In notes, handouts, tests shaded is Positive and non-shaded (white) is Negative
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1 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
2 Adding Integers with Tiles A) (+2) + (+3) = B) (-3) + (-1) = C) (+5) + (-4) = 2
3 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
4 Adding Integers with Number Lines *Start at zero. Positive integer move right, negative integer move left. I) (+4) + (-5) = J) (-2) + (+6) = 4
5 Check. Use a number line to add. K) (+7) + (+4) = L) (+6) + (-4) = 5
6 M) (-11) + (-4) = N) (+3) + (+5) = 6
7 O) (-8) + (+2) = 7
8 Subtracting Integers with Tiles A) (+5) (+3) = ADD THE OPPOSITE B) (-4) (-3) = 8
9 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
10 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
11 Check. Use tiles to subtract. A) (+2) (-7) = B) (-3) (-4) = C) (-5) (-5) = 11
12 I) (+10) (-4) = J) (-5) (+6) = K) (-3) (-5) = 12
13 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
14 Check. Use Add the Opposite L) (+6) (-5) M) (-7) (+3) N) (+9) (+4) O) (-3) (+5) 14
15 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 = 6 4 x 5 means 4 sets of = 20 15
16 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
17 Write each repeated addition sentence as multiplication. A) (-6) + (-6) + (-6) + (-6) + (-6) B) (+4) + (+4) + (+4) +(+4) 17
18 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
19 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
20 (+4) x (+3) Solution Put in 4 sets of +3 20
21 B) (+4) x (-3) Solution put in of -3 4 sets 21
22 Try these! C) (+2) x (+4) Solution D) (+2) x (-3) Solution 22
23 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
24 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
25 F) (-4) x (+3) take out of +3 4 sets Need 4 sets 3 Zero Pairs Solution: What is left in the bank?
26 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
27 H) (-2) x (+4) We need sets of zero pairs Solution: What is left in the bank? 27
28 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
29 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
30 #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
31 # 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
32 #3: (+3) x (+2) Face Right Move Forward 3 steps of size
33 #4: (+3) x (-2) Face Right Move Backward 3 steps of size
34 #5: (-3) x (-2) Face Left Move Backward 3 steps of size
35 #6: (-3) x (+2) Face Left Move Forward 3 steps of size 2 COMPLETE EXAMPLES ON PAGES
36 Try these... #7: (-4) x (+2) Face steps Move of size
37 #8: (-4) x (-2) Face steps Move of size
38 #9: (+4) x (-2) Face steps Move of size
39 Integer Multiplication on a Number Line 39
40 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
41 Multiplying Integers (+ 3) x (+ 2) Erase for answer +6 41
42 Multiplying Integers (+ 4) x (+ 2) Erase for answer +8 42
43 Multiplying Integers (+ 2) x ( 4) Erase for answer 8 43
44 Multiplying Integers ( 4) x (+ 2) Erase for answer 8 44
45 Multiplying Integers ( 5) x ( 2) Erase for answer
46 Multiplying Integers ( 3) x (+ 3) Erase for answer 9 46
47 Multiplying Integers (+ 3) x ( 2) Erase for answer 6 47
48 Multiplying Integers ( 3) x ( 1) Erase for answer +3 48
49 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
50 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
51 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 52
53 Some Properties for Multiplying: Multiplying by Zero 0 x (+3) = 0 x (-3) = Multiplying an integer by zero results in a product of. 53
54 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
55 Communative Property (+3) x (+4) = (+4) x (+3) = Did our answer change because we changed the order of the integers? 55
56 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
57 Complete Page 73 #3, 4, 5, 8 a -e, 15, 17, 18 57
58 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) = = 27 58
59 Example A 8 x 26 Same as 8 x (20 + 6) = (8 x 20) + (8 x 6) = (160) + (48) = 208 Area Model x 20 8 x Answer:
60 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 = x 30 5 x 6 = GO BACK TO SIGN 5 x -36 =
61 Example C -25 x -48 Ignore sign for area model...we know are answer is x x 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
62 Example D (-72) x (+15) Ignore sign for area model...we know are answer is x x x 10 2 x 5 72 x 15 = = + = SIGN CHECK (-72) x (+15) = 62
63 Complete Page 73 # 6 a - d # 7 a - d 63
64 Dividing Integers using Counters 64
65 Dividing Integers (+6) (+2) How many groups of +2 will make +6? 65
66 How many groups of Dividing Integers (+8) (+8) (+4) will make? The Answer? +2 66
67 Dividing Integers How many groups of will make? The Answer??? 67
68 How many groups of Dividing Integers ( 9) ( 3) will make? The Answer? +3 68
69 How many groups of Dividing Integers ( 6) ( 3) will make? The Answer? +2 69
70 How many groups of Dividing Integers ( 10) (+5) will make? The Answer? -2 70
71 How many groups of Dividing Integers ( 6) (+3) will make? The Answer? -2 71
72 How many groups of Dividing Integers ( 8) (+2) will make? The Answer? -4 72
73 How many groups of Dividing Integers (+12) ( 3) will make? The Answer? -4 73
74 How many groups of Dividing Integers (+9) ( 3) will make? The Answer? -3 74
75 How many groups of Dividing Integers (+8) ( 2) will make? The Answer? -5 75
76 Page 81 # 8 76
77 Dividing Integers using The Number Line 77
78 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
79 ( 10) (+2) 2 = 5 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
80 ( +8 ) ( +44 ) = +2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
81 ( +6 ) ( 2 ) = 3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
82 ( 9 ) ( +33 ) = 3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
83 ( 12 ) ( 4 ) = +3 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
84 ( +15) ( 33 ) = 5 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
85 ( +10 ) ( +55 ) = +2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
86 ( +12) ( 66 ) = 2 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
87 ( +11) ( 11 ) = 11 Touch me when done! Ask yourself... forwards How do I get to stepping of size? backwards Answer To get there I took steps and I am now facing
88 Extra Practice (use numberlines).. 1. (+12) (-4) (-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)
89
90 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
91 The rules for Dividing Integers are the same as Multiplying Integers. Same Signs (++) (--) Positive Or Opposite Signs (+-)(-+) Negative 91
92 Terminology (+15) (+3) = +5 Dividend Divisor Quotient We can also write a division statement as a fraction: 92
93 Practice: Divide A) (-8) (-4) B) = C) (-12) ( ) = +2 D) 93
94 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
95 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
96 G) The temperature in Nain is dropping 2 o C each hour. How many hours did it take to fall 10 o C? 96
97 Complete Pages # s 4, 5, 7a, 9, 10, 11, 12 97
98 Section 2.5: Order of Operations To ensure everyone gets the same answer for a question like 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
99 Examples A)
100 B) [(+9) ( 2)] ( 3) 100
101 C) [( 6) + ( 2)] ( 4) + ( 5) 101
102 D) 102
103 E) 103
104 F) (2 105) (2 +8) 2 104
105 G) 105
106 H) 15 [ 10 ( 2)] 106
107 I) Kathy completed the following calculations on a math test. Is she correct? Explain. 107
108 Complete Page 92 #'s 3, 5, 8, 9ad 108
109 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
110 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
111 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
112 3: An oil rig is drilling a well at 2m per minute. How deep is the well after the first 8 minutes? 112
113 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
114 5: The temperature rose 4 o C each hour for 5 hours. Use integers to find the total change in temperature. 114
115 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
116 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
117 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
118 9: The product of two integers is 20. The sum of the same two integers is 1. What are the two integers? 118
119 10: Without evaluating the quotients, which one will have the least value? Explain. ( 1428) (+84) (+1428) (+84) ( 1428) ( 84) 119
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