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# Limiting Your Options

Lesson 6 of 10

## Objective: SWBAT use systems of linear inequalities to find maximum and minimum solutions.

#### Set the Stage

*10 min*

*You will need to have copies of the lemonade information ready for this part of the lesson or be ready to project it on your board. * I begin this lesson by reviewing our discussion yesterday about limits and constraints. I thank my students for their work in helping my friend set up the model she needed to find her most profitable option. I go on to say that we have a new challenge to address today. The question is how much to charge for lemonade to make the best profit. I explain that the lemonade stand sells small and large cups of fresh lemonade. I project the lemonade information on my front board and ask my students to work individually to set up constraints for this problem like they did for others yesterday. **(MP1, MP2, MP4)** While they're working I walk around offering encouragement and reminders as necessary. When everyone is finished I tell them that today we are not just looking for the constraints, we will actually be solving for the optimal result, starting with this problem. I explain that the next step is to graph all the equations and inequalities we've written as constraints, so I distribute graph paper and rulers and let my students get to work. As always, I walk around offering encouragement and redirection as needed. When everyone has completed this section I select one good graph (I look for a graph that is clearly and correctly drawn and labeled) to project and let everyone check their work against it. I tell my students that we're ready to find how many large and how many small cups of lemonade we should sell to make a maximum profit. I circle the points of intersection on the graph and explain that these represent maximum or minimum points within the limits we've set up. You can see the equations and a sketch of the graph on my educreations video.

#### Resources

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#### Put it Into Action

*40 min*

You will need copies of the Linear Programming problems handout for this section of the class. Now that we've worked through an entire linear programming problem as a class, I tell my students that they get to work with their back-partner to solve some problems on their own. I remind them to write the constraints, then graph them, and finally use the points of intersection on the graph to find the desired solution. I distribute the Linear Programming problems handout and ask if there are any questions. **(MP1, MP2, MP4)** While my students are working I walk around offering encouragement and redirection as needed. When everyone is done, I ask them to take a "gallery walk" to check their work against the answers I've posted around the room and ask questions about any they don't understand.

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#### Wrap it Up

*5 min*

I really want my students to remember how important it is to start with good, meaningful equations so I give them a different kind of challenge to wrap up this lesson. I present them with a Linear Programming Challenge all set up with equations, a graph and a solution - except it's the wrong solution. My challenge to my students is that they figure out why it's wrong, where the mistaken assumption is. ** (MP1, MP2, MP6) **This also sets up the lesson for tomorrow, which focuses on whether the solutions are actually viable answers to the problem.

#### Resources

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- UNIT 1: First Week!
- UNIT 2: Algebraic Arithmetic
- UNIT 3: Algebraic Structure
- UNIT 4: Complex Numbers
- UNIT 5: Creating Algebraically
- UNIT 6: Algebraic Reasoning
- UNIT 7: Building Functions
- UNIT 8: Interpreting Functions
- UNIT 9: Intro to Trig
- UNIT 10: Trigonometric Functions
- UNIT 11: Statistics
- UNIT 12: Probability
- UNIT 13: Semester 2 Review
- UNIT 14: Games
- UNIT 15: Semester 1 Review

- LESSON 1: Make It
- LESSON 2: Make it More
- LESSON 3: Double Trouble
- LESSON 4: Going Graphic
- LESSON 5: Out of Bounds
- LESSON 6: Limiting Your Options
- LESSON 7: Does it Work?
- LESSON 8: Twist It
- LESSON 9: Creating Algebraically Review Stations
- LESSON 10: Creating Algebraically; Assessment