Israeli Elevations

I had mentioned last week that I had just been in Israel. Finding math applications in a foreign land is not easy. I have enough trouble doing so in the US.

Last week I looked at converting temperatures in your head from Celsius to Fahrenheit. That made me thing about negative temperatures. That made me think about negatives in general.

When encountering negative numbers for the first time, many students reject the notion that numbers can be less than zero. But of course they can. While you can't have a negative number of apples, you can have a negative amount of money, degrees, elevations, etc.

Specifically, let's look at subtraction in finding a difference of elevation. The Dead Sea is at 1,414 feet below sea level. Jerusalem is at 2,582 feet above sea level. What is the difference in elevation? 2,582 - (-1,414) = 3,996 feet.

What if we subtract in the other order? -1,414 - 2,582 = -3,996 feet. What are we to make of the negative answer? That is what the absolute value is for. If a and b are elevations, the difference in elevation is |a-b|. This is easier than writing something cumbersome like: a-b if a > b and b-a if a < b.

This shows students a practical example of integer subtraction and the absolute value concept.

One more example of this: The Sea of Galilee flows into the Dead Sea by way of the Jordan River. All of these are below sea level. The Sea of Galilee is 696 feet below sea level. Here we are subtracting two negatives. |-696 - (-1,414)| = |-696 + 1,414)| = 712 feet.

Israel Temperatures

I haven't blogged for a while because I've been in Israel. It is and will be the longest trip of my life. We went through ten time zones to get there. Not a lot of math applications on this trip, however there was one I came up with.

Most things are translated. Many speak English - at least at the touristy spots. Most of the signs are in Hebrew, English, Arabic, and might even include a picture.

However, you don't get any help on the metric system. Speeds signs were in kilometers per hour. Temperatures were done in Celsius. At the Dead Sea there was a thermometer. On television we could see weather forecasts. All were in degrees Celsius.

It was pretty warm there. It is a little closer to the Equator than I am used to. Also, were were in places below sea level. Those along with the fact that they said they were experiencing a warmer than usual November made for pretty warm days. The temperatures were usually in the 20's. My wife would see these and ask me how hot it really was.

Converting can be done with the formula F = (9/5)C+32. However, most people aren't going to be in the mood for this formula with or without a piece of paper, and usually we were without paper.

I came up with a passable method. My method was round off to the nearest multiple of five. Then divide by five, multiply by nine, then add 32. As I thought about it, I kept refining my method. My goal was to come up with something easy to use and would give a pretty good approximation for the degrees Fahrenheit.

• The above method - round off to the nearest multiple of five, divide by five, multiply by nine, then add 32.
• Multiply by two (close to 9/5) and add thirty-two.
• Multiply by two and add thirty.

I thought the last was pretty good. Adding thirty is easier than thirty-two. And it might compensate for using the larger value of two rather than nine-fifths.

So how does that work? Lets try it out for 20, 25, and 30 degrees Celsius.

• 20 degrees C:  Real temperature is 68 degrees F and with my method 72 degrees F.
• 25 degrees C:  Real temperature is 77 degrees F and with my method 82 degrees F.
• 30 degrees C:  Real temperature is 86 degrees F and with my method 92 degrees F.

I felt somewhat good about myself and my new method. Some further expansion on this application:

Graph y = (9/5)x+32 and y = 2x+30.

How close do these match up?

Are there temperature ranges this approximation works for and doesn't work for?

Are the better conversion formulas that approximate the temperature?

What would be a good approximation formula for an Israeli visiting the U.S.?

Golden Gate Bridge

I'm going to be out for a couple of weeks, so I thought I should squeeze in one more post before I take off.

Last week I ran a half marathon. I know, foolish. But the main reason I did it is because it was in San Francisco and the course crossed the Golden Gate Bridge twice. You can see the bridge from a distance in downtown San Francisco. You can also see it driving across in a car, but it goes by pretty quick. Neither of those are the same experience as crossing it on foot. The cable droops down almost all the way to the road. It was fun to get right up close to it. Its just about head level at its lowest point.

An interesting thing I noticed is that the roadway is curved. You definitely run a bit uphill then down. The highest point, though, is not in the middle. I'm sure engineers had a reason for that, but that would be beyond me.

You can get many statistics regarding the bridge on-line. I was going to cut and paste them here, but they are easy to find. A teacher could fashion math applications for anything from arithmetic to calculus. A cool one is to find the equation of a parabola approximating the cable.

Since the cable is about head-height, using the roadway as the x-axis, we can take the center of the cable to be the point (0,6). The distance between the two towers is 4,200 feet. The height of the towers above the roadway is 500 feet. So, two other points on the cable could then be (2,100, 500) and (-2,100, 500). Using a system of equations with those three points could give you an equation of a parabola.

Also, the distance from the mean high water mark to the road is 220 feet. So a parabola could be found with the Pacific Ocean represented by the x-axis.

Or it could be found using meters rather than feet.

You could probably spend a week just studying the bridge. And it would make for a cool field trip.

Election Day

This just happens to be election day. Let's follow up on my posting last week about polls. Today there won't be polls, there will be projections. They change the name, but they're really the same thing. It turns out they can be wrong.

• I read an article that said there was a primary several months ago in which pollsters took data to say that Clinton had a 99% chance of winning. Sanders ended up with a narrow victory. I guess you can't say he was wrong. Some things that are predicted to happen one percent of the time, do happen. Still probably embarrassing for those pollsters, though. 
• In 1948, Harry Truman famously held up a newspaper declaring that "Dewey Defeats Truman". He didn't. Dewey had such a lock on it. As George Gallup Jr. said about this, "We quit polling a few weeks too soon." That'll do it.
• In 1936, Literary Digest conducted a survey of its readership. It picked Alf Landon over Franklin Roosevelt. It turns out that Literary Digest (which has since gone out of business for obvious reasons) mostly appealed to a higher-income type person. That skewed Republican, thus predicting a President Landon.
• In 2000 the television networks declared Al Gore the winner of Florida. That was all he needed to be president. They had to retract that, declaring the race "too close to call". Overnight the networks declared George Bush the winner. Later, back to "too close to call". 

Last time we looked at how the polls work. A newscaster might say, "Candidate X is at 57% with a 3% margin of error". They usually don't mention the level of confidence. I thought they use a 90% confidence level. I saw something lately that said it is usually 95%. Regardless, they're pretty confident. But they aren't certain. 

Let's take that example and use a 95% confidence level. 

Candidate X is at 57% with a 3% margin of error translates to:

We are 95% sure that he is somewhere between 54% and 60%.

There is actually more to it than that. Consider a bell-shaped curve peaking at 57%. Of all the possible outcomes, 57% is most likely. Then 56%, then 55%, then 54%. Even 53% or below is not out of the question. Very unlikely, but not out of the question.

So if Candidate Y is at 45%, he/she is probably going to lose. However, it won't be because the 3% margin of error says he has to.

Pretty confusing. No wonder the pollsters get embarrassed every once in a while.