Greenwood press

22
COUNTING

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online sources for further exploration
Adventure games, permutations, and spreadsheets

Counting techniques 

English change-ringing of bells

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DERIVATIVE
. S
EE RATES
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EQUATIONS
An equation is a relationship that places equal representation to different
quantities, and is symbolized with an equals sign “ =.” All proportions are equa-
tions that are based on equal ratios. For example, Kepler’s law states that the
ratio of the cubed planetary distances from the sun is equal to the ratio of their 
squared period of revolutions around the sun, written as 
d
3
1
d
3
2
=
p
2
1
p
2
2
. (See Propor-
tions for more information about Kepler’s law on planetary motion.) Functions
can also be written as equations, because they relate two or more variables with
an equals sign. For example, the vertical height, h, of an object is determined by
the quadratic equation 
h = −0.5gt
2
+ v
0
t + h
0
, where g is the acceleration due
to earth’s gravity (9.8 m/sec
2
),
v
0
is the initial vertical velocity, and 
h
0
is the ini-
tial height of the object (see Quadratic Functions for more information about
the derivation and use of this equation). There are many other types of relation-
ships besides proportions and functions that can be represented in the form of an
equation. Some equations are bizarre and have multiple variables, making them
interesting to study or purposeful to use. Other equations simply do not belong
to a family of functions that is commonly studied in high school mathematics.
In 1622, chemist Robert Boyle showed that the product of the pressure, p,
and volume, v, of the same amount of gas at a fixed temperature was constant.
The equation to represent this relationship is 
p
1
v
1
= p
2
v
2
, where the subscript
notation represents the gas at different times. This formula indicates that as the