can the laser beam pass through a square tile’s corner point in more than one direction or not?
Can the laser beam cross a square tile in two different ways or not?
How do all of these things change when you change the number of square tiles composing the rectangle?

Excuse my complete lack of an attempt to put this into a lesson plan or anything resembling it. (I’m not a math teacher) If I were forced to do that, I think I would start with having students trace the shape of the laser beam on, say graph paper. Then maybe try different rectangle sizes? This seems to be the easiest way to quickly make this activity more visceral.

10. Gilbert Bernstein

February 13, 2010 - 1:37 am -

Also, I should mention a more esoteric and colorful modification of the above musing.

You are standing in a rectangular room whose walls are covered in mirrors. There is a guard with a laser gun standing in this room with you and they seem rather mad at you. They shoot you and you die: the end. Suppose you could have built some number of infinitely thin columns in this room (and suppose that you and the guard are essentially points). How many of these infinitely thin columns do you need to erect to keep the guard from being able to shoot you? (note that if the walls were not mirrors, one column, directly between you and the guard would suffice to block all possible shots)

11. Sue VanHattum

February 13, 2010 - 9:47 am -

The first chapter of of Harold Jacobs unusual text, Mathematics: A Human Endeavor, does something like this. He starts off talking about pool tables, and, like Bernstein did above, limits you to using a 45 degree angle. The questions become: Which corner will you end up in? Can you tell in advance? Which tables give a simple pattern for the path, and which give a complicated pattern?

The chapter is titled Mathematical Ways of Thinking. I once led an online group using this book. (We fizzled out after the first chapter.) My notes on this problem are here.

12. bmc456

February 16, 2010 - 6:40 pm -

thanks for doing a lot of the leg work. i don’t have all the technical skill to set up the short clips. how did you learn how to do this? I’d like to be able to do this on my own.

13. CLIMEguy

February 17, 2010 - 10:50 am -

Follow up to Sue’s post:
Great that you mentioned Harold Jacobs and his book: Math: A Human Endeavor. I was Dan Meyer’s age when I discovered it back in 1972. It changed my teaching from deliverer of content to explorer of what’s possible. In fact Harold was one of the pioneers of this WCYDWT theme. Instead of asking what could be done with a potentially powerful idea, he thought about the best to do it so that it would engage his high school students in the hum-drum topics of math by using engaging scenarios and stories to challenge them. The best part of the book was not the book itself but his teacher guide – at least for his first edition – which is what I used. It took me on an exciting journey through math and sharing it with my students was absolutely wonderful. (Though intended for HS, I used it with 7th/8th graders in a private school in NYC.)

I participated in Sue’s effort to help homeschooling parents learn the book. She did chapter’s 1 and 2 and I did chapter 3
For those interested in more details, here was my take on Chapter 3- Functions and their Graphs
Link

14. Dan Meyer

February 18, 2010 - 10:46 am -

@bmc456, I’ve used Final Cut Pro and Adobe After Effects for a few years now. A lot of these techniques become easy with practice, after a lot of initial fumbling. This particular exercise was way beyond my skillset, though, but became a lot easier as I Googled around for answers to my questions.

15. Alex has described the general solution.

If the distances the box travels horizontally and vertically (call them “h” and “v”) are relatively prime, then the box will always hit a corner eventually.

If the distances are not relatively prime, then it will eventually hit a corner if and only if it starts at a multiple of gcd(h,v).

So I would suggest playing with different widths and heights. “Does the box ALWAYS hit a corner eventually?” (No; you can set up a square and have it bounce around in a diamond, for instance.)

Also, you can tile the plane with copies of the screen. Then the path of the bouncing box is isomorphic to a straight line along the plane. This is another way to arrive at the same formulation; you are just asking whether the line passes through a multiple of (h,v). This is arguably an easier way to formulate the solution if the box does not move at 45 degrees.

Lots to play with here, although only your most advanced students are likely to “get” the complete solution.

16. mbs

July 1, 2010 - 10:29 pm -

Wow – great lesson and approach. Any way you could make the after effects file available?