In this chapter, students solve real-world problems involving addition, subtraction, multiplication, and division of real numbers; make use of the number line, absolute value, and square roots; and solve equations through the application of additive inverses and reciprocals. In the final lesson, students learn the methods of guess and check and working backwards.
Integers and Rational Numbers -
Addition of Rational Numbers -
Subtraction of Rational Numbers -
Multiplication of Rational Numbers -
The Distributive Property -
Division of Rational Numbers -
Square Roots and Real Numbers -
Guess and Check; Work Backward -
Problem-Solving Strand for Mathematics
The problem-solving strategies presented in this unit, Guess and Check and Work Backwards, invite students to use their mathematical intuition and common sense in making sense of everyday experiences with mathematics.
The technique of Guess and Check formalizes what many students have been doing for years: making intelligent guesses and, often, coming up with the right answer. The technique Working Backwards lays the groundwork for solving multi-step equations and helps students understand the need for using opposite operations.
Aligning with the NCTM Process Standards
In the NCTM process standards, the concepts of Connections and Representation receive special attention. In Guess and Check and Working Backwards, students recognize and use connections among mathematical ideas (CON.1), especially when encouraged to look for and think about patterns as they work through the processes. As they write equations that express the reality of the problem situation (R.2 and R.3), they use the language of mathematics to express mathematical ideas precisely (COM.4).
- COM.4 - Use the language of mathematics to express mathematical ideas precisely.
- CON.1 - Recognize and use connections among mathematical ideas.
- PS.3 - Apply and adapt a variety of appropriate strategies to solve problems.
- RP.2 - Make and investigate mathematical conjectures.
- RP.4 - Select and use various types of reasoning and methods of proof.
- 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.