FOURTH: Think of teaching as an art. Develop your own style just as an actor, dancer, painter, or writer does.

     [You have previously made two paper loops. (We'll see in a moment how they are made.) Each is about four inches wide, and about 20 inches in diameter. Along with the loops you have a pair of scissors. You begin like this.]

     "Here you see two loops of paper. I'm going to cut each one down the middle, and we'll see what we get.

     [Cut the first loop down the middle. You'll have two separate loops, each 20 inches in diameter, but only two inches wide.]

     "Now let's cut the other loop.

     [Cut the second loop down the middle, just as you did the first one. You'll have a single loop, again two inches wide, but 40 inches in diameter!]

     "The second loop looked just like the first one, but was made differently. It's called a Mobius Strip, and (in addition to the strange way it acts when it is cut) it has two wonderful properties. It has only one side; and it has only one edge.

     "At the beginning of the next class, we'll see how you can make such a loop yourself, to amuse and mystify your sisters, brothers, and parents."

     [You'll need three pieces of paper, each 4 inches wide and 45 or 50 inches long -- or longer. You'll also need some Scotch tape or paste, a pair of scissors, and two crayons or pieces of chalk of two different colors.]

     "Yesterday I showed you two paper loops. I cut one down the center, and got two separate loops. Isn't that what you would expect, if you cut a loop that way? ...

     "Then I cut a second loop down the center, and got one very large loop instead of two small ones. I told you this second loop is what's called a "Mobius strip", and that it is interesting not only because of what happens when it's cut, but also because it has only one side and one edge.

     "What does the Mobius strip have to do with math? It has to do with a branch of math called Topology, which deals with the shape of objects, where Geometry deals with their size. The Mobius strip was invented (or maybe we should say discovered) in 1858 by a German mathematician named August Ferdinand Mobius.

     "Let's make some loops. First, let's make an ordinary, non-Mobius loop. We just take a long strip of paper, and paste or tape the ends together. [Make the loop, as shown in the top, red part of the figure on the left.]

     "Now we'll make a Mobius strip. We start with an identical strip of paper [take the second strip], but before we paste the ends together, we make a single twist in the strip. [Make the Mobius strip, as shown in the bottom, blue part of the figure.]

     "First, let's see how many sides each of these loops has. [Take the simple loop, and a blue crayon. Mark one side of the loop, all around until you get back where you started. Then take the other color and mark the other side.] You see, this loop has two sides.

     "Now let's count the sides on the Mobius strip. [Take the Mobius strip, and start marking its side. You'll find that, when you get back where you started, both 'sides' of the loop are colored.] You see, our Mobius strip has only one side.

     "Next, let's see how many edges each of these loops has. [Take the simple loop, and a red crayon. Mark one edge of this loop, all around until you get back where you started. Then take the other color and in the same way mark the other edge.] You see, this loop has two edges.

     "Now let's do the same thing with the Mobius strip. [Take the Mobius loop and mark its edge in the same way. You'll find that, when you get back to where you started marking, both edges of the loop have been marked.] So the Mobius strip has only one side, and one edge.

     "And when we cut the two loops [cut them] as we did yesterday, we get two loops from the ordinary strip and one big loop from the Mobius strip.

     "Let's finish of with an even more surprising loop. [Take the third strip and make another loop, but this time twist the paper TWICE before gluing the ends together] Our ordinary loop has no twists, the Mobius strip has a single twist, and our new loop has two twists. What do you suppose we'll get when we cut this loop down the middle? ... Well, let's try it and see. [Cut it down the middle. You'll get two ordinary-sized loops, but they'll be linked together!]

     "I hope you're surprised. I was, when first I learned about and made these interesting loops."

     So there's a second example. How can you find more examples to make math fun-- enough so you have a new one every day in the week for a whole year?

     There are a lot of books full of ideas, and many math links on the Web full of ideas. Click on the Book and Link buttons down below to find them. A particularly good link is Ivar Peterson's Mathland --a site with a host of Mathematical Recreations. And two particularly useful books are The Mathematical Magpie and Fantasia Mathematica by Clifton Fadiman. Finally, if you subscribe to our hilarious free math newsletter, the Gnarly Gnews, you'll find new ideas in every issue. Just click the Newsletter button down below among our other pages to subscribe.

     Then there are our moneyback-guaranteed CD-ROM, Solid Gold Gnarly Math, and our humorous Math Ebook The Historical, Humorous Math Newsletter. Click one of the buttons down below to learn all about how these products can make math fun for your kids, supply dozens of teaching ideas, and cure any math phobias you might have. Click the Order button below for the CD, or the 'Our Ebook' button for the The Historical..., and you can have one of them working on your own computer in less than a week!

Sincerely,