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’m going with A. It asks a question, makes you curious, yet it’s simple and keeps them focused.

    B is too hard to figure out what’s going on and C has way too much chaos and it seems to have no question (although I like the bird).

  2. I like that C sort of shows that the path is always a parabola. A looks easier to try to get measurements from. (B looks like some communication with aliens.)

  3. I would say C because it has the most information in it, although Tom’s point about simplicity is good. Maybe you should show A first and C later, after the students have already done some thinking?

  4. A is best, in my opinion. We have a sense of how hard the ball was thrown, unlike in B. Unlike C, not everything was shown for us. “Will the ball go in the net?” goes out the window in C.

  5. A: I think this one is focused and clear enough both mathematically and visually. There is also a compelling question as to whether the ball will go in or not.

    B: The focus seems to be the trees and background (it’s pretty where you are). Doesn’t seem like enough to figure out anything about the ball. Perspective is also challenging to deal with in this. Although, this is the true perspective of a player.

    C: Focus is not mathematical. Students will laugh at your miss and having to chase the ball around (especially with the second you running after it). The bird in the background (as Rhett points out) is distracting and could be a focus. Although, it does show the linear (bird) vs. quadratic (ball) very clearly.

  6. What are you wanting them to understand? Picture c has linear, quadratic and exponential (if you consider rebound ratio of the bouncing ball) so I think it would generate more discussion.

  7. It’s all visual poetry. Give any of the 3 to this English teacher and I’ll dig into it for a full 45-min classroom.

    BTW, I’m still wondering how you managed to get so many balls into the air at one time…and all in a pretty line/arc. This puts the Globetrotters to shame. Will you be touring soon?

  8. Love (the first half of) Christian’s reply. Gets at SOMETHING (who knows exactly what though) of the difference between teaching quanti- and qualitative subjects.

    As for “most valuable”, the most valuable is the photo that fires you – the teacher – up. All three are an improvement over the standard clipart type drawing in a text book, but they seem to increase in complexity from A–>C. Whichever is the highest letter you can make a compelling story* (for lack of a better word in my vocabulary; I don’t teach in a classroom) with, that’s the one most valuable to student learning.

    *or maybe “question” is better. The question in the first is essentially given. The next two ask different questions that need a bit of parsing before they can be attacked, imo.

  9. I’m going with B.

    Why? It’s less helpful.

    Let them ask for the others because they will. Someone will tell you that you were kind of dumb with your choice of angle because it doesn’t show what they need to know. Then you get to ask: “What is it that you need to know.”

  10. I’m with Jake all the way. Give them B so they’re forced to tell you it’s a bad picture for answering any questions they want to ask.

    Next give them A. If you want to talk about curve fitting and extrapolation I would actually give them another shot before A that only shows the first three balls. With only a few closely spaced data points absolutely everything looks linear and their extrapolation will have your shot soaring over the hoop. Then give them A and redo the curve fit/extrapolation.

    Finally give them C and see how long it takes someone to notice, as Rhett did, that there’s something flying in the background. Estimate the distance to the mystery object and estimate its speed (either by giving them the frame rate or for extra challenge have them infer the frame rate from your shot). This will tell you if it’s a bird or a plane (I don’t have time to do this myself right now so I don’t have a vote yet).

    Basically I love all three pictures.

  11. A. While B and C could create good discussion, A is the only one where the students could really get into the math of it. B does not have enough information. C has too much (So there’s no question).

  12. Based on previous Dan posts, I’m inclined to agree with Alex’s analysis. Shot B is the closest to what you see if you shoot the ball.

    However, Kate’s recent complaint about not enough time to teach the standards begs the question of how far from the final question one should start. Can I afford the 10 min of discussion to get from B to A? Or is the question asked in A interesting enough to bypass that?

  13. I’m going to have to go with C.

    B’s perspective asks a question, but it also seems to make measuring the necessary information very difficult (How far away is the basket? How far does the ball travel between exposures? etc.)

    A is clear, asks a question, and makes it easy to measure. It’s too easy to pick out the one question it’s asking and too easy to solve it.

    C is the best, IMHO, because it has a lot going on to distract the observer. There are lots of questions you could ask & answer using it (What’s the ball’s initial velocity? How much energy is lost every bounce? What should the ball’s initial velocity be in order to make a basket? How fast is that bird flying? etc.). It’s the most like real life.

    Although you’re probably most likely to always get the questions a la John Pederson: How did you clone yourself? Is that your twin? The ball doesn’t have a shadow! Is it a vampire?

  14. If I were to show one: A

    Two: A then C

    Three: B, then A, then C.

    A and B both ask “will it go in?” But I feel that B would be difficult to actually work with. What calculation(s) would be done? You could figure out how far away the hoop is by its relative size to the basketball, but is that accessible and is it the point of the lesson?

    I think B is a good setup for a discussion of mathematical modeling as a concept itself. We don’t always choose player perspective, it may not be the most helpful. We simplify our view, make 3D into 2D, eliminate extraneous information, etc. So, as an example of doing that, flip to Image A…

    Image A: students can ask “will it go in” and students can answer it by fitting a parabola via Geogebra.

    I initially thought of C as just the “answer” slide, but I like David Cox’s idea of analyzing the successive bounces. How are they related to the first parabola’s height? Since the other bounces are coming off at an angle, tie it back into the B/A transition: how should we actually model this? “How many bounces before it starts rolling?”

    Also… with regards to the bird: Notice the equal spacing, suggesting constant velocity. Ah, but the basketball images have a varied spacing. Or does it? x spacing could be (close to) constant while y spacing is under the effect of acceleration, i.e. gravity. I don’t have a good question to ask at this point. But I’m liking all three right now.

  15. There’s a lot of talent assembled in this thread, so I’m going to give my thoughts (summary judgment: the answer is A.) but don’t take them too prescriptively.

    1. B is useless. A few people have defended B on the grounds that it’s closest to the student’s POV. This would be true if we were talking about the POV of a student shooting a ball, but what we’re really talking about here is the POV of a student watching another student shoot a ball, which means we need more distance, a wider angle, lots of room on the edges, and a neutral, flat composition.

    2. C is the most interesting. I really love C. I love the parabolas. I love the bird. The point of this post is to tease out the idea that the most interesting image isn’t always the most valuable image. I don’t know if we can challenge students here for any sustained period of time. I don’t know if we can iterate this over several similar photos. It seems to me a good closing photo just to show that, yeah, there are parabolas everywhere. Maybe we can talk about restitution (the percent decrease in bounce height) but I can think of better ways to illustrate it. Maybe we can talk about linear motion vs. quadratic motion. All we can do is talk about it, though. We can’t really do anything.

    3. Like others have said, A asks a concise question (“will it go in the basket?”) that allows students to intuit an answer without risking anything. (“Sure, you can kind of just see it.” vs. “No, it’s going short. You can tell.”) It lures students in. We can load it into Geogebra and map out the path. We can iterate it. I shot several more baskets, including one that goes way, way above the frame, and still goes in. I shot several that looked good but bounced to the side. We can show the answer by showing the rest of the shot, either confirming the student’s correct answer or talking about sources of error with an incorrect answer. The verb is do. What do students do with the multimedia? Hence, option a. Because they can do more.

  16. A reader just e-mailed to ask after some technical specs.

    I shot the video from a Flip cam mounted to a tripod. I exported the video at 15 fps (I think) to jpg images, which I imported into Photoshop, where I masked out everything but the basketballs and the birds in each frame. In the final frame, I also included myself chasing the ball.

    That’s it. These skills are difficult at first but they become very very easy. I realize hardware and software are expensive but I highly recommend the Adobe Creative Suite to math teachers. An ability to use Photoshop and AfterEffects to manipulate images and video has made a much better math teacher out of me. It’s made the job more fun also.

  17. And if you’re looking for a low-cost (ok, FREE) photo editing program that can accomplish the same thing, just download GIMP from your nearest open source mirror.

  18. @Jackie, sorry, what you see is what I got. Poor organization.

    @Rick, good reminder. I need to redouble my efforts on that front. No Photoshop tutorials. Only GIMP. Now go find us some FOSS that compares even a little to Final Cut Pro & AfterEffects.

  19. It depends on what you do.

    Choose A if you go with an exploration and C if you go for direct teaching methods. Some people actually do. Also, C can be a nice illustration to follow up ANOTHER exploration – to make a basketball connection – say, if kids just spent two hours working on parabolas with toy cannons and feel like they know them well.

    B can be used briefly in the middle of a story, but it does not set up either an exploration, like A, or a whole story/connection, like C. So, B does not contain much story power in it, but it can fit into another story.

  20. Mr. Meyer:

    I got your information from a video I watched on OSCON, I found out about OSCON on #ubuntu-california on irc.freenode.net, after a discussion about road trips. You said to get in contact with you, so here is my first attempt.

    I want to go with C, but really, I think the answer is “it depends”. I wouldn’t be surprised if some students would benefit more from A than C, and even some students might prefer B.

    I think there’s plenty of math in all of them, except the one thing that I think C has going for it is that it’s a more complete picture. I do remember drawing a picture is an important part of mathematics, I remember drawing pictures in Algebra, Geometry, Algebra II, and Trigonometry. I don’t remember drawing very much in Calculus, but maybe because Calculus in college is hazy. Did I ever draw a picture of a triple integral? Probably not, I bet my professor would have been terrified to see what I would have done on the topic.

    Kevin Demarest

  21. @David, no.

    @Kevin, thanks for getting in touch. You and Maria both point out that these are all valuable in their own ways. I suppose a better prompt here would have asked for the liabilities and advantages of each.

    I’m still going with A.

  22. Why do you think 1st period was more into C than 3rd period?

    Here’s a hypothesis: Maybe your 3rd period are more into drawing than photography? I wonder if they would be interested in drawing their own pictures of basketball?

  23. i love your series on wcydwt. it’s really challenged me to think about new ways to see the world around me and incorporate everyday things into the ways that i teach my students.

    have you seen this site? http://www.chrisjordan.com/current_set2.php?id=7

    description from the site: “Running the Numbers [the exhibit] looks at contemporary American culture through the austere lens of statistics.”

    he takes really interesting pictures that tell the story of the way Americans consume different types of products, like gasoline, cell phones, cigarettes, etc.

    i’m not sure what you could with it in a classroom b/c a lot of the analysis is already done, but i wanted to share it with you and your readers in case it could be useful :)

  24. Dan,

    I know Photoshop is fantastic, but I’ll bet the same effect can be done at about the same difficulty level with Gimp (free software).

    As for “after effects”, I’m not sure if there are any free alternatives. Does anyone have suggestions?

  25. Thanks, Bill! I will check it out.

    By the way, the bird in the background is a great item to use for constant velocity motion analysis.

  26. Hi I was wondering if the basketball shot in picture A made it into the basket. Could you please tell me.