Well, I see that you could certainly lead a question more if you had Chris put on an old-school hockey mask or you both had paper numbers on your shirts. Another question could be led if you had an explosion going on behind you both at the left end of the photo.

Here’s what comes to mind when I see these photos:

1) Start with a video clip of you alone running towards a destination with a clear clock in the video to open with the rather simple question of how long will it take you to get to the destination (I can’t think of a great way to frame the video to make this look like a real meaningful question). When students ask for specific positions and specific times you can throw up slides like what you have here.

2) Show another video clip this time a wider shot showing someone chasing you and ask will the chaser catch you before you reach your destination. Again more slides with positions and times. Hopefully this leads students to the concept of relative speed which is a nice example of an abstract mathematical concept but one that people still have some intuitive feel for.

3) Now switch to a ball rolling down an incline, but make sure there’s a plain background behind the incline and angle the camera so it looks like a flat surface. This time start with two slides showing the ball’s position and time and ask how long it will take the ball to get to the end (the end of the incline is painted or something).

4) After students make their predictions show the video and ask them why their predictions were incorrect. Ask them what the slides would look like if you took a picture of the ball every 0.1 seconds. Show them a sequence of slides and have them graph speed vs. time and distance vs. time.

5) Maybe you have students fit equations to the graphs and this is an activity about relating real world motion, graphs and equations (You could also go back and graph #2 and look for where your line and the chaser’s line intersect). Or maybe this leads into a discussion of derivatives and making the connection between pictures of the object at different moments to different points on a graph and how you use these points to approximate the slope.

Oh, Dan, please know that you have inspired me immensely this year. I am personally at a point near exhaustion, but my classes have been much improved by your ideas.

I led a lesson with a similar focus six weeks ago. See my take on my (super-new) blog: http://larkolicio.us/blog/?p=42 . I’ve been working on the idea for a while, and this is not my best lesson after your example, but I think it’s there.

I think a video is a more appropriate medium than photos here. In my class, the students could navigate to different times on the video, and I never had to focus them by picking specific times or anything – they had a TON of information and had to pick some to use, which is the thing I admire most about your approach. With only photos, you tell the kids which times you think are important, what distances are important, specify units, etc.

I don’t have a totally satisfactory way to challenge kids’ assumptions in a delayed way in this lesson yet. However, in my lesson I didn’t show the kids the end of the race, and some assumptions flew that were challenged right away. That also happened in a conflict about slope, changing speeds, units of measurement, etc, but again only in the way that happens in any discussion.

I really want to like this. They’re an improvement on the word problems we recognize under the surface. Better for helping students understand what they are dealing with and what is given and what is being asked. And I’m totally using them next time I teach Algebra 1. But if there’s a cognitive crisis lurking in there, I’m not seeing it. I don’t mean to be a pill, but I know the blowing of sunshine is not appreciated around here, so.

What would I DO? I might not bother with #1, or show it for 10 seconds. I’d leave #2 up and say very little until I heard some version of “He’s going to catch him” and hope that someone wonders when or where.

And #3, I don’t know. That would be the most important and the trickiest. I WANT to say “ask students to discuss a method for deciding where he catches him, and then share them after a few minutes” but depending on the group that might go nowhere. Failing that, and if it was a small enough group that a discussion with the whole group made sense, I would have to try to anticipate all the potential approaches, helpful and not, ahead of time, so I could keep the discussion in class making enough progress that kids didn’t lose interest.

The REALLY tricky part would be to generalize the very concrete examples to an approach that would work no matter, so that I’m teaching Algebra 1 and not 7th Grade Again. But at the same time, the concreteness is very helpful, because it keeps giving them something to grab onto. #3-8 need to be printed on handouts. That’s all I got. Someone, please do better.

First off, I’d be quick to agree with Kate and Alex’s comments. Asking when the pass occurs seems natural, or who wins the race to a specific point are useful questions. To the question of what kids do with the photo, I had to re-read your post and my first comment to think about this more concretely. What’s the acitivity? What specifically will they have to do with it?

I’d pass out the pictures in addition to projecting them, it’s nice to be able to write on this stuff.

Questions they’ll ask:
– What do you expect from me on this assignment?
– What if someone falls, speeds up, or slows down?
– What really happened?
– Am I right?
– How come there are two of the same person in the photo?
– How long is it in between exposures?

Finding the point where one person is overtaken is not going to hit the sweet spot for an eager student, for a reluctant student, it may be tricky.

I’d begin with some simple questions inviting everybody to weigh in
– What’s happening?
– Who’s going faster?
– Who’s gonna make it to the last marker (on most shots this was labeled 210) first?

Follow up with some more involved calculation questions on a handout or projected for them to complete on printouts of the photos
– How many time intervals will it take for them to reach the final marker?
– Between which two intervals do they pass?
– If this were a track, and a lap were double the distance marked, how long would it take to run x laps?

For students who breeze through, the crisis question can be stated in two ways:
– Exactly where do they meet?
– Exactly when do they meet?

For students who can do that, maybe have them extend it to a more general proof:
– Explain why if you are behind someone in a race and you are going slower than them, then you can’t catch them
– Explain how to find the exact meeting point for two racers at different points along a track at a fixed speed

General expectations that might be useful for this are:
1. Write down your process, be as clear as you can and organized, but don’t be afraid to brainstorm and think on paper as well
2. Think aloud with your neighbor. Share ideas.
3. If you can’t find an exact answer, write down an incorrect answer below and one above where you think the correct answer is.

As for a quick comparison of your pics here with Sean Sweeney’s, I’d really like Sean’s video if it were from a side viewpoint, as it is, I wouldn’t expect students to be able to generate a graph of location just watching runners run directly away from their vantage point.

My one thought stepping back from all this is that it doesn’t always need to be digital. Good questions do not require good pictures/video/digital content.

You could also try some Very Restrictive Unfair War with Dice – a game that is still not copyrighted. Give one student 15 points and another 35 The student who’s behind scores 4 each turn while the one who’s ahead scores 3. Ask the same set of questions. Iterate by giving larger numbers, decimals, fractions.

Or have students imagine a race in which each member of the class walks the a course on the perimeter of the classroom at the same time. Some go two steps at a time, some go one at a time. Have them predict when they’ll catch different members of the class, do this as a paper activity first, and then have everyone get up with their calculations and move either 1 or 2 steps at a time on your count. Have them verify their calculations at each pass.

Hmm. A passing comment from a social studies teacher…take it or leave it or throw me under the bus.

This is the first WCYDWT that I can recall leaving me positively nonplussed. The reason I always read the WCYDWTs, even though they never go into my classroom, is that I enjoy finding out how you’re going to turn an ordinary experience / easily argued about problem and scale it into all different kinds of mathematic principles.

The intellectual crises that these images lead to seem to be at their core just a repackaging of the good ole’ textbook word problem with trains leaving the station at different times. Really, who cares?

“Don’t just whinge, provide a solution.” -my last boss

A race is not static like photographs. I think that these would pack punch and hook students in if each scenario was first preceded by one second video clips from a similar vantage point, or even from a first person view.

Heck, get inside a track and put the camera inside the announcer’s box. Or spend ten minutes on youtube. Stop the tape of a 200 meter run as the athletes enter the final 50 meters. After the howls, ask the class, “Who’s going to win?”

Or put on that Jason mask. Seriously.

I found this series of photos very confusing. A lot of that confusion I think was due the fact that the wide angle view obscures the detail of the runners into near obscurity. Sean Sweeney’s video suffered from the problem of the increasing difficultly of judging the position of the runners as they get farther from the finish. These photos suffer from lack of detail caused by the wide angle view.

I was really confused by the apparent time transport that occurs between photos 3 and 4. (as well as between other adjacent photographs). The fact that photos 1 through 3 are obviously a set led me to think that the photos were all related to the same race. With the help of the comments above, I now think that you didn’t intend for the photographs to be a series but to have each stand alone as an individual problem.

The information that the strobe shots were taken one second apart definitely re-sparked my interest and led me
to consider each photo as two overlaid snapshots from a separate race from which to have students make predictions as to the entire course of the race. My confusion now subsides but my interest also does. Now each problem presented becomes a pair constant rate problems presented visually.

I liked the problem better when I was totally confused.

Here’s my suggestion on how to “be less helpful”.
1) Drop the strobes.
2) Start with a wideangle of the entire course, the position of the runners and a time reference. Ask the students, who is going to win?
3) Add closeups of sections of the race course with time references. Have lots of of these and show the students only what they ask for. Give an index of the available closeups to the students and challenge the students to predict the winner by viewing the fewest number of close ups as possible.
4) Make the first problem linear.
4a) Make the second problem linear.
4…) linear again
5) Crisis – introduce a non-linear problem (have a runner stop to tie his shoe or trip and fall.)

Optional lesson. Graphing stories. Hand out a random set of snapshots to different groups of the students and have them graph the course of the race and then discuss why different groups got the same or different results.

Optional race: Have a runner race a basket ball shot. Linear versus quadratic. Do you graph horizontal versus
vertical or position versus time?

i must be the oldest one here – or the one with the worst eyes. i need bigger. :)

i love what you’re doing dan. and more importantly – my kids do.

the frankness and authenticity of this post in particular….thank you for that as well.

When you and a hobbit are being chased by a hungry dragon, you don’t need to outrun the dragon. You only need to outrun the hobbit.

Thanks, that makes sense. I think I was too used to your usual WCYDWT formula of only showing the first image. Good lesson for me there – focus on the opener and find the good stuff there first.