# Proof

## Learn how to develop an argument, justification or proof using mathematical language.

### Curriculum

The English national curriculum for mathematics aims to ensure that all pupils reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language.

### Proof

A mathematical proof is a sequence of statements that follow on logically from each other that shows that something is always true. Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases.

### Congruent Triangles

Test your understanding of the criteria for congruence of triangles with this self-marking quiz.

### Four Colour Theorem

Show that no more than four colours are required to colour the regions of the map or pattern so that no two adjacent regions have the same colour.

### Identity, Equation or Formula?

Arrange the given statements in groups to show whether they are identities, equations or formulae.

### Line Drawings

Six line drawings that may or may not be able to be traced without lifting the pencil or going over any line twice.

### Mix and Math

Determine the nature of adding, subtracting and multiplying numbers with specific properties.

### Not Too Close

The students numbered 1 to 8 should sit on the chairs so that no two consecutively numbered students sit next to each other.

### Plus

A number arranging puzzle with seven levels of challenge.

### Proof of Circle Theorems

Arrange the stages of the proofs for the standard circle theorems in the correct order.

### Satisfaction

This is quite a challenging number grouping puzzle requiring a knowledge of prime, square and triangular numbers.

### True or False?

Arrange the given statements in groups to show whether they are always true, sometimes true or false.

### Example

State whether each of the following statements is true or false. Give reasons for your answers.

### Example

One is added to the product of two consecutive positive even numbers. Show that the result is a square number.

### Example

(a) Give a reason why 0 is an even number.

### Example

Betsy thinks that $$(3x)^2$$ is always greater than or equal to $$3x$$.

### Example

Given that $$n$$ can be any integer such that $$n \gt 1$$, prove that $$n^2 + 3n$$ is even.

### Example

Use algebra to prove that $$0.3\dot1\dot8 \times 0.\dot8$$ is equal to $$\frac{28}{99}$$.

### Example

The diagram shows a quadrilateral ABCD in which angle DAB equals angle CDA and AB = CD.

### Example

m and n are positive whole numbers with m > n

### Example

(a) Prove that the recurring decimal $$0.\dot2 \dot1$$ has the value $$\frac{7}{33}$$

### Example

(a) Prove that the product of two consecutive whole numbers is always even.

### Example

Prove that the expression below is always positive.

For Students:

For Teachers:

For All: