## Mathematics as Communication

- Lesson

This grades 6-8 activity focuses on interpreting and creating graphs that are functions of time. The first two activity sheets focus on graphs of time vs speed. The third and fourth activity sheets look at how many times an event occurred in a specific amount of time. These activity sheets are appropriate for students who have not taken algebra yet. This lesson plan was adapted from an article by Marian Moody which appeared in the December 1990 issue of the *Mathematics Teacher.*

*Introduction*

This activity was created to encourage students to observe and examine the world around them. It helps students use mathematics to model real-world problems, to reason mathematically, to communicate mathematically, and to solve problems. In particular, it helps them read and interpret graphs and organize and describe data.

*Prerequisites*

Students should be familiar with the concept of events occurring as functions of time. They should be able to set up a rectangular grid and to plot points on the grid.

*Getting Started*

The activity can be used in regular mathematics classes or cocurricular- or extracurricular-activity sessions. The time required for the activities depends on the level and ability of the students, the amount of class discussion generated by the teacher and students, and the amount of material assigned as homework.

Distribute the Graphing Functions of Time activity sheet to the students.

Graphing Functions of Time Activity Sheet |

Display an overhead transparency of the graph and make sure that students understand how the graph represents the speed of Mrs. Sung-Ki's car. Ask questions like, How fast was Mrs. Sung-Ki going at 7:15 A.M.? *(about 10 miles per hour)* What time did she get off the highway? *(about 7:42 a.m.)* How many times did Mrs. Sung-Ki come to a complete stop while she was driving? *(6 times, including arrival)* Encourage students to think of other questions that they could answer by looking at the graph. Many of their answers will not be exact. Such answers as "about eighty-eight kilometers per hour" and "between ten and twelve minutes"should be accepted.

Have students draw the graphs for problems 1, 2 and 3 on the activity sheet. After students complete their graphs, have them exchange and compare them. Encourage students to describe and explain their graphs to each other. In creating the other graphs did they use the story or the graph in the directions? Are any of the graphs related?

*Possible Solutions for the Graphing Functions of Time activity sheet:*

The following is a possible solution to problem 1:

The following is a possible solution to problem 2:

Distribute the More Graphing Functions of Time activity sheet and graph paper to each student.

More Graphing Functions of Time Activity Sheet |

Make sure that students understand how the graph represents the speed of the bus. Divide students into pairs to work on problems 1 and 2. They should cooperate in creating their graphs and stories. Both students should be prepared to discuss their work.

*Possible Solutions for the More Graphing Functions of Time activity sheet:*

1. The roller coaster will go slowly up the first long hill and will descend rapidly. From then on its speed will fluctuate until it stops. One acceptable answer is shown here.

Distribute the Collecting Data Over Time activity sheet and the Graphing Data Collected Over Time activity sheet to each student.

Collecting Data Over Time Activity Sheet |

Graphing Data Collected Over Time Activity Sheet |

Point out to students how these graphs differ from those used on the first two sheets. For example, in problem 1 on the Collecting Data Over Time activity sheet, plotting how many times the door was opened at exactly 7:00 A.M. or at exactly 4:00 P.M. would not be useful. Instead, the number of times the door was opened during an interval of time is plotted. Each space on the horizontal axis represents one hour. The Xs show how many times the door was opened during that hour. Ask students to discuss how the horizontal axis used in the graph on sheet the first sheet differs from this one.

In problem 1 on the Graphing Data Collected Over Time activity sheet, a scatterplot is used instead of a histogram. One X is used to represent how many times the lunchroom door was opened. Ask student to speculate on why we are not setting up a graph just like the one on the Collecting Data Over Time activity sheet. A helpful approach is to use the X to represent the ordered pair that is plotted. If a point is used, many students will try to connect the dots.

Students may work together or individually on these activities. After completing them, encourage students to share their results with the class.

*Possible Solutions for the Graphing Data Collected Over Time activity sheet:*

1. A few workers go in and out of the lunchroom in the morning. A few teachers visit the lunchroom between 10:00 A.M. and 11:00 A.M. and 1:00 P.M. and 2:00 P.M. for a snack. We have two lunch periods, one from 11:15 A.M. to 11:45 A.M. and another from 12:15 P.M. to 12:45 P.M. Fewer people eat during the second session.

**References**

- Graphing Functions of Time Activity Sheet
- More Graphing Functions of Time Activity Sheet
- Collecting Data Over Time Activity Sheet
- Graphing Data Collected Over Time Activity Sheet
- A set of transparencies for discussion
- Graph paper

### Learning Objectives

Students will:

- Answer questions about graphical representations of information collected over a period of time
- Create graphical representations of information collected over a period of time
- Work collaboratively and improve communication skills

### NCTM Standards and Expectations

- Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

### Common Core State Standards – Mathematics

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.