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. To find out more please visit The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.

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?

10001

What about 25?

11001

In your workbook.

Are these true?  = 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  1–0=1  10–1=1  11–1=10

Binary Sums QuestionAnswer

Answers QuestionAnswer

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) Who wins? J(n) Number of people (n) ? Who wins? J(n)

Results Number of people (n) Who wins? J(n) Number of people (n) ? Who wins? J(n)

In Binary n J(n) n J(n)

1101 n J(n)

1101 n J(n)

1 n 110

n J(n)

1 n 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  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?  = 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. To find out more please visit The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.