RATIO, PROPORTION, AND MEASUREMENT

Ratio, Proportion, and Measurement

Ratio, Proportion, and Measurement

Introduction

Many times students do learn how to solve proportion problems in classroom setting (they manage to memorize steps), but that seems to get forgotten in the flash after classroom setting is over. Maybe they only recall unclearly certain thing about traverse multiplying, but that's as far as it goes. How can we educators help them discover and retain? (Stickney & Roman 2009: 124)

Proportions and ratios are NOT some strange way-out mathematical stuff

Truly they aren't. You use them every day, constantly, whether you recognize it or not. Do you ever converse about going 55 miles per hour? Or figure how long it takes to travel somewhere with such and such the speed? Have you ever glimpsed charges such as $1.22 per pound, $4 per foot, $2.50 per gallon or alike ones? (Sapp 2006: 63) Have you ever figured how much something costs given price per pound or per gallon etc.? Ever figured your daily or monthly pay if given hourly rate? You've utilised ratios (or rates) and proportions.

Whead covering are proportions all about?

Consider problem: if 2 gallons (of something) costs this much, how much would 5 gallons cost? What is general idea to solve this problem? (Riffe & Frederick 2005: 179)Or, if vehicle travels this numerous in 3 hours, how long could it journey in 4 hours? 6 hours? 7 hours?

In proportion problems you have two things that both change at same rate. For demonstration, you have dollars and gallons as your two things. You know dollars & gallons in one situation (e.g. 2gallons charges $5.40), and you understand either dollars or gallons of another position, and are inquired missing one. For demonstration, you are inquired how much would 5 gallons cost. You know it is "5 gallons" and are asked amount of dollars.

You can make tables to coordinate your information:

Example 1:

Example 2:

2 gallons

-

5.40 dollars

5 gallons

-

x dollars

110 miles

-

3 hours

x miles

-

4 hours

In both examples, there are two things that both change at same rate. In both examples, you have four numbers (two for one situation, two for other situation), you are given three of them, and asked fourth. So, how would we explain these kinds of problems?

Ways to Solve the Proportion

If 2 gallons is $5.40 and I'm asked how much is 5 gallons, since gallons increased 2.5-fold, I just multiply dollars by 2.5 too.  (Pickar 2004: 263)

If 2 gallons is $5.40, I figure first how much 1 gallon would be, and then how much 5 gallons. Okay, 1 gallons would be half of $5.40 or $2.70, and I'll proceed five times that. 

I build the proportion like in math book and solve by cross multiplying:

5.402 gallons

=

x5 gallons

Cross-multiplying from that, I get:

5.40 × 5 = 2x

x =

5.40 × 52

=

 

I build the proportion like above but instead of cross-multiplying, I simply multiply both sides of equation by 5. 

I build the proportion but this way: (and it still works - you see, you can build two fractions ...