Thursday, October 15th 2015 Unit 2 Integers Review of Grade 7 In 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 1
Adding Integers with Tiles A) (+2) + (+3) = B) (-3) + (-1) = C) (+5) + (-4) = 2
Check. Use tiles to add. D) (+1) + (+3) = We say "positive" and "negative" E) (-2) + (-3) = F) (-4) + (+3) = G) (+4) + (-2) = H) (-5) + (+2) = 3
Adding Integers with Number Lines *Start at zero. Positive integer move right, negative integer move left. I) (+4) + (-5) = J) (-2) + (+6) = 4
Check. Use a number line to add. K) (+7) + (+4) = L) (+6) + (-4) = 5
M) (-11) + (-4) = N) (+3) + (+5) = 6
O) (-8) + (+2) = 7
Subtracting Integers with Tiles A) (+5) (+3) = ADD THE OPPOSITE B) (-4) (-3) = 8
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? 9
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? 10
Check. Use tiles to subtract. A) (+2) (-7) = B) (-3) (-4) = C) (-5) (-5) = 11
I) (+10) (-4) = J) (-5) (+6) = K) (-3) (-5) = 12
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) = 13
Check. Use Add the Opposite L) (+6) (-5) M) (-7) (+3) N) (+9) (+4) O) (-3) (+5) 14
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 15
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 16
Write each repeated addition sentence as multiplication. A) (-6) + (-6) + (-6) + (-6) + (-6) B) (+4) + (+4) + (+4) +(+4) 17
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. 18
Examples A) (+4) x (+3) The first tells us to DEPOSIT (put in the bank if positive WITHDRAWAL (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. 19
(+4) x (+3) Solution Put in 4 sets of +3 20
B) (+4) x (-3) Solution put in of -3 4 sets 21
Try these! C) (+2) x (+4) Solution D) (+2) x (-3) Solution 22
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? 23
Use zero pairs * make 4 sets of 3 zero pairs +- +- +- +- +- +- +- +- +- +- +- +- * now take out 4 sets of -3 Solution: what is left in the bank 24
F) (-4) x (+3) take out of +3 4 sets Need 4 sets 3 Zero Pairs +- +- +- +- +- +- Solution: What is left in the bank? +- +- +- +- +- +- 25
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. 26
H) (-2) x (+4) We need sets of zero pairs Solution: What is left in the bank? 27
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 28
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 29
#1: The temperature fell 3 o C each hour for 6 hours. Find the total drop in temperature. (+6) x (-3) Face Right Move Backward 6 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 30
# 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 Backward 5 steps of size 2 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 31
#3: (+3) x (+2) Face Right Move Forward 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 32
#4: (+3) x (-2) Face Right 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 33
#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 34
#6: (-3) x (+2) Face Left Move Forward 3 steps of size 2 COMPLETE EXAMPLES ON PAGES 18 24. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 35
Try these... #7: (-4) x (+2) Face steps Move of size 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 36
#8: (-4) x (-2) Face steps Move of size 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 37
#9: (+4) x (-2) Face steps Move of size 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 38
Integer Multiplication on a Number Line 39
Multiplying Integers ( + 3) x ( + 2) 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. 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 40
Multiplying Integers (+ 3) x (+ 2) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer +6 41
Multiplying Integers (+ 4) x (+ 2) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer +8 42
Multiplying Integers (+ 2) x ( 4) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer 8 43
Multiplying Integers ( 4) x (+ 2) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer 8 44
Multiplying Integers ( 5) x ( 2) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer +10 45
Multiplying Integers ( 3) x (+ 3) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer 9 46
Multiplying Integers (+ 3) x ( 2) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer 6 47
Multiplying Integers ( 3) x ( 1) 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Erase for answer +3 48
Extra Practice Multiply 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) 49
Summary Bottom page 25 For Multiplying two integers 2 positives = (+2) x (+3) = answer is positive 2 negatives = (-2) x (-3) = answer is positive negative and positive = (-2) x (+3) = answer is negative positive and negative = (+2) x (-3) = answer is negative 50
Therefore, If the signs are the SAME, the answer is POSITIVE If the signs are DIFFERENT, the answer is NEGATIVE Same Signs (++) (--) Positive Or Opposite Signs (+-)(-+) Negative 51
52
Some Properties for Multiplying: Multiplying by Zero 0 x (+3) = 0 x (-3) = Multiplying an integer by zero results in a product of. 53
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. 54
Communative Property (+3) x (+4) = (+4) x (+3) = Did our answer change because we changed the order of the integers? 55
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! 56
Complete Page 73 #3, 4, 5, 8 a -e, 15, 17, 18 57
Area Model Two Integers can also be multiplied using an area model in combination with distributive property. page 29 Distributive Property A 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 58
Example A 8 x 26 Same as 8 x (20 + 6) = (8 x 20) + (8 x 6) = (160) + (48) = 208 Area Model 20 6 8 8 x 20 8 x 6 160 48 Answer: 208 59
Example B -5 x 36 Area Model Distributive Property **Ignore the sign for 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 SIGN 5 x -36 = -180 60
Example C -25 x -48 Ignore sign for area model...we know are answer is 40 8 20 20 x 40 20 x 8 5 5 x 40 5 x 8 Complete example on bottom of page 31 of notes Textbook Page 73 #6a d Page 73 #7a d 25 x 48 = + + + = + = SIGN CHECK (-25) x (-48) = 61
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) = 62
Complete Page 73 # 6 a - d # 7 a - d 63
Dividing Integers using Counters 64
Dividing Integers (+6) (+2) How many groups of +2 will make +6? 65
How many groups of Dividing Integers (+8) (+8) (+4) will make? The Answer? +2 66
Dividing Integers How many groups of will make? The Answer??? 67
How many groups of Dividing Integers ( 9) ( 3) will make? The Answer? +3 68
How many groups of Dividing Integers ( 6) ( 3) will make? The Answer? +2 69
How many groups of Dividing Integers ( 10) (+5) will make? The Answer? -2 70
How many groups of Dividing Integers ( 6) (+3) will make? The Answer? -2 71
How many groups of Dividing Integers ( 8) (+2) will make? The Answer? -4 72
How many groups of Dividing Integers (+12) ( 3) will make? The Answer? -4 73
How many groups of Dividing Integers (+9) ( 3) will make? The Answer? -3 74
How many groups of Dividing Integers (+8) ( 2) will make? The Answer? -5 75
Page 81 # 8 76
Dividing Integers using The Number Line 77
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! 78
( 10) (+2) 2 = 5 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 79
( +8 ) ( +44 ) = +2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 80
( +6 ) ( 2 ) = 3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 81
( 9 ) ( +33 ) = 3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 82
( 12 ) ( 4 ) = +3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 83
( +15) ( 33 ) = 5 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 84
( +10 ) ( +55 ) = +2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 85
( +12) ( 66 ) = 2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 86
( +11) ( 11 ) = 11 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards 15 14 13 12 11 10 Answer... 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 To get there I took steps and I am now facing. + - 87
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)....... 88
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 89
Summary for Dividing Integers Dividing Two Positive IntegersDividing Two Negative Integers (+8) (+2) = (+6) (+3) = (+12) (+3) = What do you notice? ( 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? 90
The rules for Dividing Integers are the same as Multiplying Integers. Same Signs (++) (--) Positive Or Opposite Signs (+-)(-+) Negative 91
Terminology (+15) (+3) = +5 Dividend Divisor Quotient We can also write a division statement as a fraction: 92
Practice: Divide A) (-8) (-4) B) = C) (-12) ( ) = +2 D) 93
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. 94
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? 95
G) The temperature in Nain is dropping 2 o C each hour. How many hours did it take to fall 10 o C? 96
Complete Pages 87 88 # s 4, 5, 7a, 9, 10, 11, 12 97
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 First E Exponents (not in grade 8) D Division M Multiplication A Addition S Subtraction } } Whichever comes first in the problem Whichever comes first in the problem 98
Examples A) 9 6 + 36 4 1 99
B) [(+9) ( 2)] ( 3) 100
C) [( 6) + ( 2)] ( 4) + ( 5) 101
D) 102
E) 103
F) (2 105) (2 +8) 2 104
G) 105
H) 15 [ 10 ( 2)] 106
I) Kathy completed the following calculations on a math test. Is she correct? Explain. 107
Complete Page 92 #'s 3, 5, 8, 9ad 108
Word Problems Words/Phrases that indicate MULTIPLY Multiply by Product of Times Double Triple As much By Find the total Words/Phrases that indicate DIVIDE Quotient of Divided by Divided equally Per How many (much) each Split Equal pieces Shared 109
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? 110
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? 111
3: An oil rig is drilling a well at 2m per minute. How deep is the well after the first 8 minutes? 112
4: The temperature decreased 16 o C over a 4 hour period. Assuming the temperature decreased at a constant rate, how much did the temperature decrease each hour? 113
5: The temperature rose 4 o C each hour for 5 hours. Use integers to find the total change in temperature. 114
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. 115
7: A glacier retreated about 2m per day for 7 days. Use integers to find the total change in the length of the glacier. 116
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? 117
9: The product of two integers is 20. The sum of the same two integers is 1. What are the two integers? 118
10: Without evaluating the quotients, which one will have the least value? Explain. ( 1428) (+84) (+1428) (+84) ( 1428) ( 84) 119