What Does a System of Equations Really Look Like?

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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.

Lesson Author

James Bialasik

Amherst, NY

Grade Level

Ninth grade

Subjects

Math

Algebra

Standards

HSA-CED.A.1

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

HSA-CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

HSA-REI.C.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

HSA-REI.D.11

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

Opening

10 minutes

Investigation

10 minutes

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

10 minutes

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.

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