# Bumping Airlines

Out of 615 million airline passengers last year, half a million were bumped from flights. 9 out of 10 of those were voluntary. What percentage of all booked passengers were involuntarily bumped from a flight?

Yes, I had fun at the expense of United Airlines’ most recent viral embarrassment, but I had two serious questions that I needed answered:

1. Should I change my air travel habits?
2. How many of my Algebra 2 students could correctly answer this question?

I had my class answer both questions for me. I started class by handing them the prompt and this now famous video clip:

I then shared what I learned about the law in regards to this incident. United Airlines and law enforcement officials were legally in the right to remove the passenger from the plane. When overbooked, an airline has the right to randomly bump passengers, but they must first offer an adequate incentive for volunteers, which United did. These regulations are in the contract rules that we all agree to, but never read, when buying an airline ticket. The law also states that any passenger must comply with directions given by airline personnel or law enforcement officers. Since the unfortunate gentleman on the plane resisted the directions of the authorities, the airline and the police had the legal right to forcibly remove him.

It was the third part of the law, however, that was the most disconcerting for me. If an airline involuntarily bumps you, they must guarantee your arrival at your intended destination within 24 hours. But that is not good enough for me. I often travel to places where I am expected to be working with teachers early the next morning. A 24-hour delay would be far too late. So my new burning question is: Should I leave greater leeway in time when I am traveling? That is what I needed to answer. The students helped me think through it.

Nine out of 10 voluntarily bumped means only 10% of the 500,000 bumped passengers, or 50,000 passengers were removed involuntarily. That 50,000 out of 615 million is a whopping 0.008% of all booked passengers last year. So what does that mean in terms of my flight habits? How many times would I have to fly in order to expect being bumped at least once? 0.008% of what number equals one (1 = 0.00008x)? It turns out that I would need to fly 12,500 times. In over 40 years of an active adult travel life, I would have to board a plane nearly every day of my life to expect this to happen. Of course, probably and possibility are utterly different, so I could be bumped on my next flight, but I am not ready to start adding an extra day to every travel trip for such a small chance.

So how did my students do with this calculation? My prediction of one student was an underestimation. Five actually calculated correctly, with 3 others getting close, showing appropriate work. Why was such a simple math topic (calculating percentage) such a challenge for a group of 15 & 16 year olds? In talking with the students, I came to realize that this was the classic case of “making sense of problems.” There were multiple layers in unpacking the prompt,  as well as the added layer of interpreting such a small fraction of a single percentage point and the need to make a decision based on that numerical interpretation.

It is noteworthy to reveal that I gave this problem to four adults. All four answered correctly (0.008%), and all four struggled to make sense of what was being asked.

So how do we get students more proficient at making sense of problems that require basic math? Easy. We pose those problems more often. Which I intend to do.

# The Election Pollsters Still Got It Right

There has been a great deal of Monday morning quarterbacking about how the 2016 Presidential election polls “got it all wrong.” Radio pundits like KFI’s John and Ken have been claiming that pollsters obviously don’t know what they are doing. There are three points to consider here.

1) Did the polls get it wrong?
2) Did the pollsters do something wrong?
3) What good math activity can we generate from all this fuss?

Here are some direct answers with, hopefully, simple, clarifying mathematical (not political) explanations.

The Polls Got It Right
The poll results were within the expected margin of error. In fact, four days before the election, Harry Enten of FiveThirtyEight wrote “Clinton’s lead is small enough that it wouldn’t take more than a normal amount of polling error to wipe the lead out and leave Trump the winner of the national popular vote.” In the end, Clinton still won the popular vote, by approximately 1.5% compared to the 3.3% predicated the day before the election, well within the normal margin of error. Gallup shows that, historically, the polls have been within 2%, on average, of the actual results, and within 1% half of the time, with the victories of Reagan in 1980 and Truman in 1948 being the most notable anomalies.

In fact, Nate Silver of FiveThirtyEight noted the day after the election that a 1% swing in Clinton’s favor across all states would have flipped the Electoral College tally.

Further support that the polls got it right comes from the understanding of probability. Clinton was given a 71% chance of winning on the eve of the election. That means that Trump had a slighter better chance of winning the election than he had of flipping heads on two consecutive tosses of a coin. When heads occurs twice when tossing a coin, should we all protest that statistics and polling are unreliable? This is why Nate Silver claims that the polls missed, but he did not say that they failed.

The Pollsters Did It Right
People have been willing to give more grace to the mathematics than to the mathematicians. Pollsters (those creating the polls, not the folks on the phone) have taken a great deal of heat for poor sampling, but these pollsters have been vindicated voter turnout numbers, because the pollsters surveyed registered voters, not guaranteed voters.

PBS‘s Michael Reagan writes that the data on actual casted votes reveals that Clinton had 2 million fewer voters than Obama did in 2012, while Trump had a slight uptick over Mitt Romney. Had voter participation been similar to the 2012 election, America would have had a different 2016 result.

Liberal filmmaker Michael Moore was extremely concerned just before the election about the lack of enthusiasm for Clinton versus the overwhelming passionate support for Trump. His concern turned out to be warranted.

A Good Math Activity: Secretary Clinton Attempts A Field Goal Kick
Given the information below from FiveThirtyEight, at what distance (in yards) would a field goal kicker in 2014 have the same chance of success as Secretary Clinton in the election of 2016.

Fortunately, if an NFL kicker misses a field goal attempt from just inside the 50 yard line, I still have faith in statistics and statisticians… and America.

# Millionaires and Their Cars?

My teenage son is preoccupied with three things these days: water polo, his girlfriend and expensive cars. He has fantastic talent in water polo, and has a wonderful girlfriend. He does not have an expensive car.

Currently, he is saving for his first car and knows that he will have to start with a used, low-end model, but he dreams big. He is always talking about Ferrari’s and Rolls-Royces. We like to talk about them together and point them out on the road whenever we are driving. He is convinced that he will own one someday. When I respond to his talk of grandeur, I want to sound like the Encouraging Dad (“Terrific, what kind of successful job do see you see yourself having, so you can afford that kind of car?”), but I worry that I sound like the Practical Dad (“That’s nice, it might be more realistic to set your sights on a cheaper car.”). The truth is that my words usually come out somewhere in the middle, which led to our very interesting math conversation the other day.

On a long drive back from a water polo game, we were talking about reasonable incomes (Practical Dad ruling the moment). He is a Junior in high school, so his interest is peaking about how much money is to be made as an adult. The conversation went like this:

Me: Guess what the average annual income is in America.

Preston: I know, because we talked about this in History class. \$36,000 a year, but if I make \$100,000 a year, a can save half of that and buy a rich car in five years.

Me: Do you know what percentage of Americans make over \$100,000 a year?

Preston: 25%?

Me: It is actually about 4%. I know several people who make that kind of money. None of them drive a Ferrari, so you are going to have to make more than that. (Encouraging Dad trying to break through.)

Preston: If I made a million dollars a year, I could buy it in one year and still have enough to live on.

Me: With enough left over to care of me and your Mom. That would be awesome, but you are going to have to do something special, because less than one-half of one percent of Americans make a million dollars a year.

Preston: It has to be more than that. Look at how many rappers there are making bank.

Me: And think about how many are making just a normal living or how many are standing on a street corner singing while they hold their hat out for tips. Very few earn “Checks that look like phone numbers.”

Preston: Look at how many millionaires we know.

Me: I would say less than 5, off the top of my head.

Preston: Yeah, see?

Inspired Math Question #1: If you know 5 millionaires, what percentage is that of all the people you know?

Inspired Math Question #2: If one-half of one percent of the people you know are millionaires, how many people would that be?

Preston: I bet there are at least a million millionaires in the country.

Inspired Math Question #3: Given that there are 300 million people in the U.S., and that 75% are adults, would one-half of one percent of American adults be more than a million people? (to be estimated while driving without a calculator)

Me: I am guessing that we are both correct on this one.

Preston: I still say it has to be more than that then. (Whether I am encouraging or practical, I am still Dad, so he must win!) Look at how many expensive cars we saw just today. There was a Ferrari, a Lamborghini and a Bentley.

Me: Yes, and think of how many other cars we saw today.

Inspired Math Question #4: Approximately how many total cars might you see driving on a freeway for an hour on a Sunday afternoon? (must explain your reasoning on this one)

Inspired Math Question #5: If you see three expensive sport cars on that same trip, what percentage of all the cars would that be?

As we arrived home, Preston was still seeking victory. He is very good with mental math, so he knew where I was going with all the number crunching. In order to get the upper hand, he needed to bring in an expert, and what better expert in the world of teenagerdom to call upon than the internet? He Googled on his smart phone, “How many millionaires are in America?” and got an answer of over 3,000,000. He loudly reveled in glory. I countered with the age-old math argument of the importance of definitions. In this case, there was a difference between annual income and net worth. He was having no part of it. He was to busy flexing and bragging to Mom about how he just “owned” Dad in a math debate.

# Lesson: M&M Count and Crunch

### Have a snack and reinforce the concepts of ratios, percentages, and statistical analysis.

SUBJECT: Pre-Algebra
TOPICS: Probability & statistics, ratio & proportion, percents, unit conversion, area, bar graphs and pie charts
PAGES: 4