Further Mathematics www.fmnetwork.org.uk
the Further Mathematics network www.fmnetwork.org.uk
the Further Mathematics network www.fmnetwork.org.uk
Stretching Students: FP1
Let Maths take you Further…
Matrices
Matrices as transformations
Examining matrices with determinant zero
Matrices and Google
Challenging Transformations
Get students to work out the matrix of the projection onto y = mx.
y = 3x y x
Matrices with zero determinant
General proof that whole plane is mapped to a line.
Which lines get mapped to the points on that line?
Matrices and Google
y
2 x
1
3 z
The Mathematics
y
2 x
2 x
1 x
2 z
2 y z
2
3 z
The Mathematics
y
2 x
2 x
1 x
2 z
2 y z
2
3 z
Counting the votes
2
3
Site Votes Received
1 0.5z
0.5x + 0.5z
0.5x + y
Counting the votes
Site Votes Received
2
3 x = 0.5z
0.5x + 0.5z
0.5x + y
Counting the votes
Site Votes Received x = 0.5z
3 y = 0.5x + 0.5z
0.5x + y
Counting the votes
Site Votes Received x = 0.5z
y = 0.5x + 0.5z
z = 0.5x + y
Counting the votes
x = 0.5z
y = 0.5x + 0.5z
z = 0.5x + y
Matrices and Eigenvectors
2
The ranks x, y, z are a solution of
1
3
⎛
⎜
⎜
0 0 0.5
0.5
0 0.5
0.5 1 0 x x y y z
Iterative Computation
2
Let T
= ⎜
⎛
⎜
0 0 0.5
1 0 0.5
⎞
⎟
0 1 0
3
1
Starting with the point x
⎛
1/ 3
⎞
= ⎜ ⎟
⎟
⎝ 1/ 3 ⎠
We look at the sequence of vectors Tx , T 2 x, T 3 x, T 4 x,…..
The limit of this sequence is the gives the ranks….
The Google Equation v.2
Google needs to make sure that this process always converges and make the convergence as rapid as possible.
They came up with this equation x = (1 – δ )s + δ T x
Where s
⎛
1/ 3
⎞
= ⎜ ⎟
⎝ 1/ 3 ⎠ is the rank source and δ is the damping factor.
Notice that when δ = 1 this gives x = T x
δ = 0.95
δ = 0.85
New Sites
Site 1 2 3 4
2
4
1
0 0 1 0
2
1 0 1 1
3
0 1 0 0
3
1
Let T
=
⎜
⎜
⎛
⎜
⎜
0
1 0 0.5 1
0
0
1
0.5
0
0
0
0 0 0 0
⎟
⎟
⎞
⎟
4
0 0 0 0
Gives ranks 0.2, 0.4, 0.4, 0
New Sites
Site 1 2 3 4
2
4
1
0 0 1 0
2
1 0 1 1
3
0 1 0 0
3
1
Let T
=
⎜
⎜
⎛
⎜
⎜
0 0 0.5
0
1 0 0.5 1
0 0.5
0 0
0 0.5
0 0
⎟
⎟
⎞
⎟
4
0 1 0 0
Gives ranks 1/9, 4/9, 2/9, 2/9
New Sites
Site 1 2 3 4 5
2
4
1 0 0 1 0 0
2 1 0 1 0 0
5
3 0 1 0 0 0
1
Let T
=
4 0 1 0 0 0
3
⎜
⎜
⎛
⎜
⎜
⎜
0 0 0.5
0 0
1 0 0.5
0 0
0
0
0.333
0.333
0
0
0
0
0
0
⎟
⎟
⎟
⎟
⎞
⎟
5 0 1 0 0 0
Gives ranks 0, 0, 0, 0, 0. The problem of dangling nodes!
0 0.333
0 0 0
New Sites
Site 1 2 3 4 5
2
4
1 0 0 1 0 0
2 1 0 1 1 1
5
3 0 1 0 0 0
1
Let
4
3
T
=
⎜
⎜
⎛
⎜
⎜
⎜
0 0 0.5
0 0
1 0 0.5
1 1
0 0.333
0 0 0
0 0.333
0 0 0
⎟
⎟
⎞
⎟
⎟
⎟
0 0.333
0 0 0
5
0
0
1
1
0
0
0
0
0
0
Gives ranks 0.08 0.46, 0.15, 0.15, 0.15
Complex Numbers
Equations involving complex numbers
The proof that if z is a root then so is its conjugate
Extending work on loci
Fractal and data compression
Equations involving complex numbers
Ask students to solve simple equations which involve complex numbers.
This can become an interesting discussion point.
Conjugates
If z is the root of a polynomial equation with real coefficients then so is its conjugate.
Proof involves
(i) showing that (z
(ii) showing that (z
1
+ z
2
)*=z
1 z
2
)* = z
1
1
* + z
2
*
*z
2
*
Introduce language such as ‘ lemma ’.
Then the main result
Corollary – every polynomial equation with real coefficients with odd degree has at least one real root.
Extending work on loci
How do loci on the Argand diagram link to the coordinate geometry that students are studying in AS Core units?
Circles
Lines
Fractals and Data Compression
The Mandelbrot Set
The Mandelbrot Set is the most famous example of a fractal . The word "fractal" has two related meanings.
•In everyday life, it means a shape that is recursively constructed or self-similar, that is, a shape that appears similar at all scales of magnification and is therefore often referred to as
"infinitely complex."
•In mathematics a fractal is a geometric object that satisfies a specific technical condition, namely having a Hausdorff dimension greater than its topological dimension (yikes…)
The Mandelbrot Set
You are now going to consider the iteration z n
+
1
= z n
2 + c
Let’s work out what happens if you start with the value
0, for various different values of c .
First of all let’s take c = 1. This gives the sequence
0, 1, 2, 5, 26,…..
Now c = –1. This gives the sequence
0, -1, 0, -1, 0 ,-1,….
The Mandelbrot Set
What happens if c = 0 ?
What happens if c = i
?
What about c =
2, -2, 0.5, 1 + i, 2 i , 1 – i ?
A spreadsheet program can help us with this.
The Mandelbrot set consists of those numbers c for which the sequence starting with 0 and calculated
2 above (with ) does not tend to infinity.
z
= z
+ c
Algebra
Neat examples
Old P5 questions – have found that able students really enjoy this material.
Algebra Example
Related Example
The circle in the diagram has radius 6 cm. If the perimeter of the rectangle is 28 cm, what is its area?
Senior Team Maths Challenge 2007
Group Round
(Not to scale)
Curve Sketching and
Inequalities
Introductory Example
Different ways to think about asymptotes
Lead Example
A car travels from Trowbridge to Hatfield at an average speed of 30mph in 4hrs. At what average speed does it need to return to
Trowbridge to make its average speed for the whole journey (there and back) 60mph.
If v is the average speed for the return journey and a is the average speed for the whole journey then a = 60 – 7200/(4v + 120)
Asymptotes
Numerical approach
Different types of algebraic approach
Examples where the asymptote is quadratic
Inequalities – Examples
Get students to really understand what it means to solve an inequality
Helping students to get familiar with a case by case approach will really help them.
Solution
Series and Induction
Nice exercises with the standard summation formulae.
Non-algebra induction examples.
More advanced induction
Things to think about
Standard Summations – why are they integers .
Proof in pictures
Proof by induction without the algebra
Tiling a chessboard
Strange tilings!
Imagine I am given a square checkerboard whose number of squares in each row or column is a power of two.
So, for example:
2 by 2
4 by 4
Strange tilings!
And
8 by 8
Strange tilings!
And the list goes on forever:
16 by 16, 32 by 32, 64 by 64, 128 by 128, …, 4096 by
4096, …
For any of these infinitely many boards I am going to show how if precisely one square is removed at random from such a board, you can always tile what remains using the below tiles!
Strange tilings!
With a 2 by 2 board this is easy: whichever square is removed you are left with 3 squares which form an Lshape!
Strange tilings!
With a 2 by 2 board this is easy: whichever square is removed you are left with 3 squares which form an Lshape!
Strange tilings!
With a 2 by 2 board this is easy: whichever square is removed you are left with 3 squares which form an Lshape!
I’m now going to show you how it’s done with an 4 by 4 board.
Strange tilings!
Strange tilings!
Imagine a square has been removed at random.
I have marked this in blue.
Strange tilings!
The trick is to break the board up into four 2 by 2 boards…
Strange tilings!
The trick is to break the board up into four 2 by 2 boards… as indicated by the red lines (notice that 2 is one power of two down from 4).
Three of the 4 by 4 boards do not have a square removed.
Strange tilings!
Place a tile which covers the middle corner square of each of the 4 by 4 boards which have not yet had a square removed. Now let’s look at 8 by 8 boards….
Strange tilings!
Imagine a square has been removed at random.
I have marked this in grey.
Strange tilings!
The trick is to break the board up into four 4 by 4 boards…
Strange tilings!
Place a tile which covers the middle corner square of each of the 4 by 4 boards which have not yet had a square removed.
We now just have the problem of dealing with 4 by 4 boards each with one square removed!
Strange tilings!
But we know how to do
4 x 4 boards…etc etc.
Divisibility proofs
This example is nice because although it’s a divisibility proof, it’s algebraically less demanding than most.
