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Algebra 2
Maximize Your Profits: An Introduction to Linear Programming

Maximize Your Profits: An Introduction to Linear Programming

Algebra 2 Grades High School 8:10 Video

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Lesson Description

Learn how to solve linear programming problems to optimize real-world scenarios. This lesson covers setting up optimization equations, identifying constraints, graphing feasible regions, and finding optimal solutions.

Video Resource

Linear Programming

Mario's Math Tutoring

Duration: 8:10

Watch on YouTube

Key Concepts

Optimization Equation (Objective Function)
Constraint Inequalities
Feasible Region
Vertices of Feasible Region

Learning Objectives

Students will be able to formulate an optimization equation based on a word problem.
Students will be able to identify and write constraint inequalities from a word problem.
Students will be able to graph the feasible region defined by a system of linear inequalities.
Students will be able to find the vertices of a feasible region.
Students will be able to determine the optimal solution to a linear programming problem by testing vertices.

Educator Instructions

Introduction (5 mins)
Begin by reviewing the concept of optimization: finding the maximum or minimum value of a function subject to certain constraints. Introduce the term 'linear programming' as a method for solving these types of problems when the function and constraints are linear. Briefly explain the real-world applications of linear programming, such as maximizing profit or minimizing cost.
Video Viewing and Note-Taking (10 mins)
Play the Mario's Math Tutoring video 'Linear Programming'. Instruct students to take notes on the key steps involved in solving a linear programming problem, focusing on: Writing the optimization equation, Identifying constraint inequalities, Graphing the feasible region, Finding the vertices of the feasible region, Testing the vertices in the optimization equation.
Guided Practice (20 mins)
Work through a similar linear programming problem as a class, guiding students through each step. Emphasize the importance of carefully reading the problem to identify the objective function and constraints. Demonstrate how to graph the inequalities accurately and find the vertices of the feasible region. Show how to substitute the coordinates of each vertex into the objective function to find the optimal solution.
Independent Practice (15 mins)
Provide students with a worksheet containing several linear programming problems. Have them work independently or in pairs to solve the problems. Circulate the room to provide assistance as needed. Encourage students to check their work and compare answers with their classmates.
Wrap-up and Discussion (10 mins)
Review the key concepts of linear programming. Discuss any challenges students faced during the independent practice. Answer any remaining questions. Preview upcoming topics related to optimization and systems of equations.

Interactive Exercises

Real-World Application Brainstorm
Divide students into small groups and have them brainstorm real-world scenarios where linear programming could be applied. Have each group share their ideas with the class.
Graphing Challenge
Provide students with a set of linear inequalities and challenge them to graph the feasible region accurately. Have them compare their graphs with their classmates and discuss any discrepancies.

Discussion Questions

In your own words, explain what linear programming is and what types of problems it can be used to solve.
What are the key steps involved in solving a linear programming problem?
Why is it important to identify the constraints in a linear programming problem?
How does graphing the feasible region help us find the optimal solution?
Can you think of any real-world examples of how linear programming could be used to make decisions?

Skills Developed

Problem-solving
Analytical reasoning
Mathematical modeling
Graphing and visualization
Critical thinking

Multiple Choice Questions

Question 1:

What is the primary goal of linear programming?

To find the maximum or minimum value of a linear function.

To solve systems of linear equations.

To graph quadratic equations.

To calculate the area of a circle.

Correct Answer: To find the maximum or minimum value of a linear function.

Question 2:

Which of the following represents the 'optimization equation' in a linear programming problem?

The equation that defines the feasible region.

The equation that expresses the quantity to be maximized or minimized.

The equation that represents the constraints.

The equation of a straight line.

Correct Answer: The equation that expresses the quantity to be maximized or minimized.

Question 3:

What are 'constraint inequalities' in the context of linear programming?

Equations that define the maximum value.

Equations that define the minimum value.

Limitations or restrictions expressed as inequalities.

Equations that represent profit.

Correct Answer: Limitations or restrictions expressed as inequalities.

Question 4:

The 'feasible region' is:

The area outside the constraint lines.

The area where all constraint inequalities are satisfied simultaneously.

The area defined by the optimization equation.

The area on the negative x and y axes.

Correct Answer: The area where all constraint inequalities are satisfied simultaneously.

Question 5:

How do you find the optimal solution after graphing the feasible region?

Choose a random point within the region.

Test all points on the x-axis.

Test the vertices (corners) of the feasible region in the optimization equation.

Test the midpoint of each constraint line.

Correct Answer: Test the vertices (corners) of the feasible region in the optimization equation.

Question 6:

What does it mean to 'maximize profit' in a linear programming problem?

To find the smallest possible profit.

To find the largest possible profit while satisfying the constraints.

To make the profit equal to zero.

To ignore the constraints.

Correct Answer: To find the largest possible profit while satisfying the constraints.

Question 7:

What is a vertex of the feasible region?

Any point inside the feasible region.

The intersection of two or more constraint lines.

A point on the x-axis.

A point on the y-axis.

Correct Answer: The intersection of two or more constraint lines.

Question 8:

Why is it important to graph the constraints accurately?

To make the problem look more complicated.

To ensure the feasible region is correctly identified.

To impress your math teacher.

The graph doesn't matter.

Correct Answer: To ensure the feasible region is correctly identified.

Question 9:

What is the first step in solving a linear programming problem?

Graph the inequalities.

Write the constraint inequalities.

Write the optimization equation.

Find the vertices.

Correct Answer: Write the optimization equation.

Question 10:

In a linear programming problem, a negative value for a variable typically means:

The solution is optimal.

There is an error in the calculations.

The variable represents a loss instead of a gain.

It's not possible in real world situations.

Correct Answer: It's not possible in real world situations.

Fill in the Blank Questions

Question 1:

The equation you're trying to maximize or minimize is called the __________ equation.

Correct Answer: optimization

Question 2:

The restrictions on the variables in a linear programming problem are called _________.

Correct Answer: constraints

Question 3:

The area where all constraint inequalities are satisfied is called the __________ __________.

Correct Answer: feasible region

Question 4:

To find the optimal solution, you need to test the __________ of the feasible region.

Correct Answer: vertices

Question 5:

Linear programming is used to find the __________ or __________ value of a linear function.

Correct Answer: maximum

Question 6:

The vertices of the feasible region are found at the __________ of the lines representing the constraints.

Correct Answer: intersection

Question 7:

Before graphing, it's helpful to find the x and y __________ of each constraint line.

Correct Answer: intercepts

Question 8:

If the feasible region is unbounded, there might be no __________ solution.

Correct Answer: maximum

Question 9:

The profit equation represents the __________ we are trying to optimize.

Correct Answer: profit

Question 10:

When setting up the constriants, negative amounts of computers can't be made so x and y are __________ than or equal to zero.

Correct Answer: greater

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.A.1.9:Solve systems of linear inequalities in two variables, with a maximum of three inequalities; graph and interpret the solutions on a coordinate plane. Graphing calculators or other appropriate technology may be used. A2.A.1.7:Represent and evaluate mathematical models using systems of linear equations with a maximum of three variables. Graphing calculators or other appropriate technology may be used.

Teaching Materials

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