Unit 6 Variables and Patterns

Variables and Patterns ONLINE BOOK

Unit 6 Overview

In Variables and Patterns, your child will study some basic ideas of algebra and learn some ways to use those ideas to solve problems and make decisions.

The Investigations in this Unit will help your student learn how to:

• Recognize situations in which variables are related in predictable ways
• Describe patterns of change in words, data tables, graphs, and equations
• Use data tables, graphs, equations, and inequalities to solve problems

As your child works on the problems in this Unit, ask them questions about problem situations that involve related quantitative variables:

• What are the variables in the problem?
• Which variables depend on or change in relation to others?
• How can you use a table, graph, equation, or inequality to represent and analyze a relationship between variables?

Investigation 1: Variables, Tables, and Graphs

The goal of this Investigation is to develop students’ ability to look for quantities that change over time, especially distance and speed of moving objects, and their ability to construct and interpret data tables and coordinate graphs representing patterns in that change. These mathematical goals are pursued through work on four Problems in the context of planning and pilot-testing a three-day cross-country bicycle trip from Atlantic City, NJ, to Williamsburg, VA.

Investigation 2: Analyzing Relationships Among Variables

The goal of this Investigation is to extend students’ understanding and skill in working with relationships between quantitative variables as they are expressed in written stories, tables, and graphs. In particular, the Problems of this Investigation focus on cause-and-effect relationships where changes in one variable can reasonably be thought to cause changes in another. In these cases it is natural to call the first quantity the independent variable and the responding quantity the dependent variable (though this designation is sometimes arbitrary).

Investigation 3: Relating Variables With Equations

This Investigation continues development of student understanding and skill in analyzing relationships between quantitative variables by use of algebraic expressions and equations to represent those relationships. The first Problem focuses on algebraic equations involving only one arithmetic operation. The second Problem focuses on special properties of proportional relationships with rules in the form y=mx or y=x+a. The third Problem extends the ideas in the first two Problems to relationships expressed with two-operation rules. The relationships represented by the equations are also represented with tables and graphs. Students use numeric reasoning, tables, graphs, and equations to find the value of a variable in a given relationship. The fourth Problem returns to the Order of Operations rules in an algebraic context.

Investigation 4: Expressions, Equations, and Inequalities

The purpose of this Investigation is to address a number of CCSSM objectives in algebra. These include developing understanding of the term algebraic expression and some beginning skill in checking for equivalence of expressions. The Problems ask students to use context clues to decide when two expressions should be equivalent and then check that equivalence by inspection of values for dependent variables that arise from substitution of equal values for the independent variables. The Distributive Property is revisited and used to show equivalence of simple expressions, such as 3n+3 or 3(n+1). This is intended only as an introduction. The second major focus continues work from Investigation 3 on solving equations using tables, graphs, or equations to find the value of the a variable for equations of the form y=px and y=p+x . The third objective is to address inequalities and their solutions in simple cases.