Module 14  Proof 
Proof  Teacher Notes
Solvemymaths has produced pages like this for lots of topics
This topic can be tackled by pupils at all ability levels with increasing rigour. Thanks to @gareth_metcalfe for sharing this brilliant video 'What is a mathematical proof?'. The video explains why mathematicians spend most of their time trying things that don't work. Article on Primary Proof  Click here
ü Take Three Numbers. The idea underlying this is of taking a run of three consecutive counting numbers. The children can choose a run for themselves such as 4, 5, 6 and look at the sum of the numbers. They can choose any sequence like this that they like. It doesn't matter where they start as long as the three numbers are all 'next door neighbours'. We are asking them to explore what happens when we add these three numbers together. Is there anything special about the result? Is this special result always true? If it is, can you convince me just through examining your one example? This leads us into a 'generic proof' by careful reasoning with our chosen example.
Questions and prompts to use:
ü
Choose any number. This is going to be your particular
number for this proof. Other Examples of Proof by Generic Examples  NRICH  Click here 
Reasoning and Convincing at KS2How do you know this is a square?
Beads and Bags

‘The exception disproves the rule.’ Find a COUNTEREXAMPLE to show that each of these CONJECTURES is FALSE: ü Square numbers only end in 1, 4, 9 or 6 ü Cube numbers can end in any digit except 9 ü The product of two numbers is greater than either of the two numbers. ü The square of the number is greater than the number. ü Adding two numbers and then squaring them gives the same result as squaring them and then adding them. ü Division always results in a smaller number. ü Every whole number is either a cube number or is the sum of2, 3, 4 or 5 cube numbers. ü The sum of two numbers is greater than their difference. ü The sum of two numbers is always greater than zero. ü The product of two numbers is greater than their sum. ü The square of a number is always larger than the number itself. ü Is this identity true? 4(x  y) + 5(x + y) 7(x + y) + 2(x  2y) 
ü Two dice are thrown and the two numbers obtained are multiplied together. If the answer is even, player A scores a point, if it is odd, player B scores a point. This is a fair game. ü Counting Triangles – Click here
ü The perimeter of a rectilinear shape drawn on 1 cm squared paper is always an even number ü In any triangle, the length of each side must be less than the sum of the lengths of the other two sides ü There is no square number which has a units digit of 2 ü The mean of a set of numbers is always less than at least one of the original numbers

Primes The photograph shows the wall of my classroom. There are 18 sheets of yellow A4 card (each sheet has four numbers on it). The key features are that there are 6 numbers in each row and the prime numbers are picked out in orange.
It is handy having prime numbers for pupils to refer to, but displaying them in rows of six picks out a rather interesting result. All of the prime numbers appear to be either in the first row, or the first column, or the fifth column. A natural question for pupils to ask (or for them to be asked) is: “will the rest of the prime numbers all be in the first/fifth columns too?”. My classes have approached this question like this in the past: There can’t be any primes in the sixth column because all of those numbers are multiples of 6. The numbers in the second and fourth columns are always even, so they aren’t prime (apart from the number 2). The numbers in the third column are multiples of 3 so, apart from the number 3 itself, none of them can be prime either. The only columns that we can’t find a reason for rejecting are columns one and five. Some pupils then go on to talk about why the first row is an exception. This means the pupils have essentially proved that all primes bigger than 3 are of the form 6n ± 1. But the fun doesn’t stop there! For some pupils this can then help them with the idea that x implies y does not necessarily mean that y implies x. “All primes bigger than 3 appear in columns one and five” is not the same thing as saying “all of the numbers in columns one and five are prime”. Then there are lots of other things we can do with the number patterns involved. The sixth column is multiples of 6. How can we describe the third column? Are they the odd multiples of 3, or 6n3, or “start at 3 and go up in 6s?”. The multiples of 6 are in the sixth column. Where are the multiples of 5 and the multiples of 7? Why do they go diagonally? In columns one and five every fifth number is a multiple of 5 (starting with 25 in column one and 5 in column five). Does this pattern continue? Why? What about multiples of 7? Or multiples of 11, etc? Mark Dawes (AST Comberton College Cambridge) Original Blog
Mathematical Investigations often lead to generalisations that students can justify through trying examples leading to a reasoned argument.
ü Painted Cube: Imagine a large cube made up from 27 small red cubes. Imagine dipping the large cube into a pot of yellow paint so the whole outer surface is covered, and then breaking the cube up into its small cubes. How many of the small cubes will have yellow paint on their faces?  NRICH Click here
A number of investigations can be found here where students can be asked to justify their results and in many cases to prove their conclusion  Click here

What can this diagram be used to prove? Sum of two consecutive triangular numbers is a square number. Sum of first n consecutive odd numbers equals n^{2} The sum of two odd numbers is always an even number.
Proves that (a + b)^{2} = a^{2} + b^{2} + 2ab Perhaps better to remove the a + b labels to provide more challenge. What can you do with the diagram below?
What can this be used to prove? This is a variation of Pythagoras in that it doesn't have to be squares on the sides but any similar shapes will work, such as similar right angled triangles in this case. Area of Triangle – Video Clip

Introduction into Algebraic Proof DIFFY: there's lots of simple subtracting to be done and the tasks provide a sensible reason for introducing algebra (unlike life in general) – Click here For many pupils you might begin proof by showing some examples that work  whilst emphasising that this is not proof but merely suggests a rule works. ü The sum of 3 consecutive numbers is always divisible by 3 and the sum of 4 consecutive numbers is always divisible by 4  NRICH Click here ü The sum the first n odd numbers is always n^{2}
ü Reverse and Add 1 Write down a two digit number, e.g. 62 2 Reverse the digits to form another two digit number, e.g. 26
3
Add the two numbers, e.g.
62 + 4 Repeat for other two digit numbers. 5 What do you notice? 6 Can you explain why this happens? 7 What happens if you subtract?
Note that 62 = 10x6 + 2 and so 26 = 10x2 + 6, By using similar logic we can show that for any starting 2 digit number ab the result will be 11x(a + b)
ü Staircase Numbers
A staircase is formed by adding consecutive integers. Examples: 12 because 3 + 4 + 5 = 12 (3–step) 53 because 26 + 27 = 53 (2step) 80 because 14 + 15 + 16 + 17 + 18 = 80 (5step) Investigate staircase numbers and try to answer the following questions.
1. What do you notice about

Easier: A reason to expand brackets – Click here
Harder: Use a visual representation on a number line{ Thanks to @Mathedup )
Or if you need two different odd numbers use another set
ü Prove that the square of any odd number is always one more than a multiple of 8. ü Show that the sum of any three consecutive multiples of 3 is also a multiple of 3. ü Is the number 426 in the sequence which begins 1, 4, 7, ... ü Write down the nth term of the sequence 4, 7, 10, 13, 16, … Prove that the product of any two terms of this sequence is also a term of the sequence. ü In this question a and b are numbers where a = b + 2. The sum of a and b is equal to the product a and b. Show that a and b are not integers. ü Prove that the difference between the squares of any two odd numbers is a multiple of 8. ü Prove that (3n + 1)^{2} – (3n – 1)^{2} is a multiple of 4, for all positive integer values of n. 
Proving Algebraic Identities ü Show that (2a– 1)^{2}– (2b– 1)^{2} = 4(a–b)(a + b– 1). Difference of Two Squares
http://mathandmultimedia.com/2012/06/18/differenceoftwosquares/

Sum of Consecutive Integers What is the sum of
Proof 1 Well, if we add the first term and the second term, we have (1 + n), (2 + (n1)), (3 + (n2)), and so on.
Notice that each pair has a sum of n + 1, and we have n/2 pairs of them. Therefore, the sum of all the integers from 1 through to n, or the first n positive integers is equal to
Proof 2 A variation uses this arrangement:
Proof 3 This is a visual representation of Proof 2
Extension Method: Proof by Induction

Forming Expressions
Explain why. Make up your own expressions and try on your friends. 
Pythagoras Proof

“Which proof most closely resembles the proof that you would have produced?” “Which proof do you think your teacher would award the best marks to?”
I then gave the students in each class a list of six statements and asked them to decide whether the statements were: Always true Sometimes true Never true Producing a proof in cases where they believed the result to be true. The six statements were: The sum of any five consecutive numbers is divisible by 5. The angles of a triangle add up to 180°. The interior angles of an nsided polygon sum to180(n – 2)°. Squaring two numbers and adding them together is the same as adding the two numbers together and then squaring them. The exterior angles of a polygon always sum to 360°. The sequence of triangular numbers (1, 3, 6, 10…) has the formula ½n(n+1).
To add an additional element of
interest, students were asked to tackle the problems in
perceived order of difficulty, starting with the easiest. 
Given 4 coordinates prove that the resulting shape is a square Over 1100 Geometry Proof Problems  Click here
Beginning with the single assumption that a full turn is 360^{0} prove that: ü Angles on a straight line add up to 180^{0} ü Vertically opposite angles are equal ü Finding missing angles involving parallel lines and give clear reasoning ü Whether triangles are congruent using SAS or similar reasoning ü Prove the circle theorems

Using Algebra and Reasoning  Angle Proofs The following problems involve some simple algebra alongside some geometrical reasoning to produce a proof. 
Angle Proof
What does this prove? More than one thing? Angles in a Polygon Proofs Exterior Angles of a Polygon This lovely gif should be shown and pupils ask to draw the conclusion that exterior angles in a polygon add up to 3600 Click here Is it a proof? Great Geometric Proof  Only requires level 5 Maths topics

Geometrical Proofs This resource for more able pupils can be found in the Resources folder or Click here

Proving the area of any triangle 
Video Prove that the area of a parallelogram is base x height by cutting and rearranging the pieces to make a rectangle, can you cut in other ways? What about a triangle, kite, ...?
What about a trapezium by slicing horizontally in half and rearranging to make a parallelogram  Video

Proof involving Circles Area of a Circle  Proof?
Circle Theorem Proofs
Congruent Circles – Prove the triangle at the centre is right angled.

Proof using Vectors Proving the Midpoints of any Quadrilateral makes a Parallelogram:
A Simple Proof of an interesting fact – Click here
Recurring Decimals Proofs

Parabolic Multiplication  Video Let's say you want to multiply 5 by 8. Do the following: 1. Plot the graph of y=x^{2}. 2. Draw a line that crosses the parabola where x = 5 and where x = 8 on the parabola. (Ignore the fact that x = 5 and not +5 at the left intersection point; this calculator does not do signed arithmetic!) 3. Note the value of y where the line crosses the yaxis. 4. The value of y is 40 and indeed 5 x 8 = 40.
Can you figure out why this works? [See below]
This clever exploration, plus a number of other nice
explorations for high school students come from the book Mathematics:
A Human Endeavor by
Harold Jacobs. Other thoughts on Parabolic Multiplication  Click here 
Put these cards in order to form a proof

This video clip is Andrew Wiles explaining how he felt as he
worked through the process of proving Fermat's Last Theorem

Click here Factorising Cubic n  1 is always factor of n^{3} – 1 gif – Click here Triangle Number Differences  RISP 1 Pick two whole numbers between 1 and 10 inclusive, and call them a and b. Say that Tn is the nth triangle number. Find Ta and Tb. What is the difference between Ta and Tb? Is this a prime number? When is the difference between two triangle numbers a prime number?
When is the difference between two square numbers a prime
number? 
Proof Outside the Curriculum Stable Marriage Problem  Numberphile
Proof involving Modulus Function
Alternative: All numbers are of one of the forms 5n, 5n+1,
5n+2, 5n+3, 5n+4 and then factorise the answers.

Freedom and Constraints at KS3/4 At first glance, these contexts might appear quite constrained, and you might think that there is not much to explore. Take the time to look deeper, and by pushing the boundaries you may discover some mathematically interesting results. Combining Lengths live How many different lengths is it possible to measure with a set of three rods? Climbing the Stairs live In how many ways can I get upstairs? Going to the Cinema live A cinema has 100 seats. Is it possible to fill every seat and take exactly £100? You Can't Make an Omelette... live Can you devise the most efficient strategy for finding the highest floor from which you can drop an egg safely? 
Reasoning, Justifying, Convincing and Proof at KS3/4
What Numbers Can We Make?


