We talk a lot about problemsolving strategies at PDS, especially in the 3rd-4th grades. One of the absolute favorites among the children is the one Icall the Goldilocks Method. Some texts refer to it as guess-and-check, or predict-and-test, but the name "Goldilocks Method" seems to have a greater "stickiness" quotient for kids.
The strategy is based, of course, on Goldilocks, Goldilocks of porridge fame, Goldilocks who could have been charged with breaking and entering, Goldilocks who encountered a trio of ursine forestdwellers...okay, okay, more to the point Goldilocks, who tasted the first bowl of porridge and found that it was TOO HOT, then tasted the second, which was TOO COLD, and finally tried the third, which was JUST RIGHT, and then repeated the process, replacing hot/cold with hard/soft and porridge with beds, but still coming out with JUST RIGHT at the end.
The students I'm working with in division right now used the Goldilocks Method the other day. They were playing a game to help them work with the connection between multiplication and division, and not so incidentally to practice mental math skills. I forget the name of the game (I usually do), but hey, grab a pencil, and you can play along with us at home:
First, choose a number between 600 and 800. No round numbers. (That is, no multiples of 10, like 790, 650, or 700. You will rarely hear me ban round numbers, but the fact is they're too easy to work with.)
Next, choose an odd number between 5 and 20.
Third, write a division expression with these numbers, such as "705 divided by 15."
The quotient will be--well, we don't know yet. But we can figure it out by using the Goldilocks Method. First, take the divisor (in this example, 15). Ask yourself: what do I have to multiply 15 by to get close to 705? We'll use a little mental math here: let's see, 10 x 15 is 150, so that's not close...20 x 15? Well, that would be 300. Okay, we're not getting there very quickly, so let's try 50 x 15. We'll write that down, calculate the product with either pencil-and-paper or a calculator, and discover that 50 x 15 = 750.
All right, what would Goldilocks say? She'd say TOO HOT. Or TOO HARD. Or TOO HIGH. Or something beginning with TOO. So, we need to try a smaller number. How about 45? Well, 45 x 15 = 675. TOO COLD/SOFT/LOW. Try something that's greater than 45. 48 x 15 = 720. Getting there! But still, TOO HIGH...
You see how this works. In this example, the original three-digit number was evenly divisible by 15, so it was possible to get something that was JUST RIGHT. Go, Goldilocks! Most of the time, it isn't possible in this game. That's okay too: we get as close as we can without going over, and then take the difference as the remainder. So for 696 divided by 9, we might say:
9 x 70 = 630 TOO LOW
9 x 80 = 720 TOO HIGH
9 x 75 = 675 TOO LOW
9 x 77 = 693 TOO LOW but oh-so-close...
And so our division sentence would be that 696 divided by 9 is 77, with a remainder of 3.
Goldilocks would be so proud...
Showing posts with label third and fourth grades. Show all posts
Showing posts with label third and fourth grades. Show all posts
Sunday, March 21, 2010
Friday, February 5, 2010
Cheerful Charlie and the Cupcakes
My good friend Cheerful Charlie, I told the third and fourth graders, was having a party, at which he planned to serve cupcakes. One hundred cupcakes, to be precise. I'm not sure who-all is on his guest list (other than me, of course), but he's either inviting a lot of people, or a few big eaters, or else he just wants a lot of leftovers.
Cheerful had (wisely) decided not to make these cupcakes from scratch (his measuring skills aren't what you would call real accurate), but was having some trouble determining which store he should go to. He had three choices, I explained, and he wanted to spend as little money as he could and get as good a deal as possible, and if the students could advise him that'd be great.
So, there was Cupcakes R Us, I told the kids, which sold baskets of 50 cupcakes at a shot, and each basket cost $40 but he also had to pay a fee of $5 to park.
And there was Cupcake Depot, where cupcakes were $9 per bag for a bag of 10, plus which Cheerful had a coupon for $5 off his total purchase.
And there was Cupcake's Discount Warehouse...
Look at all the information, I told them, and do some calculations if you need to, and decide if there are other considerations Cheerful should be thinking about, and then write Cheerful a letter suggesting what he ought to do. The students could turn in a handwritten letter which we'd forward on to Mr. Charlie, I explained, or they could email him directly at cheerful.charlie@yahoo.com, an account which he checks but not as regularly as he should because he often forgets or mistypes the password.
For some students, the assignment was just a relatively straightforward problem in arithmetic. You determine how many bags, boxes, or baskets of cupcakes Cheerful needs to buy at each store so he has 100 cupcakes in all; you multiply that number by the cost per box/bag/bucket/basket; you add the parking fee or membership fee, you subtract the coupon...and if you've done it right, you inform Cheerful that the cost at all three stores is the same, eighty-five bucks, and he can go wherever he likes and it doesn't make any difference. And this is a fine way of looking at it.
But what makes the problem interesting is the real-world nature of it. Sure, price is important. But is the lowest price always the best deal? I remember when my wife and I discovered WHY generic spaghetti sauce was so cheap (hint: it contained mostly water)... So the question becomes, what other things should Cheerful be taking into account?
Well, a lot of kids came up with lots of ideas.
"You should go to Cupcake's Discount Warehouse," one student wrote. The membership fee of $5 was annoying, she pointed out, "but if you have to go back again you will have your membership card so it'll be cheaper."
"Maybe you can walk to Cupcakes R Us," said somebody else. "Or ride a bike. Then you might not have to pay to park." (Note the "might." Hedging your bets, we call it.)
"It depends," wrote another student. "How far away are the stores?" A good question. If the nearest Cupcakes R Us outlet is in Albany, Scranton, or Paramus, it isn't worth the time and the gas to get there.
"You should ask them each for a free sample," suggested one optimist. "If you say you'll buy a lot they'll probably let you have one. Buy the one that tastes best." Not much point in saving five dollars if the cheaper cupcakes tasted like sawdust or carbon paper. --Or were made primarily of water, like the generic spaghetti sauce referred to above.
"Buy them in bags of 10," someone else advised. "The prices are all the same but if you need more then you only have to buy 10 more, not 20 or 25 or 50, and that will save you money."
And another student went right to the heart of the matter: Measure the cupcakes. "Buy the ones that are biggest," he suggested.
Unfortunately, Cheerful (who likes things simple) is still uncertain what to do: he's definitely bummed at the prospect of needing to find more information. We'll keep you posted. In the meantime, it was nice to see how many students recognized that there might be other considerations besides the cost of the cupcakes. We like to remind kids that math is about real-world situations, and in particular we like to point out that answers may not be as cut and dried as the textbooks sometimes suggest they are. This was a good example of both--and a good example of how a little knowledge can be a dangerous thing.
(Though regarding "other considerations," you CAN have too much of a good thing: see one of my favorite cartoons of all time, http://xkcd.com/309/. --Cut and paste the URL into your browser window if the link doesn't work for you. The two folks on the extreme right? That would be me and my wife...)
Cheerful had (wisely) decided not to make these cupcakes from scratch (his measuring skills aren't what you would call real accurate), but was having some trouble determining which store he should go to. He had three choices, I explained, and he wanted to spend as little money as he could and get as good a deal as possible, and if the students could advise him that'd be great.
So, there was Cupcakes R Us, I told the kids, which sold baskets of 50 cupcakes at a shot, and each basket cost $40 but he also had to pay a fee of $5 to park.
And there was Cupcake Depot, where cupcakes were $9 per bag for a bag of 10, plus which Cheerful had a coupon for $5 off his total purchase.
And there was Cupcake's Discount Warehouse...
Look at all the information, I told them, and do some calculations if you need to, and decide if there are other considerations Cheerful should be thinking about, and then write Cheerful a letter suggesting what he ought to do. The students could turn in a handwritten letter which we'd forward on to Mr. Charlie, I explained, or they could email him directly at cheerful.charlie@yahoo.com, an account which he checks but not as regularly as he should because he often forgets or mistypes the password.
For some students, the assignment was just a relatively straightforward problem in arithmetic. You determine how many bags, boxes, or baskets of cupcakes Cheerful needs to buy at each store so he has 100 cupcakes in all; you multiply that number by the cost per box/bag/bucket/basket; you add the parking fee or membership fee, you subtract the coupon...and if you've done it right, you inform Cheerful that the cost at all three stores is the same, eighty-five bucks, and he can go wherever he likes and it doesn't make any difference. And this is a fine way of looking at it.
But what makes the problem interesting is the real-world nature of it. Sure, price is important. But is the lowest price always the best deal? I remember when my wife and I discovered WHY generic spaghetti sauce was so cheap (hint: it contained mostly water)... So the question becomes, what other things should Cheerful be taking into account?
Well, a lot of kids came up with lots of ideas.
"You should go to Cupcake's Discount Warehouse," one student wrote. The membership fee of $5 was annoying, she pointed out, "but if you have to go back again you will have your membership card so it'll be cheaper."
"Maybe you can walk to Cupcakes R Us," said somebody else. "Or ride a bike. Then you might not have to pay to park." (Note the "might." Hedging your bets, we call it.)
"It depends," wrote another student. "How far away are the stores?" A good question. If the nearest Cupcakes R Us outlet is in Albany, Scranton, or Paramus, it isn't worth the time and the gas to get there.
"You should ask them each for a free sample," suggested one optimist. "If you say you'll buy a lot they'll probably let you have one. Buy the one that tastes best." Not much point in saving five dollars if the cheaper cupcakes tasted like sawdust or carbon paper. --Or were made primarily of water, like the generic spaghetti sauce referred to above.
"Buy them in bags of 10," someone else advised. "The prices are all the same but if you need more then you only have to buy 10 more, not 20 or 25 or 50, and that will save you money."
And another student went right to the heart of the matter: Measure the cupcakes. "Buy the ones that are biggest," he suggested.
Unfortunately, Cheerful (who likes things simple) is still uncertain what to do: he's definitely bummed at the prospect of needing to find more information. We'll keep you posted. In the meantime, it was nice to see how many students recognized that there might be other considerations besides the cost of the cupcakes. We like to remind kids that math is about real-world situations, and in particular we like to point out that answers may not be as cut and dried as the textbooks sometimes suggest they are. This was a good example of both--and a good example of how a little knowledge can be a dangerous thing.
(Though regarding "other considerations," you CAN have too much of a good thing: see one of my favorite cartoons of all time, http://xkcd.com/309/. --Cut and paste the URL into your browser window if the link doesn't work for you. The two folks on the extreme right? That would be me and my wife...)
Saturday, December 12, 2009
The Week with Less Pizza
As you may know, the 3-4 students have been keeping track of pizza sales thus far this year. Yes, we have records stretching back as far as, let me see, September 10 or so!
For quite some time, as you'll see on the graph pictured below (in two parts), the total pizza order was a rather dull oscillation between 144 slices (18 whole pizza pies) and 152 slices (19 pies). Week after week, 144 or 152, 152 or 144. You could set your watch by it. It was like, I don't know, jazz music or Blue's Clues or driving on Interstate 65 in northern Indiana or something. As the graph shows, the median (the middle value when the data points are ordered) stayed within a very constrained band of numbers, and the range (the difference between the lowest and highest values) remained absolutely, boringly, even mindnumbingly consistent.
[Note that the number of slices actually ordered by lower school students doesn't match the number of slices we actually buy. Why is that, I wonder? Hmmm...]
Then, all of a sudden one Friday, the number of slices ordered took a nosedive. Fell off a cliff, or at least rolled down a slope, as the graph makes clear. Woke us all up, I tell you that. Boom, all the way down from the 150 region to...104. 104! Think of it! The median didn't change (why it didn't was food for thought for some of the students), but the range changed, oh boy did it ever.
Why would things be so different this week? I asked the gathered third and fourth grade children (after swearing to secrecy Ellen's class, which had handled the order and therefore knew the answer). What possibilities do you think there are?
They came up with four:
A) There were a LOT of kids out with swine flu.
B) Some of the classes were on a field trip.
C) The pizza place ran out of pizza partway through.
And
D) Not very many people were hungry for pizza that day.
I wonb't tbell ybou thbe rbeal ansbwer. But wbith anby lbuck, yobu cban gbuess.
For quite some time, as you'll see on the graph pictured below (in two parts), the total pizza order was a rather dull oscillation between 144 slices (18 whole pizza pies) and 152 slices (19 pies). Week after week, 144 or 152, 152 or 144. You could set your watch by it. It was like, I don't know, jazz music or Blue's Clues or driving on Interstate 65 in northern Indiana or something. As the graph shows, the median (the middle value when the data points are ordered) stayed within a very constrained band of numbers, and the range (the difference between the lowest and highest values) remained absolutely, boringly, even mindnumbingly consistent.
[Note that the number of slices actually ordered by lower school students doesn't match the number of slices we actually buy. Why is that, I wonder? Hmmm...]
Then, all of a sudden one Friday, the number of slices ordered took a nosedive. Fell off a cliff, or at least rolled down a slope, as the graph makes clear. Woke us all up, I tell you that. Boom, all the way down from the 150 region to...104. 104! Think of it! The median didn't change (why it didn't was food for thought for some of the students), but the range changed, oh boy did it ever.
Why would things be so different this week? I asked the gathered third and fourth grade children (after swearing to secrecy Ellen's class, which had handled the order and therefore knew the answer). What possibilities do you think there are?
They came up with four:
A) There were a LOT of kids out with swine flu.
B) Some of the classes were on a field trip.
C) The pizza place ran out of pizza partway through.
And
D) Not very many people were hungry for pizza that day.
I wonb't tbell ybou thbe rbeal ansbwer. But wbith anby lbuck, yobu cban gbuess.
Tuesday, December 1, 2009
On Family Size
It's important to connect numbers with real-life situations. Which is why I had 4th graders tell "stories" about the multiplication expression 4 x 6 as a warmup for a lesson this week. By "stories," I hasten to say, I don't mean great literary efforts, with foreshadowing and metaphor and plot twists and poetic license and all those great things. No, I mean simple situations like these:
"There were 4 glasses and each glass had 6 ice cubes in it."
"There were 4 people and each one ate 6 hot dogs."
"I saw 4 flowers. Each flower had 6 petals."
You'll note that in each case the 4 [the first number in the expression] represents the number of groups, and the 6 [the second number in the expression] represents the number in each group. Of course, 4 x 6 is equal to 6 x 4, which all the children I was working with that day knew perfectly well; but it's useful to think of the first and second numbers each playing a slightly different role in the expression.
And we were progressing swimmingly until one boy said, "There were 4 families and each family had 6..." Then his voice trailed off, and he thought, and then he said, "I mean, there were SIX families, and each family had 4 people in it."
Real-life situations indeed. No prizes for guessing how many people there were in his family!
"There were 4 glasses and each glass had 6 ice cubes in it."
"There were 4 people and each one ate 6 hot dogs."
"I saw 4 flowers. Each flower had 6 petals."
You'll note that in each case the 4 [the first number in the expression] represents the number of groups, and the 6 [the second number in the expression] represents the number in each group. Of course, 4 x 6 is equal to 6 x 4, which all the children I was working with that day knew perfectly well; but it's useful to think of the first and second numbers each playing a slightly different role in the expression.
And we were progressing swimmingly until one boy said, "There were 4 families and each family had 6..." Then his voice trailed off, and he thought, and then he said, "I mean, there were SIX families, and each family had 4 people in it."
Real-life situations indeed. No prizes for guessing how many people there were in his family!
Friday, November 13, 2009
(Why I Am Not a) Gamblin' Man (Mostly)
I have purchased one lottery ticket in my life. It was a loser. I have visited two casinos, one on a "riverboat" in Mississippi and the other on drier land in Louisiana. I put about $1.85 in slot machines, total. I lost it all. In 8th grade I was invited to a "Las Vegas" party, where I played roulette all night long (well, till 9 pm anyway) using poker chips. I bet on 36 time and again. It never came up. I lost, and lost, and lost some more. I eventually needed two loans from the bank. So, I don't gamble (mostly), because I lose (mostly, or maybe always).
Still, there are times when you just have to place that bet...
Yesterday I was in Jan's third and fourth grade class, where students are working on logic and attributes. The focus for the lesson was on Venn diagrams. You know, overlapping circle thingies, like this:
(My college roommate, Bernie, was a double major in science and philosophy. After taking a class on Tibetan Buddhism he once accidentally referred to these things as "Zen diagrams." I'm still tempted to call 'em that sometimes.)
In addition to the Zen, I mean Venn, diagram, we also had a bunch of blocks of various sizes, shapes, and colors, and labels with categories that described the blocks: "triangles," "small," "red," "yellow" and so on. I had chosen two labels at random ("What does that mean, 'at random?'" I'd asked earlier in the day, and the response was "Randomly," which was accurate if not perhaps revealing) and placed one in each circle of the Zen, I mean Venn, diagram. I got to look at the labels. The kids didn't.
The object is for the students to identify the two labels. They name blocks one by one, and I place each piece where it belongs: in the overlapping section of the diagram if it fits both labels; in one of the circles but not the other if it matches just one label; or outside both circles if it matches neither one. For instance, a small green triangle goes inside a circle marked "small," "green," or "triangle." Students use logic and the position of blocks in the diagram to determine what the labels CAN and CANNOT be.
Yesterday, after just three blocks, we had the following situation:
In the left circle, but NOT in the overlap, was a small blue rhombus. (A rhombus, for those not in the know, is not a method of transporting rhoms; it is a four-sided figure in which all sides are equal.)
In the overlap between the circles was a large blue rhombus. And outside both circles looking in was a small blue triangle.
"Talk to each other," I said. "Tell your partner what the labels COULD be and what they COULDN'T be. Then share your ideas with the rest of us."
{WARNING: SPOILERS AHEAD. You may wish to see if you can solve the problem on your own based only on this information. Remember, labels name only colors, shapes, and sizes, and we only choose two labels. Read down when you're ready...}
After the partner conversation, one of the third graders raised her hand. "I think I know what it is," she said. "This circle"--and she pointed to the one on the left--"is the circle for rhombuses. And this circle"--and she pointed to the one on the right--"has to be for big blocks."
"Why couldn't they be for blue blocks?" I asked.
"Because we tried blue with the blue triangle," she said, "and the blue triangle didn't go in either of the circles. So it can't be blue."
I asked a couple more questions like that, inquired if anyone had other ideas, and then turned back to the girl who had spoken first. "How sure are you?" I asked.
"I'm pretty sure," she told me. "Maybe 80% sure. No, 90%." (I like having kids this age express "sureness" in percentages. They seem to like it too.)
I drew a quarter out of my pocket and examined it closely. "90% sure is pretty sure," I said, "but it isn't certain. We only have three blocks so far. It's kind of early to be naming both labels, don't you think? I'm thinking it COULD be something else. I'm thinking it probably IS something else." Pause. "What do you think?"
"Umm." The girl frowned and looked back at the diagram. A classmate next to her whispered something. The girl nodded. "I still think I'm right," she informed me.
I tossed the quarter into the air and caught it nonchalantly. "I have a quarter here that says you're wrong," I said. If the labels were "rhombus" and "large," I explained, the quarter would be hers. (That got everybody's attention.) On the other hand, I added oh-so-casually, if she was wrong she would owe ME a quarter.
"Don't do it!" somebody hissed at my, ah, victim, just as someone else leaned in close to her and said "Go for it!"
"All right," she said, rolling her eyes, "you can have my allowance..."
As it turned out, of course, it wasn't necessary. We went through her reasoning, failed to find any holes (bummer, man), and revealed the labels in the Venn, I mean Zen, diagram. The girl's reasoning had been one hundred percent correct, and she had stuck to her guns despite my best attempt to rattle her. I handed over the quarter as the class cheered and surrounded her to offer their congratulations to the kid who had, if not broken the bank at Monte Carlo, at the very least outwitted the Math Guy.
I'd lost (again). But though my pocket was lighter, I was convinced that the reasoning and confidence the child had demonstrated during the lesson had been worth the very real financial hit to me...
And at any rate, now you know why I am not (generally) a gamblin' man!
Still, there are times when you just have to place that bet...
Yesterday I was in Jan's third and fourth grade class, where students are working on logic and attributes. The focus for the lesson was on Venn diagrams. You know, overlapping circle thingies, like this:
(My college roommate, Bernie, was a double major in science and philosophy. After taking a class on Tibetan Buddhism he once accidentally referred to these things as "Zen diagrams." I'm still tempted to call 'em that sometimes.)
In addition to the Zen, I mean Venn, diagram, we also had a bunch of blocks of various sizes, shapes, and colors, and labels with categories that described the blocks: "triangles," "small," "red," "yellow" and so on. I had chosen two labels at random ("What does that mean, 'at random?'" I'd asked earlier in the day, and the response was "Randomly," which was accurate if not perhaps revealing) and placed one in each circle of the Zen, I mean Venn, diagram. I got to look at the labels. The kids didn't.
The object is for the students to identify the two labels. They name blocks one by one, and I place each piece where it belongs: in the overlapping section of the diagram if it fits both labels; in one of the circles but not the other if it matches just one label; or outside both circles if it matches neither one. For instance, a small green triangle goes inside a circle marked "small," "green," or "triangle." Students use logic and the position of blocks in the diagram to determine what the labels CAN and CANNOT be.
Yesterday, after just three blocks, we had the following situation:
In the left circle, but NOT in the overlap, was a small blue rhombus. (A rhombus, for those not in the know, is not a method of transporting rhoms; it is a four-sided figure in which all sides are equal.)
In the overlap between the circles was a large blue rhombus. And outside both circles looking in was a small blue triangle.
"Talk to each other," I said. "Tell your partner what the labels COULD be and what they COULDN'T be. Then share your ideas with the rest of us."
{WARNING: SPOILERS AHEAD. You may wish to see if you can solve the problem on your own based only on this information. Remember, labels name only colors, shapes, and sizes, and we only choose two labels. Read down when you're ready...}
After the partner conversation, one of the third graders raised her hand. "I think I know what it is," she said. "This circle"--and she pointed to the one on the left--"is the circle for rhombuses. And this circle"--and she pointed to the one on the right--"has to be for big blocks."
"Why couldn't they be for blue blocks?" I asked.
"Because we tried blue with the blue triangle," she said, "and the blue triangle didn't go in either of the circles. So it can't be blue."
I asked a couple more questions like that, inquired if anyone had other ideas, and then turned back to the girl who had spoken first. "How sure are you?" I asked.
"I'm pretty sure," she told me. "Maybe 80% sure. No, 90%." (I like having kids this age express "sureness" in percentages. They seem to like it too.)
I drew a quarter out of my pocket and examined it closely. "90% sure is pretty sure," I said, "but it isn't certain. We only have three blocks so far. It's kind of early to be naming both labels, don't you think? I'm thinking it COULD be something else. I'm thinking it probably IS something else." Pause. "What do you think?"
"Umm." The girl frowned and looked back at the diagram. A classmate next to her whispered something. The girl nodded. "I still think I'm right," she informed me.
I tossed the quarter into the air and caught it nonchalantly. "I have a quarter here that says you're wrong," I said. If the labels were "rhombus" and "large," I explained, the quarter would be hers. (That got everybody's attention.) On the other hand, I added oh-so-casually, if she was wrong she would owe ME a quarter.
"Don't do it!" somebody hissed at my, ah, victim, just as someone else leaned in close to her and said "Go for it!"
"All right," she said, rolling her eyes, "you can have my allowance..."
As it turned out, of course, it wasn't necessary. We went through her reasoning, failed to find any holes (bummer, man), and revealed the labels in the Venn, I mean Zen, diagram. The girl's reasoning had been one hundred percent correct, and she had stuck to her guns despite my best attempt to rattle her. I handed over the quarter as the class cheered and surrounded her to offer their congratulations to the kid who had, if not broken the bank at Monte Carlo, at the very least outwitted the Math Guy.
I'd lost (again). But though my pocket was lighter, I was convinced that the reasoning and confidence the child had demonstrated during the lesson had been worth the very real financial hit to me...
And at any rate, now you know why I am not (generally) a gamblin' man!
Sunday, November 1, 2009
Corn, Revisited
I promised to write more about the corn project (see entry of October 13). Picking up the story from there:
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!
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!
Tuesday, October 13, 2009
Corn
How many kernels on an ear of corn? we asked the third and fourth graders the other day. They've been studying the Mayan people, who called themselves "People of the Corn," so it was a worthwhile question.
We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.
What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)
"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."
"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."
"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."
The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.
At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.
Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.
As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?
So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...
We started by having students find approximations; as you should know by now if you've been reading this blog, us Math Guys consider this a very important step. We asked students to choose a round number (a number that is a multiple of 10); the point, after all, wasn't to guess the exact number, but to use a number that makes some sense and is relatively easy to work with. You can always revise your estimate later, we assured them.
What is the estimate based on? Well, we gave them each an ear of dried corn to eyeball. Some did some quick-n-dirty calculations, fourth graders in particular. (Yes, we asked them to justify their reasoning. Some of them HATE this, but it's oh-so-good for them.)
"About 20 in each row," wrote one student. "Maybe 10 rows. 10 x 20 = 200. I estimate 200 kernels in all."
"I think there are 20 rows and 30 in each row," reported someone else, "but that might not be enough so I added a few more. I say 640."
"I think 260," wrote a third grader, who would have been happy to leave it at that, but who added, under duress from a teacher, "because it looks right. And because it's a good number." We might call this strategy "Pick-a-large-number, any-large-number, and-assign-it-great-virtue-so-critics-will-be-cowed."
The next step: Count the kernels! The classroom teachers had prepared egg cartons with ten cuplets (better them than me). Kids used their fingernails to push the kernels off the cob (great fun). Then they distributed the kernels 5 or 10 at a time into the cups, making groups of 50 or 100. Record the number, dump out the kernels, lather, rinse, repeat.
At some point along the way several students noticed that their estimates weren't looking as accurate as they had back before counting had begun. This was especially true for those whose initial strategy had been "Pick-a-large-number, any-large-number &c," but other more careful estimators ran into this difficulty too. No problem! we said. Just revise your estimate, record it--oh, and explain why you wanted to change your original prediction. (My favorite: "Because I passed my first estimate a long time ago.") You will no doubt be shocked to learn that the second set of estimates were considerably closer than the first.
Eventually, all corn kernels were off the cobs and had traveled through the eggcups and into plastic bowls or paper bags (except for a few strays which had found their way onto the floor), and everyone had an exact answer. Some were surprised to see how many there were. Others found the results unsurprising in the extreme, or claimed they did: "I knew it," crowed one boy whose answer was not, perhaps, as close as he thought.
As students finished, they compared their totals with friends and thought about questions such as Why aren't all the totals the same?, What could you do to get a better estimate next time? ("Nothing," said the young man quoted above), and About how many kernels do you think there might be in the whole class?
So, three-digit numbers, ordering, estimating, grouping by tens, fives, 50s, and 100s, and explaining reasoning. Plus, a fun project (there's something truly satisfying about flicking those kernels off the cob, and something even more satisfying about running your fingers through a nice big tub full of everyone's kernels), and one that relates to science and social studies. A worthwhile math period indeed. Next up: data analysis with these results. On Thursday we'll be in the Chapman Room calculating the median and range of the data and forming a Living Histogram. Pictures to follow, assuming my camera behaves itself...
Wednesday, September 30, 2009
Pizza!
As many of you know, the third and fourth grade classes order pizza each Friday. Children throughout the lower school put in their order; runners from the 3-4s pick up the orders and the money, determine the number of pizzas to buy, and hand-deliver it when it arrives.
Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.
How does pizza relate to mathematics? Glad you asked. Let us count the ways...
1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?
And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!
Some of the proceeds, up close and personal:
This young man is clearly enjoying himself.
Think Scrooge McDuck.
Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:
One of the "Grand Totalers," making bundles of ten for easier counting:
And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):
Pizza is a major undertaking. There are times when we teachers wonder whether it is all worthwhile, especially when we discover that one class is short $15 or that a dozen or so children neglected to sign up until the pizza, you know, arrived... BUT we continue to do it because pizza a) tastes good, b) is convenient for parents, and c) IS A GREAT TOOL FOR PRACTICING MATH SKILLS. Of the three, c) is by far the most important in my book, though your mileage may vary.
How does pizza relate to mathematics? Glad you asked. Let us count the ways...
1. Pizza order takers get good practice in counting money and determining if it matches the number of slices ordered (hint: it does only about half the time).
2. Students get practice in giving and making change.
3. Students round the number of slices ordered per class to the nearest multiple of 10 to make estimation easier.
4. Kids practice addition skills by calculating the total number of slices ordered.
5. We look at number patterns. Hmmm: when a class orders 14 slices at $1.50 per slice, we get $21. Interestingly enough, 14 plus half-of-14 equals 21--the same number, only in regular numbers rather than in money. Now why would that be?
6. Especially later in the year, we use pizza as a real-life example of multiplication--if there are 8 slices per pizza, how many slices in 5 pizzas? In 7 pizzas? In 13 pizzas?
7. Kids calculate the profit for each week's worth of pizza: if we charge a dollar-fifty per slice after buying it for [sorry, trade secrets removed--suffice it to say "less"] per slice, how much money is left over? What operation can we use to calculate it?
And there are many other ways we mathicize pizza, especially this year, but I've been typing all day and my fingers are about to fall off. So you'll have to wait for another post. Sorry! In the meantime, how about some pictures? ...Yes, yes, the very thing!
Some of the proceeds, up close and personal:
This young man is clearly enjoying himself.
Think Scrooge McDuck.
Doublechecking that the amount of money from one of the 1-2 classes actually matches the number of slices ordered:
One of the "Grand Totalers," making bundles of ten for easier counting:
And at last, the fruits of our labors--or seven eighths of them at least (did I mention that pizza and FRACTIONS go well together? No? Consider it mentioned...):
Tuesday, September 22, 2009
n (or maybe n+1) Flies on the Wall
The 3-4 classes generally begin the year with work on number sense, number patterns, place value, and number in general. This year we're starting off with some projects involving functions and some simple algebraic ideas.
Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):
Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--
Student: 38!
Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--
Students: 48.
That's right. Okay, let's make a table and try it using some other numbers for n:
n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55
Looks good. Okay, what patterns do you see? How does n change when you add 10?
Students: The ones digit stays the same.
Yeah? Always, or only most of the time?
Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.
That's right. Can you think of a number n where the ones digit would change after you add 10?
Students: several suggestions, all of them withdrawn upon further reflection.
...Why doesn't it change?
Student: The number 10 has 0 in the ones column, so it doesn't change the ones.
Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]
What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?
Students, less hesitantly than before: Always.
How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.
Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.
n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88
Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!
Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!
By how much? Let's say it as a rule.
Students, cautiously: It goes up by one in the tens column and one in the ones column.
Always, or just sometimes?
n-2 or n-3 students, where n is the total population of the class: Always.
Two or three students: Sometimes.
Why sometimes?
2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...
2 or 3 more students: Ohhh!
Let's extend the table--
59 59 + 11 70
69 69 + 11 80
n/2 students: It goes up two in the tens!
The other n/2 students: And it goes down to 0 in the ones.
Okay, let;s make the rule. Help me out here:
[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]
Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??
[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...]
Here's a fly-on-the-wall view of an introductory lesson (shh; don't let them know you're in the room):
Teacher: Suppose we choose a number from 1 to 100. We'll call that number n. We often use the letter n in math to stand for any number. Someone pick a number for n--
Student: 38!
Good enough. So if n is 38, what's n + 10? 38 + 10, right? Which is--
Students: 48.
That's right. Okay, let's make a table and try it using some other numbers for n:
n n + 10 Result
--- ------- ----
38 38 + 10 48
17 17 + 10 27
90 90 + 10 100
45 45 + 10 55
Looks good. Okay, what patterns do you see? How does n change when you add 10?
Students: The ones digit stays the same.
Yeah? Always, or only most of the time?
Students, a bit hesitantly, because you always have to watch out for trick questions: Always...always so far, anyway.
That's right. Can you think of a number n where the ones digit would change after you add 10?
Students: several suggestions, all of them withdrawn upon further reflection.
...Why doesn't it change?
Student: The number 10 has 0 in the ones column, so it doesn't change the ones.
Another student: Oh, and when you add ten on the hundreds board you just go down to the next row, so if you're in the threes column you stay in the threes column...[We use the hundred board a lot; one is pictured here.]
What happens with the tens? The tens go up? Good; by how much? By one? Always, or only sometimes?
Students, less hesitantly than before: Always.
How do you know? So, okay, let's put the rule into words: When you add 10 to a number n, the ones digit stays the same but the tens digit goes up by one.
Nice job! Okay, let's try it again, only this time we'll look at what happens when you add 11 to n.
n n + 11 Result
--- ------- ----
12 12 + 11 23
28 28 + 11 39
77 77 + 11 88
Student, bursting to be the first: I know, I know! I know the rule! It's the tens digit goes up and the ones digit goes up too!
Student, bursting to be the second: Yeah! It's the tens digit goes up and the ones digit goes up too!
By how much? Let's say it as a rule.
Students, cautiously: It goes up by one in the tens column and one in the ones column.
Always, or just sometimes?
n-2 or n-3 students, where n is the total population of the class: Always.
Two or three students: Sometimes.
Why sometimes?
2 or 3 students: Because what happens when the number is in the nines? Say you add 11 to a number like 59...
2 or 3 more students: Ohhh!
Let's extend the table--
59 59 + 11 70
69 69 + 11 80
n/2 students: It goes up two in the tens!
The other n/2 students: And it goes down to 0 in the ones.
Okay, let;s make the rule. Help me out here:
[And so we develop the rule: When you add 11 to a number n, the tens digit usually goes up one and the ones digit goes up one as well, EXCEPT that when the ones digit is 9, the tens digit goes up by 2 and the ones digit goes back to 0. We talk about why this might be the case--and then out go the students to work on developing rules for n+1, or n+19, or n-2, or perhaps even n x 5...]
Okay, class is over for the day. You can come down from the wall now! Aren't you glad none of the kids brought flyswatters today??
[Edited to add: I should note that this lesson is adapted from a set of activities in a new book by math education guru Marilyn Burns. In 2008, I spoke at a national conference of math teachers. I was disappointed to discover that I was scheduled at the same time as Marilyn, which was disappointing for two reasons...first, I didn't get to hear her, and second, hardly anybody was left to come to my workshop...
Tuesday, July 21, 2009
Of Rabbits and Math Guys
Several years ago, early in my incarnation as Math Guy, I walked into Sue's third and fourth grade classroom ready to present a lesson. I was surprised to see that a bunch of rabbits had replaced the children that day.
The class had been reading a novel about rabbits or rabbitlike creatures, Sue explained, and several children had come up with the idea of dressing like rabbits one day, and the idea had met with approval from basically everybody.
Some had done just the basics--a few face-paint whiskers, a kush ball for a tail. Others had added a carefully-stapled set of ears made from construction paper. A few had gone whole hog (whole bunny?) and dressed all in white or brown or black with socks and slippers and even gloves. They looked...different. They looked...creative.
"Greetings, rabbits," I said, and asked them to take their seats so we could begin the math instruction for the day. For rabbits, they did reasonably well sitting still, and they did an even better job of listening (must've been the big ears).
My planned lesson was on what kids often like to call "timesing." We began by reviewing some basic multiplication facts and then moved on to multiplication strategies and the link between multiplication and addition, and just before I sent them to the tables to do some independent work, it suddenly occurred to me that I was--
--that's right--
--teaching rabbits to multiply.
Bada-bing!
True story, too.
The class had been reading a novel about rabbits or rabbitlike creatures, Sue explained, and several children had come up with the idea of dressing like rabbits one day, and the idea had met with approval from basically everybody.
Some had done just the basics--a few face-paint whiskers, a kush ball for a tail. Others had added a carefully-stapled set of ears made from construction paper. A few had gone whole hog (whole bunny?) and dressed all in white or brown or black with socks and slippers and even gloves. They looked...different. They looked...creative.
"Greetings, rabbits," I said, and asked them to take their seats so we could begin the math instruction for the day. For rabbits, they did reasonably well sitting still, and they did an even better job of listening (must've been the big ears).
My planned lesson was on what kids often like to call "timesing." We began by reviewing some basic multiplication facts and then moved on to multiplication strategies and the link between multiplication and addition, and just before I sent them to the tables to do some independent work, it suddenly occurred to me that I was--
--that's right--
--teaching rabbits to multiply.
Bada-bing!
True story, too.
Monday, June 8, 2009
37 Cities, 32 States, and Lord Knows How Many Unnecessary Miles
Cheerful Charlie's Tour of the USA is at an end. Between early October and early June he visited, as the title says, 32 states plus the District of Columbia and a total of 37 cities--large ones like San Diego and Seattle, Minneapolis and Houston; smaller ones like Cedar Rapids, Iowa and Springfield, Massachusetts; and small ones indeed such as Wall, South Dakota and Virginia City, Nevada. The students got used to enormous cross-country jumps and routes without rhyme or reason--just excess mileage and wasted gas. Still, they dutifully marked in the origin of each postcard he sent, used concepts of scale and ratio to estimate the distance from one city to the next, and made helpful suggestions about ways to improve his efficiency.
We are still calculating the total mileage. But it ain't gonna be pretty.
Here is the Official T-Shirt of Cheerful's travels. Please note that the list of cities should be quite accurate, as Cheerful sent me his list for proofreading (and boy oh boy did it need it). He did NOT tell me, however, that he was going to include the three lines at the top, so I had no opportunity to do the proofing. The mistakes--OF COURSE--are his and his alone.
You can click on the image for a closer look...
We are still calculating the total mileage. But it ain't gonna be pretty.
Here is the Official T-Shirt of Cheerful's travels. Please note that the list of cities should be quite accurate, as Cheerful sent me his list for proofreading (and boy oh boy did it need it). He did NOT tell me, however, that he was going to include the three lines at the top, so I had no opportunity to do the proofing. The mistakes--OF COURSE--are his and his alone.
You can click on the image for a closer look...
Monday, June 1, 2009
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...
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...
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.
Tuesday, April 21, 2009
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!
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!
Thursday, April 2, 2009
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.
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.
Friday, March 27, 2009
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.)
(Photo credits to Jan Campbell, and to Ellen DeLong's camera.)
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