In this chapter, students discover the methods that determine solutions to systems of linear equations and inequalities. Students begin by solving systems of equations graphically, realizing that the solution for each system is the point of intersection between the two lines represented by the given equations.
Linear Systems by Graphing -
Solving Linear Systems by Substitution -
Solving Linear Systems by Elimination through Addition or Subtraction -
Solving Systems of Equations by Multiplication -
Special Types of Linear Systems -
Systems of Linear Inequalities -
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Problem-Solving Strand for Mathematics
In this chapter, Systems of Equations and Inequalities, several different methods of solving equations and inequalities are taught. Throughout, real-world problems are incorporated into the exercises, and practical applications of important concepts such as constraints, optimum solutions, feasibility regions, and maximum and minimum values are outlined.
In this chapter you might try giving students limited choices about which problems to do independently. This allows you to assess which of the problems students feel comfortable with, which they wish to avoid, and where some re-teaching might be helpful. With the section Comparing Methods of Solving Linear Systems, students might be asked to use all three primary methods on a single exercise rather than doing three different exercises with the same method. In addition to giving necessary practice it may help students become more discerning about which method is simpler in a given situation.
Alignment with the NCTM Process Standards
Using the principle of choice and, occasionally, asking students to write about a problem of their preference can address many of the NCTM Process Standards. Chief among them would be the Communications and Connections standards. Students solving and presenting solutions for equation and inequality problems must organize and consolidate their mathematical thinking (COM.1), communicate their mathematical thinking coherently and clearly to peers, teachers, and others (COM.2), and use the language of mathematics to express mathematical ideas precisely (COM.4). When students share their various insights and approaches to problems as common classroom practice, they learn to analyze and evaluate the mathematical thinking and strategies of others (COM.3).
When students are allowed to select specific problems as part of an assignment rather than to merely mimic the sample problems given in the text, they strive to recognize and use connections among mathematical ideas (CON.1) and to understand how mathematical ideas interconnect and build upon one another to produce a coherent whole (CON.2). They are encouraged to apply and adapt a variety of appropriate strategies to solve problems (PS.3) and use representations to model and interpret physical, social, and mathematical phenomena (R.3).
- COM.1 - Organize and consolidate their mathematical thinking through communication.
- COM.2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
- COM.3 - Analyze and evaluate the mathematical thinking and strategies of others.
- COM.4 - Use the language of mathematics to express mathematical ideas precisely.
- CON.1 - Recognize and use connections among mathematical ideas.
- CON.2 - Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
- PS.3 - Apply and adapt a variety of appropriate strategies to solve problems.
- R.3 - Use representations to model and interpret physical, social, and mathematical phenomena.