### How much money will "Hunger Games: Catching Fire" make?

The new "Hunger Games" movie is opening this Friday. Get ready for more Jennifer Lawrence killing everything in sight:

 Credit: Lionsgate

The first "Hunger Games" film made \$408 million at the US box office. Almost half of that total was made in just the opening week! Let's take a look at the graph of how much money the first film made on a weekly basis, to see if we can predict how the second film will do.

This looks a lot like a decaying exponential function, which can be written in the form Ce-t/τ, where C is the amplitude of the function, and τ is the time constant. Another way to think about this is that each week, the film makes a fraction of the money it made the week before. That means we can treat the weekly data as a geometric series!

The formula for the sum of a geometric series is A/(1-r), where A is the first term in the series (i.e., how much money the film makes in the first week), and r is the ratio from one week to the next. For example, if a film makes \$100 million in its first week, and \$60 million in its second week, then r = 0.6, which is pretty high for a blockbuster release. Larger values of r mean that a film has a longer "lifetime" in the theaters -- people are still buying tickets later into the film's run. After "Catching Fire" is out for a week, we'll know exactly what A is. But how can we determine what r is?

To estimate r for "Catching Fire," let's look at how some similar movies did in the past. Let's specifically take the total amount of money each film made in its first week and divide that by how much it made in total, and let's call this ratio w (for opening week). So then:

So using a film's opening week and total intake (or "gross"), we can estimate r. Let's see how a few different film franchises compared:

For each of these graphs, there seems to be a downward trend, meaning that later films in each franchise have shorter lifetimes than earlier films. But each of these franchises behaves a little differently. For example, The Harry Potter franchise steadily declined over its 8 films, while the Twilight films plummeted after the first one. Maybe after the first film, only the "Twihards" saw the remaining movies, often in the first few weeks of release (resulting in a low value of r).

With more study and more data from other films, we could generate a probability distribution for the value of r for "Catching Fire." But for now, we can guess that it'll probably be less than r for the first "Hunger Games" film, which was ~0.53. Given the positive buzz the new movie is getting, let's say r is a relatively healthy 0.5. So to figure out the total amount of money this second film makes, take the gross from its opening week, and divide it by (1-0.5). In other words, double it!

Here's our best guess for "Catching Fire":

Opening weekend (first 3 days) gross: \$160 million
Opening week (first 7 days) gross: \$220 million
Total gross: \$440 million (double the opening week)

### How to fix educational testing

Educational testing is currently a hot debate in this country. Here's a sampling of the discussion:
• In Texas, some schools are attempting to phase out the use of standardized tests for measuring accountability
• In New Mexico, hundreds of people protested the use of testing in evaluating teachers.
• In New York, 80% of parents at one elementary school opted out of a standardized test, forcing the school to cancel the exam.

Parents, educators, and students have all pointed out several key problems caused by standardized tests:
1. Testing uses time that could be better spent elsewhere.
2. Teachers are often uncomfortable with the idea of having months or years of their work evaluated by a brief exam.
3. Because of the pressure imposed by these tests, teachers feel constrained to teach to the tests.
4. Most students don't like taking the tests.
Instead of administering aggravating tests, at School Yourself we're combining learning and evaluation into a single, seamless experience.

What's the purpose of educational testing, anyway? It's a scalability issue. States can't afford to evaluate every second of a teacher's work in the classroom, but they can afford to give all the students a test once a year to see what they've learned.

Scalability is the same reason we have any tests in the classroom at all. Teachers don't have the time to sit down with every student and objectively explore everything the student does or does not know. So teachers give tests, which may not do as good a job. But tests are "good enough," and take up a lot less time than interviews.

With online learning platforms like School Yourself, we have the opportunity to do away with testing once and for all. With highly modular content and detailed analytics, we can assess what students know in real time. This eliminates the need for separate testing, and evaluation becomes part of the learning experience.

On the School Yourself platform, students are interacting every 30 seconds or so (whether it be answering a question, playing with an open-ended sandbox, etc.). Between our dynamic lessons and the practice problems we'll be adding in a few weeks, we can take high-resolution snapshots of student knowledge that are much better than what any standardized test can do.

As we fully integrate evaluation into the learning process, a challenge that digital learning platforms like ours are uniquely suited to, standardized tests will become a thing of the past.

### Seeing who "hit the wall" in the New York City Marathon

A special congratulations to our very own John Lee, who ran the New York City Marathon this past weekend (his first full marathon)!

 Photo credit: Jake Park

John trained for months, and finished the race in a time of 3:21:11 (3 hours, 21 minutes, and 11 seconds). Given that every marathon is about 26.2 miles long, we can calculate that his average pace was 7:41 (7 minutes and 41 seconds) per mile.

Smart runners will pace themselves, and run at about the same speed the entire race. But runners who don't prepare as well will often "hit the wall" around mile 20, and will slow down toward the end of the race.

Let's look at graphs of how a few runners did. We'll specifically look at John ("The Real John Lee"), another person who happened to be named John Lee and who also ran the marathon ("Fake John Lee"), and the fastest woman to run the race, Kenya's Priscah Jeptoo. Here's how they did:

The dots in the graphs represent checkpoints along the race where the runners' times were precisely measured. We don't know exactly how fast the runners traveled between the dots, but let's assume they ran at a steady pace between consecutive dots.

Of the three runners, Priscah Jeptoo (in red) finished in the least amount of time, so she was the fastest (with a time of 2:25:07). You can also see that Priscah ran a smart race, without hitting the wall, because her graph is very close to a straight line. She ran at about the same speed the entire race.

The Real John Lee (in blue) also ran a smart race, keeping a steady pace the entire time. Fake John Lee (in green), however, started off on pace with Priscah Jeptoo, but slowed down more and more as the race went on. He went out of the gates too fast, and as a result, his graph is concave down, meaning it curves downwards.

We can see this even more clearly if look at the speeds of the three runners over the course of the race. Speed is equal to distance over time, so a runner's speed between two checkpoints is the slope of the line between those points. Here's the graph of the three runners' speeds:

At certain points along the race (like around mile 13), all three runners slow down. These are probably the uphill parts of the course! And it looks like there's a nice downhill stretch around mile 22. But you can also see that Priscah Jeptoo ran a smart race, with a pretty steady speed. The Real John Lee also maintained a steady speed throughout the race. But Fake John Lee kept slowing down throughout the marathon.

So by looking at the slopes at different locations of a runner's graph of distance vs. time, we can see how quickly that runner is moving. And in this case, we can see who ran a smart, steady race, and who hit the wall.

### Why airlines don't allow cell phones

The Federal Aviation Administration (FAA) just announced that most electronic devices can now be used on airplanes, all the way from takeoff to landing. But you still can't make a call on your cell phone during a flight, and all devices must be set to "flight mode." Why can't you make calls on flights? To get a better understanding, let's use some trigonometry.

Airplanes communicate and navigate using a band of radio waves called the airband, which uses frequencies between 108 and 137 MHz (or megahertz). Radio waves are electromagnetic waves, or waves of light, which travel through space like the sine function. Waves with higher frequencies move up and down, or "oscillate," faster, while lower-frequency waves oscillate more slowly. Here are examples of waves that have different frequencies:

How do the frequencies of cell phones compare to those of the airband? Well, it depends on the carrier (Verizon, AT&T, etc.), but for the most part they're between 500 and 2500 MHz. So cell phones send and receive radio waves that are close to the Airband, but are slightly higher in frequency.

If you were to make a call on a plane, then the cell phone's radio waves and the airplane's radio waves would add together. The airplane's waves are probably a lot stronger than the waves coming out of your phone. So what happens if we add a very weak cell phone signal (say, at 700 MHz) to a strong airplane signal at 120 MHz?

The sum of the two waves looks pretty close to the airplane's signal. But what happens if the cell phone signal were a lot stronger?

Suddenly, the sum looks quite different from the airplane's signal, and that's what worries the FAA. While there are mathematical tools that tease apart signals with different frequencies that have been added together, pilots and officials are concerned that cell phone radio waves could still interfere with the communication and navigation of airplanes.

If cell phones instead used frequencies that were really far away from the airband, then the risk of interference would be even smaller (even if the phones emitted really strong signals). Why do you think that is?