Click the image for full size. You have to see it full size.
1. What questions perplex you about this photo?
What’s the perplexity score here?
davidwees: How many people there?
Peter: how many people are there?
Roz: How many people?
schwartz: is it bad that i want to know the area of the shaded regoin?
JG: How many rows can be added until the circle touches the pentagon?
Colin (@ColinTGraham): How was it built and measured out?
Sam Critchlow: what/why is the gap between the pentagon and the circle
JG: How much area is added on with each additional concentric circle?
Sam Critchlow: or what is the total open space area
Chris: how many more people to complete the circle?
Colin (@ColinTGraham): what’s the significance of the pentagon
Nick Hussain: how many more people/dwellings (?) could be added if the circle was completed?
JSR: what’s at the center?
Colin (@ColinTGraham): how many sectors if the circle was complete
We went with:
davidwees: How many people are there?
2. What is your guess? What is a number you know is too high / too low?
3. What information would you need to answer the question?
JG: Average number of people in each rectangular region
schwartz: people per square something
davidwees: I think we can estimate people per square
schwartz: radius of part circle?
Barb: Was admission charged? If so, who sold the tickets? They could give a ballpark figure
Colin (@ColinTGraham): is each sector evenly divided and how many sit in each of the eleven concentric rings?
davidwees: and get the scale from the size of the tracks shown
Roz: we definitely need scale
David’s response is right on point:
davidwees: so I think we could get a pretty good estimate without much more information
You don’t need anything more here. (I wonder what it takes to get students comfortable with imaginary units, as in “the radius of the circle is 500 burningmans,” etc.)
Nevertheless, here are two images that are interesting, if not useful also:
4. Submit your work.
I knew we wouldn’t have time for this. Here’s the Evernote page, though, where Colin Graham posted his work:
5. Show the answer.
[BTW: Though the photo is clearly timestamped 2009, various commenters have outed themselves as serious Burning Man attendees to tell me that this is 2010’s photo. I have adjusted the news clipping accordingly.]
Download High Quality
Mr. KOctober 6, 2010 - 6:37 am -
>>How was it built and measured out?
By hand, believe it or not. Outer arc measured with a transit and laser range finder, everything else involves lining up premeasured chains.
Your answer, btw, does not match the year for the image. (The red arrow points at my art project that has far outlived my attendance at the event.)
Sorry if this doesn’t help the math at all.
Elizabeth SOctober 6, 2010 - 6:39 am -
I’m with Mr K.
Dan MeyerOctober 6, 2010 - 6:52 am -
Sure? The satellite image says 2009. The newspaper clipping says 2009.
josh g.October 6, 2010 - 7:08 am -
What’s up with the little white-with-black-outline CG spikes scattered throughout the high-res image? It kinda of looks like it corresponds with population density in the camp.
Great lesson idea, btw.
Mr. KOctober 6, 2010 - 7:55 am -
I’m pretty sure the image is from 2010. It probably says a lot about my geek level that I can recognize individual camps from that.
For verification, the base of the central sculpture is different each year, and the shadows from your image match up with the shadows from here.
The difference in actual population between the two years is about 20%, so it actually matters, depending on how accurate you want your guesses to be.
AveryOctober 6, 2010 - 9:10 am -
Wasn’t sure where to post this as it pertains to the broader concept of WCYDWT, but I’ve realized all of the examples so far have something in common: a numerical answer.
I guess I worry a little that problems where the process is really the answer (the classic goat/cabbage/fox puzzle is the first example that popped into my head) are getting shortchanged.
Don’t get me wrong, I love the idea of WCYDWT. Just starting to think about how we can expand this beyond the scope of problems with nice number answers.
Of course, I don’t have any answers yet…
Dan MeyerOctober 6, 2010 - 9:28 am -
The numerical answer lends itself to guessing. Students can develop number sense by guessing the answer while at the same time making a painless investment in the problem.
Which isn’t to say you’re wrong. I thought I’d outline the incentive towards numerical answers, though.
josh g.October 6, 2010 - 9:50 am -
re: which year this image is from, Geoeye has another image from 2009 which people who feel like figuring this out could use for comparison:
It’s not as large of an image, doesn’t show the whole site … looking at the structures around the circle-thing looks like the same year to me. But the very central-thingy is hard to tell (maybe one image had a lot of people there at the time while the other didn’t?)
Would people have had buildings / campsites set up that identically between years? Never been there, wouldn’t know.
Dan MeyerOctober 6, 2010 - 10:16 am -
BTW: the article Dave mentioned about crowd density is this one. It’s really good:
How Do You Measure A Crowd?
RandyOctober 6, 2010 - 4:04 pm -
Before I read the article, I thought the photo was of some sort of solar power array system.
Carl MalartreOctober 7, 2010 - 6:51 am -
@ColinGraham: I find it funny how you and I did not use any circle math to do our estimates (all regions are equal in our world). I was at about 60 000 people, a bit over! I did not think that finding a precise formula was more important than finding the right number of people per meter (or your favorite area unit).
Do most people solving this problem just guess the number of areas and the number of people per area?
Anybody got better results with a different formula?
For example, let’s say that you have your formula and you just adjust the number of people per meter , we could see who got the best formula in the classroom?
Would it be pertinent to then ask students “who got the best formula” instead of “who got the best people per meter estimate”?
Dan DawsonOctober 7, 2010 - 7:38 am -
For those questioning if this image was from 2009 or 2010, it was definitely 2010. The shape of the temple (just North of the man) was very unique this year. I only got to stay for the first couple of days this year, but it was still worth the trip :-)
Got quite a few images of the buildings, cars, art and people:
People asked why the space between the circle and the pentagon, the pentagon is a trash fence with it’s goal to catch any MOOP (Matter out of Place = trash) that might get away from the campsites. They do add additional circles as the crowd increases, they even added a circle on Wednesday this year during the event. The dimensions of the streets vary depending on their goals, they made several blocks wider this year so the camps could have more room.
What, nobody wanted to estimate # of attendees by the number of potties provided? Where’s the creativity? *grins*
BrianOctober 7, 2010 - 9:04 am -
Since my class had fun being challenged by the area of the Pentagon problem in one of Dan’s slides, I would also ask as a challenge problem for those who finish first… Based on our estimate, what is the maximum number of people who could camp inside the pentagon.
There might be some interesting images of ultra crowded festivals that we could show for fun too.
AnonProfOctober 7, 2010 - 1:17 pm -
This is a great problem!
My observation is that this is a Fermi problem. In other words, it’s a back-of-the-envelope estimation (order-of-magnitude estimation) problem, which feels a little different than many of the math problems you’ve been using in the past. It’s in a similar vein to, say, “how many gas stations are there in the US?”.
I think back-of-the-envelope estimation is a fantastic skill to teach and I suspect you’ve barely begun to mine the potential of this particular area.
There’s lots of stuff written on back-of-the-envelope estimation among the physics community. Any physicists want to suggest favorite estimation problems that might be appropriate in a classroom?
Dan MeyerOctober 7, 2010 - 1:41 pm -
If this were some generic scrum of attendees, I’d be inclined to go the Fermi route. This is an incredible shape, though, one that lends itself to something more rigorous. Why not actually calculate the area covered by the campsites? It’s like a sector of annulus. (If there’s a name for this, someone let me know.) Colin goes the Fermi route, saying there are 176 regions, but the largest region looks at least 250% bigger than the smallest region. It’d be a shame to let all that geometry go to waste.
Bill BradleyOctober 7, 2010 - 2:07 pm -
@AnonProf I do Fermi Problems with my Physics students, but it doesn’t have to be Physics related. I use “How many text messages are sent in the US in a year” “How many gallons of gasoline are used in the US in a year” “How many people watched [recent televised event]?” and other things where it’s easy to find a “right” answer as warm-ups just to show them that with even a small amount of knowledge and reasonable assumptions you can get a good ballpark answer. I also use it to teach process, since I grade based on them explaining their assumptions and estimations.
Mr. KOctober 7, 2010 - 3:30 pm -
“”the largest region looks at least 250% bigger than the smallest region.””
The question that the city planners still can’t answer is:
“If the ring roads are 200 ft apart, how much error is there in modeling the blocks as rectangles?”
(And the people who deal with the resulting chaos can’t answer whether the error stays the same or changes as you go further out)
LauraOctober 7, 2010 - 6:47 pm -
So…the whole time I was reading this (especially the part where you listed the questions participants had) I kept thinking, “Am I the only one wondering what the HECK is this?”
I don’t think I could get beyond “What the heck is going on?” to move on to more mathematical questions/musings.
And thanks to this post, I did stay up til 2 last night reading about Burning Man instead of planning/grading. Lots of coffee was consumed this a.m.
Dan MeyerOctober 8, 2010 - 11:24 am -
Okay, fine. I fixed the attendance.
Really feels like I ought to include quotation marks around “fixed,” though.
DaveOctober 8, 2010 - 1:47 pm -
Laura, you are not the only one. I find the image fascinating, but completely overwhelming. It induces sensory overload, and I don’t understand what I’m looking at. If I were asked to make sense of such a picture as a high school student then I would probably give up in frustration.
But at least I can find the area of the pentagon in square pixels. I found the coordinates of the vertices using Paint, and plugged the numbers into the polygon area formula (http://mathworld.wolfram.com/PolygonArea.html). I don’t really know how to estimate the scale, since I don’t know what I’m looking at.
DaveOctober 8, 2010 - 1:53 pm -
Of course, any classroom will have students who have a wide range of skills. Some students will have high visual intelligence, but some of us will have great difficulty interpreting a complex picture. How would you teach the rest of us how to read a picture in a math class?
gasstationwithoutpumpsOctober 8, 2010 - 4:32 pm -
I think that the assumption Dan was making was that every student already knew all about Burning Man. Remember that he taught in Santa Cruz County, where is sometimes seems that Burning Man happens year round.
Colin GrahamOctober 8, 2010 - 5:22 pm -
Ok, some points…
The burning man was the third of three WCYDWT examples that Dan pushed us through. Knowing, I hope, a little about how Dan thinks, I abandoned my actual knowledge and tried to put myself in the situation of the average 15 yo in a maths class in the UK – not an easy task for maths teachers to do, and we need to remember this!
@Carl I worked on the basis that it would be possible to come up with a reasonable estimate based on treating each of the defined areas as being rectangles or, at least, as being of equal area, since I think most students would make the same assumption. I submitted my results based on this (incidentally none of the other participants did this, so I am not going to get defensive because I have nothing to compare myself against!).
@Dan Yes, I did adopt a “Fermi” approach, if you want to give it that particular label but, in the 4 or 5 minutes we had, I did not consciously think: “Ooh, I can apply a Fermi approach here…”, I just thought “OMG, I have 5 minutes to give this maths teacher geek an answer that doesn’t make me look like a complete moron…” OK, slight exaggeration! And I did at least go as far as qualifying my answer in terms of crowd density, which many/most students would not even bother about…. ;-P
I knew nothing about the burning man, I also knew nothing about the ESPN ‘football’ baby we looked at earlier, as our first problem…. In the UK, football could be rugby or soccer, and rugby could be rugby union or rugby league… This is nteresting from the point of view of other cultural issues that have been raised elsewhere. Is the number of cones on a dancer’s skirt something of cultural significance to me…? No. It’s just another arithmetic/algebra problem, therefore there is no cultural interference… Hmm.
As a maths teacher, would I want to go back and investigate the idea that each of the segments/sectors I assumed in my calculation was of equal size? Yes. Is it important for students to do this in a ‘rough guess’ scenario? No. Does it bother me that I couldn’t give a more exact answer based on using geometric formulae for calculating the areas of each of the individual segment? Yes, but we didn’t have time to do that and, as Barb pointed out, why not just ask the people who sold the tickets how many were there…? Pseudocontext? ;-P
Did I make a comment about the segment at 6 o’clock in the picture being circular, and so disrupting the equilibrium? Yes, but Dan edits the conversations, to distill the essence of what is important about the problem, for his blog entries, which is legitimate, but some of the comments people make here in response to Dan’s blog were already covered in the live session.
What I think is important about these types of “guess the answer” problems is that the majority of students should be within a certain percent, or order of magnitude, when they do the “back of the envelope” calculations. The application of some ‘logic’ to narrow down the crazy maximum and minimum answers is what needs to be identified and, in my view, taught or encouraged.
It’s interesting that we never discussed what would be reasonable or unreasonable answers in any of the three problems we did together, apart from a brief exploration with the “football baby”. “Reasonable” here was very much teacher-led = Dan. Is 60,000 way too high or a near miss for the burning man? How about 25,000… too low? How about challenging assumptions, what effect does that have on calculations? There is a lot of opportunity for exploration here beyond the session we had together.
In the 40 minutes we had with Dan, we went through 3 problems. Not everyone got their calculations right or close or in time, but I’d like to think we all came away with a better understanding of two things: how students might approach this type of problem; and, not everyone gets their “back of the envelope” calculations right under a time constraint (do they, Dan? ;-P )
Colin GrahamOctober 8, 2010 - 6:06 pm -
Just an after-thought
@Carl the meter is not my ‘favorite area unit’, neither is the metre my favourite area unit. Oops, spelling difference there… I made comments about metric measurements because the majority of people outside the US would be using metric estimations, and asking non-imperialists to participate in these kinds of session adds another dimension — potentially!
I just find it intensely annoying that the US continues to use Imperial (=colonial UK) measurements. The US, Liberia and Burma are the only countries which continue to do so. The metric system is more straightforward for students to handle, given the logic under which the original standards were defined…
And, I would have been much happier with weights expressed in stones, instead of pounds. 240 pounds means nothing to me, but 17+ stones does, being an ancient colonizer…! ;-P
Colin GrahamOctober 8, 2010 - 6:09 pm -
@Dan I hardly think this photo was taken at midnight on Friday… whatever hardened attendees might attest…
Dan DawsonOctober 8, 2010 - 7:39 pm -
Not that it relates to the math in any way, but for Colin, the image was taken Sept 2 at 11:41 am. :-)
Carl MalartreOctober 9, 2010 - 8:22 am -
Yahoo! With this “fix”, 60,000 was not too far :-)
@Dan is it useful to get the students immersed in the scenario with a video, for example:
@Colin Sorry for the meter typo. I’m Canadian, we work with… both units all the time ;-)
“Yes, but we didn’t have time to do that and, as Barb pointed out, why not just ask the people who sold the tickets how many were there…? Pseudocontext? ;-P”
As soon as it becomes truly interesting for the targeted students, I guess the job is done. They could use a web browser and just find the number, but they would not be too *impressed* by that path.
@Dan can we limit the resources of the students? Can students “pretend” if it’s interesting enough to not take the easy path?
Dan MeyerOctober 9, 2010 - 1:24 pm -
Sure. I mean, with the Big Baby lesson, I removed identifying information about the team, so students couldn’t Google the team roster.
During this particular lesson, I didn’t mention that it was Burning Man.
There seems to be rampant confusion about pseudocontext, specifically regarding the ease with which someone could derive an answer through non-mathematical means. (ie. “I could just ask the kid’s age.” or “I could just ask the ticket taker how many people came.”)
The real-world verifiability of an answer is a crucial feature of WCYDWT-style problems. After we do our work, I play the end of the water tank video. After we do our work, I reveal how many tickets are on the roll. After we do our work, I reveal how long it took me to walk up the down escalator. These make the problems more satisfying not less. The question is, do the water tank, ticket roll, and escalator lead naturally to the math, or is the teacher forcing the math through pseudocontext?
BrianOctober 9, 2010 - 3:03 pm -
Since the biggest region is about 250% bigger than the smallest… I would want my students to estimage using the size of the middle regions. After that, making this an area problem in the geometry would be where I would be taking it.
JenOctober 9, 2010 - 8:25 pm -
@Carl: “Do most people solving this problem just guess the number of areas and the number of people per area?”
As a Geometry teacher, I immediately went to the sector of an annulus approach. I estimated the outer radius at 1mile and the inner radius at half a mile. I also assumed the sector covers two-thirds of a circle. Once I had the overall square footage, I estimated that each person would comfortably require around 400 sqft by visualizing how crowded it seems in pictures I’ve seen of BM. ALL that estimation/assumptions led me to an estimate of 54,739 people. Pretty decent!
Great problem! By the way, I know of no better term for “sector of an annulus.”
TelanniaOctober 10, 2010 - 7:32 pm -
Sorry, I am new to following these posts. What does WCYDWT stand for? Great mathematical discourse problems!
Dan MeyerOctober 10, 2010 - 8:20 pm -
Hi Telannia, it stands for What Can You Do With This?
Alex EckertOctober 10, 2010 - 10:20 pm -
I haven’t read all comments, but in just browsing I didn’t see anyone mention where my class ended up taking this. But before I get to that…
I’m hooked. It took me a while to get hooked on this WCYDWT idea, but I officially am. Probably because it took me a while to actually administer a WCYDWT lesson somewhat correctly. Which didn’t happen until the last period of geometry on Friday. Which brings me to…
We squared a circle (circumscribed a square around a circle and used the radius of the circle to find the area of the square). Then pentagoned the circle, then hexagoned the circle, then octagoned, and so on. Eventually the kids see that a regular polygon with an infinite amount of sides is a circle, which is pretty cool because pi gets developed.
So Burning Man was awesome for us. The circle was pentagoned, and even though we didn’t go that route they still recognized the shapes and the beauty of the picture. My first geometry class found out how many people they thought were in the campsite. They didn’t even come close. My second geometry class got much closer. Then it hit me. “How much space do you think those 75,000 people (their estimate) would need to be comfortable?” “Here’s a square foot of space. How much space do you think 75,000 people would need?”
And they were off. Arguing and coming up to the board and drawing and arguing some more. Their guesses weren’t even close (I think the highest guess was 1,000,000 sq ft, when it turned out that the campsite portion of Burning Man was actually 42 mil or something like that). But they found the square footage of the campsite and loved doing it. I even videotaped one of my students, a former drug addict, saying, “I LOVE GEOMETRY!”
Further extensions to come…given the square footage allotted per person at Burning Man 2010, how many people would comfortably fit at the campsite if the “C” were extended to be inscribed inside the pentagon?
BrianMOctober 22, 2010 - 4:14 pm -
I gave this a try today… on a Friday afternoon, 7th period, after a midterm, during one of our first raining days. So that adds to the picture of what I was up against.
I was only able to get a small amount of buy-in. One problem I ran into was that they figured out it was burning man, and after I told them what year it was, one student googled the answer. This was problematic, because I initially had them guess what they thought the answer was, plus a guess that is definitely too high, and definitely too low. Having them guess at the answer is how I get a lot of buy-in for these WCYDWT. But once one student shouted the answer, I noticed a definite drop in interest. If you do this lesson, I recommend not giving them the year till you give them the solution.
We didn’t come up with a solution, because there was not enough time, so I presented it as a “let’s think about this”, and then we will solve it Monday. I had about 15 minutes available at the end of class. I think 15 minutes is too much for a “let’s think about this”, and not enough time for a “let’s solve this”.
I am going to pick the lesson up again Monday morning, and try to run with Alex’s suggestion of “How much space do you think those 75,000 people (their estimate) would need to be comfortable?”. I like this because I can get them back to guessing at an unknown quantity, which is an element I lost with my google searching student.