Purposeful Ramblings Through My Mind

This is actually "Last Week In Math," because I didn't get to it, but I have fifteen minutes right now and I know you're on the edges of your seats waiting for it. I wouldn't want to disappoint!

I realized at the beginning of last week that I didn't have enough marks for the report cards; I was short measurement and geometry marks, though I had taught all the other strands in third term. I decided measurement was the more crucial of the two, and also the one that I'd touched on at another time, so I figured I could get a mark for area quite easily.

Thursday was a simple review of area of a rectangle, which I was pretty sure they already knew. I was right; only two or three kids needed to discover the formula for area, and by the end of the lesson all of them had it. That was the grade five expectation met. So I looked ahead, because I like to give A's, and discovered that the grade six expectation was area of triangles and parallelograms, using the formula for area of a rectangle as the starting point.

My kids can do this, thought I.

So I started them on Friday with a sheet of grid paper and asked them to draw a 3cmx4cm rectangle. We figured out the area with no problem. Then I asked them to draw a diagonal line from one corner to an opposite corner, creating two triangles. With some demonstration of what I meant, they accomplished this.

I got them to tell me about the triangles. They were right-angled triangles. The other two angles were both acute. (We discussed why the other two angles in a right-angled triangle have to be acute at this point.) The length of one side of each triangle was 3 cm, and the length of another side was 4 cm. So far, so good, but nobody had yet come up with the word I was looking for, so I primed them: during the quilt unit, we talked about shapes like this. Compare these two triangles. Someone said they were the same, and then I managed to draw the word out of them: they're congruent, and they're rotated 180 degrees.

Then I asked them to look at the rectangle with its triangles and describe it using a fraction. I shaded one of the two triangles and asked for a fraction; without too much pulling, they said it was half.

Without belabouring the point, I asked: If the entire rectangle is 12cm^2, what is the area of each of the triangles?  They got the answer - 6cm^2.

I asked them to draw a few more rectangles, bisect them the same way, and test this; when they counted squares and parts of squares, did they get that the triangles were in fact half the area of the rectangles?  Next I asked them to draw a right-angled triangle and use what they knew about rectangles to come up with the area of the triangle.  I was looking for the kids who could do it without completing the rectangle first; a few kids needed me to draw the congruent triangle for them so they could see what I meant by it.  But within another ten minutes, kids were showing me their work and explaining it, so I knew they got it.

Then I started offering the challenge question.  I took their grid paper and drew a triangle on it that wasn't a right-angled triangle, and asked them to use what they'd just learned to find the area of that triangle.

A couple of them started by finding the height, thereby creating two right-angled triangles, and then building rectangles on each of those.  One or two drew a rectangle without creating the two right-angled triangles first; to those ones I asked how they knew that the triangle I'd drawn was exactly half of the rectangle they'd drawn, since the other triangles in their rectangle weren't congruent with the first one.  Then they figured out that they could prove that the triangle was half of the rectangle if they drew in the height and made right-angled triangles.

At the end of the lesson, I gathered all the kids together who had completed the extension activity (I didn't force anyone to do it after they got the right-angled triangle, because that was already an A.)  I explained the different terminology: when we're talking about rectangles, we use the terms length and width, but when talking about triangles, we use base and height instead; they mean the same thing in terms of their drawings but it's important to know the terminology so you know what other mathematicians are talking about.  I also showed them a couple versions of the algebraic formula for area of a triangle: 1/2 bh, bh/2, and the formats they're more familiar with involving symbols for multiplication and division.

Next up: area of a parallelogram.

The 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.

I think maybe I'll make this a weekly feature, just for the heck of it. It reminds me that I'm doing a good job.

So, this week in math, we started by reviewing what little we'd learned about equivalent fractions before the March break. Since exploratory learning is starting to become de rigeur for my students (as it should be) I took them to the computer lab and let them play around with the fractions manipulatives on the national library of virtual manipulatives website. The differentiation was easy. Don't get this at all? Try the basic fractions manipulative. You've already figured out the pattern rule? Great, do the grade six one.

On Monday, I gave them an assignment to do. They would get four or five pieces of regular paper in different colours, their choice. On each one, they'd write a different common fraction. The example, which I made up with my six-year-old on Sunday, was one half. Then they'd come up with three or four equivalents for it, and represent those equivalents in pictures. They had to explain the fractions and how those fractions were connected to the original fraction. There is a rubric - let me know if you want it.

The kids who were having trouble even accessing one half came for some guided math with me. We did fraction strips, and played around a bit with blocks, until they got to the point where they understood that if you divided all the pieces the same way, you had an equivalent fraction.

Then, because an abstract understanding is important to work towards, we took the equivalent fractions they'd come up with and analyzed them. How do you get from 1/2 to 4/8? If they expressed it as adding, I told them they were right, but they'd see the pattern faster if they thought of it as multiplication. So what would they multiply by? Some figured out the pattern independently; others needed to be shown several times, and finally got it with the help of more manipulatives. Then I encouraged them to use the pattern rule they'd found to come up with what they thought was an equivalent fraction. They had to come up with the proof that they were right by drawing a representation they could connect back to the original fraction.

As of this writing, most of my kids have finished two or three sheets of equivalent fractions. The rubric gives an A to kids who can connect their equivalents to decimals or percents, so I've explained the idea of repeating fractions to the kids trying that for 1/3. I suspect most will finish on time - the assignment is due on Monday.

Next Week: comparing and ordering fractions is so much more interesting when the denominators are different.