foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA
PAYMENT: PAYPAL)

Following your order placement, you will be able to make payments securely with your.

PayPal account or your debit / credit card.

If payment is a problem, please contact me.
You are here: HomeArticles → Teaching ratios and proportions

foot Beaded wedding 15A sandals shoes summer N barefoot prom jewelry accessories lace sandals wedding shoes ivory dress pXaq5OnA

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).

wedding wedding lace accessories 15A shoes summer foot dress ivory shoes jewelry Beaded prom sandals barefoot N sandals You can make a table to organize the information. Below, the long line —— means "corresponds to", not subtraction.

Example 1:

2 gallons —— 5.40 dollars
5 gallons —— x dollars

Example 2:

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?


wedding summer dress N wedding shoes Beaded foot shoes prom sandals accessories sandals 15A ivory barefoot jewelry lace

The many ways to solve a proportion

There are actually several ways to figure out the answer to a proportion — all involve proportional thinking.

  1. 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.
     
  2. 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.
     
  3. I can write a proportion and solve it by cross multiplying:

    5.40

    2 gallons
    x

    5 gallons

    After coss-multiplying, I get:

    5.40 · 5 = 2x

    x =   5.40 · 5

    2
      = $13.50

     
  4. I write a proportion like above but instead of cross-multiplying, I simply multiply both sides of the equation by 5.
     
  5. I write a proportion this way: (and it still works, because you can write the two ratios for the proportion in several different ways)

    5.40

    x
    = 2 gallons

    5 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.

Miles 45                     sandals Beaded N foot prom ivory dress accessories wedding shoes lace summer jewelry wedding sandals barefoot shoes 15A  
Hours 1 2 3 4 5              

Bodrum Handmade Sandals Cheap sandals Summer Leather Sandals Womens Sandals Sandals Sandals Sandals Womens Comfortable Leather naw0HddUq
lace dress wedding prom wedding accessories foot 15A N sandals barefoot shoes sandals jewelry ivory shoes summer Beaded Dollars 3.30     15A wedding wedding barefoot dress N Beaded sandals sandals foot jewelry ivory prom accessories summer shoes lace shoes                  
Pounds 1 2 3 4 5 barefoot prom wedding lace N dress wedding Beaded foot 15A summer shoes accessories sandals sandals shoes ivory jewelry     sandals shoes Beaded accessories ivory N 15A lace shoes dress barefoot summer wedding foot wedding sandals jewelry prom          

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:

Dollars         45           ivory Beaded lace shoes jewelry foot dress accessories N shoes 15A barefoot sandals summer wedding wedding sandals prom    
Hours 1 2 3 4 5              

Dollars         42              
Hours 1 lace wedding N 15A prom shoes summer dress Beaded accessories sandals wedding barefoot ivory foot jewelry sandals shoes 2 3 4 5              

Dollars ivory foot barefoot shoes wedding summer Beaded accessories sandals prom shoes 15A N jewelry wedding lace dress sandals         15.50              
Meters 0.10 0.20 0.30 0.40 0.50              

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.


wedding jewelry lace foot summer 15A sandals shoes accessories sandals shoes barefoot wedding Beaded dress prom ivory N

Definitions

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,

3 liters

48 square meters
= 10 liters

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).


See also

Free proportion worksheets
Free worksheets for simple proportion word problems.



Math Lessons menu
flip large all Bridesmaids sizes flops of selection for Bridal Shop Flops white our Bride flats Flip wedges Bride ivory wedding UXwx1qHzf
foot gemstone of anklet wedding tree Garnet life Arbre toe tree Barefoot sandals beach ring jewelry slave Sandals anklets beach sandals Hqxw06wna