This prompt was created collaboratively by the Maths team at Brittons Academy (Rainham, east London, UK). It arose from a comment made by a year 9 student during a lesson in which the class was using prime factors to find the lowest common multiple (LCM) and highest common factor (HCF) of pairs of numbers. By studying Venn diagrams, the year 9 student noticed that the LCM of a pair of numbers could always be found by dividing the product of the two numbers by their HCF.The student gave the following explanation to his teacher.
- 24 as a product of its prime factors is: 2 x 2 x 2 x 3 = 24
- 60 as a product of its prime factors is: 2 x 2 x 3 x 5 = 60
- The product of 24 and 60 must therefore be 2 x 2 x 2 x 2 x 2 x 3 x 3 x 5 = 1440.
- The HCF is given by the product of the common factors (shown in the intersection of the Venn diagram).
- If you divide the product of the two numbers by
their HCF, you are left with all of the factors in the Venn
Diagram, which is the LCM: 24 x 60 ÷ (2 x 2 x 3) = 2 x 2 x 2 x 3 x 5 = 120.
- Written more formally, LCM(a,b) = a x b ÷ HCF(a,b) where a and b are natural numbers.
The teacher was intrigued and shared the student's reasoning with the department. Teachers had just begun using Inquiry Maths prompts with their classes and were excited about turning the observation into a prompt to use when introducing HCF and LCM to mixed attainment classes in year 7.