Intent

This activity gives students more experience in writing rules for situations and in finding the common solution to a pair of equations.

Mathematics

Students continue their exploration of linear functions. In particular, they examine ways to find the input for which two linear functions give the same output.

Progression

Students work on the activity in groups and then discuss their work as a class. The equation developed by the class for Question 5 will be returned to later in the unit when students have the symbol-manipulation tools necessary to solve it algebraically.

Approximate Time

30 Minutes

Classroom Organization

Groups, followed by whole-class discussion

Doing the Activity

Tell students that paying attention to water supplies was an important aspect of life in the Overland Trail era, as it still is today. This activity explores some questions about monitoring water supplies. Point out the setting, northeast Nevada, on the map.

As students work, give various groups transparencies to use in preparing for the class discussion.

Discussing and Debriefing the Activity

Most of this activity’s content should be familiar to students. This discussion will serve as the context for bringing out some new ideas. After presentations of Questions 1 and 2, record the expressions students developed (50 – 3X and 100 – 8X) for use following the discussion of Questions 4 and 5.

If students have difficulty with Question 4, remind them that one of the lines represents the amount of water the Stevens family has on a given day and the other line represents the amount of water the Muster family has on a given day. They should recognize that to find a day when the two families have the same amount of water, they want to look for a point that is on both lines.

Ask how the answers to Question 5 show up on the graph. Bring out that the solutions are the x-coordinates of the points where the graphs cross the x-axis. Introduce the idea of x-intercept

as being the point on the graph where the y-value is zero.

Now is a good opportunity to set the stage for solving these types of problems algebraically. Ask how students might use their expressions from Questions 1 and 2 to write a single equation that could be used to identify the day when the families have equal amounts of water. If needed, emphasize that the two expressions represent the amount of water that the two families have after X days, so for the families to have the same amount of water, the two expressions must represent the same value.

The idea of using these expressions to create an equation is often a conceptual leap for students. You might use a guess-and-check approach to help them see this connection. For instance, ask whether the families had the same amount of water after six days and how they know. Students should see that they can answer this by evaluating the expressions 50 – 3X and 100 – 8X at X = 6 and noting that the results are different. For the families to have the same amount of water, the results must be the same—that is, 50 – 3X and 100 – 8X must be equal for a specific value of X.

Students should have noted from the graph that the lines intersect at (10, 20). Ask what this means about the equation 50 – 3X = 100 – 8X. Bring out that when X is 10 (the first coordinate), both sides of the equation equal 20 (the second coordinate).

Post the equation 50 – 3X = 100 – 8X for further discussion after More Mystery Bags.

Key Questions

How can you use your expressions from Questions 1 and 2 to get an equation for finding the day when the families have equal amounts of water?

Supplemental Activity

The Perils of Pauline (extension) is a well-known but challenging puzzle problem. Students are given information about the speed of an oncoming train and the position of a person in a tunnel that the train is approaching and are asked to determine the person’s speed given that she made it out of the tunnel on time.