Getting from Solving a Proportion to
a Proportional Relationship
The Common Core State Standards for Mathematics
(CCSSM) has a domain in grades 6 and 7 called:
Ratio and Proportional Relationships
• In grade 6 the focus is on ratios.
• In grade 7 the focus is on proportional relationships.
• But where is setting up and solving a proportion?
It is not there at all !
Even the term proportion is not there.
This means that in implementing the CCSSM, there is
a lot of work to be done connecting the following:
A. setting up and solving a proportion
(which has always been emphasized in school)
B. working with proportional relationships
(which sometimes falls by the wayside in school)
A key to the connection is the idea of a constant of
proportionality.
From physics
The gravitational force between two objects is proportional
to the product of their masses and inversely proportional
to the square of the distance separating them.
Notice the big difference in the way the term “proportional” is used
in these two cases.
The CCSSM explains how to represent a proportional relationship:
(Ratio and Proportional Relationships, 7-2c)
Represent proportional relationships by equations.
For example, if
total cost t is proportional to
the number n of items purchased
at a constant price p,
then the relationship can be expressed as
t = pn
The constant p is called the constant of proportionality of the
proportional relationship.
This CCSSM treatment is good. But a connection needs to be made
between this treatment and the familiar setting up and solving a
proportion.
equivalent ratios
constant ratio
Are these terms synonyms?
If not, what is the difference?
They are not synonyms.
The difference is a key to understanding the difference between
A.
a classical approach to "ratio and proportion"
B.
a modern understanding of proportional relationships
and
The tank problem:
total volume
= 30 liters
20 cm
exposed surface area
6 cm
volume = ?
A.
A procedural solution:
Take the 3 numbers from the problem and set up a proportion, with
the unknown volume v occupying the fourth position:
v = 30 liters
6 cm 20 cm
Then solve this equation. (Solution: v = 9 liters)
total volume
= 30 liters
20 cm
exposed surface area
6 cm
volume = ?
B.
A conceptual solution:
The basic geometry of the water in the tank is
volume = base area • depth
The cross sectional area of the tank is the same for all depths.
So this ratio is constant:
volume
depth
There are two cases of this ratio that we know about for the tank:
filled to 6 cm, and filled to the top. So we have the proportion:
v = 30 liters
6 cm 20 cm
Solving as above: v = 9 liters
total volume:
V = 30 liters
total depth:
d = 20 cm
exposed surface area
partial depth d
partial volume v
In the tank filling situation:
The volume v is proportional to the depth d.
In the tank filling situation:
The volume v is proportional to the depth d.
It is also true that
The depth d is proportional to the volume v.
In the tank filling situation:
The volume v is proportional to the depth d.
It is also true that
The depth d is proportional to the volume v.
However, according to a modern view of proportionality,
the relationship between two volumes is not proportional.
- The total volume V is not proportional to the partial volume v.
(The ratio is not constant.)
Similarly, the relationship between two depths is not
proportional.
- The total depth D is not proportional to the partial depth d.
(The ratio is not constant.)
total volume:
V = 30 liters
total depth:
d = 20 cm
exposed surface area
partial depth d
partial volume v
A.
Based on the CCSSM, a general form for expressing a proportional
relationship between variables x and y is:
y = kx
The constant k is the constant of proportionality.
B. In the tank problem, the volume v and the depth d are proportional.
So for some constant k, there should be an equation of the form:
v=k•d
What is k?
A.
Based on the CCSSM, a general form for expressing a proportional
relationship between variables x and y is:
y = kx
The constant k is the constant of proportionality.
B. In the tank problem, the volume v and the depth d are proportional.
So for some constant k, there should be an equation of the form:
v=k•d
What is k?
k= v
d
C.
The basic geometry of the water in the tank is
volume = base area • depth.
The base area of the tank is the same for all depths, so this ratio is
constant:
volume
depth
30 liters
D. Evaluated for a volume
full tank
= this is = 1.5 liters per centimeter
depth
20 cm
E.
So, the proportional relationship between volume and depth can be
represented mathematically as
v = 1.5 d
Here, k is the constant of proportionality.
Note that the units check out:
(But, there is more to say about this!)
total volume:
V = 30 liters
total depth:
d = 20 cm
exposed surface area
partial depth d
partial volume v
a.
solving a proportion
From the solution to the tank problem we have the proportion
v = 30 liters
6 cm 20 cm
Solving as above:
v = 9 liters
b.
a proportional relationship
The proportional relationship is represented mathematically as
v = 1.5 d
The constant of proportionality, 1.5 liters per cm, is found from
30 liters = 1.5 liters per cm
20 cm
a.
solving a proportion
From the solution to the tank problem we have the proportion
v = 30 liters
6 cm 20 cm
Solving as above:
v = 9 liters
b.
a proportional relationship
The proportional relationship is represented mathematically as
v = 1.5 d
c.
The constant of proportionality, 1.5 liters per cm, is found from
30 liters = 1.5 liters per cm
20 cm
the connection between (a) and (b)
The value of the ratio in the proportion (a) is
the constant of proportionality in the relationship (b).
d. a crucial point
The trouble is, the value of the ratio in the proportion is typically
not found in school problems. Therefore, the proportional
relationship is not seen.
e. a very important point:
Once the proportional relationship is represented mathematically:
v = 1.5 d
the original problem, and any other related problem, can be solved
directly:
For example: if d = 6 cm, then
v = 1.5 • 6 = 9 liters
The proportional relationship allows us to give a general solution.
f. a third important point:
Notice that in a proportion such as
9 liters = 30 liters
6 cm
20 cm
the two ratios are equivalent. This is a static situation dealing with
4 numbers.
But in a proportional relationship, the ratio is constant:
v =1.5 liters per cm
d
This is a dynamic situation dealing with variable quantities.
The difference between the two approaches is essentially the
difference between
a classical theory of proportions, which developed long before
the scientific revolution and before functions,
and
a modern approach to proportionality, used throughout the
STEM disciplines.
total volume:
V = 30 liters
total depth:
d = 20 cm
exposed surface area
partial depth d
partial volume v
a. The constant of proportionality is
volume = 30 liters = 1.5 liters per centimeter
depth
20 cm
b. In the physical tank situation, this means that adding 1.5 liters of
water increases the depth by one centimeter.
But there is a better way to look at this. Recall that:
1 liter = 1,000 cubic centimeters (cc)
So, another way of writing the constant of proportionality is
volume = 30,000 cc = 1,500 sq cm
depth
20 cm
But 1,500 sq cm is the constant cross sectional area of the tank!
The new way of writing the proportional relationship is then:
v = 1,500 • d
v (cc) = 1,500 (sq cm) • d (cm)
c. In summary:
The constant of proportionality represents an
invariant of the physical tank situation.
If the cross sectional area was not constant, there
would be no proportional relationship between
volume and depth.
5.
How does the basic mathematics of proportionality
connect to the mathematics of simple functions?
This is easy:
In general, a proportional relationship is expressed in a form
such as
y=k•x
Our example was
v = 1,500 • d
This form is the very simplest function type.
6. In general, if we know that P is proportional to Q, what sort of entities
might P and Q be?
A.
In a modern view of proportionality, as seen in STEM disciplines:
1.
P and Q are variable quantities
2.
P and Q have a constant ratio
In the tank problem, the variable quantities were v and d:
v = 1,500 • d
v = 1,500 sq cm (v and d have a constant ratio)
d
B. However, in the classical theory of proportions, when a sequence of
four numbers a, b, c, and d form a proportion
a c
=
b d
it is said that the four numbers a, b, c, and d are proportional.
Here, there are no variables.
C. In school mathematics materials, and especially in the
internet sites devoted to them, there is emerging a new and
illogical meaning of "proportional" in which it is said that the
two ratios in a proportion are proportional.
v = 30 liters
6 cm 20 cm
This makes no sense. The ratios are equivalent, not
proportional.
It is the two quantities in the ratio that are proportional.