The Seven-Sided Hexagon; or, Epic Geometric Fail

So there I was in the local Rite Aid the other day, picking up grape juice and glue (don't ask), and my path toward these goodies took me down the Seasonal aisle. Where I discovered, much to my surprise, that though it was already early June the Back to School sales had not yet started and there was not a single Halloween mask anywhere in sight. (I wonder if the Home Office knows about this?)


More amazing still, I spotted a few assorted pieces of summer merchandise. Notably, a large box that contained a "Hexagonal Canopy." Which had a helpful explanatory diagram on the side of the box, for those who weren't quite sure what "hexagonal" meant:



Oops.

Pentagon, Hexagon, Heptagon...

So there I was in the kindergarten, and the children were showing me how well versed they had become in the shapes of the pattern blocks (thanks, Robbie and Bill!).

"This shape has three sides," said Robbie, holding a triangle so the kids couldn't see it, and the children chorused "It's a triangle!"

"This shape has four sides, and they are all equal," she continued, and "Square!" shouted the class.

"And this one has six sides..." "Hexagon!"

"Oh, oh!" called out a little guy upon seeing the shape displayed (and yes indeed, it WAS a hexagon--phew!). "I know another shape! It's LIKE the hexagon! It's a--a--" He screwed up his face, thinking hard... "It's an OXagon!"

All I can tell you is, I would dearly love to see an oxagon in the wild. Wouldn't you?

Finding the Center

One of the perks about being a member of NCTM (http://nctm.org, the National Council of Teachers of Mathematics) is that you get a subscription to a journal called, what else, Teaching Children Mathematics. This journal has a monthly feature called "Problem Solvers," which presents an open-ended problem and encourages teachers to try it with their classes. Teachers are then invited to write up their experiences and send 'em in. From time to time I've tried these problems out, and once I even got around to sending in my reflections.

Anyway, a recent Problem Solvers challenge caught my eye: How would you go about finding the geographic center of the United States (minus Alaska and Hawaii)? O-ho! I thought. This will be an interesting problem to do with all the grade levels I work with! But then field trips and special events got in the way, and so did division and fractions and 3-d geometry and other such valuable topics--so in the end I managed to do the problem only with a few 4th graders and a few 1st graders.

At some point I'll talk more about the 4th graders, who generally did quite well--they showed some sophisticated thinking about the assignment, and made use of a number of different mathematical skills to come up with an answer. This post, though, will be about the 1st graders, whose work was...um...

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For the children , this was one of the easiest questions I'd asked all year. “It’s right here,” said one girl, touching the middle of the border between Kansas and Nebraska. The others nodded agreement and, not to be outdone, put their fingers on roughly the same spot themselves. That part of the Great Plains has never been so crowded.

This was a good estimate—a very good estimate, in fact, but I was looking for an explanation of how they'd figured it out too. When no clear explanation was forthcoming--in fact, when there was no explanation of any kind--I asked whether there were any tools they could use to show me what they were thinking. When I said tools I had in mind, oh, rulers, or some other kind of measuring device. They did not.

“A jackhammer?” suggested one boy.

"You could use a compass," said the girl who had made the initial estimate. "You would walk with the compass. You can start anywhere, like in California. Then you walk this way.” She put her finger near San Francisco and slid it eastward on the map. “When you get there, you stop.”

“How do you know when you’re there?” I asked.

She shrugged. “Because you’ll get to that place, and then you’ll be there.” She was too polite to say Duh!, but you could hear it all the same.

“What do the rest of you think?” I inquired. A chorus of “I agree”s and “Uh-huhs” rose from the other children. I believe this is called proof by intimidation.

I decided we'd better back up. “How about this table?” I asked. “Where’s its center? And how do you know?” Several hands slapped down in a place reasonably close to the center, if not the exact spot. The center, they explained, had to be in the middle of the lines that divided the table in half. Duh! Again, politeness reigned, but I knew what was what.

“So now we know about the center of the table,” I said. “I wonder if that might help us find the center of the country.” I opened up the map again. “What do you think?”

There was brief discussion. One child pointed out that the United States wasn’t a nice regular shape, such as a circle or a square, so it didn’t really have a center. Another argued that the whole world would have a center, “because that would be a sphere and then you could find the middle of it.” But they all deferred to a girl who cut to the chase. "The center would be right here," she said, stabbing a forefinger at a spot in the middle of Kansas, just south of the original place chosen. “That’s the center.”

Back we’d come to our starting point. “How do you know?” I asked once more, feeling like the twenty-first century version of a broken record and hoping she'd say something about lines that divided the country in half...

Nope. The child looked at me with something resembling pity. “You go up in a plane,” she said, “and then you can see where the center is and you go there.” Duh!

From which I conclude one or more of the following:

*First graders seriously underestimate the size of the country.
*First graders see no reason to calculate the exact position of a center when eyeballing it will do.
*Sometimes it’s really hard to explain your thinking, especially just before lunch on a Monday morning.

Oh well--onward!

P. S. If you'd like to know more about the geographic center, here's a rundown: http://en.wikipedia.org/wiki/Geographic_Center_of_the_Contiguous_United_States.

Fractions + Transformations = ?

This is a recipe.

Start with a small 4x4 square.

Sketch a continuous series of line segments (no curves) to divide the square neatly in half.
(Be interesting, please: no fair drawing a straight vertical or horizontal line or a simple diagonal.)

Prove that the two sections do indeed take up the same area.


Check to see if the figure has rotational symmetry. (That is, if it looks exactly the same when it's rotated any distance less than 360 degrees.)

Color the two sections contrasting colors.

Repeat the process 3 times. You may transform the original design by a) rotating (turning) the design 90, 180, or 270 degrees, or b) reflecting (flipping) the design as if it were appearing in a mirror. You may also keep the design oriented exactly the same as the original.


Arrange the four squares into a larger square.

Repeat this larger square four times. Place these together to create a sixteen-square unit.

Write a description of what you did.


The pictures show the results.

And the answer to the equation? Well, we could say "Fractions + Transformations = An Example of Applied Mathematics." Or, we could also say simply "Fractions + Transformations = Art." Your choice!

Pi Day

One of the great mathematical holidays of the year is Pi Day, which occurs every March 14. (It may be the only mathematical holiday of the year, in fact, but never mind.) The third and fourth grade classes celebrated Pi Day this year by deriving this number--or as close as we could get.

We began by distinguishing, of course, between pi (the number) and pie (the food). Several students said they'd heard of pi (the number). All students said they'd heard of pie (the food). Once we had that out of the way, we introduced the concept of DIAMETER (distance across a circle, through the center) and CIRCUMFERENCE (the distance around a circle). "Okay, here's the question," I said. "Are these distances related? If you know the diameter of a circle, can you use that to calculate the circumference without measuring?"

To explore, we had pairs of students use tape measures to determine the diameter and circumference of various circular objects in their classrooms--clocks, round tables, woven mats, stools, and more, rounded to the nearest whole centimeter. Here's a sampling of their results:

Bowl d = 9 cm C = 28 cm

Stool d = 21 cm C = 66 cm

Wheel d = 13 cm C = 45 cm

Garbage can d = 49 cm C = 150 cm

"Any patterns in the data?" I asked. Ye-es, the students said tentatively; the circumference is always more than the diameter. "By a lot, or by a little?" I queried. By a lot, they agreed. Usually, one cautious soul hastened to add. "So maybe if I add the same number to each diameter, I'll get the circumference," I suggested. "What number would I have to add to get the circumference? Talk with a partner and see what you come up with."

They considered the question and then they shook their heads. The numbers didn't work. You need to add a small number to the diameter of something small, like a magnifying glass, they pointed out, but you have to add 100 or even more to the diameter of something big. Clearly, addition was NOT the way to go.

If addition didn't work, the obvious answer was multiplication, and the students quickly moved in that direction. (Multiplicative thinking at work! Hurray!) "The circumference is, like, double the diameter," one student suggested. "More than double," a classmate countered. "Triple the diameter," said someone else. We tripled a few diameters using mental arithmetic. "When you triple it," someone concluded, "you get just a little bit less than the circumference." "Usually," piped up our cautious friend from before. "Oh, I know!" shouted an excited third grader. "You triple the diameter and then you add one! Oh, wait a minute--"

This, as it turned out, was a job for a calculator. We keyed in circumferences and divided by the diameter, then recorded the ratio (a new word for most students) on the board, cutting the endless stream of decimals to two places. 3.33, 3.00, 2.97, 3.24... "They're mostly around 3," students noted. I nodded. "We can't measure exactly with our tape measures," I explained, "but if we could, we'd discover that the ratio is always the same--a little over 3. This number is pi. The digits of pi go on forever, but are there any guesses about what pi would be if we just used two decimal places?" There were plenty of guesses, of course, there always are, but one exceptionally observant fourth grader had a reason for her answer. 3.14, she said, for why else would Pi day be March 14?

Why else, indeed?!

A footnote: This year, Congress passed a resolution officially declaring March 14 to be Pi Day; the resolution was also an attempt to highlight the importance of math education. Incredibly enough, ten representatives voted against the resolution. It's fun to speculate why. Perhaps they have had bad experiences in the past with irrational numbers, or maybe they think the value of pi should be determined by the open market, not the government--oh, wait... More information about the bill can be found here.

(Photo credits to Jan Campbell, and to Ellen DeLong's camera.)