Once all the students had the complete and accurate number of kernels, we assembled in the Chapman Room. "Who thinks they have the MOST kernels of anyone in all three classes?" I asked. Several people were pretty sure the honor was theirs, but one young man from Jan's class took the prize: he had 644 kernels on his ear of corn, a full 43 more than the next runner-up.
"Okay, how about the LEAST?" We had a few who coulda been contendahs, but again one student won out--another of Jan's students, down at 289.
All right. We had the greatest and the least values. One way of describing a set of numbers, I explained, is to find the range: the distance between the least and the greatest. (This tells you roughly what kind of a spread you have in the data: are the numbers generally pretty far apart, or are they mostly close together?) As a group, we estimated the difference, then subtracted to find out. "Close together, or far apart?" I asked when we had our result.
"FAR APART," chorused 48 voices.
How right they were. The range was--quite large. Taken together, the two lowest figures were less than the highest. There's plenty of variation among ears of corn, evidently.
Next we turned our attention to the median, or the center value when the numbers were all ordered. We had the students sit in a line--well, technically a curve--arranged from 289 up to 644. When everyone was in order I had them all stand and look around. "Where do you think the median value is?" I asked. "Point to the person who you think had the median amount of corn kernels."
Fingers waved toward the middle of the line. Most people in the middle of the line pointed to themselves. To find out the real answer, we started at the outside of the line and had students sit down two by two: 644 matched with 289, 601 matched with 293, and so on. Like a very slow row of falling dominoes, or perhaps like spectators doing the wave at a baseball stadium, they sat down, or fell down, depending on their level of coordination and their penchant for dramatics. Little by little, the number of children standing diminished. The 500s disappeared altogether, so did the 300s. The upper 400s took their seats. People began revising their predictions.
Before long, we were down to two students. One had amassed a total of 408 kernels. The other had--412. There was an even number of people. The answer, someone realized, was to split the difference, and that's exactly what we did. The median was 410. If you wanted to choose one number to stand for all the numbers in the group, you could do a lot worse than choose 410.
(The picture below shows the Final Two. Everyone else has been eliminated from contention as the Merry Median of the Corn Kernels. Thanks to Jan for the photo.)
One more project remained. You've heard of the Living Flag? Well, this was to be a Living Histogram. (A histogram has nothing to do with allergies--it's a bar graph in which the bars stand for a range of numbers rather than a single figure or response.) We had the students divide themselves into groups, according to the number of kernels: up to 299 over here, 300-349 over there, 350 to 400 in that corner. Then we called the groups over one by one and had group members sit in a line, creating eight lines of varying lengths in all. "What do you notice?" I asked, and they noticed quite a lot. The longest line was in the middle, they explained, the shortest lines on the outside. It was like stairs, someone said; it was like a roller coaster, said someone else. They were quite right, too. It was about the normal-est curve I'd encountered in the last few months--the nice bell shape you read about.
(Here are the lines. You might recognize the two almost-median-winners, smack dab in the center of the longest line there in the middle of the photo. See how neatly all these things work out?)
So, a nice way to spend a misty, mathy morning. The kids enjoyed getting their minds around the concept of range and median--and did it very well, I might add. They were surprised to see how big the range actually was, and they very much liked using their own bodies to locate the median. And while some of the players were beginning to get a bit restless toward the end, they kept their sense of curiosity about the graph and loved the idea of constructing it themselves. We'll continue to explore range and median--and who knows, we may get back out to the Chapman Room with a different set of data someday!
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