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. Some pictures and details can be found at the following sites:
Students may 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?
- What are the coordinates of A? Explain your answer.
- What are the coordinates of C? Explain your answer.
[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.]
- 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.]
