Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
A Natural Way To 48 27 49 29 52 30 14 55 Summerschool 2015 17 8 18 41 21 20 19 60 40 38 65 64 39 63 62 37 36 59 42 22 7 35 58 23 1 Algebra 43 9 6 34 57 10 0 16 44 24 2 5 33 56 25 3 15 45 11 4 32 54 26 13 31 53 46 12 28 50 51 47 61 Martin Kindt Freudenthal Institute Algebra the art of ‘thingification’ And now is a good moment to open up Pandora’s box and explain one of the most powerful general weapons in the mathematician’s armory, which we might call the ‘thingification of processes’. Ian Stewart in Nature’s Numbers Algebra at school RESTRICTIONS equations inequalities linear programming PROCESSES (CHANGE) operations functions graphs PATTERNS & FIGURES sequences figured numbers ICME Berkeley 1980 1 1 2 3 5 8 13 Guess the rule ! ? ? ? ? 2 3 5 8 13 2 3 5 8 89 13 21 5 34 5 1 1 1 1 f0 = 1 f10 = 89 f20 = 10946 f1 = 1 f11 = 144 f21 = 17711 f2 = 2 f12 = 233 f22 = 28657 f3 = 3 f13 = 377 f23 = 46368 f4 = 5 f14 = 610 f24 = 75025 f5 = 8 f15 = 987 f25 = 121393 f6 = 13 f16 = 1597 f26 = 196418 f7 = 21 f17 = 2584 f27 = 317811 f8 = 34 f18 = 4181 f28 = 514229 f9 = 55 f19 = 6765 f29 = 832040 Another pattern ? 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... Even or Odd? Explain your strategy! A E C D B F From ‘PATTERNS and SYMBOLS’ (MIC) 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... ODD + ODD = EVEN ODD + EVEN = ODD choose five consecutive Fibonacci-numbers .... 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... + 24 = 3 8 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... + 39 = 3 13 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... + 699 = 3 233 representation of any 5 consecutive Fibonacci-numbers first in the style of second third fourth fifth Euclid PROOF! + a a b b + a+b b+a+b a+b+b+a+b a b b a a b b b a a + (a + b + b + a + b) = (a + b) + (a + b) + (a + b) b + a a + 2a + 3b b = a+b 3a + 3b = 3 (a + b) a + 2b 2a + 3b Progressive symbolization more ‘Fibonacci-exercises’ * Take any subsequence of nine consecutive numbers. The sum of the first and the ninth number equals .... 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ... * Compare the sum of any six consecutive numbers with the 5th in the row .... Which relation do you observe? Prove that it’s valid for all groups of six. * Design your own Fibonacci-exercise PATTERNS & FIGURES Ian Stewart A natural number is an idea that has long ago been thingified so thoroughly that everybody thinks of it as a thing. figured numbers Which pattern has the biggest number of dots? Same question .... Productive exercises * Design a dice-pattern with 625 dots * Design another dice- pattern with a number of dots between 100 and 1000 Nikomachos of Gerasa (ca. 100 AD) sum of ‘odds’ = square number sum of ‘evens’ = oblong number Introduction to Arithmetic sum of consecutive numbers = triangular number square number n2 oblong number oblong number n (n + 1) or n2 + n triangular number 1 --- n 2 (n + 1) Air-show a squadron flied in a ‘W-formation’ W-numbers 0 1 2 3 pattern-number 0 1 2 3 4 5 6 number of dots 1 5 ... ... ... ... ... Formula? 0 1 2 W=4n+1 3 0 1 2 W = [2n] + [2n + 1] = 4n + 1 3 W = double V 1 = + W = (1 + 2n) + (1+ 2n) 1 = 1 + 4n - W-squadrons sometimes can transform in square(dron)s * Which ones? Why? Sequence of W-numbers 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 0 2 6 12 Sequence of W-numbers 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 Is every odd square a W-number ? 2n + 1 2n + 1 2n + 1 4n2 + 2n + 4n2 + 4n + 1 = 4(n2 + n) + 1 oblong number 4 oblong number + 1 = square number 4 n n + 1 + 1 = 2n + 1 2 number-strip 0 1 2 3 4 5 6 7 8 9 Which colour has 2015 ? 10 0 1 2 3 4 5 6 7 8 9 10 Add a white number to a blue one, what do you get? Does the colour of the answer depend on the chosen white and blue number? + 2 1 3 6 9 12 4 6 9 12 7 9 12 10 5 8 11 14 17 20 23 15 18 21 24 15 18 21 24 27 15 18 21 24 27 30 12 15 18 21 24 27 29 31 13 15 18 21 24 27 30 33 36 16 18 21 24 27 30 33 36 39 19 21 24 27 30 33 36 39 42 22 24 27 30 33 36 39 42 45 using dot patterns White + Blue = Red + = = + R W B R R W B W W B R B B R W 0 1 2 3 4 5 6 7 8 9 10 What is the colour-pattern of : 0, 10, 20, 30, 40, ... ? And what of : 0, 100, 200, 300, 400, ...? And of : 0, 1000, 2000, 3000, 4000, ... ? What is the colour of 2015 ? 2000 + 10 + 5 has same colour as 2 + 1+ 5 =8 The colour of a number = the colour of the sum of its digits Formulas 0 1 2 3 4 5 R = 3n 6 7 8 W = 1 + 3n 9 10 11 B = 2 + 3n 12 13 14 15 16 17 3n 1 + 3n 2 + 3n n = 0,1,2,3, ... start number 2 +5 7 +5 arithmetic sequence 12 +5 17 +5 22 +5 27 2 + 5n expression n = number of steps equal steps Adding number-strips (‘thingification’) 4 1 9 4 14 7 19 10 24 + 13 29 16 34 19 4 + 5n + 1 + 3n = = 1 4 +5 4 9 14 19 24 +5 7 5 +8 +3 12 +3 19 10 + 13 = 26 33 29 16 40 34 19 47 4 + 5n + 1 + 3n = 5 + 8n +8 2 5 8 5 11 14 = 17 20 5 = 2 10 +3 5 25 +15 +3 5 5 8 40 11 55 14 = 70 17 85 20 100 2 + 3n +15 = 10 + 15n 0 0 0 1 1 2 1 7 4 6 3 19 9 12 6 37 16 20 10 61 0 0 0 1 1 2 1 7 4 6 3 19 9 12 6 37 16 20 10 61 25 30 15 91 36 42 21 127 n2 n2 + n nn + 1 ---------------------2 ??? 0 0 0 1 1 2 1 7 4 6 3 19 9 12 6 37 16 20 10 61 25 30 15 91 36 42 21 n2 n2 + n nn + 1 ---------------------2 ? 127 ??? 1, 7, 19, 37, 61, ... +6 1 +12 +18 +24 +30 7 19 37 Hexagonal numbers 1, 7, 19, 37, 61, ... +6 +12 +18 +24 +30 ‘honey-comb-numbers’ 7 19 37 32+1=7 3 6 + 1 = 19 3 12 + 1 = 37 7 19 37 32+1=7 3 6 + 1 = 19 3n(n + 1) + 1 3 12 + 1 = 37 7 19 37 61+1=7 6 3 + 1 = 19 6 1--- n n + 1 + 1 = 3n n + 1 + 1 2 6 6 + 1 = 37 7 19 37 1 + 6 + 6 2 + 6 3 + ... + 6 n = 1 + 6 (1 + 2 + 3 + ... + n) = 1 + 6 1--- n n + 1 = 3n n + 1 + 1 2 48 27 49 29 52 30 14 54 26 13 55 34 57 18 35 58 8 7 6 41 21 20 19 60 40 38 63 62 61 65 64 39 37 36 59 42 22 9 1 17 43 23 10 0 16 44 24 2 5 33 56 11 4 32 45 25 3 15 31 53 46 12 28 50 51 47 Three propositions about algebra education 1 During the first two years of secondary education, teachers should pay a great deal of attention to the arithmetic side of algebra, i.e. algebra in relation to (mental) arithmetic, number patterns, number theory and combinatorial counting problems. 2 It is advisable to use historical contexts. Babylonian, Egyptian, Greek, Indo and Arabic mathematics have a great deal to offer in the area of concrete (and discrete) algebra 3 Once students attain reasonable mastery over enough tools or techniques, algebra can and should - be used to make proofs , for example of special properties of natural numbers. Homework (1) * Make two numberstrips, one with the oblong numbers and the other with the hexagonal numbers. Adding these strips pairwise, you will find a strip of ... square numbers! How to prove this? Homework (2) * Check and prove the three given formulas to the diagonals. Give the three missing formulas. 48 3(n + 1)2 27 49 29 52 30 14 54 34 23 8 7 6 18 35 42 22 9 1 17 43 24 2 5 44 10 0 16 33 25 3 15 45 11 4 32 55 26 13 31 53 46 12 28 50 51 47 21 20 19 36 41 3n2 + 4n + 1 39 38 37 40 3n2 + 3n Welcome to Holland 3 0 4 1 5 2 9 12 15 18 7 10 13 16 19 8 11 14 17 20 6