2. Proof at
Key Stage 4
Jo Sibley

3. Terry Pratchett from ‘Thief of time’
Headmistress:
[Algebra is] far too difficult for seven-year olds!
Teacher:
Yes, but I didn’t tell them that, and so far, they haven’t found out.

4. Choose four terms: Integer Sum Algebraic Consecutive Difference Sweets
Elephant
Multiple
Statement
Even
Write down
Assumption
Square
True
Miliband
Symmetry
Convention
Congruent
Chord
Odd
Geometric

5. Warm-up: A B 2 3
Each card has a letter on one side and a number on the other.
From this starting position, which two cards must you turn over to prove that all the cards with a vowel on them have an even number on the other side?

6. Some things to think about:
Most GCSE-examined proof questions are algebraic or geometric.
Use the recipe:
Start with the basic ingredients
Work through the steps in order
Don’t leave anything out
‘Reveal’ the result

7. Some basic ingredients:
odd number
even number 2n
multiple of 3 3n
different odd number
2k + 1
different even number 2k
different multiple of 3 3k

8. Some basic ingredients:
odd number
2n + 1
even number 2n
multiple of 3 3n
different odd number
2k + 1
different even number 2k
different multiple of 3 3k

9. Some basic ingredients:
Convention: Mathematicians use n or k to represent an integer
odd number
2n + 1
even number 2n
multiple of 3 3n
different odd number
2k + 1
different even number 2k
different multiple of 3 3k

10. Using the ingredients:
Prove that the difference between any two odd numbers is always even.

11. Using the ingredients:
Any odd number:
2k+1
A different odd number:
2n+1
Find the difference:
(2k+1)-(2n+1)
=2k+1-2n-1
=2k-2n
=2(k-n)
The difference between two integers is always an integer so this is always even.
Stir the algebraic mixture:
Present the result:
Anything here which might prompt discussion?

12. Using the ingredients:
Prove that the difference between consecutive square numbers is always odd.

13. Using the ingredients:
A square number: n2
A consecutive square number:
(n+1)2
Find the difference:
(n+1)2 - n2
=n2+2n+1-n2
=2n+1
This is always odd for integer n.
Stir the algebraic mixture:
Present the result:

14. Loudmouth Len group exercise:
I bet you a pint that all these things are true!

15. Loudmouth Len claims:
If one of (n+m) and (n-m) is even then so is the other.

16. Loudmouth Len claims:
The sum of two consecutive triangle numbers is always a square number.

17. Loudmouth Len claims:
n3 +2n, where n is a positive integer, is always divisible by 3.

18. Loudmouth Len claims:
If a and b are distinct positive integers then the mean of their squares is always greater than the square of their mean.

20. Opportunity to correct a common error…

21. Opportunity to correct a common error…
…and reinforce an underappreciated method.

22. Geometric proofs
Labelling conventions – pair task

23. Geometric proofs
Labelling conventions – pair task

24. Geometric proofs Labelling conventions – pair task
Reflecting triangle BDE in line BD gives triangle BDF.
Since BDE = ADC, line FDC is straight.
Angles BFC = BAC are subtended by the same chord BC and so are equal.
Hence BED = BAC = 50

26. What’s missing here?

28. Which is the valid proof?

30. Re-order correctly?

31. Re-order correctly?

32. Recognise this?

33. Yup, Hannah’s Sweets

34. Yup, Hannah’s Sweets

35. Hannah’s sweets - 2015 Malcolm’s sweets 2005
A bag contains a mixture of red sweets and blue sweets.
There are seven more blue sweets than red sweets.
Let the number of red sweets be n.
The probability of selecting a red sweet followed by a blue sweet without replacement is
Show that
Show that the probability of choosing two sweets of the same colour is
Malcolm’s sweets 2005

36. Proof at Key Stage 4 Jo Sibley josibley@furthermaths.org.uk
…oh… wait… who had ‘Elephant’?

37. Proof at
Key Stage 4
Jo Sibley