In this activity, students reason about distance, elapsed time, and speed in the context of two animals moving toward each other at a constant speed. 

To find how far apart the two camels are on a trail after different amounts of time, students are likely to find how far each camel travels each hour and then consider the combined effect on the distance between the two camels. To determine when the camels will meet, some students may reason incrementally about how long before the distance apart goes to 0 miles. Others may use a combined unit rate—the distance covered by both camels per hour (6.4 miles)—and divide it into the total distance (24 miles).

The last question presents students with a situation in which only one camel is moving toward the other. It prompts students to reason about whether some quantities in that situation—distances between the camels in 1, 2, and 3 hours, the elapsed time until the camels meet, and the point where they meet—could be the same as in the original situation.

To answer the questions, students need to make sense of the situation—by drawing diagrams, reasoning with tables, or using multiple representations—and persevere in solving problems (MP1).  As they interpret given quantities, make estimates, or check that their strategies and solutions make sense in context, students practice reasoning quantitatively and abstractly (MP2).

Select previously identified students to share their responses to the question of when the two camels will meet. Start with students who use multiple steps and move toward those using more efficient strategies (as shown in the student responses).

Then invite other students to share their agreement or disagreement about the three statements in the last question and why they agree or disagree.

If not mentioned by students, highlight how the unit rate of 6.4 miles per hour applies in both situations. The rate at which the distance apart shrinks each hour is the same when the two camels were walking toward each other (at 3.4 miles per hour and 3 miles per hour) and when one was walking (at 6.4 miles per hour) and the other was stationary.

In this activity, students continue to practice working with rates in the context of constant speed. They can apply previously gained insights to solve problems about two people moving away from each other. 

As in the previous activity, students can approach the problems in many ways, reason about different unit rates, and make sense of the situation by creating a drawing or diagram. No representations are given, though students may create tables—or double, triple, or quadruple number line diagrams—to support their thinking. (See student responses for examples.)

Monitor for students who use different rates to find out who—Jada or her cousin—travels faster. Students may compare:

• Elapsed time for the same distance (1 or 2 miles)
• Distance traveled in the same amount of time (12 or 24 minutes)
• Pace of each person (in minutes per mile or hours per mile)
• Speed of each person (in miles per hour or miles per minute)

When quantifying how much faster one person is than the other, some students may compare additively (7.5 more miles per hour or mile farther per minute) while others compare multiplicatively (4 times as fast). Both are valid ways to show understanding.

To find out how far apart the two people will be after a duration and when they will reach a certain distance apart, students may use individual pace or speed, or calculate the combined speed (the combined distance traveled per minute or hour). 

Specifying the units and explaining the context for a rate gives students an opportunity to attend to precision (MP6).
 

Invite previously selected students to share how they determined whether Jada or her cousin traveled faster and how much faster one is than the other. Sequence the strategies from most common to least common. If any student used a drawing or diagram to represent the situation, consider having them share first.

One key takeaway of the discussion is that students can find and use different rates or unit rates to solve problems. Some rates can be more helpful than others, depending on the question we are trying to answer. To emphasize these ideas, consider recording the rates that students mention, organizing them in a table, and displaying them for all to see:

To connect the discussion to the learning goals, consider asking questions such as:

• “We found the distance between two people who were moving away from each other. How is that different from finding the distance between a moving person and a fixed point?” (The former involves combining two individual distances and thinking about two speeds.)
• “We figured out the speed at which Jada and her cousin each traveled from the pier. Can we find out how fast they are traveling away from each other? How?” (Yes, it is the sum of their individual speeds.)
• “Which rate or unit rate may be especially helpful for answering how far apart Jada and her cousin will be after 15 minutes? Why?” (their combined speed in miles per minute, which can be multiplied by 15, or the combined speed in miles per hour, which can be multiplied by or divided by 4)
• “Which rate or unit rate may be especially helpful for answering how long it will take them to be 5 miles apart? Why?” (individual distances traveled in 24 minutes, because they are 1 mile and 4 miles, which add up to 5 miles)