Paul Salomon’s “imbalance problems”. You know how I love a thought-provoking picture.
We interrupt our regularly scheduled long patch of radio silence to share with you an arresting, mathematically rich visual:
Thanks to Josh Kershenbaum for the tip.
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.)
Okay, one more shout out.
My colleague Japheth Wood (News from the Math Wizard), with whom I’m delighted to be co-teaching a class for preservice teachers, is an awesome problem composer. He’s also been dipping into the sea of math and math ed blogging, one toe at a time. He’s finally got one whole foot in:
In particular, check out that image.
Don’t move past it to the text (mine or Japheth’s) until you’ve sat with it long enough to absorb everything. If you find yourself with a mathematical question, don’t move on till you’ve tried to answer it.
Seriously; stop reading and go look.
There’s a lot of difference, pedagogically and content-wise, between this image and Dan’s boat-in-the-river video but there’s something very important and very exciting in common. Both manage to ask a very specific and mathematically rich question without whispering a single word. I think the natural current could be strengthened a bit by putting little venus flytraps on top of the square numbers, but that’s the only improvement I can think of.
So, questions for you:
a) What question, if any, does that image leave you with? (Am I right that there is a natural question you can’t help but have once you’ve absorbed the image?)
b) What are the features of the image that lead to the question? Given a mathematical question, how do we go about turning it into a wordless image that asks it?
Japheth and I passed out slips of paper to our class of preservice math teachers a week ago. On the slips were either Japheth’s original image, or tieandjeans’ modification. We asked them to write down a mathematical question that the paper provoked, and then try to answer it. We didn’t give them a ton of time. (Less than 10 min.) Interestingly, while I thought the sense of danger in the modification would make the gravitational pull toward our intended question (will the grasshopper manage to avoid all the squares?) greater, our students’ knowledge of Super Mario Bros was a distraction, because the up/down motion of the plants, and the question of Mario’s specific trajectory, became relevant considerations. (You can see below that one group, perhaps reading what we were going for, explicitly ruled out those considerations.) So I think the students that got the original grasshopper image actually gravitated toward the intended question more predictably. I still think the sense of danger would help, but maybe we just keep the grasshopper and add venus flytraps that appear static and aren’t close to the trajectory?
Anyway, here’s what they came up with. As you can see, for all of the above, the natural current is still pretty strong.
Grasshopper / Mario Problems
If the placement of fly traps continues, and Mario times his jumps so the trajectory never hits the plant, will he ever land on a fly trap and die?
Does Jiminy land on a box?
When does Mario’s jump not clear the venus fly trap?
Will the grasshopper land on one of the empty boxes (perfect squares)?
Hopping by 4. What is the mathematical formula to determine where Mario lands?
Mario will never land on a perfect square.
How many times will the grasshopper land between consecutive perfect squares?
For # between , , if odd then , if even then .
Will he ever land on a perfect square?
Can the grasshopper keep jumping without hitting the black box? If no, then when will he hit the black box?
Grey boxes increase by +4
Black boxes increase by consecutive odd #s, +3, +5, +7, +9
Do you notice a pattern between the grey boxes? How about the black? Can you predict what # Mario will land on next?
The next plant will appear at 36. Mario will land at 26, then 30.
When will Mario land on a flower (perfect square)?
Will Mario die and when?
When will Mario land on a plant?
Can he change the size of his jumps and still ensure he will not land on a plant?
Will the grasshopper ever land on a black square?
Can (4n-2) be a perfect square?
Will the grasshopper land on a boxed number (i.e. a perfect square)?
Sum of squares must be a multiple of 4 or odd.
(Ed. note: they’re misquoting a result they found the previous week. The result was about the difference of squares.)