This Week In Math
Apr. 27th, 2010 11:30 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
velvetpageThe problem:
You've been pretty lazy about laundry lately. You've been throwing socks into the drawer without matching them up first. You also haven't taken your laundry downstairs to wash it, so at the moment, there's only two unmatched pairs of socks in your drawer - one black, one red. Most of the time this isn't a problem because you've got lots of time to get dressed, but this morning, you're being picked up to go on a trip with a friend, and your alarm doesn't go off. Your friend knocks on the door expecting you to be ready, and you need to get dressed in a flash. You reach into the sock drawer and pull out two socks.
what is the probability that they match?
Now, I can't take credit for this problem. It's straight out of the Ministry of Education document entitled "Guide to Effective Instruction in Mathematics: Probability," for grade four to six. I do take some credit for how far we took it, however.
We started with paper bags, in which were two pairs of construction-paper socks. Kids did thirty trials and then predicted the actual probability based on their experiment. Then we figured out how many possible pairs there were, and how many of those pairs gave them a matching set. We got two matching pairs out of six possible pairs, or 2/6. Using what we knew of equivalent fractions, we reduced that to 1/3.
Then we extended it. What happens if we've got three pairs of socks? Is the probability of a matching pair better or worse or the same? I asked them to predict what they expected to find, and write down their prediction; then I asked them to prove it. Well, with three pairs, the probability is 3/15, which is 1/5. I reiterated, as I have many times, that it didn't matter too much if their prediction was wrong; what mattered was that, when they realized they were wrong, they went back to figure out why they'd made that prediction and checked to make sure they had it right now. I made sure we were using good scientific language for this process - hypothesis, experiment, proof.
Okay, so what about four pairs of socks? They figured out that the probability then was 4/28, and reduced that with help to 1/7. That was the end of day one.
On day two, I took the information we'd gathered the day before for two, three, and four pairs of socks, and organized it into a chart. Then I asked them to find the patterns, and use the patterns to predict the next term.
They came up with two patterns, only one of which I'd found myself. The first group noticed the pattern in the reduced fractions - 1/3, 1/5, 1/7 - and predicted that the next reduced fraction would be 1/9. Then they worked backwards to figure out the unreduced fraction of 5/45. The other group took the unreduced fractions - 2/6, 3/15, 4/28 - and figured out that the distance between 6 and 15 is nine, and the distance between 15 and 28 is 13 which is 9+4, and they postulated that the next term would be 13+4 more than 28, which is 45.
Anyone who got as far as seven or eight terms and gave a pattern rule that worked got a B. If they could use the phrase "theoretical probability" in their answer, that was bumped up to an A-, because the distinction between experimental and theoretical probability is a grade six topic according to the curriculum. Those who continued to develop the pattern for many more terms got an A.
Then I asked those who clearly understood that pattern to come to the carpet, and I introduced them to the concept of the nth term - when you don't know the term number, you can replace it with the variable n. If we could figure out how to get from the term number to the reduced fraction, consistently, then we could come up with any term even if they were out of order. So we looked at it, and realized that 1/3 is one less than two times two; 1/5 is one less than three times two; 1/7 is one less than four times two; and so on. So the denominator was two times the term number minus one. I showed them how to write this; 1/2n-1. To get an A+, all they had to do was show me that they could apply this to fill in two lines of the chart that were out of order, because the ability to solve an algebraic equation is a grade seven topic - two years above grade level.
In my class of twenty-five, I gave out exactly two B's. Everyone else got an A. My students on IEPs ALL got A's, and I didn't even have to adjust their expectations downwards; I just had to make sure they had access to support to clarify the patterns they saw.
The A's here are for both probability and patterning, so that's two A's on most report cards for my kids.
You've been pretty lazy about laundry lately. You've been throwing socks into the drawer without matching them up first. You also haven't taken your laundry downstairs to wash it, so at the moment, there's only two unmatched pairs of socks in your drawer - one black, one red. Most of the time this isn't a problem because you've got lots of time to get dressed, but this morning, you're being picked up to go on a trip with a friend, and your alarm doesn't go off. Your friend knocks on the door expecting you to be ready, and you need to get dressed in a flash. You reach into the sock drawer and pull out two socks.
what is the probability that they match?
Now, I can't take credit for this problem. It's straight out of the Ministry of Education document entitled "Guide to Effective Instruction in Mathematics: Probability," for grade four to six. I do take some credit for how far we took it, however.
We started with paper bags, in which were two pairs of construction-paper socks. Kids did thirty trials and then predicted the actual probability based on their experiment. Then we figured out how many possible pairs there were, and how many of those pairs gave them a matching set. We got two matching pairs out of six possible pairs, or 2/6. Using what we knew of equivalent fractions, we reduced that to 1/3.
Then we extended it. What happens if we've got three pairs of socks? Is the probability of a matching pair better or worse or the same? I asked them to predict what they expected to find, and write down their prediction; then I asked them to prove it. Well, with three pairs, the probability is 3/15, which is 1/5. I reiterated, as I have many times, that it didn't matter too much if their prediction was wrong; what mattered was that, when they realized they were wrong, they went back to figure out why they'd made that prediction and checked to make sure they had it right now. I made sure we were using good scientific language for this process - hypothesis, experiment, proof.
Okay, so what about four pairs of socks? They figured out that the probability then was 4/28, and reduced that with help to 1/7. That was the end of day one.
On day two, I took the information we'd gathered the day before for two, three, and four pairs of socks, and organized it into a chart. Then I asked them to find the patterns, and use the patterns to predict the next term.
They came up with two patterns, only one of which I'd found myself. The first group noticed the pattern in the reduced fractions - 1/3, 1/5, 1/7 - and predicted that the next reduced fraction would be 1/9. Then they worked backwards to figure out the unreduced fraction of 5/45. The other group took the unreduced fractions - 2/6, 3/15, 4/28 - and figured out that the distance between 6 and 15 is nine, and the distance between 15 and 28 is 13 which is 9+4, and they postulated that the next term would be 13+4 more than 28, which is 45.
Anyone who got as far as seven or eight terms and gave a pattern rule that worked got a B. If they could use the phrase "theoretical probability" in their answer, that was bumped up to an A-, because the distinction between experimental and theoretical probability is a grade six topic according to the curriculum. Those who continued to develop the pattern for many more terms got an A.
Then I asked those who clearly understood that pattern to come to the carpet, and I introduced them to the concept of the nth term - when you don't know the term number, you can replace it with the variable n. If we could figure out how to get from the term number to the reduced fraction, consistently, then we could come up with any term even if they were out of order. So we looked at it, and realized that 1/3 is one less than two times two; 1/5 is one less than three times two; 1/7 is one less than four times two; and so on. So the denominator was two times the term number minus one. I showed them how to write this; 1/2n-1. To get an A+, all they had to do was show me that they could apply this to fill in two lines of the chart that were out of order, because the ability to solve an algebraic equation is a grade seven topic - two years above grade level.
In my class of twenty-five, I gave out exactly two B's. Everyone else got an A. My students on IEPs ALL got A's, and I didn't even have to adjust their expectations downwards; I just had to make sure they had access to support to clarify the patterns they saw.
The A's here are for both probability and patterning, so that's two A's on most report cards for my kids.
(no subject)
Date: 2010-05-01 01:54 pm (UTC)-- hendrik
(no subject)
Date: 2010-05-01 04:33 pm (UTC)