Intent

This activity, a follow-up to More Fair Share on Chores that builds on students’ work on Fair Share for Hired Hands, will provide information on students’ understanding of the connection between graphs and equations.

Mathematics

Students write and graph an equation for a linear situation. Next they determine a solution to a pair of conditions represented by linear equations and are asked to find the solution to a system of two linear equations in two unknowns.

Progression

Students will work on this activity individually and then share their techniques as a class. The activity concludes with a discussion of graphical methods that emphasizes that every point on a line fits the equation for that line.

Approximate Time

5 minutes for introduction

20 minutes for activity (at home or in class)

15 minutes for discussion

Classroom Organization

Individuals, followed by whole-class discussion

Materials

Students’ work on Fair Share for Hired Hands

Doing the Activity

Tell students that this activity extends Fair Share for Hired Hands in the same way that More Fair Share on Chores extended Fair Share on Chores. To encourage students to use a graphing approach, you might suggest they use the graph from Fair Share for Hired Hands to begin this work.

Discussing and Debriefing the Activity

Begin the discussion by having students compare results and graphs in their groups.

As before, there are several ways to approach the problem, and students may not have solved it by graphing the two equations. They should have the graph of the equation Y = X + 1 for Question 2, but they might have answered Question 3 by a guess-and-check method.

Observe the group interactions to gauge how well students can plot graphs and write rules and to get a sense of how well they understand that the points on a graph are solutions to the equation or to the situation that the graph represents.

Ask groups for an equation that represents the additional information provided in Question 3. If they come up with 3X + 4Y = 30, ask how they would put this equation in a form that would allow them to graph it on the calculator. You may want to suggest that they think about how they would find the rate for each more-experienced hand if they knew the rate for each less-experienced hand.

If needed, have them go through the arithmetic with a specific value for X and then analyze the arithmetic steps (in a manner like that described for Fair Share on Chores). They should be able to express the condition that the total pay is $30 in a form similar to Y = (30 – 30X)/4.

Have all students plot both graphs on the calculator, use the Trace key to locate the point of intersection, and confirm that the solution they found earlier matches this work. Use this work to help emphasize that points on the graph are solutions to the equation and that the intersection point is a solution to both equations. Ask students to articulate how they know that the point of intersection yields the solution for both conditions of this problem. How might you convince someone that the coordinates of the point where these two lines intersect tells where both conditions in this problem are satisfied? Their replies might include the notion that if you check that point in each equation or in each situation, it makes a true statement.

Key Questions

What equation represents the information in Question 3?

How can you rewrite the equation 3X + 4Y = 30 so you can graph it on the calculator?

How would you find the rate of pay for each more-experienced hand if you knew the rate for each less-experienced hand?

How might you convince someone that the coordinates of the point where these two lines intersect tells where both conditions in this problem are satisfied?