Intent

Students examine how to use graphs to find the solution that fits two linear conditions.

Mathematics

Students write and graph an equation representing a new linear condition for the situation first encountered in Fair Share on Chores. Next, they are asked to find a solution that fits both this new condition and the original condition; that is, they are asked to find the solution to a system of two linear equations in two unknowns (although that formal mathematical terminology is not used).

Progression

Students will work on the activity in groups and then share their solution methods with the class.

Approximate Time

30 minutes

Classroom Organization

Groups, followed by whole-class discussion

Materials

Students’ work on Fair Share on Chores

Doing the Activity

Tell students that this activity is similar to the work they have been doing recently.

Students should not have much trouble with Question 1. If they do, help them recall the methods they have used recently.

Allow groups to struggle to find their own ways to solve Question 2. Here are three strategies they might use.

When each group has found a solution, pick two or three to present their methods to the class.

Discussing and Debriefing the Activity

Begin, if you think it is needed, with a discussion of the individual parts of Question 1, or just have a student give the equation from Question 1c, B=G+12B=G+12 size 12{B=G+ { {1} over {2} } } {}

Then let the chosen groups present their work on Question 2. If no group uses a graphical method to solve the system of equations, lead the class through a graphical solution. Ask,

How did you represent graphically the pairs of shift lengths in which each boy’s shift is half an hour longer than each girl’s shift?

How did you represent graphically the pairs of shift lengths that total ten hours?

Students should recognize that each condition, by itself, is represented graphically by a line. If necessary, emphasize that every point on a given line fits the equation for that line. It shouldn’t be hard to get someone to articulate that we want to find a point that is on both lines.

Because the point of intersection is a point on both graphs, it must satisfy both conditions of the problem. Ask several students to state this observation in their own words.

Since students have both equations in a form that gives B in terms of G, they can draw the two graphs simultaneously. They can use the trace feature to estimate the coordinates of the point of intersection. The point is exactly (1.7, 2.2), which translates to 1 hour, 42 minutes for each girl and 2 hours, 12 minutes for each boy. Have students verify that this pair of times fits both conditions of the problem.

Figure 1

Key Questions

How did you represent the first condition graphically?

How did you represent the second condition graphically?

What points fit both conditions?