Lesson 8

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. As students work, notice students using each strategy.

• By finding or estimating the speed of the train in miles per hour and comparing this to the speed of the car
• By finding the distance the car travels in 4 hours and comparing it to the distance the train travels in 4 hours

Give students 3 minutes of quiet think time. Ask students to calculate the answer mentally and to give a signal when they have an answer and explanation. Follow with a whole-class discussion. 

Invite students to share their strategies. Make sure to highlight different strategies, such as calculating the train's speed from the information and calculating how far the car would travel in 4 hours.

Record and display student explanations for all to see. To involve more students in the conversation, consider asking:

• Did anyone solve the problem in a different way?
• Does anyone want to add on to _____’s strategy?
• Do you agree or disagree? Why?

Here, students use a scale and a scale drawing to answer a speed-related question. The task involves at least a couple of steps beyond finding the distance of travel and can be approached in several ways. Minimal scaffolding is given here, allowing students to model with mathematics more independently (MP4).

As students work, notice the different approaches they use to find the actual distance and to determine if the driver was speeding. Some likely variations:

• Comparing the speed in miles per minute (calculating the car’s speed in miles per minute and converting the speed limit to miles per minute).
• Comparing the speed in miles per hour (finding the car’s speed in miles per minute and converting it to miles per hour so it can be compared to the speed limit in miles per hour).
• Comparing the time it would take to travel the same distance at two different speeds (the car’s and the limit).
• Comparing the distance traveled in the same amount of time at two different speeds (the car’s and the limit).

Identify students using each method so they can share later.

For classrooms using the digital version of the activity, students will be measuring with the Distance or Length tool  The tool will measure the shortest distance between two points or the length of a segment.

Students might not realize that they need to compare two quantities (either two speeds, two distances traveled in the same amount of time, or two durations of travel) in order to answer the question about speeding. Remind them that there are two potential scenarios here: the driver is obeying the speed limit or the driver is not obeying it.

Once students have found the distance between A and B to be about 8.5 miles, they might be inclined to divide 55 by 8.5 simply because 55 is a larger number. Using double number lines or a table to show the relationship between miles traveled and number of hours might be helpful, as might using friendlier examples of distances (e.g., “How long would it take to travel 110 miles? 11 miles?”).

Ask students to indicate whether they believe the driver was speeding or not. Invite students who approached the task in different ways to share, highlighting methods that focus on:

• Calculating or estimating the speed the driver is traveling (in miles per minute or miles per hour)
• Finding how long it would take to make the trip at the speed limit
• Finding how far the driver would travel in 8 minutes going at the speed limit

Display their work or record or summarize it for all to see.

Ask students if they thought any method seems more efficient than others and why. Highlight that all the methods involved finding the distance traveled, and that the scale drawing and scale enabled us to find that distance.

One method for solving this problem which avoids the decimals that approximate fractional quantities is to observe that 55 miles per hour is the same as 55 miles in 60 minutes, so that is less than one mile per minute. So it will take more than 8.5 minutes to travel 8.5 miles at 55 miles per hour, and the driver must have been speeding.

In the previous activity, students were given a map with a scale and the amount of time it took to get from one place to another. They used this to estimate the speed of the trip. In this activity, students work with the speed and a map with scale to find the amount of time a trip will take.

The main strategy to expect is to measure the distance between the two locations on the map and use the scale to convert this to the distance between the actual cities. Then students can calculate how long it will take at 15 mph.

Tell students that they will now use a scale drawing (a map) to solve a different problem about travel, this time focusing on how long it will take. 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.

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.

First, have students compare answers with a partner and discuss their reasoning until they reach an agreement.

Next, 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 students estimated the trip to be. What are some reasons for the differences? Possible explanations include:

• Measurement error
• The road is not straight and so needs to be approximated
• For students who lay out the scale over and over again to cover the distance, it is difficult to estimate the fraction of the scale at the last step

Because of these different sources of inaccuracy, reporting the distance as 50 miles is reasonable; reporting it as 52 miles would require a lot of time and measurements; and reporting it as 51.6 miles is not reasonable with the given scale and map.