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Building a Mathematics Model

Disclaimer:

The comments in this essay are restricted to the simple kinds of application problems studied in elementary algebra classes.  These problems lead to one or more equations (or inequalities) to be solved.  Problems requiring calculus or more advanced mathematics are not considered here.

I. Introduction:

There are three phases to solving application problems with mathematics.  The first phase is to translate the application problem into a strictly mathematics statement(s).   These statements are called the mathematics model and will be equations or inequalities.   The second phase is to solve the equations or inequalities.   The third and final phase is to translate the solution into a meaningful response to the application problem.  The first phase is by far the most difficult of the three, phase three is generally not too difficult, and the second phase is easy provided the solver has learned a bit of elementary algebra.  At this point in the essay we will cease writing “or inequalities”.
When variables are assigned to the quantities in a particular problem and an equation is derived, the problem has been divorced from the particular application and is at that point a purely mathematical problem. All peculiarities of the application are stripped away so as not to confuse or interfere with the solution process.
The algebraic skills required to solve the equations are the subject of courses like Elementary Algebra, Intermediate Algebra, College Algebra, Trigonometry, and PreCalculus.   Throughout these same courses application problems (the dreaded word problems) are presented.  The traditional presentation frequently results in a hodge-podge of ideas and methods rather than a coherent view of the process.  This essay is an attempt to teach a more unified approach to creating the mathematics model.

II. The Model is a Known Formula

In many instances the model is a well established formula.  In such cases, the information in the problem must be interpreted in terms of the variables in the formula.  The known values are substituted for the appropriate variables and the resulting equation is solved for the unknown variable.  For example, to determine the height of a 24 cu.ft. cylinder with radius 2 ft., the radius and volume are substituted into the familiar formula V = πr²h to obtain 24 = 4πh which is solved for h to obtain .

Of course a well established formula is completely worthless to the investigator if he/she is not aware of the formula and the meaning of the variables in the formula.  An educated adult is aware of and understands many basic formulas about geometric shapes as well as formulas like d= rt and is therefore expected to be able to answer simple questions involving those formulas.  On the other hand, most people do not know much about resistors in electrical circuits.  So if the question is what single resister is equivalent to three resistors in parallel, the solver must find the proper formula.  A simple Google search for “resistors in parallel” quickly produces the desired formula and the question is easily answered.

Formulas Illustrate the Detached Language Nature of Mathematics.

One reason mathematics is used to answer questions in every discipline is because it is a detached language. Mathematics is called a detached language because when a problem from any discipline has been modeled with mathematics, all reference to the original discipline has been removed. The model is detached from the peculiarities and imprecision of language and the discipline itself. The question can be answered without reference to the original discipline. Consider the following incomplete list of very simple formulas together with a very brief explanation of their meanings in the discipline where they arise.

When viewing the formulas in the left column, it should be clear that in each case one quantity is equal to the product of two other quantities.  From a mathematical perspective it doesn’t make any difference whether the quantities are mass, resistance, percent, or index of refraction. From a mathematical perspective all of those formulas could be replaced with z = xy or A = BC or l = sx. The only thing the formula expresses is a mathematical relationship between the quantities. The formula is detached from the particular discipline which gave rise to the formula. With an elementary understanding of equations, one can see that in each of the above cases if two of the quantities are known the third can be calculated.

For example, you do not need to know the physics of springs or the research work done by Robert Hooke (1635 – 1703) to determine the amount of force required to stretch a spring with a known constant a prescribed distance. You simply substitute the known numbers into the formula Hooke so generously left for us and compute the product.

It certainly is not true that all mathematical models are as simple as those listed above, but the principle remains the same, the mathematical model is detached from everything non-mathematical.

Problems involving percent deserve special attention because they are undeservedly the most misunderstood and maligned problems in our society.

Percent means parts per hundred. The word comes from the Latin phrase per centum, which means per hundred. In mathematics, we use the symbol % for percent. This definition leads us to conclude that percent is a rate just as speed is a rate. Compare the following two statements:
      60 mph means 60 miles per 1 hour
      60%     means 60 parts per 100 parts
Percent problems are no different than distance, rate, time problems.

Some confusion is created by careless and incorrect use of the words percent and percentage.  The correct use of these words is defined by the formula:

Percentage = (Percent)(Base)

Percent is the rate while percentage is the number obtained by multiplying the base by the percent.

If you remember this formula and the meaning of the terms, then a “percent problem” is as easy as a distance, rate, time problem.

The major cause of confusion about percent problems is caused by the multitude of goofy methods of teaching the subject. A cursory examination of books or of Google will uncover many different ways to present the topic of percent. While a good many of these methods will produce “the right answer” very few will produce an understanding of the concept. Moreover, they invariably unduly complicate the process.

III. The Transitive Property of Equality

The Transitive Property for equality of real numbers states that if a, b, and c are real numbers such that a = b and c = b, then a = c.

A less cryptic statement of this property might be: If two real numbers a and c are individually equal to a third number b, the original two numbers a and c are equal.

An even less precise, but correct, statement of this property which is quite useful for constructing mathematical models is:

It two expressions represent the same quantity, the two expressions must be equal.

Subject to the conditions of the very first sentence of the Disclaimer section of this essay, The key to every “word” problem, or “application” problem, or “modeling” problem is this interpretation of the Transitive Property.  However, whenever the model is a well established formula it is wise to use the formula (what was a key step in deriving that formula? – make a guess).

IV. Building a Mathematical Model

Translating a verbal statement into an equation is always accomplished by finding two algebraic expressions for some single quantity in the verbal statement. Since these two expressions represent the same quantity, they are equal. Expressing that relationship (equality) with an = symbol produces the desired equation (model).

It is important for you to understand that all “translations” must be from normal English into the language of Algebra.  The final statements must be algebraic statements which lend themselves to algebraic manipulation.  For example it is not good enough to say p is 3 times as long as w.  You must write p = 3w.

Sketches help to organize your thinking.  Sometimes those sketches are actual drawings of the situation and in other cases the sketches may simply be diagrams (Venn, hierarchy, flow, tree, network, maps, mind map, etc.) to show relationships.  A good definition (not stipulative) of diagram is:
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique.

Problem Solving Strategy:

• Understand the problem.
• Read and reread the problem understanding every word.
• Make sketches and drawings even if only symbolic.
• Assign a variable to the quantity to be determined.
• List the known quantities.
• List the unknown quantities.
• Observe/list relationships between quantities.
• Determine which of the unknown quantities is to be determined.
• Express all other unknown quantities in terms of this variable.
• Translate the problem into an equation.
• Use known and unknown quantities to express some single quantity in two different ways.
• Since these two expressions represent the same quantity, they are equal.
• Solve the equation. (Forget about the application problem during this step)
• Interpret the solution in terms of the problem.
• State the conclusion.