Algebraic
Expressions
Education's purpose is to replace an
empty mind with an open one.
Malcolm Forbes
Expressions
Math expressions represent a
convenient way to translate verbal
expressions
What is the area of a rectangle?
Length times Width
If the length is 3 meters and the width
is 2 meters, what is the area?
A=LxW
A = 3 x 2 = 6 m2
A, L and W are the variables. It is
any letter that represents an unknown
number.
Algebraic Expression
An algebraic expression is a quantity that
contains numbers and variables.
x + y , 3a2  a , 3x + 2y  z
Terms
A term is a number, a variable, or a product of
numbers and variables.
Components of an
Algebraic Expression
 Constant
 Variable
term: fancy name for a number
term: terms with letters
3xy – 4z + 17
 Variable expression with 3 terms:

3xy, -4z, 17
 2 variable terms and 1 constant term
 Example:
Variable Terms
 Consist
 The
of two parts
variable(letter) part
 The number part
 Example:
 2xy has a coefficient of 2
 -6j has a coefficient of –6
 W has a coefficient of 1
In expressions, there are many
different ways to write
multiplication.
1)
2)
3)
4)
5)
ab
a•b
a(b) or (a)b
(a)(b)
axb
We are not going to use the multiplication
symbol any more. Why?
Division, on the other hand, is
written as:
1)
x
3
2) x ÷ 3
Example of evaluating an
expression.
Evaluate 3xy – 2x + 7y
when x = 2 and y = 3
3(2)(3) – 2(2) + 7(3)
18 – 4 + 21
14 + 21
35
The value of the expression is 35.
Word Phrases as Algebraic
Expressions
Addition
(+)
sum
plus
added to
more than
increased by
total
Subtraction
(–)
difference
minus
subtract
less than
decreased by
less
Multiplication
(·)
product
times
multiply
of
double/triple
Division
()
quotient
divide
shared equally
among
ratio of
Word Phrases as Algebraic
Expressions
Write as an algebraic expression. Use x to represent
“a number.”
5 decreased by a number
In words:
Translate:
5
5
decreased by
–
a number
x
Word Phrases as Algebraic
Expressions
Write as an algebraic expression. Use x to represent
“a number.”
The quotient of a number and 12
The quotient of
In words: a number
Translate:
x
and
12

12
x
or
12
Write an algebraic expression
for the following
m increased by 5
m+5
7 times the product of x and t.
7(xt) or 7xt
Write an algebraic expression
for the following
11 less than 4 times a number.
4n - 11
two more than 6 times a number.
6n + 2
Which of the following expressions
represents:
7 times a number decreased by 13
1.
2.
3.
4.
7x + 13
7x - 13
13 - 7x
13 + 7x
Which one of the following expressions
represents:
28 less than three times a number
1.
2.
3.
4.
28 - 3x
3x - 28
28 + 3x
3x + 28
Which of the following verbal
expressions represents:
2x + 9
1.
2.
3.
4.
9 increased by twice a
number
a number increased by nine
twice a number decreased
by 9
9 less than twice a number
Which of the following verbal
expressions represents:
x2 + 2x
1.
2.
3.
4.
the sum of a number squared
and twice a number
the sum of a number and twice
the number
twice a number less than the
number squared
the sum of a number and twice
the number squared
Which of the following expressions
represents:
four less than the cube of a number
1.
2.
3.
4.
4 – x3
4 – 3x
3x – 4
x3 – 4
Terms
 Like
terms
 Terms
with the same variable part
 Same means same letter(s) and power(s)
2x, -5x
¾x2, 7x2
31xy, 4xy
Terms
 We
simplify variable expressions by
combining like terms.
 To
combine like terms, work with the
coefficients of the like terms
Terms
 We
simplify variable expressions by
combining like terms.
 To
combine like terms, work with the
coefficients of the like terms
Combining Like Terms

t + t + t+ t + t

There are five variables which are like
terms therefore we simply add them like
we would if they were numbers.

t + t + t+ t + t = 5t
Combining Like Terms
4t + 3t + t = (4+3+1)t = 8t
2t2 + 8 – 5t2

Rearrange the variables so that all like terms are
side by side.
2t2 – 5t2 + 8 (notice that the sign in front of the
number came with the number)
2t2 – 5t2 + 8 = -3t2 + 8
Combining Like Terms
2t + 3t2 – 2t –t2
3t2 – t2 + 2t – 2t (collecting like terms)
2t2 + 0
2t2
Which figure below models the
simplification of - 4x - 5 + 7x + 7 using
these tiles?
1.
2.
3.
4.
Combining Like Terms
3x + 5 – 9x =
– 6x + 5
-5 +3b – 7 – 5b =
– 2b - 12
3b – 5b = -2b
-5 – 7 = -12
Simplifying Algebraic
Expressions
Rewrite using as few symbols as possible
 Use the distributive property if necessary
to remove parentheses.
 Combine like terms
 More often than not will have numbers and
letters in the final answer.

Distributive Property
Distributive Property
a(b  c)  a  b  a  c
or
a(b  c)  a  b  a  c
Objective - To use the distributive property
to simplify numerical and variable
expressions.
Order of Operations
3(4  5)  3(9)  27
Distributive 3(4)  3(5)
Property
It works!
12  15
27
Why use the distributive property?
3(x  2)  3(x)  3(2)  3x  6
Simplify using the distributive property.
1) 5(x  3)
4) 4(3  y)
5 x  53
43 4 y
5x  15
12  4y
2) 6(y  7)
6 y  6 7
6y  42
5) 10(x  7)
10  x  10  7
10x  70
3) 3(m  8)
3 m  38
3m  24
6) 4(k  2)
4k  42
4k  8
Geometric Model for Distributive Property
A  wl
4
5
2
Two ways to find the area of the rectangle.
As a whole
As two parts
4   5  2
Geometric Model for Distributive Property
4
45
4 2
A  wl
5
2
Two ways to find the area of the rectangle.
As a whole
As two parts
same
45  4 2
4   5  2
4   5  2  4  5  4  2
Scientific Notation
A short-hand way of writing
large numbers without
writing all of the zeros.
The Distance From the Sun to
the Earth
93,000,000
Step 1
Move decimal left
 Leave only one number in front of
decimal

Step 2

Write number without zeros
Step 3

Count how many places you moved
decimal
 If
moved to left, make it positive
 If moved to right, make it negative

Make that your power of ten
 93,000,000
--Standard Form
x 107 --Scientific Notation
 9.3
Practice Problem
Write in scientific notation.
Decide the power of ten.
1)
2)
3)
4)
5)
10?
98,500,000 = 9.85 x
64,100,000,000 = 6.41 x 10?
279,000,000 = 2.79 x 10?
4,200,000 = 4.2 x 10?
0.000013 = 1.3 x 10?
9.85 x 107
6.41 x 1010
2.79 x 108
4.2 x 106
1.3 x 10-5
Complete Practice Problems
Write in scientific notation.
1)
2)
3)
4)
50,000
7,200,000
802,000,000,000
0.000000000631
1) 5 x 104
2) 7.2 x 106
3) 8.02 x 1011
4) 6.31 x 10-10
Write in Standard Form
Positive exponent → move to right
Negative exponent → move to left
x 106
 9.01 x 104
 3.95 x 10-3
 6.27
 6,270,000
 90,100
 0.00395