# Lesson 10

Measuring Long Distances Over Uneven Terrain

## 10.1: How Far Is It? (5 minutes)

### Optional activity

Students have experienced measuring short distances with a ruler or a measuring tape. In this activity, students start to think about how they can measure longer distances over uneven terrain. This activity is intended to set the stage for the upcoming activities, not to completely resolve the question. Students have an opportunity to think about the limitations of methods that may work for short distances but not for long distances. They also consider real-world situations that involve the measurement of long distances.

### Launch

Arrange students in groups of 3–4. They will stay in these groups throughout this 4-lesson unit. Ask students how they have measured the length of objects in school (with a ruler, yardstick, or measuring tape). Where else in real life do people measure distances, especially longer ones? Brainstorm some situations together (distance driven in a car, length of a garden fence, length of a hiking trail, etc.). Give students 2–3 minutes of quiet work time, followed by small-group discussion.

### Student Facing

How do people measure distances in different situations? What tools do they use? Come up with at least three different methods and situations where those methods are used.

### Anticipated Misconceptions

Some students may get stuck thinking about classroom situations. Prompt them to think about other situations outside of school, such as driving in a car, measuring the distance between cities, measuring the length of a fence around a yard, etc.

### Activity Synthesis

Invite students to share some ideas of how to measure with their group.

Speaking, Representing: MLR8 Discussion Supports. Give students additional time to make sure that everyone in their group can explain all three different methods and situations they created. Prompt groups to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to speak and clarify their thinking. This will also help students improve the quality of their explanations during the whole-class discussion.
Design Principle(s): Optimize output (for explanation)

## 10.2: Planning a 5K Course (10 minutes)

### Optional activity

In the previous activity, students started to think about how to measure distances in different situations. The activity introduces the context of designing a course for a 5K fundraising walk. Students will continue working with this context in future lessons. In this activity, they come up with a method for measuring the walking distance of a path that is too long to measure with a measuring tape. Students will try out their method in the next activity.

Students get a chance to engage in many aspects of mathematical modeling (MP4). The modeling cycle starts with formulating the question that we want to answer, clarifying which quantities are involved, and how to measure them. In many problems, this step is done for students. Here, we are giving them the opportunity to think about how to set up the problem and what tools are appropriate to measure distances.

### Launch

Keep students in the same groups. Provide access to measuring tools, such as yardsticks, meter sticks, and tape measures. Ask students if they have ever participated in or watched a walk-a-thon or race. Explain that sometimes a race is done by repeating a shorter course several times, e.g. a mile is about 4 laps around a track. For this activity, they should plan for a course that is about 500 meters long that walkers can go around multiple times.

Give students 5–6 minutes to work with their group.

Conversing: MLR8 Discussion Supports. When preparing a plan for measuring the course, invite students to use a sentence frame such as: “One method for measuring the course is . . . ." Encourage students to consider what details are important to share and to think about how they will explain their reasoning using mathematical language. This will help students to converse while they design the course and decide how to measure its distance.
Design Principle(s): Optimize output (for description)

### Student Facing

The school is considering holding a 5K fundraising walk on the school grounds. Your class is supposed to design the course for the walk.

1. What will you need to do to design the course for the walk?

2. Come up with a method to measure the course. Pause here so your teacher can review your plan.

### Activity Synthesis

Students check with the teacher about their method of measurement and then move on to the next activity.

## 10.3: Comparing Methods (20 minutes)

### Optional activity

In this activity, students use the method they came up with in the previous activity to measure the length of a path chosen by the teacher. Each group can begin working on this activity as soon as they have finished the previous activity and checked in with the teacher.

It is not important that students’ results are very accurate. They will measure the distance again with a trundle wheel in a later lesson. The main point of this activity is to think about measurement methods and to discuss the advantages and disadvantages of different methods.

### Launch

Keep students in the same groups. Provide access to measuring tools. Show students the path they should measure. Give students time to measure with their group.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer for data collection and organizing information about methods, lengths and average between two measurements.
Supports accessibility for: Language; Organization

### Student Facing

Let’s see how close different measuring methods are to each other. Your teacher will show you a path to measure.

1. Use your method to measure the length of the path at least two times.
2. Decide what distance you will report to the class.
3. Compare your results with those of two other groups. Express the differences between the measurements in terms of percentages.
4. Discuss the advantages and disadvantages of each group’s method.

### Anticipated Misconceptions

Some students may need to be reminded how to use the measuring tools accurately, such as starting at the 0 mark and keeping the measuring tool going in a straight line.

### Activity Synthesis

Invite the different groups to share their solutions. Ask them to:

• Compare how close their answers are.
• Compute the approximate relative error (difference/total length).
• Discuss the advantages and disadvantages of their methods and sources of discrepancies in their measurements, and how a small error can propagate. ​

The takeaway should include:

• We can use proportional reasoning to find longer distances. If we know it takes 10 steps to walk 8 meters, then it will take 20 steps to walk 16 meters.
• Small errors can magnify over longer distances.
• Methods were either not very precise (prone to introduce error), or they were precise but cumbersome to implement.
Representing: MLR3 Clarify, Critique, Correct. Display an incorrect statement about the percentage difference between measurements such as: “One group compared their measurement of 1000 meters to another group’s measurement of 992 meters and calculated the percent difference as 8%.”. Prompt students to clarify and critique the error, and then write a correct version. This helps students evaluate, and improve on, the written mathematical arguments of others.
Design Principle(s): Cultivate conversation; Maximize meta-awareness