INCHES AWAY
Grades 7-10

Students are anxious to become drivers but few students understand how to use highway or street maps to plan an itinerary. This lesson introduces students to maps, scale representations, ratios and cartography.

MATHMEDIA: Module 8
FUTURES WITH JAIME ESCALANTE: Cartography

Students will:

• define and give an example of ratio
• demonstrate how to use ratios on maps to determine distance
• determine distance between two points on a map

• several copies of maps of your area showing city streets or general city areas
• ruler
• pencil
• paper
• calculator
• a transparency of the map
• overhead projector
• "Inches Away" student sheet

Ask students how far it is from their house to a friend's (or from their city to another local city or town.) Discuss ways they can determine distances using a map.
Lead the discussion into "scales", and how 1 inch on the map might equal ten miles in reality. (Check your map for the actual scale and use those figures in your explanation.)
Pick two points on the transparency of the map, and measure the distance in inches. Ask students to determine the distance.

Record student responses in equation form on the overhead.
Example: 1 inch = map distance
10 miles actual distance
Ask students to predict the distance if the scale was 1 inch = 20 miles (20 miles.)
Next, move the projector further from the screen to enlarge the image. Then go to the screen on which the image is projected and measure that distance. Let's say it is 1 foot. Then ask "What is the actual distance between these two places? Has it changed?" They will respond "No."
Explain that the measured map distance has changed but not the actual distance between the points, even though they look farther apart and we are taking measurements from the screen instead of the transparency.
Lead students in a discussion of why the actual distance is the same but the numbers have changed. A ratio provides for a constant relationship between quantities so that when one quantity changes, so does the other one. When the scale line equals 1 inch, the distance between the two points is 10 times the length; however, when the length of the line increases as a result of enlargement, the distance between the two points increases proportionately.

Ask students what they would need to know to determine how long a trip between the two points would take. (You need to know the speed.)
Discuss with students the use of maps for determining distance between two places and the length of time trips take.

Each Mathemedia video module is divided into three sections. CUE tape to the third section by using the fast forward function. It features teenagers who must decide how far it is to a chosen destination, and whether they can make it there in a given amount of time.
Questions are asked of the audience throughout the video. Be prepared to pause the tape and allow for discussion and written answers. This fragmenting technique focuses students' attention and provide immediate validation of prior understanding.
Pass out the student handout. PAUSE tape as the video poses these questions:
1. If 1 inch represents two miles, what do 6 1/2 inches represent? What will the driving time be? At 35 miles an hour, how much time will they spend driving in the city? What calculation is used? Discuss the answer.
What other factors determine driving time? (traffic, road conditions.)
RESUME video.
PAUSE:
2. Can they make it in 2 1/2 hours? Should they go? What other factors would affect whether they should go or not? (Detours, weather, uphill driving.)
Chuckie and Brian used a ruler to measure the distance on the map. When would this be a good tool to use and when would it not be? (With straight roads it would be fine, but with others it would not be as accurate.)
View "Cartography" from "Futures with Jaime Escalante." Jaime's class learns about various careers in the field of cartography.
Discuss the varied skills necessary to create maps. Direct students to list different careers in which cartography skills are used.

On the map you have provided, create questions which require students to determine the actual distance between pairs of points that you have already measured. Assume a given speed and challenge students to calculate how long each trip might take.
Student will record there calculations and answers on the student sheet.

Plot the itinerary for your family's dream vacation or a trip across the United States. Using maps, determine the short route and second longer route by car. Calculate your mileage and travel time. Then contact the American Automobile Association and request verification of your information. If you or your family are members, AAA will provide you with detailed maps and itinerary. See if you can improve on their recommended travel plans.

1. Find a street map of your home and school. Draw an enlarged (or smaller) scale map of this same area.
2. Log onto the Internet and search for MAPS, CARTOGRAPHY, or USGS (United States Geological Survey.) Investigate these web sites to see which types of maps are available (topographical, weather, street, relief, travel, earthquakes, satellite) and the type of units of measurement that are used. Find out what skills are necessary to be a map maker.

Master Teacher: Randall Lam