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"But this is grade seven math!"
Feb. 16th, 2010 07:37 pmWe've been doing multiplication in math lately.
Now, when I teach multiplication, it's embedded in a visual context that is also easy to represent in a tactile way, so as to reach as many students as possible. So we start with arrays, or area models, more commonly known as rectangles. In the question 3x5, for example, I'd get graph paper and draw a line three blocks across, then five blocks down. The kids can easily visualize that the answer is 15 because they can finish the rectangle and count the blocks inside it. If they can't conceive of it visually, they can use blocks to do the same thing.
Well, we realized while doing the quilt blocks that when you have the same number of blocks across as you do down, the shape you've made is a particular kind of rectangle - a square. We further realized that you could subdivide many squares into smaller squares and still have squares; for example, 6x6 gives you a square of 36, and that grid can be further broken down into nine blocks of four squares each or four blocks of nine squares each.
In the grade five curriculum, the only appearance of exponents is in units of measurement for area and volume; so the kids learn centimetres squared as cm^2 and centimetres cubed as cm^3, but otherwise never see an exponent. But when we were doing square numbers anyway, I figured there was no harm in showing them how people write square numbers.
Fast-forward to about two weeks ago. I was coming up with division questions for my class, focusing on those with no remainders or decimals for simplicity's sake. I gave my kids 196/7. They were to figure it out using manipulatives (blocks) or graph paper arrays (drawing the seven squares across the top and figuring out how many rows they'd have to have in order to have 196 boxes in their rectangle.) Most did this by doing 7x10, and subtracting 70 from 196, then doing that again, then figuring out how many were left over and how many times 7 would give them 56. It's a visual representation of a standard division algorithm.
One kid finished really fast, so as an extension, I asked him: can you use the array you drew to figure out if 196 is a square number?
He cut out his rectangle, cut it in half, put the two halves next to each other, and realized he now had a 14x14 array, so he answered yes, and explained it to the class.
Then I gave them 256/8; how can I tell that 256 is a square number? What is it the square of?
Well, we did a bunch of them, playing with the arrays as the kids developed their understanding of friendly numbers to include square numbers. (Friendly numbers are those that are easy to work with, like multiples of 10 or 25. They're numbers you round to and calculate with when you don't want to do it the long way.) Eventually, we figured out that 7^2x3^2=21^2.
Today, I introduced them to the idea of a mathematical proof. I told them, "We figured out on Friday that seven squared times three squared equalled twenty-one squared. What I want to know is, does that pattern hold true all the time? If I pick two random numbers and multiply their squares, will my answer be equal to the product of those two numbers, squared? I want you to do a bunch of examples and show that this is true in every single one of them." Meanwhile, I asked three of my brightest kids - including the original 14x14 kid - to try the same thing, only with cubes instead of squares.
They did it. A few of them needed to check their multiplication on calculators, but they did it, and they understood it.
Then they asked me why it was important, so I showed them the basics of scientific notation. They grasped that really fast, and the idea that you could express seventy million as 7x10^7 was intriguing to them.
One of them asked if this was still grade five math. I got out my curriculum document and went on a hunt, because, while I was sure we were well beyond grade five math, I wasn't sure how far; it's been five years since I taught grade seven. I discovered that scientific notation isn't an expectation until grade eight, and facility with exponents isn't an expectation until grade seven. So I told them that if I gave them a quiz on this material, and they all did fairly well at it, I could give ALL of them an A - because they're working beyond grade level on this topic. Furthermore, if they didn't quite get it, that was absolutely fine - because I knew that every single person I'd asked to do that problem was fairly adept at grade five multiplication, so I could still give them a B. (There was another group working on multiplication facts, because they've managed to miss them up to this point. If I can get them up to speed on their facts, they may still get a B, but they're almost out of the running for an A because we're running out of time.)
I've come to a few conclusions with this.
First, if you're determined to ask big questions with big connections in math, you're not going to stick to your curriculum.
Second, sticking to the curriculum is boring.
Third, artificial divisions of concepts aren't worth the paper they're printed on. If my kids are able to handle it, and interested in doing it, and coming across the ideas on their own but needing help expanding their understanding of it, it would be bad teaching for me to hold them back because it's not in the curriculum. The curriculum is a guideline, not a Bible. Nobody is going to care if I cover too much math, anymore than they've ever cared if I didn't cover quite all of it. (I'm in no danger of that. I still have fractions and probability to do, but they won't take me four months and I'll be through the curriculum in plenty of time to do some grade six or seven stuff.)
Next up: x^2 times 2^2=36. Solve for x. (The grade five curriculum doesn't call for variables in expressions requiring multiplication either. I don't care.)
Now, when I teach multiplication, it's embedded in a visual context that is also easy to represent in a tactile way, so as to reach as many students as possible. So we start with arrays, or area models, more commonly known as rectangles. In the question 3x5, for example, I'd get graph paper and draw a line three blocks across, then five blocks down. The kids can easily visualize that the answer is 15 because they can finish the rectangle and count the blocks inside it. If they can't conceive of it visually, they can use blocks to do the same thing.
Well, we realized while doing the quilt blocks that when you have the same number of blocks across as you do down, the shape you've made is a particular kind of rectangle - a square. We further realized that you could subdivide many squares into smaller squares and still have squares; for example, 6x6 gives you a square of 36, and that grid can be further broken down into nine blocks of four squares each or four blocks of nine squares each.
In the grade five curriculum, the only appearance of exponents is in units of measurement for area and volume; so the kids learn centimetres squared as cm^2 and centimetres cubed as cm^3, but otherwise never see an exponent. But when we were doing square numbers anyway, I figured there was no harm in showing them how people write square numbers.
Fast-forward to about two weeks ago. I was coming up with division questions for my class, focusing on those with no remainders or decimals for simplicity's sake. I gave my kids 196/7. They were to figure it out using manipulatives (blocks) or graph paper arrays (drawing the seven squares across the top and figuring out how many rows they'd have to have in order to have 196 boxes in their rectangle.) Most did this by doing 7x10, and subtracting 70 from 196, then doing that again, then figuring out how many were left over and how many times 7 would give them 56. It's a visual representation of a standard division algorithm.
One kid finished really fast, so as an extension, I asked him: can you use the array you drew to figure out if 196 is a square number?
He cut out his rectangle, cut it in half, put the two halves next to each other, and realized he now had a 14x14 array, so he answered yes, and explained it to the class.
Then I gave them 256/8; how can I tell that 256 is a square number? What is it the square of?
Well, we did a bunch of them, playing with the arrays as the kids developed their understanding of friendly numbers to include square numbers. (Friendly numbers are those that are easy to work with, like multiples of 10 or 25. They're numbers you round to and calculate with when you don't want to do it the long way.) Eventually, we figured out that 7^2x3^2=21^2.
Today, I introduced them to the idea of a mathematical proof. I told them, "We figured out on Friday that seven squared times three squared equalled twenty-one squared. What I want to know is, does that pattern hold true all the time? If I pick two random numbers and multiply their squares, will my answer be equal to the product of those two numbers, squared? I want you to do a bunch of examples and show that this is true in every single one of them." Meanwhile, I asked three of my brightest kids - including the original 14x14 kid - to try the same thing, only with cubes instead of squares.
They did it. A few of them needed to check their multiplication on calculators, but they did it, and they understood it.
Then they asked me why it was important, so I showed them the basics of scientific notation. They grasped that really fast, and the idea that you could express seventy million as 7x10^7 was intriguing to them.
One of them asked if this was still grade five math. I got out my curriculum document and went on a hunt, because, while I was sure we were well beyond grade five math, I wasn't sure how far; it's been five years since I taught grade seven. I discovered that scientific notation isn't an expectation until grade eight, and facility with exponents isn't an expectation until grade seven. So I told them that if I gave them a quiz on this material, and they all did fairly well at it, I could give ALL of them an A - because they're working beyond grade level on this topic. Furthermore, if they didn't quite get it, that was absolutely fine - because I knew that every single person I'd asked to do that problem was fairly adept at grade five multiplication, so I could still give them a B. (There was another group working on multiplication facts, because they've managed to miss them up to this point. If I can get them up to speed on their facts, they may still get a B, but they're almost out of the running for an A because we're running out of time.)
I've come to a few conclusions with this.
First, if you're determined to ask big questions with big connections in math, you're not going to stick to your curriculum.
Second, sticking to the curriculum is boring.
Third, artificial divisions of concepts aren't worth the paper they're printed on. If my kids are able to handle it, and interested in doing it, and coming across the ideas on their own but needing help expanding their understanding of it, it would be bad teaching for me to hold them back because it's not in the curriculum. The curriculum is a guideline, not a Bible. Nobody is going to care if I cover too much math, anymore than they've ever cared if I didn't cover quite all of it. (I'm in no danger of that. I still have fractions and probability to do, but they won't take me four months and I'll be through the curriculum in plenty of time to do some grade six or seven stuff.)
Next up: x^2 times 2^2=36. Solve for x. (The grade five curriculum doesn't call for variables in expressions requiring multiplication either. I don't care.)