Intent

Students examine possible solutions to a linear equation in two variables and explore how to express that equation as one variable in terms of the other. The activity focuses on the connection between an equation and its graph to further set the stage for considering the graphical meaning of the solution to a system of equations.

Mathematics

Students continue work with all four representations of linear functions. Most importantly, they write a rule in Y= form and consider the meaning of specific points on the line.

Progression

Students use a context to derive a set of ordered pair solutions, plot these pairs, and determine an equation for the linear relationship. Again, students are asked to convert the equation into Y= form. Finally, a class discussion emphasizes the meaning of the points along the graph of the line. This sets the stage for the next activities, in which students consider the meaning of a point of intersection.

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

Doing the Activity

Let students know that this activity is similar to the previous activity, Fair Share on Chores. Although students have graphing calculators available, suggest they initially draw the graph on paper.

Discussing and Debriefing the Activity

Have students compare ideas and answers in their groups. You might have several students put their solutions for Question 1 and 2 on the board. After a few minutes, gather the class and have some discussion of the responses.

Next, turn to Question 4, eliciting verbal descriptions of how to get Y from X, and comparing the different versions.

Discuss the equations students wrote for Question 5, which probably look much like Y=20−3X4Y=20−3X4 size 12{Y= { {"20"-3X} over {4} } } {}. As a class, test some of the points students found from their graphs in Question 3.

At this point you can ask the class, What is the relationship between fitting the equation and being on the graph? Students should come to this conclusion:

Every point on the graph represents a solution to the equation, and every solution to the equation corresponds to a point on the graph.

Essentially, this is the definition of a graph.

Then ask, Is every solution to the problem on the graph? Does every point on the graph represent a solution to the problem?

The discussion should bring out the fact that a point on the graph, which represents a solution to the equation, does not necessarily represent a solution to the problem. For example, the point represented by X = 4 and Y = 2 is on the graph and fits the equation, but is not a solution to the problem, because the experienced workers should have a higher pay rate than the inexperienced workers.

This tells us that even though the graph represents the equation, the equation itself does not precisely represent the problem. The graph for the actual problem is only a portion of the line. In other problems, only points with whole-number coordinates will fit the problem.

Key Questions

How did you get Y from X ?

What is the relationship between fitting the equation and being on the graph?

Is every solution to the problem on the graph?

Does every point on the graph represent a solution to the problem?