Intent

Students consider how changes in starting value and rate affect graphs and rules.

Mathematics

This activity requires students to develop algebraic rules for given situations. They will use graphs or other representations of the resulting linear functions to answer questions about rate and solutions to a system of two linear equations.

Progression

Students will work collaboratively in their groups to put to use the connections they have made among situations, tables, graphs, and rules. Some class discussion will follow, primarily to emphasize the role of rate and starting value in each type of representation of the situations.

Approximate Time

30 minutes

Classroom Organization

Groups

Doing the Activity

Draw students into the activity by reading as a class the brief history of James Beckwourth in the student book. Groups that finish early can search classroom reference materials or the Internet to learn more about James Beckwourth.

The activity poses some mathematical challenges, beginning with interpreting the given information and determining how to structure the graph. Expect students to wrestle with these parts of the activity with their groups.

While monitoring, be prepared to help groups to interpret the graphing instructions in Question 1. Ask, What clues do the instructions in Question 1a give you about the labels for the x- and y-axes? Suggest that students assign Day 0 to July 28. If a group is stumped trying to draw the graph or write a rule, remind them that an In-Out table can be helpful.

For groups uncertain about how to approach Questions 2b or 2c, encourage them to draw a graph or a table so they can see more information about the situation.

It is important that everyone gets through Question 2b. You might have students who don’t finish do so as homework or as an extension activity.

Discussing and Debriefing the Activity

There is no need to debrief Question 1. Request volunteers or have some groups prepare in advance to present Question 2, asking that they create a graph and write a rule as part of their response.

Focus the discussion of Question 2 on the connection between the numbers in the problem and the graph and rule that students develop. You can return to Question 1 to emphasize how and where student choices for rate and starting value affect the various representations.

During the discussion, it is useful to mention points on the graph and to record their coordinates using conventional notation—for example, (0, 23). If necessary, ask students to interpret the meaning of this notation.

If time allows, have the students in a group present their work on Question 3 to prompt further discussion. Because every group’s response will be different, students will have to communicate carefully the methods they used to solve the problem, rather than simply stating the solution.

Key Questions

What clues do the instructions in Question 1a give you about the labels for the x - and y -axes?

How does the 12 show up on the graph? In the rule? Why?

How does the 23 show up on the graph? In the rule? Why?