Module 14 Proof
Module Content Lesson Resources Lesson Plans Exam Questions  

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

 

 

 

 

Proof by Generic Examples

  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:
Can you tell me three consecutive numbers?
Make multilink towers or draw a picture to represent each.
What is their total?
Why?
How can you re-organise your numbers and their representations to show this?
Can you carry out this exploration yourself? Can you prove that your result holds for any three consecutive numbers by unpacking this one? -
Click here for complete article

  Choose any number. This is going to be your particular number for this proof.
Square your chosen number.
Subtract your starting number.
Is the number you're left with odd or even?
Create a model or a picture of your calculation, using your chosen number, and examine this model carefully.
Can you use this one model to prove that your result is always true and not just true for the particular number that you chose to start with?

Other Examples of Proof by Generic Examples - NRICH - Click here

Reasoning and Convincing at KS2

How do you know this is a square?

Beads and Bags
How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

Fifteen Cards
Can you use the information to find out which cards I have used?

Spot Thirteen
Choose 13 spots on the grid. Can you work out the scoring system? What is the maximum possible score?

Cuisenaire Spirals
Can you make arrange Cuisenaire rods so that they make a 'spiral' with right angles at the corners?

Money Bags
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?

Sealed Solution
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

Magic Vs
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

 

Proof by counter example

The exception disproves the rule.

Find a COUNTER-EXAMPLE to show that each of these CONJECTURES is FALSE:

  Square numbers only end in 1, 4, 9 or 6

  Cube numbers can end in any digit except 9

  The product of two numbers is greater than either of the two numbers.

  The square of the number is greater than the number.

  Adding two numbers and then squaring them gives the same result as squaring them and then adding them.

  Division always results in a smaller number.

  Every whole number is either a cube number or is the sum of2, 3, 4 or 5 cube numbers.

  The sum of two numbers is greater than their difference.

  The sum of two numbers is always greater than zero.

  The product of two numbers is greater than their sum.

  The square of a number is always larger than the number itself.

  Is this identity true?   4(x - y) + 5(x + y)  7(x + y) + 2(x - 2y)

 

Proof by exhaustion

  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.

  Counting Triangles Click here

Proof by Reasoning

  The perimeter of a rectilinear shape drawn on 1 cm squared paper is always an even number

  In any triangle, the length of each side must be less than the sum of the lengths of the other two sides

  There is no square number which has a units digit of 2

  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 cant 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 arent 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 cant 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 doesnt 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 6n-3, 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

 

 

Proof by Investigation

Mathematical Investigations often lead to generalisations that students can justify through trying examples leading to a reasoned argument. 

 

  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

 

 

Visual Proofs

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 n2

The sum of two odd numbers is always an even number.

 

 Proves that (a + b)2 = a2 + b2 + 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.

  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

  The sum the first n odd numbers is always n2

 

  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 +
                                                     
26
                                                      
88

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,
so when we add them we get 11x6 + 11x2 = 11x(6 + 2)

By using similar logic we can show that for any starting 2 digit number ab the result will be 11x(a + b)

 

  Staircase Numbers

 

 

5

 

4

 

3

 

 

 

 

 

 

 

 

 

 

 

A staircase is formed by adding consecutive integers.

Examples:-   12 because 3 + 4 + 5 = 12                    (3step)

                   53 because 26 + 27 = 53                    (2-step)

                   80 because 14 + 15 + 16 + 17 + 18 = 80 (5-step)

Investigate staircase numbers and try to answer the following questions.

1.       What do you notice about
         
(i)  3-step numbers,        (ii)  5-step,  (iii)  7-step numbers?
2.       What about odd-step staircase numbers?
3.       Can you explain this result?
4.       Do even-step staircase numbers behave in the same way?
5.       Explain.
6.       Describe how to find a staircase number.
7.       Is it possible to find more than one staircase?
8.       Are there any numbers which are not staircase numbers?  If so, which?

 

 

Algebraic Proofs

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

Prove that the square of any odd number is always one more than a multiple of 8.

Show that the sum of any three consecutive multiples of 3 is also a multiple of 3.

Is the number 426 in the sequence which begins 1, 4, 7, ...

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.

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.

Prove that the difference between the squares of any two odd numbers is a multiple of 8.

Prove that (3n + 1)2 (3n 1)2 is a multiple of 4, for all positive integer values of n.

Proving Algebraic Identities

Show that (2a 1)2 (2b 1)2 = 4(ab)(a + b 1).

Difference of Two Squares

http://mathandmultimedia.com/2012/06/18/difference-of-two-squares/

 

 

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 + (n-1)), (3 + (n-2)), 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:

Source

 

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?

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 n-sided 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 n(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.
Mathematics in School, January 2005

 

Geometric Proof

Given 4 coordinates prove that the resulting shape is a square

Over 1100 Geometry Proof Problems - Click here

Proof involving Angles

Beginning with the single assumption that a full turn is 3600 prove that:

Angles on a straight line add up to 1800

Vertically opposite angles are equal

Finding missing angles involving parallel lines and give clear reasoning

Whether triangles are congruent using SAS or similar reasoning

Prove the circle theorems

http://nrich.maths.org/6007

 

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

Numberphile

Prove Pythagoras' Theorem


Alternatives - NRICH
Click here

 

Geometrical Proofs

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

 

Area of 2D Shapes

Proving the area of any triangle - Video
Proving the area of a Trapezium -
Video
Does the Trapezium need to have a line of symmetry?

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

Proofs
Student Resource

 

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=x2.

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 y-axis.

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

Proving Cosine Rule

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 n3 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?
Between two cubes?
    

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.
Eg:  5n+2: (5n+2)+(5n+2)+1 = 5(5n+5n+1)+2 so not a multiple of 5 etc

 

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?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Magic Letters
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Summing Consecutive Numbers
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

Marbles in a Box
In a three-dimensional version of noughts and crosses, how many winning lines can you make?

1 Step 2 Step
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

What's it Worth?
There are lots of different methods to find out what the shapes are worth - how many can you find?

Take Three from Five
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

Number Pyramids
Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Tilted Squares
It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Painted Cube
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Arithmagons
Can you find the values at the vertices when you know the values on the edges?

Odds and Evens
Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers?

Route to Infinity
Can you describe this route to infinity? Where will the arrows take you next?

Diminishing Returns
In this problem, we have created a pattern from smaller and smaller squares. If we carried on the pattern forever, what proportion of the image would be coloured blue?

Seven Squares
Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100?

What's Possible?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

Attractive Tablecloths
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?