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Maths Magic CTYI Review Session David Goulding

Maths Magic CTYI Review Session David Goulding. These slides were presented to the parents during the final class of the course Main aim is to give an overview of the different topics covered during the eight weeks

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Maths Magic CTYI Review Session David Goulding

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  1. Maths Magic CTYI Review Session David Goulding David Goulding

  2. These slides were presented to the parents during the final class of the course • Main aim is to give an overview of the different topics covered during the eight weeks • CTYI is geared towards students demonstrating high academic achievement over a range of ages • Maths Magic was run for students ranging in ages between 8 and 12 • While classes were planned in advance, it was sometimes necessary to change course during the class in order to keep interest levels high David Goulding

  3. Pascal’s Triangle Natural Numbers 1 1 1 1 21 1 33 1 1 4 6 4 1 1 510 10 5 1 1 615 20 15 6 1 1 7 21 35 35 21 7 1 1 828 56 70 56 28 8 1 Triangular Numbers David Goulding

  4. Odds of Getting an Exact Number of Heads or Tails 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 HT HH HT THTT HHHHHTHTHTHHTTHTHTHTTTTT Odds of getting a head or tail is 1/2 Rosencrantz - Heads??? David Goulding

  5. Rolling dice Well there are six numbers on a die, 1-6, if the die is fair (that is not biased towards any one number), then the chances of getting any one of the numbers is 1/6 What about if I roll a pair of dice and look at the sum?? 36 12 18 36 David Goulding

  6. Odds of Winning Dave’s Lottery Dave’s lotto is a simple game, there are six numbers (1-6) you must choose four of them: what are the chances of winning? Let’s look at the combinations (1,2,3,4) (2,3,4,5) (1,2,3,5) (2,3,4,6) (1,2,3,6) (2,3,5,6) (1,2,4,5) (2,4,5,6) (1,2,4,6) (3,4,5,6) (1,2,5,6) (1,4,5,6) (1,3,4,5) (1,3,4,6) (1,3,5,6)

  7. There are 15 ways to choose 4 numbers from 6, we can see this from Pascal’s triangle 1 6 15 20 15 6 1 The general formula for calculating combinations is in fact given by Where n!=nx(n-1)x(n-2)x…x3x2x1 Let’s look at example above, want to choose four numbers from 6, so that’s David Goulding

  8. Odds of Winning The Irish Lotto In the Irish Lotto there are 45 numbers and we must choose the correct six numbers, using the same formula as before we have This works out to be 8,145,060 different combinations (again trust me on this one!!) Conclusion: Don’t play the Lotto David Goulding

  9. Functions and Graphs Let’s suppose we have some sort of magic box, which we’ll call the squaring box, we put a number into the box and it give back the square of the number, for example 24 We then generalised this to the following function: David Goulding

  10. Functions and Graphs This lead us to be able to draw our first graph with the students, so we drew a graph of “t-squared” – we looked at this then in relation to graphs of running times for the students David Goulding

  11. My Perfect Birthday A perfect number is a number whose divisors (excluding itself) add up to the number! Let’s look at my birthday : June 28th Divisors of 6 are 1,2,3,6. Now 1+2+3=6 So 6 is perfect!! Divisors of 28 are 1,2,4,7,14,28. Now 1+2+4+7+14=28 So 28 is perfect!! The next two perfect numbers are 496 and 8128 – you can trust me on those ones!! (unsolved) Are there any odd perfect numbers? David Goulding

  12. Prime Numbers A prime number is a number divisible only by itself and one. (We do not include one as a prime) Let’s look at primes, well 2,3,5,7,11,13,17,19 are all primes…..is there a way of generating the primes, well yes we can use the famous Sieve of Eratosthenes

  13. Now the Sieve is all well and good but it is extremely time consuming to carry out….is there another way Well there are tests for certain types of primes but there is no magic way of writing done all the prime numbers!! How many primes are there? The answer is that there is an infinite number of primes, this was proven in the class as follows. Suppose we had only a finite number of primes, say 2,3,5,7 and 11. Now we know that any number is either prime or composite (that is it is divisible by some number-in fact divisible by primes). What about the number (2x3x5x7x11) + 1 = 2311 well this is not divisible by any of 2,3,5,7,11 therefore it is either prime or we have not included a prime in our list!! (Turns out this is prime as well) David Goulding

  14. Seven Bridges of Konigsberg First introduction to Graph Theory! David Goulding

  15. One possible solution that the class came up with: David Goulding

  16. 4 Why does this work? Well Euler said that if we have either 0 or 2 nodes of an odd degree then the problem will have a solution – Try find another possible solution, simply by adding additional paths and then counting the number of lines entering each node 5 3 So the Bridges of Konigsberg turned out to be the forerunner of modern day graph theory 4 David Goulding

  17. Monty Hall Problem The Monty Hall Problem is based on the game show “Let’s Make a Deal” which was a very popular tv show in the States in the 1960s and 1970s. In the problem, there are three doors 1, 2 and 3, behind one of the doors is a car but behind the other two there are goats!! Haven chosen a door, Monty Hall would then open a door for the contestant to reveal a goat. Monty Hall would then offer the chance to swap doors. The problem was then whether it was more beneficial to stick or swap My favourite demonstration is by Professor Marcus du Satoy David Goulding

  18. Switching Loses Switching Wins Switching Wins So 2 out of 3 times you win by switching, so probability of winning by switching is 2/3, while it is only 1/3 by staying David Goulding

  19. Summing Consecutive Numbers When Gauss was fourteen, his teacher suffering from a heavy night at the opera the evening before decided that the best way to keep the students quiet was to get them to add all the numbers from 1 to 100. Unfortunately for him, Gauss was a bit too clever even by our standards this is what he did 1 + 2 + 3 + … + 98 + 99+100 100+99+98+ …+3 +2 +1 101+101+101+…+101+101+101 David Goulding

  20. Summing Consecutive Numbers So we have that the sum is simply (100*101)/2 (since we counted from 1 to 100 twice!) We can extend this to the generalised formula Which in simply terms, means if we want to sum all the numbers from 1 up to a certain number, we simply take the number, add one and multiply by the original number and finally divide by 2…hey presto we have the sum Incidentally it’s 5050 in the original problem David Goulding

  21. Number of Squares on a chess board? Well exactly how many squares are there on a chessboard? No, not 64!! You forgot about the squares of size 2,3,4, etc. How many of size 1? 64 Size 2? 49 Size 3? 36…etc. See the pattern? It’s the sum of the square numbers from 1 to 8 # of squares = 1+4+9+16+25+36+49+64 = 204 David Goulding

  22. Matrix Multiplication Matrix Multiplication was introduced in order to show the students that sometimes the order of multiplication is important.

  23. Codes and Ciphers In this class, the students were introduced to some simple codes and the corresponding ciphers These included a simple numerical substitution, where A=1, B=2, C=3, etc. The Caesar Cipher in which the letters are moved along the alphabet by a fixed number of positions for example A=H, B=I, C=J, etc Sometimes these are easy to break…especially if somebody is a VAXTMXK David Goulding

  24. Another example was the well known Pigpen cipher, shown to the left with coded and decoded message below This class made use of Simon Singh’s Code Book David Goulding

  25. Final Problems Parents you are all to enter into a deal with your kids, I am going to propose a problem to you all and you are going to try and solve it yourselves. If you fail to do so by tomorrow you must give 1c to your child, if you can not solve it on Monday you must give them 2c, 4c on Tuesday if you haven’t solved it and so on until 30 days have elapsed Sounds like a pretty fair deal, don’t you agree? Here’s the puzzle so: David Goulding

  26. Being an extremely cruel and vicious maths teacher, I have decided to take ten of the students and stand them in a line facing the wall, one behind the other. On each of their heads I will place a hat, either black or white, I don’t know how many of each colour that is up to me to choose when I see fit. Now the kids are not allowed to speak or communicate once they are placed in the line. When I say so the kids starting from the back must tell me what colour hat they are wearing, if they pick the wrong colour unfortunately they will die. However if they pick the right colour I will let them live (this time!). Your puzzle is to come up with the best strategy for the kids in order to save as many as possible…and just so you know you can definitely save at least nine of the kids!!

  27. And just for completion, if you don’t manage to solve the problem, in 30 days time you will owe your child €10,737,418.23 Best get working on that problem I guess!! David Goulding

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