Proportional Reasoning

What is proportional reasoning? Are “proportions” truly a different topic than “functions”? If our students master linear functions, will that enable them to reason with proportional quantities? Stated another way: Does knowing the mathematical object imply that a person can reason with that object to solve problems?

I think we, as mathematicians, tend to see proportionality as a special case of functions … or as a particular type of equation involving two ratios or rates. Why does a person need to be able to reason with proportions … or proportionality?

Take a look at this problem, which is paraphrased from my “Math – Applications for Living” course:

A car is driven at an average speed of 44 miles per hour. At this speed, the car averages 33 miles per gallon. How much gasoline does the car use per hour? Follow-up question: If the tank is full with 14 gallons of gasoline, how long could the car be driven (in hours)?

I’ve seen students struggle with this problem, which I use to highlight proportional reasoning. Each rate is a statement of proportionality … miles driven is proportional to the time, the gallons of gasoline are proportional to the miles driven, etc. Further, each statement of proportionality (rate) is equally true in two forms — as stated, and the inverted rate.

By looking at the units we need (like gal per hr in the first question), we set up the rates to provide that answer:

To the extent that our students take basic science classes, this proportional reasoning is very valuable … and has no direct connections to the concepts of functions. Bringing up concepts of input and output only complicates these problems … because each statement of a rate allows any of the quantities to be the output; identifying an ‘output’ quantity is done by looking at the nature of the output needed for the question at hand.

Instead of addressing proportional reasoning, we often ‘help’ students by teaching them keywords to indicate multiplication or division (if it says “how many pieces can you get” it is division, etc). You may have noticed how limited this approach is, because problems are phrased differently. If we look at proportional reasoning with the rates, it becomes much easier.

Proportional reasoning comes up, in a natural way, when we start studying probability. Take the classic type of problem to introduce probability:

A container has 6 red marbles, 3 blue marbles, and 1 white marble. What is the probability that a marble, chosen randomly from this container, will be blue?

The concept here is something like “the probability of something happening is proportional to the number of those ‘things’ that are in the entire group”. Many simple probabilities are based on this, as is the question of ‘drawing two blue marbles’ from this container.

Unfortunately, what many students remember about proportions has limited value (and is often mis-applied) … “cross products”. Given a proportional situation, the important thing is being able to write two rates or ratios which make the same comparison (they follow the proportionality involved); as described above, it is also important to be able to write products of rates to produce the desired units.

Although a slope is a rate, which might suggest proportionality, the use of linear functions is not proportional reasoning (especially as experienced by novice learners). The connection between proportionality and linear functions is not an equivalence; it is more of an issue of ‘shared concepts’. We should not assume that knowledge of linear functions has much to do with proportional reasoning.

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