Content Objective
While teaching my Pre-Calculus and AP Calculus students, I have realized that using mathematical symbols to create algebraic expressions and to solve text problems is usually a big problem for them. They struggle to translate a verbal problem statement into symbolic mathematical expressions and equations.
Let’s say students should solve the following problem: “Find the surface area of a sphere at the instant when the rate of increase of the volume of the sphere is nine times the rate of increase of the radius.”1 (Actually, this problem statement is not very good. The problem mentions “a sphere’, but it is not about a single sphere. It is about a family of spheres whose radius is varying, in what way or with respect to what variable is not explained.) As you can see, there is not a single math symbol in this problem. It is just one sentence written in English. Students should be able to “translate” it into a mathematical language using formulas and create a model based on the given data. This is an extremely challenging task for my students. Most of them are ready to give up without even trying.
The origin of this struggle goes all the way back to arithmetic and simple algebra. They should have been taught how to write algebraic expressions and create math models of a word problem. These skills are essential for their success in upper level math, physics and chemistry. Unfortunately, most of them have very modest, if any, skills of that kind. So, my goal in designing this unit is to help teachers and students to fill in this gap.
The fear of Mathematics is similar to the fear of speaking a foreign language. Some words in a foreign language we just do not know and, therefore, struggle with translation. To be successful in Math, we should be able to “translate” a problem from English into Mathematical language. Part of this is knowing what each word means. A deeper difficulty is that translation cannot be word for word. It must convey the overall meaning.
Coming back to the above mentioned related rates problem, students will learn, that the word combination “the rate of increase of the volume” is translated into mathematical language in a very simple way: dV/dt. Analyzing the problem, students should come to the conclusion that the volume is changing with respect to time. They should introduce variable t (time) and emphasize, that t was the unspecified variable, as a result of the incompleteness of the problem. The act of introduction of variable t should be made explicit. In similar fashion, the “rate of increase of the radius” is dr/dt.
To translate it further, we will need to create an algebraic expression:
dV/dt = 9 dr/dt
This expression looks complex and “all Greek” to Algebra students. However, the concept we have used while writing this expression is the same we would use to write a very simple algebraic expression “number A is nine times greater than number B”.
A = 9B
So, students should be taught writing algebraic expressions as early as possible.
Comments: