## What is a proportional relationship?

The Common Core State Standards (CCSS) use the term “proportional relationship” throughout the standards.  Fortunately, the standards clearly describe what is meant by that term.  To put the definition into proper context, we need some background terms.  Based upon the definitions in the Appendix of the Ratio and Proportional Relationships Progression:

• A ratio is an ordered pair of numbers, A:B, which are not both zero.
• The value of a ratio A:B is the quotient A/B as long as B is not zero.
• Two ratios A:B and C:D are equivalent if there is a number, c, such that C=cA and D=cB.
• (Easily inferred from the Progressions document)  A ratio relationship is a collection of ratios that are mutually equivalent to each other.

A ratio A:B determines a ratio relationship C.  For example, if C is the ratio relationship determined by 2:3, then

C = {X:Y such that X:Y is equivalent to 2:3}.

In particular, C is the set {2:3, 4:6, 6:9, …} while 2:3 is a ratio in that set.  Adults often blur the distinction between a ratio and a ratio relationship; we often use the ratio 2:3 to refer to the ratio relationship.  Thus it is important for teachers to make the distinction between ratio and ratio relationship until students can derive which meaning is being used in the context on their own.  Note: Students learn what a ratio relationship is first, and use that knowledge to understand its sister term proportional relationship.

Notice that the set C is also equal to the set:

C = {x:y such that y = (3/2)x},

where k = 3/2 is called the constant of proportionality.  When a ratio relationship is thought of in this way it is called a proportional relationship.

A set of ordered pairs is a proportional relationship if there is a number $k$ so that every ordered pair $(x,y)$ in the set satisfies the equation $y=kx$.

Finally, let’s compare the use of the terms proportion” and “proportional” in the CCSS and the progression documents.  The term proportional by itself sets up a proportional relationship.  Here are some examples of how that is done:

• It specifies a ratio A:B, which can then be used to specify a proportional relationship.
• It specifies a the constant of proportionality, which then determines a proportional relationship.
• It doesn’t specify a ratio or a constant of proportionality, but does state that a proportional relationship exists.  For example, “The number of total number of candies is directly proportional to the number of boxes b.

proportion, on the other hand, is an equation stating that the values of two ratios are well-defined and equal.  Like all equations, a proportion involving only numbers (no variables) can be either true or false.  So, thinking of ratios, 2:3 and 3:5, the proportion “2/3=3/5” is a perfectly acceptable false proportion!  In solving problems, however, we are interested in finding solutions to equations, that is, finding the value of x for which 2/x = 3/5 is true.

CHANNEL: Engineering School Mathematics