Sunday, November 11, 2012

Living Math Monday: The Eclipse

We've been reading Life of Fred now for about a month, and we're a little more than halfway through the first book. In the sixth chapter, we learn about a mathemetician named Archimedes who once attempted to determine how many grains of sand it would take to fill the universe. Vigintillion, a word Archimedes invented to give name to such a quantity has become quite the buzz word around here. "I wuv you vigintilwon," Brian says. Archimedes also loved to draw circles, so much so in fact, he died protecting his beloved circles. Besides learning a little about thoughtful Archimedes, we've also learned about ellipses, set theory, number families, time, counting by fives, negative integers, and algebra, seasons, and geography. We've even learned a small smattering of German and Latin. Of all the concepts that Fred has dabbled with, the ellipse seems to be a favorite with our favorite five-year-old math professor. (Fred, of course!) Given Fred's preoccupation with the ellipse, I decided we'd spend a little time with the ellipse as well.

Before I share our elliptical journey with you, I want to mention something else first. When we first began with Fred, we were busy memorizing addition math facts so that we could progress to the next section in our Singapore curriculum. I think by this time, Dorothy would have been able to draw a number bond tree for any number between 1 and 10. She not only knew all the ways to break apart the number 8, but she could break apart each numeral and reduce them all to the number 1. She had it down pat.

Then we began with Fred. In the first chapter, Fred introduces the number 7, giving one way to make seven. Chapter Second, Fred again gives us 7 albeit by a different route. Chapter Third, 7 again. We are more than halfway through Book 1, and we have yet to work with a number other than 7. Strange, I thought. Very strange. I loved everything else we were learning through Fred. But I failed to understand where he was taking us with the repetition.

Then one day, we were reading in Fred about how Fred went to the dock of a lake, wanting to rent a rowboat. There were 4 boats to the left of the boathouse and 3 boats to the right of the boathouse. Dorothy was reading along, and here she stopped. She became very excited and shouted out, "It's just like the number bonds! 7 is the Whole (the total number of boats), and the 4 on the left is a Part, and the 3 on the right is a Part!"

After spending I don't know how many hours on games and worksheets (and yes we used manipulatives), she finally got it that those numbers stood for something real. It finally clicked. And now I get it. That is, now I understand why it is that Fred stays with one number so long.

Now moving on to the ellipse.

If you haven't heard of the Young Math series, you must look it up. Some of the books are quite pricey, so beware. But in my opinion, they're worth the cost. I can't help but wonder if I had had access to such books as a kid, whether I might have come to love math instead of dreading it as I did for so long. The book starts out with a simple demonstration, using a drinking glass, of how the circular opening of the cup viewed from different perspectives, will flatten into an ellipse which itself flattens more and more until all that can be perceived is a straight line. We tried this with cups, coins, bowls, toilet paper rolls, and tuna fish cans. At one point, Dorothy and I, our noses resting on the edge of the table, were looking at a coin across the flat table plane, and I was instantly reminded of the opening paragraphs of Flatland.

Then we set about drawing ellipses using a length of yarn, two thumbtacks and a pencil. First we marked off 4 inches. Then we marked the halfway marks between each inch, for a total of 7 marks. (4 marks plus 3 marks equals 7 marks).

Then we put a thumbtack in the 1 mark and the 7 mark. (With a thick padding of newspaper under our paper as suggested by the book.) Then we made a loop with our yarn that would fit almost but not quite snugly around the thumbtacks. Then with our pencil inside the yarn loop, we traced an ellipse letting the yarn guide our pencil around. Then after we had our first ellipse, we moved the thumbtacks to mark 2 and mark 7, thus moving our foci inward (I love giving my kids such a rich mathematical vocabulary). When we finished, this is what it looked like.

This was very difficult. In fact, I ended up taking over while Dorothy watched. The yarn kept slipping under the pencil tip and tugging the thumbtacks inward, causing the ellipses to be skewed. So for the next days exercise, I used foam board instead of newsprint for backing, thinking the extra firmness would better hold the tacks.

For this exercise, we marked off seven marks just like before, but instead of moving the thumbtacks, we made the loop bigger with each new ellipse.

As you can see from our shaky drawing, the foam board didn't work any better than did the newsprint. But, one exercise left to go, and for this one, we dropped any pretense of Dorothy trying to draw the ellipse. I did them all myself, trying not to curse the cursed yarn and tacks.

For this exercise, we moved the right thumbtack inward by one mark each time we drew an ellipse, while the left tack stayed in mark 1. Notice how the ellipse narrows to the right of the center, but widens to the left? Neato beato.

On the third day with The Ellipse, we read about how the earth follows an elliptical pattern around the sun.

And we discovered that an ellipse can narrow to such a point that it becomes a parabola, then a hyperbola, which is what led to our playing in the darkened bathroom with flashlights. Which is just too much fun to be called math. Well, real math anyway. I almost feel like we're cheating somehow.

Anyway, Dorothy and Brian are so intrigued by the ellipse (now all carrots, cucumbers, bananas, etc. must be sliced into ellipses), that I wondered if there existed such a device that would make the drawing of an ellipse a little easier than the ol' thumbtack and yarn routine.

So Dorothy and I googled elliptical device or machine.

You know, I forgot about that elliptical-exercise-fitness thingy from forever ago.

So I narrowed our search. I added geometry to the parameters, and bingo-bango I found a wealth of information about geometrical elliptical machines. We learned about the Do-nothing machine, so called because it literally does nothing. We learned that there is an elliptical saw that allows a person to cut . . . well an ellipse. And we learned that there is a patent pending on a device for drawing ellipses. But nothing available to order now. At least not anything we could find. So we're stuck with the thumbtack and yarn.

But we did learn something else. Something very interesting. And satisfying in an elliptical sort of way.

While we were searching for a geometrical elliptical drawing machine, we learned that such a machine had already been invented. Long, long ago. Thousands of years ago, in fact.

By a man who loved to draw circles.

A man named Archimedes.

Which brings us back to


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