What can you do with this?

We interrupt our regularly scheduled long patch of radio silence to share with you an arresting, mathematically rich visual:

[source]

Thanks to Josh Kershenbaum for the tip.

UPDATE 7/17:

To clarify something: this image was originally created as an answer to a question, but I didn’t create a WCYDWT tag for that angle on it. To me, the image lands as (a) very beautiful; relatedly, (b) haunting, hard to get out of my head; therefore, (c) incipiently highly narrative – it is asking us to surround it with story, whether the original story conceived by the maker of the image or some other; and, lastly, (d) unavoidably mathematical. I don’t have a clear sense of the next move, but I do think this image opens a rich vein of something for a math teacher to use. The question “What can you do with this?” isn’t rhetorical: what’s your next move?

(See exchange with Dan Meyer below.)

20 thoughts on “What can you do with this?

  1. Yeah, super fun. Check me on my methods here, would you? This image has all the shock value of a good third-act plot twist. Like, “My word, that’s what all the water looks like?” So the first act is the question, “How large of a sphere would all the water in the world fit into?” (My right arm for an image that’d prompt that question.) The second act is the information listed on the page you linked plus the tool of spherical volume. Then the third act is the visual reveal. The sequel?

    The three-act story isn’t any less arbitrary a template for lessons than Madeleine Hunter’s seven steps. Maddy enervates me, though, where this math storytelling thing energizes me.

    1. Far be it from me to sap and impurify the WCYDWT brand. You’ve always been hard on images that aren’t posing a question. It’s not my style to open with a closer, either, though, and for this image I stand behind both the claim (“arresting, mathematically rich visual”) and the question (“what can you do with this?”). The visual was originally created (and the person who sent it to me saw it) as an answer, not a question, but my own take is that it’s just too haunting not to provoke curiosity, and too mathematics-rich for that curiosity not to be exploitable by math teachers. I have no idea what the next move is. I’d love any thoughts about it.

      Of course it could work as a (counterintuitive?) reveal, if we can find a different approach to the question it was originally created to answer – Gilbert Bernstein makes a first pass below. But actually, that’s not how it fell for me when I looked at it. Maybe it’s all the haunting, morally complicated sci fi I’ve watched in the last week (had just come home from X-Men: First Class last night when I found the link in an email, and it’s only been 6 days since I watched the first three episodes of The Sarah Connor Chronicles back to back…), but when I look at it I get an inarticulate but distinct sense of melancholy mixed with a hint of foreboding. The narrative isn’t clear to me but I know it’s there, it and 80 others. And because of the spheres, all 81 are unavoidably full of math.

    2. This is indeed a haunting image!

      When Dan suggested a sequel, this immediately came to mind. Of course there’s less water, but how much scarcer is it? . . . oh, and is that a question of volume or of surface area?

      (I’m only an ex-GTA but i hope to teach again and i’m so glad i found these great math teaching blogs! I’m paying attention.)

  2. Perhaps you could pose the question visually with the water from the globe draining off into a pan or trough that runs into some sort of spherical container?

    Try putting this into fixed-width text display to see what I’m getting at:
    ooo OOO OOO ooo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oOO OOo
    oO OOo
    oOO OOo
    oOO OOo
    ooo OOO OOO ooo

    O

    O

    +—————–O————————+
    / /|
    +——–/ ( ( O ) ) / +
    / +——/ / /
    |__| +——————————————+ /
    o | |/
    o +——————————————+
    o

    [ many different sized spheres here ]

    (credit here: http://ascii.co.uk/art/circle for the circle used as the globe)

    1. arghh, here’s a version with periods in place of most of the spaces that should display correctly, if less nicely:

      …………………………………ooo OOO OOO ooo
      ……………………………..oOO……………..OOo
      ………………………….oOO…………………….OOo
      ……………………….oOO………………………….OOo
      ……………………..oOO……………………………..OOo
      ……………………oOO…………………………………OOo
      …………………..oOO…………………………………..OOo
      ………………….oOO…………………………………….OOo
      …………………oOO………………………………………OOo
      …………………oOO………………………………………OOo
      …………………oOO………………………………………OOo
      …………………oOO………………………………………OOo
      …………………oOO………………………………………OOo
      ………………….oOO…………………………………….OOo
      …………………..oOO…………………………………..OOo
      ……………………oOO…………………………………OOo
      ……………………..oOO……………………………..OOo
      ……………………….oO………………………….OOo
      ………………………….oOO…………………….OOo
      ……………………………..oOO……………..OOo
      …………………………………ooo OOO OOO ooo
      ……………………………………….
      ……………………………………….O
      ……………………………………….
      ……………………………………….O
      ……………………………………….
      ……………………….+—————–O————————+
      ………………………/……………………………………/|
      ……………..+——–/……………( ( O ) )………………/ +
      ……………./ +——/……………………………………/ /
      ……………|__|…..+——————————————+ /
      …………….o…….|……………………………………|/
      …………….o…….+——————————————+
      …………….o…….

      [ many different sized spheres here ]

  3. The first question that hits me when I see it is, “No way. Really?” And I reach for my nearest envelope or napkin to start Fermi-ing on. A lot of mathematics is driven by posing surprising conjectures and then trying to verify or disprove them; maybe this is in that category.

    The next questions that arise for me are, how many miles high is the drop? how much water per person is that? How much of it is fresh? And so on. Another question that might arise: “So if you spread it all back out, the oceans must be really thin – but I thought they were deep.” (The aspect ratio of width-to-depth of the ocean has been compared to a piece of typing paper, or the thickness of the chalk if you draw a circle on the board with radius of approximately your arm.)

    Of course what you DO with this is see if any of those questions, or others, arise from your class. If they don’t, put it away, move on, and rethink the intro.

  4. This is cool. I’m intrigued and want to know what people come up with. I agree with Dan about the ideal version of this problem…the only thought I have is to make a few images like this one with water droplets of different sizes: equal to the earth, bigger than the earth, half the diameter, way smaller, etc.
    Reading this whole discourse reminded me of a series of problems Larry Zimmerman introduced to our department a few years ago. I couldn’t find my notes on his original problems, but when I searched online I found this series (http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/rope-around-earth)…Problem A and D1 are the ones that I’m familiar with, and the juxtaposition of them is what makes it so interesting. People’s intuition is consistently wrong if they are unfamiliar with the problems.
    Rock on folks.

  5. Uhmmmm, not to mention the value of this on day one this coming August, if…

    you’re a Marine Biology teacher in Missouri. Perfect. Thanks for posting.
    😉

  6. Aliens. This picture makes me think of a giant UFO come to suck up all our water. Assuming no water-condensing technology, how big would their container need to be?

    But also… this could be an extension of a problem involving fruit. Get a kiwi, a lime, an orange, an apple, a potato, peel them, and squash their peels into spheres. Finding the volume of the next fruit’s peel given some information about volume and thickness of peel would be a natural question. And then this picture becomes an extension.

    Alternately, since we can play movies backwards, start with a squished up ball of apple peel and play it backwards onto the fruit, then show this picture. How much of the Earth’s surface is covered with water? If it’s really 3/4 of the Earth’s surface, how deep is the water.

    Extra points if you could decide if the thickness of the water “peel” is most like which fruit, or even replicate the experiment in fruit, carving away only the peel that doesn’t represent land mass and squishing it up.

    1. I was waiting for somebody else to have a sci-fi angle! I can’t get the questions

      WHY IS THERE A GIANT WATER DROPLET SUSPENDED OVER THE AMERICAN WEST?

      and

      WHAT WILL HAPPEN WHEN IT FALLS?

      out of my head!

      (Love the fruit connection as well, btw.)

      1. That’s how perplexing the respondents found their own questions. A one means they don’t care all that much about their question. A four means they’re dying to know, etc.

  7. So after reading the source material, my first question is what would be the size of the fresh water sphere compared to the all water and similarly the fresh water sphere to a total of saline sphere?

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