What is a Mathematical proof and justification?

A ‘proof’ is a logical sequence of statements used in Mathematics to establish the certainty of a mathematical state, formula or equation. Broadly speaking, there are two different types of proofs in IGCSE Maths: algebraic proofs and geometric proofs. Algebraic proofs involve the manipulation of algebraic expressions, as well as problems that concern number properties and identities, while geometric proofs are proofs of vectors, congruence and circle theorems. IGCSE students often find these questions tricky, but they form an essential part of the syllabus. Those studying Mathematics are expected to not only understand proofs and justifications, but also be able to replicate the proper steps in response to problems posed on the exam paper.

The key step in writing proofs in IGCSE is to determine the key assumptions before starting any calculations. When structuring arguments in a statement, ensure that you are always logical, clear, and concise. Once the premises have been established, then calculations can follow until the conclusion is reached.

Students can struggle with these types of questions, as the statements being proven are often ones that they have been instructed to remember, and use as rules by which to solve an arrangement of mathematical problems. For instance, one of the earlier things we are taught when learning about triangles, is that the internal angles always add up to 180°. In Circle Theorems, we are told that angles in a semicircle are always 90°.

Going back to something more basic, we are taught from a young age that numbers such as 2, 4, 6, 8 and 10…are even numbers, and that 1, 3, 5, 7 and 9…are odd numbers. But how can we prove that adding two odd numbers together will always give us an odd number? And how can we prove that the sum of 3 consecutive odd numbers will always be a multiple of 3? These are key questions that IGCSE students are expected to be able to answer.

Example of a Mathematical geometry proof

Let’s consider the following examples.

Example 1

A, B and C are points on a circumference.

AOC is the diameter of the circle.

Prove that angle ABC is 90°.

This is a type of geometric proof question, where students are expected to prove the provided statement using their knowledge of the properties of circles and triangles. To solve this problem, a line should be drawn from the center of the circle (O) to point B. Since OA, OB and OC are all radii of the circle, they are all equal in length. This means that triangle AOB must be an isosceles triangle, and the two angles that are touching the circumference must be the same. By the same logic, BOC is also an isosceles triangle, and the angles at B and at C are also the same.

Consequently, angle ABC must be the sum of the other two angles (i.e. a + b), and since the internal angles in a triangle has to add up to 180°, we have the following equation:

a + b + (a + b) = 180°

This can be rearranged and simplified as follows:

2a + 2b = 180°

a + b = 90°

Since ABC is a + b, ABC must be 90°.

Example of an algebraic proof

Example 2

Prove algebraically that the sum of two consecutive numbers is odd.

A number can be represented as ‘n’. This means that the next consecutive number can be represented as ‘n+1’.

The ‘sum’ of two numbers means the numbers need to be added together, so,

n + n + 1

= 2n + 1

‘2n’ will always be even, as it’s a multiple of 2. Therefore, ‘2n + 1’ will always be an odd number, and the sum of two consecutive numbers is odd.

Further help in Mathematics

IGCSE students often find mathematics proofs and justifications to be tricky, and the best way of getting past this particular block in revision is through diligent practice. We always recommend to our IGCSE candidates that they improve their understanding of Mathematics through a combination of worksheets, devised problems, and selected questions from past papers. This is the best way to ensure that an IGCSE student is ready to take on their exam, including questions on proofs and justifications!

If you need help in IGCSE Mathematics, or any other subject, BartyED’s team of highly experienced and qualified tutors are available to support you. For many IGCSE students, this is the first time that they will be faced with an ‘official’ examination, and that can be an intimidating prospect. The extra support and resources provided by our expert team has ushered countless young learners to top grades in IGCSE and beyond.

If you or your child would like to learn more about the unparalleled support offered by BartyED, reach out by email (enquiries@bartyed.com) or phone (+852 2882 1017) today to find out more.