Time for a Hero
With so much competition for the very top grades in GCSE mathematics, it can be very worthwhile to reflect on the ‘armoury’ of strategies we offer our more able students as they prepare for the more challenging questions at A* and A level GCSE mathematics.
It has been fascinating to have the opportunity of stepping outside of the syllabus in order to use an innovative strategy to solve a higher level GCSE question that is frequently examined by all GCSE mathematics exam boards.
Consider a question that has asked candidates to find the area of a triangle given three sides for example A = 40m, B = 60m and C = 80m. If the vertical height of the triangle is not given, this question can be rather daunting and time consuming for some students as it requires rearranging the cosine rule to first obtain an angle and then to use this result to find the height of the triangle and, finally, obtain the area of the triangle.
Step forward Hero’s Formula!
Although not given to students on the formula sheet, more able students can easily remember the stunning elegance and innovation of the formula without having to know the magic of its discovery. More than two thousand years ago Hero proved that we can find the area of a triangle given three sides by:
(1) Finding the semi-perimeter of the triangle – in this case (40m + 60m + 80m)/2 = 90m
(2) Remembering Hero’s Formula:
Area of a triangle = the square root of S(S – A)(S – B)(S –C)
As if by magic, 5 or 6 marks can be gained in just a few seconds by calculating:
Area = square root of 90 times (90 – 40) times (90 – 60) times (90 – 80)
Which equals the square root of 90 times 50 times 30 times 10
Or 1161 . 9 square metres to 1 decimal place.
If a student obtains the correct result, full marks are awarded even if the method is not the one expected by the examiner.
Once able students become use to using Hero’s Formula, it becomes a powerful and useful piece of additional armoury which can give an elegant way of securing just a few more marks and, concurrently, giving some additional welcome time to tackle other challenging questions in the exam paper.
Chris Curtis
Head of Mathematics
