Math

 

 

Here are two items to give you an idea of my approach to teaching math. First, an essay that I wrote some years ago for my school’s website. And second, a chapter from a book I’m working on.

 

Why do I love teaching math to young students?

Well, first of all, I love teaching these classes because of my students! They are full of excitement, wonder and curiosity. They’re very willing to work hard, and they love learning. Each year, beginning in September, I put a lot of effort into building a positive classroom atmosphere of mutual support and helpful collaboration. And my young students do make this happen. I love watching them work together, and I rejoice each time one says to another, “I respectfully disagree.”

I also need to thank the teachers who’ve worked with my math students in earlier grades, both at PFS and at other schools. My students almost always come to me with good math skills and a positive attitude towards math. I am also grateful to the parents of my students, who have prepared them to be active learners and helpful class citizens, and who support their learning in many ways.

Finally, I need to say a big thank you to my boss, Jane Fremon, who, since the founding of our school, has encouraged me to teach math my way. And “my way,” I shamefacedly admit, is to make class more fun for me. (I hope this benefits the students too!) Naturally we cover much the same topics any third or fourth grade class will cover. But I do look for ways to teach the standard skills and practice the standard drill inside discovery lessons. A few examples follow.

Subtraction with borrowing! Most young people need some repetition and drill on this skill. I have learned to use Kaprekar’s Constant (named after the Indian mathematician D. R. Kaprekar) as a great way to practice borrowing. We start with any four digit number where not all four digits are the same. We arrange the digits to make the biggest possible number and the smallest possible number, and then we subtract the latter from the former. Then we iterate, using that answer to begin the process again. We all start first with a “seed” I select, so we can all help each other. I tell the students that if they work carefully and check their work with each other, after a few iterations they’ll make an amazing discovery. And they do! After the first discovery, and a few sentences describing it, students are allowed to pick their own seed and see what happens with that starting number. We generally continue this work over two or three days, and it’s always an engaging and enjoyable lesson. And oh, thanks to the way it’s set up, every subtraction has borrowings!

Number facts! One of the ways we practice our multiplication facts is with the whole-class game Buzz. We go around the room and students simply say counting numbers in order, but they substitute “buzz” for any multiples of the times table we’re practicing. (For example, “1, 2, buzz, 4, 5, buzz, 7, 8, buzz…”) Great drill in and of itself, and a good opportunity to reinforce clear speaking and careful listening. But then come my questions. “Did the buzz move? Or did the same kids say buzz as we kept going around?” This leads to interesting conjectures, and eventually the idea of factors. After some experience with the game, I show them how we can model our buzz games by drawing star polygons. (With a class of 12 students we can model the class by 12 equally spaced points on a circle, and see how the buzz travels around.) This usually leads to some initial confusion and then bursts of creativity. Children often turn their star polygons into colorful art works, and many delight in exploring “funny” situations. Like the ones times table. Or the six times table with twelve points. Or the thirteen times table with twelve points! And of course we get to practice using our rulers and protractors…

The Invention of Chess! Young children seem to love doubling, and they love large numbers. There is an ancient story about the invention of chess, in which the inventor asks for a reward of rice, one grain for the first square, two for the second, four for the next, and so on, doubling each time. This lesson gives third graders practice adding a number to itself, or multiplying a number by two (and seeing that these processes are equivalent). There are lots of opportunities for discovery here: for example the repeating pattern in the ones column of the powers of two, or the fact that the sum of the powers of two up to any power of two (for example, 16) is always one less than the next power of two (1 + 2 + 4 + 8 + 16 = 31 = 32 – 1). And there are also lots of opportunities to learn about estimating. (What could we fill with 1023 grains of rice?) It’s also an opportunity to keep a neat table with three columns:

 

square                         grains                          total so far

1                                   1                                   1

2                                   2                                   3

3                                   4                                   7

4                                   8                                   15

 

A Table of Factors! This is yet another way to practice multiplication facts, but it offers much much more. As the students work I ask them to list the factors of each counting number in a chart, along with how many factors the number has, and then a place for comments. We learn that one is a unit, and we learn to label the other numbers as prime or composite. We also learn that six is called a perfect number because 1 + 2 + 3 = 6. I let them know that the next perfect number is smaller than 30 and ask them to search for it. I also ask them to keep track of square numbers. This project goes on for several days, with lots of team-work, checking and correcting. And students do make discoveries. (And I need to say my students are wonderful at writing down a discovery and never calling it out, thus allowing their classmates the chance to make the same discovery.) By the end of the lesson we’ve all learned (I hope!) that square numbers, and only square numbers, have an odd number of factors, and that 28 is the second perfect number (1 + 2 + 4 + 7 + 14 = 28). Students groan when I tell them the next perfect number is bigger than 400. But then I tell them there’s a shortcut. We observe that 6 = 2 x 3 and 28 = 4 x 7. And I tell them that only numbers with this form can be even perfect numbers. (No one knows if there are any odd perfect numbers.) And amazingly we’ve already run into these numbers with the Invention of Chess story! (See the chart above.) The next candidate is 120 = 8 x 15. I let them know that 120 isn’t perfect because its factors add up to way more than 120. But the following one works! 496 = 16 x 31 is perfect! Hey guys, I ask, what makes 8 x 15 different? Look!

2 x 3 is perfect!          4 x 7 is perfect!         8 x 15 is not perfect!        16 x 31 is perfect!

We brainstorm for a while. And almost every year a little third grader raises a hand and says, often timidly, “Is it because 15 is composite and all the others are prime?” Gotta love it!

 

 

Chapter Ten: YBC 7289, The Chocolate Theorem, & Goldilocks

Each class, each group of students, is different. And the same class is different on different days. There were days when I felt my students needed to do quiet review or skill practice, without needing to listen to me or absorb new ideas. There were times when I felt a math story, or a math game, would be the best thing for that day. And there were days to explore new ideas. I mention this because I’m now on the third chapter about square numbers, Pythagorean triples and right triangles. These chapters give one way to organize and present these ideas. But in real life, I followed the abilities, the interests and the feel of a particular class, and presented these ideas in various ways and different orders.

Often I would begin our investigation into the Pythagorean Theorem by drawing a scalene right triangle on the board, along with the squares on each side of the triangle. (Drawing 1) I’d label the squares small, medium and big. Then I’d tell my students, “Imagine these are squares of chocolate. You can have the big square, or you can have both the small and medium squares. Which way will you get more chocolate?” This usually led to a lively discussion.

To help us decide the question, I would ask students to draw a 3 cm by 4 cm right triangle on cm graph paper. Then I’d ask them to draw the squares on the two legs and label them with their area. Easy enough, 9 cm2 and 16 cm2. But the hypotenuse is slanty, so it’s not so easy to draw its square. I’d ask the students to measure the hypotenuse with their rulers. Hmm, seems to be about 5 cm long. So then I’d ask them to draw a 5 cm by 5 cm square on another sheet of graph paper, cut it out, and scotch tape it onto that hypotenuse. Label its area. 25 cm2. So in this drawing, which way would we get more chocolate? Ah, we’d get the same amount either way. (Drawing 2)

I’d let my students know that mathematicians have proved that this will always be true, in any right triangle. The usual name for this fact is the Pythagorean theorem, but we can call it the chocolate theorem if we wish! All the Pythagorean triples we found can be the three sides of a right triangle. So I’d ask them to go back to their lists of Pythagorean triples, choose one of them, and draw a right triangle with those sides, along with the squares on the sides.

One reason I teach them about the “chocolate theorem” is to reinforce the idea that this is a statement about areas. The drawing and cutting and pasting also reinforces that. It’s been my experience that some high school students, who can answer test questions on the Pythagorean theorem successfully, really haven’t absorbed that. If you ask them the question about the squares of chocolate, they might say the big square gives you the most, or that they’re not sure.

Of course we haven’t proved the theorem. And we haven’t looked at triangles with some sides not whole numbers. Was that hypotenuse really exactly 5 cm? To touch on these two issues, I tell the class a little “just so” story that I made up. Imagine a girl in ancient Mesopotamia whose family makes square tiles for flooring. She likes to help her parents, and one day she has an idea. She draws a diagonal on a bunch of the square tiles, and glazes the triangles created in two different colors. She experiments and sees that one thing she can do now is create a checkerboard pattern of bigger squares, each one formed by four isosceles right triangles. (Drawing 4a) Her parents like the idea, and so do their customers!

But our little girl is also curious and interested in math ideas. She knows the tiles they make are one foot long on each side. How long is that diagonal? Now over the years I’ve found that this is a difficult question for many students, even some older students. “I don’t know,” “One foot?” and “Two feet?” are all common answers. I think “I don’t know” is a pretty good answer! If a student says “two feet,” I may introduce them to my “toddler theorem.” If a toddler is at one corner of a square, and mom is at the corner diagonally opposite, and the toddler needs his mom, will he scurry to her along the diagonal or follow two sides of the square? The toddler clearly knows the diagonal is shorter than two sides added together. If a student says the diagonal is “one foot,” I may bring up my “thirsty traveler theorem.” You’ve been walking in the desert, and there’s a stream with good drinking water! Would you rush to it along one side of the square (following a line perpendicular to the stream) or would you go along the diagonal of the square? Clearly we know the diagonal is longer than a side.

So now we know the diagonal of that 1 x 1 square is longer than 1 and less than 2. But how long is it? Here’s where our young mathematician from ancient Babylonia can help us. I draw a diagram on the board (Drawing 4b) and ask students to notice the bigger square that she invented, formed by four triangles. What’s its area? Almost always a few students can see that the area must be two square feet, since it’s formed by four halves of the smaller squares. Well, if the area is two, then the length must be the square root of two. And what is the square root of two? That question opens a big can of worms, and how far we look into it depends on the age, abilities and interests of the class.

One thing to notice is that now we’ve proved the Pythagorean theorem for this particular case, isosceles right triangles. It’s clear that the squares on the two legs add up exactly to the area of the square on the hypotenuse. Keeping with our Mesopotamian theme, this would be a good time to show the class the remarkable tile called YBC 7289, from the Babylonian collection at Yale University. (Drawing 4c)

It shows a square with its two diagonals. The length of one side is given, and so is the length of the diagonal. And there is a third number on the tablet, written in the center: 1: 24 , 51, 10. (Like us the Mesopotamians used a place value system, but they used sixty as their base, rather than ten.) Using fractions that would be

1 + 24/60 + 51/3600 + 10/216000

Or, in our notation, approximately 1.414213. And that’s the square root of two, correct to six decimal places! Those ancient Mesopotamian mathematicians were no slouches! Clearly they knew that the diagonal of a square could be found by multiplying its side by the square root of two, and clearly they had an algorithm for extracting square roots.

With some classes I might just mention that the square root of two is a little smaller than one and a half. Or about 1.4. Usually I mention that the square root of two is a funny kind of number. It’s between one and two, but it’s not equal to one plus any fraction. (I’m always amazed that students don’t rebel at this suggestion, jump up and down, scream that that makes no sense, that that’s impossible!) Often I’ll mention the technical name: it’s an irrational number. To get a feel for an irrational number, I sometimes give older classes an activity I call Goldilocks and the Square Root of Two. It involves making a chart that begins like this:

fraction   fraction squared      as a mixed number   as a decimal     too big or too small

3/2                9/4                                  2 1/4                       2.25                       too big!

 

7/5              49/25                               1 24/25                   1.96                       too small!

 

17/12          289/144                             2 1/144                    2.006944…         too big!

One puzzle for my students is how to find the next fraction in this pattern. If they don’t figure that out, I give them the secret, and ask them to do another few lines if they’re able. This exercise gives them the experience of finding better and better approximations for the square root of two. But no fraction is “just right”!

For older students, who are interested in proofs, we can look at a proof that the square root of two is irrational. The one that I think has worked best is to look at prime factorizations. If you square a number, all of its prime factors will have a partner. For example,

60 = 2 x 2 x 3 x 5   and 602 = 2 x 2 x 2 x 2 x 3 x 3 x 5 x 5

If we simplify a fraction and end up with exactly two, that means we’re left with two in the numerator and one in the denominator. For example,

36/18 = (2 x 2 x 3 x 3)/(2 x 3 x 3) = 2/1 = 2

But if top and bottom numbers are both square numbers, how could we ever end up with just a single 2 on the top?! If a 2 from the bottom canceled its partner, what happened to the partner of that 2 on the bottom? Or to quote a one sentence proof from the mathematician Lagrange about the square root of two: “It cannot be found in fractions, for if you take a fraction reduced to its lowest terms, the square of this fraction will again be a fraction reduced to its lowest terms, and consequently cannot be equal to the whole number 2.”

And since we’re discussing proofs, let’s circle back to the Pythagorean theorem, which has well over a hundred proofs. I think the easiest one to present to young students might be the proof represented by Drawings 3a and 3b.

 

This demonstrates the theorem for one particular triangle, a 5, 12, 13 one, but if we substitute a, b, and c for the sides, we can see that the theorem is true for any right triangle. For students with an interest in proofs, there are several others (suitable for young students) that rely on other concepts, such as similar triangles. Occasionally, I taught high school geometry to 8th graders in my school who were a bit ahead in math. I made sure they learned Euclid’s proof from Book I of The Elements. The Pythagorean theorem is Proposition 47 (of 48) in Book I, so it seems Euclid viewed this all-important theorem as the goal of Book I. (Proposition 48 is the converse of the Pythagorean theorem.) I also made a point of teaching the Law of Cosines to my geometry students, so they could see there was a way to generalize the Pythagorean theorem to all triangles.