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Transcript
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