Standard 5.21a,b / Patterns, Functions and Algebra
Page 1
Variables and Expressions
(Study Guide) A
variable is:
a symbol that can stand for any one of a set of numbers or other objects.
a quantity that can have different values. Any letter can be used as a variable.
a symbol that represents an unknown number (can be letters, boxes, or more).
A variable expression:
is a variable or a combination of variables, numbers, and symbols.
represents a mathematical relationship.
does not have an equal sign (=).
is like .a word phrase ("in the house" without a subject and verb).
The following are examples of variable expressions:
x
n-3
t+8
Variable expressions can also be written for a verbal expression such as "5 more than a
number." Which one of the variable expressions below represents the verbal
expression "seven less than a number?"
n+7
n-7
7xn
n+7
There is only one way to write expressions involving addition and subtraction, but we
can write expressions involving multiplication and division in two different ways. Here
are the ways that variable expressions can be written:
Addition
Subtraction
Multiplication
Division
a+5
x-3
6y or 6 x y
d 8 or d/8
You may have noticed that using the variable x can be confusing because the
multiplication sign (x) is very similar to the letter "x". Because of this possible confusion,
in algebra you might see multiplication indicated in different ways- with the multiplication
sign, with parentheses, with a small dot, or with nothing at all.
3b
3xb
3(b)
3 b
All of these examples mean exactly the same thing: three times a number, b.
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Standard 5.21a,b / Patterns, Functions and Algebra
Page 2
Variables and Expressions
(Study Guide)
Sometimes, math problems can be challenging, especially if they are word problems
and represent events that happen in real-life. One strategy in problem solving is to
guess, but guessing can be too difficult sometimes. Mathematicians use algebra to help
them solve problems. With algebra you are able to turn real-life problems into
mathematical expressions and sentences that help you solve problems.
When we use a variable in math, we say "n represents (a number of something)." This
number might be the number of apples your mom wants to buy at the store, or the cost
of a movie tiqket, or the length of the side of a rectangle. Consider this problem:
Mrs. Smith went to the store to buy some apples and some bananas. She spent $5.00.
The apples were $2.00 a pound and the bananas
a dollar less a pound than the
apples. Mrs. Smith needed at least 2 pounds of the apples for her pie. How many
pounds of each did she buy?
To solve this problem, we can start by saying, "Let p (or any other letter) represent the
number of pounds."
If p represents the number of pounds, then $2.00 x p stands for the total cost of the
apples. If the bananas are $1.00 less than the apples, then they cost $1.00 a pound. In
algebraic language is would be written $1.00 x p. Both $2.00 x p and $1.00 x p are your
variable expressions.
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Standard 5.21c / Patterns, Functions and Algebra
Page 3
Open Sentences
(Study Guide)
An open sentence is:
a statement that contains a variable and an equal sign.
Open sentences can also be written for a verbal expression such as "The number of
crayons is 15." That sentence would be written as c = 15, where c is a variable that
stands for the number of crayons.
Remember how we compared a variable expression to a written phrase? We can also
compare an gpen sentence in math to a complete sentence in writing. A complete
sentence in writing must have a subject and verb. An open sentence in math must have
a variable as a subject and an equal sign as a verb.
The following are examples of open sentences:
7 + n = 15
32-x = 24
36-y = 9
px7 = 28
How would you write "5 less than a number is 20?"
5-x = 20
20-5 = x
x-5 = 20
5 + x = 20
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Standard 5.22 / Patterns, Functions and Algebra
Page 4Open Sentences in Problem Solving
Remember, it helps to take problems with numbers and translate them into
mathematical sentences to solve them. The variable is a letter that stands for the value
that you are looking for.
How many potato chips are in the bag if the bag plus five more equals 43 potato chips,
where b stands for the number of chips in the bag?
In this problem, b is the number we are looking for. It is the number of chips in the bag
Let's try another problem.
Jack's dog weighs twice as much as his cat. If his dog weighs 28 pounds, how much
does his cat weigh?
Let p stand for the weight of the cat. Then we will put the rest of the problem into a
mathematical sentence to include an equal's sign.
2p = 28
The 2 is right next to the variable p, and that means it is 2 times the variable. There
doesn't have to be a multiplication sign (x) when a variable and a number are places
side by side.
How about this problem: Tickets to the concert cost $12. Which of the following
sentences shows the cost (c) of 8 tickets?
•
12c = 8
• 12xc = 8
• 8c=12
• 12+ 8 = c
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