Teaching ratios and proportions
Often, students learn how to solve proportions by memorizing the steps, but then they also forget those in a flash after school is over. They may remember faintly something about cross multiplying, but that's as far as it goes. How can we educators help them learn and retain how to solve proportions?
Ratios and proportions are NOT some way-out math stuff
Truly they aren't. We use them constantly, whether we realize it or not. Do you ever talk about going 55 miles per hour? Or figure how long it takes to travel somewhere with such-and-such a speed? You have seen unit prices, such as $1.22 per pound, $4 per foot, or $2.50 per gallon. Have you ever figured how much something costs given the unit price or what is your monthly pay if given the hourly rate? You've used ratios (or rates) and proportions.
What are proportions?
The following two problems involve a proportion:
- If 2 gallons of gasoline costs $5.40, how much would 5 gallons cost?
- If a car travels a certain distance in 3 hours, what distance could it travel in 7 hours?
The general idea in these problems is that we have two quantities that both change at the same rate. For example, in the top problem we have (1) gasoline, measured in gallons, and (2) money, measured in dollars. We know both quantities (both the dollars & gallons) for one situation (2 gallons costs $5.40), we know ONE quantity for the other situation (either the dollars or the gallons), and are asked the missing quantity (in this case, the cost for 5 gallons).
You can make a table to organize the information. Below, the long line —— means "corresponds to", not subtraction.
2 gallons —— 5.40 dollars 5 gallons —— x dollars
110 miles —— 3 hours x miles —— 4 hours
In both examples, there are two quantities that both change at the same rate. Both situations involve four numbers, of which three are given and one is unknown. How can we solve these types of problems?
The many ways to solve a proportion
There are actually several ways to figure out the answer to a proportion — all involve proportional thinking.
- If two gallons costs $5.40 and I'm asked how much do 5 gallons cost, since the amount of gallons increased 2.5-fold, I can simply multiply the dollars by 2.5, too.
- If two gallons costs $5.40, I first figure how much 1 gallon costs, and then multiply that by five to get the cost of 5 gallons. Now, 1 gallon would cost $5.40 ÷ 2 = $2.70, and then $2.70 × 5 = $13.50.
- I can write a proportion and solve it by cross multiplying:
After coss-multiplying, I get:
5.40 · 5 = 2x
x = 5.40 × 5
- I write a proportion like above but instead of cross-multiplying, I simply multiply both sides of the equation by 5.
- I write a proportion but this way: (and it still works, because you can write the two ratios for the proportion in several different ways)
= 2 gallons
My point is that to solve problems like above, you don't need to remember how to write a proportion or how to solve it — you can ALWAYS solve them just by using common sense and a calculator.
And this is something students should realize, too. Make them understand the basic idea so well that they can figure proportion problems out without using an equation, if need be. However, I feel you should also teach cross-multiplying as it is a very necessary "trick of the trade" or shortcut when solving equations.
One basic idea that always works for solving proportions is to first find the unit rate, and then multiply that to get what is asked. For example: if a car travels 110 miles in 3 hours, how far will it travel in four hours? First figure out the unit rate (how far the car goes in 1 hour), then multiply that by 4.
How to teach proportions
To introduce proportions to students, give them tables of equivalent rates to fill in, such as the one below. This will help them learn proportional reasoning.
Work with these tables (first using easy numbers) until the students get used to them. You can tie in some of them with real-life situations. For example, you can take a situation from a proportion word problem in your math curriculum and make an equivalent rate table from it.
As you advance, give students tables of equivalent rates to fill in where the "givens" are in the middle:
Of course the students should notice that it is easy to fill in the table if you first figure out the unit rate then find the other amounts.
The next step: proportion problems and thinking
After studying tables of equivalent rates, the students are ready to tackle word problems. Choose simple ones at first, and let them think! They might very well come up with an answer on their own by making a table or by figuring out the unit rate. So... you don't actually need to write an actual proportion to solve a proportion word problem.
However, I don't want to put down equations or cross-multiplying; students studying algebra and pre-algebra courses still need to learn to solve proportions with cross-multiplying. It's just that learning to use common sense is even more important.
Did you notice I didn't give definitions of the terms ratio and proportion? Well, I didn't want to confuse. Sometimes you don't have to learn the exact definitions up front, but you can start by learning to solve word problems — even real-life problems.
A RATIO is two "things" (numbers or quantities) compared to each other. For example, "3 dollars per gallon" is a ratio, and "40 miles per 1 hour" is another. Here are some more: 15 girls versus 14 boys, 569 words in 2 minutes, 23 green balls to 41 blue balls. Your math book might say it is a comparison of two numbers or quantities.
A related term, RATE, is defined as a ratio where the two quantities have different units. Some people differentiate and say that the two quantities in a ratio have to have a same unit; some people don't differentiate and allow "3 dollars per gallon" to be called a ratio as well.
PROPORTION is an equation where two ratios are equal. For example, "3 dollars per gallon" equals "6 dollars per two gallons". Or, 2 teachers per 20 students equals 3 teachers per 30 students. Or,
48 square meters
160 square meters
Of course, for it to be a problem, you need to make one of those four numbers to be an unknown (not given).
Ratio and Proportion is applicable to other areas in math. For example: finding the lowest common multiple of two numbers. Try LCM[62,217]. The lowest common proper fraction is relatively prime. So 62/217 = 2/7 (i.e. the ratio is 2:7. Thus one simply cross multiplies the original proper fraction by the relatively prime one. 62 x 7 = 217 x 2. Or 434 = 434. This is the LCM of 62 & 217.
Or try LCM[20, 50]. 20/50 = 2/5. 20 x 5 = 50 x 2 = 100.
LCM[15, 20]. 15/20 = 3/4. 15 x 4 = 20 x 3 = 60. Etc.
John Shotzbarger, BSEE 6/3/06