A proportion is a statement of equality between two ratios. It is an equation whose members are ratios. A ratio is a comparison of any two quantities. The ratio of 2 to 4 can be stated as 2 out of 4, 2:4, or . A ratio in which the comparison is between quantities with different units is a rate. Miles per hour, miles per gallon, square yards covered per gallons of paint are all examples of rates. When the rate is expressed as a comparison between x units and 1 unit, the rate is a unit rate. A speed limit of 55 mph is an example of a unit rate. The comparison is .

To develop the concept of proportion, begin by investigating simple problems built around ratios as factors of change or unit rates. The idea of a ratio as a rate is the underlying concept behind proportional reasoning. Proportional reasoning provides the underpinning for algebra and beyond. Change and rate of change is a key concept for all algebraic reasoning. Let's explore ratios in both contexts—as factors of change and as unit rates.

Three apples cost 70¢. How much will a dozen cost?

Three dozen pencils cost $0.72. How much will 7 pencils cost?

In both examples it is important that the method used is determined by the compatibility of the numbers involved. Also note that when we are comparing quantities using ratios as factors or unit rates, we are using a multiplicative comparison rather than an additive comparison. Most students up to this point have thought of comparisons as additive. For example:

In October there were 300 girls and 250 boys on the honor roll. In November there were 350 girls and 300 boys on the honor roll. Did the number of girls or the number of boys grow more?

If you are making an additive comparison, you could say they both grew by the same amount—50 students.
Based on a multiplicative comparison, the rate of growth for girls on the honor roll was and the rate of growth for the boys on the honor roll was . Based on the greater rate of growth, the number of boys on the number roll increased proportionally more because . Both answers are correct. The second answer, however, shows a proportional increase.

Students may have more experience in their mathematical background using additive comparisons and will need to investigate many examples to feel comfortable with multiplicative reasoning. Often times we use proportional reasoning and not realize it. More examples of proportional reasoning include finding the missing side with similar triangles, converting measurements, and using scales to enlarge or reduce a shape.

An important property of proportions is that , if and only if a • d and b • c. Is ?
Is 14 • 3 = 21 • 2? Yes, 42 = 42. We can use this property when determining a missing number in a proportion. This property is commonly called the Cross-Product Property.

If the ratio of girls to boys in a math club is 2 to 3 and there are 21 boys, how many girls would you expect to be in the club? We will solve this problem using the Cross-Product Property.

A 2-ounce bag of M&M's contains 4 red M&M's. Using this ratio, how many red M&M's would you expect to be in a 16-ounce bag?

Method 1: Cross-Product Property

Method 2: Factor of Change

The scale on a map is 1 inch : 20 miles. The distance between your house and grandma's house is 5 inches on the map. How far, in miles, is your house from grandma's house?

Method 1: Cross-Product Property

Method 2: Factor of Change

The Smith family is traveling cross country. They drive 396 miles in 9 hours. At this rate, how far will they drive in 24 hours?

Method 1: Cross-Product Property

Method 2: Unit Rate

If a horse gallops 3 miles every 6 minutes, how many miles will he gallop in 30 seconds?