Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Problem Solving, Reasoning
School: Okemos High School; 1,300 students
Location: Okemos, Michigan
Teacher: Jacqueline Stewart
Years Teaching: 20
Students in Classroom: 20
Students are introduced to a modeling activity in which they will investigate the mathematics of bungee jumping. The class discusses the materials, variables, and strategies for conducting the investigation. Students work in groups to gather and graph data. As they work, they discuss how differences in the weights and the stretch of a rubber band create a pattern in their graphs. At the end of class, students present their graphs and Ms. Stewart asks questions about assumptions and how the graph patterns inform bungee jump safety issues.
The Class Challenge
Student groups use fishing weights, paper clips, and rubber bands to perform the following tasks:
- Make a model representing bungee jump equipment.
- Simulate several jumps using different weights and measure the rubber band's stretch in each simulation.
- Create a table and a graph displaying the data for each simulation.
- Consider how greater and lesser weights affect the rubber band's stretch.
Course and Curriculum
This course, called Integrated Mathematics One, integrates algebraic concepts and skills with geometric and statistical concepts. Students study exponential functions and use technology to create graphs and to solve inequalities from tables and graphs. Students are required to take either this course, which is based on the first course in the four-year college preparatory Core-Plus Mathematics Project curriculum, or Algebra I.
Prior to this lesson, students studied scatterplots, measures of central tendency, measures of variability, and how different statistical graphs carry different information. This lesson on rate of change and modeling was the first in a unit on patterns of change. The unit introduced the use of table and graph models to predict relationships between variables. This bungee jump model, for which each group used slightly different materials, did not include the force exerted by the jumper.
Following this lesson, students further discussed how all the groups' graphs were the same because they were all roughly linear. Students also investigated other situations that produce graphs and tables with constant or nonconstant rates of change, and then modeled these situations with recursive and closed algebraic symbolic models. In the following unit, students studied slope and intercept in linear models and the connections among graphs, tables, and rules.
A Pre-Viewing Exploration for Teacher Workshops
Measuring the Stretch
Objective: To collect data and to determine a function that represents the data.
Materials: One overhead projector for the whole group. One large rubber band, five balloons, five paper clips, scales to measure balloons' weight (or graduated beaker to measure displacement in water), water, measuring tape, coordinate graph on overhead acetate sheets, and an overhead pen for each small group. Each group's rubber band should be a different length, but have the same width and be the same type.
Solve this problem in small groups.
Measure the length of the balloon and rubber band with no weight (0, length). Fill the balloons with different amounts of water. Weigh each balloon and record its weight on a chart. Use a paper clip to attach the balloon containing water to the rubber band. Now measure the length of the balloon and rubber band when stretched. Record the stretched length next to the appropriate weight on your chart. Repeat this process for all five balloons and graph your data (weight, length) on the overhead sheets.
- Is there a pattern in the graph?
- What type of function is represented by the data?
- Can you write a function to represent the data? What is the function?
- Compare your group's graph to other graphs. What is the relationship between your graph and the other graphs? Do they all behave identically?
- How are the graphs similar or different?
- What can you conclude about the slope of your graph? How does the slope compare with the slope of other graphs?
- What is the general function for the stretch of the balloon and rubber band represented in the graphs? (The stretch does not include the length of the rubber band.)
- What would happen if you added other variables to the activity, such as more than one rubber band?
Exploring Teaching Issues
- What is the value of doing this activity before assigning it to students?
- How would you organize and manage this lab activity in your classroom?
- The points appear to form a straight line.
- A linear function.
- Yes. Answers will vary, but the function would be of the form, y = ax + b (where y is stretched length, x is weight, and b is length of the rubber band without weight).
- On each graph, the intercept is the length of the rubber band without weight. All the graphs form lines, but not necessarily the same line.
- There is a proportional relationship between the weight and the stretch, but that number is not the same for different lengths of rubber bands.
- The slope changes for each different length of rubber band. Answers will vary to the second question. If the same type of rubber band is used, the relationship between the stretch length and the size of the rubber band can be noted. The larger the rubber band, the more the stretch and the greater the slope.
- The stretch s is proportional to the added weight w; s = kw, where k is constant or the slope of the line.
- Answers will vary depending on the variable. With two rubber bands, k would be half as large.
Topics for Discussion
What are the benefits of engaging students in simulations?
Discuss Ms. Stewart's decision to vary the materials used by groups.
- How does the investigation allow students to determine a formula or rule that explains the data and can be used for predictions?
- How might the discussion have been different if students had made their graphs with graphing calculators?
- What other functional relationships would be appropriate for secondary students to simulate?
Compare the plotted data and discuss differences.
- How might the differences between the groups' materials encourage students' understanding of the concept of variable?
- If each group used the same materials, how would the discussion and the issues addressed have been different?
- How would you design a follow-up lesson?
- Compare the students' techniques for collecting the data.
- Since the bungee jump problem is a linear function, what factor contributes to the student graphs being nonlinear?
- Are the data continuous or discrete? How do you know?
- The data indicated that the stretch increased more rapidly than a linear relationship, s = kw (where s = stretch, k = constant to be determined by data, w = weight). How would students determine s when s = k1w + k2w2 and s = k1w + k2w2 + k3w3?
- Cite evidence that students were able to use their graphs to make predictions for bungee jump safety.
- Ms. Stewart challenged students to consider factors such as the scale of the graphs in examining data across the groups. How did her focus on differences promote student interest and learning? What else might they discuss in comparing the data?