by
Bernadette Dvorscak
is awesome.
This lesson is for students in grades 5 through 7.
The objectives are:
Identify a proportion.
Test an equation to see if it is a true proportion and find the missing term in a proportion.
Use proportions to solve problems:
Find reduced scale sizes for a set of structures (skyscrapers, bridges, and tunnels).
Find proportional amounts of ingredients for a recipe.
A ratio a comparison of two quantities. The ratio of dogs to cats is 3 to 2. The ratio of gerbils to guinea pigs is 9 to 6. A proportion is a comparison of two quantities. The ratio of 3 to 2 is the same as the ratio of 9 to 6.
For every three animals of one species there are two of the other species. These two ratios are equal to each other: This is a proportion.
3/2 = 9/6
Hey! That look's like a pair of equivalent fractions. Right, fractions are rational numbers. When two fractions represent the same amount, they are equivalent fractions. Two ratios that are equivalent to each other are a proportion.
How do you know if two ratios are a proportion?
An equation such as 5/8 = 60/96 is a proportion if the two ratios are equivalent to each other. Either reduce one ratio to lowest terms or raise the other ratio to equivalent terms.
5 x 12 = 60 8 x 12 = 96 therefore 5/8 = 60/96
60 / 12 = 5 96 / 12 = 8 therefore 5/8 = 60/96
Another way to test a proportion for is to use cross multiplication.
7:3 = 28:12
Multiply 7 by 12. 7 x 12 = 84
Multiply 3 x 28. 3 x 28 = 84
Both products are 84, therefore 7:3 = 28:12
What if one term in a proportion is missing? How can you find the missing term?
One way to find a missing term in a proportion is to find the equivalent fraction.
4/9 = 96/?
4 x 24 = 96 therefore 9 x 24 = 216.
4/9 = 96/216
A second way to find a missing term in a proportion is to use cross multiplication.
27/5 = 8/?
27 x = 5 . 8
27 x = 40
x = 40/27
x = 1.48
There are many real-life applications for proportions. Some of these are:
Maps, blueprints, and scale models. An appropriate ratio, or scale, is chosen. Large objects or areas can then be shown in a considerably smaller size or space. Movies frequently use scale models of ship, planes, buildings.
Medicine dosages are frequently related to the weight of the patient, so the dosage must be increased proportionally to weight.
Recipes or formulas may be increased or decreased by using proportions.
Economic and sociological predictions based on the relationship of two quantities.
In Activity 1, you will find the reduced scale for buildings, bridges, and tunnels. For a printable worksheet, click here.
Activity 1
In this activity, you will convert the actual height of skyscrapers or the actual length of bridges and tunnels into a reduced scale of inches. The scale you will use is 50 feet = 1inch; this is the ratio of 50/1. Round all answers to the nearest hundredth. To find the actual height or length, click on the name of the structure. Remember to keep the order of the terms the same in both proportions.
Bonus: Using the data for either skyscrapers, bridges, or tunnels, create a graph comparing all the structures in the category.
(Building Big on the PBS website is an excellent resource for lessons.)
In Activity 2, you will either increase or decrease the amounts of a selected set of ingredients for a recipe. For a printable worksheet, click here.
Activity 2
Use the recipe for Salad Nicoise and determine the amount of each of the given ingredients you will need to make the quantities that are indicated.
Number of Servings
Ingredient
3
7
15
29
Anchovy fillets Balsamic vinegar Eggs Cherry tomatoes Nicoise olives Garlic
Other Resources
Other math problems that involve proportion may be found at the following sites.
Use proportions to find the quantities of each ingredient needed to make lemonade.
Use proportions to find the new dimensions of a picture whose area has been enlarged.
Find out how long it will take the math teacher to bicycle the width of Wisconsin
Compare cat steps to dog steps
Use proportion to compare the volume of two objects.
Use proportion to find data about continental drift
The following unit deals with scale, ratio, and proportion through model building.
http://www.cis.yale.edu/ynhti/curriculum/guides/1981/5/81.05.06.x.html
For information and/or activities on special effects go to the following sites:
© Bernadette Dvorscak