Last week, in Number Sense 003, we discovered a straight line in a stretched out piece of string. We also found that we could use that string to draw circles. So our humble piece of string gives us both a straight edge, and a circle drawing device, commonly called a compass. Let's see what we can do with these two tools.
We ended last week with a piece of a line, a line segment:
We also saw that, by comparing the stretched string (IE: the line segment) to a number line, we could measure the length of the line segment. So the line segment could represent some actual number.
This particular line segment looks like it represents a number slightly smaller than 11.
Can we divide this line segment in half?
The simplest way to do it might be to get a string the same length as the line segment, and fold the string in half. Voila: done. But let us suppose that we want to use our straight edge and compass tools to do the job. How can we accomplish this feat?
We can try placing our compass center in the middle of the line segment, and trace a circle from one end of the line segment to the other, if the circle touches the other end of the segment, then we have found the middle.
Oops: not quite in the middle of the line segment. We could continue trying, and would eventually get close, but let's try another way. The problem we have with this approach is that we don't know where the middle of the line is. This strategy for finding it is: guess, then check. Repeat until we guess correctly.
Rather than guess and check, let's see if we can't figure this out.
What do we know about the line segment? We don't know where the middle is (that's what we are trying to find.) We don't know exactly how long it is (a bit less than 11 isn't close enough, we want an exact answer.) What's left? The ends of the segment. We know where the line segment ends on the left and on the right.
So lets put the point of our compass on one end of the line, and draw an arc. (If you are trying this at home, using a loop of string as a compass works better than holding one end down with your finger. Use two pencils – use one pencil point to hold the loop at the center of the circle, stretch the loop of string out to the edge of the circle with the other pencil, and draw an arc.)
Okay, that was fun. Let's do the same thing on the other end, making sure to keep the circle the same size.
Where the two arcs cross each other, we have a point that is the same distance from each end of the line segment. That is somewhat like the middle of the segment. After all, we are looking for a point on the line segment that is the same distance from both ends. Our point isn't on the line segment, but it is the same distance from both ends – we are halfway there!
Seems like we could get closer to putting our point on the line segment if the circle we used to draw the arcs were smaller.
It worked: our second point is closer to the line. Let’s try this again.
We are getting closer still. It seems, though, like we are still guessing and checking. However, if we take a look at the work we've done so far, we might notice a pattern.
All the points we found are on the same line. Since the points on that new line are the same distance from the ends of the line segment, we can see there is a point that is both the same distance from the ends of the line segment AND on the line segment itself. It is where the constructed line crosses our line segment.
So, we have successfully divided our original line segment in half. Since the length of the line segment can represent a number, we have also successfully divided that number by 2.
People who learn arithmetic the traditional way may object: “But what's the answer?” because they expect all math problems to end up with a single number after an equal sign. That didn't happen in this case. We didn't start by knowing exactly what number we were working with. All we knew is that it was more than 10 and less than 11, and that it was some fixed number (fixed by the length of the line segment.) We ended up with a new number (a new line segment) that is more than 5 and less than 6, and even if we don't know it's value, we do know it is exactly half of our original number.
So, we solved the problem we set about to solve – we divided the line segment in half. We also solved a related problem – we divided the number represented by that line segment by 2.
Have fun in the comments...