Rules of Thumb |
TASK. Some drivers learn the rule of thumb, "Follow two car lengths behind for every 10 miles per hour." Others learn, "Stay two seconds behind the car ahead." Do these two rules give the same results? Is one safer than the other? Is one better for roads with speed limits of 45 or 50 miles per hour and another for highways on which the speed limit is 65 or 70 miles per hour?
COMMENTARY. Obtaining a driver's license has become one of the "rites of passage" in the U.S. On almost every written driver's test, applicants are asked how closely one driver should follow another on the highway. We all appreciate the dangers of tailgating--not enough stopping time and not enough space to avoid an accident. However, it is not clear that there is agreement about what actually constitutes tailgating--how far apart cars should be.
Rules of thumb are helpful guidelines--sometimes derived from experience--that are calculated using easily available measurements. Often they are developed under particular conditions and may be extremely inaccurate if those conditions are not fulfilled. The existence of two rules of thumb for the same situation suggests a natural question: Are the two rules simply two different ways of saying the same thing or are they offering different advice? As stated, the rules may provide visual images of how far to stay behind another car, but translating that understanding into practice on the road may be quite a different matter. The exercise of interpreting rules of thumb and comparing their results with real data could help students realize that the rules they use have implications for their actions. Also, there is the reality of high incidences of automobile accidents among new drivers. This exercise may help students examine and improve their driving habits.
In order to do the task, students need to know what it means to make a comparison. They have to identify the quantities needed in order to calculate the following distances given by the two rules and represent the rules mathematically. There are many ways to do this--written descriptions, tables, equations, or graphs, all basic tools of mathematical literacy. A comparison requires that the two representations use the same units of measurement--hence some conversions are necessary from the units used in the original rules of thumb. Such conversions are an essential part of many everyday situations, both at work and at home.
MATHEMATICAL ANALYSIS. To begin, students might be well advised to consider the case in which two automobiles are traveling at a steady rate. The information presented is not complete and students will find that they have to seek out missing data. Naturally, what students seek will depend on their interpretation of the task. One necessary piece of information may be average car length.
The units for the car-length rule are miles per hour and car lengths, and the units for the two-second rule are miles per hour and seconds. To compare the two rules, both need to be written in the same units. A typical sedan is about 14 feet, so the car-length rule might be translated as "follow about 28 feet behind for every 10 miles per hour" or as the equation y = 28(x/10), where x is the speed of the car in miles per hour and y is the following distance in feet.
If a car is traveling at x mph, then it travels x miles in one hour--in other words, x/3600 miles in one second. The two-second rule is then "if your speed is x mph, follow about 2x/3600 miles behind." As an equation, it is
z = 2(x/3600), where x is again the speed of the car in miles per hour, but this time z is the following distance in miles (not feet as in the previous equation), and we use a different letter to distinguish it from y above.
Now the rules are both in terms of miles per hour and units of distance but not the same units of distance. The car-length rule is as follows:
y = 28(x/10),
where y is the following distance in feet. The two-second rule is
z = 2(x/3600),
where z is the following distance in miles. Simplifying the car-length rule gives
y = 2.8x,
where y is the following distance in feet. Simplifying the two-second rule gives
z = x/1800,
where z is the following distance in miles.
Now it's a matter of converting z to feet (or y to miles). There are 5,280 feet in a mile, so x/1800 miles is 5280(x/1800) feet. That's about 2.93x feet--very close to the distance given by the car-length rule!
Some driver's manuals give data on the distance cars travel before they are able to come to a complete stop. Often the distance is broken into two components, the reaction distance and the braking distance. The reaction distance is the distance traveled while the driver reacts to a situation and hits the brakes. The braking distance is the distance traveled from the time the brakes are applied until the car comes to a stop. A simplified version is given in Table 1.
Table 1:
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This table allows a comparison of the distances given by the rules of thumb with actual stopping distances. But the stopping distances are the distances required for a car to stop before hitting an immovable object blocking the road, whereas the rules of thumb assume that the car in front is also moving forward. This table suggests some questions about the rules of thumb: How much reaction time does each rule allow? Why are the rules of thumb linear and the stopping distances non-linear--and does this matter?
EXTENSIONS. In 1977, a National Observer article stated, "The usual rule of thumb in the real-estate business is that a family can afford a house 2 to 21 2 times its income." Incomes and housing prices have changed considerably since 1977, and real-estate agents' rules of thumb may have changed as well. Every subject--from shop to physics, from auto mechanics to economics--introduces rules of thumb that work well in appropriate situations. Even in mathematics, practices that students don't understand may acquire the status of rules of thumb for them and may be misapplied.
The original rule of thumb gave the measurement of a person's waist in terms of the measurements of their thumb, wrist, or neck. "Twice around the thumb is once around the wrist. Twice around the wrist is once around the neck. Twice around the neck is once around the waist." (The Dutch refer to "rules of fist," possibly for similar reasons.) The differences in body proportions at different ages (see Figure 1) suggest that this rule may have been developed for adults and may not be useful in designing clothes for young children. Students can be asked to create a rule that would work for young children. Because children's proportions change so rapidly with age, such a rule might include age as a variable.
Figure 1: Changes in shape between
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There are numerous other rules of thumb: "The rule of 72" in finance, "Double the tax to get the tip" in a restaurant, "Magnetic north is true north" in navigation, and so on. Students can compare the results of these rules with actual data or investigate the accuracy and derivation of such rules in their areas of interest. For instance, The Joy of Cooking provides the following rule of thumb for cooking turkeys, "allow 20 to 25 minutes per pound for birds up to 6 pounds. For larger birds, allow 15 to 20 minutes per pound. For birds weighing over 16 pounds, allow 13 to 15 minutes per pound. In any case, add about 5 minutes to the pound if the bird you are cooking is stuffed" (Rombauer & Becker, 1976). Students could explore the reasonableness of such predictions: might one conclude that a 5.9 pound bird requires (5.9) x (25) = 147.5 minutes, while a 6.1 pound bird requires no more than (6.1) x (20) = 122 minutes?
There are many other natural variations on the original problem as well. How sensitive is the car-length rule to what is assumed about the length of a car? Is the difference in average length of European versus U.S. sedans important to this rule of thumb? How should the two rules be modified for use on wet pavement? Questions might be raised about what happens if one car is traveling faster than the other or about the relationship between age and reaction time. In a state with a large number of retirees such as Florida, should the rules of thumb be the same as those in states with younger populations?
Another issue concerns the usability of the two rules for following distance. If the two rules give essentially the same advice, is one easier to use in practice than the other? Is it easier to think in terms of distance measured in car lengths, picturing the space filled with cars, or to pick a marker such as a road sign or billboard, and count seconds? Opinions will vary as to which is the easier method.
References
- Peitgen, H.-O. et al. (1992).
- Fractals for the classroom. New York: Springer-Verlag.
- Rombauer, I. S. & Becker, M. R. (1976).
- The joy of cooking. New York: Bobbs-Merrill Company.
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