### 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.

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

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.

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

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

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

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

A number arranging puzzle with seven levels of challenge.

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

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

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

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### Example

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

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### Example

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

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### Example

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

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

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### Example

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

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

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### Example

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

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### Example

m and n are positive whole numbers with m > n

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

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### Example

Express as a single fraction and simplify your answer.

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### Example

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

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### Example

Prove that the expression below is always positive.

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