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:

• Copy the graphic scale onto another paper and use it to mark 4-mile increments along the route
• Measure the straight-line distance between the cities with a ruler
• Measure the route with a ruler, adjusting the angle of the ruler as they go
• Use a flexible tool (for example, a strip of paper or a piece of string) to approximate the route and then straighten out the tool to measure the route’s distance with a ruler

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.

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:

• There is error inherent in the process of measurement.
• The road is not straight and so distance 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 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.

In this activity, students use a map with a graphic scale to find the distance of travel and then calculate the speed of travel. They compare two different trips and determine which vehicle was traveling faster. Minimal scaffolding is given here, allowing students to persevere in problem solving (MP1).

There are several different approaches that students could use to compare the speed of the car with the speed of the helicopter. Monitor for students who:

• Compare the time it would take each vehicle to travel the same distance.
• Compare the distance each vehicle could travel in the same amount of time.
• Compare the speed of each vehicle in miles per minute.
• Compare the speed of each vehicle in miles per hour.

Plan to have students present their strategies in order from most common to least common. 

In the digital version of the activity, students use an applet to measure a map with a graphic scale. The applet allows students to add points and line segments and to measure distances. The digital version may help students measure quickly and accurately so they can focus more on the mathematical analysis.

The purpose of this discussion is to remind students of various strategies for reasoning about situations involving constant speed. First, ask students what distance the driver traveled from Point A to Point B. Make sure students are in agreement that this distance is about 8.5 mi before continuing the discussion.

Invite previously selected students to share their method for comparing the speeds of the car and helicopter. Sequence the discussion of the strategies in order from most common to least common. (It is not necessary to demonstrate every possible method. The goal is for students to see several different ways to apply ratio and rate reasoning to analyze this problem.)

After each student shares, consider organizing their reasoning into a table of equivalent ratios, displayed for all to see. This will make it easier for students to see connections between the various methods. Connect the different responses to the learning goals by asking questions such as:

• “What do the approaches have in common? How are they different?”
• “Did this method compare the distance, speed, or elapsed time for the two vehicles?”
• “What role did scaling up (multiplication) play in each approach?”

The main takeaway is that to determine which of two objects is moving faster, you can identify which one travels more distance in the same amount of time or which one travels the same distance in less time.