Three Skills
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Chapters and Appendices
Would you like to show yourself or others how to be algebra power users?
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Chapter 10
Describing & Changing Calculations
Previous Section: 10 Shorthand Notation (formulas) for calculations
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,
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To begin our next line of reasoning, we will group the multiplication as follows.
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In shorthand notation, the volume V of a box is given by
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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.
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.
2.2 Second Formula
where the letter A is our shorthand for the result of the product L
·W. But expression L ·W equals the area of the base.
Therefore, an alternate formula for the volume is V = H·A
where A stands for the result of L ·W or the area of the
base. The alternate formula can be used if the dimensions L and W
are given or measured. The alternate formula can also be used if the base area A
is given, but the values of W and L are unknown (or forgotten).
But whether unknown, known or forgotten, their product L·W must
equal the area A.
V = H ·A
2.3 Back to the First Formula
Chapter Sections:
Next Section: 10. Replacement Principle
Next Chapter: 11 Why Shorthand