## Taking Its Toll

- Lesson

In this lesson, students will compare the price of a toll to the distance traveled. Students will investigate data numerically and graphically to determine the per-mile charge, and they will predict the cost if a new tollbooth were added along the route.

Prior to the lesson, attempt to obtain a ticket for a toll road in your state. Ask a toll booth attendant while traveling, or write your state’s department of transportation or turnpike commission. Explain that you are an educator who will be using the ticket for a classroom activity. (The Federal Highway Administration maintains a list of state transportation web sites. A quick internet search should return viable results.)

You can also find information about toll rates via the Internet. Most states have a turnpike web site. Enter your state name and the word "turnpike" or "toll road" into a search engine.

Several states do not have state toll roads. Other states have toll roads that charge a constant fee, regardless of distance. If this is the case in your state, you may wish to use the data from a neighboring state for this lesson, or you can use the Pennsylvania Turnpike Toll Rates Overhead to provide data for your class.

Pennsylvania Turnpike Tolls Overhead

Another effective introduction is to pose the following problem:

The [State] Turnpike Commission is planning to add a toll plaza in [City]. They have asked for your help in determining the amount of the toll that should be charged at this exit.

Of course, this introduction could be embellished to include reasons why an exit is being added, such as the construction of a new sports stadium, easier access to local shopping, and so forth.

Assign students to groups of three. Make all materials easily accessible, and designate one member to get the Taking Its Toll Activity Sheet, while another member gets calculators, rulers, and graph paper.

Taking Its Toll Activity Sheet

Students should then gather data for the Toll Data Recording Sheet in either of two ways:

- Many states include a toll calculator for their turnpike on a web site. These calculators often give both the amount of the toll as well as the distance between toll plazas. Use a search engine to find your state’s turnpike web site; then, use the toll calculator to gather the data, or allow students to use the web site to gather data on their own.
- Alternatively, students can use the state’s toll ticket in conjunction with a state map to find the necessary information. The ticket will provide toll information, and the map will provide distance. Discuss with students how to determine the necessary information from the ticket. There may be different tolls for travelers depending on the state vehicle classification system that is used.

Have the students work together in their groups on the student activity sheet. If necessary, have them record data on the last page of the activity sheet, or have them enter the information into the Toll Data (Excel) Spreadsheet.

The first column is only needed to identify each exit. In the second column, students should record the total distance from the start of the toll road. This information may be easily obtained if the exit numbers correspond to mileage; if not, students will need to determine this information from another source. In the third column, students should record the cumulative toll.

Beware of common student errors when recording data. Computation errors often occur in the third column. In addition, some plazas do not have a toll, so students may be confused as to how the cumulative total is affected. Be sure that students verify their data before proceeding.

Using their data, students should then create a scatterplot. Students can create this scatterplot on the graph of Question 3 of the Taking Its Toll activity sheet, or you may wish to have them use the chart feature if they entered their data in a spreadsheet program.

Results will vary from state to state, but the scatterplot for most toll roads generally indicates a linear pattern with a positive slope. For instance, the tolls on the Pennsylvania Turnpike result in the following:

After gathering data and making a graph, students should analyze the results. Gathering the data and creating graphs are skill-based tasks. While it is important for students to develop these skills, it is more important for students to conduct an analysis of their completed graphs to develop conceptual understanding of rates of change.

Questions 4-10 on the activity sheet require students to analyze the graph, and you may wish to have students work on these questions in their groups. After they complete these questions, conduct a follow-up discussion to review their answers and to ensure a high level of student understanding. Allow each group to present their line of best fit and how it was determined.

Alternatively, you may wish to have students consider Questions 4-10, as well as other questions that you devise, as part of a whole-class discussion. If you choose this option, do not distribute the second and third pages of the activity sheet. Instead, ask these questions one at a time, and allow students to discuss them as a class. The conversations that occur between students are often beneficial. (To ensure that all students participate in this discussion, and to prevent just a few students from monopolizing the conversation, use a random selection technique to call on students, such as putting all student names in a hat and drawing them at random.)

- Taking Its Toll Activity Sheet
- Pennsylvania Turnpike Tolls Overhead, or a toll ticket from a state toll road
- State map
- Computers with Internet access
- Web site address for state toll road(s)
- Calculator
- Graph paper
- Rulers

**Assessment Options**

- Questions 9-10 on the Taking Its Toll activity sheet are appropriate as
homework. By providing an opportunity for individual student
reflection, you will be able to determine if all students understood
the lesson.

- Students should be given opportunities to share their graphs and
answers with others in a small group. As students work and discuss,
circulate and listen to conversations. Use this opportunity to address
misconceptions, and provide validation for correct student discoveries.

Require students to submit a graph. Graphs should be evaluated on accuracy, appropriate labeling, and completeness. The Taking Its Toll Activity Sheet could also be submitted for evaluation.

**Extensions**

- Have students consider the cost of using toll roads for various vehicle
classes. (For instance, the toll for a motorcycle is typically less
than that for a tractor trailer.) Further, have them consider the
discounted fares for using Smart Tag, EZ Pass, and other commuter
payment options. These graphs will likely vary from the one found in
the lesson.

- Allow students to research the average toll per mile for the road since it was first opened. Because the amount of the toll is tied to inflation, a graph showing the average toll over time will likely indicate that the pattern of increases is not linear but exponential.

**Questions for Students**

1. What does the slope of the line of best fit mean? How can you interpret this given the x- and y-axes labels?

[The graph shows distance on the x-axis and amount of the toll on the y-axis. The slope of the line represents cost per mile.]

2. Is there a pair of consecutive points that would result in a slope of zero? What does a zero slope indicate about the toll between two plazas?

[A slope of zero indicates that the toll does not change. That is, the same amount is charged at each plaza.]

3. Explain how the line of best fit could be used to determine the toll if a new plaza were added somewhere along the toll road.

[Along the x-axis, students can find the distance (mile marker) for the new plaza. The corresponding y-value of the line of best fit is a good approximation of the toll at the new plaza.]

**Teacher Reflection**

- Were clear expectations discussed so that students knew what was expected of them? If not, how can you make expectations more clear 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?
- How did this lesson address different learning styles? What improvements could be made to make the lesson better suited for all students?
- Were students able to make a connection between the graph and what it means for a traveler?
- Did students have the proper foundation in working with linear equations, slope, and intercepts? If not, what knowledge were they lacking, and what could have been done to have them acquire the necessary skills?
- Did you use this lesson as a guided introductory lesson on graphing equations, or did you use it as the culminating lesson of a unit on linear equations? Which approach would be better?
- How did students show that they achieved the objectives of the lesson?

### Learning Objectives

Students will:

- Determine slope given two ordered pairs.
- Determine a line of best fit based on given data.
- Determine the average cost per mile for passenger vehicles.
- Model, analyze, and make predictions for adding a new toll plaza.

### NCTM Standards and Expectations

- Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.

- Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.

- Use graphs to analyze the nature of changes in quantities in linear relationships.

- Make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit.

### Common Core State Standards – Mathematics

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8, Functions

- CCSS.Math.Content.8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.