Algebraic Expressions
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 WidthIf the length is 3 meters and the width is 2 meters, what is the area?
A = L x WA = 3 x 2 = 6 m2
A, L and W are the variables. It is any letter that represents an unknown number.
An algebraic expression is a quantity that contains numbers and variables.
x + y , 3a2 a , 3x + 2y z
A term is a number, a variable, or a product of numbers and variables.
Terms
Algebraic Expression
Components of an Algebraic Expression
Constant term: fancy name for a number
Variable term: terms with lettersExample: 3xy – 4z + 17
Variable expression with 3 terms: 3xy, -4z, 172 variable terms and 1 constant term
Variable Terms
Consist of two partsThe variable(letter) part The number part
Example:2xy has a coefficient of 2-6j has a coefficient of –6W has a coefficient of 1
In expressions, there are many different ways to write multiplication.
1) ab 2) a • b 3) a(b) or (a)b 4) (a)(b) 5) a x b
We are not going to use the multiplication symbol any more. Why?
Division, on the other hand, is written as:
1)
2) x ÷ 3
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 + 2114 + 2135The value of the expression is 35.
Word Phrases as Algebraic Expressions
Addition
(+)
Subtraction
(–)
Multiplication
(·)
Division
()sum
plus
added to
more than
increased by
total
difference
minus
subtract
less than
decreased by
less
product
times
multiply
of
double/triple
quotient
divide
shared equally
among
ratio of
Word Phrases as Algebraic Expressions
Write as an algebraic expression. Use x to represent “a number.”
In words:
5 decreased by a number
5
Translate: 5decreased 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
In words: a number
Translate: x
and 12
12 or 12x
The quotient of
Write an algebraic expression for the following
m increased by 5
7 times the product of x and t.
m + 5
7(xt) or 7xt
Write an algebraic expression for the following
11 less than 4 times a number.
two more than 6 times a number.
4n - 11
6n + 2
Which of the following expressions represents: 7 times a number decreased by 13
1. 7x + 132. 7x - 133. 13 - 7x4. 13 + 7x
Which one of the following expressions represents:28 less than three times a number
1. 28 - 3x2. 3x - 283. 28 + 3x4. 3x + 28
Which of the following verbal expressions represents:
2x + 91. 9 increased by twice a
number
2. a number increased by nine
3. twice a number decreased by 9
4. 9 less than twice a number
Which of the following verbal expressions represents:
x2 + 2x1. the sum of a number squared
and twice a number
2. the sum of a number and twice the number
3. twice a number less than the number squared
4. 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. 4 – x3
2. 4 – 3x3. 3x – 44. x3 – 4
Terms
Like termsTerms with the same variable part
Same means same letter(s) and power(s)
2x, -5x¾x2, 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 + 02t2
Which figure below models the simplification of - 4x - 5 + 7x + 7 using these tiles?
1.
2.
3.
4.
Combining Like Terms
3x + 5 – 9x =
-5 +3b – 7 – 5b =
– 6x + 5
-5 – 7 = -12
3b – 5b = -2b
– 2b - 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
a(b c) a b a c
a(b c) a b a c or
Distributive Property
Objective - To use the distributive property to simplify numerical and variable expressions.
3(4 5) 3(9) 27Order of Operations
Distributive Property
3(4) 3(5)12 15
27It works!
Why use the distributive property?3(x 2) 3(x) 3(2) 3x 6
Simplify using the distributive property.1)
2)
3)
4)
5)
6)
5(x 3)5 3
5x 15
6(y 7)6 7
6y 42
3(m 8)3 8
3m 24
4(3 y)4 3 4 y
12 4y
10(x 7)10 x 10 7
10x 70
4(k 2)4 k 4 2
4k 8
5 x
6 y
3 m
Geometric Model for Distributive Property
Two ways to find the area of the rectangle.
4
5 2
As a whole As two parts
A w l
4 5 2
Geometric Model for Distributive Property
Two ways to find the area of the rectangle.
4
5 2
As a whole As two parts
A w l
4 5 2 4 5 4 2
4 5 4 2
same
4 5 2 4 5 4 2
Scientific Notation
A short-hand way of writinglarge 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
9.3 x 107 --- Scientific Notation
Practice Problem
1) 98,500,000 = 9.85 x 10?
2) 64,100,000,000 = 6.41 x 10?
3) 279,000,000 = 2.79 x 10?
4) 4,200,000 = 4.2 x 10?
5) 0.000013 = 1.3 x 10?
Write in scientific notation. Decide the power of ten.
9.85 x 107
6.41 x 1010
2.79 x 108
4.2 x 106
1.3 x 10-5
Complete Practice Problems
1) 50,0002) 7,200,0003) 802,000,000,0004) 0.000000000631
Write in scientific notation.
1) 5 x 104 2) 7.2 x 106 3) 8.02 x 1011 4) 6.31 x 10-10
Write in Standard Form
6.27 x 106
9.01 x 104
3.95 x 10-3
6,270,000 90,100 0.00395
Positive exponent → move to rightNegative exponent → move to left