Variables: Variable Expressions, Interpreting
Variable Directions & Variable Substitution Class
Notes:
Variables are letters or symbols that represent a quantity that can change
or vary. They are typically lower case, like a, b, c, x, y, z, etc. For
example, "t" changes each month because it represents the number of txt
messages sent or received during the month over my 500 included txt
msgs. This month, I sent 537 txt msgs, so my “t” is 37 (537-500=37).
A constant is a quantity that does not change. For example, my txt plan
costs $9.99 per month for 500 txt msgs. The constant in this example is
$9.99. The price per message beyond 500 messages is also a constant,
and is $0.20 per message.
An algebraic expression contains one or more variables and may contain
operation symbols. We can create an algebraic expression to express: “I
will pay $9.99 plus twenty cents per txt message “t” beyond my 500
included messages.”
$9.99 + $0.20t
A numerical expression is an expression that only contains numbers.
So 85  5 and 10 + 3 + 5 are numerical expressions.
Interpreting variable directions is a problem solving skill that means
translating words into math, specifically the language of algebra.
Did you know…?
…the word ALGEBRA is derived from
the word al-jabr, which appeared in the
title of al-Khwarizmi’s treatise on
algebra. Al-Khwarizmi lived during the
late eighth and early ninth centuries.
You can translate English into Spanish, and you
can translate mathematical situations
expressed as words into symbols, and vise versa.
In word problems, you may need to identify the
action to translate words into math, specifically
algebra. In some word problems, word order
may be confusing.
1. First, we look for actions that signal each operation—addition,
subtraction, multiplication and/or division.
2. Second, we translate words into algebraic or numerical expressions,
or write numerical or algebraic expressions with words.
3. Third, we assign a value to each variable and find the value of
each expression (see variable substitution.)
Page 1
Variables: Variable Expressions, Interpreting
Variable Directions & Variable Substitution Class
Notes:
Here is a chart to help you remember which words refer to mathematical
operations:
ACTION
Put together or
combine
Find how much
more or less
Put together
groups of equal
parts
Separate into
equal groups or
fair share
Addition
Subtraction
Multiplication
Division
37 + 28
90 – 12
8  48 or 48  8
Or (8)(48) or 8(48) or
(8)48
 The difference of
90 and 12
 12 less than 90
 take away 12
from 90
Plus…
 decreased by
 fewer than
 the product of 48
and 8
 the product of 8
and 48
 8 groups of 48
Plus…
 Factor of
k - 12
8  w or (8)(w) or 8w
 The difference of
k and 12
 12 less than k
 take away 12
from k
 the product of 48
and 8
 8 groups of w
OPERATION
Numerical
Expression
Words
Algebraic
Expression
Words
Page 2
 37 increased by
28
 The sum of 37
and 28
 28 more than 37
Plus…
 Increased by
 More than
 Combined,
together
 Total of
 Sum
x + 28
 x and 28
 The sum of x and
28
 28 more than x
327
3
4
4  12 or
12
327  3 or
 the quotient of
327 and 3
 the ratio of 327
and 3
 4 out of 12
Plus…
 per (month, day,
etc.)
 out of
 percent (divide
by 100)
n  3 or
n
3
 the quotient of n
and 3
 the ratio of n and
3
Variables: Variable Expressions, Interpreting
Variable Directions & Variable Substitution Class
Notes:
Examples:
Write an algebraic expression for each phrase.
STRATEGY-Remember FITT.
Remember—you reverse the number and the variable when you see:
F (subtracted from)
I (divided into)
T (more than or less than)
T (added to)
1. the product of a number (x) and 3
2. the difference between three times a number (k)
and 4
3. 15 more than the cube of a number (m)
4. the product of two and three divided into twentyfour
5. Rita drove an average of 55 miles per hour on her trip to the
mountains. You can use the expression 55h to find out how many
miles she drove in h hours. If she drove for 5 hours, how many miles
did she drive?
Variable Substitution
We can also evaluate algebraic expressions by substituting given values
for the variable. “Plug in” the given value of the variable for each
expression.
Write the algebraic expression in word form. Then simplify the expression
by substituting the value given for the variable.
Page 3
Variables: Variable Expressions, Interpreting
Variable Directions & Variable Substitution Class
Notes:
Algebraic expression in word form
Simplify the expression by substituting
the value for the given variable.
6.
r + 288
When r = 12
7.
64
u
When u = 16
8.
7z-5
When z = 5
9.
3(w + 4)
When w = 4
10. 21 + 3 •m)
When m = 1
11. 3n + 15
When n = 4
12. e2 – 7
When e = 8
13. 10x – 4y
When x = 14 and y = 5
Let’s go back to our txt msg conversation. This month I sent 537 txt
messages. If my txt plan includes 500 minutes, the number of
messages,“t”, more than 500 included minutes is 537 – 500 = 37. Last
month, I sent 525 txt msgs, so t was 25. Two months before, I sent 515 txt
msgs, so t was 15. I can represent these values in a
“Since my parents agreed to
table, and find the value of each expression.
Finding an expression is similar to finding a rule for a
pattern. For my cell phone, my txt msg plan is $9.99
each month. For all the txt msgs I send over 500, I
have to pay $0.20 per txt msg. Find an algebraic
expression that describes my txt msg plan.
pay $9.99 each month for 500
txts, you can see that I owe
them money every month.”
…Anonymous
Each row of the table individually will satisfy several possible expressions;
you must find an expression that works for all rows in the table.
$9.99 + $0.20t
Month
t 9.99 + 0.2t
This month
37
$17.39
Last month
25
$14.99
Two months ago 15
$12.99
Page 4
9.99 + 0.2(37)
9.99 + 0.2(25)
9.99 + 0.2(15)
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