Lesson: Proportions
Introducing the Concept
Your students will have seen maps in their classes and will have seen their parents use a map when they traveled. Discuss with students how real distances that are large are represented by short distances on a map. However, they can use the map to figure out how far apart those places are by measuring the distance on the map, and using proportions. A day spent using a map to find distances would be well spent.
Materials: several copies of Student Map Worksheet (PDF file); overhead ruler; ruler for each student
Preparation: Make enough copies of the map to give each student a copy. Make an overhead transparency or a large copy to put on the board in the front of the class.
Prerequisite Skills: Students will need to be able to measure distances with a ruler. They will also need to be able to find equivalent fractions by multiplying the numerator and denominator by the same number. Students should also be familiar with solving equations.
Put your overhead transparency map on the overhead projector or the large map on the board up front.
- Say: We are going to use this map to find out how far it is between these cities, traveling on the roads indicated.
Pass out a map and a ruler to each student or to pairs of students. - Ask: Using this map, how far is it from Franklin to Hamilton?
Have the students measure the distance with their rulers and have someone come to the overhead to measure it so all can see. The distance should be 5 inches. - Ask: If every two inches on the map represents 24 miles, how can we find how far is it from Franklin to Hamilton?
If a student suggests setting up a proportion, have him or her come to the board and set it up. If no student suggests setting up a proportion, set one up. The proportion would be
, where n is in miles. - Say: Now we need to find a value for n. How do you solve a division equation?
If students don't remember to multiply both sides of the equation by the same number, remind them. - Say: Let's multiply both sides by 24n.
What will we get on the left side of the equation? (2n).
What will we get on the right side?(24 x 5, or 120)
So what equation do we have now? (2n = 120)
Point out that this is the same as multiplying the numerator of the left side of the equation by the denominator of the right side and setting that product equal to the product of the other numerator and denominator. - Ask: Who can solve this equation? (n = 60)
How far apart are the cities? (60 miles) - Say: Now solve for the other distances between the cities indicated on your maps. Be sure to give both the distance in inches on the map and the distance in miles on the road.
After students have found their answers, discuss the answers with the students to see if they make sense. The distance that was 3.5 inches on the map, in miles, should be halfway between the distance that was 4 inches and the distance that was 3 inches.