Overview

The purpose of this lesson is to give students an opportunity to apply systems of equations to a real life situation.  Students start by using javabars to visually represent an exchange of goods on a remote island.  Students then use Graphing Calculator to explore the graph of the system and the possible solution set.  Lastly, students solve the system using a matrix and develop a currency system for the island based on their findings.  This lesson allow students to explore the given system from many different angles.

Objectives

1. Students will set up a system of equations, given a word problem.
2. Students will interpret the graph of a system of equations in three variables by telling how many solutions the system will have and describing those solutions.
3. Students will interpret the solutions of a system of equations, and apply the solutions in a real way. 

Standards Addressed

NCTM Principles and Standards

• apply and adapt a variety of appropriate strategies to solve problems

• recognize and apply mathematics in contexts outside of mathematics

• select, apply, and translate among mathematical representations to solve problems

• use representations to model and interpret physical, social, and mathematical phenomena.

• draw reasonable conclusions about a situation being modeled

• use symbolic algebra to represent and explain mathematical relationships

Georgia Performance Standards

• Build new mathematical knowledge through problem solving

• Solve problems that arise in mathematics and in other contexts

• Students will make connections among mathematical ideas and to other disciplines

• Select, apply, and translate among mathematical representations to solve problems

Prerequisite Skills

Students should understand how to solve a system of equations using matrix algebra.
Students should be familiar with describing the solutions of systems of equations in two variables.
Students should be familiar with Java Bars and Graphing Calculator.

Materials Needed

Pairs of Students (recommended but optional)
Java Bars
Graphing Calculator (computer program)
TI-84 Graphing Calculator (optional but recommended)
Activity Web site
Pencil
Notebook Paper

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Activity

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Rationale

The first objective of this lesson is for students to set up a system of equations given a word problem.  Word problems in general can be difficult for many students.  Matrix word problems are even more difficult since they often involve several variables.  Having students explore the island economy in Javabars allows them visualize the economic exchanges by physically manipulating the goods.  We feel this process helps students in setting up their system of equations.  The second objective is for students to describe the solutions of a system of equations in three variables.  While solving systems of equations using matrix algebra is often emphasized in schools, interpreting those solutions is not.  Graphing Calculator enables students to graph the system in three dimensions and rotate the graph to explore the solutions.  Students are able to identify the solution on the graph and describe it.  The graph provides a rationale for the algebraic solution the students are then asked to find.  When to matrix is solved, students can look back at the graph to interpret the solution.  The last step of the problem is to develop a currency system for the island economy.  Students take their findings from the previous steps and replace the island's bartering system.  We feel this final step reinforces the solutions of the system.  Students are able to see the relationship between the values of the different goods as an interpretation of the solution.

Assessment

Students worksheets will be graded on thoroughness and accuracy.  Although students may work as a pair, each student is required to complete and turn in

1. answers to the questions asked in the assignment
2. printout of Javabars exploration
3. printout of graphing calculator graph
4. currency system for the island economy
5. homework assignment

Each of these components will be worth 10 points.  This project is worth 50 points total.

Evaluation

Throughout the lesson we will continually evaluate technology effectiveness, ease of students' use of the technology, time management, class participation, and possible activity improvements.


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