## Walking to Class: Modeling Students' Class Schedules with Time-Distance Graphs

- Lesson

Students use their class schedules to create
time-distance graphs by counting the number of walking strides they take from
their lockers and timing themselves as they walk through their class schedules.
They will use their graphs to answer questions about slope, *x-* and *y*-intercepts, and the meaning of horizontal and vertical lines.

In this lesson, students will collect, use, graph, and interpret data to represent linear time-distance relationships.

Be sure to have enough copies of the Activity Sheets, graph paper, enough watches, and make sure the distances between classes are reasonable. Students will be counting the number of steps between classes, and recording the times associated with those steps. They will then graph the data and interpret elements of their graphs.

Matching Graphs Activity Sheet

Use the Matching Graphs Activity Sheet to introduce the lesson to students. Have students work alone or in pars.

Explain to students that they will be creating a time-distance graph based on their class schedules. Beginning at their lockers, they will measure the distance to their classes (in steps). They will record the time it takes to travel as well as the time spent in class.

The data collection in this activity enables students to envision the distances they walk to collect the data, thus giving them a concrete experience to relate to the resulting graph. The data collection process takes about 15 − 20 minutes. If time is a concern, the data collection could be assigned as an out-of-class activity, or students can use a pre-existing set of data in the Sample Data Activity Sheet.

Have students read through the Class Schedule Activity. To collect the data, students should work alone or in pairs. If working in pairs, select one student's schedule as the source of the data to be collected. The two types of data to be collected should be the number of walking strides from the student's locker to each classroom, and the amount of time it takes the student to walk to and from each classroom. If students are working in pairs, they should decide who will walk, count steps, and record time.

Before students begin, show a sample on the board. Explain that when the data collection process is complete the data collected should look something like this:

Time (in minutes) Number of Walking Strides

0 0

2 67

45 67

47 0

49 132

92 132

Notice that this describes a student's path from the start of the school day (0, 0). It takes 2 minutes for the student to walk to first period, 67 walking strides away from his or her locker. This is represented as (2, 67). The student is in class for 43 minutes before walking back to the locker, and so on. After graphing the first horizontal segment (which represents the student being in class), ask students to keep the scale constant with a break in the graph to represent the time spent in class. This way, more emphasis can be placed on seeing the time-distance relationships going to and from class from the student's locker.

Depending on the amount of time you have, the student's schedule could be followed through only a few classes, instead of all of the classes. Students should have a minimum of 8 data points.

For homework have the students graph the collected the data they collected. Refer students to number 2 on the Class Schedule Activity Sheet. It may be appropriate to remind them about the importance of selecting intervals when graphing. The next day, have students display their graphs in the classroom, so they can compare each other's work. Students could even answer questions such as: "What scales did other people choose and why? Who has to walk the farthest from their locker during the day? Who walks the fastest? Slowest? What information about student course schedules cannot be obtained from the time/distance graph? What errors can occur in data collection?" Have students complete the rest of the Class Schedule Activity Sheet.

### Ideas for Differentiation

- Struggling students may need to focus only on graphing the data or may need only to focus on answering questions about data already graphed.
- Students achieving at a higher level could explore the relationship between slope and rate of change. Furthermore, they could investigate realistic ranges of slopes.

- Graph paper
- Matching Graphs Activity Sheet
- Matching Graphs Answer Key
- Class Schedule Activity Sheet
- Class Schedule Answer Key
- Sample Data Activity Sheet
- Stop watches or regular watches

**Assessment Options**

- Give students time-distance graphs, with no scales on the
*x*- and*y*-axes. Ask them to write scenarios that the graphs could model and a matching graph. Determine a scale for each axis. Display their stories on one side of the room and the graphs on the other side of the room. Ask students to match each story with its graph. - Give students
a pre-made time-distance graph about a similar situation (e.g., traveling to
and from a friend's house with a constant rate of travel). Have them answer the following questions:
a. What does the

*y*-intercept represent in this situation?b. What does the

*x*-intercept represent?c. What does the slope tell you about the trip?

d. What do the axes represent on the graph?

**Extensions **

- Use the motion detector
to create graphs described in the articles
*Mission Possible! Can You Walk Your Talk?*(Mathematics Teaching in the Middle School, 2000) and*Teaching about Functions through Motion in Real Time*(Mathematics Teacher, 2006). - Have students select sections of their graphs and write linear equations to represent the sections.

**Questions for
Students **

1. What parts of the graph did you use to help match the stories to the graphs?

[Students use the slopes, scales on the axes,x- andy-intercepts.]

2. Explain which graphs were the easiest to match and why.

[The graphs with large changes according to the slope.]

3. Explain which graphs were the hardest to match and why.

[The graphs with the only difference being in the scale factors.]

**Teacher Reflection **

- If you did not use the provided set of data, how might the effectiveness of this lesson have changed?
- How did students demonstrate that the opening activity helped prepare them for the lesson?
- Were students' levels of enthusiasm and involvement high or low? What might be some reasons for this?
- Identify the difficulties students had in creating a graph from their data. What knowledge did your students need to have, regarding graphing data of this nature, to be successful?

### Learning Objectives

Students will:

- Collect appropriate data for creating time-distance graphs.
- Graph data from a table of values.
- Calculate the slope of diagonal and horizontal lines.
- Explain the meaning of
*x*- and*y*-intercepts, given the context of their data.

### NCTM Standards and Expectations

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

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

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

### Common Core State Standards – Mathematics

Grade 8, Expression/Equation

- CCSS.Math.Content.8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

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.

Grade 8, Functions

- CCSS.Math.Content.8.F.B.5

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.