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?
- 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
- 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
[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.]