How can systems of linear equations model real-world situations?

What are the limits of mathematical representation and modeling?

When is the “correct” answer not the best solution?

Notes:

The intent of this unit is to develop in students an understanding that systems of linear equations can be used to solve real world problems.

In order for all students to successfully complete the learning activities, academic prompts and Differentiated Instruction should be integrated throughout the unit.

·       Interest can be developed through a variety of real-world problem situations.

·       Readiness can be determined by teacher observation and/or pre-assessment.

·       Learning styles and readiness levels can be addressed through visual presentations, such paper-and pencil graphs and calculator-generated graphs; kinesthetic presentations, such as those utilizing graphing calculators; auditory presentations, such as group reporting and teacher-directed instruction.

·       Content can be differentiated by readiness as students are provided with open-ended academic prompts designed to provide insight into the current level of student understanding.

·       Process can be differentiated through the use of whole class, small group, paired and individual activities.

·       Product can be differentiated as students are prompted to demonstrate their understanding with written explanations and by creating visual displays.

·       Scaffolding can be provided through the use of exit slips, which inform the teacher of the immediate learning needs of individual students.

Unit Summary:

In the context of a real-world situation, students create two linear equations that model the conditions.  Students act out/model the situations that are simultaneously imposed in the problem.  Students progress to using tables and graphs to model the situation.  They find values that make both equations true and come to understand that the intersection of the two lines provides a solution to the system of equations and to the real-world problem.

In small groups, students investigate solutions to systems of linear equations.  They interpret and describe solutions to systems that have no solution (inconsistent), one solution, or infinitely many solutions (redundant).

As students work in small groups to solve problems, watch for student ability to collect data in tables, write linear equations in slope-intercept form, and determine the solution. 

Students then progress to algebraic methods of solving systems of linear equations in two variables:  the substitution method and the elimination method.   These methods allow students to determine the exact solution of a system without graphing or constructing a table. 

The elimination method provides an opportunity to review the value/usefulness of “standard form.”   Students develop their skills in using and choosing among these methods as they use systems of linear equations to model real-world situations (see Air Traffic Controller, http://smartskies.arc.nasa.gov.).

Asking students to create Frayer models of these three methods of solving systems of linear equations (graphs, substitution, elimination), gives them an opportunity to compare and contrast the advantages and disadvantages of each.