So we were working on time in the 1-2s this week. Today, the kids named and ordered various units of time, from milliseconds and seconds up to centuries and millenniums, and explained what they knew of the relationships between them, using nice NUMBER SENTENCES (60 sec = 1 min, 1 day = 24 hours, etc). On the whole, they did quite well, though why "half an hour" and "5 minutes" don't count as separate units of time was a bit mysterious for a few of our first graders. Next year--
Anyhow, after this lead-in, I asked children to fill out a sheet about time units. The structure was simple enough. "It takes about one SECOND to..." was the first one, and kids were supposed to think of an activity that takes about one second. Then they followed it with one minute, one hour, and one day.
The responses were fun and revealing of children's understanding: one second to "squash a bug," "pick up a feather," or "say four letters of the alphabet," one hour to "clean my room" or "draw a perfect picture" (quick, tell Picasso!), one day to "make a really good sculpture."
My personal favorite, though? "It takes about one minute to do my homework." Given that mathups, reading, and spelling alone are supposed to take at least 20-25 minutes each night, this is the sort of statement that is perhaps better left unsaid. Ah well--by high school I'm sure this child will have figured that out!
Friday, May 29, 2009
Thursday, May 28, 2009
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...
*************************************************
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.
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...
*************************************************
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.
Wednesday, May 20, 2009
Psst! Rules!
PDS is a progressive school. right? Right. So it doesn't bother with silly things like rules, right?
Well--no. We do bother with rules. As well we should. The first rule of the school is, or used to be, "No one may interfere with the learning of another." A good rule, and a sensible one, which recognizes the important distinction between rights and responsibilities, even if it lacks the pizazz of what used to be the school's second rule, "Windows are not doors."
We have plenty of other rules regarding behavior, too, many of them developed by the children themselves (most of these begin with the word No, as in No hitting/No kicking/No biting/No poking people with sharp sticks/No knocking over other people's block buildings) or by the children in conjunction with a teacher (these are stated much more positively--Walk in the halls/Be nice to people/Share classroom materials/Treat other people's block buildings with respect). My favorite of these from my classroom-teacher days was Eat regularly, which had nothing to do with having frequent snacks but was one second grader's attempt to condense No burping at lunch and No opening your mouth while you're eating to show people what's inside into one relatively positive statement...
But the rules I want to discuss in this post are math rules. Yes! Math rules, as I use the term, are statements about math that are always true. We encourage children to come up with these rules as they work on problems and as they talk about math. Here are 3 examples, named for the children who first suggested them:
These may seem basic to you, and of course they are. None of these will make the pages of American Mathematical Monthly. They've all been discovered before. On the other hand, let's keep in mind that these are first and second grade children who are just beginning to make sense of the number system. The idea that numbers will always behave in a certain way is by no means obvious. Most things about school aren't that predictable. Think about reading: the letter o can stand for several different sounds, some words add an s in the plural while others add es, a word like set or right can have multiple meanings. Math is different. Math is a little more--reliable.
So we ask children to look for these always situations in math. We ask them to think carefully about how numbers work, about whether a particular result will happen no matter what or whether it will happen only sometimes. The benefits, I think, are clear. A child who recognizes that an odd number plus an even number will always be an odd number is thinking hard about numbers and their properties. She's doing some basic number theory; she's acting like a scientist, making and then testing a conjecture. A child who sees that adding or subtracting zero never changes the original number is finding a pattern, generalizing from special cases, and boldly going where (to the best of his knowledge, anyway) no man has gone before.
Yes, we could tell children that odd + even is always odd, and very often we do just that. But it's more powerful, and the effects are longer-lasting, if children come to discover these rules themselves. Now if you'll excuse me, I need to go determine whether every even number greater than 2 can be written as the sum of two primes...
Well--no. We do bother with rules. As well we should. The first rule of the school is, or used to be, "No one may interfere with the learning of another." A good rule, and a sensible one, which recognizes the important distinction between rights and responsibilities, even if it lacks the pizazz of what used to be the school's second rule, "Windows are not doors."
We have plenty of other rules regarding behavior, too, many of them developed by the children themselves (most of these begin with the word No, as in No hitting/No kicking/No biting/No poking people with sharp sticks/No knocking over other people's block buildings) or by the children in conjunction with a teacher (these are stated much more positively--Walk in the halls/Be nice to people/Share classroom materials/Treat other people's block buildings with respect). My favorite of these from my classroom-teacher days was Eat regularly, which had nothing to do with having frequent snacks but was one second grader's attempt to condense No burping at lunch and No opening your mouth while you're eating to show people what's inside into one relatively positive statement...
But the rules I want to discuss in this post are math rules. Yes! Math rules, as I use the term, are statements about math that are always true. We encourage children to come up with these rules as they work on problems and as they talk about math. Here are 3 examples, named for the children who first suggested them:
These may seem basic to you, and of course they are. None of these will make the pages of American Mathematical Monthly. They've all been discovered before. On the other hand, let's keep in mind that these are first and second grade children who are just beginning to make sense of the number system. The idea that numbers will always behave in a certain way is by no means obvious. Most things about school aren't that predictable. Think about reading: the letter o can stand for several different sounds, some words add an s in the plural while others add es, a word like set or right can have multiple meanings. Math is different. Math is a little more--reliable.So we ask children to look for these always situations in math. We ask them to think carefully about how numbers work, about whether a particular result will happen no matter what or whether it will happen only sometimes. The benefits, I think, are clear. A child who recognizes that an odd number plus an even number will always be an odd number is thinking hard about numbers and their properties. She's doing some basic number theory; she's acting like a scientist, making and then testing a conjecture. A child who sees that adding or subtracting zero never changes the original number is finding a pattern, generalizing from special cases, and boldly going where (to the best of his knowledge, anyway) no man has gone before.
Yes, we could tell children that odd + even is always odd, and very often we do just that. But it's more powerful, and the effects are longer-lasting, if children come to discover these rules themselves. Now if you'll excuse me, I need to go determine whether every even number greater than 2 can be written as the sum of two primes...
Thursday, May 14, 2009
Unringable Numbers
To succeed in doing division, especially with large numbers, there are two prerequisites. The first is that you have to know a lot of multiplication facts like THAT . Our 3-4 students--fourth graders in particular--have done a great job with this. The children, as a rule, seem to recognize the progress they've made in committing these facts to memory, and also in understanding the broader relationships between numbers (if you know that 2x8=16, how does that help you find 4x8?). It's been gratifying to get to the point in the year where I say "7x5" to a group and hear a grand chorus of "35" shooting back at me.
The second prerequisite is to think flexibly. Now that children have gotten good at telling me the product for a given expression, we need to switch the task (isn't that always the way??). So these days I give them the product and have them tell me the factors (the numbers to be multiplied to make my number). When I say "35," then, the students need to respond "7x5" (or 5x7); when I say "36," they should tell me "6x6" or "9x4." The language I use with children is that numbers such as 35 should "ring a bell"--that is, they should spark an immediate connection in children's minds with 5x7.
This ability is critical for doing division. One thing that makes it hard, though, is that not all numbers ring bells. An example is 17--the only multiplication expression that goes with 17 is the dull-and-boring 17x1/1x17. 43 won't ring any bells, either. Children not only have to be able to shoot back "3x9" when I say 27; they have to be able to recognize which numbers don't go with any expressions. That's new and different, and it takes children a while to figure out what's going on. Fortunately, young brains are quite malleable, and when they've gotten the idea, they're usually quite good at it.
So we were practicing some of these relationships this afternoon in one of the classes. 49, I said to the fourth graders I was working with, and they quickly responded with 7x7. How about 30? I asked, and got, variously, 5x6, 2x15, and 3x10. Twenty-five? That was easy, they scoffed--5x5. Then up came 37. Ring any bells? I asked.
There was silence for a moment as the students considered. At last, one child raised her hand. "It's unringable," she said confidently.
The usual mathematical term for this concept is prime, a term that this child knew but had temporarily forgotten. I must admit, though, that I have a certain fondness for unringable. Perhaps we can get the mathematicians' union to recognize it as an alternative. In a discipline that features the Pigeonhole Principle and the Generalized Ham Sandwich Theorem, I'd say it isn't entirely out of the question...
The second prerequisite is to think flexibly. Now that children have gotten good at telling me the product for a given expression, we need to switch the task (isn't that always the way??). So these days I give them the product and have them tell me the factors (the numbers to be multiplied to make my number). When I say "35," then, the students need to respond "7x5" (or 5x7); when I say "36," they should tell me "6x6" or "9x4." The language I use with children is that numbers such as 35 should "ring a bell"--that is, they should spark an immediate connection in children's minds with 5x7.
This ability is critical for doing division. One thing that makes it hard, though, is that not all numbers ring bells. An example is 17--the only multiplication expression that goes with 17 is the dull-and-boring 17x1/1x17. 43 won't ring any bells, either. Children not only have to be able to shoot back "3x9" when I say 27; they have to be able to recognize which numbers don't go with any expressions. That's new and different, and it takes children a while to figure out what's going on. Fortunately, young brains are quite malleable, and when they've gotten the idea, they're usually quite good at it.
So we were practicing some of these relationships this afternoon in one of the classes. 49, I said to the fourth graders I was working with, and they quickly responded with 7x7. How about 30? I asked, and got, variously, 5x6, 2x15, and 3x10. Twenty-five? That was easy, they scoffed--5x5. Then up came 37. Ring any bells? I asked.
There was silence for a moment as the students considered. At last, one child raised her hand. "It's unringable," she said confidently.
The usual mathematical term for this concept is prime, a term that this child knew but had temporarily forgotten. I must admit, though, that I have a certain fondness for unringable. Perhaps we can get the mathematicians' union to recognize it as an alternative. In a discipline that features the Pigeonhole Principle and the Generalized Ham Sandwich Theorem, I'd say it isn't entirely out of the question...
Tuesday, May 12, 2009
POW!
Three fifth graders came by the office the other day to visit at lunchtime. Two of them were there mainly to show off the shirts they had recently picked up at a museum. (The third, as you'll see from the picture, had something else in mind altogether.)
"POW!" I said cheerfully, hoping they'd congratulate me on my ability to read ALL CAPS. "Great shirts. Are they Batman-related?" I was not much of a Batman fan as a kid, but I do dimly remember the TV show, with the fight sequences that would pause so the POW! WHAM! OUCH! BOP! EEK! sound effects could appear on the screen. The style of the writing on the shirts reminded me of the series. (Right??)
"No," they said in that exasperated tone that only early middle school girls can achieve. "Look! This is what it stands for!" And they pointed one by one to the letters. "Problem Of the Week!"
POW! Problem of the Week! For two whole years these girls dealt every, well, week with the POW, which their teachers and I sometimes called the Problem of the Week but sometimes simply called the POW. The Problems of the Week, stored in my computer under the name of (what else) "POW stuff."
"Oh," I said, embarrassed. "Problem of the Week! Of course it stands for Problem of the Week. I knew that!"
And if I say it often enough, maybe it'll become true.
"POW!" I said cheerfully, hoping they'd congratulate me on my ability to read ALL CAPS. "Great shirts. Are they Batman-related?" I was not much of a Batman fan as a kid, but I do dimly remember the TV show, with the fight sequences that would pause so the POW! WHAM! OUCH! BOP! EEK! sound effects could appear on the screen. The style of the writing on the shirts reminded me of the series. (Right??)
"No," they said in that exasperated tone that only early middle school girls can achieve. "Look! This is what it stands for!" And they pointed one by one to the letters. "Problem Of the Week!"
POW! Problem of the Week! For two whole years these girls dealt every, well, week with the POW, which their teachers and I sometimes called the Problem of the Week but sometimes simply called the POW. The Problems of the Week, stored in my computer under the name of (what else) "POW stuff."
"Oh," I said, embarrassed. "Problem of the Week! Of course it stands for Problem of the Week. I knew that!"
And if I say it often enough, maybe it'll become true.
Wednesday, May 6, 2009
Practical Arithmetic?
While doing research on something completely unrelated to math, I ran into a cute (and tongue-in-cheek) article in an Indiana newspaper of 100 years ago, give or take.
Having trouble with his homework (sound familiar?), young John, a seventh grader, seeks out his father for help (sound familiar again?). It's a word problem, of course. "Asked how much money he has in the bank," the problem reads, "[a man] replies, 'If I had $10 more I would have $1,000 more than half what I now have." John is supposed to find out how much the man has in his account.
Dad is no help. That's because he's a card-carrying curmudgeon. Anyone who wants to know the actual amount, he snarls, should simply ask the teller. "In my day," he announces, unable to resist a dig at "modern" methods of teaching math, "we had practical problems in our arithmetic." [Kids, get off my lawn.]
Curious, John now repairs to the library to examine arithmetic books from the previous century. Each of them, it turns out, is chock-full of the sorts of problems his father is enthusiastically disparaging. John eventually tracks down an 1805 text, where he finds the following gem:
"A good man driving his [geese] to market was met by another, who said, 'Good morrow, master, with your 100 geese.' Says he in reply, 'I have not 100 geese, but if I had half as many as I now have and two and a half geese besides the number I now have already I should have 100.' How many geese had the man?"
At which point the writer of the article temporarily abandons both John and his father and addresses the reader directly. "How long," he demands, "would you permit a man to live if he made such an answer to you?"
[Well, let me think. What's the best way to express the answer, I wonder? Ah--got it. If the time interval were increased by seventeen seconds, it would be precisely four times one third the actual number of seconds diminished by the cube root of...]
Having trouble with his homework (sound familiar?), young John, a seventh grader, seeks out his father for help (sound familiar again?). It's a word problem, of course. "Asked how much money he has in the bank," the problem reads, "[a man] replies, 'If I had $10 more I would have $1,000 more than half what I now have." John is supposed to find out how much the man has in his account.
Dad is no help. That's because he's a card-carrying curmudgeon. Anyone who wants to know the actual amount, he snarls, should simply ask the teller. "In my day," he announces, unable to resist a dig at "modern" methods of teaching math, "we had practical problems in our arithmetic." [Kids, get off my lawn.]
Curious, John now repairs to the library to examine arithmetic books from the previous century. Each of them, it turns out, is chock-full of the sorts of problems his father is enthusiastically disparaging. John eventually tracks down an 1805 text, where he finds the following gem:
"A good man driving his [geese] to market was met by another, who said, 'Good morrow, master, with your 100 geese.' Says he in reply, 'I have not 100 geese, but if I had half as many as I now have and two and a half geese besides the number I now have already I should have 100.' How many geese had the man?"
At which point the writer of the article temporarily abandons both John and his father and addresses the reader directly. "How long," he demands, "would you permit a man to live if he made such an answer to you?"
[Well, let me think. What's the best way to express the answer, I wonder? Ah--got it. If the time interval were increased by seventeen seconds, it would be precisely four times one third the actual number of seconds diminished by the cube root of...]
Sunday, May 3, 2009
George's Excellent Adventure
Sometimes the best lessons are the ones you don't plan.
Friday morning, Ellen poked her head into the office as I was preparing for a fraction lesson with the 1-2s. "Elizabeth found a Where's George dollar in her lunch money," she said. "Okay if we take a few minutes to enter it at the beginning of math time today?"
Where's George, I should explain, is a lovely internet project that tracks paper money as it moves across the country (www.wheresgeorge.com). Since the 3-4 classes handle lots of money in their capacity as Pizza People, they occasionally run into Where's George bills, which are recognizable by special markings. We log onto the site, enter the bill's serial number, note our location, and press Enter. If the sound on my laptop is turned on, we'll then hear a cash register noise and the bill's previous location(s) will appear. Most of the bills we've found thus far have come from nearby places such as Pennsylvania and Massachusetts, Brooklyn and Kingston, but we've had bills from Missouri, Tennessee, and Texas as well. It's fun, and suspenseful, and teaches a bit about geography--and you never know when someone will find "our" bill and put it in again.
When we entered Elizabeth's bill, the screen showed that the bill was now 1128 miles from its original location. I had a sudden brainstorm. Instead of scrolling down and telling the class where the bill had come from, I'd have them narrow the possibilities by using math--specifically, their measuring and estimation skills. They'd been working on maps all year long, after all, filling in states that Cheerful Charlie had visited in his round-the-US tour. Ellen got one of the students' maps, and we hung it up. We determined that 1128 was very close to 1100, in double-round numbers, and at 200 miles to the inch, the class quickly calculated that the starting point was about 5.5 inches away from us.
It was clear to most of the students that the possibilities would form the arc of a circle, and so we did a little measuring. We ended up with a curve beginning at the western end of Michigan's Upper Peninsula and then zagging through Wisconsin, Minnesota, Iowa, Missouri, Arkansas, and Mississippi--all of them marked on the students' maps--before catching a piece of south-central Florida and disappearing over the Atlantic Ocean. "Why can't the bill have started here?" I asked, indicating where the arc crossed the Gulf of Mexico. That was obvious. "It's too wet for money in the ocean," a third grader answered (unless, he added, there were islands he "didn't know about"). As for why we didn't go north of Michigan, that was obvious too: Canada has its own money.
I scrolled down on the webpage and revealed the answer: the bill had originated in Florida. ("I knew it!" half the class exclaimed.) I named the town, which I'd never heard of. But Ellen had: her brother lived there. She asked if there was any way to find out who started the bill on its travels. Well, yes, there was; I clicked on the profile button and found a first name, Bob.
It wasn't Ellen's brother. But that was all right. Bob had provided us with a nice map of the US, each state filled in with one of six colors. Now I had my second brainstorm. We'd done a little real-life estimating and measuring with scale; it was time for some real-world data analysis.
"What do you think this map shows?" I asked, turning the computer so the students could see. Temperatures, guessed one boy. Good thought, but no. How many people live in each state? asked a girl. Close, I said. Think about what website this is, Ellen suggested, and suddenly hands were flying up all over the meeting area. Bob, they realized, had marked dozens and dozens of bills and sent them into the wild. The colors showed how many of those bills had turned up in each state.
[Here is Bob's Hit Map, by the way:]
Right on the money! (So to speak.) The only question now was which colors stood for the most bills and which for the fewest. To help, I had them identify a few key states on Bob's map, and then I gave them a little extra information. California, I explained, had the most people of any state. Texas, New York, and Florida were next. Wyoming took up a lot of space, but it had fewer people than any other state.
Working as a group, the class swiftly came up with a sensible schematic for the colors. Red, the color of Florida, California, and New York, would be the most. Bright green, it seemed apparent, would be next, judging from what the children knew of population and distance, and so on, down to lowly Wyoming, the only state that was colored gray.
The guesses were in. It was now time for the Great Unveiling. I had everyone's full attention: they were deeply invested in the outcome by now. And the results were entirely satisfactory. The class had four out of six colors right; the only error had been reversing the orderof the fourth- and fifth-most colors.
Not bad, not bad at all, I told them, and we moved on to the regularly scheduled lesson on division.
Friday morning, Ellen poked her head into the office as I was preparing for a fraction lesson with the 1-2s. "Elizabeth found a Where's George dollar in her lunch money," she said. "Okay if we take a few minutes to enter it at the beginning of math time today?"
Where's George, I should explain, is a lovely internet project that tracks paper money as it moves across the country (www.wheresgeorge.com). Since the 3-4 classes handle lots of money in their capacity as Pizza People, they occasionally run into Where's George bills, which are recognizable by special markings. We log onto the site, enter the bill's serial number, note our location, and press Enter. If the sound on my laptop is turned on, we'll then hear a cash register noise and the bill's previous location(s) will appear. Most of the bills we've found thus far have come from nearby places such as Pennsylvania and Massachusetts, Brooklyn and Kingston, but we've had bills from Missouri, Tennessee, and Texas as well. It's fun, and suspenseful, and teaches a bit about geography--and you never know when someone will find "our" bill and put it in again.
When we entered Elizabeth's bill, the screen showed that the bill was now 1128 miles from its original location. I had a sudden brainstorm. Instead of scrolling down and telling the class where the bill had come from, I'd have them narrow the possibilities by using math--specifically, their measuring and estimation skills. They'd been working on maps all year long, after all, filling in states that Cheerful Charlie had visited in his round-the-US tour. Ellen got one of the students' maps, and we hung it up. We determined that 1128 was very close to 1100, in double-round numbers, and at 200 miles to the inch, the class quickly calculated that the starting point was about 5.5 inches away from us.
It was clear to most of the students that the possibilities would form the arc of a circle, and so we did a little measuring. We ended up with a curve beginning at the western end of Michigan's Upper Peninsula and then zagging through Wisconsin, Minnesota, Iowa, Missouri, Arkansas, and Mississippi--all of them marked on the students' maps--before catching a piece of south-central Florida and disappearing over the Atlantic Ocean. "Why can't the bill have started here?" I asked, indicating where the arc crossed the Gulf of Mexico. That was obvious. "It's too wet for money in the ocean," a third grader answered (unless, he added, there were islands he "didn't know about"). As for why we didn't go north of Michigan, that was obvious too: Canada has its own money.
I scrolled down on the webpage and revealed the answer: the bill had originated in Florida. ("I knew it!" half the class exclaimed.) I named the town, which I'd never heard of. But Ellen had: her brother lived there. She asked if there was any way to find out who started the bill on its travels. Well, yes, there was; I clicked on the profile button and found a first name, Bob.
It wasn't Ellen's brother. But that was all right. Bob had provided us with a nice map of the US, each state filled in with one of six colors. Now I had my second brainstorm. We'd done a little real-life estimating and measuring with scale; it was time for some real-world data analysis.
"What do you think this map shows?" I asked, turning the computer so the students could see. Temperatures, guessed one boy. Good thought, but no. How many people live in each state? asked a girl. Close, I said. Think about what website this is, Ellen suggested, and suddenly hands were flying up all over the meeting area. Bob, they realized, had marked dozens and dozens of bills and sent them into the wild. The colors showed how many of those bills had turned up in each state.
[Here is Bob's Hit Map, by the way:]Right on the money! (So to speak.) The only question now was which colors stood for the most bills and which for the fewest. To help, I had them identify a few key states on Bob's map, and then I gave them a little extra information. California, I explained, had the most people of any state. Texas, New York, and Florida were next. Wyoming took up a lot of space, but it had fewer people than any other state.
Working as a group, the class swiftly came up with a sensible schematic for the colors. Red, the color of Florida, California, and New York, would be the most. Bright green, it seemed apparent, would be next, judging from what the children knew of population and distance, and so on, down to lowly Wyoming, the only state that was colored gray.
The guesses were in. It was now time for the Great Unveiling. I had everyone's full attention: they were deeply invested in the outcome by now. And the results were entirely satisfactory. The class had four out of six colors right; the only error had been reversing the orderof the fourth- and fifth-most colors.
Not bad, not bad at all, I told them, and we moved on to the regularly scheduled lesson on division.
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