MTH 232 Section 8.1 Algebraic Expressions, Functions, & Equations

Algebra: So Soon? In grades 3 – 5, all students should: represent the idea of a variable as an unknown quantity; express mathematical relationships using equations. Source: Principles and Standards for School Mathematics by NCTM, page 158.

Section Topics: the meaning and uses of variables; how to form algebraic expressions involving variables; the definition and visualization of functions; the solution of an equation by evaluating the unknowns, by making a graph, or through algebraic means.

Constants and Variables Constants are fixed values—e.g., the number of feet in a mile, the number of sides in a triangle, the sum of 5 and 7. Variables are changeable quantities, usually denoted by a symbol (typically letters). Variables are used in at least four different ways in algebra:

1. Variables Describe Generalized Properties A generalized variable represents an arbitrary member of a set for which a property holds. Example: the Associative Property of Real Numbers states that, for all real numbers a, b, c; (a + b) + c = a + (b + c)

2. Variables Express Relationships Mary has seven more marbles than John. Let M equal the number of marbles Mary has, and Let J equal the number of marbles John has. Express the relationship between the number of marbles each child has in three different ways.

3. Variables Serve as Unknowns In Relationships In earlier grades, students might be asked to find a number that makes a particular sentence true: By middle school, the questions become more formal (and comprehensive): find all values of x for which (2x – 8)(x + 3) = 0.

4. Variables Express Formulas Indicate what each formula represents in the following examples: 1.d = rt 2.P=2L + 2W 3.F=(9/5)C + 32

Some Important Definitions A numerical expression is any representation of a number that involves numbers and operation symbols. An algebraic expression is a representation that involves variables, numbers, and operation symbols. An equation is a statement in which two algebraic expressions are equal. The domain of a variable is the set of values for which the expression is meaningful.

More About Equations Every equation falls into three categories: 1.Identity (true for all values) 2.Contradiction (true for no values) 3.Conditional (true for certain values) The solution set to a given equation is the set of all values in the domain that satisfy (make true) the equation.

Functions In many cases, the value of one variable is dependent upon the value(s) of another variable (or other variables). The rule that connects these variables is called a function. Functions can be used to show relationships between quantities, describe change, and make predictions.

Ways To Express Functions 1.Formulas 2.Tables 3.Arrow Diagrams 4.Machines 5.Graphs