Fahrenheit to Celsius: An exploration in college algebra

Mathematics and Computer Education, Spring 2003 by Fay, Temple H, Hardie, Keith A

1. INTRODUCTION

Part of the teaching "reform" movement established in the undergraduate curriculum has included an emphasis on graphical interpretation, and the study of linear equations and their inverses plays an important role. One of the interesting examples of "applied linear equations" is the pair of formulas for converting between degrees Celsius and degrees Fahrenheit. In this article, we suggest that the classical "exact" formula for this conversion is not "user-friendly", and we offer an "approximate" linear transformation that is easier to remember and use. Indeed, we call our formula a "real-time" formula, since the arithmetic is easy to perform mentally and one does not need a calculator or computer.

By investigating both the exact conversion and the approximate conversion, one gets to discuss some important concepts that will be useful to the student in an introductory algebra course. These include thinking about what makes an approximation "good" or, more precisely, what "accuracy" is interpreted to mean "good"; what is a "working range of a variable"; what are the "accuracies" for both converting from Celsius to Fahrenheit and vice versa, should they be the same? This model provides interesting and relevant problems for small group investigation and discussion. Furthermore, it lays the foundation for understanding numerical approximations that will arise in other classes in other disciplines.

2. FAHRENHEIT TO CELSIUS

Consider the everyday problem for the international traveler: how to convert from temperature given in degrees Fahrenheit to temperature given in degrees Celsius and vice versa. Of course, this problem has a well-known solution. Once it is recognized that the relationship between Fahrenheit and Celsius is linear, that water freezes at 32 degrees Fahrenheit and at 0 degrees Celsius, and that water boils at 212 degrees Fahrenheit and at 100 degrees Celsius, then the linear equation

gives the conversion from degrees Fahrenheit T^sub F^ to degrees Celsius T^sub C.

But while this formula is exact and practical, it is difficult to remember unless one does such conversions often; also, mentally calculating five ninths of some number is not particularly simple. We offer a different model to convert Fahrenheit to Celsius that is easy to remember and easy to perform mentally, and thus may be considered more "user-friendly":

1. Subtract 32 from T^sub F^

2. Divide by 2 (2.2)

3. Add 10 %

Suppose we wish to convert 72 degrees Fahrenheit to Celsius. Subtracting 32, we get 40; dividing by 2, we get 20; adding 10%, we get 22. Thus we claim that 72 degrees Fahrenheit is 22 degrees Celsius (approximately). Using the exact formula, we calculate five ninths of 40, which is 200/9 or about 22. The true conversion is 22.222 (rounded to three decimal places); but in casual use, we seldom measure degrees in decimal or in fractional parts. Indeed, we round to the nearest degree; we do not truncate.

Choosing 72 degrees to convert may have been too easy. Suppose we choose to convert 47 degrees Fahrenheit to Celsius. We subtract 32 to get 15; divide by 2 to get 7.5; and add 10% or 0.75 to get 8.25, or 8 degrees Celsius (approximately). The exact conversion is 8.333.

Note that we have refrained from writing down the analytical formula for this approximation. The three steps above are simple and easy to remember. Putting this is an equation form is a good student exercise, but the equation form visually disguises how simple the model is:

This formula might also be interpreted as "user-unfriendly" since multiplying by 0.55 is as difficult as multiplying by 5/9.

There are several natural questions to ask that can be discussed by the class or investigated in small groups. These include:

* Perhaps 72 and 47 worked accidentally; what about other values? Suppose the temperatures are very high or below zero; what is the accuracy of the approximation? How should we decide if this accuracy is good or bad? Sufficient or insufficient?

* Give arguments pro and con for the model (2.2) to be preferred to Equation (2.3).

* Over what range of Fahrenheit temperatures are the two formulas within two degrees of each other? Within one degree of each other?

* How can we graphically display these two ways of converting from T^sub F^ to T^sub c^? Since we round to the nearest degree, perhaps a step function approach would be useful.

* The exact formula to convert from Celsius to Fahrenheit is

What is the inverse formula for the approximation? Is it more "user-friendly" than the exact formula?

* If one plots the two formulas (2.1) and (2.3) as shown in Figure 1, we see that both straight lines pass through the point (32,0). We could have instead considered an approximation to pass through the point (212,100). Find a "user-friendly" formula for such an approximation. Draw a figure similar to our Figure 1. Is this approximation better than ours? What do you mean by better? Are there circumstances where one would be preferred over the other?

The usefulness of an approximation to a linear equation is an idea that is not often discussed. There are a number of critical notions relating to linear equations that are brought out in thinking about this approximation. Slopes, equations of lines passing through given points, graphical interpretation, what is a good approximation and why, and operating ranges for variables are just a few of these notions. The authors believe that this model provides a meaningful example to help explain these concepts to beginning students and provides an introduction to approximations at the same time.