PROOFS #1: Aspect Ratios, the Golden Ratio, and Z’s TV
(April 16, 2013) Your cousin has just gotten a new job as an announcer on the Golf Channel, and is so excited for you to watch every broadcast. While you may enjoy playing golf, you don’t really enjoy watching it. So you decide to place a small flat-screen TV between 2 windows above your kitchen sink. Now you can multi-task. The space is 12 inches wide. TVs are sold by diagonal measurement. What size TV should you order?
Sounds simple enough, right? But, as both G and P commented later in this first Math Circle session on proofs, “This is turning out to be pretty complicated!”
Almost immediately, the students realized that there is some geometry formula that would be helpful here. R, P, and C discussed this and collaboratively remembered the Pythagorean Theorem, a2+b2=c2.
“We did a proof of that in my math class,” said P.
“Why would someone want to prove that?” I asked.
P answered, “We would want to be sure that it worked, and also it can help us to understand better what we’re doing when we use it.” Here today, we didn’t prove it (yet!), but instead put it on the board with a diagram. N and G labeled the appropriate triangle sides. Then the questions began:
- “Wait a minute; we only have the width and not the height. Is there a certain ratio TV screens come in?”
We discussed the concept of aspect ratio, and its history. The standard American screen came in a width to height ratio of 4:3, while widescreen TVs have an aspect ratio of 16:9. We diagrammed these sizes and talked about our preferences. R, P, N, and C all preferred the 16:9 screen, while G preferred the 4:3. She posited that we prefer what we’re used to, which led to a lengthy conversation about the Golden Ratio (phi, or Φ). The group decided to test the two TV aspect ratios, and choose whichever was closer to Φ, which is equal to 1.6180339887….
“Stop writing!” said someone. So I did.
“Why did you stop?” protested N. I threw that question out to the class to answer.
“Because she would go off the board,” replied someone else.
“Isn’t that like the square root of 2, where someone was pushed off a cliff because of it? asked R.
“It is,” I answered, “but I’m not yet ready to go into numbers that might have led to deaths.” So, with that topic temporarily tabled, the students decided to round Φ to 1.6, and got to work converting the other ratios to decimals. The students did it collaboratively as I, as usual, served only as secretary, documenting their work on the board. When 4/3 turned out to be a repeating decimal, they debated how many decimal places to use before inserting the repeat bar. R made a case for placing it on the second 3, but was outvoted by the rest of the group. They argued that a single 3 after the decimal point was sufficient. When R still disagreed, I started to ask her whether it’s okay to compare a number with one decimal place with a number with two. P announced “sig fig!” Not everyone was familiar with significant figures, so I made the analogy between this concept and the grammar concept of not comparing apples to oranges.
“But you can compare apples to oranges!” claimed C. The group debated whether she was right, then returned to the problem. They discovered that the wide screen ratio was closer to Φ than was the standard. Playing devil’s advocate, I asked whether this conclusion would still hold had they carried each calculation to another decimal place. They tried it, rounding correctly, and stuck to their conclusion: 16:9 is the way to go. Then another question arose:
- “Wouldn’t it look better with a little bit of room on each side of the screen?”
N suggested a distance of 1 inch between the screen and each window, leaving a 10 inch width to work with for the TV screen.
- “But don’t flat-screen TVs have borders?” asked someone else.
I told them that I read online that the average border was 2.5 inches. “But that would only leave us 5 inches for the screen!” they protested.
- “Does it have to have a border, and if it does, can we make the assumption for this problem that it doesn’t?” asked G.
“We can assume anything we want as long as we agree and state it as an assumption,” I answered. R pointed out that Z (a friend and frequent Math Circle participant, who unfortunately couldn’t be part of this Circle) has a TV without a border.
“Z’s screen does have a tiny thin border,” countered N. When I asked whether we should factor that into our calculations, the group immediately agreed with G that we should assume no border at all. Debate then moved to a (not the) method to calculate the height of the screen, given the width and the aspect ratio. Various methods emerged, but the group used C’s algebraic approach. (You can see the method in the photos below.)
“I never knew why you can do that,” commented someone when we had to divide 10 by 1.7 and someone else suggested moving the decimal one space to the right in both numbers to ease calculation. It turns out that no one knew why you can do that, despite knowing this shortcut. I showed them the reason this works, and hopefully made the point that an algorithm should always be backed up by a reason.
Nearly an hour and a half after we started, we finally had the number we needed to be able to apply the Pythagorean Theorem (the height). Time was up, and more so, and I had to send them home. P was so disappointed because she’s going out of town next week and would “miss the fun” when we actually answered the question. I told her to take a photo of our board work and try to finish at home, and compare her work to the write-up I do of next week’s session.
Next week, we’ll finish this problem and start to prove some things. In the meantime, don’t be surprised to see your kids calculating the aspect ratios of your laptops, phones, and other devices.