In this chapter, students are introduced to inverse variation. They graph rational functions and divide polynomials. After being introduced to rational expressions they learn to add, subtract, multiply, and divide them. Students then solve rational equations. The chapter ends with surveys and sampling methods.
Inverse Variation Models -
Graphs of Rational Functions -
Division of Polynomials -
Rational Expressions -
Multiplication and Division of Rational Expressions -
Addition and Subtraction of Rational Expressions -
Solutions of Rational Equations -
Surveys and Samples -
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Problem-Solving Strand for Mathematics
The first portion of this chapter cements connections between algebraic and geometric ways (graphical displays) of representing functions and relationships in mathematics. Vertical, horizontal, and oblique asymptotes visually confirm what students have learned earlier in their studies: division by zero is undefined in our number system. The connection between inverse variation models and the graphs of rational functions is presented in the examples and review questions which pose real-world problems using rational functions.
In the mid-lesson of this unit, the division of polynomials is related back once again to graphing and the horizontal asymptote studied previously. In the lesson, Rational Expressions, values that are excluded when simplifying rational expressions are shown to be those very values that are vertical asymptotes, values that cannot exist for . Students who become adept at moving between the algebraic and the graphic will have more insights into problems they encounter and thus more tools for tackling future challenges.
The last lesson of the unit looks closely at topics in statistics. Students are asked to identify biased questions as well as biased sample populations and to display, analyze, and interpret statistical data effectively. Let students know they will have a survey project to complete before introducing the Surveys and Samples materials lesson and before reviewing Examples 3 and 4, Designing a Survey, and Examples 5 and 6, Display, Analyze, and Interpret Data, since the information needs to be seen in context in order for its value to be recognized.
Alignment with the NCTM Process Standards
The first lessons of this chapter consistently recognize and use connections among mathematical ideas (CON.1), encourage students to understand how mathematical ideas interconnect and build on one another to produce a coherent whole (CON.2), and—especially in real-world problem solving scenarios—recognize and apply mathematics in contexts outside of mathematics (CON.3). In the lessons, Rational Expressions and Solutions of Rational Equations, classic work and motion problems are presented so as to put these problem-solving techniques into meaningful perspective (PS.1, PS.2, and PS.3). In the Surveys and Samples lesson, students create and use representations to organize, record, and communicate mathematical ideas (R.1); select, apply, and translate among mathematical representations to solve problems (R.2); and use representations to model and interpret physical, social, and mathematical phenomena (R.3).
- 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.
- CON.3 - Recognize and apply mathematics in contexts outside of mathematics.
- PS.1 - Build new mathematical knowledge through problem solving.
- PS.2 - Solve problems that arise in mathematics and in other contexts.
- PS.3 - Apply and adapt a variety of appropriate strategies to solve problems.
- R.1 - Create and use representations to organize, record, and communicate mathematical ideas.
- R.2 - Select, apply, and translate among mathematical representations to solve problems.
- R.3 - Use representations to model and interpret physical, social, and mathematical phenomena.