- Why should I bother learning this?
Much of the mathematics used every day is "relational" in nature -- that is, it is the relationship between the numbers which is of importance, not necessarily the numbers themselves. Some examples of these relationships are common in everyday happenings, such as 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 for miles when reading a map. These relational aspects of mathematics lead to the idea of proportions and the need for students to do proportional reasoning. For example, when planning a trip, the map 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 via division. A proportion is a statement that two ratios are equal to one another. A proportion is an equation and solving a proportion usually involves solving for some unknown in the equation.
- Why does cross multiplication work when solving proportions?
When students cross multiply when solving a proportion, they are really multiplying both sides of the equation by the products of the two denominators. What happens is that the denominators divide out with one of the factors on each side, looking like you have just cross multiplied. For example, in the proportion , when we multiply both sides by , the 35's divide out on the left side of the equation leaving . On the right side of the equation, the n's divide out leaving .