In high school I enjoyed writing about conspiracy theories. Who killed JFK? Did aliens crash land in Roswell? Does Chipotle put crack in their burritos to make them so addictive and delicious? The last one is more of a personal pondering. The government seems to be the central focus of the vast majority of conspiracies, likely due to its imposing size, societal ubiquity, and all the money they feel the need to take out of my diminutive paycheck. But I think the government is a bit too scatterbrained to be able to pull off all these feats of conspiratorial prestige. Let us look at a former top secret government endeavor, the Manhattan Project.
Blowing things up is intriguing, and from my experience with fireworks, it is pretty exciting. Ignoring the social and ethical implications, the atomic bomb is a scientific wonder. But perhaps even more impressive is the work of G.I. Taylor in exposing the carelessness of the government. The government tested the first atomic bomb on July 16, 1945. The iconic photographs of the explosion were published Life Magazine. Taylor managed to calculate the yield of the bomb (which was the equivalent of 20 kilotons of TNT) within 10% of the actual value using data from conventional bombs, measurements from the photographs, and basic dimensional analysis. At the time, the yield was considered highly classified information. Naturally, the government was not a happy camper.
Not exactly a Watergate scandal, but it is a powerful demonstration of the importance of dimensional analysis. Dimensional analysis is as simple as unit conversion and as complex as the development of laws for physical principles. I believe it is one of the most important concepts anybody can learn about math and the world. My girlfriend says that it is the reason why she passed all her classes. Math is conceptually scary; all the symbols and theories and laws and equations and relations can be overwhelming. Dimensional analysis can be a helpful way to teach students how to analyze just about any problem and determine how to solve it. I am a big fan of not teaching students a specific method to solve a particular problem, but rather teaching them how to critically analyze any problem and come up with a method for finding the solution. Dimensional analysis can provide the framework.
Dimensional analysis can be taught in its most basic form, unit conversion. Say I have a pet T-Rex who has the swine flu. Luckily, I have left over medication from when I had the swine flu. The directions read: Take 1 drop per 10 lbs of body weight per dose, 3 doses a day. How much does T-Rex need? Let us use dimensional analysis. The first step, what do we want to find? We want to know how many drops per dose T-Rex needs. What do we know? We know that we need 1 drop/10 lbs/dose, and that T Rex weighs 7 tons. The final units we need are drops/dose. When we do dimensional analysis, bottom cancels with top. First, we should get the weight in pounds. (7 tons/T-Rex)(2,000 lbs/1 ton) We can cancel the tons. Then we just do the math, multiply anything on top, and divide by anything on bottom. 7*2,000 = 14,000 lbs/T-Rex. Next, we need to find how many drops. 1 drop/10 lbs/dose is the same as 1 drop/(10 lbs * 1 dose) which is easier to work with. We have drops on top and dose on bottom, but we also have lbs on bottom. So get rid of pounds. (14,000 lbs/T-Rex)(1 drop/(10 lbs * 1 dose) All we need to do is divide 14,000 by 10 which is 1,400. So we now have 1,400 drops/dose for T-Rex. So T-Rex needs 1,400 drops per dose 3 times a day, which means I probably would have to go to the doctor and see if I can get the medication wholesale. Luckily, dinosaurs are extinct making the scenario irrelevant.
What I love about dimensional analysis is that not only is it applicable to any math or science field, it also can have real world implications. You can relate it easily to situations in every day. You want to throw a party for 20 guests. How many pizzas should you order? You have a quarter tank of gas left and the next gas station is 70 miles away. Will you make it? You can even use it as a starting point for error analysis. If a student can understand dimensional analysis and how to effectively use it, I firmly believe that student will be more successful in any math or science course taken.