Probability and Percentages

Probability reared its random head in the 3-4s today. We investigated vocabulary such as impossible, unlikely, equally likely, likely, and certain, in addition, of course, to random, defined by one third grader as "not moving your hand around and around and around trying to find exactly the right one." We also introduced various ways of using numbers to write probabilities. If there is just one red card in a group of 5, then the chances of getting a red card (at random, of course), are "1 out of 5," or "1 in 5," or "1/5."

Or "20%." Percentages can be tricky and often require some serious numbercrunching to carry out. At the same time, they can be extremely useful in comparing two probabilities (it's hard to tell by looking whether 3 out of 7 is better or worse than 12 out of 29) and in getting a rough idea of how likely an event actually is (a percentage is easier to interpret than a fraction like 57/243). So we did some fairly straightforward percentages, using what students already know about fractions and division. If the probability of drawing a red card is 1/5, that's also 20%, because 20 is one fifth of 100. And if there were two red cards out of 5, the probability would be 40%--double the previous one. We also looked at more complicated situations such as 1/7, dividing 100 by 7 to get an approximate equivalent of 14%. Not good odds, the classes agreed.

Later, on their own, they found the probabilities of various events, expressing them in both fraction form and as a percentage. Conversion was easy enough when the denominator of the fraction was 10; most students recognized right away that 7 out of 10, say, was 70%. Other denominators were a bit more complicated. Still, the students persevered, and in the end 110% of them thoroughly understood percentages...wait...

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!

Bad at Fractions

I just about always wear a collared shirt with buttons to school, so several kids noticed when I showed up in a T-shirt today. "I don't think I've ever seen you in a half-sleeve T-shirt," one fourth-grade girl commented. "I have," a classmate said proudly. "Really?" I asked. "Here at school?" "No," he said. "In a restaurant."

There was a reason for the shirt. The third and fourth graders are working on fractions, and the shirt's message is, well, fractional. It proclaims:

5 out of 4 people are bad at fractions.

I used the shirt's message as a very informal way of checking students' understanding of fractions and fractional thinking. My hope was that they'd lodge a complaint, and fortunately I was right.

"Your shirt's wrong," one student stated flatly after she read it. "It should be '4 out of 5 people are bad at fractions,' not 5 out of 4."

"Yeah," a classmate agreed. "It doesn't make sense this way. If there are only 4 people, you can't take 5."

"It can't be more than the whole," someone in another class pointed out. "It's 1 and one fourth, but that doesn't make sense when you're talking about people."

"The shirt is bad at fractions," somebody said. "It's a bad-fraction shirt. It's complaining about people being bad at fractions, but the person who made it is the one that's bad at fractions."

"They're trying to disguise the fact that they're bad at fractions," noted a fourth grader.

"So I guess I should take it back to the store and exchange it for a shirt that's mathematically correct," I said. "What do you think?"

A few nodded slowly, but the bulk of them shook their heads. "It's a joke," someone explained. "People will see the shirt in the store and say, 'Oh, that's wrong!' and then they'll buy the shirt to make other people confused."

That settled, we moved on to the rest of the lesson.

There's a lot more to fractions, obviously, than determining what's wrong with a T-shirt statement. Still, it's kind of fun to use something as mundane as a T-shirt to do a brief informal assessment--and nice to know that the kids could see the error, and even, perhaps, the irony.

Photo credit to Rhiannon P. in Jan's class.