Geometric Bites

Proving that the sum of three consecutive integers is divisible by 3. Algebraic proof questions in Edexcel International GCSE (Higher) often look difficult because they are written in words. The quickest way to handle them is to translate the words into standard algebraic forms that guarantee the number property you need (even, odd, multiple, consecutive, square, etc.). Once the translation is correct, the rest of the proof is usually straightforward simplification, factoring, and a clear final statement. This post gives a compact “toolkit” of the most common forms, presented in a table you can reuse, plus a small set of extras and techniques that frequently appear in Higher-tier proof questions. The core principle In an algebraic proof, represent the numbers so the required property is built in. For example: If a number is even, write it as 2n for some integer n. If a number is a multiple of 3, write it as 3n for some integer n. The phrase “for some integer n” matters...

An improper fraction being turned into a mixed number. Fractions are used to represent parts of a whole, but once a fraction becomes larger than 1, there are two common ways to write it: as an improper fraction or as a mixed number . These two forms often represent exactly the same quantity; the difference is simply how the number is written. Improper fractions An improper fraction is a single fraction where the numerator is greater than or equal to the denominator. This means the fraction is at least 1 whole (and possibly more). Examples: 7/4 12/5 9/9 In 7/4, there are 7 parts, and each whole is made from 4 parts. Since 7 is larger than 4, the value is greater than 1. Mixed numbers A mixed number is written as a whole number followed by a proper fraction. The proper fraction shows the leftover part after counting whole units. Examples: 1 3/4 2 2/5 3 1/6 In 1 3/4, the “1” shows one whole, and “3/4” shows three extra quarters. Same value, differe...