Linear Relationships

Vocabulary

Multiplying a number by -1 always gives you the opposite of that number

-4 = 4 -1

12 = -12 -1-1

Multiplication Property of -1

Multiplying a number by 0 always

gives you ZERO

-4 = 0 0

12 = 0

Multiplication Property of 0

0

12 x 3 x -3 x 5 x 0 x 8 = 0

Examples of Dimensions

1 foot

5 feetLength, 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 5 x 8 = 8 x 5You 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!

( 2 x 3) x 4 = 2 x (3 x 4) 6 x 4 = 2 x 12 24 = 24

FUNNEL METHOD!

Use the ORDER OF

OPERATIONS!

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 Term 1 Term 2

12 a + 13 b-8 – 12b7 + 8 2x – 3y

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: List the coefficients here:

x3a-y4a + 6b

Like terms – numbers and/or variables which can be combined due to their “likeness”

-3

1.45

-1

Example 1

5

Example 2

-4y

y

8.2y

0.5y

Example 3

5.2x

4x

-3x

x

Combining or Collecting Like terms – you can combine/collect like terms by adding them up.

-3

1.45

-1

= 2.45

Example 1

5

Example 2

-4y

y

8.2y

0.5y

= 5.7y

Example 3

5.2x

4x

-3x

X

= 7.2x

Unlike Terms – numbers and/or variables which can not be combined.

-4a

1.45

-1c

Example 1

15z

Example 2

-4b

y

8.2

4.9d

Example 3

5.2x

4y

-3

12c

Repeated Addition Property of Multiplication: every multiplication problem can be written as an addition problem.

Product What it means Addition Problem

2 x 5 2 sets of 5 5 + 54x 4 sets of “x” x + x + x + x3a 3 sets of “a” a + a + a

Any number multiplied by 1, gives you that same identical number.

-4 = -4 1

12 = 12 1

Multiplicative Identity Property of 1

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)

Problem a b

X + 2 = 9

B + -9 = -27

14 + c = -92

-100 = -42 – (-d)

x + a = bIdentify the “a” and “b” in each equation.

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:

x + 2 = 9LEFT RIGHT

-2 -2Add -2 to both sides of the equation.

X = 7

These are two equivalent equations.