Teaching ratios and proportions
Often, students learn how to solve proportions by memorizing the steps, but then they also forget that 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?
Ratios and proportions are NOT some way-out math stuff
Truly they aren't. We use them every day, 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? Have you ever seen unit prices, such as $1.22 per pound, $4 per foot, $2.50 per gallon? Have you ever figured how much something costs given the unit price or what is your daily or monthly pay if given the hourly rate? You've used ratios (or rates) and proportions.
What are proportions?
Consider the problem: if 2 gallons (of something) costs this much, how much would 5 gallons cost? What is the general idea to solve this kind of problem? Or, if car travels this distance in 3 hours, what distance could it travel in 4 hours? 7 hours?
In proportion problems you have two quantities that both change at the same rate. For example, let's say you have dollars and gallons. You know the dollars & gallons in one situation (e.g. 2 gallons costs $5.40), and you know either the dollars or the gallons of another situation, and are asked the missing quantity. For example, you may be asked how much would 5 gallons cost.
You can make a table to organize the information:
|Example 1:||Example 2:|
In both examples, there are two quantities that both change at the same rate. In both examples, you have four numbers (two for the one situation and two for the other), you are given three of them, and asked the fourth. How can we solve these types of problems?
The many ways to solve a proportion
- If 2 gallons costs $5.40 and I'm asked how much does 5 gallons cost, since the amount of gallons increased 2.5 times, I can simply multiply the dollars by 2.5, too.
- If 2 gallons costs $5.40, I first figure how much 1 gallon would cost, and then use that to figure out the cost of 5 gllons. Now, 1 gallon would cost 1/2 of $5.40 = $2.70, and then I take that times five.
- I write a proportion and solve it by cross multiplying:
Cross-multiplying from that, 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 fractions 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 the proportion or how to solve it -- you can ALWAYS solve these kinds of problems just by using common sense and a calculator. And this is the best approach to teach them to students, too: make them understand the basic idea so well that they can figure the problems out without using an equation. However, you should cover cross-multiplying, too, for completeness' sake.
One basic idea for solving proportion problems is to think about the unit rate, and then multiply 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 it goes in 1 hour, then multiply that by 4.
Or, if 1 meter costs $5.44, how about 0.40 meters? You can divide the price by 10 to get the price for 0.10 meters, then multiply that by four.
How to teach proportions
I hope by this point you understand the basic idea in these kinds of problems. To introduce them to your students, give then a few tables of equivalent rates to fill in, like the one below. This will help them learn proportional reasoning.
Make enough of these tables (first using easy numbers) so that the students get used to them. You can tie some of them in with real-life situations - check the proportion word problems in your math book for those.
Then, give students a few tables 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 filling in tables of equivalent rates, the student is ready to tackle word problems. Choose simple ones at first, and let them think! Students might very well come up with an answer on their own. They might make a table. Or, they might figure out how to calculate the unit rate and go from there.
If the student succeeds in figuring out how to do these type problems, then that method is likely to stick with them much better than a rote-memorized procedure that they don't understand why it works.
That is all there is to these proportion word problems. You don't actually need to build an equation to solve them.
Now, I don't want to put down equations or cross-multiplying; it's just that understanding should come first, and is more important. It is still needful to learn to solve proportions with cross-multiplying in algebra and pre-algebra.
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.
RATIO is two "things" (numbers or quantities) compared to each other. For example, "3 dollars per gallon" is a ratio. "40 miles per 1 hour" is another. Here are some more: 15 girls versus 14 boys, or 569 words in 2 minutes, or 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 things 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 kids equals 3 teachers per 30 kids. 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).
A self-teaching worktext for 6th-7th grade that covers ratios, proportions, aspect ratio, scaling, and various kinds of problem solving with the help of a bar (block) model.
Download ($5.00). Also available as a printed copy.
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