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 division. Show all posts
Showing posts with label division. Show all posts
Sunday, March 21, 2010
Thursday, July 16, 2009
SummerMath, Part 3: The Bikepath, the Ballpark, and Beyond
My family went to the ballgame the other day, attracted by among other things a "Henry Hudson Bobblehead" giveaway (see picture). My son is quite eager to show off his Hudson Valley roots with this, um, iconic image when he heads west for his next college semester, and as for the rest of us, well, how could we pass up such a quality and historic freebie??
(Ol' Henry)
Anyhow, the game put me in my mind of Sports and Math. I spent most of my childhood free time engaged in one of five activities:
1) eating
2) reading the classics, mainly the Hardy Boys books
3) writing short stories with meandering plots and lots of unnecessary characters
4) playing board games and card games (see SummerMath Parts 1 and 2)
and
5) playing, watching, reading about, or thinking about baseball.
Baseball and math are closely linked, and in fact I learned quite a lot about math from my interest in baseball. My 1972 Sports Illustrated baseball board game (see 4 and 5 above) helped inform me about probability. I can remember the power I felt when I realized that I could use what (little) I knew about ratios to compare teams' won-lost records in my head--was it better to be 38-37 or 37-36, and how could I prove it? And I developed some facility with division, if not comprehension of WHY it worked, by virtue of calculating my batting average every day back when I was ten or so. (My batting average was very good. I counted it as a hit, of course, if someone muffed a ball I'd put in play. Or if the umpire mistakenly called me out when I was CLEARLY safe at first--don't laugh, it happened all the time. Or if I hit a line drive or a deep fly ball that somebody managed to corral, but which clearly SHOULD'VE been a hit--why should I be penalized just because my opponents had good hands? That was in addition to the occasional, you know, REAL hits I got. As I said, my batting average was very good.)
In any case, there are lots of ways to combine math with sports, for those of you whose children like to watch baseball, play soccer, ride bikes, or mess around with balls and such in the back yard after dinner. Here are some ideas of questions you can ask and projects you can do:
*Counting and estimating. "I wonder how many pitches the pitcher will throw this inning. Do you think it'll be more than 15 or less than 15?" "Take ten shots on goal from right here. Let's see how many go in...Now let's move you back a few feet. How many do you think will go into the net now?" "Good job! We just did 6 throws back and forth in a row without dropping a single one. Think we can beat that record? Let's keep track."
*Adding and subtracting, multiplying and dividing. "The scoreboard says the Renegades are winning 7 to 2. How many runs are they winning by?" "That's your third basket. Each basket is worth 2 points. How many points do you have so far?" I'll just add that I have taught many primary graders who could count rapidly by twos, fives, and tens when they came to my class, and a few who could rattle off threes, fours, and nines; but the only one I ever had who could count fluently by sevens was the one who lived and died with the NY Giants. Sevens...football...hmm!
*Measuring. "You sure hit that one a long way! I wonder how far it went.." You can measure with "nonstandard units," such as steps or rake lengths, which tends to be a little more meaningful for younger children, or with standard units--feet, yards, meters. "14 rake lengths--that's a lot. Whoa, that one went even further! Would you say 15, or 20, or even more?" How long does it take to run around the yard or the perimeter of the park? Time your child; let your child time you. Write it down. Try it again another day. Look at the map of one of the local bike paths. "It's 10 and a half miles long! How far do you think we'll get before I'll be ready to turn around?...I see another mileage marker up ahead--4 miles and still going!"
*Graphing. These take a little more time and energy, but they're great for kids who really love sports, especially team spectator sports. Work with your child to make a bar graph showing his or her favorite team's wins and losses.
(A sample bar graph)
Update it daily; use the internet or the newspaper to get the scores.
Or, make a line graph showing the number of runs your team scores on a daily basis. Look how the line moves around. What has the trend been? More runs over time, or fewer or about the same? How could you show the number of runs they gave up each day on the same graph?
(A sample line graph)
Can you make a graph showing how many times you go swimming/bicycling/hiking this month? We'll write the words down here; put up a blue sticker for the water whenever we swim, a red sticker for the color of your bike to show each time you go for a ride, a green sticker for the color of the leaves to stand for a hike.
(A sample picture graph)
Which has the most so far? The fewest? How many more bike rides have you taken than hikes?
Of course, I don't mean to reduce sports and physical activity to numbers. Nor is the point for kids to quantify their outside play. Be sure that timing and measuring are just for fun, a nice way of bringing a little math into children's lives, not an opportunity for frustration and embarrassment because they can't seem to beat their old record; be sure that a graph is a cute little add-on, not another chore that has to be done or the sole reason for taking a bike ride or going out for a hike. Sports are their own reward. Though, now that I think about, the ability to hit .658 (and calculate it properly!) might be its own reward, too...
(Ol' Henry)Anyhow, the game put me in my mind of Sports and Math. I spent most of my childhood free time engaged in one of five activities:
1) eating
2) reading the classics, mainly the Hardy Boys books
3) writing short stories with meandering plots and lots of unnecessary characters
4) playing board games and card games (see SummerMath Parts 1 and 2)
and
5) playing, watching, reading about, or thinking about baseball.
Baseball and math are closely linked, and in fact I learned quite a lot about math from my interest in baseball. My 1972 Sports Illustrated baseball board game (see 4 and 5 above) helped inform me about probability. I can remember the power I felt when I realized that I could use what (little) I knew about ratios to compare teams' won-lost records in my head--was it better to be 38-37 or 37-36, and how could I prove it? And I developed some facility with division, if not comprehension of WHY it worked, by virtue of calculating my batting average every day back when I was ten or so. (My batting average was very good. I counted it as a hit, of course, if someone muffed a ball I'd put in play. Or if the umpire mistakenly called me out when I was CLEARLY safe at first--don't laugh, it happened all the time. Or if I hit a line drive or a deep fly ball that somebody managed to corral, but which clearly SHOULD'VE been a hit--why should I be penalized just because my opponents had good hands? That was in addition to the occasional, you know, REAL hits I got. As I said, my batting average was very good.)
In any case, there are lots of ways to combine math with sports, for those of you whose children like to watch baseball, play soccer, ride bikes, or mess around with balls and such in the back yard after dinner. Here are some ideas of questions you can ask and projects you can do:
*Counting and estimating. "I wonder how many pitches the pitcher will throw this inning. Do you think it'll be more than 15 or less than 15?" "Take ten shots on goal from right here. Let's see how many go in...Now let's move you back a few feet. How many do you think will go into the net now?" "Good job! We just did 6 throws back and forth in a row without dropping a single one. Think we can beat that record? Let's keep track."
*Adding and subtracting, multiplying and dividing. "The scoreboard says the Renegades are winning 7 to 2. How many runs are they winning by?" "That's your third basket. Each basket is worth 2 points. How many points do you have so far?" I'll just add that I have taught many primary graders who could count rapidly by twos, fives, and tens when they came to my class, and a few who could rattle off threes, fours, and nines; but the only one I ever had who could count fluently by sevens was the one who lived and died with the NY Giants. Sevens...football...hmm!
*Measuring. "You sure hit that one a long way! I wonder how far it went.." You can measure with "nonstandard units," such as steps or rake lengths, which tends to be a little more meaningful for younger children, or with standard units--feet, yards, meters. "14 rake lengths--that's a lot. Whoa, that one went even further! Would you say 15, or 20, or even more?" How long does it take to run around the yard or the perimeter of the park? Time your child; let your child time you. Write it down. Try it again another day. Look at the map of one of the local bike paths. "It's 10 and a half miles long! How far do you think we'll get before I'll be ready to turn around?...I see another mileage marker up ahead--4 miles and still going!"
*Graphing. These take a little more time and energy, but they're great for kids who really love sports, especially team spectator sports. Work with your child to make a bar graph showing his or her favorite team's wins and losses.
(A sample bar graph)Update it daily; use the internet or the newspaper to get the scores.
Or, make a line graph showing the number of runs your team scores on a daily basis. Look how the line moves around. What has the trend been? More runs over time, or fewer or about the same? How could you show the number of runs they gave up each day on the same graph?
(A sample line graph)Can you make a graph showing how many times you go swimming/bicycling/hiking this month? We'll write the words down here; put up a blue sticker for the water whenever we swim, a red sticker for the color of your bike to show each time you go for a ride, a green sticker for the color of the leaves to stand for a hike.
(A sample picture graph)
Which has the most so far? The fewest? How many more bike rides have you taken than hikes?
Of course, I don't mean to reduce sports and physical activity to numbers. Nor is the point for kids to quantify their outside play. Be sure that timing and measuring are just for fun, a nice way of bringing a little math into children's lives, not an opportunity for frustration and embarrassment because they can't seem to beat their old record; be sure that a graph is a cute little add-on, not another chore that has to be done or the sole reason for taking a bike ride or going out for a hike. Sports are their own reward. Though, now that I think about, the ability to hit .658 (and calculate it properly!) might be its own reward, too...
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...
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