10 + 10 = 100

“Any questions?”

I asked this at the end of class today*, and one of my students thought he’d get smart and ask “Why is two plus two equal to four?”

Luckily for the student, the bell was about to ring, and I did not get a chance to work out my modeling thing on him and his smart-alecky question. (That’s what tomorrow’s class is for ;) My only response was “Because we said so.”

Here’s what I’m thinking… the words two and four, and the symbols 2 and 4 are just models we use to represent some quantities. Think back before we had symbols to when Og, the recently ex-hunter/gatherer wants to marry Uga who is also a recent farmer/herder. Og says, “Me have this many egg-layers,” pointing to his two chickens. “You have same,” he continues, pointing to Uga’s chickens, “so together us have this many.” Og does not have symbols for the number of chickens, so he actually refers to them. Uga, being slightly more advanced, and not particularly fond of Og, says “You marry Moga, and you get this many egg-layers.” She holds up five fingers, showing that she has created a new model, representing the chickens with her fingers. Mistaking Uga’s model for something entirely different, Og clubs her on the head, and drags her and her chickens off to his cave.

Fast forward to a civilization with bartering and without my corny dialogue. Now, traders know they cannot bring their chickens with them everywhere, so they must have some way of representing the chickens. Maybe the trader carried with him bagful of small pebbles, a model for the hen pen he has back home. Or, better yet, he has learned how to write – a very powerful and in some cases magical model – and represents his chickens with a bunch of hash marks. Now, we start seeing verbal and symbolic rather than concrete models for quantities.

So, to ancient Romans, and much of Europe in the Middle Ages, there was no 2+2=4. There was only II+II=IV, only without the + or = signs; I’m not sure how they represented addition. The Arab world at this time was much more savvy about their models, and could write something more like 2+2=4.

Now, here’s where we see the power of a good model. While Albertus is trying to represent the population of his town with the numeric model MMMDXXXVII, al-Khwarizmi can use 3087 for the same number, and can also add and subtract his accounts much more quickly than Albertus. A merchant named Leonardo of Pisa (also known as Fibonacci) is one of the people to figure out that Arabic numerals are so much nicer than the Roman ones, and starts using them, being one of the guys who helps start the Renaissance in Italy.

Now a good model is only fun if we can mess with it, and see where it leads, right? So we use the Arabic numerals to represent our accounts, and as we pay our bills, we find that we have to subtract the amount we owe (325 lira) for the gilding on our Renaissance chair from the amount we have in our account (268 lira) and we suddenly realize that we still owe 57 lira! If we tweak the model ever so slightly, we have negative numbers – no longer are we restricted to our model representing some physical quantity! We can now model a more abstract concept, like money owed.

If we keep tweaking our model, we eventually end up with lots of algebraic symbols, like the superscript for raising to a power (which Rene DesCartes and Francois Viete invented, since they decided that xx was too cumbersome and x^2 was much prettier**), which led to all sorts of properties of exponents.

Or, even more fun, we start using the square root symbol (which is actually just a check-mark shape; the horizontal bar part we always write is actually a grouping symbol called a vinculum – it’s the same grouping symbol we use when writing fractions or repeating decimals). So we merrily use our symbols to model the square root of 9 (3) and the square root of 2 (approximately 2.414, which irritated the Greeks because it turned out to be irrational), and then some smart-alecky student asks “What’s the square root of negative nine?” So we tweak the model a little more, and “EUREKA!” we have complex numbers!

So, why is 2+2=4? Because this is our current model for Og’s chickens. If you don’t like it, use a different model, like base2, in which 10+10=100.



* I actually think this is a pretty lame way to end a class, especially when the bell is about to ring. I’m not really expecting any questions, and even if there are any, I won’t really have time to answer them adequately. It just kind of slipped out today, but I’m glad it did, since it gave me a side track for my train of thought.<\teacher mode>

** Do you see my problem with algebraic notation here? x^2 just isn’t as pretty as if I were actually able to figure out how to get superscripts and other math symbols.

I love models...

I was thinking about what I last wrote about modeling, and I realized that the area and circumference formulas I mentioned were also models. This time, the models are algebraic and represent some other physical model. Consider the variable r, which last time represented the radius of the circle. Physically, it represents the length of some line segment. This diagram is another way to model our generic line segment. Note that even though this diagram has some thickness, we are only measuring one dimension. Actual line segments don’t have thickness or width, only length.

Area cannot be measured the same way as length, since area is the “filling in” of some space. We cannot measure area in inches, or meters, or miles, or arm lengths; we cannot fill up some space with line segments, since they have no width. So we must measure in some units that fill up space, like squares, or triangles, or circles. Squares are convenient, since they can fill up a space without overlapping or leaving space between. (This is called “tessellating”, and it cannot be done with circles alone, and some triangles don’t tessellate as uniformly as squares.) So we measure in “square units”. This diagram shows a square with area equivalent to r-squared, since each side of the square has length r.

Here is a diagram showing four r-squared’s, or 4r^2. If we include the circle with radius r, we see that the area of the circle is less that 4r^2. I’m not going to go into the proof of why the area is pi*r^2, as I am feeling like this is rambling a bit too much, and I am suddenly feeling like I am trying to teach someone something about area, when I really want to talk about models.




One of the advantages of models is that if we play with the model – and see how far our tether can reach:) -- we can often come up with some further insights into the thing we are modeling. Let’s look at the circumference formula for example. We know that C=2*pi*r, and we can model that physically with pieces of string, each the same length as our radius. If we lay the pieces on the circle, marking where each piece stops, and continue as far as we can around the circle, we find that we can fit a little over six whole pieces of string. (See the model … er, diagram?...) The great thing about this is that it doesn’t matter what size circle we start with, we can always lay down a little more than six radius-lengthed strings. In fact, this 6.something or other is what the ancient Greeks figured was 2*pi. (They actually did this with a diameter-lengthed string, and were able to fit a little more than three of these around the circle – pi diameters around the circle.)

Here’s the further insight – If we animate our mental picture, sweeping the line segment radius around, from one end of a piece of string to the other, we get an angle that is the same no matter what size circle we have. And we can fit 2*pi of these angles into the circle. This special angle is called one radian. (Just like you can measure length using different units, like inches or miles, you can measure angles in different units.)

Okay, I’m still feeling all teachery, so I’m going to stop now. This whole idea of models intrigues me, and, since it is a different way of looking at math, I am exploring ground that is really familiar to me. I do want to get to some more unfamiliar topics like game theory and graphs on other kinds of grids, but I still have some ideas about models to explore (and I have not yet completely figured out how to get really good diagrams and algebraic notation on this blog).

Tethering Goats

I have been thinking about what it means to “do math” and I’ve decided that it is more than solving equations/problems. That’s finding answers, and math is really so much more than that. We had a professional development meeting the other day, and an expert in math literacy spoke to the department. We have been focusing on literacy quite a bit the last few years at my school, but not specifically what that means in a math class. One of the things that struck me after the recent meeting was that when we read, we construct pictures in our minds about what is happening. When we read narratives it is okay if those pictures don’t match the words exactly, as long as we have the action straight. When reading mathematics, on the other hand, the need for those pictures to be exact and precise, and in many cases animated, is very important.

So I have a feeling that doing math means constructing models, both physical and abstract. It’s not about the questions and answers, but about the pictures we make in our heads. Once we have a good picture, there are lots of questions for which we can find answers.

For example, there is a classic problem in which we consider tethering a goat with a rope of a particular length, staked to the ground in some position. The picture in your head probably resembles the image here, and you can even animate it, imagining the goat wandering around eating whatever grass is within reach.

As a mathematician, I start wondering about how much grass is available for the goat to eat? How far can the goat reach? I’m sure you can figure out that the goat can reach all the grass within a circle centered at the stake in the ground. The radius of the circle is the length of the tether. How do I know that this is a circle? Because one definition of a circle is the set of points equidistant from a given point. The given point is the stake, and all the points at the farthest reach of the goat are the length of the tether (and the distance the goat can stretch its neck) from the stake. All of a sudden, I can create an abstract model in my head – I no longer need the picture of the goat to think about where it can graze. My picture is just a point (the stake) and the circle showing the limit of the goat’s range. If I want, I can include a radius (the tether) in the model as well. I can also animate my model, picturing the radius sweeping around the circle and the goat tests the boundary all the way around.

The other aspect of an abstract model is that I am not interested in the actual length of the tether, so I use a variable to represent that length; I’ll use t. Now, it does not matter whether the goat is on a tether that is 5 feet long, or 18 feet long. I can even consider a tether that is a mile long, if I assume the field is really big and has nothing in the way that the tether could get wrapped around. (I’ll think about those possibilities later.)

Why is the model important? Because it allows me to answer lots and lots of questions about the situation! Here are a few:
1. If the tether is 8 feet long, what is the area in which the goat can graze?
2. Suppose we want to fence in the grazing area, just in case the goat chews through the tether. How many feet of fence would we need if the tether were 10 feet long?
3. If a goat needs 50 square feet of grazing space each day, how long should I make the tether?

Since I have the model of a circle in mind, I can answer all these questions by using what I know about circles: The area within a circle is equal to pi times the square of the radius, and the perimeter or circumference of the circle is equal to pi times twice the radius, or pi times the diameter.

Here’s another question: Suppose the tether were 10 feet long, and that the goat wanders randomly within its grazing area. What percentage of the time is the goat within five feet of the stake? For this question, I need to modify my model slightly. I am no longer thinking of just the circle with a radius, but two circles, one inside the other, and a point (the goat) wandering around inside. I am making a big assumption here: the goat wanders randomly. A particular goat may be very obstinate about this, and decide to sit in one spot all day; another goat may decide to constantly try the limits of the tether, and remain at the outer circle all day. As a mathematician, however, I am not interested in these special cases, but in the general wandering goat. Anyway, my model now has a circle with radius five inside a circle of radius 10, and a point moving around inside. The probability that the goat is within the inner circle is the ratio of the area of the inner circle (the part I’m interested in) to the area of the outer circle (the total allowable wandering space). In other words: (5^2*pi)/(10^2*pi). Squaring, we get (25pi)/(100pi), and simplifying, we get ¼ or 25%.

Now, being a mathematician, this is not where the problem ends. Answer getting is not the purpose. Completely understanding the model is what this is all about. So, I start generalizing a little. Remember that I was going to use a variable for the length of the tether? So the question now becomes “what percent of the time does the goat spend within a region halfway to the edge of his grazing area?” Using the same formula as above, but using t for the length of the tether, and ½t for half that length, we have:
(½t)^2*pi / t^2*pi
= ¼ t^2 / t^2
= ¼ or 25%
In other words, no matter how long the tether, the area of the inner circle is always 25% of the area of the outer circle, if the radius of the inner circle is ½ the radius of the outer circle. I keep jumping between the physical model of the goat and the abstract model of the concentric circles.

As I write this, I am thinking about my model, and I realize that if the inner circle is ¼ the area of the outer circle, then I can divide the circle into quarters differently than I usually do. I bet you think about dividing the circle into quarters like the diagram on the left. But the diagram on the right also has the circle divided into four equal parts! Again, I start animating the pictures. The left diagram starts rotating around, and folding along the lines, and it's pretty clear to me that all four quadrants have the same area. The diagram on the right is not so easy. I can rotate the picture and prove to myself that the three sections in the outer ring are all equivalent, but how can I manipulate the model to make that center circle, which I know has the same area, match up with the other three sections? This will take some thought...

Here are some variations on the inital situation that also require some changes to the model. (I'm going to have to explore these in a later post.) What if the tether were staked along the side of a building? What if the stake was at the corner of a building? What if the tether was long enough so that the goat could walk all the way around the building?

I am beginning to feel like a goat on a really long tether, exploring the outer reaches of my model's boundaries!

By the way, here are the answers to the questions above: (I apologize for the format; I am still learning the capabilities of posting pictures and math symbols here.)
1. 64*pi which is about 201.062 square feet of grass
2. 20*pi which is about 62.832 feet of fence
3. sqrt(50/pi) or approximately 3.989 foot long tether

Just Starting Out...

I have been thinking about starting a blog for about a year now, and decided it would be best to just jump in. The idea first came to me when a student asked about what kind of math I do. He had some experience with his science teachers doing research, his art teachers producing some beautiful pieces, and his English teachers publishing articles or even books.

But what do Math Teachers DO??

I thought about this a while, then attended a meeting of the Metropolitan Mathematics Club where the speaker was talking about what the graph of a line might look like if the grid were not on a flat surface. I was really intrigued by the possibilities, and spent several days making sketches and trying to prove a couple of ideas I had. Life and teaching intruded however, and I had to leave the wanderings behind.

A few months later, I was rummaging through my bookshelves and came across a book on Game Theory. I remembered that I had wanted to pick this up at one point, but, again, was distracted by other details.

Last week, I attended another MMC meeting at which the speaker talked about Dynamic Systems and ergodicity. I took lots of notes, and revelled in the brain activity. Chores at home the next day left ergodicity in the dust.

Now, the art teachers at my school have taken over several of the display cases to show off some of their own work. I was really blown away when I saw this stuff. I was reminded that even though we are most often identified as teachers, we have degrees in our fields, and know and understand the material we teach at a really deep level. I was again reminded of that question from last year What do math teachers do and decided it was definitely time to "show off" my own art.

Math teachers do MATH!!

So, my purpose is posting to this blog is to demonstrate some of the math I enjoy and want to learn. I am not sure whether to start with non-planar graphs, game theory, or ergodicity, but wherever I start, I look forward to the trip.