"I am going to have a party," read the question given to a number of our 1-2 students the other day. "I want to invite ___ people." (The blank is standard: everybody gets a different number, which a) cuts down on the Problem of Roving Eyes and b) allows us to give somewhat harder numbers to kids who are ready for a challenge while keeping the same problem frame for everyone.)
"I have ____ tables where my guests can sit," the problem continues. "Each table has room for _____ people. Do I have enough tables, or do I need to get more?"
Different kids had different ways of attacking the problem, as usual. Some sketched the tables, drew chairs around them, and counted by ones. Others dispensed with the chairs and simply wrote the number at each table, then counted by that number if they knew how. A couple didn't bother with a sketch at all. One or two made groups with checkers or other materials--7 groups of 6 checkers, for instance, to represent 7 tables with 6 people at each--and then checked the number of people to see if they'd gone over or not. The strategies were generally quite accurate, if not consistently efficient: the next step will be to move kids away from the pictures and toward more abstract skip-counting and other strategies.
At any rate, children needed to show or describe their work and then answer the question (which, if you recall, had something to do with whether there were enough tables or whether we needed to get more). Several children didn't recall--they needed a reminder to do this part--but eventually we had the answers we sought.
"I have enough tables," one child wrote confidently and accurately. (Actually, she wrote "enuff," but let that pass...)
"You have enough tables. Am I rite?" wrote another child, perhaps a little less confident than the first. (Yup, I told him, you're rite. Um, right.)
"You need to get more tables," wrote a third responder, "because seven tables is going to be a smaller number of people. You need 8 tables." She included a careful sketch with the correct number of heads jowl by jowl at each table: an arrow then pointed to the last table, with the helpful label "extra."
"I have to sell one more chair," wrote still another girl. A somewhat convoluted way of saying that she not only had enough tables--she had an extra seat. I'm not entirely clear whether the sale would be an auction for the right to attend the party, or simply an attempt to convert an unwanted and unnecessary item into cold hard cash. Either way, this is a girl who knows the value of a buck.
And perhaps my favorite: the boy who discovered that he had space for 52 when he only needed to seat 49. After showing his method, he concluded: "You need more people."
Friday, October 30, 2009
Monday, October 26, 2009
Virtual Manipulatives
We sometimes haul out the laptops during lower school math times and have kids work with Utah State University's National Library of Virtual Manipulatives website. We've made good use of this site for projects with both the 3-4 and the 1-2 classes, but my personal favorite is the subtraction.
See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.
--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.
Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.
Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...
[Down, boy.]
[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]
Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).
The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.
See, you get these rods and cubes, just like base blocks only they're on the screen and exist only in pixel form, so they don't fall off the table and get lost and they can't be used as hockey sticks and pucks, or as drumsticks or grenade launchers or whatever else creative minds have in store for them.
--Oh, and then when you model regrouping (which I prefer not to call "borrowing," as I've said before, because you don't ever give it back--I prefer to use the phrase "stealing") you actually grab one of the virtual tens rods and bring it over to the ones column and let go and watch as it separates itself into ten little ones cubes.
Then you hear the kids saying WHOA! and COOL! and NEATO TORPEDO! (well, not that one, maybe) and the like.
Then you get to separate hundreds into tens the same way and thousands into hundreds and the whole thing is utterly charming and truly awesome and the best thing next to...
[Down, boy.]
[The picture below isn't actually from the virtual manipulatives website--it's from a powerpoint presentation I made dramatizing the process. What you see here is the ones stealing a ten, in the dead of the night of course, dragging it back to Ones Street, and breaking it into ten little ones cubes so there'll be enough ones to carry out the subtraction.]
Anyhow, children sometimes ask how they can get to the site at home. Unfortunately, the address isn't straightforward. If you google "virtual manipulatives," it's the first site that comes up (as of today, anyway).
The whole site's URL is http://nlvm.usu.edu/en/nav/vLibrary.html. If you're interested, take a spin around the site with your child(ren). It may not be the equivalent of a medieval European cathedral, but as the Michelin guide would put it, it's quite definitely worth a visit.
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...
Friday, October 9, 2009
p-a-t-t-e-r-n-s
What do you call it
When things repeat?
We call it...
A pattern.
Head, shoulders, knees, and feet,
Head, shoulders, knees, and feet,
That
Is a pattern.
A, B, C, A, B, C, A, B, C, A, B, C
That
Is a pattern,
1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3
That
Is a pattern.
What do you call it
When things repeat?
We call it...
A pattern.
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...):
Getting to School
If you're like most of us, you've always wanted to know how kindergarteners get to school. Lucky you! Now's your chance to find out--because the kindergarten recently put together a graph showing that information. Voila--!
(You can click on the picture to enlarge it.)
The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.
There are the reading-the-graph questions, of course:
*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?
And the interpreting-the-graph questions too:
*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?
But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.
(You can click on the picture to enlarge it.)
The kids enjoy the drawing, of course (some of them enjoy it quite a lot--I'm very fond of the multicolored, creatively shaped cars at the top of the cars column, along with the 3-wheeled truck on the far right). But it's also a good learning experience for these newly minted K students.
There are the reading-the-graph questions, of course:
*Which way of getting to school was the most common?
*Which was second most common?
*How many children came to school by truck?
*Which way of getting to school was used by 7 children?
And the interpreting-the-graph questions too:
*Suppose we asked the same question tomorrow and made a graph about that. Would the graph look exactly the same? Mostly the same? Not at all the same? Why?
*What if we made a graph showing how people got home? What do you think that would look like? Why?
*Why do you think no one got here on a skateboard? A surfboard? A motorboat?
*Do you think the school needs a bigger parking lot? Why?
But mostly, there's the notion that you can take information and show it in a way that makes it available to everybody who comes along. You can tell, just by looking, that more people in your class come to school by bus than come by van, and that a LOT more come by car than come by truck. You can locate your own name (or your own truck) on the graph and show a friend how you got to school that day. You can find out how a friend arrived. The information will be there today--and tomorrow--and the next day--and it will remain available forever, or at least as long as the teachers choose to hang it in the hall. Knowledge is power, we like to say; and graphs, I believe, are a really good example of that adage.
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...
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