Teaching percentages and proportions through cross-multiplication

Percentages can often be a difficult concept to grasp for students, particularly if they have only recently learned proportions. Fortunately, there is a very easy way to conceive of percentages, which makes use of a basic understanding of ratios. Essentially, students must be told that percentages are simply a special kind of proportion; one in which the denominator is always 100.

The key to any lesson using ratios as a launching point, and percentages as the end point, is understanding cross-multiplication. An exercise taking this approach can really make use of virtually any objects to aid in visualization, but M&Ms are always good, because they give the students a tasty reward when the less on is complete.

Tell the students to take 4 red M&Ms out of the wrapper and place them on their desk. Ask them, "How many of the 4 M&Ms on your desk are red?" The answer, which should be obvious to most students, is 4. Next, ask them to express this as a ratio, with the number of red M&Ms on top, and the number of total M&Ms on the bottom -- 4/4.

Next, ask the students to take out one blue M&M and place it with the other four. Now ask, "How many of your 5 total M&Ms are red?" Then, once again, ask the students to express the number of red M&Ms to the number of total M&Ms as a ratio -- 4/5.

The next part may be tricky for some students. Ask them, "If you had twice as many M&Ms on your desk, but the same ratio of red to blue, how many blue M&Ms would you have?" Allow them to take out as many more M&Ms as they need to solve the problem. Now the students can be shown that this can be expressed as a proportion, and can be solved using cross-multiplication: 4/5 = x/10. It is essential to show the students that no matter which individual number is unknown, cross-multiplication can be used to solve it.