SWBAT represent a problem situation using a system of inequalities and represent their solution as an area bounded by linear equations in the coordinate plane.

Life is full of constraints and usually more than one at a time. So what is allowed? Students use systems of inequalities to determine feasible values on the coordinate plane.

5 minutes

At the start of class I will show students Slide 2 from System of Inequalities Day 1. I will give students time to process this comparison with a partner. As students discuss the inequalities, I listen for misconceptions such as misreading the inequality (x + 2 is less than 5) or misunderstanding the solution set.

15 minutes

As we begin our work, I display Slide 3 of System of Inequalities Day 1. The question on this slide asks students to begin the process of trying to understand the problem. I allow students time to work quietly by themselves. I ask them to write down everything they can in terms of possible solutions that would make the constraints true.

**Teaching Note**: Some students will invariably decide that they have “the list” of all lengths and widths that would satisfy both constraints. For these students try to find out (without giving them too much information) if they had considered widths such as 4.1 or 8.2, etc.? This will get them thinking about all of the decimal possibilities and will help them realize that there are actually an infinite number of solutions that satisfy both of the constraints.

Once all students have thought through the problem completely and made a list of some possible solutions, I ask each student to pair up with their partner and share results. I encourage them to focus on both commonalities and differences in the lists they had constructed. I ask them to justify why they chose the values that they did** (MP3)**.

Next, I ask pairs to determine a way to visually represent their solution to this problem. As students discuss their options, I listen for students that are suggesting the use of the coordinate plane. I keep those groups in mind as “go to” groups when it is time to share out with the whole class.

- My students have some experience with graphing inequalities. Nonetheless, it is important to take time in this lesson to review the choice of inequality symbols and the representation of boundary lines (dotted or solid).
- When graphing the two inequalities, determine the boundary line first (ex: 2x + 2y = 20) and ask students how to graph the line (either using intercepts, slope intercept, or some other viable method).
- Determine if the points on the boundary line would be viable points or if there are others.

Typically, students will name integer values that are below the line that would also make the inequality true. Now would be a good time to bring students that you extended in the beginning (those thinking about rational values) into the conversation to help push other students thinking.

Once students realize it would be impossible to plot every point that makes the inequality true, they should brainstorm another method to show all of the solutions to the inequality. s When graphing the second inequality y > 4. I ask students to plot the line first. Then, I ask students, "Why are points on that line not solutions?" Finally I ask, "How will we represent a boundary that does not contain any points in the solution set?"

**Teaching Note**: It is a good idea to use two different colors to shade graphs to help the solution area stand out.

Before we move on, I ask pairs to choose at least two test points in the solution area and explain to their partner why they are viable solutions to the problem.

20 minutes

As we prepare for the Don't Sink the Boat problem I give each student graph paper or whiteboards. I intend for them to investigate this question by modeling the constraints. I anticipate that my students will employ both numeric and algebraic methods.

If students struggle to understand the question, I may ask them to work on Part 2 of the question first. Perhaps by substituting values, students will obtain a better understanding of the problem and then work towards a generalization and model of the problem.

5 minutes

For today's Ticket Out students need to prove to someone why they are incorrect. I remind students that proving and telling are not the same. In order to prove something is incorrect they must demonstrate, using mathematics, why the solution would not be true. This assessment will give me a good idea of how students are thinking about the solution set to a system of inequalities in two dimensions at this point in the unit.

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