Intent

In this activity, students extend their work with linear functions beyond the first quadrant, giving meaning to the x-intercept of a graph. This activity also generates one more equation obtained by setting two linear expressions equal, setting the stage for the development of methods for solving simple linear equations.

Mathematics

Students continue their work with representing situations by equations. In this case, they consider the starting values and rates related to the profit a miner might make using two different methods for collecting gold. They compare the associated linear functions both graphically and symbolically, considering questions that can be answered by the x-intercepts and the point of intersection or by setting up an equation for finding when the two functions are equal.

Progression

Students work individually to explore two methods for collecting gold and the profit associated with each, using graphs to answer questions about break-even and equal-profit points. In a class debriefing, they develop and solve linear equations to answer the same questions.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, then groups, followed by whole-class discussion

Doing the Activity

After students read The California Experience to put the gold rush in perspective, introduce them to this activity. Make sure they understand that profit is income minus expenses. If the activity is assigned as homework, it is a good idea for students to start in class, even briefly, to ensure they know how to approach Question 1.

Discussing and Debriefing the Activity

Have students debrief their work in their groups. Invite a few students to post their expressions from Question 3a and 3b on the board, as these will be used in discussing Questions 4 and 5. Also ask a group or two to prepare transparencies to lead discussion of Questions 4 and 5. One group’s solution for each will be sufficient, especially if students are becoming proficient with questions like these.

After Questions 4 and 5 are presented, review the graphical and algebraic ways to approach these types of questions.

First, note that the starting values are both negative and give the places where the graphs cross the y-axis.

Then point out that the break-even point is the value where each graph crosses the x-axis. Bring out that finding the break-even point is like solving an equation in which the expression for profit is set equal to 0. Thus, the panning method breaks even at the point where the value of x gives a solution to the equation 15x – 60 = 0.

Finally, emphasize once again that finding the equal-profit point is the same as finding the point of intersection of the two graphs and that this point can be determined by solving the equation formed by setting the two algebraic expressions equal to each other. By now, this will be routine thinking for some students. You may want to pose it as a question for students who need to give this relationship additional thought.

Students can verify that the x-coordinate of the point of intersection on the graph fits the equation 15x – 60 = 30x – 420. You can also bring out that the y-coordinate of this point is the value of each side of this equation.

Post this equation, 15x – 60 = 30x – 420, for discussion after More Mystery Bags.