Two Tasks on The Structure of a Proof - Part 2

 

Part 2 Read the following question.

 

     Prove that the difference between the squares of any two odd numbers is a multiple of 8.
(You may assume that any odd number can be written in the form 2r – 1, where r is an integer).
Now attempt the matching exercise. In the first matching exercise you have to match the student to what they have done to attempt to answer the question.
In the second matching exercise you have to match the student to the mistake they have made.

 

Student 1:
From above the difference between the squares of any two odd numbers = 4(a-b)(a+b-1)
If either (a-b) or (a+b-1) is even then 4(a-b)(a+b-1) must be a multiple of 8.
If a and b are both odd or both even then a-b is even
If a is odd and b is even or the other way round then (a+b-1) is even
So we have 4 times an even number times something else which must be a multiple of 8

Student 2:
(2r-1)(2r-1)=4r2-4r+1
(2x-1)(2x-1)=4x2-4x+1
(4r2-4r+1)-(4x2-4x+1)
4r2-4r-4x2+4x
4(r2-r) – 4(x2+x)

Student 3:
(2x3)-1=5
32=9   52=25   difference =16 (8x2)
(2x7)-1=13
72=49   132=169   difference=120 (8x15)

 

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Now you have completed this task you should have a good idea of how to set out a mathematical proof. This is important because proof is the bedrock of mathematics. Many other subjects you might be studying use mathematical theories as part of their language and they need to know that these theories are reliable.
If you wish to study proof further then you could look at:

  1. http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php
  2. http://mathforum.org/isaac/problems/pythagthm.html
  3. http://nrich.maths.org/6333