Intent

In this activity, students begin to study straight-line graphs in more depth. They identify connections between the coefficient in the symbolic form of a linear function and the related graph and situation.

Mathematics

Students’ work shifts from fitting lines to data to working explicitly with a linear function of two variables. Students create and compare graphs involving constant rates, focusing on how the starting values and rates affect the graphs. They also create rules for constant-rate situations and examine how such rules depend on the rate and starting value.

Progression

Students work individually and then prepare in their groups to present methods and solutions to the whole class.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

20 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Materials

Grid transparencies

Doing the Activity

Initiate the activity by having students read the scenario and Question 1a. Ask a few volunteers to summarize the information. Then ask for a volunteer to state what students are to do for Question 1a.

Tell students to rename July 12 as Day 0. This will eliminate differences in student approaches, which is not the emphasis in this activity.

Discussing and Debriefing the Activity

Ask each group to prepare a transparency of one of Questions 1 to 4. This preparation will provide a context for group members to answer one another’s questions and to compare methods.

Begin with presentations for Questions 1a–c and 2a–c, focusing comments and inquiry on parts b and c by asking students to articulate how the starting values and rates affect the graphs.

For Question 1b, be sure students articulate that the Buck family graph “starts higher” than that for the Woods family.

For Question 1c, students will likely say that the graph for the Woods family is “going up faster” or something similar. Be sure to insert the word rate into this discussion.

For Question 2c, the discussion should make use of the term parallel to describe the two lines.

Ask the class, Why do the graphs from Question 1 both rise as they go to the right while the graphs from Question 2 both fall as they go to the right? Students should recognize that graphs that go “up to the right” represent situations in which the dependent variable increases as the independent variable increases, while the opposite is true for graphs that go “down to the right.” The use of phrases such as “as time passes” rather than “as the independent variable increases” is appropriate while students are still making sense of these mathematical observations.

Have another member of the group present the group’s responses to Questions 1d and 2d. Ask questions to heighten awareness of the connections among the situations, graphs, and rules. How did the group use the information in the situation to write this rule?What information did the group use to write this rule?Is there information in the rule that shows up in the graph? How? Why?

Questions 3 and 4 introduce the idea of using a pair of graphs to find a “common value.” Solicit at least one presentation that demonstrates how a student used the graph to answer each question.

While the goal is to seed students’ ideas about using a graph, discuss all methods used to answer these questions. For instance, in Question 3, someone might say that because the Woods family is gaining 5 miles per day on the Bucks and are 40 miles behind to start, they will catch up in 8 days.

For Question 4, students should see that the two families never have the same amount of coffee, because they are consuming coffee at the same rate. Make sure the graphical connection is made and uses the term parallel lines.

Key Questions

Why do the graphs from Question 1 both rise as they go to the right while the graphs from Question 2 both fall as they go to the right?

How did this group use the information in the situation to write this rule?

What information did this group use to write this rule?

Is there information in the rule that shows up in the graph? How?

Supplemental Activity

What We Needed (reinforcement) asks students to figure out how long it took for their families to travel from Ft. Laramie to Ft. Hall and how much of two commodities they would need to bring with them.