ABSOLUTE VALUE
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For each value, write it opposite, then its
absolute value.
opposite absolute value
1) -3
+3
3
2) 4
-4
4
3) 15
-15
4) -7
+7
7
5) 0
0
0
6) 1
-1
1
7) -1
+1
1
8) -50
+50
50
9) 10
-10
10
10)-2.5
+2.5
2.5
15
ABSOLUTE VALUE
1.
|17|
|–29|
2.
17
5.
29
–|35 + 76|
–|
–
101
101
–101
3.
|
6.
–|41 – 19|
–| 22 |
– 22
–22
–|45|
– 45
–45
7.
– | – 2 – 1|
–| – 3 |
–
3
–3
–|–247|
– 247
–247
4.
8.
–| –(–7)|
–| 7
–7
|
Integer Addition - With Counters
1.
•
•
•
•
–3
+9
Draw 3 negative
counters ...
.. then 9 positive
counters
Cancel opposites.
What’s left? +6
2. 2 + –7
•
•
Draw 2 positive
counters ...
. then 7 negative
counters
Cancel opposites.
What’s left? –5
3.
–6
•
•
•
•
•
•
+ –4
Draw 6 negative
counters ...
. then 4 negative
counters
.No opposites, so ...
–
What’s left? 10
Integer Addition - Number Line
1.
–3
+9
Start at 0
2.
2 + –7
Start at 0
3.
–6
+ –4
Start at 0
The first integer, -3, is
the first move.
The first integer, 2, is
the first move.
The first integer, -6, is
the first move.
The second integer, 9, is
the second move.
The second integer,–7, is
the second move.
The second integer,–4, is
Where did you end up?
Where did you end up?
Where did you end up?
.
-3 + 9
=6
2 + –7
= -5
the second move.
-6 + –4
=
-10
Integer Addition - The Rules
1.
–3
+9
9
–
3
--6
+6
2. 2 + –7
–
–5
2
--5
+
.
– 10
1. What’s the sign on the bigger absolute value?
** NEGATIVE so the answer is NEGATIVE **
2. Now, signs are DIFFERENT, so ...
3. SUBTRACT the absolute values.
4. Write down that number.
7
3. –6 + –4
1. What’s the sign on the bigger absolute value?
** POSITIVE so the answer is POSITIVE **
2. Now, signs are DIFFERENT, so ...
3. SUBTRACT the absolute values.
4. Write down that number.
6
4
--10
1. What’s the sign on the bigger absolute value?
** NEGATIVE so the answer is NEGATIVE **
2. Signs are SAME, so ...
3. ADD the absolute values.
4. Write down that number.
.
Integer Addition - Practice
–16
–6
–12
–2
4
4
0
–6
–2
–8
–1
–10
–5
1
–7
–10
5
–8
–4
–2
–9
5
–11
–7
–17
–6
–16
–10
–3
–3
–2
–2
More Integer Addition - Practice
= –34
=1
=6
= –7
= 14
= –11
.
= –15
=6
= –31
= –7
= –12
= 65
= –26
= –28
=9
= –8
= 25
= –31
=–80
= 41
Integer Subtraction - Using Counters – (No Notes, Just Watch)
For high school and college level math, it’s easier to think of
subtraction as adding the opposite sign.
I want to
take
away 14
positives!
SUBTRACTION
is the same as
ADDING THE OPPOSITE
11 – 14
–8 – 5
I want to
take away
7
negatives!
–4 – (–7)
or
or
or
11 + (–14)
–8 + (–5)
–4 + (+7)
11 – 14
–8 – 5
–4 – (–7)
or
or
–4 + (+7)
11 + (–14)
.
I want to
take
away 5
positives!
–3
–8 + (–5)
–13
or
or
–4 + 7
+3 or
3
Integer Subtraction - Using Number Line– (No Notes, Just Watch)
1.
11 – 14
2.
Start at 0.
–8
–5
Start at 0.
The first integer, +11, is
the first move.
The first integer, –8, is
the first move (left).
The subtraction sign
means to move to the
left...
The subtraction sign
also means to move to
the left...
...the second integer,
+14, means
move 14 to the left.
...the second integer, +5,
means move 5 spaces to
the left.
Where did you end up?
11 – 14 =
.
3.
-3
Where did you end up?
–8
– 5 = –13
–4
– (–7)
Start at 0.
The first integer, –4, is
the first move.
The subtraction sign
means move to the
left...
... but the negative sign
reverses it, so...
...move to the right 7
spaces.
Where did you end up?
–4
– (–7) = +3
KEEP  CHANGE  CHANGE
the 1st
integer
subtraction to
addition
the sign of the
2nd integer
7 + (–9)
–4 + (–1)
3 + (+5)
–6 + (+8)
9 + (+4)
1 + (–7)
–3 + (–8)
–5 + (+2)
–1 + (+9)
3 + (–4)
5 + (+6)
–8 + (–7)
.
Integer Subtraction - The Rules
1.
11 + –14
–3
2.
–8
–8
–
14
11
--3
–5
+ –5
–1 3
3.
1.
11 – 14
–4
– (–7)
–4
+ +7
+. 3
8
5
--13
What’s the sign on the bigger absolute value?
** NEGATIVE (–14) so the answer is NEGATIVE**
3.
Now, signs are DIFFERENT, SUBTRACT the absolute values.
4.
Write down that number.
7
4
--3
a.
b.
c.
CHANGE the subtraction to addition, then...
CHANGE the sign of the 2nd integer [ “– 5” --> “+ (–5)” ]
IGNORE the original problem.
2.
What’s the sign on the bigger absolute value?
** NEGATIVE (–8) so the answer is NEGATIVE**
3.
Now, signs are SAME, so ADD the absolute values.
4.
Write down that number.
1.
–
CHANGE the subtraction to addition, then...
CHANGE the sign of the 2nd integer [ “– 14” --> “+ (–14)” ]
IGNORE the original problem.
2.
1.
+
a.
b.
c.
a.
b.
c.
CHANGE the subtraction to addition, then...
CHANGE the sign of the 2nd integer [ “– (–7)” --> “+ (+7)” --> +7 ]
IGNORE the original problem.
2.
What’s the sign on the bigger absolute value?
** POSITIVE (+7) so the answer is POSITIVE **
3.
Now, signs are DIFFERENT, SUBTRACT the absolute values.
4.
Write down that number.
.
Integer Subtraction - Practice
.
11
13
1
5
0
17
3
0
–1
2
–8
4
2
–2
–5
–2
–4
3
–8
13
0
–11
12
13
–1
6
3
3
–1
4
–2
–1
More Integer Subtraction - Practice
= –85
1.
= – 15
2.
= 60
3.
.
4.
= –54
7.
5.
= – 84
8.
6.
= 93
9.
= –65
– 16 – ( – 95)
10.
= –56
= 79 11. – 5 – ( – 9) = 4
= 98
12.
= 97
13.
14.
15.
= –59
= –37
= 36
Integer Multiplication - Using Counters
1.
–4
•5
•
Draw 4 negative
counters ...
... 5 TIMES
•
–4
•5=
–20
Integer Division - Using Counters
2.
–28
•
Draw 28 negative
counters ...
•
... then DIVIDE
them into 7 groups
7
–28
.
7
=
–4
Integer Multiplication - Using Number Line
1.
–6 • 3
–18
•
Draw a dot at zero
•
The 1st integer, –6 , tells you the size of the jump.
•
The 2nd integer, 3 , tells you how many TIMES.
•
So, jump 6 backward 3 TIMES.
Integer Division - Using Number Line
2.
–42
3
–14
.
•
Draw a dot at zero
•
The 1st integer, –42 , is the total
•
The 2nd integer, 3 , tells you into how many pieces to DIVIDE it.
•
So, DIVIDE the –42 into 3 pieces.
•
How big is each piece?
Integer Multiplication - The Rules
1. –7 • 8
When you multiply or divide integers, it’s easy:
1.
– 56
Look at the signs: • If they’re the
SAME ...
• then, the answer’s POSITIVE
Signs are different,
so the answer is
negative.
2.
• If they’re
DIFFERENT
• then, the answer’s NEGATIVE
Multiply the absolute values.
Integer Division - The Rules
–90
–9
2.
When you multiply or divide integers, it’s easy:
1.
Look at the signs:
Signs are the same , so
the answer is positive.
+ 10
SAME ...
• then, the answer’s POSITIVE
• If they’re the
DIFFERENT
• then, the answer’s NEGATIVE
• If they’re
.
2.
Divide the absolute values.
Integer Multiplication and Division - Practice
.
36
–21
–32
21
–6
7
–5
–9
–4
–6
6
2
–32
–9
–6
63
–7
–7
4
–35
16
4
–8
–8
6
–1
–4
12
72
–81
–7
–1
More Integer Multiplication and Division - Practice
= –48
1.
= 168
2.
5.
= –50
9.
6.
= –105
10.
= – 24
= 70
11.
= – 96
3.
= – 70
7.
4.
=288
8.
13.
= – 21
15.
= – 11
17.
14.
= –5
16.
= – 20
18.
.
= 24
= 143
12.
=– 200
= 17
= 13
1
–15
1
–13
0
–15
–1
2
3
6
–13
–1
2
0
–1
0
–4
–10
.
1
3
–10
–10
0
– 16
– 12
– 10
1
–13
0
–10
–11
–1
Integers Operations
4
-10
1
7
-50
-125
-1
-5
-8
-3
4
3
1
-3
-9
-1
-7
96
-4
-3
.
Order of Operations
2
Simplify 3(4 - 2) - 8 ÷ 2 + 9
3 (4 - 2) - 8 ÷ 22 + 9
3 (2) - 8
3 (2)
6
-
─
4
8
2
2 +9
÷
÷
4 +9
2
+9
+9
13
1. First, simplify inside parenthesis
2. Next, evaluate the exponent.
3. Then, multiply or divide.
* Which one first?
Always work from left to right.
4. Finally, add or subtract.
Always work from left to right.
So, to simplify an expression using order of operations, you should:
Simplify
Parenthesis
Evaluate
Exponents
Multiply
Add
Divide
Subtract
or
or
Order of Operations
1.
11 – 32 ÷ 8 • 3
11 –
4
11 –
2.
• 3
– 7 + ( – 9 )( 1 2 ) ÷ 2
– 7 +
12
– 7 +
–1
3.
18 +
(–2)
– (–12)(–4)
[48 –
18 +
(–2)
–
48
[48 –
–
48
–23 + 7 – (–8 + 2)
( – 5 + 9 )3 + ( – 2 )
(
.
4
( –54)
4 . [48 – (12 – 14) • 2] + 8 ÷ (–8)
–32
5.
÷ 2
–61
18 + (–6) ÷ 3 – (–12)(–4)
16
( –108)
)3 + ( – 2 )
12
+ (–2)
denominator (bottom) = 10
(–2)
• 2] + 8 ÷ (–8)
(–4)
] + 8 ÷ (–8)
52
+ 8 ÷ (–8)
52
+
(–1)
51
–23 + 7 –
(–8 + 2)
–23 + 7 –
(–6)
–16
–
numerator (top) =
(–6)
–10
–10
10
=
–1
.
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