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curiosity & conviction curiosity & conviction hmmm mmmm aha ha thanks to Rob Eastaway play around; explore a pattern begins to emerge and a relationship seems to hold reasons are sought and may be found ×2 +3 +3 ×2 7 ×2 +3 +3 10 14 17 ×2 20 ? ×2 +3 +3 ×2 37 ×2 +3 +3 40 74 77 ×2 80 you are testing out the generalisation (rule) that the difference is always try out two easy starting numbers, two harder numbers and two mega - hard starting numbers 3½ ×2 +3 +3 ×2 . 2 1 ×2 +3 +3 47.52 ×2 10.2 × 2 – 10 –5 +3 +3 –7 –2 ×2 –4 ×2 +3 +3 65 × 2 68 m ×2 +3 +3 ×2 d ×2 +3 +3 ×2 Beth’s number b ×2 +3 +3 2b + 3 ×2 2b + 6 2(b + 3) that’s a proof found a rule (generalisation) checked it out with a range of numbers tested it widely; thrashed it in fact done reversing (inverses) to get back to the starting number built algebraic expressions with ‘m’ for a million proved that the rule always works using someone’s (unknown) number here’s another proof any old number 2m + 3 ×2 +3 +3 ×2 5p – 1 ×2 +3 +3 ×2 ×2 + 10 + 10 ×2 ×2 +b +b ×2 ×5 +3 +3 ×5 ×t +a +a tn + a ×t tn + ta ta – a a(t – 1) 8n + 1 numbers can’t ever be the square of an even number for what integer values of n is 8n + 1 a square number? 0 1 3 8n + 1 = 25 8n + 1 = 36 and a relationship seems to hold 6 10 getting all the squares of the odds for what integer values of n is 8n + 1 a square number? 3 for what integer values of n is 8n + 1 a square number? 3 for what integer values of n is 8n + 1 a square number? 3 n = 10 n=3 n=6 n = 10 numbers 2 7 – 2 5 24 2 3 – 2 1 8 2 10 – 2 8 36 2 11 – 40 2 9 2 6 – 2 4 20 notice? algebra (n + 2 2) – 2 n (n + 2)(n + 2) – 2 n 2 – n (n + 2)(n + 2) 2 n 2 – n + 2n + 2n + 4 2 n 4n +4 – 4( 4n + 1) 4 + 2 n proof diagram diagram diagram diagram proof utilising this diagram: use this multi-purpose diagram to try to prove that: 1 x ≥ 0, x + x ≥ 2 1. for 2. where 8n + 1 is a square number, n is a triangular number 3. the arithmetic mean of two numbers, ½ (a + b) greater than the geometric mean, √ab 2 a–b 2 4. a+b 5. a2 – b2 = (a – b)(a + b) 2 – 2 = a.b Babylonian multiplication is for x ≥ 0, 1 x+ x ≥ 2 1 x x x+ x+ 1 x 1 x 2 ≥ 4 ≥ 2 √b √a a.m. > g.m. (√a + √b )2 > 4√ ab Babylonians a b b a–b they used this to multiply and to solve a quadratic equation (a + b)2 – (a – b)2 = 4ab 8 × ½ (1 × 2) + 1 9 8 × ½ (2 × 3) + 1 25 8 × ½ (3 × 4) + 1 49 8 × ½ (4 × 5) + 1 81 8 × ½ (5 × 6) + 1 121 8 × ½ (6 × 7) + 1 169 ….. n+1 n a – b = 2w a b area and perimeter of rectangles the area is 18 the perimeter is 22 the area is 24 the perimeter is 28 the area has the same value as the perimeter the area has double the value of the perimeter perimeter area = 24 cm² = 20 cm = 22 cm = 28 cm = 50 cm = 97 cm = 35 cm = 21.4 cm area and perimeter this shape can be described as (20 , 18) can you see why? (1) try to find some rectangles (or squares) that fit the rule: P–A=4 (2) try to find some rectangles (or squares) that fit the rule: P – 2A = 2 (3) try to find some rectangles (or squares) that fit the rule: 3P – 2A = 18 (A, P) area and perimeter P – 2A = 2 2PP––2A A = 48 3P – 2A = 18 next? P–A= 4 and in general? 2P – A = 16 what size (dimensions) rectangle fits the rules 3P – 2A = 18 and P– A= 4 ? 2 wide rectangles ‘A’ squares A – 4 single units plus 4 lots of 2 P=A–4+8 P–A= 4 2 wide rectangles, by induction start with 4 squares P=A+4 every time you add in 2 squares you add 2 onto the perimeter so P still = A + 4 P–A= 4 3 wide rectangles P=⅔A+ 6 4 × 2 + A – 4 – (⅓A – 2) ⅔A + 8 – 4 + 2 P = ⅔A + 6 4 wide rectangles P=½A+ 8 4 × 2 + A – 4 – 2(¼A – 2) ½A+8–4+4 P=½A+8 P A area and perimeter (i) this shape can be described as (20 , 18) can you see why? (a) try to sketch or write down the dimensions (length and width) of the following rectangles (or squares): (1) (1 , 4) (9) (9 , 12) (2) (2 , 6) (10) (6 , 14) (3) (3 , 8) (11) (10 , 14) (b) which rectangles fit the rule: P–A=4 ? (c) which rectangles fit the rule: P – 2A = 2 ? (4) (4 , 8) (12) (12 , 14) (d) (5) (4 , 10) (13) (7 , 16) (6) (6 , 10) (14) (12 , 16) (7) (5 , 12) (15) (15 , 16) (8) (8 , 12) (16) (16 , 16) which rectangles fit the rule: 2P – A = 16 ? (e) which rectangles fit the rule: 3P – 2A = 18 ? in general? (A, P) area and perimeter (ii) (1) this shape can be described as (20 , 18) can you see that P = A – 2 ? or that A – P = 2 ? there is only one more rectangle (with whole number sides) that fits this rule can you find it? (2) what size (dimensions) would the rectangle, (24 , 20) be? this fits the rule A – P = 4 try to find the other rectangle (with integer sides) that fits this rule (3) what size (dimensions) rectangles fit the rule 16A = P2 ? (4) what size (dimensions) rectangles fit the rule P = 4A + 1 ? (5) what size (dimensions) rectangle fits the rules 3P – A = 36 and 2P – A = 16 ? (6) what size (dimensions) rectangle fits the rules 3P – 2A = 18 and P – A = 4 ? angle bisector angle bisector angle bisectors and the incircle d in terms of a , b and c? b d in terms of b? d a c 90 + ½a 90 + ½b 90 + ½c b + 270 + ½(a + b + c) 90 + ½c 90 + ½a 90 + ½b a c angle bisectors what is the relationship between the (angles) dots? two interior bisectors and an exterior bisector intersect the lines (one extended) of the triangle what is the relationship between the red dots? any quadrilateral does this always happen? when you bisect all of the angles show that the resulting quadrilateral must be cyclic place all of the digits 1 to 9 in the cells so that, the rows the columns and the two diagonals all sum (add) to the same number 1 2 3 4 5 6 7 8 9 the total of the digits 1 to 9 is 45 the total of the three lines is 45 so the total of each line must be 45 3 = 15 each of the three lines has the same total 4 lines passing through the central cell the total of these four lines is 4 × 15 = 60 but there is only a total of 45 available the central cell is counted 4 times (3 too many) so 3 × central cell = 60 – 45 central cell = 5 9 5 1 the 9 must go somewhere – try it in a corner cell need the 1, to make 15 6 9 5 1 consider the 6 it can’t go in the same line as 9 6 9 8 7 5 1 consider the 6, 7 and 8 neither of them can go in the same line as 9 6 9 8 7 5 1 two places for three numbers eh? 9 5 1 the 9 cannot go in a corner the 9 must go in one of the middle cells 9 5 1 need to sum to 6 only 4 and 2 4 9 2 3 5 7 8 1 6 there is essentially only one magic square using the 1 to 9 the Lo Shu magic square a magical turtle emerged from the water with the curious and decidedly unnatural (for a turtle shell) Lo Shu pattern on its shell early records are ambiguous, referring to a "river map", and date to 650 BCE but clearly refer to a magic square by 80 CE, and explicitly give one since 570 CE