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Integers
Order of Operations
Substitution MATH 7 REVIEW DAY 1
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Ex: Consider the addition 3 + (-2)
We can illustrate the addition using solid and hollow dots
3 + -2 = 1
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Ex: Now consider the addition -3 + 2
Again illustrate the addition using solid and hollow dots
-3 + 2 = -1
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To recap: 3 + 2 = 5 same sign addends
(-3) + (-2)= (-5) = -5
3 + (-2) = 1 different sign addends
-3 + 2 = - 1
Can we describe a general rule for adding integers?
We see two cases: same sign addends
different sign addends
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Addition of Integers
When the addends have the same sign:
Add the numbers. Keep the sign.
When the addends have different signs:
Which sign is bigger? Use that sign for the sum.
Subtract the numbers.
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Subtraction of Integers
Let a and b be integers.
Then a – b = a + (-b).
Change subtraction to
addition and change the sign
of what follows.
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Subtraction of Integers
We can also use the number line and direction arrows to illustrate
subtraction of integers. Let a positive number be represented by a
right-facing arrow and a negative number
be represented by a left-facing arrow.
The operation of subtraction acts to flip the direction of
the number being subtracted’s arrow.
positive
negative
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Ex: Model the subtraction 3 – (-2) using the number line to find the
difference.
0
Start at zero and draw the first addend, 3
Positive
From where the first arrow ends, draw the
second addend, - 2 Negative
Where the second arrow ends is the difference
5
Remember, subtraction flips the arrow!
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Ex: Model the subtraction -3 – (-2) using the number line to find the
difference.
0
Start at zero and draw the first addend, -3 Negative
From where the first arrow ends, draw the second addend, - 2
Negative
Where the second arrow ends is the difference
-1
Remember, subtraction flips the arrow!
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Multiplying Integers:
3 x 2 = 6 same sign factors
-3 x (-2) = 6
-3 x 2 = -6 different sign factors
3 x (-2) = -6
Can we describe a general rule for multiplying integers?
We see two cases: same sign factors positive
different sign factors negative
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Dividing Integers:
6/3 = 2 the same sign
-6/(-3) = 2
-6/3 = - 2 different sign factors
6/(-3) = - 2
Can we describe a general rule for dividing integers?
We see two cases: same sign factors positive
different sign factors negative
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Order of
Operations
( ) +
X -
43
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Please Excuse My Dear
Aunt Sally
This will help
to you to
remember
the order of
operations.
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Add +
Subtract -
Multiply x
Divide
Please Excuse My Dear Aunt Sally
P
E
M
D
A
S
Parentheses ( )
Exponents 43
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Please Excuse My Dear Aunt Sally
Parentheses ( )
Always do
parentheses
1st.
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Please Excuse My Dear Aunt Sally
Exponents 43
Always do
Exponents
2nd.
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Multiply x
Divide
Please Excuse My Dear Aunt Sally
Do
multiplication
and division
3rd, from left to
right.
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Add +
Subtract -
Please Excuse My Dear Aunt Sally
Do addition
and
subtraction
4th, from left
to right.
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PEMDAS
3+23- (9+1)
3+23- 10 3+8-10
11-10
1
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PEMDAS
3 (9+1) + 62
3(10)+62 3(10)+36
30+36
66
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PEMDAS
4+5 x (6-2)
4+5 x 4
4+20
24
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PEMDAS
4+ 10 x 23 -16 4+10 x 8 -16
4+ 80 -16
84-16 68
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PEMDAS
21 + 102 10
21+10010 21 + 10
31
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PEMDAS
10+72-2 x 5
10+49–2 x 5 10+49- 10
59 - 10
49
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PEMDAS
64 (9 x 3-19) 64(27 –19)
64 8
8
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Evaluate a Variable Expression – write the expression,
substitute a number for each variable, and simplify
the result.
Value of a Variable – A number that may be
substituted or assigned to a particular variable; such
as n = 3; or x = 5.
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Example 1: Evaluate each expression when n = 4
Substitute 4 for n. Simplify
Simplify (means to solve the problem or perform as
many of the indicated operations as possible.)
7
343
nSolution:
3 b. n Substitute 4 for n. Simplify
1
343
nSolution:
3 a. n
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Example 2: Evaluate each expression when x = 8
Substitute 8 for x. Simplify
Simplify (means to solve the problem or perform as
many of the indicated operations as possible.)
Solution:
4 b. x
2
484
xSolution:
x5 a.
40
)8(55
x
Note: No operation sign
between a variable and
number– indicates
multiplication problem.
Using parenthesis is the preferred method
to show multiplication. Additional ways to show
multiplication are:
85 ;85 ;85);8)(5(
Substitute 8 for x. Simplify
Recall that division problems are also
fractions – this problem could be
written as:
44
2;
4
8
4
xx
because
x
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Example 3: Evaluate each expression when x = 4, y = 6, z = 24.
xy5 a.Substitute 4 for x; 6 for y. simplify
solution
Recall: No
operation sign
between
variable(s) and
a number–
indicates
multiplication
problem.
Xy means 4(6);
5xy means
5(4)(6)
)6()4)(5(5 xy
)6()20(
120yz b.
Solution: 624 yz
4
Recall that:
46
24624
so,
y
zyz
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Evaluate each expression when a = 6, b = 12, and c = 3
ac4 1.
ca 2.
cba 3.
ba 4.
cb 5.
bc 6.
A
A
A
A
A
A
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Evaluate each expression when a = 6, b = 12, and c = 3
ac4 1.
)3()6)(4(4 ac
Notice that all the numbers and letters
are together and that there are no
operation symbols which indicates
that this is a multiplication problem.
Substitute the value for a = 6 and c = 3
into the problem and multiply
)3()24(
72
multiply
Simplified
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Evaluate each expression when a = 6, b = 12, and c = 3
ca 2.
36ca
Division Problem
Substitute the value for a = 6 and c = 3
into the problem and divide
2 Simplified
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Another way to
solve division
problems is to
write them as
fractions and
simplify. 23
6
c
aca
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Evaluate each expression when a = 6, b = 12, and c = 3
cba 3.
3126 cba
Addition problem
Substitute the value for a = 6, b=12,
and c = 3 into the problem, then add
318
Simplified 21
Add
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Evaluate each expression when a = 6, b = 12, and c = 3
ba 4.
)6)(12(ba
multiplication problem
Substitute the value for b=12 and a = 6
into the problem, then multiply
72 Simplified
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Evaluate each expression when a = 6, b = 12, and c = 3
cb 5.
312cb
Subtraction problem
Substitute the value for b=12 and a = 3
into the problem, then Subtract
9 Simplified
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Evaluate each expression when a = 6, b = 12, and c = 3
bc 6.
123bc
Division problem
Substitute the value for c=3 and b = 12 into
the problem, then Divide
Note: It is better to rewrite this division
problem as a fraction.
This fraction can now be reduced to its
simplest form.
12
3
Simplified
3
3
12
3
4
1
Divide both
numerator and
denominator by
the GCF = (3) to
reduce this
fraction.
It is OK to have a fraction
as an answer.
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middle of the
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each answer
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