Pedal Power

• Lesson
9-12
1

In this lesson, students investigate slope as a rate of change. Students compare, contrast, and make conjectures based on distance-time graphs for three bicyclists climbing to the top of a mountain.

As suggested by the activity "From Graphs to Stories" (NCTM, Navigating Through Algebra in Grades 6-8, 29), students should be asked to associate real-life meaning to situations represented graphically. This lesson provides students with a graph of an authentic situation. Students can use any method they like to determine the meaning of points and slope on the graph as it relates to the situation of three people biking uphill.

Distribute the Pedal Power Activity Sheet, which shows the distance-time graph for three cyclists.

To begin the lesson, present students with the following situation:

Bicyclists claim that the longest steep hill in the world is in Haleakala National Park, and they have the sore muscles to prove it! The hill leads up a volcano on the island of Maui, Hawaii. Over the course of a 38-mile road, this hill rises from sea level at the coast to over 10,000 feet.

Three proficient cyclists—Laszlo, Cliantha, and Joseph—rode this entire hill to the top. They started together at the bottom of the volcano, and they reached the top at the same time. The graph shows the distance of each cyclist with respect to time.

To heighten interest in the problem, you may wish to show students pictures of the Haleakala National Park or provide some background information. Use a simple internet search to find these images.

Students may also want to use the Internet to find information on bicycles and speeds that can be maintained when riding uphill.

To get students thinking about the situation, ask the following warm-up questions:

• Lance Armstrong’s average speed in his six Tour de France victories from 1999-2004 was about 24 miles per hour. Assuming that he pedals at his average speed and takes no breaks, how long would it take him to get to the top of the volcano?
• People who aren’t Lance Armstrong can travel at about 12 miles per hour on a bike. At that speed, how long would it take to reach the top of the volcano?

[At 24 miles an hour, it would take 38/24 hours, or about 1 hour, 35 minutes, for Lance Armstrong to climb the hill. At 12 miles an hour, it would take 38/12 hours, or about 3 hours, 10 minutes, for an average biker to climb it. However, both of these estimates are probably too low, as all bikers travel slower going uphill.]

After a brief discussion, allow students to consider each question on the activity sheet individually. Then, have students share their thoughts with a small group. Each group should reach consensus and then present their results to the class. A whole-class discussion should follow, focusing on the question groups below:
• Estimate the vertical coordinate of B. Justify your guess.
• Estimate the horizontal coordinate of B? Justify your guess.
• What are the coordinates of B?

[A biker on flat ground can average about 12 miles per hour. Climbing Haleakala, an average biker would likely go much slower, maybe 5-9 miles per hour. Therefore, it would take about 4-7 hours to complete the ride up Haleakala, so a reasonable estimate for the x‑coordinate of B is about 5 hours. The distance to the top of Haleakala is 38 miles, so the y‑coordinate of B is 38 miles. With B at (5,38), then students might estimate the coordinates of A to be roughly (2,26) and the coordinates of C to be approximately (3,13).]

• Which cyclist had a steady speed all the way up the hill? How do you know?
• Which cyclist was slow at first and then sped up? How do you know?
• How would you describe Laszlo’s speed?

[Cliantha held a consistent pace up the hill, because the slope of her line never changed. Joseph started slowly and then increased his speed, which is evident by an increase in slope. On the other hand, Laszlo started very quickly but then slowed down, because the slope of his line decreased.]

• The three cyclists started together at the bottom, and they reached the top at the same time. Is there any other time that Laszlo, Cliantha, and Joseph were at the same height at the same time? How do you know?

[No, there are no other times when they were at the same height. If they were, their lines would cross at locations other than O and B.]

• Find the slope of each line segment on the graph. What does each slope mean in the context of the problem?

[Assuming a time of 5 hours to travel the 38 miles up Haleakala, the slope of Cliantha’s line is 38/5 = 7.6. This means that Cliantha’s speed was 7.6 miles per hour for the entire trip. From the bottom to A, the slope is 26/2 = 13, meaning that Laszlo’s average speed for the first portion of the ride was about 13 miles per hour. He then slowed down, and his speed dropped to (38 ‑ 26) / (5 ‑ 2) = 4 miles per hour for the remainder. From the bottom to C, the slope is 13/3 ≈ 4.3, and from C to the top, the slope is (38 ‑ 13) / (5 ‑ 3) = 12.5. This indicates that Joseph’s speed increased from 4.3 miles per hour to 12.5 miles per hour.]

Assessment Options

1. In whole-class discussions, ask students to present their answers and explain why slope represents something other than speed. Encourage and validate a variety of appropriate responses. Ask students to write an entry in their journals that includes the following pieces:
• How slope is determined.
• Examples of graphs for which slope represents speed but for which the units differ.
• Examples of graphs for which slope does not represent speed.
2. Allow students to write authentic problems to fit graphical models that you provide. Ask them to interpret slopes in the context of the problems. For example, you might provide the following graph for students to consider. Because the axes indicate time and volume, students might suggest that the slope of each line segment represents the rate at which water fills a swimming pool.

Extension

Ask students to consider models that are not linear and to discuss slopes of non-linear curves. For example, you might provide a parabolic curve, which could represent the path of a baseball or softball.

Questions for Students

1. Is speed a rate? Explain.
2. Is slope a rate? Explain.
3. Give an example of a situation where slope does not represent speed.
4. What kind of graphs appear in the first quadrant only?

Teacher Reflection

• Were students able to make guesses without a great deal of guidance? Were they able to get the information they needed to feel comfortable in their guesses?
• Do you think they have a sense of speed? Would an introductory activity about speed, possibly using a Calculator-Based Laboratory (CBL), have helped?
• Did students know what was expected of them? If not, how can you make expectations clearer in the future?
• Were students focused and on-task throughout the lesson? If not, what improvements could be made the next time this lesson is used?
• This lesson is very open-ended. Once given the scenario, students are expected to make progress by themselves, with very little guidance. How might this lesson be structured for low-ability students to minimize frustration?
• What indications were there that students had achieved the objectives of the lesson?

Learning Objectives

Students will:

• Determine the slope for a distance-time graph.
• Interpret slope as a rate of change.
• Interpret a graph in the context of the situation.

NCTM Standards and Expectations

• Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
• Draw reasonable conclusions about a situation being modeled.
• Approximate and interpret rates of change from graphical and numerical data.