Speed, Function, and Critical Thinking

By Gene Fiorini, Ph.D. in mathematics

How often has this happened to you? You’re driving down the interstate obeying the speed limit when someone barrels past you at a high rate of speed. Being a conscientious driver you refrain from using sign language. Twenty miles down the road you exit the interstate. Stopped right beside you at the next red light is none other than the speedster from twenty miles ago. How is that possible?

There are, of course, very good reasons why we should not drive with excessive speed. No matter how much these reasons are emphasized people still seem to drive fast under the misguided principle that they will get there sooner. Yes, that’s right, driving faster does not necessarily mean we will get there sooner. How can that be?

So, how do we convince drivers that speeding does not pay in terms of time? Anyone with an understanding of basic mathematical formulas knows the answer to that question. Yes, that’s right, Mathematics! It can be shown using basic mathematical principles that the benefits of speeding diminish with increasing speeds.

For this scenario begin with the assumption: “The greater the speed, the less time it takes to get there.” True enough. To have a better understanding of what this expression actually means in mathematical terms we concentrate on the relationship between distance, speed and time: D = S x T

Most think of the relationship as linear. That is, if speed is increased from 10 miles per hour to 20 miles per hour the same amount of time would be saved as when speed is increased from 60 miles per hour to 70 miles per hour.

The best way to test the validity of this explanation is to run some numbers. Since we are interested in the relationship between speed (S) and time (T), we will keep distance (D) fixed. For the sake of our simple experiment, suppose we fix distance at 20 miles. That is: 20 = S x T

To spare the reader the agony of actually doing any calculations, some of these values appear in table 1 and are plotted in figure 1.

It is clear from the table and the graph that the time saved diminishes as speed increases. Increasing speed from 10mph to 20mph saves one hour, but increasing speed from 60mph to 70mph saves less than three minutes. Factor in all the reasons for stopping and slowing down and hardly any time at all is saved increasing from 60mph to 70mph. Even less time is saved by traveling at even higher speeds.

Many people believe that if mathematics is useful at all, then it is only useful at the highest echelons of science. Not true. There are many simple everyday scenarios that are good examples of how mathematics fits into the fabric of our lives. The definition of a function (A rule which assigns to each element of the domain exactly one element from the range.) is often confusing and meaningless. But replace the words “element of the domain” and “element from the range” with common objects and it begins to make a little more sense:

• Each element of the domain is assigned exactly one element from the range.
• Each item in the supermarket is assigned exactly one price.
• Each book in the library is assigned exactly one call number.
• Each student is assigned exactly one identification number.
• Each player is assigned to exactly one team.

All of these examples suggest that the assignment process serves the purpose of organizing the system in question so that it functions properly. A good mental exercise would be to see how many of these “real world” assignments you can find. It’s a good way to determine just how common, and important, the functional assignment is for organizing systems. Besides, playing this game at family functions and holiday parties will put you at the top of everyone’s invite list!

You may think of math and math teachers as ogres of mythic proportions. But if you leave out the mathematician, you invite problems of even greater orders of magnitude.