Unit 1, Lesson 8
Scale Drawings and Maps (Optional)
Grade 7
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This Warm-up serves two purposes. It refreshes the concept of distance, rate, and time of travel from grade 6, preparing students to use scale drawings to solve speed-related problems. It also allows students to estimate decimal calculations.
Students are likely to approach the question in a few different ways. Monitor for students who:
Give students 3 minutes of quiet think time. Ask students to give a signal when they have an answer and explanation. Follow with a whole-class discussion.
Two cities are 243 miles apart.
It takes a train 4 hours to travel between the two cities at a constant speed.
Which is traveling faster, the car or the train? Be prepared to explain your reasoning.
Invite students to share their strategies. Make sure to highlight different strategies, such as calculating the train's speed and calculating how far the car would travel in 4 hr.
Record and display student explanations for all to see. To involve more students in the conversation, consider asking:
MLR7: Compare and Connect
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In this activity, students use a map with a graphic scale to find the distance of travel and then calculate the amount of time a trip will take. As students think about how to use the graphic scale to get the information needed to solve the problem, they are reasoning quantitatively and abstractly (MP2) and choosing tools strategically (MP5).
Monitor for students who use these approaches to determine the distance between the two cities:
Tell students that they will now use a scale drawing (a map) to solve a problem about biking a long distance. Ask students what is the farthest they have ever biked. How long did it take? Do they know someone who has biked farther or for longer? If so, how far and how long?
Keep students in the same groups. Give students 4–5 minutes of quiet work time followed by partner and whole-class discussion.
A cyclist rides at a constant speed of 15 miles per hour. At this speed, about how long would it take the cyclist to ride from Garden City to Dodge City, Kansas?
Map with a line, trending downward and right. Endpoints are Garden City and Dodge City. Scale with line segment labeled 4 miles. The distance between the endpoints is about 12 and one half of the scale line segments.
The goal of this discussion is to highlight different strategies for estimating the distance between the two cities and dealing with the fact that the road is not straight. Invite students to share how they estimated the distance between the two cities (and how long it takes the cyclist to travel this distance). Ask students to consider the different distances that students estimated the trip to be. What are some reasons for the differences? Possible explanations include:
Because of these different sources of inaccuracy, reporting the distance as 50 mi is reasonable; reporting it as 52 mi would require a lot of time and measurements; reporting it as 51.6 mi is not reasonable with the given scale and map.
MLR7 Compare and Connect. After several strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask:
Maps with scales are useful for making calculations involving speed, time, and distance. Here is a map of part of Alabama.
A map of a part of Alabama. Three points indicate three different cities on the map. The bottom point is labeled Montgomery, the middle point is labeled Centreville, and the top point is labeled Birmingham. A scale on the map shows 1 inch equals 20 miles. The distance between Birmingham and Montgomery is approximately 4 point 5 inches on the map.
Suppose it takes a car 1 hour and 30 minutes to travel at constant speed from Birmingham to Montgomery. How fast is the car traveling?
To make an estimate, we need to know about how far it is from Birmingham to Montgomery. The scale of the map represents 20 mi, so we can estimate that the distance between these cities is about 90 mi.
Table with arrows. Time, hours. Data, from top to bottom, 1.5, 3, 1. Distance, miles. Data from top to bottom, 90, 180, 60. On the left and right of the table, two downward arrows labeled times 2 and times one third.
Since 90 miles in 1.5 hours is the same speed as 180 mi in 3 hours, the car is traveling about 60 mi per hour.
Suppose a car is traveling at a constant speed of 60 miles per hour from Montgomery to Centreville. How long will it take the car to make the trip? Using the scale, we can estimate that it is about 70 mi. Since 60 miles per hour is the same as 1 mile per minute, it will take the car about 70 minutes (or 1 hour and 10 minutes) to make this trip.
Help us improve by sharing suggestions or reporting issues.
The road from Garden City to Dodge City has many twists and bends. Students may not be sure how to treat these. Tell them to make their best estimate. Measuring many small segments of the road will have the advantage that those short segments are straight but it is time consuming. A good estimate will be sufficient here.
Tell students that they will now use a scale drawing (a map) to solve a problem about speed of travel. Arrange students in groups of 2 and provide access to geometry toolkits. Give students 6–7 minutes to work on the problem either individually or with their partner.
Select students with different strategies, such as those described in the Activity Narrative, to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially from students who haven't shared recently.
A driver is traveling at a constant speed on Interstate 90 outside Chicago. If she traveled from Point A to Point B in 10 minutes, at what speed was she driving? Explain your reasoning.
Students may think that the car drove in a straight line from A to B. Point out the path of the interstate on the map. Consider asking whether this path is longer or shorter than the straight line distance.