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.
2.
3.
4.
5.
Formulas
Tables
Arrow Diagrams
Machines
Graphs