Wednesday, February 18, 2015

Section 14-2: Lengths in Right Triangles (Day 114)

Well, last night I was able to watch The Theory of Everything. The subject of this movie is the 20th (and 21st) century physicist, Stephen Hawking.

Here is a link to a biography of Stephen Hawking:

In some ways, Hawking is to physics as Turing was to computer science. Just as Turing sought out a Universal Turing Machine that could solve any problem, Hawking sought a Theory of Everything, a single equation that could describe any force. As the scientist describes in the movie, there are two major theories -- Einstein's Theory of Relavity (which models the very fast) and Quantum Mechanics (which models the very small). Before those theories, Isaac Newton's mechanical laws ruled. But what Hawking wanted was a single equation that would subsume all of the currently known theories, but as of today, a Theory of Everything remains elusive.

Notice that this marks the difference between physics and mathematics. We can mathematically prove that a Universal Turing Machine is impossible. But we can't actually prove things impossible in physics, like a Theory of Everything.

A Theory of Everything would describe all forces. There are four known forces -- electromagnetic, gravity, strong, and weak. Forces are usually described mathematically as vectors, since they have a magnitude and a direction. For example, the force that gravity exerts on us is called our weight. It has a magniude -- our weight in pounds -- and a direction -- down, towards the center of the earth. Recall that vectors (second half of Chapter 14) will be our main topic here on the blog next week.

SPOILER ALERT: Hawking is best known for astrophysics, and the movie discusses how he wrote his first book, A Brief History of Time, in the late 1980's. END SPOILER. But Hawking also wrote a book about mathematics -- actually, he compiled it. God Created the Integers is an anthology of 31 of the most famous mathematicians of all time, from Euclid up to -- surprise, surprise -- Alan Turing! I point out that the title God Created the Integers refers to a quote by the 19th century German mathematician Leopold Kronecker: "God created the integers; all else is the work of man." Like many students today, Kronecker preferred integers to rational and especially irrational numbers -- he was what we call a constructivist. Mathematical constructivists also reject some use of indirect proof.

SPOILER ALERT: It seems interesting that the first word of a book written by Hawking would be "God," considering how his atheism was at odds with his wife's Anglicanism during the movie. This is why I point out that the word "God" refers to that Kronecker quote. END SPOILER.

I hope that the two movies about famous scientists, Imitation Game and Theory of Everything, clean up at the Oscars this weekend. I believe that one way that we can convince our students to be more interested in math and science is to show them these famous scientists whenever we can. Even today's Google Doodle is an opportunity to tell our students about Alessandro Volta, the Italian physicist who invented the first battery at the end of the 18th century. (So now you know why we refer to batteries as having nine volts!)

Section 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.

Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)

Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.

In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:

Given: Right triangle
Prove: a^2 + b^2 = c^2

Statements                                Reasons
1. Right triangle                       1. Given
2. a geometric mean of c & x,  2. Right Triangle Altitude Theorem
    b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy)       3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy                 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy               5. Addition Property of Equality
6. a^2 + b^2 = c(x + y)              6. Distributive Property
7. x + y = c                                7. Betweenness Theorem (Segment Addition Postulate)
8. a^2 + b^2 = c^2                    8. Substitution (step 6 into step 7)

It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices. We'll find out soon enough.

Notice that one of the questions that I included from the text involves the Girl Scouts -- in particular, a girl scout troop leader who calculates the height of a tower using notebook paper (the sole purpose of which is to ensure that the angle is actually 90 degrees). This time of year is Girl Scout Cookie season, so of course I had to include a Girl Scout problem.

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