What Does a System of Equations Really Look Like?
Objective: The objective of this lesson is to create a link between the algebraic representation of a situation and the graphical representation. This lesson will focus on the appearance of systems of equations in two variables. Students will build meaning by determining the solution to a system by an inspection of the coordinates that make up each equations graph.
Big Idea: This lesson serves as the link between understanding constraint equations algebraically and understanding them graphically. Students will investiagte the appearance of systems of equations in the coordinate plane.
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Investigation
Without a graph, students will determine some coordinates that would lie on two lines by using the equations. Students should be encouraged to determine points by looking at the structure (MP7) of the equation (for example x + y = 3, what two numbers add up to 3?) Students are asked to list seven points to encourage them to see the pattern extending into the negative values as well as the positive values. Then, by inspection students can determine the solution point (if possible from their values) and verify their findings by graphing the two equations.
Students will also look at the three cases for a system of linear equations, namely, no solutions, one solution, and infinite solutions. Again, this can be done both by looking at the structure of the equations (parallel lines, lines that intersect, or the same line) and then students can use a graphing calculator to see how the graphs of each of these cases appears. Lastly, students will be modeling a situation algebraically that involves two constraint equations set in a real world context.
Closure
As a ticket out the door, students will have a choice of two assessments of learning. Both will give you valuable information about how to structure groups of students for the next days lesson. The target level question requires students to see how the stucture of the system leads to no solution (parallel lines). The more complex question assesses students understanding that lines extend infinitely in both directions and that if two lines do not have the same slope they will eventually intersect.
