Chapter 10 Describing &
Changing Calculations

Volume 2, Three Skills for Algebra

Talking about Two More Skills

Besides identifying numbers and quantities or talking about them, we can also describe and change calculations which include them or which say how to compute them. Simple calculations can be described with words alone or with shorthand notation. More complicated computations need the shorthand notation. Manipulating, changing or massaging (term learnt from a fellow instructor in 1984) of calculations, is best done with shorthand notation. Any description of a calculation with or without this shorthand notation gives what is called a formula. There is more to mathematics than just doing arithmetic precisely.

Examples of Formulas and Shorthand Notation

1.1  Rectangles

The computation of the area of the rectangle can be rewritten with shorthand notation as follows. To introduce shorthand notation, we say the area A of a rectangle is given by its width W times its length L. Here, we use A as shorthand for the area of a rectangle, L as shorthand for its length and W as shorthand for its width. The formula (recipe) for calculating the area A of a rectangle can be written more briefly as

Shorthand notation provides a code for the description of calculations. Formula decoding is required. The shorthand formula A = L ·W is more compact (takes less room) than the word-only description. This formula is meaningless for us when the role of the letters in this shorthand description is not explained. To understand and to use the shorthand description or formula, you need information. You need to know or find what numbers or quantities the symbols mean or represent. In the above formula, L stood for the length of a rectangle. This has to be said to you or you have to ask. To anyone without this information, the formula remains mysterious.

Talking about and describing computations almost gives us the power to do them. In the area calculation, the area A is obtained from the recipe A = L×W provided the length L and W are given or can be found. Without this information, we can describe or understand a calculation but not use it. The above rectangle example reminds us of the following:

1. We can talk about quantities or numbers without doing any arithmetic. We can speak about numbers and quantities even if we have not measured them or do not know their values exactly.
2. We can describe calculations without performing them. This description can be done with words alone (throw out the letters) or with mathematical shorthand notation, as convenient.

1.2  Triangles

In words, the area of a triangle is given by one half the length of a base of the triangle multiplied by the height of the triangle. This formula can be justified but at this moment we will not worry about why it holds. We may also write more briefly

We have used single letters in this shorthand description of the calculation. Any mark or squiggle or symbol you can draw and name can serve as shorthand for some number or quantity.

Perhaps, we should use A_triangle or another symbol, since we have already used the letter A in the previous rectangle example. Alternatively, we adopt the following rule: while you are reading this triangle example, we use the letter A here as shorthand for the area of the triangle only. More will be said on using and reusing (recycling) shorthand symbols (for example, letters) and the roles they take. Think of them as actors which can perform many parts. They may take only one role in any scene, except for stories and scenes involving identical twins or cases of mistaken identities.

1.3  Circles

The symbol for the Greek letter called Pi is p. In words, the area of a circle is given by the number p times the square of the circle's radius. The square of a number or quantity refers to the number or quantity times itself.

¹⁰ Geometrically, the numerical value of 5-squared is the number of unit squares in a square whose sides are of length 5 units. Similarly, the value of 5-cubed is the number of unit cubes in the cube with edges of length 5 units. The ancients thought of numbers in geometric terms involving lengths, areas and volumes, and not in terms of decimal notation.

The square¹⁰ of 5 for instance is 5² = 5 ×5 = 25. We can also more briefly write

To rewrite or encode this formula in shorthand form, we will first describe the code. Let A be shorthand for the area of a circle.

¹Here we must forget the previous meanings and roles of the letter A as the area of a rectangle or the area of a triangle.

Let r be our shorthand for the radius of the same circle.

 Then the previous word-only formula for the area of a circle is written A = p·r ·r or as

A = pr²

2  Changing Calculations

The compact description of formulas using shorthand notation is useful for changing the way calculations are done. Note that when two calculations give the same result, one can be done or written instead of the other. This is the replacement principle. The rules of algebra (more precisely rules which say when two different calculations give the same result) tell us when one calculation can be replaced by another. These rules, to be seen later, are also stated or described with shorthand notation.

2.1  First Box Volume Formula

The volume of a box is given by the height times the width times the length of the box in question. More precisely,

To begin our next line of reasoning, we will group the multiplication as follows.

In shorthand notation, the volume V of a box is given by

2.2  Second Formula

The base of the box is a rectangle with area A = L·W. This gives

The symbol A and the product L ·W both represent the area of a rectangle. Here A gives the result of computing the product L ·W. The product tells us the value of A. So in describing the volume calculation, we can replace the symbol A by the product W·L, or vice-versa, as convenient.

2.3  Back to the First Formula

Our second and alternate formula for the box volume is V = H·A where A represents the base area. Suppose you met someone who accepted this alternate formula but who doubted our original formula for the volume. What can we do to convince him or her that our original formula says how to compute the volume as well? The following words may help.

To convince the person, we first recall and try to use the base area formula A = L ·W. Let's hope this is accepted. Now if some one gives us the width and length of the base, we can calculate from the rectangle area formula A = (L ·W) and then compute V using the equality V = H·A . This suggests that the original calculation V = H·(L·W) for the volume of the box because the single symbol A and the computation L·W both represent and both can be viewed as shorthand for the same quantity, namely the area of the base. So the symbol A and expression W ·L can each replace the other, whether or not the values of A, L and W are known or not.

In closing, this suggests, we can go back and forth between these two ways of computing the volume of the box. We can use whatever is the most convenient - requires the least amount of work.

3  To Find A Rectangle's Dimensions

The rectangle area formula is easy to compute if you are given the width W and the length L. But can we use the area formula A = L·W to find the width W when the area A and length L are given? The rectangle area formula says

If you multiply W by a non-zero number called it, and then divide by the same number called it you get back your original quantity W. This is a rule or property of arithmetic with whole numbers and fractions, etc. A description of these properties will be given later. So multiplying and then dividing W by the number L gives the same result as doing nothing to W. This suggests the expression [( W·L)/(L)] when calculated, gives you the width W. Now we can write W = [( W·L)/(L)]. The equality sign is used to signal that the expression on either side of it gives the same result. But the expression W ·L whenever computed is the same as the area A. So we replace the computation of W ·L by A with the understanding that A = W ·L always represents this product W ·L.

Now W can be obtained by calculating [(( W ·L))/(L)]. The latter gives the same result as [(A)/(L)]. So we have a new width formula W = [(A)/(L)] for computing W whenever L and A are given. This formula is correct if A = W ·L for every rectangle you meet. Similarly, the length formula L = [(A)/(W)] can be obtained by interchanging the roles of the actors L and W above.

4  Formulas as Potential Calculations

We have discussed or described two recipes or formulas for calculating areas and volumes without doing any arithmetic. Given the heights, lengths and widths involved, we could compute the areas and volumes. That is easy to do by hand. It is also easy or easier to use a calculator to do the arithmetic for us. Think in terms of potential calculations: formulas describe calculations that could be done (or avoided) as needed. We can postpone calculations, unless we need to do them. Note that when you see a formula for the first time, you may need to practice using it.

5  Further Readings

The following books (and others) cover ideas not included above.

1. Mathematics Made Simple by A. Sperling and M. Stuart, Doubleday 1981 edition, ISBN 0-385-17481-0.
2. Algebra, the Easy Way by D. Downing, 1989, Barron's Educational Series, Inc, 250 Wireless Boulevard, Hauppauge, New York 11788. ISBN 0-8120-4194-1.
3. How to Solve Algebra Word Problems by W. A. Nardi, Simon & Shuster Inc, Gulf+Western Building, One Gulf + Western Plaza, New York, NY 10023. ISBN 0-6680-06574-5.

6  Two Notions of What is a Variable

6.1  With and Without Symbols

Numbers and quantities which may change or vary are said to be variables. This first notion of a variable does not involve or require the presence of shorthand notation (symbols) to represent the number or quantity in question.

But there is a second notion of a variable employed in mathematics. A symbol or letter which represents a number or quantity is also be called a variable if the number or quantity concerned may change or vary, that is if the number or quantity represented is a variable according to the first notion. While a symbol or letter may be called a variable, not all variables are given or represented letters or symbols. We can talk about numbers and quantities without employing a written symbol for each one.

Remark.   A change may be required in mathematics texts and dictionaries to recognize both notions and not just the second.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science