Proportionality (mathematics)
Proportionality in mathematics refers to the relationship between two ratios that are equivalent, expressed as a proportion. For example, if the ratio of two quantities is represented as a fraction, like 2/7 being equal to 6/21, this indicates proportionality. Understanding proportions is essential in everyday life, such as when calculating travel times based on speed or determining the better value between different product sizes.
In practical scenarios, such as comparing cereal prices, one can use unit rates to identify the cheaper option by analyzing the cost per ounce. A key concept related to proportionality is the constant of proportionality, which represents the ratio that connects two proportional quantities, like hourly wages.
Proportional thinking also applies to qualitative situations where specific quantities are not provided, allowing for logical reasoning, such as spacing nails on a board based on length and quantity. While proportional thinking is useful in many contexts, it’s important to recognize situations where it may lead to incorrect conclusions if the quantities are not consistently proportional.
Subject Terms
Proportionality (mathematics)
A proportion is a statement that two ratios are equal. For example, 2/7 = 6/21 is a proportion. Proportions also show a relationship between two ratios.
Proportions are very important in everyday thinking, for example, realizing that it has taken you 15 minutes to drive 10 miles can allow you to predict that it will take you 1 hour to drive 40 miles at that same rate. Proportional thinking is also used to determine the better value when buying food items, as when comparing a 20-ounce box of cereal that costs $2.00 with a 28-ounce box that costs $3.25. In this example, the larger box is more expensive by 50%, yielding 30 ounces for $3.00. The relative cost, $2.00 for 20 ounces, is in proportion to $3.00 for 30 ounces. By comparing this new ratio ($3.00 for 30 ounces) with the 28-ounce box, it becomes apparent that the $2.00 box is the better deal (more cereal for less money). A more traditional way of solving this problem is to find a unit rate for each box of cereal. In this case we get:
$2.00/20 ounces = 10 cents/ounce
$3.25/28 ounces = 11.6 cents/ounce
It is important to note that in the above situation the two boxes ware not in proportion regarding the ratio of cost to amount. If they were, then neither of the boxes would have been a better deal. However, because the ratio of the 20-ounce box was less than that of the 32-ounce box, the ratios were not in proportion, and one was a better deal.
Ratios
A ratio is a relationship between two quantities. Ratios convey the relative sizes of the two quantities in a particular situation. For example, a particular room could contain a ratio of 5 men to 7 women. We usually write this ratio as 5:7, or 5/7. A rate is a ratio that can be extended to a broad range of situations, such as 70 miles per hour. There are several ways of articulating a ratio with words; the ratio a/b could be said as "a to b," "a per b," "a for b," "a for each b," "for every b there are a," "the ratio of a to b," and "a is to b."
Constant of Proportionality
A key aspect of proportional situations (when the two ratios are equal) is identifying what is known as the constant of proportionality. This is a ratio that describes the way in which two quantities that are in proportion are related. Hourly wages provide most workers with experience of a constant of proportionality. For example, a rate of $10.00 per hour, 10 is the constant of proportionality that relates the two quantities of hours worked and money earned. Table 1 shows various amounts of money made by working various amounts of hours, and it is clear that the two columns are related by a factor of 10. This shows how the two quantities are in proportion.
Another way of seeing this is to take any related pair of numbers in the table (e.g., $40 and 4 hours) and find their ratio—the result will always be the same, in this case 10, the constant of proportionality.
Many situations allow the use of proportional thinking. These are situations where the ratios between two quantities change in a constant manner (in other words, where a constant of proportionality exists). For example, a 16-ounce bottle that is 30% alcohol in water has a ratio of alcohol to water that is 3/7 (30% alcohol, 70% water). Removing a spoonful would leave the ratio as 3/7, as this mixture is assumed to be uniformly mixed. Note that 3/7 is the constant of proportionality in this situation. If an office has a total computer supply of 9 Macs and 15 PCs, the ratio of Macs to PCs is 3/5. If 8 computers were removed from one of the cubicles, it would not be possible to tell the ratio of Macs to PCs for this selection without knowing if the computers in the cubicle are in the same proportion as those in the office as a whole. Therefore, while proportional thinking can be used in many daily situations, there are some situations for which proportional thinking may lead to false conclusions.
Qualitative Proportional Thinking
Proportional thinking can also be used in situations that do have specific quantities. These are often called qualitative proportional situations, as the comparisons are made without the use of numbers. For example, suppose Bill and George have two pieces of wood, and that Bill’s piece is longer. They each need to hammer nails into the board in a straight line so that the nails are equally spaced apart. If Bill has fewer nails to hammer than George, then qualitative proportional thinking can be used to tell which board will have the nails farther apart. Specifically, because Bill has fewer nails and a longer piece of wood, he would need to space them out more than those on George’s board. This idea of more for less is a common way of thinking about these kinds of situations, which are not proportional, but use proportional thinking.
Bibliography
Brousseau, Guy, Virginia Warfield. Teaching Fractions Through Situations: A Fundamental Experiment. New York: Springer, 2013.
Ercole, Leslie K., Manny Frantz, and George Ashline. "Multiple Ways to Solve Proportions." Mathematics Teaching in the Middle School 16.8 (2011): 482-490.
Lobato, Joanne, and Amy B. Ellis. Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics: Grades 6–8. Reston, VA: National Council of Teachers of Mathematics, 2010.