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The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students. To find out more please visit www.furthermaths.org.uk A Level Maths & Further Maths Highly regarded and popular A levels Facilitating subjects for universities Many A level and university subjects require maths knowledge Opens doors to a variety of careers, many of which are well paid Enjoyable Challenging Useful Careers that need maths qualifications Engineer Scientist Teacher Computer programmer Games designer Internet security Logistics Doctor Dentist Meteorologist Finance Accountancy Business management Nursing Veterinary Sports science Physical Therapist Pharmacist Medical statistician And many more… Josephus Flavius Magic Cards What does 495 mean? What about 3287? Suppose instead of having a number system based on 10s, we used 2s. What would that look like? What would 17 look like? What would 17 look like? 4 3 2 2 2 2 1 0 0 1 0 2 2 0 1 What about 25? What about 25? 24 23 22 21 20 1 1 0 0 1 In your workbook. Are these true? 1 + 1 = 10 10 + 1 = 11 10 + 11 = 101 Binary Sums Key Addition Results for Binary Numbers 1 + 0 = 1 1 + 1 = 10 1 + 1 + 1 = 11 Key Subtraction Results for Binary Numbers – 0 = 1 10 – 1 = 1 11 – 1 = 10 1 Binary Sums Question 111+100 101+110 1111+111 111-101 110-11 1100-101 1110+10111 1110+1111 11111+11101 Answer Answers Question 111+100 101+110 1111+111 111-101 110-11 1100-101 1110+10111 1110+1111 11111+11101 Answer 1011 1011 10110 10 11 111 100101 11101 111100 Josephus Flavius… • • • • Jewish historian 1 century AD Trapped with 40 soldiers Preferring suicide to capture they decided to kill themselves • They formed a circle and starting from one, every remaining alternate person was eliminated… Josephus, not keen to die, quickly found the safe spot in the circle and thus stayed alive. An example We will start with a circle of 10 people. The arrow in the diagrams will show the person to be left alive at that stage. Conclusions • If we always start with 1 then even numbers will not survive! • The survivor for a group of 10 people is person number 5. • We shall say J(10) = 5. The Challenge Can you quickly determine J(41), the position of the survivor in a group of 41 people? What about groups of any size? Complete the table Number of people (n) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 41? Who wins? J(n) Number of people (n) Who wins? J(n) Results Number of people (n) 1 2 3 4 5 6 7 8 9 Who wins? J(n) 1 1 3 1 3 5 7 1 3 Number of people (n) 10 11 12 13 14 15 16 17 41? Who wins? J(n) 5 7 9 11 13 15 1 3 19 In Binary n 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 J(n) 1 01 11 001 011 101 111 0001 0011 n 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001 0101 0111 1001 1011 1101 1111 00001 00011 J(n) n J(n) 1 110 n J(n) 1 110 n J(n) 1 110 n 1 110 J(n) 110 1 n 1 110 J(n) 110 1 What is actually happening when we are moving the first digit to the end? Consider what is happening when 495 changes to 954. First we subtract 400, to give us 95. Then we multiply by 10, to give us 950. Then we add 4 to give 954. What is happening in binary? Consider 1011. Moving the first digit to the end gives us 0111: First subtract the highest power of two (removing the one in the far left column, which is worth 8). Then multiply by two, to move everything left one place value column. Then add 1. When there were 41 people, Where did Josephus stand? 41-32 = 9 (subtract the highest power of 2) 9x2 = 18 (multiply by 2) 18+1 =19 (add one) Stand in place 19. In summary! There are 10 types of people in the world; those who understand binary and those who don’t. A Level Maths & Further Maths Highly regarded and popular A levels Facilitating subjects for universities Many A level and university subjects require maths knowledge Opens doors to a variety of careers, many of which are well paid Enjoyable Challenging Useful Careers that need maths qualifications Engineer Scientist Teacher Computer programmer Games designer Internet security Logistics Doctor Dentist Meteorologist Finance Accountancy Business management Nursing Veterinary Sports science Physical Therapist Pharmacist Medical statistician And many more… The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students. To find out more please visit www.furthermaths.org.uk