Objective

Opening

Investigation

Closing + Homework

SUPPLEMENT: Linear Programming Application Day 1 of 2

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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.1a

Interpret parts of an expression, such as terms, factors, and coefficients.

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

10 minutes

If students have worked on a previous linear programming problem and written about their work, I like to share some student work at the beginning of class that highlights particularly good writing about math thinking. Students can comment on what they notice in the student example and highlight which sentences help the reader understand how the writer found his/her solution.

I have found that students really like to see other students’ work. I leave the name off the student work and allow the author to remain silent about his/her work or allow the author to take credit. Seeing other students’ work allows students to get clearer about what is expected of them in terms of mathematical writing. I have students come up to the Smartboard and highlight sentences that they think show strong mathematical writing.

Next, I introduce today’s task. Students will be working in groups to solve another linear programming problem. This time, they are looking to minimize cost, rather than maximize profit. I have students read the problem aloud in the whole group and ask for clarifying questions.

This lesson is a good opportunity for students to practice their skills in service of reviewing for an assessment or before working on a final project like creating their own linear programming problems.

Investigation

40 minutes

Differentiation

At this point in the lesson, students will break down into pre-determined homogenous groups. This will allow students to grapple with the problem where they are in the learning process. I think about where I position the groups in the classroom so I can be near groups I know will need more help and I let more advanced groups try to figure things out for themselves. Students will have a chunk of time to work on the problem and I circulate to answer clarifying questions and/or provide guiding hints. I usually position myself to sit with the group(s) I know will need the most guidance. I try to let the other groups wrestle with the problem and use each other as resources.

This problem provides students the opportunity to practice SMP 4: Model with Mathematics. The complexity of this problem requires students to map relationships between different variables graphically and then analyze those relationships to draw conclusions.

Things I watch for:

Some students will have trouble figuring out the scale for their graphs. I encourage them to try to find the highest point that will occur on each axis based on their inequalities. This should help them to realize how high they have to go on each axis.
Students may have trouble writing the inequality that reflects they have to admit as many out of district students and Boston students. I have them try out numbers to see which pairs would fit to help them decide about the inequality.
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.

At the end of this section, students should have graphed the three inequalities and be ready to shade the feasible region. If some groups are further along or behind, I may adjust the homework assignment accordingly.

Closing + Homework

10 minutes

Out of class work

For today's end of lesson reflection, I ask students to verbalize how solving the Summer School problem compared to the previous Linear Programming problems they worked on. I ask them specifically: