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Linear Relationships
Vocabulary
Multiplication Property of -1
Multiplying a number by -1 always
gives you the opposite of that number
-4

12

-1
-1
=4
= -12
Multiplication Property of 0
Multiplying a number by 0 always
gives you ZERO
12 x 3 x -3 x 5 x 0 x 8 = 0
-4

12

0
0
=0
=0
Examples of Dimensions
1 foot
5 feet
Length, Width and Height
are dimensions
These are examples of
rectangular arrays for the
product of 8.
Area Model for Multiplication
4 x 3 can be represented by this rectangle
because 4 x 3 = 12. See the 12 boxes?
Commutative Property of Multiplication
To “commute” to work means you
go to work one way and you come
home the other way.
X

Y= Y

X
5x8=8x5
You will always get the same product, no
matter which way you solve it.
Area Model for a Right Triangle
Area of a right triangle = half the
area of the rectangle.
4
3
The area of this right triangle is half of the
area of 4 x 3 which would be 6 square units.
Associative Property of Multiplication
The 3 numbers don’t switch
places!
Only the
parentheses do!
FUNNEL
METHOD!
( 2 x 3) x 4 = 2 x (3 x 4)
Use the
6 x 4 = 2 x 12
ORDER OF
OPERATIONS!
24 = 24
Terms are numbers and/or variables
which can stand alone or are separated
by “+”, “-”, “x” or “ “

Examples of Terms
X
-y
0.5
12a
Find the Terms
Problem
12 a + 13 b
-8 – 12b
7+8
2x – 3y
Term 1
Term 2
Coefficients
Coefficients are numbers
which are directly in front of a
variable.
3x
The “3” is the coefficient! The
“x” is the variable!
Coefficients
Problem:
x
3a
-y
4a + 6b
List the coefficients here:
Like terms – numbers and/or variables
which can be combined due to their
“likeness”
Example 1
Example 2
Example 3
5
-4y
5.2x
-3
y
4x
1.45
8.2y
-3x
-1
0.5y
x
Combining or Collecting Like terms – you
can combine/collect like terms by adding
them up.
Example 1
Example 2
Example 3
5
-4y
5.2x
-3
y
4x
1.45
8.2y
-3x
-1
0.5y
X
= 2.45
= 5.7y
= 7.2x
Unlike Terms – numbers and/or variables
which can not be combined.
Example 1
Example 2
Example 3
15z
-4b
5.2x
-4a
y
4y
1.45
8.2
-3
-1c
4.9d
12c
Repeated Addition Property of
Multiplication: every multiplication
problem can be written as an addition
problem.
Product
What it means
2x5
4x
3a
2 sets of 5
4 sets of “x”
3 sets of “a”
Addition Problem
5+5
x+x+x+x
a+a+a
Multiplicative Identity Property of 1
Any number multiplied by 1, gives
you that same identical number.
-4

12

1
1
= -4
= 12
Equations of the Form:
x+a=b
One variable, two numbers.
To solve for x, add the opposite of “a” to both sides of the
equation.
Examples:
X+2=9
B + -9 = -27
14 + c = -92
-100 = -42 – (-d)
x+a=b
Identify the “a” and “b” in each equation.
Problem
X+2=9
B + -9 = -27
14 + c = -92
-100 = -42 – (-d)
a
b
Addition Property of Equality:
Given an equation, to keep the equation
balanced, if you add something to the
left hand side of the equation, you must
add that same thing to the right hand
side of the equation.
Example:
LEFT
RIGHT
x+2=9
Add -2 to both sides of
the equation.
-2
-2
X = 7
These are
two
equivalent
equations.