Patterns & Algebra 2 AP Book 4.2

Patterns & Algebra 2 AP Book 4.2 page 1 AP Book PA4-15 page 175 1. 8 km (6 km + 6 km = 12 km; 20 km 12 km = 8 km) 2. 5 km (15 km 3 hours = 45 km; 50 km 45 km = 5 km) Teacher to check number lines for Questions 3 & 4. 3. 4 blocks (After 3 minutes she ll have travelled 4 blocks 3 min = 12 blocks, so she will be 16 12 = 4 blocks from home.) 4. 6 minutes (12 blocks 2 blocks per minute = 6 minutes) AP Book PA4-16 page 176 Teacher to check number lines for all questions. 1. 25 km (After three days James will have travelled 75 + 75 + 75 km = 225 km, so he will be 250 225 = 25 km from the finish) 2. 10 minutes (250 words 25 words per minute = 10 minutes) 3. 200 metres (100 + 50 + 50 = 200 m from start) 4. 30 metres (10 + 5 + 5 + 5 + 5 = 30 metres) 5. The 6th and 12th steps have both red and blue paint on them (since 6 and 12 are multiples of both 2 and 3). AP Book PA4-17 page 177 1. a) RYYRYYRYY b) YRYRYRYRY c) RYYRRYYRRYYR f) YYRYYYRYYYRYY 2. a) Yes b) Yes c) No d) No e) Yes f) Yes 3. a) Yes b) Yes c) No d) No e) Yes f) No 4. a) YRR b) RRRY c) YYRR d) YRRY e) RYRYYY f) RY 5. Yellow 6. Yellow 7. Red 8. YRY 35 th block = red 9. RWBB 41 st block = red 10. Down since the core is UD, all evennumbered triangles (like the 22 nd ) will point down. 11. 5 coin 12. a) 1 coin since the core is three coins long (5, 1, 1 ) and 15 is divisible by three, the 15 th coin will be the final coin in the core. b) Value of core: 7 so the value of first 15 coins = 7 5 sets of 3 coins = 35 AP Book PA4-18 page 180 d) +,, + e), +, f) +, +, + g) +,, + h) +, +, i),, + j) +, +, + k) +,, + l) +, +, + 2. a) +,, + C,, B b),, B,, + C c),, + C,, B d),, B +,, + C AP Book PA4-19 page 181 1. b) + 3, 1, + 4, 3 c) + 3, + 1, + 4, + 2 d) 2, + 4, 4, + 7 e) + 6, + 1, 5, + 8 f) 3, 4, + 2, 1 g) 5, + 2, + 5, 3 h) + 3, 7, + 7, + 7 2. a) A: 9, 12, 15, 18, 21 B: 7, 10, 13, 14, 19 b) B: 6, 10, 14, 17, 21 A: 5, 9, 13, 17, 21 c) A: 35, 30, 25, 20, 15 B: 30, 25, 20, 15, 5 d) A: 10, 9, 8, 6, 5 B: 11, 10, 9, 8, 7 e) C: 17, 22, 28, 32, 34 B: 17, 14, 10, 9, 6 A: 14, 19, 24, 29, 34 f) C: 21, 19, 15, 13, 9 A: 10, 13, 9, 7, 5 D: 19, 17, 15, 13, 11 B: 9, 12, 15, 18, 21 3. b) Start at 3, add 2 c) Start at 19, subtract 3 4. a) Start at 8, add 3 b) Start at 14, subtract 4 c) No rule d) Start at 61, add 4 5. a) Increasing b) Repeating c) Decreasing d) Increasing e) Repeating f) Decreasing 6. a) 6, 9, 12, 15, 18 b) 26, 22, 18, 14, 10 c) 39, 44, 49, 54, 59 7. Answers will vary. 8. Answers will vary. 9. Answers will vary. AP Book PA4-20 page 183 1. a) b) c) d) d) RYRRYRRYRRYR 1. b) +, +, e) YYRYYRYYRYYRY c),, +

page 2 2. a) b) c) 6. Each column decreases by 3. Each row increases by 3. One diagonal repeats; the other diagonal decreases by 6. 7. x 1 2 3 4 5 6 1 1 2 3 4 5 6 2 2 4 6 3 3 6 9 12 15 18 4 4 8 12 16 20 24 5 5 10 15 20 25 30 2. Answers may vary Sample: A A B B A A B B B B A A B B A A 3. a) L R T T L R T T L R T T L R T T L R T T L R T T L h) 5 i) 15 j) 5 k) 5 l) 5 m) 70 n) 8 o) 6 p) 13 5 8 2 3 5 q) 17 d) 3. a) 2 4 6 b) 2 4 6 c) 2 4 6 d) 2 4 6 4. a) Start at 8; add 2 b) Start at 2; add 6 c) Start at 6; add 6 d) 1st diagonal (given): Start at 2; add 8 2nd diagonal: Start at 6; add 4 5. Both rows and columns are increasing by 2. One diagonal increases by 4; the other repeats. 6 6 12 18 24 30 36 a) Corresponding rows and columns are equal. Each row & column increases by its row / column number (for example, the 5 th column increases by 5). The main diagonal increases by 3, 5, 7, 9, 11. Each diagonal above and below the main increases by patterns of increasing numbers (for example: the diagonal 1, 2, 6, 12, 20, 30 increases by 4, 6, 8, 10). NOTE: Students may observe other patterns teacher to check. b) Answers will vary AP Book PA4-21 page 185 1. Answers may vary slightly. Sample: A B C B C A C A B R T T L R T T L R T b) Row 6 4. a) There are two whole numbers in between each shaded square. The shaded squares form diagonal lines, right to left. b) 5, 10, 15, 20, 25,, 100 c) 15, 30, 45, 60, 75, 90 The ones digits of these numbers alternate between 5 and 0. The tens digits increase by 2, then by 1, then by 2, 1 etc. in a repeating pattern. 5. < 11 > 11 by 3 3, 6, 9 12, 15, 18 Not by 3 1, 2, 4, 5, 7, 8, 10 11, 13, 14, 16, 17, 19, 20 6. The top number is the sum of the two numbers in the boxes below. 7. a) 6 b) 8 c) 11 d) 11 e) 1 f) 3 g) 4 10 7 6 4 3 r) 15 7 8 2 5 3 s) 32 14 18 11 3 15 t) 45 26 19 17 9 10 u) 123 49 74 32 17 57 AP Book PA4-22 page 187 December 1. a) S M T W T F S 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 b) The shaded boxes show the days when Rona has guitar lessons. 2. If April 1 st is a Sunday, there are 4 Tuesdays in the month: Huyan will have $20. 3. Alex and Dan both have piano lessons on the 18 th of October (a Friday).

page 3 4. Teacher to check that calendars are completed properly. a) Start at x, add 7. b) Again, the pattern will be an increase of 7. This occurs because there are 7 days in a week. AP Book PA4-23 page 188 2, 4, 6, 8, 10, 12,, 100 2. The 2 nd, 4 th, 6 th, 8 th and 10 th columns are shaded. Each row has 5 multiples of 2 evenly spaced apart. 3. All the ones digits for the multiples of two will be one of 0, 2, 4, 6, or 8, and they will cycle through in order. 4. If the ones digit of a number is 2, 4, 6, 8 or 0 then the number is a multiple of two. 5. Circle: 18, 32, 76, 30, 94, 82 6. Circle: 5, 75, 37, 83 7. When you add two even numbers, the sum will always be an even number. AP Book PA4-24 page 189 5, 10, 15, 20, 25,, 100 2. In every row there are two shaded numbers. There are two shaded columns in the whole chart the 5 th and the 10 th. 3. The ones digits of the multiples of 5 will always be 5 or 0, and will alternate between these two numbers. 4. For any number, if the ones digit is a 5 or a 0, the number is a multiple of 5. 5. Circle: 45, 60, 90, 85, 25, 50 6. No the multiples end in 0 are even, but multiples that end at 5 are odd. 7. Circle: 205, 225, 385, 755 AP Book PA4-25 page 190 8, 16, 24, 32, 40,, 96 2. 0 8 4 8 1 6 5 6 2 4 6 4 3 2 7 2 4 0 8 0 The ones digits start at 8 and decrease by 2 down each column: 8, 6, 4, 2, 0. 3. The tens digits increase by one down each column: 0, 1, 2, 3, 4. 4. 88 128 96 136 104 144 112 152 120 160 AP Book PA4-26 page 191 1. 49 17 37 6 36 86 8 24 96 61 10 50 40 25 15 65 b) Answers will vary, though will be two of 60, 70, 80 or 90. c) Answers will vary 5 79 2. 57 62 5 25 60 40 35 80 75 30 AP Book PA4-27 page 192 32 24 72 48 16 56 8 17 1. a) Gaps: + 2, + 3, + 4, + 5, + 6 2, 4, 7, 11, 16, 22 b) Gaps: + 1, + 2, + 3, + 4, + 5, + 6 3, 4, 6, 9, 13, 18, 24 c) Gaps: + 3, + 5, + 7, + 9, + 11 11, 14, 19, 26, 35, 46 d) Gaps: + 2, + 4, + 6, + 8, + 10, + 12 6, 8, 12, 18, 26, 36, 48 e) Gaps: 1, 2, 3, 4, 5 17, 16, 14, 11, 7, 2 f) Gaps: 2, 4, 6, 8, 10 32, 30, 26, 20, 12, 2 g) Gaps: 1, 3, 5, 7, 9 31, 30, 27, 22, 15, 6 2. Fig 3. Fig h) Gaps: 5, 10, 15, 20, 25, 30 110, 105, 95, 80, 60, 35, 5 of Squares 1 1 Added 2 4 3 3 9 5 4 16 7 5 25 9 6 36 11 of Squares 1 1 Added 2 5 4 3 11 6 4 19 8 5 29 10 AP Book PA4-28 page 193 1. a) 2 b) 32, 64, 128, 256 c) In each case, the gap is equal to the most recent number in the pattern. 2. a) Start at $15 and add $5 each week. b) Start at $1 and multiply by 2 each week. c) Answers will vary. d) W O K 1 $1 $15 2 $2 $20 3 $4 $25 4 $8 $30 5 $16 $35 6 $32 $40 7 $64 $45

page 4 Olivia will save more money. (Teacher to check if this matches student predictions.) 3. a) The gap alternates between + 3 and 2. b) 6, 9, 7 4. Figure of Dots Gap 1 3-2 6 3 3 10 4 4 15 5 5 21 6 6 28 7 5. Monday: 10 min Tuesday: 12 min Wednesday: 14 min Thursday: 16 min And 16 + 14 + 12 + 10 = 52, so Jane trained for a total of 52 minutes in the first four days. AP Book PA4-29 page 194 1. a) Min Sec 1 60 2 120 3 180 4 240 5 300 b) Years Weeks 1 52 2 104 3 156 4 208 c) Years Days 1 365 2 730 3 1095 4 1460 2. There are 48 months in 4 years: 12 4 = 48. 3. The rabbit s heart beats 1000 times in 5 minutes. 4. 1500 2 = 750, so a goose can fly 750 km in one day. It can fly 4500 km (750 6) in 6 days. 5. 18 + 16 + 16 + 16 + 16 = 82, so Miguel earns $82 after 5 hours of work. 6. 1986 + 76 = 2062 2026 + 76 = 2138 2102 + 76 = 2214 So, Halley s Comet will return on earth in 2062, 2138 and 2214. 7. a) 999 2 = 1 998 999 3 = 2 997 999 4 = 3 996 999 5 = 4 995 999 6 = 5 994 b) 6 x 9 = 54 6 x 99 = 594 6 x 999 = 5 994 6 x 9 999 = 59 994 6 x 99 999 = 599 994 8. Answers will vary AP Book PA4-30 page 195 For Questions 1 & 2, teacher to check that apples are drawn properly. 1. a) 2 b) 1 c) 2 d) 6 2. b) 7 = 3 + 4 c) 3 + 1 = 4 d) 4 + 4 = 8 3. b) 10 = 4 + c) 9 = 4 + d) 5 = 3 + e) 7 = 2 + f) 10 = 4 + g) 12 = 5 + h) 5 = 3 + AP Book PA4-31 page 196 For Questions 1 & 2, teacher to check that apples are drawn properly. 1. a) 6 b) 4 c) 3 d) 6 2. a) 7 3 = 4 b) 7 2 = 5 3. b) 2 c) 3 d) 2 e) 2 f) 4 4. a) 3 + 2 = 5 b) 3 + 5 = 8 c) 15 3 = 5 5. Answers will vary AP Book PA4-32 page 197 1. a) 5 b) 6 c) 8 d) 4 e) 3 f) 9 g) 2 h) 6 i) 5 j) 3 k) 4 l) 4 m) 3 n) 5 2. a) 1 + 1 + 5 = 7; or 2 + 2 + 3 = 7; or 3 + 3 + 1 = 7 b) 1 + 1 + 6; or 2 + 2 + 4; or 3 + 3 + 2 3. 1 + 1 + 3 = 5 and 2 + 2 + 1 = 5 4. a) 11, 12, 13, 14 b) 9, 8, 7, 6 c) 10, 20, 30, 40 5. a) Increases by 1 b) Decreases by 1 c) Increases by 10 6. Inner ring = 4 points Outer ring = 2 points (2 + 2 + 4 = 8) AP Book PA4-33 page 198 1. 13 triangles will be needed for the 6 th figure. 2. a) She will need 16 triangles. b) Answers will vary 3. At the 5 th wall, Hank will be 400 metres from the start: 200 + 50 + 50 + 50 + 50 = 400. 4. a) b) 10, K, Q c) 010, 100, 001 d) 001, 011, 111 e) 020, 030, 010 f) AE, AF, AG, AH g) M O M h) T 2222 T 22222 T 5. The core of the pattern consists of 5 shapes. We know that 5 4 = 20, so a complete pattern ends on the 20 th shape. A new pattern starts on the 21 st shape, so to find the 23 rd shape in the pattern look at the third element in the core: a square. 6. a) 24 b) 32 c) 12 7. a) 3 727, 3 827, 3 927 b) 7 237, 8 238, 9 239 c) 4 567, 5 678, 6 789 8. a) The 12 th step will have both.

page 5 b) His left foot will land on the 6 th and 12 th steps. 9. The 6 th and 12 th people receive both a free pen and book 10. Emma will need 21 blocks to build a stairway 6 steps high: 1 + 2 + 3 + 4 + 5 + 6 = 21.