Making Maths Interesting
Few would argue that mathematics education could be greatly improved. Countless is the number of math teachers who've been asked "When am I ever going to use this?" When I was an adolescent, the stock answer to this query was, "You're not going to carry a calculator with you everywhere you go," but smartphones and other mobile devices have made this rebuttal obsolete. For now, mathematicians can still retreat to the "There are practical uses in science" defense, but I've noticed even this is slipping. Many popular science books, for which mathematics would seem to be very relevant, feature a distinct lack of equations. A quick glance through the bestselling science texts quickly reveals that even the most theoretical of theoretical physics books offers little more than a cursory glance at the mathematics which made it all possible. The end result of this (at least for me) has been numerous students asking to take advanced physics classes "without the maths." After all, what does it say about the importance of mathematics when the world's leading scientists don't put any in their bestsellers?
Fortunately, there seems to be a mini-movement afoot that's attempting to bring actual mathematics back to the forefront. There has been a rush of books that use estimation in an attempt to make the maths itself interesting and relevant. Sanjoy Mahajan's Street Fighting Mathematics, Lawrence Weinstein's and John A. Adams' Guesstimation and Guesstimation 2.0, and my own Ballparking and How Many Licks? are just a few of the recently published books that teach the art of estimation. Far from typical boring rote calculations, estimations are an easy way to show that maths can be funny, thought-provoking, gross, sexy, and a variety of other adjectives you never thought you'd see in the same sentence as the word "mathematics." As such, estimations are an ideal means of engaging students in a way that speaks to them.
Since the Olympics was recently in London, here are three sports examples pulled from my own book that illustrate how even simple maths (and perhaps a little bit of physics) can be used to entertain and educate:
1. Mexican High Jumpers
The 1968 Summer Olympics in Mexico City saw record-setting leaps in the high jump, long jump, and triple jump, not to mention several throwing events. It's been hypothesized that the vat of broken records was due to Mexico City's high altitude, which would create a lower acceleration of gravity.1
So how much of a difference would the added altitude make on a jump? This is a problem best tackled using a technique that physicists call "scaling." Introductory physics tells us that the range R and maximum height h of a projectile both scale as the inverse of the gravitational acceleration g. By "scales as" we simply mean the two are proportional:
The gravitational attraction between objects is weaker when the objects are far apart. This fact can be expressed through another scaling relation:
where r is the distance between the objects. We can rearrange these scaling relationships as we would for a normal algebraic equation to get
For terrestrial objects, r is the distance from Earth's center to its surface. For the most part, the gravitational acceleration on Earth is roughly constant at about 9.8 m/s2. This is because Earth is very spherical, so r is fairly constant at around 6,378 kilometers. However, in places like Mexico City, the altitude is very high, which can make g change slightly. Mexico City is located 2.2 kilometers above sea level, which would represent a 0.034 percent increase in the radius r. Using our scaling relations, we can see that a 0.034 percent increase in r results in a 0.069 percent increase in both the jump range R and jump height h. Putting this into perspective, you would expect increases of about 1.6 millimeters in the high jump and 6.3 millimeters in the long jump.2 Jumping records are generally broken by centimeters. Although the decreased gravity would increase the distance jumped, it doesn't have a very large effect.
2. Remember to Breathe
Jack Handey might be happy to know that during my extensive research conducted over the last two minutes, I have found no such provision in the International Weightlifting Federation (IWF) rules. Moreover, it's unclear why the IWF would support any such rule, as uncontrolled urination would seem to provide no discernible unfair advantage for lifters in a competition. In contrast to urination, one might expect that uncontrolled flatulence may provide a slight advantage because the gas would help propel the lifter upward. How much extra weight could a lifter with uncontrollable flatulence squat?
Farts come in a variety of shapes and sizes.3 Assuming they have a volume of about ten cubic centimeters and the same density as air (about 1.2 kg/m3), each fart will weigh about ten milligrams. Let's assume they come out at a speed of 1.0 m/s and last only 1.0 seconds. Using this data and dimensional analysis, we can estimate the total force:4
Continuous flatulence would produce a 0.01-Newton reaction force upward, which is equivalent to lifting an extra gram (~0.002 pounds) of weight.
3. There's No "I" in STD
Recently USA basketball star Kobe Bryant's wife Vanessa was in the news for allegedly getting furious after seeing pictures of a shirtless Bryant partying in Barcelona. Given Bryant's highly publicised history of infidelity, her complaints are not at all unreasonable.
After all, you don't have to be one of Tiger Woods's 19 mistresses to know that athletes love sex. In the professional sports world, athletes are much more likely to be a Tiger Woods than a Tim Tebow, but even by these heightened standards, one athlete stands out from the crowd. In his autobiography A View from Above, NBA basketball "Hall of Famer" Wilt Chamberlain claimed to have slept with 20,000 women.5 If this were the case, how many STDs would Chamberlain have contracted?
Chamberlain was about 55 years old when his autobiography was published. Assuming he started having sex at age 18, he would need to have sex with an average of 1.5 new women each day. There are a variety of STDs lurking out there. Some, like genital herpes, can infect as many as 1 in 6 people. Others, like HIV, are more deadly but less common, infecting about 1 in 700 people.6 We can take the higher number because we know at least this many people are infected with an STD. This means that at least 1 out of 6, or roughly 3,300 of Wilt's "acquaintances," likely had an STD.
The rate at which STDs are transmitted depends on a variety of factors (disease type, overall health of the individuals involved, whether or not a condom was used, etc.). Wilt's serial philandering occurred largely before the discovery of AIDS, so he likely didn't practice safe sex. As a rough order of magnitude estimate, I'm going to assume a 10 percent transmission rate for unprotected intercourse between infected and uninfected individuals.7 At this rate, Wilt would have contracted 330 STDs.
Even if Wilt practiced safe sex 100 percent of the time, his sheer volume of partners still puts him at high risk. Condoms are advertised to be about 98 percent effective. At this rate, even with condoms, Wilt would have contracted anywhere from 7 to 70 STDs depending on whether you assume the 2 percent figure means the condom user was infected 2 percent of the time or merely that 2 percent of the time he is as likely to be infected as he would without wearing a condom. Either way, claiming you've slept with 20,000 people is the logical equivalent of running through the street screaming, "I HAVE HERPES!!!"
What can we take away from this? Even more than some random bits of numerical trivia, estimations can be a fun way of making math interesting and relevant.
 The lighter air may also make a significant difference, but I'm ignoring this for simplicity.
 This assumes a 2.36-meter high jump and an 8.95-meter long jump.
 Not to mention flavors.
 Dimensional analysis is a technique physicists use to estimate quantities. It works by taking all the parameters that go into a calculation and combining them in such a way as to give the correct units. In this example, we're calculating a force which has units of Newtons or, equivalently, "kilograms × meters / seconds2".
 FUN FACT: Being tall and lanky was not the only reason Wilt was called "The Stilt."
 Statistics obtained from Book of Odds at
 For some diseases like HIV, the infection rate can be significantly lower.