For the last month, I have had this single image banging around in my head, hogging valuable CPU cycles. I couldn’t find it anywhere else so I shot it myself. Click the photo for high quality. See the pilot for instructions.
BTW: The comments feature no fewer than two dozen lesson inspirations, at which point the questions become (I think) which lesson inspiration a) will sustain the most interesting math the longest? and b) which prompt can be summarized the most succinctly, the most viscerally? I think those are two of the most important metrics for evaluating these ideas.
Under that light, you have Ben Wildeboer: “Calculate time before impact with the ground.” It’s visceral. The student wonders first how she’ll find that information in a static image. It seems impossible. The result is a page of physics function work.
Also: “How high was the ball when it was dropped?”
Or: “How long has it been in the air?”
My work for both of those questions:
The first answer is off by nearly a meter. That’s just under 100% error.
BTW: A reader writes to let me know I blew the math here.
It seems that the work shown is using different reference points. At the top of the diagram the top part of the ball is used, and at the bottom of the diagram, the bottom part of the ball is used. I think the top of ball should be used for both or the bottom of ball; either of which would require knowing the diameter of the ball.
He’s absolutely right, which would explain the 100% error.
17 Comments
Alby
November 23, 2008 - 9:43 am -There’s a fantastic lesson on projectiles hidden here.
Just the fact that the ball is more solid towards the top of the image shows that it’s slowing down and there’s all sorts of speed calculations possible if you know that the image was taken over 0.04s.
dan
November 23, 2008 - 10:58 am -So you can see why I couldn’t dislodge this picture once it got a clawhold in my head. The ball is actually falling, though, and was dropped, more to the point of the problem.
Scott Elias
November 23, 2008 - 11:42 am -So you dropped it. Which means Alby’s observation still applies, of course… A great starting point for discussion. “Can you even tell if it’s going up or down? Would it matter?”
OK. So the ball was moving more slowly at the top, but not because it was slowing down on the way up, because it was speeding up on its way down.
You could extend this one from Algebra through Calculus…
Starting with Alby’s ideas, you could examine the image and pretty closely estimate the distance traveled in 1/25s to find the average velocity of the ball over the “blur.” Velocity at the top would be 0 regardless, but the average velocity would need a + or – depending on whether it’s moving up or down.
Bring in some kinematics and you could calculate the velocity at the bottom. Thanks to Dan tipping his hand, we know it’s vf at the bottom and vi at the top, but what if we didn’t know? Would it make a difference if we let the bottom be vi and the top be vf?
For Algebra, explore the symmetry of the parabola on a height vs. time graph – why would the picture look the same regardless of which side of the vertex we were on?
Bust some Calculus and do some differentiation to look at the instantaneous velocity at any point. Bring in vectors to talk about what makes things “slow down” (a and v pointing in opposite directions) or “speed up” (a and v pointing the same way).
That’s just off the top of my head.
Dang I miss teaching.
Ben Wildeboer
November 23, 2008 - 11:52 am -Lots of good physics-y stuff here.
1. Hide the EXIF data and calculate the exposure time.
2. Calculate time before impact with the ground (though you wouldn’t need the blur for that).
3. Calculate acceleration due to gravity (since you can figure out the distance the ball has traveled in the specified time).
4. Calculate how long the blur of the ball would be if the shutter was opened after the ball had already fallen 50cm. Then, if you’re really tricky, take that picture and compare how accurate your calculations were.
5. Similar to above, you could calculate how long the blur of the ball would be if you used different shutter speeds.
6. You could do some stuff with the optics (focal length, etc.), but I think there would be more effective ways to get at that.
7. Have them figure out the setup to even get this shot. My guess is it involved a tripod and a remote shutter button, but the ball (assuming it was touching the ledge originally) appears to only have fallen about 2-3 cm when the shutter was opened. This means if you dropped it by hand you got your hand out of there pretty dern quick. I’m guessing there must’ve been some other way to release the ball.
I think I’d have them preform various calculations and them give them a tennis ball and see if they can verify their calculations experimentally.
dan
November 23, 2008 - 2:41 pm -Scott, nice. Good stuff. Now here’s the thing that excites me lately. I’m past the point of training my eyes to correlate interesting images & video to math standards. I’ve got that. Where I’m at now is, once I experience that initial spark, how do I develop it into a list of possible lessons, something taught and assessed, and then how do I select the ideal lesson from that list?
Like Ben says about his optics prompt, “I think there would be more effective ways to get at that.”
You’ve tossed out lesson ideas across the high school math spectrum. How do we now decide which of your lesson prompts has the most potential? How do we then develop it?
I feel close to some kind of grand unification theory for math curriculum design.
Ben,
re #1: without knowing the height from which the ball was dropped, how do you calculate exposure time? I could’ve dropped the ball from a ladder with a very fast shutter speed to match or from right above the meter stick with a slower shutter speed.
likewise #2: without the blur, how do you know if the ball was dropped from a mile in the air (and is consequently moving very very quickly) or from a inch above its current position (and is only just picking up speed)?
I like #5, though I can say from experience that reproducing the estimate with a camera, tripod, and tennis ball would be very, very difficult.
Ben Wildeboer
November 23, 2008 - 4:30 pm -re #1: I was making an assumption (which in viewing the picture more critically was obviously erroneous) that the tennis ball was dropped from the top of the meter stick. You would need the initial height to figure this one out.
#2: Again, this was based on the assumption that you knew the height of the release. However, with the blur, this is a nice three-stepper (at least the way I calculated it).
According to my rough measurements, you dropped the ball from a height of about 2.75 m (just about 9 ft.). Is it fair to guess the room where this picture is taken has 9 foot ceilings?
Maybe there’s a little CIA in this lesson as well. You hear about experts who look closely at photos and video to determine the location of individuals (I’m thinking along the lines of bin Laden, kidnappers, etc.) based upon subtle clues in the recording. For example, I’d bet you took this picture at school? Schools tend to have taller ceilings than the standard 8 ft. residential ceiling. Or perhaps your residence simply has cathedral ceilings. Not sure if this tangent is too applicable to a math-specific class. I certainly see where it could fit into a forensic science class.
Mr Follett
November 23, 2008 - 5:40 pm -Derive formula for velocity of ball based on length of streak, ball diameter, and shutter speed. In fact, I want to try that…
Kate
November 23, 2008 - 5:43 pm -My first thought seeing the picture was “Time Warp“! Most of it consists of filming things with expensive cameras that look cool when you slow them way down. Then they spend 10 seconds on a token physics lesson, then they show more cool stuff. Kind of like Mythbusters or Good Eats, I’m always looking for excuses to incorporate clips into class.
Tom
November 23, 2008 - 6:19 pm -I’d give them the height of the drop and have them calculate the exposure time. The for extension have them graph exposure time vs height to achieve the same length of blur.
I think that’d work.
Bryan Cook
November 23, 2008 - 7:07 pm -Sorry that this is an unrelated comment, but I’m just getting started with a teaching blog of my own. As part of my learning process, I’ve been advised to “audit” some well-written blogs, and I chose yours as my first audit. Your site is recommended on my blog, and I’d love to have you take a look at it and comment with your thoughts on my page at your convenience.
Unfortunately I don’t think I have ANY readers yet, but perhaps your goodwill can pay off in the future once I become rich and famous in the blogosphere ; )
Scott Elias
November 23, 2008 - 7:25 pm -Dan – The last thing I typed in version 1.0 of my response was, “Given some time, I’m sure I could make an actual lesson out of this…”
That’s the hard part, indeed.
Alby
November 24, 2008 - 9:34 am -Ooh ooh! Introduce (spontaneous) symmetry breaking! You can’t tell whether the ball is rising or falling so it’s symmetric in time.
Jason Dyer
November 25, 2008 - 1:57 pm -I’m most interested in the fact that the amount of exposure the ball had was longer at the top than the bottom (giving a visible effect) because the the acceleration changed the velocity.
dan
November 25, 2008 - 9:43 pm -FYI to the commenters here: I added a summary judgment to the post itself. Thanks for the ideas, everyone.
Mr. K.
November 27, 2008 - 6:23 am -It must be a middle school math thing.
The question I want to ask is “How long after the ball was dropped was this picture taken?”
David
November 29, 2008 - 10:27 pm -Am I the only one who had difficulty perceiving the image? All I could see was a meter stick and a green blur until I scrolled down and read the discussion. :(