The Proof Materials Project (PMP)
Can it be expected that yr 12 AS mathematicians already have a good
experience and expertise in the use of mathematical proof ?
Will they demonstrate skills that show them to have developed proof concepts
at Proof Level 5, as described by Sue Waring 1
‘Pupils are aware of the need for a generalised proof, can understand the
creation of some formal proofs, and are able to construct proofs in a variety of
contexts, including some unfamiliar.’
Aware of deficiencies in my teaching of proof in past AS courses as a ‘bolt on
extra’ with little connection with other AS topics taught. I am very keen to
examine methods for improved teaching of the concepts and language of proof
within AS topics. Ideally I would like to enable ‘pupils to perceive proof as an
integral part of the mathematics curriculum, and not as a separate strand.’ Sue
Waring 1
Objective 1
To assess the ‘proof level’ of each individual pupil by an initial assessment.
Objective 2
To build up pupil competency in use of the correct language of mathematical
proof.
Objective 3
To build up pupil competency in use of the different proofs, including proof by
contradiction, deductive proof, proof by exhaustion and the idea of counter
examples.
Objective 4
To enable deductive reasoning and questioning to become the normal learning
experience of AS maths.
Resource materials
1
Can you prove it ? Sue Waring
Developing reasoning through algebra and geometry
QCA Publications 2004
MEI Pure 1 materials
SMP 16 – 19 Mathematical Structures
MEP materials
www.sasked.gov.sk.ca/docs/math30
1
Evaluation of the ‘mini project’
I administered a very informal assessment at the start and end of year 12, using
the set of questions labelled ‘Learning to think Mathematically’.
A very quick and superficial assessment determined that all the students had
made improvements in their ability to use the correct language of mathematical
proof and in their ability to use different proofs. (Objectives 3 and 4).
However, I do wonder whether this is a significant improvement and better than
expected for the general improvement in their Mathematical skills after a year
of learning AS Mathematics.
Objective 1 was not achieved through lack of time available.
Objective 4 was achieved and enabled me to enjoy the teaching of proof (for
the first time!)
I will certainly use this approach of teaching proof within AS topics for future
courses, hopefully enabling pupils to perceive proof as an integral part to the
mathematics curriculum.
C. Meadows-Smith
08.08.05
2
Learning to think mathematically
Please try to answer as many of these questions as possible. There is no problem if you
fail to finish them.
1
What do you understand by the following terms, you might like to
examples or describe the differences.
a
verify
b
prove
c
fallacy
d
solve
e
counter example
f
proof by contradiction
g
proof by deduction
h
proof by exhaustion
2
give
What is wrong with this proof ?
Let
Multiplying by a
Subtracting b2
Factorising
Dividing by (a – b)
Substituting for a
Dividing by b
A ‘proof’ that 2 = 1
a=b
a2 = ab
2
a – b2 = ab – b2
(a – b) (a + b) = b(a – b)
(a + b) = b
2b = b
2=1
3
Try to prove that when you multiply two odd numbers together the answer is
always odd.
4
A 3 digit number is such that its second digit is the sum of its first and
third digit. Prove that the number must be divisible by 11.
5
Try to prove that no square number ends with ‘2’
6
Try to prove that the base angles of an isosceles triangle are equal
A perpendicular bisector drawn from the base may help.
7
Try to prove that the angle at the centre of a circle is twice any angle at the
circumference
You can assume that base angles of an isosceles triangle are equal
that the exterior angle of a triangle is equal to the sum of
the interior opposite angles.
3
RESOURCES
LESSONS/STARTERS
EVALUATIONS
4
Context
Aim
Type of proof
Prove Cosine rule
Prove formula for
area of any triangle
Prove Sine rule
Vocabulary
introduced
Revising Cosine and Sine Rules
Group work
Using proof cards as a lead to students generating
their own proofs
Proof by deduction
Diagram and proof statements printed on cards *
Students working in pairs to use cards to present a
coherent proof
Volunteers to present proof to class
Students to criticise proof constructively
Students working in pairs to present coherent proofs
Volunteers to present proofs to class
Students to criticise proof constructively
Axioms
Deduction
Evaluation
Good discussion – axioms
assumptions Pythagoras & SOH CAH TOA
Card approach to Cosine Rule went well enabled all students to be involved
Presentation of proofs to class was good but dominated by the more confident
mathematicians
Criticism of proofs was reasonable but not all students had the confidence to
contribute
Proof cards*
In ∆ BCN
CN2 = a2 – ( c – x) 2
In ∆ ACN
x = b Cos A
a2 = b2 + c2 – 2cx
b2 = a2 + c2 - 2ac Cos B
In ∆ ACN
CN 2 = b2 – x2
a 2 – c2 + 2cx – x2 = b2 – x2
a2 = b2 + c2 - 2bc Cos A
c2 = a2 + b2 - 2ab Cos C
5
Context
Reinforce degrees and radians
Introduce formulae for area and arc
length of sectors
Aim
Build up students’ confidence in
presenting coherent proofs to each
other
Using axioms to deduct facts about
sectors
Type of proof
Proof by deduction
Assumptions
 c = 180o
Area circle =  r2
Circumference of circle = 2  r
If P and Q lie on the circle centre O,
radius r, then the triangle OPQ has area
½ r2 Sin θ where θ = POQ
Proofs based on radians and sectors Students were presented with a set of
radian cards (see attached)
Working in pairs they were asked to use
the information on card A to work out
the answers to the other cards
Each student took turns to present a
formula to the class (their own choice)
Discussions and constructive criticisms
were encouraged
Evaluation
Discussions were of good standard
Excellent way to encourage logical thought processes
All students were involved and appeared to gain confidence in discussing proofs
Idea of using axioms to build up proofs reinforced effectively
Excellent method of clearing up misunderstandings
Needed more than one lesson
6
7
Context
Aim
Type of proof
Prove opposite angles
in cyclic quadrilateral
are supplementary
Prove angles in the
same segment are
equal
Assumptions
Vocabulary used
Starter to lesson consolidating formulae for sectors
Students to attempt proof with no guidance
Proof by deduction
Introduction was a simple diagram of a cyclic
quadrilateral and instructions to prove the theorem.
Students to work in groups
Gradually provide hints & suggestions if required
Students to present proof to class
Using proof of cyclic quadrilateral to lead onto proof of
angles in same segment
Angles in quadrilateral add to 360o
Base angles of isosceles triangle are equal
Axioms
Therefore
Guidance provided when required
Consider four points A,B,C,D lying on a circle with
centre O
Connect the four points to O.
This gives four isosceles triangles as shown.
Angles in a quadrilateral add to 360o
2w + 2x + 2y + 2z = 360
Consider the angles at A and C
A + C = z + w + x + y = (2z + 2w +2x + 2y)/2 = 180
Since ABCD and ABCE are both cyclic quadrilaterals
what does this tell you about the angles at D and E?
8
Evaluation
Good starter for the lesson
It was encouraging to note that the students were able to attempt the proof
without any guidance
Once the 2nd diagram was drawn most were able to make a good start and
realised that isosceles triangles were key to the proof
One of the confident mathematicians (who can dominate discussions) assumed
wrongly that lengths AB and DC were identical
This led to a very useful discussion
One of the less confident mathematicians was able to present the proof
coherently
A useful session, good discussions, students are gaining confidence
The same segment proof was achieved easily
9
Context
Aim
Type of proof
Prove that the
tangent is
perpendicular to the
radius at point of
contact
Vocabulary used
Starter to a lesson consolidating Sectors
To introduce the idea of proof by contradiction
Proof by contradiction
Teacher directed
Step by step approach
Make an assumption that is the contradiction of the
statement to be proved
Therefore
Implies
The Proof
OTA  90o
90o must be at S
OST = 90o
 OT is the hypotenuse to ∆ OTS
OT  OS
But T is a point on the circumference of the circle
 S must be inside the circle
 straight line TS cuts the circumference at another point
But this is impossible as ST is the tangent with point of contact T
 original statement OTA = 90o must be wrong
 OTA = 90o
The tangent is perpendicular to the radius at the point of contact
Evaluation
Too much input from teacher
Only the more confident mathematicians were able to follow the logic
Only an introduction, will need follow up and consolidation
A useful consolidation of proof language
10
Context
Aim
Type of proof
Prove that if a2 is
even then a is even
Assumptions
Vocabulary used
Starter to lesson introducing sequences and series
To follow up the concept of proof by contradiction
To consolidate some of the proof language
Proof by contradiction
Proof by deduction
Class discussions, teacher directed
Assumption made that is a contradiction of the
statement to be proved
Step by step approach
Odd and even numbers
Therefore
Implies
Implied by
Necessary and sufficient
The Proof
There is an odd number a such that a2 is even
a = 2n + 1 ( as a is any odd number)
 a2 = (2n + 1)2
= 4n2 + 4n + 1
= 4n(n + 1) + 1
2
 a must be odd
 there is not an odd number a such that a2 is even
If a is odd then a2 is odd
If a is even then a2 is even
a is even  a2 is even
a is even  a2 is even
a is even  a2 is even
Evaluation
Better student involvement
A good discussion on the contradictory statement
Students grasped the concept quickly
Students followed the logic of the proof competently
All students were involved in creating the proof and discussions
Regained their confidence
Good introduction to Proof language and the concept of necessary and
sufficient
11
Context
Aim
Type of proof
Prove that √2 is
irrational
Assumptions
Vocabulary used
Starter to lesson consolidating sequences and series
To consolidate the concept of proof by contradiction
To consolidate generating the contradictory statement
To consolidate using appropriate connecting language
To demonstrate how this proof was built from the
previous one : a2 is even  a is even
Proof by contradiction
The contradiction statement was generated by class
discussion
Students then worked in pairs
Each step to the proof written on a card
Students to place the steps in the correct order
Volunteer to present the proof to the class
even numbers
a2 is even  a is even
Therefore
Necessary and sufficient
The proof cards
√2 is rational
 can be expressed as √2 = a/b where a and b have no common factors
2 = a2 / b2
2b2 = a2
a2 is even  a is even
a is even  a = 2n ( n is an integer)
a2 = 4n2
2b2 = 4n2
b2 = 2n2
b2 is even  b is even
as a and b are both even there is a common factor of 2
This a contradiction of original statement
 √2 is not rational
 √2 is irrational
Evaluation
Students were far more confident with the concept
Students were more familiar with the use of appropriate language, discussions were good
All were involved when putting the proof together
A good confident presentation of the proof by one of the less confident mathematicians
The exercise took about 10 mins
12
Context
Aim
Type of proof
To prove that if
acute angles a and b
are supplementary
then Sin a = Cos b
To prove that
Sin(a + b)≠Sina +Cosb
Assumptions
Vocabulary used
Part of lesson on trigonometry
To consolidate the idea of counter example
To consolidate the idea of difference between
conjecture and proof
Proof by deduction
Students to investigate Sin a – Cos b for acute
supplementary angles a and b
Encouraged to suggest a conjecture
Discussion of difference between conjecture and proof
Encouraged to prove that Sin a = Cos b
Encouraged to check to see if it works for other angles
Students to investigate the statement
Discussion of counter examples
SOH CAH TOA
Counter example
Conjecture
Evaluation
Good discussion of difference between conjecture and proof
Students generated the proof with confidence worked well in pairs
Used SOH CAH TOA in proof
Understood the concept of counter example
13
Context
Aim
Type of proof
Prove Pythagoras’s
theorem
Prove the trig
identity
Cos2θ + Sin2θ = 1
Vocabulary used
Starter to lesson on trigonometry identities
Introduction to trig identities
To build up confidence with generating proofs
To demonstrate the difference between demonstrate
and prove
Proof by deduction
Class discussion about the difference between
demonstrating the theorem and proving it
Students shown diagram of square encouraged to
consider the area of the square
Students working in pairs to generate proof
Investigate whether the identity is correct
Discussions of the difference between conjecture and
proof
By considering acute angles and writing Cos θ as a/h
and Sin θ as b/h use the previous proof to generate
the new proof
Conjecture
Identity
Evaluation
Students demonstrated confidence in generating proof, did not require extra
guidance
Good discussions of difference between conjecture, demonstrate and proof
Good student questions about extending the trig identity to all angles
14
15
Context
Aim
Practice and
consolidation
Vocabulary used
Lesson on using appropriate mathematical language
To consolidate the use of appropriate mathematical
language
Class discussions, teacher led
Practice exercise ( MEI Pure Mathematics 1)
Necessary
Sufficient
Necessary & Sufficient
   
Evaluation
Students appeared confident when working through the consolidation
exercises
All were involved
Good discussions
16
17
18
19