As the close of another school year approaches, children and youth across the country trudge through end-of-year testing. Teachers work to prepare their students both academically and mentally. A concept some teachers work to explain and students struggle to understand is exponential and quadratic growth rates. Part of the problem is that this math is often taught on paper with a simple x/y axis; it doesn't seem to have any relationship to the real world. But if you look at something like Angry Birds or blockbuster summer movies, these concepts become easier to understand.
A quadratic is a function of the form y = ax + bx + c, where the values of a, b, and c define the curve. Where are parabolas? One location is in the game Angry Birds, as discussed in my Huffington Post blog post "Frustrated With Math? Try Angry Birds!" and in more detail in my book Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing. The red bird in the introductory levels follows the path of a parabola, as seen below.
Between being released from the slingshot to the apex of the trajectory, the bird experiences quadratic growth in altitude. That red angry bird moves pretty quickly. What difference would it be to have that angry bird's altitude grow exponentially? Let's ask a broader question first and return to this in a bit.
So how fast can exponential growth really be? First, it is fast -- very fast. Frankly, exponential growth is fast enough to make the difference between a summer blockbuster and a bomb at the box office. This summer, films like Godzilla, Maleficent, and the next installments of the Transformers and Planet of the Apes films will be released. Let's see these growth rates in this context, which is another topic detailed in a similar way in Math Bytes.
So what would be worse for a film: having a million people tweet poor reviews of the movie over an hour, or having one person tweet in the first minute, and from that point on having twice as many people tweet each minute as in the minute before? Clearly, having a million people people bash a film is not great. But let's see what happens in this doubling. In the first minute, one poor review is posted. In the second minute, two more poor reviews appear. In the third minute, an additional four poor reviews are posted, with a total of seven negative tweets having been written, as seen below:
Simple enough. Not much harm. But it doesn't take long to see that this rate continuing produces a problem -- a blockbuster problem. At 10 minutes, we still only have a bit over 500 people writing a poor review that minute, with just over 1,000 total having tweeted over those 10 minutes. Not bad. But notice that after another 20 minutes, half a million people are tweeting at that time, with over a million having crafted the 140-character thumbs-down reviews. We still have 40 minutes to go before the hour is up! In fact, we need less than 30 minutes to reach the entire population of our planet, which would be great for the volume of Twitter's subscribers but terrible for the film.
Now let's compare this to quadratic growth. At minute 2 we have 2*2, or four people tweeting. At minute 10 we have 10*10, or 100 people tweeting. Then, after 30 minutes, we have less than 10,000 people tweet in total! In fact, at this growth rate, even after 180 minutes, under 35,000 people are tweeting at that moment! While this is a lot of tweets, clearly it is far from the rate of our doubling, which was exponential.
Finding examples can be difficult, as growth is often not that fast. Yet we frequently use the term "exponential growth" when we in fact simply mean fast growth. An enlightening activity is searching Google News for the term "exponential growth" and discussing whether the claims for such growth are indeed true. Quadratic growth is fast. But exponential growth is very fast and rather deceptive. For our poor reviews, it doesn't look like anything is happening over the first 10 minutes. Keep in mind that if those million tweets were distributed evenly over an hour, we'd have about 16,000 tweets per minute, or about 250 tweets per second. In the doubling, this rate doesn't appear for a bit. Even so, the number of tweets is doubling each and every second.
So let's return to our earlier question. Why does an angry bird not follow exponential growth as it is shot out of the slingshot? It wouldn't take long for the bird to find itself in space. The bird may accelerate quickly, but "quickly" doesn't mean exponentially. In fact, maybe it wouldn't take too, too long to have that angry bird in a galaxy far, far away.
Soon, students will be beyond being tested on such ideas. We will be in the slumber of summer. Yet in all likelihood, at some point in those months, a song, image, or movie will go viral on the Internet and the world will ask the same question, "How can this happen?" Hopefully, a group of students now working to hone their skills in exponential and quadratic growth will remember just enough of their school studies to explain how such seemingly impossible rates can occur!