# Pulling the Systems Concepts All Together

## Objective

Students will model a situation using systems of equations and draw connections between solving the system algebraically and graphically.

#### Big Idea

This lesson serves as a closure to linear systems. Students will make connections between solving a system graphically and by both algebraic methods (elimination and substitution). The system of equations is used to model a real world situation.

## Opening

15 minutes

This task requires very little instruction.  Students will work in pairs in order to solve the system and interpret the solution to determine whether the dollar belongs inside or outside of the cash box (p.s. It belongs inside!).  As mentioned above, most students will be able to solve the system but many students will struggle with determining where the dollar should go.  It all comes down to students understanding that their can't be an odd number of "couples" tickets to the dance.

Extention/scaffolds: The opening activity for this lesson involves a deep understanding of systems.  Most students will be able to solve the system of equations but stronger students will be able to realize whether or not the dollar belongs inside or outside of the box based on number sense.

Environment: Students will work in pairs on the opening activity and individually on the investigation portion of the lesson.  Once students all have an opportunity to complete all parts of the investigation students can get back into their partnerships to compare their work and discuss their ideas.

## Investigation

25 minutes

Have students work by themselves to solve this system of equations graphically and by two methods algebraically.  In doing this, students will gain an understading of how a problem can be solved in a variety of ways.

What to watch out for: It is important that you continually push students to give meaning to their answers.  When solving the system either by graphing or algebraically they are answering a question about the number of freshman and sophomores at the meeting.  Each time they write their answer they should write it as 6 freshman and 4 sophomores.  Moreover, when students label their axes on the graph they should indicate which axis is the "freshman" axis and which is the "sophomore" axis.

The final question can give you some insight into student understanding.  When students finish and are able to work in pairs to compare their responses, I listen to their explanation of the last question and why there could not be 5 freshman and 5 sophomores.  What you want to hear is somthing about the point (5,5) being on one of the equations but not the other, this means that since that combination will not work for both equations it cannot be a solution to the system.

## Closure

5 minutes

Ask students to compare all three methods of solving a system of equations.  For this task, which method was the most efficient?  Explain why you think so.