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# What is the "Point" of Solving a System?

Lesson 3 of 13

## Objective: Students will be able to connect to the meaning of solving a system of equations. Namely, that a line is made up from an infinite number of points that make that line's equation true. If we find a point that is on both lines that it has to make both equations true. Also that the other points on the line do not make both equations true.

#### Opening

*15 min*

**Environment: **Students should be reorganized for this task so that they are grouped homogeneously**. **This way each partnership can work at a pace that is suited for both members and I can sit with students who need more direct help.

This powerpoint should be used to demonstrate the meaning of the solution to a system of equations. Students are usually great at graphing lines and finding out where they cross but don't connect this with the fact that the solution makes both equations true. In this opening students are walked through this understanding step by step so that they can see and understand this connection.

Slide 3: Allow students a few minutes to come up with a series of points for each equation on their own. Then do a** think pair share** so that students can compare strategies around how they came up with the points (e.g. by looking at the graph and noticing a pattern, using the formula or equation, some other way). When students share out as a class you can use non-verbal cues (thumbs up/thumbs down) to determine how many other students used a similar strategy. All students to see, through their sharing of ideas, that when the x and y values for each point on the line are substituted into the equation it makes the equation true.

Slide 4: If students complete the task in slide 3 successfully, they will see how the point that the two functions have in common will make both equations true. However, I also like to have students pick a couple of points on the red line and show that they do not make the blue equation true. This serves to solidify the fact that the point where the two lines cross is the only point whose coordinates will satisfy both equations.

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#### Investigation

*15 min*

Students can have the opportunity to practice solving systems of equations graphically and then using their solutions to show that it makes both equations true. The repetition in this investigation allows students to deepen their understanding of this concept while extending their understanding to systems with no solutions and infinite solutions.

**Extention/scaffolds: **During this lesson, I will often pull a small group of students who is having difficulty graphing equations and work with them on both the graphing aspect and the substitution of the solution into each equation.

#### Resources

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#### Closure

*15 min*

This ticket out the door question will allow you to see quickly which students have acquired the understanding from the lesson and the explanation that accompanies their solution will help you guage the depth of the understanding. This information can be used in order to form groups for the next two days of modeling investigations.

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- LESSON 1: Introduction to systems of equations
- LESSON 2: What Does a System of Equations Really Look Like?
- LESSON 3: What is the "Point" of Solving a System?
- LESSON 4: Fitness Center Question
- LESSON 5: Cell Phone Plans
- LESSON 6: How are Systems of Equations related to Equations and Functions?
- LESSON 7: Solving Systems of Equations Without a Graph
- LESSON 8: Practice with the Substitution Method
- LESSON 9: Penny Problem
- LESSON 10: Practice Solving Systems Algebraically
- LESSON 11: Pulling the Systems Concepts All Together
- LESSON 12: An Interesting Lottery
- LESSON 13: Don't Sink The Boat