Part 1 - Two Tasks on the Structure of a Proof

 

A proof is defined as - a formal series of statements showing that if one thing is true something else necessarily follows from it. In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true.[1][2] Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproven proposition that is believed to be true is known as a conjecture.


The statement that is proved is often called a theorem.[1] Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.


Proof is something that is touched on very little at GCSE level. As a result it is something students often find challenging. The purpose of this resource is to help you think about how to set out a proof, what a proof requires, the type of process a mathematician might go through in order to prove something and the sort of language that should be used.


You will be working towards an ‘if – then – so – therefore- QED’ structure (see table 1 below) of setting out a proof because a clear structure makes it easy for the reader to understand the process you have gone through to complete the proof. You will also be thinking about how to set out your proofs or explanation in a clear sequential manner so that it is easy to see your mathematical thinking and the process you have been through to get to your answer.

1

If

The statement you are trying to prove is true

2

Then

It logically follows from this that

3

So

As a results of this

4

Therefore

The statement you were trying to prove is true

5

QED

 

For example: Prove that the sum of two odd numbers is an even number.


Words

 

If

Odd + odd=even

 

Then

Even+1+even+1=even

 

So

Even+even+2=even

 

So

Even+even+even=even

 

Therefore

As even numbers are made up of 2s added together even numbers added together must give an even answer

 

Therefore

Odd+odd=even

 

In algebra this could be expressed as:


Algebra

Odd+odd-even

(2n+1) + (2m+1) is even

2n+2m+2 is even

2(n+m+1) is even

2x is even

Odd+odd=even

 

Matching Task 1 (Matching Exercise)
Now  try to arrange the following proof in the correct order:


Words

If

Odd × even= even

Then

(even + 1) × even = even

So

Even × even + even= even

So

Even +even = even

Therefore

Odd  × even = even

Click here for a suggested answer

 


 

Task 2 - This question is from the . After reading the question read these three responses and decide which of these students has answered the question and why.
6.       (a)     Show that       (2a – 1)2 – (2b – 1)2 = 4(a – b)(a + b – 1)

Student 1 answer:
(2a – 1)2 – (2b – 1)2 = (4a2-4a+1)-(4b2-4b+1) = 4a2-4b2-4a+4b
4(a-b)(a+b-1)= (4a-4b)(a+b-1)
=4a2+4ab-4a-4ab-4b2+4b=4a2-4b2-4a+4b

Student 2 answer:
(2a-1)(2a-1)-(2b-1)(2b-1)=(4a-4b)(a+b+1)
4a2-4a+1-4b2-4b+1=4a2-4a-4b2+4b
4a2-4a-4b2+4b=4a2-4a-4b2+4b

Student 3 answer:
(2a-1)(2a-1)=2a2-2a-2a+1=2a2-4a+1      so (2a – 1)2 – (2b – 1)2= 2a2-4a-2b2-4b
(2b-1)(2b-1)=2b2-2b-2b+1=2b2-4b+1
4(a-b)= 4a-4b
(4a-4b)(a+b-1)= 4a2+4ab-4a-4ab-4bb-4b= 2a2-4a-2b2-4b


Click here for a suggested answer

Now go to Part 2