## Proportions: When Students Ask

• Why should I bother learning this?
Much of the mathematics used every day is in the form of rates and proportions. Some examples of everyday use of rates and proportions are the price of gasoline per gallon, the speed of a bike in miles per hour, the rate of pay per hour, and the number of inches to miles when reading a map. These examples lead to the idea of proportions and the need for students to do proportional reasoning. For example, when you plan a trip, the map you use may have a scale in which 1 inch equals 12 miles. Thus, if the distance you plan to travel is 8 inches long on the map, the actual distance is 96 miles. If you can average 50 miles per hour, you will get there in a little less than two hours. Encourage students to come up with other ways they might use proportions.
• How do ratios and proportions differ?
Students will often mix up ratios and proportions. A ratio is a comparison of two quantities by division. A proportion is a statement that two ratios are equal to one another. A proportion is an equation. You solve a proportion to find the value of an unknown in the equation.
• Why does cross multiplication work when solving proportions?
When students cross multiply while solving a proportion, they are really multiplying both sides of the equation by the products of the two denominators. What happens is that each denominator cancels one of the factors on each side, so that it seems as if you have just cross multiplied. For example, in the proportion , when we multiply both sides by , the 35s divide out on the left side of the equation leaving . On the right side of the equation, the ns cancel, leaving .