Mathematical proof worksheet

Mathematical proof at a glance

A mathematical proof is a logical sequence of mathematical statements that proves that something is always true. For example, we could use algebra to prove that the sum of two consecutive odd numbers is always a multiple of 4 or we could use circle theorems to prove a statement about angles. 

Algebraic proofs involve working from a starting point, which we write on the left hand side. They often require the expansion of brackets followed by the manipulating/simplifying/ factorising of the algebraic expression, which we write in a logical step by step way on the right hand side. We continue until a conclusion can be made about the final expression. When working on algebraic proofs, we generally use 2n when we want to represent an even number (integer) and 2n+1 to represent an odd integer.

Geometrical proofs could involve using angle facts or circle theorems to prove statements about angles or using our knowledge of vectors to prove geometric statements. Sometimes, problems might involve using our knowledge of geometry and algebra together to form a proof.

We could also be asked to prove something by counterexample. This means finding an example which doesn’t work for the given statement, therefore proving it is not alway true. For example, if you are given the statement that if you add 1 to a multiple of 4 you always get a prime number, you could prove this is not true by finding an example where it doesn’t work. 

Mathematical proof is studied further in A Level maths, and other methods of proof, such as proof by induction and proof by contradiction, are introduced. 

Looking forward, students can then progress to additional proofs in maths worksheets or more algebra worksheets, for example a sequences worksheet, simultaneous equations worksheet or straight line graphs worksheet.

For more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts.