Objective

Opening

Investigation

Closing

SUPPLEMENT: Linear Programming Application Day 2 of 2

Objective

SWBAT write inequalities using constraints, graph inequalities and identify a feasible region, write a profit function, and find maximum profit.

Big Idea

This lesson gives students the opportunity to synthesize what they have learned before they begin to create their own linear programming problems.

Lesson Author

Amanda Hathaway

Boston, MA

Grade Level

Ninth grade

Tenth grade

Subjects

Math

Algebra

Systems of Equations and Inequalities

Graphing linear inequalities

Standards

HSN-Q.A.1

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.*

HSN-Q.A.2

Define appropriate quantities for the purpose of descriptive modeling.*

HSA-SSE.A.1

Interpret expressions that represent a quantity in terms of its context.*

HSA-CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

HSA-REI.C.5

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

HSA-REI.C.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

HSA-REI.D.10

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

HSA-REI.D.12

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

HSF-IF.B.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

HSF-IF.B.5

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

MP1

Make sense of problems and persevere in solving them.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

Opening

5 minutes

I remind students that they are working on the Summer School Linear Programming Problem again today. The goal for this lesson is for students to finish their work on this problem and be ready to present it to the class.

Students began working on this problem in the previous lesson. I like to start the class by having them reflect on what worked well in their groups yesterday. This can help set the tone for a second productive day. I find that sometimes two days of group work in the same group can lead to some off track behavior or a loosening of classroom norms. So, the real purpose of this Lesson Beginning activity is to set the tone for the day’s work. Students could report out from their groups or I might have each student free write individually: what made their group successful on the previous day?

Investigation

25 minutes

Intervention and Extension

At this point in the lesson, students will go back to their homogenous groups to continue their work. Students should have graphed the three inequalities and found the feasible region by the end of yesterday’s lesson (perhaps with some homework) and are ready to graph the profit line (or in this case, cost line).

Things I watch for:

Students often struggle when they get to the profit line (or in this case, cost line) piece of the work. I remind them that they can choose any number of total costs that the school will spend. They may initially choose a number that is too small; I help them find a better line by encouraging them to choose a larger amount of cost. A number that works nicely for the cost line is $144,000.

As groups finish, I have them prepare presentations on this assignment.

Closing

25 minutes

Students will close the class today by presenting their work on this problem. This is a good opportunity for students to explain the process, how they understand the problem, and use mathematical language. I like to focus on SMP3: Construct viable arguments and critique the reasoning of others here. For the presenters, I focus on the progression they describe to explain their problem solving process. For the observers, I focus on asking clarifying questions that help deepen understanding.

I prompt students to:

Explain how they wrote their inequalities. What inequalities were confusing to write? We spend some time on D > B.
Explain how they sketched their graphs. How did they decide on an appropriate scale? How did they know which side of the line to shade? How did they graph the inequality D > B.
Explain how they knew where the feasible region was.
Explain how they came up with their profit/cost line. How did they decide on a total number of profit?
Explain how they knew where the minimum cost would be. Explain the difference between finding a minimum and a maximum.

At the end of class I let students know that they will be starting to write their own linear programming problems in the next class. This work was in preparation for that process.

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