Grade 9 Math Notes
Class Z83
Review: Addition and Subtraction with Integers
0-8
Multiplication with Integers
(-5) x (-3) = Positive Product
(-3) x (-3) x (-1) = Negative Product
(-2) x (-8) x (-1) x (-4) = Positive Product
(-2) x (-8) x (-1) x (-4) x (1) = Negative Product
(-2) x (-8) x (-1) x (-4) x (-1) x (-3) = Positive Product
(-2) x (-8) x (-1) x (-4) x (-1) x (-3) x (5) = Negative Product
When there are an even number of negative factors in a multiplication expression, the product
will always be POSITIVE. When there are an odd number of negative factors, the product will
always be NEGATIVE.
-5 x -3 = +15
,-3 x -2 x -4 = -24
-3 x -2 x +5 = +30
The number of positive factors in a multiplication expression DOES NOT impact whether the
product is positive or negative.
+5 x -2 x +5 = -50
-3 x -1 x -1 x -1 x 3 = +9 or 9
-2 x (-2) x (-1) = -4
Division with Integers
Division is the inverse or opposite operation to multiplication. There are three parts to a division
equation:
12 ÷ -3 = -4
The number that is being divided (12) is called the dividend.
The number which the dividend is being divided by (-3) is called the divisor.
The answer is called the quotient.
Because division and multiplication are so closely related, multiplying the divisor and quotient
must equal the dividend. In this example, the divisor is (-3) and the quotient is (-4), therefore if
they’re multiplied together, they should equal 12 or +12:
-3 x -4 = +12
There’s an even number of negative factors in the above multiplication expression, therefore the
product is +12. This strategy may enhance your ability to comprehend division expressions
which include integers (positive and negative whole numbers).
, -12 ÷ -3 = +4
+36 ÷ -4 = -9
-18 ÷ +9 = -2
-30 ÷ -10 = 3
Order of Operations with Integers
An operation is when you do something to a number. The most common types of operations
include: addition, subtraction, multiplication and division.
When an expression has more than one operation, there is an order to which the operations are
completed. Along with addition, subtraction, multiplication and division, mathematicians need to
also consider expressions which include brackets and exponents.
Many of you are familiar with the acronyms BEDMAS or PEDMAS.
Brackets
Exponents
Division/Multiplication
Addition/Subtraction
The first step when solving expressions with multiple operations is to seek and solve anything
within a bracket. Once that’s complete, solve anything which includes an exponent. Things
become slightly more complex when encountering division and multiplication. Although division
comes before multiplication in the acronym BEDMAS, it’s important to remember that we always
work left to right. Therefore, if in the expression multiplication arrives before division, then we do
the multiplication first. Again, division and multiplication are solved together before addition and
subtraction. Let’s have a look at some examples:
(-5 + -3) x -2
= (-8) x -2
= +16
, +3 + -2 x (-5 + +2)
= +3 + -2 x -3
= +3 + +6
= +9
(+8 - -2 x -3) + -4 ÷ -2
= (+8 - +6) + -4 ÷ -2
= +2 + -4 ÷ -2
= +2 + +2
= +4
-15 x -2 ÷ +3 x -1 + -18
= +30 ÷ +3 x -1 + -18
= +10 x -1 + -18
= -10 + -18
= -28
+8 x -3 + (-5 x -2 x 4) ÷ -2
= +8 x -3 + (+10 x 4) ÷ -2
= +8 x -3 + 40 ÷ -2
= -24 + 40 ÷ -2
= -24 + -20
= -44
-20 ÷ -2 x -3 + (-4 x -2)
= -20 ÷ -2 x -3 + +8
= +10 x -3 + +8
= -30 + + 8
= -22
Class Z83
Review: Addition and Subtraction with Integers
0-8
Multiplication with Integers
(-5) x (-3) = Positive Product
(-3) x (-3) x (-1) = Negative Product
(-2) x (-8) x (-1) x (-4) = Positive Product
(-2) x (-8) x (-1) x (-4) x (1) = Negative Product
(-2) x (-8) x (-1) x (-4) x (-1) x (-3) = Positive Product
(-2) x (-8) x (-1) x (-4) x (-1) x (-3) x (5) = Negative Product
When there are an even number of negative factors in a multiplication expression, the product
will always be POSITIVE. When there are an odd number of negative factors, the product will
always be NEGATIVE.
-5 x -3 = +15
,-3 x -2 x -4 = -24
-3 x -2 x +5 = +30
The number of positive factors in a multiplication expression DOES NOT impact whether the
product is positive or negative.
+5 x -2 x +5 = -50
-3 x -1 x -1 x -1 x 3 = +9 or 9
-2 x (-2) x (-1) = -4
Division with Integers
Division is the inverse or opposite operation to multiplication. There are three parts to a division
equation:
12 ÷ -3 = -4
The number that is being divided (12) is called the dividend.
The number which the dividend is being divided by (-3) is called the divisor.
The answer is called the quotient.
Because division and multiplication are so closely related, multiplying the divisor and quotient
must equal the dividend. In this example, the divisor is (-3) and the quotient is (-4), therefore if
they’re multiplied together, they should equal 12 or +12:
-3 x -4 = +12
There’s an even number of negative factors in the above multiplication expression, therefore the
product is +12. This strategy may enhance your ability to comprehend division expressions
which include integers (positive and negative whole numbers).
, -12 ÷ -3 = +4
+36 ÷ -4 = -9
-18 ÷ +9 = -2
-30 ÷ -10 = 3
Order of Operations with Integers
An operation is when you do something to a number. The most common types of operations
include: addition, subtraction, multiplication and division.
When an expression has more than one operation, there is an order to which the operations are
completed. Along with addition, subtraction, multiplication and division, mathematicians need to
also consider expressions which include brackets and exponents.
Many of you are familiar with the acronyms BEDMAS or PEDMAS.
Brackets
Exponents
Division/Multiplication
Addition/Subtraction
The first step when solving expressions with multiple operations is to seek and solve anything
within a bracket. Once that’s complete, solve anything which includes an exponent. Things
become slightly more complex when encountering division and multiplication. Although division
comes before multiplication in the acronym BEDMAS, it’s important to remember that we always
work left to right. Therefore, if in the expression multiplication arrives before division, then we do
the multiplication first. Again, division and multiplication are solved together before addition and
subtraction. Let’s have a look at some examples:
(-5 + -3) x -2
= (-8) x -2
= +16
, +3 + -2 x (-5 + +2)
= +3 + -2 x -3
= +3 + +6
= +9
(+8 - -2 x -3) + -4 ÷ -2
= (+8 - +6) + -4 ÷ -2
= +2 + -4 ÷ -2
= +2 + +2
= +4
-15 x -2 ÷ +3 x -1 + -18
= +30 ÷ +3 x -1 + -18
= +10 x -1 + -18
= -10 + -18
= -28
+8 x -3 + (-5 x -2 x 4) ÷ -2
= +8 x -3 + (+10 x 4) ÷ -2
= +8 x -3 + 40 ÷ -2
= -24 + 40 ÷ -2
= -24 + -20
= -44
-20 ÷ -2 x -3 + (-4 x -2)
= -20 ÷ -2 x -3 + +8
= +10 x -3 + +8
= -30 + + 8
= -22
