Chapter 6: Addition and Subtraction of Integers

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1 Chapter 6: Addition and Subtraction of Integers Getting Started, p a) 8 > b) 12 < 14 c) 24 > 6 d) 99 < a < b is true, since a is farther to the left than b on the number line. 3. a) b) c) d) a) b) c) d) c) Tomorrow s temperature: 5. 4 C 5 C 1 C +1 d) 6 C < 4 C, because it is lower a) +4 on the thermometer than 4 C. 6.1 Comparing Positive and Negative Numbers, p a) +3 > +2 b) +2 > 3 c) +3 > d) > 3 5. a) b) 7,,,, +1, 6. a) < since is to the left of b) > 1 since is to the right of c) +2 > 3 since +2 is to the right of 3. b) 1 e) There are three spaces between C and +1 C so C is 3 C colder than +1 C. 7. a) C 6. a) +25. For example, it is positive because Tara has a positive bank balance. b). For example, it is negative because Takuya must pay the library $2. c). For example, it is negative because it is below the reference height of sea level. d) 38 C. For example, I chose ( ) because it is below the reference point of C. e) +41 C. For example, I chose (+) because it is above the reference point of C. 7. Number Countdown Term 4 T minus four seconds +4 T plus four seconds takeoff 8. a) For example, positive is greater than zero. Negative is less than zero. b) For example, a dictionary definition for positive states: not negative or greater than zero. For example, a dictionary definition for negative is: not positive or smaller than zero. b) c) For example, numbers on the left of the number line are less than numbers to the right on the number line. a) 8, 3,, +2, +7 d) 14, 12, 1, b) 7,, +1, +4, +7 e) 37,,, 9, +11 c) 1, 11, +1, +14, For example, when comparing two negative numbers, the more negative number is less than the other number. when comparing two positive numbers, the more positive number is greater than the other number. When comparing a positive number and a negative number, the positive number is always greater than the negative number. a) +1 > 4 b) +13 > c) 1 < d) +3 > 3 e) 15 > 16 f) 1 < For example, numbers on the left of the number line are less than numbers to the right on the number line. e) b) d) a) c) Nelson Mathematics 7 Solutions 6-1

2 11. a) Two of, 6, 7,... b) Two of 9, 8, 7, 6, 5, or 3 c) One of 4, 3,, or 1 and one of 1, 2, 3, 4 d) Any negative integer from 16, 17, 18,... e) For example, 12. For example, 4 is greater than 1 because 4 is to the right of 1 on the number line. 4 is less than +1 since it is to the left of +1 on a number line For example, 1 is less than +1 because 1 is to the left of +1 on a number line Order the daily high temperatures from least to greatest: +5,, +9, +1 Order the daily low temperatures from least to greatest: 7, 6, 3, a) March 26: +1 b) March 27: +5 c) March 26: d) March 28: The message is: SMART. S M A R T Order the player s scores from greatest to least: Name Score Ming +4 Barbara +3 Toni +3 Raj +3 Monica 1 Riki Jan Arvin 4 a) Ming will be captain since she has the highest score. b) The five players with the highest scores are the starting players. They are: Ming, Barbara, Raj, Toni, and Monica. 17. Rank the scores from lowest to highest: Player Country Final Score Mike Weir Canada 7 Phil Mickelson United States Jim Furyk United States 4 Vijay Singh Fiji 1 Retief Goosen South Africa +1 Angel Cabrera Argentina +2 Mike Weir won because he had the lowest score. 18. a) i) +1 (ii) (iii) +2 (iv) 8 b) 8,, +2, +1 c) For example, the integer represents neither an income nor a debt. 19. a) True. For example, all integers less than are to the left of on a number line. These are all negative numbers. 4 is an example. b) False. This is true if you are talking about the positive side on a number line. The opposite occurs on the negative side. (The farther from, the smaller the integer is.) For example, 5 < 1. c) True. All negative integers are to the left of 1 on a number line. These are all less than 1, and 1 is not less than itself. d) True. Any integer that is positive lies to the right of. These are all greater than is the integer Adding Integers Using the Zero Principle, p Expression Using counters Using numbers a) ( 3) + (+2) ( ) + ( ) ( 1) + () + (+2) 1 b) ( 4) + (+6) ( ) + ( 4) + (+4) + (+2) ( ) +2 c) (+5) + ( 6) ( ) + (+5) + ( 6) 1 ( ) d) () + (+7) ( ) + ( ) () + (+5) + (+2) +2 e) (+2) + ( 8) ( ) + ( (+2) + () + ( 6) ) 6 f) ( 1) + ( 9) ( ) + ( 1) + ( 9) ( 1 ) 4. Use the zero principle. a) (+3) + ( 3) b) ( 7) + (+7) 5. a) ( 3) + () () b) (+2) + () c) ( 4) + (+1) ( 3) d) ( 7) + (+6) ( 1) e) () + () ( 7) f) () + (+2) ( 3) 6. For example, (5) + (+25) because they are opposite integers. When you add opposite integers the result is. 7. a), 1,, 3, 4,, 6, 7; For example, add 1 to the previous term. b) 3,, 1,, +1, +2, +3; For example, add +1 to the previous term. 8. a) (+1) + ( 1) + ( 1) + ( 1) ( 1) b) ( 1) + (+1) + (+1) + (+1) (+1) c) (+1) + ( 1) + (+1) + ( 1) + ( 1) + + ( 1) ( 1) d) (+1) + ( 1) + ( 1) + ( 1) + (+1) + ( 1) + ( 1) 6-2 Chapter 6: Addition and Subtraction of Integers

3 9. a) ( 3) + (+3) + (+5) + (+5) +5 b) ( 1) + () + ( 1) ( 3) + ( 1) 4 c) (+2) + (+1) + ( 4) ( 1) 1. a) ( 1) + () > ( 4) b) (+2) + () ( 3) c) ( 3) + (+6) > (+2) d) (+5) + ( 7) () e) () + ( 4) < () f) () + (+1) < 11. For example: a) (+1) + (+1) + (+1) + (+1) + ( 1) +3 b) ( 1) + ( 1) + ( 1) + (+1) c) (+1) + (+1) + ( 1) + ( 1) d) (+1) + ( 1) + (+1) + ( 1) + ( 1) a) For example, using numbers: (+1) + (+1) + ( 1) + (+1) + ( 1) + ( 1) + (+1) + (+1) + ( 1) (+1) (+1) b) For example, I chose to use numbers because I find it easier to add the pairs of opposite integers to get zero. 13. For example: (+3) + ( 7) + ( 1) (), () + ( 1) + (+1), (+1) + ( 4) + () 14. For example, to end up with (+1) the boxes must add to. Only pairs of (+1) and ( 1) can add to, but there are 3 boxes to fill in, so you cannot pair one of the terms and the boxes cannot add to. 15. a) All the rows, columns, and diagonals should add to 6. Rows: (+1) + ( 6) + ( 1) ( 6) ( 4) + () + ( 6) ( 3) + (+2) + () ( 6) Columns: (+1) + ( 4) + ( 3) ( 6) ( 6) + () + (+2) ( 6) ( 1) + + () ( 6) Diagonals: (+1) + () + () ( 6) ( 3) + () + ( 1) ( 6) b) The middle row shows the Magic Sum to be +9. This means that that bottom right entry must be +4. The remaining entries can be found by completing each column, row, or diagonal to get Rows: (+2) + (+1) + (+6) (+9) (+7) + (+3) + ( 1) (+9) + (+5) + (+4) (+9) Columns: (+2) + (+7) + (+9) (+1) + (+3) + (+5) (+9) (+6) + ( 1) + (+4) (+9) Diagonals: (+2) + (+3) + (+4) (+9) + (+3) + (+6) (+9) c) Use trial and error to complete the square The sum is. Rows: (+3) + ( 4) + (+1) () + + (+2) ( 1)+ (+4) + ( 3) Columns: (+3) + () + ( 1) ( 4) + + (+4) (+1) + (+2) + ( 3) Diagonals: (+3) + + ( 3) ( 1) + + (+1) d) Students may use the answer to a previous part and add the appropriate integer to modify the Magic Square to meet the requirements. For example, adding 6 to each term of the answer for part c) creates a Magic Square whose values range from 1 to a) True. For example, when you add a positive integer to a positive integer the result is a larger positive number. E.g. (+25) + (+1) (+35) b) True. For example, when you add two negative integers the result is a more negative number. E.g. () + ( 8) ( 13) c) False. For example, if you add a smaller positive number to a bigger negative number the result is a negative number. E.g. ( 4) + (+2) () 17. a), +1, 1, +2,, +3, 3, +4, 4, +5 For example, one possible rule is add +1, then add, then add +3, then add 4, and so on. b) 1,, 1, 1,, 3,, 8, 13, 1, 34 For example, the pattern is to add the two preceding terms to get the next term. 18. The sum of the integers from to 5 is. For example, the sum is because we are adding 5 pairs of opposite integers. Each opposite pair adds to. 19. Four-day weather forecast High temperature ( C) Wednesday +1 Thursday +5 6 Friday +9 7 Saturday 3 Total a) Average high (+32) 4 or 8 C b) Average low ( 16) 4 or 4 C c) The range is from +1 C to 7 C, or 17 C. Low temperature ( C) 6.4 Adding Integers That Are Far from Zero, p a) b) 7 c) 7 d) +7 e) a) 39 red (positive) counters and 26 blue (negative) counters. b) For example, 26 red counters and 26 blue counters will add to since each red counter represents (+1) and each blue counter represents ( 1). c) There will be 13 red counters left over. d) (+39) + (6) (+13) Nelson Mathematics 7 Solutions 6-3

4 7. a) () + () ( 7) b) () + (+2) ( 3) c) () + (+5) (+3) d) () + () ( 7) 8. a) (+5) + (+3) (). 5 red counters and 3 red counters. In total there are 8 red counters. b) () + ( 3) ( 8). 5 blue counters and 3 blue counters. In total there are 8 blue counters. c) ( 4) + ( 4) ( 8). 4 blue counters and 4 blue counters. In total there are 8 blue counters. d) ( 1) + ( 15) (5). 1 blue counters and 15 blue counters. In total there are 25 blue counters. e) ( 15) + (+1) (). 15 blue counters and 1 red counters. There are 5 blue counters left over. f) (+11) + ( 3) (). 11 red counters and 3 blue counters. There are 8 red counters left over. 9. a) (+5) + ( 1) () 11. a) (5) + (+38) (+13) ( 15) + (+38) (+23) Since (+23) (+13) +1, the second sum is 1 more than the first sum. b) For example: (+125) + (2) (+73) b) (+1) + () (+5) (+125) + ( 32) (+93) c) ( + ( 15) ( 15) d) ( 1) + (+1) e) ( 3) + ( 35) ( 65) f) (+35) + () ( 15) a) () + () ( 7) b) ( 4) + (+5) (+1) c) ( 6) + (+2) ( 4) d) (+4) + ( 7) ( 3) e) () + () ( 7) f) (+1) + ( 8) (+2) Since (+93) (+73) (+2), the second sum is 2 more than the first sum. 12. a) Always true. For example, when you add two negative integers the result is always a bigger negative number. E.g. ( 4) + ( 6) ( 1) b) Not always true. For example, (+5) + () (+3). c) Always true. For example, in order for two integers to add to zero they have to be the same distance away from but in the opposite direction. E.g. () + (+5) Player Goals for Goals against +/ Score Heidi Rana Meagan Sonya Indu Order the players from highest to lowest score: Heidi, Rana, Sonya, Indu, Meagan. 15. Starting temperature ( C) Temperature change ( C) Final temperature ( C) a) +1 () + (+1) 4 b) 1 6 ( 1) + ( 6) 16 c) 8 + ( 8) 8 d) +5 (+5) + () e) +7 9 (+7) + ( 9) f) (+18) + ( 1) 6-4 Chapter 6: Addition and Subtraction of Integers

5 Sum Students must find a strategy to create a Magic Square. One way is to add 4 to each row of the Magic Square in question 16. Adding 1 to each entry does this For example, both processes involve moving to the left on a number line. E.g. (+25) + ( 15) +1 is the same as If the move does not cross, that is, if you are left with a positive umber, then adding a negative number to a positive number is like subtracting. 6.5 Integer Addition Strategies, p a) ( 4) + (+55) + (+5) + ( 4) + ( 1) ( 4) + ( +5) + (+55) + () ( 4) + (+6) + () (+2) + () ( 3) b) ( 13) + () + ( 12) + (+1) + (+9) ( 13) + ( 12) + ( ) + (+1) + (+9) (5) + (+27) (+2) c) The answer is 73. Keystroke sequence: / / a) () + ( ) + ( 3) + (+5) () + ( ) + (+5) () + () b) ( ) + (+5) + ( ) + ( 3) + () () c) ( ) + ( 3) + ( +5) + () () + () d) ( +5) + () + ( 3) + () ( +3) + ( 3) + () + () () 6. a) ( +4) + ( 3) + (+1) + (+6) + () ( +1) + (+1) + (+6) + () () + () (+6) b) ( +4) + (+1) + (+6) + ( 3) + () (+11) + () (+6) c) For example, you can add numbers in any order and you will always get the same result. 7. For example, a) regrouping: ( 12) + (+2) + () ( 1) + () ( 15) b) regrouping: (+23) + ( 14) + ( 7) (+23) + (1) (+2) c) regrouping ( 18) + (+5) + (+18) ( 18) + (+18) + (+5) + (+5) (+5) d) regrouping: ( +7) + ( 3) + ( 13) + (+6) (+4) + ( 7) ( 3) e) regrouping: ( 1) + ( 3) + ( +5) + (+1) (1) + (+6) (+9) f) regrouping: ( 1) + (+48) + ( 38) + ( 9) ( +38) + ( 38) + ( 9) + ( 9) ( 9) 8. For example, (+28) + ( 12) + (+19) + (1) + ( 1) (+4), (+31) + ( 15) + (+29) + ( 17) + (4) (+4) Nelson Mathematics 7 Solutions 6-5

6 9. For example, a) separation of positive and negative sums: () + ( 4) + (+3) + () + ( 6) + (+4) + (+1) + (+5) ( ) + (+3) + (+4) + (+1) + (+5) + ( 4) + () + ( 6) (+21) + ( 15) (+6) b) regrouping: ( 1) + ( 15) + (+15) + (+2) ( 1) + (+2) + ( 15) + (+15) (+1) + (+1) c) regrouping: ( +45) + ( 35) + ( +15) + (5) + ( +2) + () 12. First calculate the depth of the submarine at each time: 11:: ( 3) + (3) ( 353) 12:: ( 353) + ( 31) ( 384) 13:: ( 384) + (+18) ( 366) 14:: ( 366) + ( 64) ( 43) 15:: ( 43) + (5) ( 345) Time 11: 12: 13: 14: 15: Change in depth (m) Depth (m) Depth of submarine at 15: ( 3) + (3) + ( 31) + (+18) + ( 64) + (5) ( 345) The depth of the submarine at 15: is 345 m. ( +1) + ( 1) + (+15) 13. a) S3 + (+15) (+15) d) calculator: (78) + (+415) (+137) e) calculator: (+426) + ( 42) + ( 318) (+426) + (6) ( 134) 1. Game / score Player s score separation of positive and negative sums: (+11) + ( 1) + (+4) + ( 3) + () + (+5) + + (+1) ( +11) + (+4) + (+5) + (+1) + ( 1) + ( 3) + () (+21) + ( 6) (+15) 11. First calculate the prices of the stock each week: Week 1: price (+21) + (+1) (+31) Week 2: price (+31) + (+3) (+34) Week 3: price (+34) + ( 12) (+22) Week 4: price (+22) + ( 6) (+16) Week 5: price (+16) + ( 15) (+1) Week 6: price (+1) + () (+9) Week Price Change ($) Price ($) Samantha sold the stock in Week 5, during which the stock fell to $1. +9 ADD S b) For example, the final sum is +5 because the 4 integers that were added (+4,, +1, 3) have a sum of. c) ADD Mid-Chapter Review, p For example, a) A profit is a positive gain for a company. A loss is negative since its leads to a debt for the company. b) (+5) C is warmer than () C because it is above zero. c) A hockey player on the ice earns a negative point when there is a goal against her team and a positive point when there is a goal for her team. d) A driver travelling at 5 km/h wants to accelerate by 5 km/h. The driver adds (+5) km/h to the speed. If the driver wants to decelerate, the speed would decrease by 5 km/h. This is like adding () to the speed. 2. a) b) Chapter 6: Addition and Subtraction of Integers

7 c) 9 d) Use regrouping: ( 34) + ( +17) + (+18) + ( ) + ( 15) + () ( 34) + ( +35) + ( 35) + () ( 34) + + () ( 36) 3. a) () > ( 1) b) ( 14) < ( 1) c) () < d) (+5) < (+1) 4. a) 7, a) ( +11) + (6) + ( 15) ( 15) + ( 15) ( 3) b) ( 33) + ( ) + (+12) ( 33) + ( 8) ( 41) 12. At the end of June, Georgina s investment is worth $1154. (1125) + ( 15) + (5) + (+137) + (+91) + () + () $1154 b) 3, c), d), a) (+3), ( ) + ( ) ( ) b) ( 7), ( ) + ( ) ( ) c) (+1), ( ) + ( ) ( ) d), ( ) + ( ) + ( ) 6. a) ( 8) + (+3) () b) (+2) + () c) () + (+1) + (+7) (+12) 7. a) For example, the sum of two integers is positive when the sign of the integer with the largest magnitude is positive. E.g. () + (+7) (+2). b) For example, the sum of two integers is negative when the sign of the integer with the largest magnitude is negative. E.g. ( 7) + (+5) (). 8. ( 45) + (+1) (+55) Anthony was 55 m above sea level. 9. a) ( 15) + (+5) ( 1) b) ( 11) + () ( 13) c) (+123) + ( 92) (+31) 6.6 Using Counters to Subtract Integers, p a) ( 4) (+2) ( 6) b) (+3) (+2) (+1) c) (+3) () (+5) d) ( 3) () ( 1) e) () ( 3) (+1) 6. a) Parts a), c), and e) require adding zeros. b) part a) ( ) ( ) ( ) ( ) ( 6) part b) ( ) ( ) (+1) part c) ( ) ( ) ( ) ( ) (+5) part d) ( ) ( ) ( 1) part e) ( ) ( ) ( ) ( ) (+1) 7. ( ) ( ) ( ) ( 3) No zeros are needed because there are enough blue counters on the left to permit the subtraction. 8. ( ) ( ) ( ) ( ) ( ) (+3) Three zeros must be added because there are not enough blue counters on the left to permit the subtraction. Nelson Mathematics 7 Solutions 6-7

8 9. ( 3) (+4) ( ) ( ) ( ) ( ) ( 7) ( 3) + ( 4) ( ) + ( ) ( 7) Therefore, ( 3) (+4) ( 3) + ( 4) ( 7) 1. (+6) ( 4) ( ) ( ) ( ) ( ) (+1) ( 4) (+6) ( ) ( ) ( ) ( ) ( 1) The expressions do not have the same value, so they are not equal. 11. a) ( ) ( ) (+1) b) ( ) ( ) ( ) ( ) ( 1) c) ( ) ( ) ( 1) d) ( ) ( ) ( ) ( ) (+1) e) ( ) ( ) ( ) ( ) (+7) f) ( ) ( ) ( ) ( ) ( 7) g) ( ) ( ) ( ) ( ) (+7) h) ( ) ( ) ( ) ( ) ( 7) 12. a) ( ) ( ) (+4) b) ( ) ( ) ( ) ( ) ( ) ( 4) c) ( ) ( ) ( 4) d) ( ) ( ) ( ) ( ) (+4) e) ( ) ( ) ( ) ( ) () f) ( ) ( ) ( ) ( ) ( 8) g) ( ) ( ) ( ) ( ) () h) ( ) ( ) ( ) ( ) ( 8) 13. a) ( ) ( ) ( ) (+4) b) ( ) ( ) ( ) ( ) ( ) (+5) c) ( ) ( ) ( ) (+5) d) ( ) ( ) ( ) ( ) ( ) (+5) e) ( ) ( ) ( ) ( ) () f) ( ) ( ) ( 1) Parts b), c), and d) have the same result. 14. a) (+5) (+4) (+1) (+5) (+3) (+2) (+5) (+2) (+3) (+5) (+1) (+4) b) () (+4) ( 9) () (+3) ( 8) () (+2) ( 7) () (+1) ( 6) c) () ( 9) (+4) () ( 8) (+3) () ( 7) (+2) () ( 6) (+1) d) ( 1) () (+1) ( 1) ( 3) (+2) ( 1) ( 4) (+3) ( 1) () (+4) 15. a) No. The answer is ( 1). b) ( ) ( ) ( ) ( ) ( 1) c) Use the zero principle to add 6 zeros. Now there are enough positives on the left to perform the subtraction. You are left with 1 negatives. The answer is a) (+1) (+1) does not require zeros to be added to the first term because there are enough positives to perform the subtraction. b) (+1) (+1) requires zeros to be added to the first term because there are not enough positives to perform the subtraction. c) ( 1) ( 1) does not requires zeros to be added to the first term because there are enough negatives to perform the subtraction. 6-8 Chapter 6: Addition and Subtraction of Integers

9 d) ( 1) ( 1) requires zeros to be added to the first term because there are not enough negatives to perform the subtraction. e) (+1) ( 1) requires zeros to be added to the first term because there are not enough negatives to perform the subtraction. f) ( 1) (+1) requires zeros to be added to the first term because there are not enough positives to perform the subtraction. g) (+1) ( 1) requires zeros to be added to the first term because there are not enough negatives to perform the subtraction. h) ( 1) (+1) requires zeros to be added to the first term because there are not enough positives to complete the subtraction. 17. Day 1 ( C) Day 2 ( C) Difference a) 3 1 ( 1) ( 3) (+2) b) +5 4 ( 4) (+5) ( 9) c) 1 +6 (+6) ( 1) (+16) d) 1 ( 1) (+1) e) +1 +7(+7) (+1) ( 3) f) 8 1 ( 1) ( 8) (+7) g) 1 ( 1) ( 1) h) 1 +2 (+2) (1) (+41) 18. Golfer Day 1 Day 2 Change (Day 2 Day 1) Ming 1 +4 Kaitlyn Omar Anthony Braydon Tynessa Rana 7 9 Ming ( 1) () (+4), Kaitlyn (+1) (+5) (+5) Omar (+6) ( 1) (+16), Anthony ( 18) ( 1) ( 8) Braydon () ( 15) (+1), Tynessa: (+7) (+5) (+2) Rana: ( 9) ( 7) () 19. a) Sometimes true: () ( 1) ( 1), but () ( 3) (+1) b) Sometimes true: (+2) (+1) (+1), but (+2) (+3) ( 1) c) Always true: (+2) ( 1) (+3) d) Never true: () (+3) e) Sometimes true: () ( 3) (+1). is greater than 3, but the difference is positive. 2. a) () (+3) (); () + ( 3) () So () (+3) () + ( 3). This shows that subtracting a positive integer is the same as adding the opposite of that integer. b) Yes, subtracting an integer is always the same as adding its opposite, as this example shows: () (+3) ( ) ( ) ( ) ( ) () () + ( 3) ( ) + ( ) () In both cases the result is three more negatives than (). 21. a b a b a) b) 15 1 c) a) (+4) + (+2) (+3) ( ) + ( ) ( ) ( ) ( ) (+3) b) ( 4) + ( 3) () ( ) + ( ) ( ) ( ) ( ) () c) (+3) ( 8) + ( 1) ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) + ( ) (+1) 23. a) For example, (+1) + ( 1) (+1) ( 1) b) Possible solutions: + () (+5) ( 1), (+2) + ( 12) ( 1) 6.7 Using Number Lines to Subtract Integers, p Each arrow goes from the second number in the subtraction question to the first number. a) (+1) () b) ( 1) (+15) c) (5) () d) (+5) ( 1) e) (+9) (+2) 5. a) +15 b) 5 c) 35 d) +15 e) a) +4 b) 35 c) ( 35) (+4) Both distances have a magnitude of 2 but the direction from to 4 is negative and the distance from 4 to is positive. 8. The starting point is 5 units to the right of the ending point. For example: ( 3) (+2) Nelson Mathematics 7 Solutions 6-9

10 9. ( 3) (+4) ( 7) g) () (+2) ( 3) + ( 4) ( 7) h) (+1) ( 1) (+2) Both of these expressions have the same value because subtracting +4 is the same as adding (+36) ( 34) (+7) ( 34) (+36) ( 7) a) () ( 4) (+2) a) Start of arrow End of arrow Subtraction statement a) 1 ( 1) () b) (+1) (+15) c) 1 16 ( 16) ( 1) d) () () e) 8 ( 8) f) +2 8 ( 8) (+2) g) (+15) ( 15) h) ( 85) ( 15) b) (+3) (+7) ( 4) 4 b) c) (3) (1) () c) d) d) (+35) (+32) (+3) e) (+1) ( 1) (+2) +2 e) f) S1 S8 S6 S4 S f) () () g) h) Chapter 6: Addition and Subtraction of Integers

11 14. (+6) ( 4) ( 4) (+6) They represent the same distance, but from different perspectives. From ( 4) to (+6) is (+1), whereas from (+6) to ( 4) is ( 1). 15. Last year s gain/loss ($) Current gain/loss ($) Change in value this year ($) a) 3 35 ( 35) ( 3) () b) ( 15) (+2) ( 35) c) (+2) (+15) ( 13) d) (+15) (95) (+7) e) ( 95) ( 15) (+55) f) ( 111) (+537) ( 648) g) ( 121) ( 97) (4) h) (+128) ( 32) (+16) h) a b a b a) b) c) a) (+4) + (+2) (+3) (+3) b) ( 45) + ( 35) () ( 6) c) (+37) ( 85) + ( 1) (+112) 45 a) b) d) (+12) ( 9) + ( 1) (+2) c) d) e) f) g) e) ( 1) (1) + (2) ( 11) f) (+31) (5) + ( 153) (+353) a) (+15) ( 9) (+15) + (+9) (+24) (+15) + (+9) (+24) Nelson Mathematics 7 Solutions 6-11

12 b) (+15) () (+15) + (+2) (+35) a) By looking at the maze, you can see that the path from the entrance to Exit B is much shorter than the path to Exit A c) The predicted difference is (+5) Subtracting a negative number is the same as adding a positive number, so I added (+15) and (+35) to get (+5). 6.8 Solve Problems by Working Backwards, p Yes. The original number is always one more than the result. For example, after the first step you have one more than you started with. After the second step you have the number you started with subtracted from 9. After the third step you have the number you started with subtracted from 1. After the final step you have one less than the number you started with. So the end result is that the four steps result in a subtraction of one from the original number. To reverse the process, you just need to add one to the result. 4. The original number is 14. The steps are as follows: Start with +1. Subtract from (+1) (+2) Find the opposite. () Subtract 9. () ( 9) (+7) Add 7. (+7) + (+7) (+14) 5. The original number is 28. The steps are as follows: Start with 12. Add 36. Subtract 18. Add 9. Subtract For example, follow the steps below: Think of a number. Add +3 to your number. Subtract from the number. Add to the result. Subtract 6 from the result. Find the opposite of the result. Add twice the original number to the result. 7. The steps should have the overall effect of adding. For example: Think of a number. Take the opposite of your number. Subtract 6 from the result. Add to the result. Subtract 2 from the result. Add twice the original number to the result. Exit A Enter Exit B b) Working backwards through a maze is probably no different than working forwards in a maze. 9. You must work backwards to find out how much he lifted during the first week. Start with the final weight of 8 kg and subtract +2 kg for each of the 8 weeks that he lifted weights. () (+2) (+2) (+2) (+2) (+2) (+2) (+2) (+2) 64 kg Lloyd lifted 64 kg in his first week. 1. Work backwards by multiplying the amount of grapes by 2 each day: Friday 8 grapes, Thursday grapes, Wednesday grapes, Tuesday grapes, Monday grapes Assuming Heidi ate half the grapes on Monday, she bought 128 grapes. 11. Work backwards as follows: Day 8: $2.5 Day 7: $5. Day 6: $1. Day 5: $2. Day 4: $4. Day 3: $8. Day 2: $16. Day 1: $32. The original price was $ a) For example: b) For example: 13. For example: Problem: Janet is training for a race. Every week, she runs 3 km more than the previous week. During her 7 th and final week of training, Janet runs 24 km. How far did she run during the first week? Solution: Solve by working backwards. Start with 24 km and subtract 3 km for each of the 6 weeks before her final week of training. (+24) (3) (3) (3) (3) (3) (3) +6 Janet ran 6 km in the first week. 14. Referring to the discs in order from smallest to largest as A, B, C, D, and E, and the posts from left to right as 1, 2, and 3, one solution is: A to 3, B to 2, A to 2, C to 3, A to 1, B to 3, A to 3, D to 2, A to 2, B to 1, A to 1, C to 2, A to 3, B to 2, A to 2, E to 3, A to 1, B to 3, A to 3, C to 1, A to 2, B to 1, A to 1, D to 3, A to 3, B to 2, A to 1, C to 3, A to 1, B to 3, A to Chapter 6: Addition and Subtraction of Integers

13 Chapter Self-Test, p b) a) a) +4 b) 1 2. a) (+3) > () b) ( 4) > ( 7) c) ( 4) < 3. a) (+5) + ( 4) (+1) b) ( 3) + (+1) () c) () + ( 3) () d) () + (+7) (+2) e) (+2) + ( 35) ( 15) 4. ( ) + ( ) ( )+ (+1) Use the zero principle to combine the positive and negative counters. You are left with one positive counter Draw an arrow from to +3. Draw another arrow that is 2 units long to the left of +3. The second arrow ends at ( ) ( ) ) ( ) ( 7) Use the zero principle to add 4 zeros so that there are enough positive counters to perform the subtraction. 7. a) ( 3) (+2) () b) (+3) ( 1) (+4) c) (+3) () (+5) d) ( 1) () (+1) e) ( 4) () () 8. To input a negative number, input its opposite then press +/. a) / b) 117 +/ a) (+5) + ( 8) ( 3) b) ( 1) + ( 3) ( 13) c) ( 7) (+2) ( 9) d) ( 8) ( 4) ( 4) e) ( 4) + () + (+6) ( 3) f) (+1) + ( 15) + () ( 1) 1. (+3) + ( 193) + (5) + (+51) +? (1) (+351) + (18) +? (1) (+133) +? (1)? (1) (+133) ( 343) Friday s loss was $343. Chapter Review, p. 221 c) b) a) a) +4 b) +2 c) 2. a) ( 1) > ( 3) b) (+12) > ( 1) c) (+2) > () 3. Yes. +9 and 9 are 18 units apart and are one-digit integers For example: (+4) + ( 6) (); ( 5) + (+3) ( 2); ( 7) () () In the circle with ( 7) and (+4), the remaining two regions must add to (+3) to make the circle. This means (+1) and (+2) must be the numbers. If (+2) is in the centre region then it will not be possible to make the left circle sum to. Therefore, (+1) is in the central region and (+2) is in the remaining region in the top circle. Now, in the left circle, the remaining two regions must add to give () because (+4) and (+1) are in the uppermost regions. The only way this can happen is if () and are in the left circle. The remaining regions in the circle on the right must sum to give ( 3) and this is only possible if ( 3) and are in the two remaining regions. Since must be in both the left and right circles, it must be in the region shared by them. This leaves () in the region belonging only to the circle on the left and ( 3) in the region belonging only to the circle on the right 6. regrouping: (+4) + (+5) + () + ( 1) + (+5) + (+1) (+4) + () + (+1) + ( +5) + ( 1) + (+5) (+4) + () + (+1) + (+3) The stock was up $3 from its value on January using a calculator: a) ( 128) + ( 65) + ( 38) (31) b) ( 373) + (8) + (+12) (69) 8. a) (+7) ( 4) + (+5) (+16) b) ( 1) + ( 3) (+4) ( 17) c) (+5) ( 8) (+13) d) (+4) + () (+38) 9. a) + 1 is greater than 1. If is a positive integer then + 1 is greater than 1 because the result is more positive. If is a negative integer then + 1 is still greater than 1 because the result is less negative. Nelson Mathematics 7 Solutions 6-13

14 1. a) (+4) (+2) (+2) b) ( 7) (+4) ( 11) (least) c) (+6) ( 3) (+9) (greatest) d) ( 3) () (+2) 11. For example, (+4) () ( 18) + ( 11) + ( 3) (+6) ( 18) + ( 11) + ( 3) (+24) + ( 11) + ( 3) (+13) + ( 3) (+1) 12. For example: ( 1) + ( 1) + (); ( 4) + (+4) + ( 4); ( 1) + (+5) + (+1) 13. For example, subtracting an integer is like adding the integer with the opposite sign. When subtracting, you find the space between two integers. When adding, you start at one integer and then move a distance that corresponds to the other integer 14. a) ( ) + ( ) (+5) b) ( ) ( ) () For example, since the two numbers have a difference of 5, I looked at pairs of numbers that have a difference of 5 and tried to find the pair that sum to ( 13). Since the sum I was looking for is negative, I started with ( 1) and ( 6), then moved in the negative direction. I found that ( 4) and ( 9) are the integers since: ( 9) + (+5) ( 4) and ( 9) + ( 4) ( 13) 16. a) I am adding two negative numbers. I add the whole number parts. The sign of the result is negative, 413. b) I am adding a positive number and a negative number, so I subtract the smaller number from the larger number. The negative number is larger so the sign of the result is negative, 666. c) The arrow points left so the sign of the result is negative. Both numbers have the same sign so subtract the whole number parts c) ( ) + ( ) ( ) ( 1) +6 7 d) The arrow points left so the sign of the result is negative. The numbers have opposite signs, so add the whole number parts d) ( ) ( ) ( ) ( ) () Chapter 6: Addition and Subtraction of Integers