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...)
Friday, February 5, 2010
Saturday, January 9, 2010
Sleepovers
(Warning: This is not about math.)
Back when I was a classroom teacher, I once taught a girl we will call Claudia. Claudia was new to the school and was a member of my 1-2 class. She was shy and quiet, but she took school seriously and was quite eager to please.
Writing skills are of course a major part of the 1-2 curriculum. Most children try out different subjects and themes as they are learning to write. Not Claudia. Her first three pieces of writing were all essentially the same. Allowing for some variation in wording and the name of her host, they all read like this:
"I slept over my friend Kari's house. First we played. Then we ate dinner. After that we watched TV. Then we played some more. We went to bed. In the morning we had breakfast. Then we watched TV. We played. Then I went home."
At first, I simply worked with Claudia to improve her writing given this basic framework. Could she add some details? What show did she watch, what did she have for dinner? How about putting in some describing words? Was the bed hard or soft, was the TV show funny or scary? I offered some ideas for varying the sentence structure and helped her explore other ways to organize the information. "Conferencing," we call it (though I'm not sure we called it "conferencing" back in the day), and Claudia dutifully added the information I told her to include, and each of the first three stories wound up a little bit different in final form, though not, I must admit, a lot different.
Perhaps, for deep psychological reasons, it was very necessary for Claudia to tell this same story again and again, but where her writing was concerned the constant repetition seemed to be getting in her way. Ideas, after all, were important too. I decided that the "Claudia and the Sleepover" series, Vols 1-3, was borne less of emotional turmoil and more of being in a rut. So when it came time for her to start her next piece of writing, I called her aside.
"You've been writing a lot about your sleepovers," I told her. "I know you like writing those stories, but I think you're ready to write about something new. Let's try this. What's the weirdest, strangest, most unbelievable thing you can think of?"
Claudia thought for a moment. "You mean," she said slowly, "like, my stuffed animals came to life?"
This was a greater leap than I had expected--a highly imaginative response, and given almost immediately. YES!! I thought. "Sounds great," I said, smiling. "So, here's your first line: 'Last night my stuffed animals came to life!' Got it?"
Claudia nodded.
"Then go get 'em," I said, steering her back to her seat. She picked up her pencil and started writing. Excellent, I thought as I moved off to consult with the next child. I'd said the right thing at the right time. I'd handled Claudia well; I'd started her down the road toward more creativity and a greater interest in writing. This was why the school paid me the big bucks. (So to speak).
That warm glow lasted only ten minutes, however, because when I returned to check in with Claudia this is what she had written on her paper:
"Last night my stuffed animals came to life. And they had a sleepover. First, they played..."
Back when I was a classroom teacher, I once taught a girl we will call Claudia. Claudia was new to the school and was a member of my 1-2 class. She was shy and quiet, but she took school seriously and was quite eager to please.
Writing skills are of course a major part of the 1-2 curriculum. Most children try out different subjects and themes as they are learning to write. Not Claudia. Her first three pieces of writing were all essentially the same. Allowing for some variation in wording and the name of her host, they all read like this:
"I slept over my friend Kari's house. First we played. Then we ate dinner. After that we watched TV. Then we played some more. We went to bed. In the morning we had breakfast. Then we watched TV. We played. Then I went home."
At first, I simply worked with Claudia to improve her writing given this basic framework. Could she add some details? What show did she watch, what did she have for dinner? How about putting in some describing words? Was the bed hard or soft, was the TV show funny or scary? I offered some ideas for varying the sentence structure and helped her explore other ways to organize the information. "Conferencing," we call it (though I'm not sure we called it "conferencing" back in the day), and Claudia dutifully added the information I told her to include, and each of the first three stories wound up a little bit different in final form, though not, I must admit, a lot different.
Perhaps, for deep psychological reasons, it was very necessary for Claudia to tell this same story again and again, but where her writing was concerned the constant repetition seemed to be getting in her way. Ideas, after all, were important too. I decided that the "Claudia and the Sleepover" series, Vols 1-3, was borne less of emotional turmoil and more of being in a rut. So when it came time for her to start her next piece of writing, I called her aside.
"You've been writing a lot about your sleepovers," I told her. "I know you like writing those stories, but I think you're ready to write about something new. Let's try this. What's the weirdest, strangest, most unbelievable thing you can think of?"
Claudia thought for a moment. "You mean," she said slowly, "like, my stuffed animals came to life?"
This was a greater leap than I had expected--a highly imaginative response, and given almost immediately. YES!! I thought. "Sounds great," I said, smiling. "So, here's your first line: 'Last night my stuffed animals came to life!' Got it?"
Claudia nodded.
"Then go get 'em," I said, steering her back to her seat. She picked up her pencil and started writing. Excellent, I thought as I moved off to consult with the next child. I'd said the right thing at the right time. I'd handled Claudia well; I'd started her down the road toward more creativity and a greater interest in writing. This was why the school paid me the big bucks. (So to speak).
That warm glow lasted only ten minutes, however, because when I returned to check in with Claudia this is what she had written on her paper:
"Last night my stuffed animals came to life. And they had a sleepover. First, they played..."
Saturday, January 2, 2010
A PalindromemordnilaP A
John McAdam, professor of education at Marist College and a fellow Math Guy in spirit if not in title, reminds me that today is a palindrome: when written in month/day/year format, the digits are the same read forward as read backward:
01/02/2010
Okay, okay, you have to put a couple of leading zeroes in there, but pretty darn good, wouldn't you say?
And FYI, the next one will be 11/02/2011. A loooong way off. So as Paul Simon (the singer not the senator) never put it,
enjoy it while you can, Nan
it ain't all that bad, Dad
there's no need to sob, Bob
just sit down and chill, Lil
and eat a banana, Hannah
(or some other food, Dood)
Just leave it to me... .
01/02/2010
Okay, okay, you have to put a couple of leading zeroes in there, but pretty darn good, wouldn't you say?
And FYI, the next one will be 11/02/2011. A loooong way off. So as Paul Simon (the singer not the senator) never put it,
enjoy it while you can, Nan
it ain't all that bad, Dad
there's no need to sob, Bob
just sit down and chill, Lil
and eat a banana, Hannah
(or some other food, Dood)
Just leave it to me... .
Saturday, December 19, 2009
On Homework
"Okay," says the teacher, relentlessly cheerful as teachers of youngish children are expected to be, "does everyone understand this chapter?"
"YES!" bellow the assembled students.
"Have you all finished the sample problems?" the teacher continues.
"YES!" the kids chorus again.
"Are there any questions about the homew--"
"NO!"
You probably can guess the punch line. But if not, check it out (just copy and paste):
http://www.babyblues.com/archive/index.php?formname=getstrip&GoToDay=11/29/2009
"YES!" bellow the assembled students.
"Have you all finished the sample problems?" the teacher continues.
"YES!" the kids chorus again.
"Are there any questions about the homew--"
"NO!"
You probably can guess the punch line. But if not, check it out (just copy and paste):
http://www.babyblues.com/archive/index.php?formname=getstrip&GoToDay=11/29/2009
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.
Wednesday, December 9, 2009
What to Wear in Winter
As a high school student, I came up with a foolproof (and very mathematical) way to determine what winter clothes I needed.
I attended a PK-12 school, and the estimate depended on the behavior of two very different student groups: kindergarteners and sixth/seventh graders.
The K students were sent in (by parents) with masses of winter protection--coats, boots, hats, mittens, earmuffs, scarves, alpenstocks, beeveils, etc--and sent out (by teachers, into the elements) the same way.
The 6th/7th graders, regardless of what they were sent in wearing or sent out wearing, very quickly removed as much outer clothing as possible. There was something truly cool about wearing short sleeves as the mercury dipped down to the single digits Fahrenheit. (Also something truly frostbitten about it, but when you're in middle school you don't care.)
My formula was simple: to determine what level of clothing I needed, I found the halfway point between the overdressed kindergarteners and the underdressed middle school students. That was what I put on before leaving school.
Worked every time. And to judge by what I see out on our playground this winter, the formula continues to work today!
I attended a PK-12 school, and the estimate depended on the behavior of two very different student groups: kindergarteners and sixth/seventh graders.
The K students were sent in (by parents) with masses of winter protection--coats, boots, hats, mittens, earmuffs, scarves, alpenstocks, beeveils, etc--and sent out (by teachers, into the elements) the same way.
The 6th/7th graders, regardless of what they were sent in wearing or sent out wearing, very quickly removed as much outer clothing as possible. There was something truly cool about wearing short sleeves as the mercury dipped down to the single digits Fahrenheit. (Also something truly frostbitten about it, but when you're in middle school you don't care.)
My formula was simple: to determine what level of clothing I needed, I found the halfway point between the overdressed kindergarteners and the underdressed middle school students. That was what I put on before leaving school.
Worked every time. And to judge by what I see out on our playground this winter, the formula continues to work today!
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!
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