Proportional Reasoning
Much of science and mathematics have a great deal to do with finding patterns.
Many scientists spend a lot of time trying to determine how various variables are related
mathematically for a given physical phenomena. If a relationship does exist then one can
make predictions for various situations. Ones does not then need todo all the possible
combinations that may not easily be done in the laboratory.
This unit is designed to help one understand some of the simpler mathematical
relationships that exist in nature. The unit will also point out some of the simple patterns
associated with the various relationships.
The following relationships will be developed:
A) Direct
B) Inverse
C) Square
D) Square Root
E) Inverse Square
F) No relationship
Patterns that will be investigated for each are:
1) Factor Effect
2) Characteristic Plot 3) Constant Ratio
4) Straight Line Plot 5) Same Factor Effect
Each of the relationships have their own patterns. Many of the patterns are
similar and easy to recall.
To better understand relationships, only two variables will be considered at one time.
a) the independent variable - the one that you can often decide upon actual values
before actually collecting data.
b) the dependent variable - the one that changes as the independent variable is
varied. The result of this variable depends on what the independent variable is.
Tables: the first or left hand column of all tables will always be the independent
variable, the second column will be the dependent. Always provide the unit of
measurement under the variable. Mathematical operations (multiplying or raising to a
power) on variables will always take place on the independent variable.
Ratios: will be set up so the dependent variable is divided by the independent variable
(∆y/∆x).
Graphs: the independent variable will be plotted on the X (horizontal) axis and the
dependent variable on the Y (vertical) axis. Always provide the unit of measurement
under the variable.
The Factor Effect
The following data pairs represent the mass and volume quantities for some material. In
this case the volume is the independent variable while the mass is the dependent variable.
A) Direct Relationship - this section illustrates patterns that show how the dependent
variable changes as the independent variable is changed.
SUMMARY: The factor effect pattern associated with the Direct Relationship is that
whatever factor change occurs for the independent variable, the same factor change
occurs for the dependent variable
B) Inverse Relationship - the following data pairs represent the Period and
Frequency of a rotating object. The frequency has been selected as the independent
variable.
SUMMARY: The factor effect pattern associated with the Inverse Relationship is that
whatever factor change occurs for the independent variable, the inverse of that factor is
the factor change for the dependent variable.
Self Quiz:
Assume that there is a direct relationship between the position a car has and the
time it is on the road. Fill in the blanks of the following tables:
Assume that there is an inverse relationship that exists between the following
two variables, x and y, which have no units. Fill in each of the blank boxes in the
following problems.
C) Square Relationship - the following data pairs represent the Area and Length
of various rectangles that have the same width. The Length has been selected as the
independent variable.
SUMMARY: The factor effect pattern associated with Square Relationship is that
whatever factor change that takes place for the independent variable will cause the
dependent to change by the Square of that factor change.
Self Quiz: Assume a Square Relationship exists for each of the following tables.
D) Square Root Relationship - the following data pairs represent the Period and its
corresponding Radius of an object that is attached to the end of a string and rotating
about a circular path. The radius is the independent variable.
SUMMARY: - the factor effect pattern associated with the Square Root Relationship
is such that whatever factor change occurs for the independent variable the dependent
variable will change by a factor which is the Square Root of that factor change.
Assume a Square Root Relationship exists for each of the following tables of data.
The Characteristic Plot
A) Direct Relationship - the following Mass-Volume data is from the Factor Effect
section and will be used here to determine the characteristic plot for a Direct
Relationship.
SUMMARY: the characteristic plot for
an Inverse Relationship is a curve of
the type illustrated above (Hyperbola).
As one of the variables increases the
other decreases
C) Square Relationship
SUMMARY: the characteristic plot
for a Square Relationship is a curve
of the type illustrated (parabola). The
Dependent variable
increases more rapidly than the
independent variable does.
SUMMARY: the characteristic plot for
a Square Root Relationship is a curve
of the type illustrated above. The
independent variable increases more
rapidly than the Dependent variable.
The Constant Ratio
Direct Relationship
-----Review------
As can be noted from the above table, the factor change for both variables is the same
and the graph has a constant slope. Is there anything else that is constant? For this
example a test of the product and ratio of the two variables will be illustrated.
As indicated in the above table the ratio is a constant value.
SUMMARY: the pattern found when there is a Direct Relationship is that the ratio of
the dependent variable over the independent variable is a constant value. In this case,
the Mass/Volume is constant.
Which of the following sets of data represent an example of a Direct Relationship?
Inverse Relationship
-----Review------
This time the factor change is not the same and the graph is not a straight line. Recall
that the factor change of the dependent is the inverse of the factor change of the
independent variable. The following illustrates a ratio of the two variables and a ratio of
dependent over the inverse of the independent variable.
The inverse Relationship also demonstrates a Constant Ratio. In this case the ratio of
the period over the inverse of the frequency is a constant.
SUMMARY: The pattern of a constant ratio continues. For the inverse relationship,
the ratio of the dependent variable over the inverse of the independent variable is a
constant.
Which of the following sets of data represent an example of an Inverse Relationship?
Square Relationship
-----Review------
The square relationship does not show a straight line plot nor does it show a factor
change that is the same for both variables. Recall that the factor change of the
dependent variable is the square of the factor change of the independent variable. The
following two ratios are tested.
The square Relationship also demonstrates a Constant Ratio. In this case the ratio of
the Area over the square of the Length is a constant.
SUMMARY: The pattern of a constant ratio continues. For the Square Relationship,
the ratio of the dependent variable over the square of the independent variable is a
constant.
Which of the following sets of data represent an example of a Square Relationship?
Square Root Relationship
-----Review------
The Square Root relationship does not show a straight line plot nor does it show a
factor change that is the same for both variables. Recall that the factor change of the
dependent variable is the square root of the factor change of the independent variable.
The following two ratios are tested.
The Square Root Relationship also demonstrates a Constant Ratio. In this case the
ratio of the Period over the square root of the Radius is a constant.
SUMMARY: The pattern of a constant ratio continues. For the Square Root
Relationship, the ratio of the dependent variable over the square root of the independent
variable is a constant.
Straight Line Graphs
Recall how the Direct relationship data had a constant ratio as well as a straight
line plot. The constant ratio has also been found for each of the relationships. Since
constant ratios can be determined for each relationship, is it possible to generate a
straight line plot using the dependent and independent variable? The answer to this
question can be found in a pattern already shown in the direct relationship results.
Those results illustrated a constant of the dependent variable over the independent
variable and also a straight line of the dependent variable versus the independent
variable. The following demonstrates the plots of the dependent variable versus the
independent variable raised to the power used to generate the constant ratio.
Inverse Relationship
Square Relationship
Square Root Relationship
Indeed straight line plots do exist for each type of relationship. Note the pattern:
The Same Factor Change
One more pattern needs to be discussed. Each relationship had a constant ratio
and a straight line plot both of which were first demonstrated in the direct relationship.
In addition, whenever the independent variable doubled the dependent variable also
doubled. Is it possible to find a quantity which as it is doubled the dependent variable
also doubles? The pattern found in the last two sections will be used to develop the
idea of a constant factor change.
Inverse Relationship
Square Relationship
Square Root Relationship
The same factor increase can be determined for each relationship. Note the pattern:
Which relationship does each of the following sets of data represent?
Proportions
When each of the following can be found for a set of data pairs
1) same factor change 2) constant ratio 3) straight line plot
scientists summarize those phenomenon by saying that a direct proportion exists
between the two variables.
The words “is proportional to” will be used several times in the following
paragraphs and will be symbolized with the symbol  (Dependent  Independent)
Direct Relationship
The results of the direct relationship are summed up by saying that the mass is
proportional to the volume or the mass  the volume. The statement is summing up
each of the following:
I. When the volume doubles so does the mass
II. There is a constant ratio of mass/volume
III. There is a straight line plot of mass versus volume.
Also the constant ratio expression (k = (mass/volume)) can also be written
Mass = k * Volume
Inverse Relationship
The results of the inverse relationship are summed up by saying that the period
is proportional to the inverse of the frequency or Period  1/Frequency
The statement is summing up each of the following:
I. When the inverse of the frequency doubles so does the mass
II. There is a constant ratio of the period over the inverse of the frequency
III. There is a straight line plot of period versus 1/frequency.
k = (Period/(1/frequency))or Period = k * (1/frequency)
Square Relationship
The results of the square relationship are summed up by saying that the Area is
proportional to the square of the Length or Area  Length2
I. When the square of the length doubles so does the area
II. There is a constant ratio of the Area over the square of the Length
III. There is a straight line plot of the Area versus Length squared
k = (Area/Length2) or Area = k *Length2
Square Root Relationship
The results of the square root relationship are summed up by saying that the
Period is proportional to the square root of the radius or Period  Radius
I. When the square root of the radius doubles so does the period
II. There is a constant ratio of the Period over the square root of the radius
III. There is a straight line plot of the Period versus the square root of the radius
k = (Period/
Radius
) or Period = k *
Radius