8.1: A Train and a Car (5 minutes)
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.
Two cities are 243 miles apart.
It takes a train 4 hours to travel between the two cities at a constant speed.
- A car travels between the two cities at a constant speed of 65 miles per hour.
Which is traveling faster, the car or the train? Be prepared to explain your reasoning.
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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?
8.2: Driving on I-90 (15 minutes)
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.
Tell students that they will now use a scale drawing (a map) to solve a problem about speed of travel. Survey the class on their familiarity with highway travel and speed limits. If some students are not familiar with speed limits, ask those who are to explain.
Arrange students in groups of 2 and provide access to geometry toolkits. Give students 5 minutes to work on the problem either individually or with their partner.
In the Digital Activity, students have choices about the number of points to plot along the route and whether or not they want to draw segments. Students need to pay attention to the map legend; turning on the grid helps them see that one unit on the grid is equivalent to 0.5 miles.
Supports accessibility for: Social-emotional skills; Conceptual processing
- A driver is traveling at a constant speed on Interstate 90 outside of Chicago. If she traveled from Point A to Point B in 8 minutes, did she obey the speed limit of 55 miles per hour? Explain your reasoning.
- A traffic helicopter flew directly from Point A to Point B in 8 minutes. Did the helicopter travel faster or slower than the dri