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Linear Inequalities Lesson Plan
 

Abstract

This lesson is designed to discuss concepts of Linear Inequalities as use them to maximize profit in a business they create.

Standards (NCTM)

Algebra

Represent and analyze mathematical situations and structures using algebraic symbols.
  • Develop an initial conceptual understanding of different uses of variables;
  • Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope.
  • Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships;
  • Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.

Use mathematical models to represent and understand quantitative relationships.
  • Model and solve contextualized problems using various representations, such as graphs, tables and equations.

Student Prerequisites

  • Arithmetic:

    Students must be able to:

    • perform basic operations with numbers such as addition, subtraction, multiplication, and division
  • Algebra:

    Students must be able to:

    • solve for a variable in linear equations.
  • Technological:

    Students must be able to:

    • perform basic mouse manipulations such as point, click, and drag.
    • use a browser such as Safari for experimenting with the activities.

Teacher Preparation

Teacher will need:
  • Article The Walmart You Don't Know (Fast Company, Issue 77 December 2003 by Charles Fishman) or another article that relates pricing with profit.
Students will need:
  • access to a browser.
  • pencil and paper.


    • Lesson Outline

    1. Focus and Review

      • Read the portions of the article that pertain to Vlasic pickles.
      • Explain that this article exemplifies how pricing a product affects profit and reminds consumers that the objective of a business is to maximize profit.

    1. Objectives

      2. Students will work with and solve linear inequalities in two variables using a combination of algebra and graphing. Students will also get to see a practical application of linear inequalities by creating a business and maximizing their profit.

    1. Guided Practice

      • Ask the students if they have seen a problem like 10x+5y &le 25. Ask them if they know how to solve this algebraically and graphically. Have students solve this problem on their own and then have one of them demonstrate their solutions in front of the class.
      • Ask students if they know how to solve a system of linear inequalities like:

        10x+5y &le 25
        12x+3y &le 27

      • What does it mean to solve a system of linear inequalities? How would you graph these? Have the students graph these on their own using graph paper.
    1. Teacher Input

      • Make sure the students understand that solving a system of linear inequalities means to find the graph of all ordered pairs of real numbers that simultaneously satisfy all the inequalities in the system.
      • Using a graphing program on your computer, preferably one that graphs inequalities, graph this set of inequalities. If you don't have such a program, use Graph Sketcher on Interactivate's website.
      • Make sure a few definitions and concepts are made clear:
        • Solution or feasible region: the graph of all the ordered pairs of real numbers that satisfies all the linear inequalities simultaneously.
        • Corner Points: a point in the solution region that is the intersection of two boundary lines.
      • Explain that we are going to use this knowledge of Linear Inequalities to talk about Linear Programming.
        Linear Programming: a mathematical process that has been developed to help management in decision making.
      • Describe the activity: Using Linear Programming, students are going to create a business where the purpose is to maximize profit!! They will design the products and determine the labor departments, hours, and profit. This is a challenging activity so go over an example so students see how everything is played out.
        • In your business, you decide to produce M&Ms and Snicker Bars. Your two labor departments for production are the Chocolate Coating department and the Packaging Department. It takes .7 hours to chocolate coat the M&Ms and 1.2 hours to package them. For Snickers, it takes 1.1 hours to chocolate coat and .8 hours to package. With this information, develop two linear equalities if the maximum labor hours for the chocolate coating and packaging departments are 500 and 456 hours.
        • If M&Ms are sold for a $1.00 a package, your profit might be about $.30 and if the Snickers are sold at $.75, your profit would be about $.25 (these numbers are made up). Thus, your profit equation would be: P=.30X+.25Y Once again, the objective is to maximize profit!
      • Steps to follow using your example:
        1. Determine your variables. X: # of M&Ms produced monthly. Y: # of Snickers produced monthly.
        2. Determine Labor Departments: Chocolate Coating and Packaging
        3. Determine Labor Hours: .7 hours for M&Ms and 1.1 hours for Snickers in the Chocolate Coating Department; 1.2 hours for M&Ms and .8 hours for Snickers in the Packaging Department. We also decided that the hours restrictions per month were 500 and 456 hours for the two departments.
        4. Write inequalities and other restrictions:
          .7X+1.1Y &le 500
          1.2X+.8Y &le 456
          X &ge 0
          Y &ge 0
        5. Write Profit Equation: P=.30X+.25Y (For every M&Ms sold, you make $.30 and for every Snickers sold you make $.25).
        6. Graph the inequalities using Graph Sketcher and determine the feasible region.
        7. Find the Corner Points:
          Fundamental Theorem of Linear Programming:If the optimal value of the objective function in a linear programming problem exists, then the value must occur at one (or more) of the corner points of feasible region.
          8. Therefore, the optimization (maximum profit) will occur at one of the corner points.
        8. Plug corner points into Profit equation (you should have three) and determine which pair maximizes profit (this is under the assumption that all the M&Ms and Snickers that you produce are sold).
    1. Guided Practice

      • Have the students, in pairs, create their own business modeling it similar to yours. Let them choose what products they will produce, labor departments, labor hours, and the profit equation. They will want to use Graph Sketcher to graph their inequalities and a mixture of both the graphs and algebra to solve for the corner points. Although the students don't know exact values for labor hours and profit, let them come up with their own as the point of the assignment is to learn how to use linear inequalities to maximize profit.
    1. Teacher Input

      • Because students are developing their own equations, their lines may not intersect on the graph. Therefore, help them develop, if necessary, equations that when graphed, intersect each other in the first quadrant.
    1. Guided Practice

      • When students are done, have them share their results with the class, including the graph they found and how much of each product they should sell to maximize profit.
    1. Closure

      • Review the definitions, concepts, and procedures discussed in class. Ask them to share any thing they found was interesting/frustrating and discuss their thoughts.


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