Lesson 8

Students have had experience determining speed given a ratio of time and distance. This task prompts students to use more than one strategy to solve speed-related problems (minutes passed and miles traveled) and practice reasoning in multiple ways, enabling them to see the connections across strategies.

There are several ways students can calculate how many miles Andre’s dad could run in 30 minutes if traveling at a speed of 12 miles in 75 minutes. A few strategies:

• Using the speed. The ratio \(12:75\) has an associated unit rate of \(\frac{12}{75}\) or 0.16 miles per minute. To find the distance traveled in 30 minutes, multiply 0.16 miles per minute by 30. (\(0.16) \boldcdot 30= 4.8\), so he can run 4.8 miles in 30 minutes. Note that the rate per 1 associated with this unit rate is called speed.
• Using the pace. \(\frac{75}{12}\) or 6.25 minutes per mile is also a unit rate for the ratio. To find the distance traveled in 30 minutes, divide 30 by 6.25. \(30 \div 6.25= 4.8\). Note that the rate per 1 associated with this unit rate is called pace.
• Using a scale factor: Noticing that in the “time” column of a table, 75 multiplied by \(\frac{30}{75}\) is 30, and \(\frac{30}{75}=0.4\). The unknown number of miles is 4.8, because \(12 \boldcdot (0.4) = 4.8\).

• Scaling up: Noticing that, going up in the “time” column, 75 is \(30 \boldcdot (2.5)\). The unknown number of miles is then \(12 \div 2.5 = 4.8\). As students work, notice different strategies being used so that they can be represented during discussion later.

Arrange students in groups of 2 and give them 3 minutes of quiet think time, followed by sharing with a partner and whole-class discussion. Ask students to be mindful of how they are thinking about the questions and be prepared to share their reasoning. 

Invite students to share with their partner: their solution ideas and their explanations of how unit rates and scaling can be helpful. Then, give each student a new partner to repeat the process. Ask students to practice using mathematical language to be as clear as possible when sharing with the class, when and if they are called upon [see MLR 1 (Stronger and Clearer Each Time)].

Select students with different strategies to share with the class. Record their methods and display them for all to see. If any relevant strategies are missing, demonstrate them and add them to the display. Help students notice how unit rates and scaling can be helpful in solving similar problems.

Tell students when we find how the number of miles per minute or meters per second an object is moving, we are finding the speed of the object. When we find the number of minutes per mile of seconds per meter, we are finding the pace of the object.

This task asks students to answer questions in the context of constant speed. If you wish to have students engage in more aspects of mathematical modeling (MP4), have students keep their books or devices closed and only display the stem that establishes the scenario. Ask students what they notice and wonder, and select questions to answer that are established through class discussion.

In this activity, students reason about the distances between the two friends, elapsed time, and speed. Students reason both quantitatively and abstractly (MP2); students can estimate some of the solutions or check that they make sense in the given context (e.g., an earlier meeting time of the two friends would mean that one or both of them are traveling faster). As students work in groups, monitor the strategies they use to solve the last three problems, such as detailed diagrams of the path with marked-off distances, different ways of using tables, and so on.

Share some information about the system of Rail Trails in the United States, as some students may be unfamiliar with non-motorized trails. Explain to students that, since they are built on old railway lines, these trails have very little gain or loss in elevation—making it reasonable to maintain a constant speed while walking, running, or cycling. Tell students about a Rail Trail near the school, if there is one.

If you (optionally) decide to take a less structured approach and compel students to engage in more aspects of mathematical modeling, have students keep their books or devices closed and only display the stem that establishes the scenario. Ask students, “What do you notice? What do you wonder?” Record the things they notice and wonder for all to see. Select questions that students posed for the class to explore that are similar to the questions in the task.

Arrange students in groups of 3–4. Give students a few minutes of quiet think time to complete the first two questions, and then time to discuss their responses with a partner. Encourage students to look at their partner’s approach and choices (e.g., how they work out the values in the table or sets up the table, how calculations are done, etc.). Ask students to pause for a brief whole-class discussion afterwards.

To make sure students are on the right track, display at least one student solution for each of the first two problems for all to see before they move on to complete the activity. Give students 8–10 minutes to work and discuss in groups, and tell them that each group will be assigned a problem to explain to the class. Provide each group with tools for creating a visual display.

Encourage students who are struggling to make sense of the mathematics to make a picture of the path and mark off distances after certain time periods.

Look for students misinterpreting expressions of time. For example, 2.5 hours after 8 a.m. is 10:30 a.m., not 10:50 a.m.

Students who are unsure about how to calculate distance apart in the table may benefit from creating a table with 4 columns: time in hours, how far Kiran has traveled, how far Clare has traveled, and the distance between them.

Invite selected groups to present the solutions to their assigned problems. If possible, start from the most common strategies and move to the least common. Highlight effective uses of unit rates, equivalent ratios, scaling, and table representations in students’ work.

This optional activity is more opportunity to practice working with rates, in a new situation that involves constant speed of multiple people moving at the same time. This problem has less scaffolding than the previous activity. There are many different unit rates students may choose to calculate while solving this problem. Specifying the units and explaining the context for a rate gives students an opportunity to attend to precision (MP6).

Monitor for students that use different strategies to solve the problem:

• Creating a drawing or diagram that represents the situation
• Finding how far each person travels in the same amount of time (such as: In 24 minutes, Jada bikes 4 miles while her cousin swims 1 mile.)
• Finding how long it takes each person to travel the same distance (such as: To go 2 miles, it takes Jada 12 minutes and her cousin 48 minutes.)
• Calculating the pace of each individual
• Calculating the speed of each individual
• Calculating the combined speed of how fast they are moving away from each other

Give students a few minutes of quiet think time to complete the two questions, then pause for a brief whole-class discussion afterwards.

Some students may confuse the meaning of the speed and pace, thinking that 24 minutes per mile is faster than 6 minutes per mile. Make sure they have labeled the units on their rates, and prompt them to consider what the words “per mile” tell us about the situation.

The key takeaway of this discusion is the idea that students can find and use different unit rates to solve the problem, so it is important to specify what a particular unit rate measures. Invite students who used different strategies to share how they solved the problem. 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. Some possible strategies to highlight include:

• Creating a double, triple, or quadruple number line diagram showing the elapsed time and distances traveled
• Finding how far they will travel in the same amount of time
• Calculating their individual paces in minutes per mile (6 minutes per mile for Jada, 24 minutes per mile for her cousin)
• Calculating their individual speeds in miles per minute (\(\frac16\) mile per minute for Jada, \(\frac{1}{24}\) mile per minute for her cousin)
• Calculating their individual speeds in miles per hour (10 miles per hour for Jada, 2.5 miles per hour for her cousin)
• Calculating the combined speed of how fast they are moving away from each other (\(\frac{5}{24}\) mile per minute or 12.5 miles per hour)

Some unit rates can be more helpful than others, depending on the question we are trying to answer. Consider asking discussion questions like these:

• “Which unit rate was most helpful for answering how far apart they will be after 15 minutes?” (their speed, either in miles per minute or miles per hour)
• “Which unit rate was most helpful for answering how long it will take them to be 5 miles apart?” (their pace)
• “How did the fact that they were traveling away from each other affect the problem?” (The total distance between them at any point was the sum of the distance each person had traveled. The rate at which they were moving away from each other was the sum of their individual rates of travel.)