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Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
Unit 2 Integers Review of Grade 7In your text book Yellow is Positive and Red is Negative
In notes, handouts, tests shaded is Positive and non-shaded (white) is Negative
Remember that one Positive and one Negative form a Zero Pair. (+1) + (-1) = 0
Thursday, October 15th 2015
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Adding Integers with TilesA) (+2) + (+3) =
B) (-3) + (-1) =
C) (+5) + (-4) =
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Check. Use tiles to add.D) (+1) + (+3) =
E) (-2) + (-3) =
F) (-4) + (+3) =
G) (+4) + (-2) =
H) (-5) + (+2) =
We say "positive" and "negative"
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Adding Integers with Number Lines*Start at zero. Positive integer move right, negative integer move left.I) (+4) + (-5) =
J) (-2) + (+6) =
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Check. Use a number line to add.K) (+7) + (+4) =
L) (+6) + (-4) =
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M) (-11) + (-4) =
N) (+3) + (+5) =
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O) (-8) + (+2) =
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Subtracting Integers with TilesA) (+5) – (+3) =
B) (-4) – (-3) =
ADD THE OPPOSITE
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C) (-5) – (-1) =
D) (-2) – (-6) =
We only have 2 negatives, and need to remove 6 negatives. So we ADD 4 zero pairs because adding zero pairs is the same as adding ZERO. Now we can remove 6 negatives, what are we left with?
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E) (-3) - (+1) =
We have 3 negatives, and need to remove 1 positive. So we add 1 zero pair. Now we can remove 1 positive, what are we left with?
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Check. Use tiles to subtract.A) (+2) – (-7) =
B) (-3) – (-4) =
C) (-5) – (-5) =
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I) (+10) – (-4) =
J) (-5) – (+6) =
K) (-3) – (-5) =
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Remember the rule for subtracting integers ?ADD THE OPPOSITE: the first integer stays the same, change the subtract sign to addition sign, then switch the second integer to its opposite.
(-4) – (-3) → (-4) + (+3) =
(-5) – (-1) → (-5) + (+1) =
(-2) – (-6) → (-2) + (+6) =
(-3) – (+1) → (-3) + (-1) =
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Check. Use Add the Opposite
L) (+6) – (-5) →
M) (-7) – (+3) →
N) (+9) – (+4) →
O) (-3) – (+5) →
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Section 2.1 Using Models to Multiply Integers
Multiplication can be viewed as repeated addition.
From earlier grades:
2 x 3 means 2 sets of 3
3 + 3 = 6
4 x 5 means 4 sets of 5
5 + 5 + 5 + 5 = 20
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Try these:
2 x -3 means ___ sets of ____
4 x - 5 means ___ sets of ____
(+3) x (-5) means ___ sets of ____
(+3) x (-2) means ___ sets of ____
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Write each repeated addition sentence as multiplication.
A) (-6) + (-6) + (-6) + (-6) + (-6)
B) (+4) + (+4) + (+4) +(+4)
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What if the first integer is negative?
In grade 8, we will be taking the idea of repeated addition and combine it with making deposits and withdrawals from a bank as positive and negative.
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Examples
A) (+4) x (+3)
The first tells us toDEPOSIT (put in the bank if positiveWITHDRAWAL (take out of the bank) if negative
The second integer tells us what to "put in" or "take out"
The bank will start as a circle with zero value.
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(+4) x (+3) Solution
Put in4 sets
of +3
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B) (+4) x (-3) Solution
put in of -34 sets
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Try these!
C) (+2) x (+4) Solution
D) (+2) x (-3) Solution
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E) (-4) x (-3)
withdrawal of -3(take out)4 sets
* If the bank starts at zero, how can we take out four sets of -3 ?
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Use zero pairs* make 4 sets of 3 zero pairs
* now take out 4 sets of -3Solution: ____ what is left in the bank
+- +-+- +-+- +-
+- +-+- +-+- +-
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F) (-4) x (+3)
take out of +34 sets
Need 4 sets 3 Zero Pairs
Solution:What is left in the bank?
+- +-+- +-+- +-
+- +-+- +-+- +-
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Try these!
G) (-2) x (-5)
We need ___ sets of ____ zero pairs
Solution: ______What is left in the bank?
Complete all examples in notes upto and including page 14.
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H) (-2) x (+4)
We need ___ sets of ____ zero pairs
Solution: ______What is left in the bank?
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Whenever the first integer is negative, we need to "take out", so zero pairs need to be added to the bank first.
Complete Page 68 #'s 5, 6, 9, 10
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Multiplying Integers using Number Lines
* Start at zero* The first integer indicates which direction to face and how many steps to take
Positive = Face Right Negative = Face Left
* The second integer indicates which direction to move and the size of the steps
Positive = Move forward Negative = Move Backward
* Follow these steps to land on the answer
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#1: The temperature fell 3oC each hour for 6 hours. Find the total drop in temperature.
(+6) x (-3)
Face Right Move Backward6 steps of size 3
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
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# 2: Kyle spends $2 a day for recess at school, 5 days of the week. Use a number line to represent how much Kyle spends a week?
(+5) x (-2)
Face Right Move Backward5 steps of size 2
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
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#3: (+3) x (+2)
Face Right Move Forward3 steps of size 2
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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#4: (+3) x (-2)
Face Right Move Backward3 steps of size 2
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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#5: (-3) x (-2)
Face Left Move Backward 3 steps of size 2
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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#6: (-3) x (+2)
Face Left Move Forward3 steps of size 2
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
COMPLETE EXAMPLES ON PAGES 1824.
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Try these....
#7: (-4) x (+2)
Face _____ Move __________ steps of size ___
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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#8: (-4) x (-2)
Face _____ Move __________ steps of size ___
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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#9: (+4) x (-2)
Face _____ Move __________ steps of size ___
10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10
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IntegerMultiplication
Number Lineon a
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
( ) ( )
Tells us which direction the man is facing.
Tells us the number of steps to
take.
Tells us to step
backwards or forwards.
Tells us the size of the steps.
+ 3 x + 2
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
(+ 3) x (+ 2)
+6Erase for
answer
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
(+ 4) x (+ 2)
Erase for answer +8
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
(+ 2) x ( 4)
Erase for answer 8
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
( 4) x (+ 2)
Erase for answer 8
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
( 5) x ( 2)
Erase for answer +10
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
( 3) x (+ 3)
Erase for answer 9
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
(+ 3) x ( 2)
Erase for answer 6
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Multiplying Integers
10 2 3 4 5 6 7 8 9 1012345678910
( 3) x ( 1)
Erase for answer +3
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Extra PracticeMultiply the following using number lines.
1. (+2) x (+5)
2. (+3) x (1)
3. (4) x (+2)
4. (2) x (5)
5. (+3) x (3)
6. (4) x (1)
7. (+5) x (2)
8. (3) x (+1)
9. (2) x (2)
10. (1) x (+8)
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Summary
For Multiplying two integers
2 positives =(+2) x (+3) =
2 negatives = (-2) x (-3) =
negative and positive = (-2) x (+3) =
positive and negative =(+2) x (-3) =
Bottom page 25
answer is positive
answer is positive
answer is negative
answer is negative
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Therefore, If the signs are the SAME, the answer is POSITIVE
If the signs are DIFFERENT, the answer is NEGATIVE
Same Signs (++) (--)PositiveOrOpposite Signs (+-)(-+)Negative
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Some Properties for Multiplying:
Multiplying by Zero
0 x (+3) = 0 x (-3) =
Multiplying an integer by zero results in a product of ________.
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Multiplying by One (Multiplicative Identity)
(-3) x (+1) = (+3) x (+1) =
Since multiplying by 1 does not change the number, we call 1 the multiplicative identity.
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Communative Property
(+3) x (+4) = (+4) x (+3) =
Did our answer change because we changed the order of the integers?
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NO
multiplication is commutative because we can change the order of multiplying the integers but it didn't change the answer.
(-3) x (+4) = (+4) x (-3) =
Addition is also commutative!
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Complete Page 73#3, 4, 5, 8 a -e, 15, 17, 18
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Area ModelTwo Integers can also be multiplied using an
area model in combination with distributive property.
Distributive PropertyA property multiplication stating that a product
can be written as a sum of 2 products
For example: 3 x (4 +5)= ( 3 x 4) + (3 x 5) = 12 + 15= 27
page 29
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Example A 8 x 26 Same as 8 x (20 + 6)
= (8 x 20) + (8 x 6)= (160) + (48)= 208
Area Model20 6
8 8 x 20 8 x 6 160 48
Answer: 208
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Example B -5 x 36
Area Model Distributive Property**Ignore the signfor area model to start -5 x (30 + 6)
= (-5 x 30) + (-5 x 6)30 6 = -150 + -30
5 5 x 30 5 x 6 = -180 150 30
150 + 30 180
GO BACK TO SIGN5 x -36 = -180
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Example C -25 x -48
Ignore sign for area model...we know are answer is _____
40 820 20 x 40 20 x 8
_____ _____
5 5 x 40 5 x 8_____ _____
25 x 48 = _____ + _____ + _____ + ______
= _____ + _____
= _____
SIGN CHECK (-25) x (-48) = _______
Complete example on bottom of page 31 of notes TextbookPage 73 #6adPage 73 #7ad
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Example D (-72) x (+15)
Ignore sign for area model...we know are answer is _____
10 5
70 70 x 10 70 x 5_____ _____
2 2 x 10 2 x 5_____ _____
72 x 15 = _____ + _____ + _____ + ______
= _____ + _____
= _____
SIGN CHECK (-72) x (+15) = _______
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Complete Page 73# 6 a - d# 7 a - d
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Dividing Integersusing
Counters
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(+6) ÷ (+2)
+6+2 ?will makeHow many groups of
Dividing Integers
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(+8) (+4)+2
(+8)
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Dividing IntegersThe Answer?
?will makeHow many groups of??
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(9) (3)+3
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(6) (3)+2
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(10) (+5)-2
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(6) (+3)-2
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(8) (+2)-4
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(+12) (3)-4
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(+9) (3)-3
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Dividing IntegersThe Answer? ÷
?will makeHow many groups of
(+8) (2)-5
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Page 81 # 8
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Dividing Integersusing
The Number Line
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Directions:Grab the first number in the division and place it in the first blank in the "Ask yourself..." question. Touch just below the second integer in the question. Look at the sign of the second integer. Grab and move "forward" (for +) or "backward" (for -) and place it in the second blank in the question. Then grab the second number in the division and place it at the end of the question.
Use the correct man to do answer the question on the number line. Then fill in the first blank in the "Answer..." with the number of steps and the second blank with the direction the man is facing (+ or -). This give you the quotient!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
(10) (+2)
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
(10) 2
Answer...
To get there I took steps and I am now facing .+
-
= 5Touch me when done!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+8 +4+8 4 = +2Touch me when done!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+6 22+6 = 3Touch me when done!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
9 +39 3 = 3Touch me when done!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
1212 44 = +3Touch me when done!
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0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+15 3+15 3 = 5Touch me when done!
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October 26, 2015
0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+10 +5+10 5 = +2Touch me when done!
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October 26, 2015
0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+12 6+12 6 = 2Touch me when done!
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October 26, 2015
0 1 2 4 5 6 7 8 9 10 11 12 13 14 153123456789101112131415
( ) ( )
Ask yourself...
How do I get to stepping of size ?
forwards
backwards
Answer...
To get there I took steps and I am now facing .+
-
+11 1+11 1 = 11Touch me when done!
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Extra Practice(use numberlines)
1. (+12) (-4)
2. (-8) (-2)
3. (+6) (-3)
4. (-15) (+5)
5. (-15) (-5)
..........
6. (+8) (-2)
7. (+9) (-3)
8. (+10) (+2)
9. (-12) (-3)
10. (-15) (+3)
..........
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Summary for Dividing Integers
Dividing Two Positive Integers
(+8) ÷ (+2) =(+6) ÷ (+3) =(+12) ÷ (+3) =What do you notice?
Dividing Two Negative Integers
(8) ÷ (2) =(6) ÷ (3) =(10) ÷ (2) =What do you notice?
Dividing a Positive by a Negative
(+8) ÷ (2) =(+6) ÷ (3) =(+12) ÷ (3) =What do you notice?
Dividing a Negative by a Positive
(8) ÷ (+2) =(6) ÷ (+3) =(10) ÷ (+2)=What do you notice?
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The rules for Dividing Integers are the same as Multiplying Integers.
Same Signs (++) (--)PositiveOrOpposite Signs (+-)(-+)Negative
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Terminology(+15) ÷ (+3) = +5
Dividend Divisor QuotientWe can also write a division statement as a fraction:
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Practice: DivideA) (-8) ÷ (-4)
B)=
C) (-12) ÷( ____ ) = +2
D)
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E) Shannon made withdrawals of $7 from her bank account, for a total withdrawal of $63? Use integers to find how many withdrawals she made.
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F) Chris and his 3 friends together owe $12. They agree to share the debt equally. What is each person‛s share of the debt?
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G) The temperature in Nain is dropping 2oC each hour. How many hours did it take to fall 10oC?
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Complete Pages 87 – 88 #‛s 4, 5, 7a, 9, 10, 11, 12
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Section 2.5: Order of Operations
To ensure everyone gets the same answer for a question like 9 × 6 + 36 ÷ 4 1 we follow the proper order of operations.
B – Brackets FirstE – Exponents (not in grade 8)D – Division M MultiplicationA AdditionS – Subtraction
}}
Whichever comes first in the problem
Whichever comes first in the problem
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Examples A) 9 × 6 + 36 ÷ 4 1
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B) [(+9) – (2)] × (3)
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C) [(6) + (2)] ÷ ( 4) + (5)
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D)
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E)
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F) (2 × 105) – (2 +8) ÷ 2
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G)
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H) 15 ÷ [ 10 ÷ (2)]
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I) Kathy completed the following calculations on a math test. Is she correct? Explain.
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Complete Page 92 #'s 3, 5, 8, 9ad
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Word Problems
Words/Phrases that indicate MULTIPLY Words/Phrases that indicate DIVIDE
Multiply byProduct ofTimesDoubleTripleAs muchByFind the total
Quotient ofDivided byDivided equallyPerHow many (much) eachSplitEqual piecesShared
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1: You have no money, instead you borrow $2 each day for 3 days. What is the total debt by the end of the third day?
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2: Matthew has agreed to donate to a local charity for 2 years. If he gives $25 per month, how much has he donated at the end of the 2 years?
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3: An oil rig is drilling a well at 2m per minute. How deep is the well after the first 8 minutes?
Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
4: The temperature decreased 16oC over a 4 hour period. Assuming the temperature decreased at a constant rate, how much did the temperature decrease each hour?
Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
5: The temperature rose 4oC each hour for 5 hours. Use integers to find the total change in temperature.
Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
6: The water level in a well dropped 5cm each hour. The total drop in the water level was 30cm. Use integers to find out how long it took for the water level to change.
Math 8_Unit 2 Integers_NOTES.notebook
116
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7: A glacier retreated about 2m per day for 7 days. Use integers to find the total change in the length of the glacier.
Math 8_Unit 2 Integers_NOTES.notebook
117
October 26, 2015
8: Winnie used the money in her savings account to pay back a loan from her Mother. Winnie paid back her Mother in 12 weekly payments. Over the 12 weeks, the balance in Winnie’s savings account decreased by $132. By how much did her balance change each week?
Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
9: The product of two integers is 20. The sum of the same two integers is 1. What are the two integers?
Math 8_Unit 2 Integers_NOTES.notebook
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October 26, 2015
10: Without evaluating the quotients, which one will have the least value? Explain.(1428) ÷ (+84)(+1428) ÷ (+84)(1428) ÷ (84)
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