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.

Proof by Generic Examples

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:
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? -

ü  Choose any number. This is going to be your particular number for this proof.
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

### 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: 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) Proof by exhaustion 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 Proof by Reasoning 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

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 n2

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)

ü  Staircase Numbers

 5 4 3

A staircase is formed by adding consecutive integers.

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

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:

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 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:   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
 Using Algebra and Reasoning - Angle Proofs The following problems involve some simple algebra alongside some geometrical reasoning to produce a proof. 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
 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   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
 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 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)²+(5n+2)+1 = 5(5n²+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.

How many different lengths is it possible to measure with a set of three rods?

In how many ways can I get upstairs?

A cinema has 100 seats. Is it possible to fill every seat and take exactly £100?

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

### 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?