Changing Calculations

from Chapter 10, 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. Manipulating, changing or massaging calculations, is best done with shorthand notation.

 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.

  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,

The order in which the multiplication is performed does not affect the result. That is a property of or rule for arithmetic.

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