SWBAT solve a system of inequalities to identify the feasible region of the solutions.

The point of this activity is for students to solve a system of inequalities to find a treasure, and be able to provide reasoning and feedback to other students.

10 minutes

The focus of today's lesson will be the ability for students to complete the paired activity. In this task they apply solving a system of inequalities on a map to find a hidden treasure. Prior to this task, a Warmup is intended to take 10 minutes. The warmup reviews solving a system of inequalities. It reinforces how to shade and interpret the meaning of the solutions.

For the warm up, I provide the students with 4 questions. The first question requires the students to think backwards to identify the x and y coordinate that satisfies the equation. The second task is to graph the line using the two coordinates from Question 1. Questions 3 and Question 4 review with students that the solutions or side to shade of the line can be found by either testing a point or interpreting the meaning of the inequality in slope intercept form.

**Materials needed for this lesson**:

- copies of the Warmup
- copies of the Exit Slip
- copies of the Treasure Map (one per pair or group of 3 if necessary)
- Instructions for the map activity may be posted on the projector
- ruler
- colored pencils

25 minutes

In this cooperative activity, I use a map off of this website with my instruction sheet. I changed some of the inequalities being used in the activity, and the questions for students to complete. I modified the worksheet to just use the Treasure map only. I give each pair of students or group of 3 if necessary, a treasure map. I give every student Instructions for the map activity, and state that my expectation is for every student to show the work and turn in the answered questions from the instruction sheet. Every pair or group is to put all members names on the map and complete solving the inequalities by graphing with colored pencils on the treasure map. One student example of the treasure activity is available in the resources of this lesson.

In the cooperative activity, all of the inequalities have greater than or less than results, therefore none of the points on the lines are solutions. Some of my students used markers, but the intersection was difficult to determine, so colored pencils are recommended.

Some common mistakes during this cooperative activity are the following:

- The slope is graphed incorrectly.
- The y intercept is graphed incorrectly.
- The shading is on the wrong side of the line.
- The algebra is done incorrectly when using inverse operations to solve for y, or the direction of the sign is not changed when multiplying or dividing by a negative.
- The math is done incorrectly to check the point as a solution.

Students will critique the findings of other students and exchange reasoning in order to form a consensus about the correct location of the treasure in the Peer Feedback. Students will join other groups and provide feedback and make corrections based on the skills listed above. I will assist students that are unable to agree, and use questioning to help them continue moving forward in their productive struggle.

15 minutes

The Peer Check is intended to provide the students immediate feedback on their work, so that they may correct it if needed. During the peer check, I post the following questions on the projector to guide students and to continue to push them forward in their own productive struggle.

- How do you check the slope of a line?
- How do you check the y intercept of a line?
- How do you check the x intercept of a line?
- What method was used to identify the side of the line that should be shaded? Where is the work shown to support the shaded solutions?
- Where is the treasure?
- How have the solutions to the system of inequalities been checked algebraically?
- If the treasure has been found in different locations, identify mistakes, and agree upon where the treasure is located.

10 minutes

I use this Exit slip as a formative assessment to monitor individual student progress. This exit slip states 4 inequalities in the system that students are required to graph, and use the intersection to solve and show the solutions to the system. If time permits, the exit slip is meant to be checked or handed in before students leave class. To provide students immediate feedback, sometimes I have students trade papers to check the exit slip or self-check it. If time is a factor, I assign the exit slip as homework that students are to turn in the following day of class.

The exit slip provides me with a quick check on any remediation needed, common mistakes, or individual student needs. It helps guide me in what I still need to teach or present. During this lesson, I have tried to emphasize that inequalities are formed from the constraints and boundaries created in real world situations. Also, that the solutions to the system of inequalities is in the intersection called the feasible region. I am preparing students for lessons on linear programming that I will present later this school year.

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