About the Further Mathematics
Support Programme Wales
Further Mathematics
Support Programme
Wales
WJEC FP1
• The Further Mathematics Support Programme (FMSP) Wales
started in July 2010 and follows on the success of the Further
Mathematics Support Project in England.
• The FMSP Wales is managed by WIMCS in partnership with
MEI and funded by the Welsh Assembly Government.
Sofya Lyakhova
FMSP Wales
WJEC FP1 topics
•
•
•
•
•
• 3-year pilot project. All schools in Wales are invited to register
with FMSP Wales to obtain a free access to online database of
FM resources. Tuition is available in South Wales only.
Ideas came from
Complex numbers
Polynomials (quadratics, cubics, quartics)
Series and Proof by Inductions
Matrices
Differentiation
• Jean van Schaftingen (Louvain-la-Neuve)
• Ben Sparks www.bensparks.co.uk
• Vitaly Moroz (Swansea)
Integral FP1 WJEC resources
www.integralmaths.org
• Peter Gordon (NJIT)
Identities – Complex Numbers,
Identities
Polynomials, Differentiation
Expressions, equations, formulae and identity
( a  b)( a  b)  a  b
2
2
x2  7
4x
( a  b) 2  a 2  2ab  b2
( a  b) 2  a 2  2ab  b2
( a  b)3  a 3  3a 2b  3ab2  b3
( a  b)3  a 3  3a 2b  3ab 2  b3
a 3  b3  (a  b)( a 2  ab  b2 )
a  b  (a  b)( a  ab  b )
3
3
2
2
is an expressions
x2  7
2
4x
is an equation
A  r 2
is a formulae
Some equations, however, are true for all value of x,
and are called identical equations, or identities
3( x  5)  3x  15
3( x  5)  3x  15
1
WJEC FP1 – Complex Numbers
WJEC FP1 – Complex Numbers
Not all numbers have been around a thousand years
ago. Which of these do you think older than
others? Which are the newest?
•
•
•
•
•
•
Etc…
Fractions
Positive integers
Zero
Negative integers
Roots and surds
π
-4
-3
-2
-1
•
Solve these equations:
1.
2.
3.
4.
5.
6.
WJEC FP1 – Complex Numbers
0
1
2
3
4
5
6
7
8
9
10 11 12
x + 7 = 10
7x =10
x² = 10
x + 10 = 7
x² + 7x = 0
x² + 10 = 0
WJEC FP1 – Complex Numbers
-7 or 0
-3
10/7
3
√(-10 ) ???
√10
Etc…
-4
-3
-2
-1
•
Solve these equations:
1.
2.
3.
4.
5.
6.
0
1
2
3
4
5
6
x + 7 = 10
7x =10
x² = 10
x + 10 = 7
x² + 7x = 0
x² + 10 = 0
7
8
9
10 11 12
Etc…
-4
-3
-2
-1
0
1
x² + 10 = 0
1.
2.
3.
4.
5.
6.
2
3
4
5
6
7
8
9
10 11 12
x = √(-10 ) ???
x=3
x = 10/7
x = ± √10 (= ±3.162…)
x = -3
x(x + 7) = 0 → x =0 or -7
x = √(-10 ) ???
WJEC FP1 – Complex Numbers
WJEC FP1 – Complex Numbers
• Can you solve this equation
x3  7 x  6  0
-3
1
2
2
WJEC FP1 – Complex Numbers
•
WJEC FP1 – Complex Numbers
You will have already come across the
Quadratic Formula:
If ax 3  bx 2  cx  d  0
ax 2  bx  c  0
x
•
 b  b 2  4ac
2a
This will solve ANY quadratic equation
(and this is only really a part of it…)
WJEC FP1 – Complex Numbers
•
WJEC FP1 – Complex Numbers
Cardano (and some others) in the 16th century
were trying to find a general formula for CUBIC
equations like the one already known for
quadratics.

So if it is a number what is it like?
let i   1
(Imaginary bit)
i 
2
(square both sides)
i3 
i4 
i5 

This “i” has some funny properties… but it does
follow the normal laws of algebra.
5i
WJEC FP1 – Complex Numbers
This point is 3+2i
4i
Imaginary Part (2)
3i

Real Part (3)
2i
So if it is a number what is it like?
i
let i   1
i  1
(square both sides)
i  i
Since i³=i² x i
i4  1
Since i4=i³ x i
i5  i
And off we go again…
3

(Imaginary bit)
2
Etc…
-4
-3
-2
-1
0
-2i
This “i” has some funny properties… but it does
follow the normal laws of algebra.
-3i
-4i
-5i
1
2
3
4
5
6
7
8
9
10 11 12
• The complex numbers have actually turned out
to be stunningly useful for very practical subjects
such as Engineering, Physics and Computing.
• They are now found to be at the heart of
equations of Quantum Theory which have
massively broadened our understanding of our
universe.
-6i
-7i
Etc…
3
WJEC FP1 – Complex Numbers
• A complex number z is of the form x + yi, where x
and y are real numbers. The real part of z is denoted
Re(z) (= x) and the Imaginary part of z is denoted Im(z)
(=y). Notice therefore that Im(z) is actually REAL!
(The set of Real Numbers is therefore a subset of the
set of Complex Numbers.)
WJEC FP1 – Complex Numbers
•How can we make sure that the complex numbers behave like
numbers? Do they obey the normal rules of algebra? Can they
be added, subtracted, multiplied and divided?
Complex Number Arithmetic: - Examples:
(3 + 4 i)+(-2 + 7 i)= x+iy ?????
(3 + 4 i) - (-2 + 7 i) = x+iy ?????
• How can we make sure that the complex numbers
behave like numbers? Do they obey the normal rules
of algebra? Can they be added, subtracted, multiplied
and divided?
WJEC FP1 – Complex Numbers
(3 + 2i)
÷
(4 – 3 i)
= x+yi ?????
(2 – 5 i) (-3 + 4 i) = x+yi ?????
(3 + 2i) ÷ (4 – 3i ) = x+yi ?????
WJEC FP1 – Complex Numbers
Teaching and learning resources
www.integralmaths.org
5 3
1 2
WJEC FP1 – Polynomials
WJEC FP1 – Polynomials
ROOTS OF QUADRATICS
Ax 2  Bx  c  0
Which of the graphs below correspond to
D<0
A>0
A<0
D=0
two different real roots
two equal real roots
a
b
ROOTS OF QUADRATICS
D>0
no real roots
Ax 2  Bx  c  0
True or False:
A. A quadratic equation always has two roots
B. A quadratic equation can have two real roots
C. A quadratic equation can have two complex roots
D. A quadratic equation always has two real roots
c
d
E. A quadratic equation can have one real and one complex
root
F. If α is a root of the equation (complex or real) then x-α is a
2
factor of the polynomial Ax  Bx  c
4
WJEC FP1 – Polynomials
Th 1. If  ,  are the roots
WJEC FP1 – Polynomials
ROOTS OF CUBICS and QUARTICS
Ax 4  Bx 3  Cx 2  Dx  E  0
of a quadratic equation ax 2  bx  c  0 ,
b
c
then α  β  - and   .
a
a
p. 147, Gaulter&Gaulter, Further Pure Mathematics
Th 2. If  ,  ,  are the roots
Ax 3  Bx 2  Cx  D  0
A. How many roots?
B. What combinations of complex and real roots are
possible?
of a cubic equation ax 3  bx 2  cx  d  0,
b
c
d
then α  β    - ,       ,   
a
a
a
p. 149, Gaulter&Gaulter, Further Pure Mathematics
Identities - Polynomials
Example 1
Example 2
a and b are unknown.
a
Identities - Polynomials
2
 b2  4
ab  1
Find
a
a
a and b are unknown.
a  b  2
ab  2
Find
2
2
b
 b



1
1


a2
b2
WJEC FP1 – Polynomials
Teaching and learning resources
www.integralmaths.org
a
 b
a
 b
2
a
2
2


 b 
1
1


a 2
b2
2
WJEC FP1 – Proof and
Proof by Induction
Go through different types of proof in one lesson,
see Algebraic Proof powerpoint
5
WJEC FP1 – Series
WJEC FP1 – Series
• Any even number can be defined by 2n
2,4,6,8,…100,102,…
2n where n=1,2,…
• Any odd number can be defined by 2n+1
1,3,5,7,…
2n+1 where n=0,1,…
• Any odd number can be defined by 2n-1
1,3,5,7,…
2n-1 where n=1,2,…
n

1  2  3  4  5  ...  n 
r
r 1
2  4  6  ...  2 n 
n

2r
r 1
n

1  3  5  ...  ( 2 n  1 ) 
( 2 r  1)
r0
n

1  3  5  ...  ( 2 n  1 ) 
( 2 r  1)
r 1
• Any square number can be defined as n²
1,4,9,25,…
n² where n=1,2,…
1  4  9  ...  n
2

n

r
2
r 1
WJEC FP1 – Series
WJEC FP1 – Series
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
10 11
2
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55
11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 = 110
10
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
= (10 x 11) / 2 = 55
1 + 2 + 3 + 4 + ….. + (n -1) + n
= n(n + 1) / 2
11
WJEC FP1 – Matrices
WJEC FP1 – Series
Conjecture:
1  2  3  ...  (n  1)  n 
n(n  1)
2
for n = 1, 2, 3, ….
or
n
r 
r 1
n(n  1)
2
for n = 1, 2, 3, ….
Let us talk about
numbers again!
Commutative law of addition: m + n = n + m .
A sum isn’t changed at rearrangement of its addends.
Associative law of addition: ( m + n ) + k = m + ( n + k ) = m + n + k .
A sum doesn’t depend on grouping of its addends.
Commutative law of multiplication: m · n = n · m .
A product isn’t changed at rearrangement of its factors.
Associative law of multiplication: ( m · n ) · k = m · ( n · k ) = m · n · k .
A product doesn’t depend on grouping of its factors.
Distributive law of multiplication over addition: ( m + n ) · k = m · k + n · k .
This law expands the rules of operations with brackets (see the previous
section).
6
WJEC FP1 – Matrices
WJEC FP1 – Matrices
Task 1. Solve 3x=5. Write your solution carefully step
by step.
Moving to
simultaneous
equations
Ax  B , A  3, B  5
3x  5
x  A 1 B ,

Task 2. Solve 3x=5. Write your solution step by step,
avoid using division .
2 x1  x 2  3
3 x1  5 x 2  11
Task 3. Solve 3x=5. Write your solution step by step,
avoid using division or fractions.
What would change if the commutative law of
multiplication did not hold??
2 x1  x 2  3 x 3  12
3 x1  5 x 2  2 x 3  7
x1  2 x 2  x 3  1
WJEC FP1 – Matrices
WJEC FP1 – Matrices
Moving to
simultaneous
equations
If we want to manipulate matrices like we manipulate
number, we must be able to:
Ax  B, A  3, B  5
3x  5
 x  A1 B,
x 
 3
 2  1
AX  B, A  
, X   1 , B   
11
3 5 
 x2 
2 x1  x 2  3
3 x1  5 x 2  11
If P and M two 2x2
matrices, is it always the
case that P+M=M+P?
1) add matrices
Is it always the case
that PM=MP?
2) multiply matrices
??
 x  A 1 B,
12 
 x1 
2 1 3 
 
 


AX  B, A   3 5  2 , X   x2 , B   7 
1
x 
1 2  1
 
 3


2 x1  x 2  3 x 3  12
3 x1  5 x 2  2 x 3  7
??
x1  2 x 2  x 3  1
Px0=0,
P+0=P
3) have a zero matrix
4) have an analogue of 1
5) divide matrices??
 x  A 1 B,
Identities - Differentiation
Example 2
Identities - Differentiation
Q2.
a) Factorise
x  h 3  x 3
x  h 4  x 4
1

1
x3
1

1
x4
 x  h 3
a) Simplify
 x  h 4
b) Using the result in a) simplify
x  h 3  x 3
b) Using the result in a simplify

h
4
4
x  h   x 
h
What happens when
1
x  h 3

1
x3
h
h 0
?
1
x  h 4
h

1
x4


What happens when
h0
?
7
ALGEBRAIC PROOF
Further Mathematics
Support Programme Wales
Sofya Lyakhova
sofyalyakhova@furthermaths.org.uk
www.furthermaths.org.uk/wales.php
Starter
Further Mathematics Support
Programme Wales aims to encourage
more students to take Further
Mathematics AS/A level qualification
• Revision sessions (online, face-toface, video conferencing)
• Enrichment activities
• Careers in Maths talks
Starter
1. Any even number can be
defined by 2n
a) Always
b) Sometimes
c) Never
Starter
Starter
2. When you square a number,
the answer is positive
3. x 2  y 2  ( x  y )( x  y )
a) Always
b) Sometimes
c) Never
a) Always
b) Sometimes
c) Never
1
Starter
Starter
4. If two lines are each perpendicular
5. x 2  4 x  5  2( x  7)
to a third line, they must be
parallel to each other
a) Always
b) Sometimes
c) Never
a) Always
b) Sometimes
c) Never
Starter
Starter
6. An odd number can be defined
as 2n+1
7. An odd number can be defined
as 2n-1
a) Always
b) Sometimes
c) Never
a) Always
b) Sometimes
c) Never
… proofs are chains of logical steps, where
every next step is based on a previous step
and every step must be true!
… distinguish between practical
demonstrations and proof
2
PROOF
Part 1. Algebraic proof
Part 1. Algebraic Proofs
Part 2. Use of a counter-example
• In this section a number of general results
about properties of numbers will be proved
using algebra
Part 3. Proof by contradiction
Part 4. Proof by Induction
Part 1. Algebraic proof
Part 1. Algebraic proof
Find the mistake in the proof below
Part 1. Algebraic proof
•
•
•
•
•
•
•
•
•
•
Proof?
Let a = b
Then a² = ab
(multiply by a)
a² + a² = a² + ab
(add a²)
2a² = a² + ab
(simplify ‘a²’s)
2a² - 2ab = a² + ab - 2ab
(subtract 2ab)
2a² - 2ab = a² - ab
(simplify ‘ab’s)
2(a² - ab) = 1(a² - ab)
(factorise)
2=1
(cancel (a² - ab))
Hmm…
We get nonsense because we’ve actually divided by
zero. We can’t let that happen.
Part 1. Algebraic proof
Bertrand
Russell,
mathematician and
philosopher
Example 1. Prove that the sum of squares of
two consecutive integers is always odd.
Example 2. Prove that the product of an
even number and an odd number is
always even
3
Part 1. Algebraic proof
Part 1. Algebraic proof
Solution
(a)
5n
Question 1
(a) Write down an expression, in terms of n,
for the nth multiple of 5.
(b) (i)
Let the first number be 5n
so the second number is 5(n+1)
(b) Hence
the sum is 5n + 5(n+1) = 5n + 5n + 5
= 10n + 5
(i) prove that the sum of two consecutive
multiples of 5 is always an odd number,
(ii) prove that the product of two consecutive
multiples of 5 is always an even number.
= 5(2n + 1)
Which is odd since 2n + 1 is odd for
all integer values of n.
so we have odd x odd = odd
Part 1. Algebraic proof
Solution (continued)
(b) (ii) Using 5n and 5(n + 1) again
product 5n x 5(n + 1) = 25n(n + 1)
if n is odd then n + 1 is even
Part 1. Algebraic proof
A few important remarks:
• A demonstration is not a proof,
unless you demonstrate all cases!
if n is even then n + 1 is odd
as 25 is odd, we will always have
odd x odd x even which is always even
Part 2. Use of a counter-examples
Example 1. All prime numbers are odd.
Example 2. Charlie says “ If x is a positive
2
integer, then x  x  1 is always prime.”
Show that Charlie is wrong.
• Demonstration is useful to
understand the nature of the result
Part 2. Use of a counter-examples
• Sometimes you may met a conjecture, that
is an unproven claim.
• If a conjecture turns out to be true, it may
be quite difficult to prove it for all possible
cases.
• On the other hand, if a conjecture is false,
you only need to find one case where it is
fails in order to demonstrate its falsehood.
Such a falilure is called a counter-example
4
Part 2. Use of a counter-examples
Part 3. Proof by contradiction
Q1. James says “If you add two prime numbers
together you always get another prime number”.
Show that James is wrong.
Q2. If x is positive, then 1+10x-x² is also positive.
Show that this statement is false.
Q3. Petra says “If n is a positive integer, then the
value of n²+n+41 is always prime”. Show that
Petra is wrong.
Proof by contradiction
• Sometimes to prove a conjecture one can
start with stating the opposite.
• Assume the opposite is true and call it our
assumption.
• Start manipulating with the assumption.
• You may end up with a conclusion which
contradicts your assumptions. In this case
your assumption was wrong.
• This proves the original statement!
Proof by contradiction
Rational Numbers
Rational Numbers
Etc…
1
2
3
4
5
6
7
8
9
?
1
10 11 12
1
• The followers of Pythagoras thought that every number
could be written as a fraction.
• The cult of the Pythagoreans was quite insistent on this
point.
• But a man called Hippasus challenged Pythagoras…
Proof by contradiction
•
Hippasus asked the question about the length of this
diagonal.
•
Pythagoras’ own theorem said the length had to be √2,
but they couldn’t find the fraction which represented it.
There’s a good reason why not…
•
Proof by contradiction
Rational Numbers
Rational Numbers
If we assume √2 is rational then it
can be written
as a fraction:
?
1
a
(where a and b have no common factors)
b
2
a
2= 2 1
b
2
2
So a² is an even number 2b = a
?
2=
so a is an even number
so
a = 2c
1
1
• So what happened to Hippasus
who first challenged this idea that
every number was rational?
• They drowned him…
• Or so the legend goes.
2b 2 = 4c 2
b 2 = 2 c 2 So b² is an even number
and a = 4c
so b is an even number
so we can substitute 4c²
But if a and b are both even they have a common factor of 2
for the a² in this equation
So we have a contradiction
2
2
√2 ≠
a
b
5
Part 4. Proof by induction
Part 4. Proof by induction
Why natural numbers are so special?
1
2
3
4
5
6 7 8 9 10 .......
Why natural numbers are so special?
1
2
3
4
5
6 7 8 9 10 .......
• Every number has a successor
• 1 is not a successor of any number
• No two numbers have the same successor
•
Part 4. Proof by induction
Any property which belongs to 1 and also to the
successor of any number that also has the same
property, belongs to all natural numbers. 
Natural numbers are inductive
1
Giuseppe
Peano,
Italian
mathematician
Part 4. Proof by induction
Suppose we want to prove that something is true for all
numbers......
... It would be enough to show that
1)the statement is true for 1, and
2)if it is true for an arbitrary number n, then it is true for its
successor n+1.
Example. 1x2 is even. Does it mean that every natural number
multiplied by 2 is even?
If nx2 is even, then (n+1)x2 = 2n+2 = even +2 = even,
so true for every successor.
So, by induction is true for all natural numbers! EASY!
2
3
4
5
6 7 8 9 10 .......
Any property which belongs to 1 and also to the successor
if any number that also has the same property, belongs
to all natural numbers.
Suppose we want to prove that something is true for all
numbers......
... It would be enough to show that
1)the statement is true for 1, and
2)if it is true for an arbitrary number n, then it is true for its
successor n+1.
Part 4. Proof by induction
Example. If there are N pigeonholes and N+1 object to be placed in them , then one pigeonhole must have two or more objects in it. (Pigeonhole Principle)
It would be enough to show that
1) the statement is true for 1, and
If n=1 then we have one pigeonhole and two objects. So there are two objects in the same pigeonhole. 2) if it is true for an arbitrary number n, then it is true for its successor
n+1.
n: there are n pigeonholes and n+1 objects and one of the pigeonholes has two or more objects in it. n+1: What can we say about n+1 pigeonholes and n+2 objects? Can we find a pigeonhole which has two or more objects? Take one pigeonhole at random…
6
Part 4. Proof by induction
Suppose we want to prove that something is true for all
numbers......
... It would be enough to show that
1)the statement is true for 1, and
2)if it is true for an arbitrary number n, then it is true for its
successor n+1.
PROOF
When we do not agree
with a conjecture we
try to find a counterexample
• Algebraic Proofs
• Use of a counter-example
• Proof by contradiction
Use it when you
actually agree with
the conjecture
• Proof by Induction
Only used when
dealing with positive
integers
What is next? - students
1. Your teacher will receive a list of
exercises for you to try.
2. If you enjoyed this session consider
studying A-level Maths and Further
Maths. www.furthermaths.org.uk
What is next? - teachers
1. You will be emailed list of exercises
and solutions for your students to try
on their own.
2. Your feedback would be highly
appreciated. Please email me
ACFMSPWales@wimcs.ac.uk
3.If you enjoyed this talk please mention
it to other schools in your area.
More video talk organised by FMSP
Wales:
1. Careers in Mathematics
2. Infinity and Beyond
For more information
ACFMSPWales@wimcs.ac.uk
www.furthermaths.org.uk
3. The History of Numbers
Let Maths take you Further…
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