Math Strand

Come Scale Away

Lesson Overview:

Proportional reasoning is important for the development of algebraic thinking. A critical goal of middle school mathematics is to connect experiences of everyday life with fundamental ideas of proportionality. In this activity, students estimate distances, craft ways to check the accuracy of their distances and then create a map on graph paper that represents the area for which they estimate and check distances. With the help of the maps, students develop a mini-course modeled after a full-fledged orienteering course in the same area.

In the course of the lesson, students are likely to become aware of the fact their personal perception of distance is skewed from the actual distance between landmarks. Students may have trouble making reasonable estimates in these situations. With practice, however, students improve their estimate and develop a practical, lifelong skill.

Connection to Core Lesson:

The core lesson (Steppin' into Fitness to Find an Energy Balance) involves making many calculations related to time and distance. It creates a scenario in which students figure out how long it might take to hike the entire Appalachian Trail. It also requires students to make other calculations, such as how many calories they would burn while hiking. This lesson extends that idea of calculations based on proportionality/scale.​

Common Core State Standards

National Standards

Learning Objectives:

By completing this lesson, students should:

• Use and understand proportional reasoning to determine actual distances from one place to another while utilizing regional maps.
• Be able to come up with ways to measure distances without benefit of surveying equipment.
• Acquire experience using a compass.

Materials Needed:

•  Markers and pencils.
•  Clipboards.
•  Graph paper.
•  String.
•  Compasses (directional and drawing).
•  Calculator.
•  Ruler or marked card.
•  Protractors.

Background:

You should have a general idea about what the sport of orienteering is. Typically, orienteering involves using a detailed map and a compass to navigate your way around a course with designated points drawn on the map. The International Orienteering Federation and Orienteering USA have great background information.

Students should have some experience estimating distances and working with scales on maps. It is helpful, but not necessary for them to have worked with directional compasses. Students will need to practice working with a compass, so if you are unfamiliar working with a compass, practice ahead of time. Additionally, you will need to figure out where your students will design their courses. Your local/community orienteering club may also help facilitate this activity. They are often very helpful in building interest in their sport by supplying speakers or instructors.

Suggested Lesson Steps:

1. Explain to students that they are going to have the opportunity to apply skills they have been studying to a popular outdoor physical activity -- "orienteering." Ask them what they think that term might mean, if they have ever used a compass, and how they think it might relate mathematics. Through the discussion of orienteering, make sure that students understand its basics, including the items that would be necessary for orienteering (e.g., map and compass), as well as those that might be helpful (e.g., GPS)

2. Divide the class into pairs or small groups -- depending on the number of compasses you have. Instruct students on the use of a compass or GPS. Have students practice reading bearings (the direction of travel, as indicated by the compass) pointed toward a few familiar "landmarks." This can be done in the classroom or at an adjacent outdoor area.

3. Ask each pair/small group of students to estimate distances between different landmarks in the classroom or an adjacent outdoor area. If all of the students estimate the distance between the same landmarks, find out the highest and the lowest estimate for each of the distances they estimate. If each pair/group determines its landmarks, have students record their estimate for later comparison.

4. Challenge students to come up with a more accurate way to determine actual distances. For example, students might use string to get the straightest route between two points, walk the distance a few times to ensure accuracy and then measure the distance of each step. Perhaps students will walk a yardstick or meterstick the length of the string. Encourage creative solutions.

5. Students should record the more accurate actual distances and compare them with their estimates. Ask each pair/small group to share its most accurate and least accurate estimates. Was their accuracy generally affected by the distance -- i.e., more accurate for shorter distances? If the space between landmarks was flat?

6. Using graph paper, each pair/small group should create a map of the area containing the landmarks. Students should figure out what scale they want to use to present the map. Plus, using their compass, each pair/small group should ensure that its map is accurate directionally. Students should make two copies of their maps.

7. With one of the maps, each pair/small group should design an orienteering course. The course they design should consist of at least five different directions and distances to follow. That is, from a starting point, the course should prescribe going a certain number of feet/yards/meters in a particular direction to a spot in the area, then from that point a certain distance in a different direction, and so on. Students should sketch the course on their map, but also record directions and distances on a separate piece of paper. You can also lead the activity into a discussion of caloric balance by having students estimate the number of calories required to walk their distances.

8. Have each group exchange the list of directions and distances for its orienteering course with another group. Set a time limit and then instruct each group to follow the directions and record the path of its movement on the additional map. Ideally, students would run the course, trying to complete the course within the given timeframe (or before).

9. As a group completes an orienteering course and records its path, have each group it meet with the group that designed the course. The two groups should compare and contrast the maps. Did the group arrive at the correct end spot within the time limit? Did they err in following the directions? In depicting the directions on the map?

10. Reconvene as a whole class for students to share their experiences and to show how closely the two maps of the orienteering course compare with one another.

Learn Orienteering provides detailed instructions on using compasses with maps.

The International Orienteering Federation and Orienteering USA are comprehensive sites about the sport of orienteering and contain information about using maps and compasses. Includes resources and links.