Lesson 7: Expressions
Now that you can identify real numbers and perform basic arithmetic operations, you can write
expressions. An expression is a group of numbers, symbols, and operators (such as + and ×) that show
the value of something. For example, 3 × 6 is an expression. So is 9 + 2.
Lesson Objectives
After completing this lesson, you will be able to:


Describe numerical expressions.
Describe algebraic expressions.
Numerical Expressions
“Numerical” means involving numbers. Therefore, a numerical expression is an expression that involves
a combination of numbers with one or more operational symbols (such as + or –). Examples of numerical
expressions include 4  8 , 22  9 , 4(8  3) , 9  4  6 , and 3  2  4  .
2
Numerical expressions can be evaluated, or simplified. When you evaluate an expression, you perform
calculations according to the order of operations in order to find the value of the expression.
Here are some examples in which numerical expressions have been simplified:
5  (9  2)  5  18  13
4(6  3)2  4(3)2  4(9)  36
32  9(6)  9  54  45
Algebraic Expressions
Expressions can also be algebraic. An algebraic expression is a mathematical sentence (essentially) that
includes one or more values (terms). At least one term includes a variable. A variable is a letter or
symbol that represents some unknown value. The variable is the part of the expression that may
change. (In other words, the variable varies.)
Algebraic expressions can look complex when you are first reviewing them. Some examples of algebraic
expressions include a  2 , 3a2 , 2a  9 , and 4(a  6) .
Algebraic expressions are useful for representing mathematical and real-world situations. Figure 1
shows a few examples of statements that can be written using algebraic expressions.
Verbal Statement
2 more than a
5 less than 2 times b
The sum of 2 squared and c
Algebraic Expression
a2
2b  5
22  c
Figure 1—Examples of Algebraic Expressions
Sets of real numbers can be modeled with algebraic expressions. Figure 2 illustrates a few of these
representations.
Number Set
Positive, even numbers
Positive, odd numbers
Square numbers
Algebraic Expression
2a , where a represents a
counting number
2a  1, where a represents a
counting number
a 2 , where a represents a
counting number
Figure 2—Number Sets Modeled with Algebraic Expression
Finally, real-world situations can be modeled with algebraic expressions. Figure 3 provides several
examples.
Real-World Situation
Algebraic Expression
Cost of two sandwiches
Cost of a soda and a candy bar
Cost of four popcorns and six sodas
Sum of two rolled dice
Product of two rolled dice
2s
s c
4 p  6s
xy
xy
Figure 3—Real-World Situations Shown as Algebraic Expressions
Remember, the variable in an expression is the part of the expression that may change. That is the part
of the expression represented by the letter, such as s, c, p, x, y, etc.
Evaluating Algebraic Expressions
Algebraic expressions can be evaluated by replacing the unknown variable with a numeric value.
Suppose the items shown in Figure 3 were valued as follows:





Sandwich: $3.95
Soda: $1.95
Candy Bar: $2.25
Popcorn: $2.75
Two rolls of a dice show a 3 and a 5
When these values are plugged into the algebraic expressions, they make more sense. Review Figure 4.
This table shows the same information as Figure 3, but with values plugged in.
It also shows how the expression can be evaluated once its values are known.
Algebraic
Expression
What does the Substitution
variable mean?
Values
2s
s = cost of one
sandwich
2(3.95)
$7.90
How much do a soda and
candy bar cost?
s c
s = soda
c = candy bar
1.95  2.25
$4.20
How much do four
popcorns and six sodas
cost?
4 p  6s
p = popcorn
s = soda
4(2.75)  6(1.95)
Real-World Situation
How much do
sandwiches cost?
two
What is the sum of two
rolled dice?
xy
What is the product of
two rolled dice?
xy
Value on first
dice: x
Value on second
dice: y
Value on first
dice: x
Value on second
dice: y
of
Evaluation
$22.70
35
8
35
15
Figure 4—Real-World Situations with Values Applied
Now that you are familiar with exponents and expressions, take your next examination, Lessons 6-7
Examination. Then, move on to the next lesson.