Thursday, March 26, 2009

Circle Circumscribing a Triangle

Triangle ABC Given a triangle ABC in R2, there is exactly one circle that goes through the points. How do you find the circle?

I'll assume you know how to construct the perpendicular bisector of a line segment.

Any circle that touches both A and B must have its center on the perpendicular bisector of AB.

Similarly, any circle that touches both A and C must have its center on the perpendicular bisector of AC.

The circle we are looking for has its center at the intersection of those two bisectors, and the radius is determined from that center point to any of the three original points.

Tuesday, March 24, 2009

Triangle Inequality on R

A bug lives on a line. The bug starts at point A and crawls to point B. Then she crawls to point C. We know that she always crawls at least as far going from A to B to C as if she had just crawled straight from A to C. How can we prove this fact mathematically?

If the points are identified as real numbers along the real line, then we have the distance between A and B defined to be

  • d(A, B) = B-A if B ≥ A, and
  • d(A, B) = A-B otherwise.

So now restating what we want to prove, given A, B, C in R, prove that d(A, B) + d(B, C) ≥ d(A,C).

One way to prove this is by a case analysis. There are six possible orderings of A, B, C on the line:

  1. A ≤ B ≤ C
  2. A ≤ C < B
  3. B < A ≤ C
  4. B ≤ C < A
  5. C < A ≤ B
  6. C < B < A
Question: Did we cover all the possible orders? How do you know?

I'll do two cases:

  • Case 1 is easy. If A ≤ B ≤ C then d(A, B) + d(B, C) = B-A + C-B = C-A = d(A, C).
  • Case 5 is a little trickier.
    If C < A ≤ B, then d(A, B) + d(B, C) = B-A + B-C = 2B-A-C.
    How does that relate to d(A, C) = A-C?
    We can approach this problem by observing what we want. We want d(A, B) + d(B, C) ≥ d(A, C),
    so we want 2B-A-CA-C
    so we want 2B-2A ≥ 0 (by adding A-C to both sides)
    so we want B-A >=0 (by dividing both sides by two)
    which is true because A ≤ B. So it will work out. We just have to run it backwards:
    We know A ≤ B, therefore B-A ≥ 0 therefore 2B-2A ≥ 0 therefore 2B-A-CA-C therefore d(A,B) + d(B,C) ≥ d(A,C).
The fact that's being proved seems obvious, but it requires a little creativity to actually do the proof for all six cases.

Teaching discussion:

  • Sometimes you just have to do all the cases. Students, especially kids, are reluctant to do a case analysis. Six cases seems like too much. But sometimes it's just the right way to solve a problem: enumerate the cases and solve each one.
  • I've seen people teach by doing something like:
    "We know A ≤ B, therefore B-A ≥ 0 therefore 2B-2A ≥ 0 therefore 2B-A-CA-C therefore d(A,B) + d(B,C) ≥ d(A,C)."
    It's just magic. How did you come up with that? Better when teaching is to show how to figure out what the magic sequence is. The "magic" is especially prevalent on proofs in analysis: You have to show "for every ε there is a δ" and you write down some magic formula for δ as a function of ε, and the students wonder where in the world that came from.

Monday, March 23, 2009

vector addition

A vector is an arrow. What is important about a vector is
  • its length, and
  • its direction.
The starting point of a vector is not important. You can slide an arrow around, and if you don't stretch it or turn it, it's the same vector. (Note, I'm not representing vectors as coordinate pairs, but as graphical things that I draw on the white board. They are arrows.) The rule for adding two vectors, X and Y is to slide Y until Y's starting point is on top of X's ending point. Now draw an arrow from the beginning of X to the end of Y. That's the result of adding X to Y.

limit goes to e

We've done symbolic derivatives for polynomials with nonnegative exponents. We've also done a few simple limits. So this 5-minute math item is more advanced than some of the others Topic: What is the limit as n goes to infinity of (1+1/n)^n, proceeding by example, and using a calculator.

Exposition: Suppose you lent $1 at a 100% loan for 10 years.

  • Compound once you get (1+1)^1 = $2
  • Compound twice you get (1+1/2)^2 = 2.25
  • Compound thrice you get (1+1/3)^3 = 2.3737
  • Compound 10 times you get 2.5937

What happens as you compound more? The kids' initial guesses were that you would get an unbounded return or that you would get $3 eventually. We saw that it seemed to zero in on 2.718 by doing a million times then a billion times (at which point the calculators started producing random values in low order digits). Then I asked them to find e^1 on their calculator.

Plotting Some Polynomials

Plot the following functions
  • y=x
  • y=x+1
  • y=2x
  • y=2x+1
  • y=x*x
  • y=x*x*x

Formulas for Lines

Lines that don't go straight up and down (i.e. those that have well-defined slopes) can be written as y=mx+b. This is called slope-intercept form because m is the slope of the line and b is the y value when the line crosses the y axis. Figure out the slope-intercept form for the problems from the previous posting.

Slope

The slope of a line is the rate of change in the y axis divided by the rate of change in the x axis.
  1. Draw Cartesian coordinate system, and draw a line that goes through points (0,0) and (1,1). What is the slope? (Change in y is 1, change in x is 1, so the slope is 1.
  2. Draw a line that goes through (2,2) and (3,2). What is the slope? (Change in y is 0, change in x is 1, so the slope is 0.
  3. Draw a line that goes through (2,2) and (3,1). What is the slope? (Change in y is -1, change in x is 1. Slope is -1.
  4. Draw a line that goes through (-1,-1) and (-1,-2). What is the slope? (Change in y is -1, change in x is zero. The slope is -1/0 which is undefined.
Discussion topics. Is the slope for #4 infinite? Maybe it's minus infinity.

Five Minute Math

We've got a whiteboard in our dining room, and I've found that the kids can handle about five minutes of math after dinner. This blog is the list of topics that I've covered. I started the 5-minute math a few years ago, and today's topics assume some background. The key is to limit the discussion to 5 minutes (unless the kids really want to go longer). This blog's ground rules. I'm assuming that you, the reader, already know the math. So I'm going to write down what I tell my kids, and you can connect the dots.

About Me

I'm research faculty at MIT, and Chief Architect at Tokutek.