## Like Ripkin

I haven’t been sleeping well so I was almost relieved when I got back to my CHU after teaching math for two hours at the end of the day to find that the Internet had gone down and I wouldn’t have to make a post. Then of course nature kicked in and I had to fiddle with it until it started working again.

The math class had an interesting problem on some of the homework I had assigned. I knew how to figure it out using a brute force-like approach but I stumbled for a good ten or fifteen minutes trying to figure out an elegant way to explain the problem: something simple enough to be understood while providing a good foundation for future reasoning, rather than a simple trick based on memorization or a pseudo-formula magic incantation. The problem was “If it takes 5 men working 4 days to load 5000 tons, how long does it take 8 men to load 10,000 tons?”

Well, the elegant way is to figure it takes one man four days to load 1000 tons, or “1000 tons per man per 4 days” which is trivially rewritten as “250 tons per man per day.” Now, you multiply the whole thing by 8 men: 2000 tons per day. Then 10,000 tons divided by 2000 tons/day is (10,000 tons/2000 tons) * 1 day, or 5 days. It looks a lot better written out because you can see the units build up and then cancel out, which is how you know you have the right answer. The units in this case are: tons, days, and men.

May 10, 2010 Comments Off

## Three corners, three sides, infinite possibilities

Kids, even grown up ones, ask the darnedest things. During class yesterday we talked a little bit about geometry, and one asked, “How many kinds of triangles are there?” We had been discussing the concept of congruency, and I drew some examples of right triangles, equilateral triangles, and isosceles triangles.

I was actually stumped by this question. I vaguely remembered that triangles were sometimes called obtuse and acute, but I wasn’t sure if that was a common use, or if it was more typical to describe them as having “an obtuse angle” or “all acute angles.” Somebody said, “I think one is type is scalene.” Right, there is that. I was able to do a simple proof in class that a triangle could only have one obtuse angle. That is, since the inside angles of a triangle come to 180º, and since an obtuse angle is one that is greater than 90º, it follows that the other two angles must be acute (less than 90º). I still wasn’t sure if I could enumerate all types of triangles, or if it was even possible to do so. I promised them I would give them an answer. Here it is—or at least, close enough.

Triangles can be classified according to the size of their interior angles. Based on the fact that the interior angles must add up to 180º, it follows that there are three kinds:

- All of the angles are less than 90º (acute). For example, 80º—60º—40º. This is known as an
*acute triangle*. - One of the angles is obtuse, that is, greater than 90º. The other two angles must be acute. This kind of triangle is an
*obtuse triangle*. - One of the angles is exactly 90º. As expected, the other two must add up to 90º (for example, 1º and 89º), so they are acute. This is a
*right triangle*.

It turns out you can also classify triangles according to the relative lengths of their sides.

- Suppose your triangle’s three sides are the same length. This is an
*equilateral triangle*. It turns out that there is only one way to make such a triangle to work out, and that is by having all of the interior angles the same. Since they must sum to 180º, each angle is 60º. - Perhaps only two sides are the same length. It works out here that the angles of the “odd” side are identical to each other. This shape is an
*isosceles triangle*. - Finally (since there are only three sides to consider!), there is the possibility of having a triangle where all three sides are different. This rogue is the
*scalene triangle*.

So, pop quiz. How many types of triangles are there?

April 27, 2010 Comments Off