Last year I devoted some Spring blogs to English (as an examiner of GCSE English Literature for this subject looms large.) Here I shall look at Maths - and especially 'border issues', points of knowledge or know-how that candidates commonly neglect to revise, or put into practice with sufficient care.
My sources are recent examiners' reports on GCSE Maths Foundation from the CCEA (Northern Ireland) and WJEC (Wales). These came up quickly in my search - and provide typical material from which prospective candidates (and their teachers) might profit. Here are some findings in a 5-point checklist.
1. Recognition of mathematical terms ... and common numbers that illustrate them. Make sure none of these are perplexing or unfamiliar:
vertex - (plural vertices) - a corner, a point where the lines of a 3D shape meet. Questions sometimes ask for the number of vertices on, for example, a cube (Answer 8)
square number, cube number, prime number. The first is the outcome of multiplication of a number by itself (eg. 3 x 3 = 9. 9 is the square number. The second is the outcome of multiplication of a number by itself, and once again - or 3 x 3 x 3 = 27. 27 is the cube number. A prime number can be exactly divided only by the number 1 - and itself (eg. 7. 7 divided by 1 = 7 and 7 divided by 7 = 1.)
Merely knowing the meaning is not quite enough - the well-prepared will know the identities or values of the square, cube and prime numbers up to 100, so that they can read, in such a list as follows - 12 19 36 64 80 100 that 19 is a prime; 36, 64 and 100 are squares; and 64 is a cube (as well as a square).
2. Fractions without a calculator - (1) how to add and subtract them without a calculator. Since this skill will always be tested, it must be worthwhile to learn how to do it. 1/3 + 1/5 does not equal 2/8 !
The proper method takes at least 4 mini steps (3 multiplication and 1 addition) - these will be rewarded with marks. The method is first multiply the denominators: 3 x 5 = 15. This will be the new common denominator. The numerator will be found by cross-multiplying first numerator and second denominator, then second numerator and first denominator, and adding: (3 x 1) + (5 x 1) = 8. So the answer is: 8/15.
- (2) how to simplify them. Simplifying fractions, or giving them in their 'lowest terms' - is basically a test in division. 24/54 for example is an inelegant way to leave a fraction. Many candidates will see that, with 24 and 54 being even numbers, both top and bottom of this fraction can be divided by 2: 12/27. Some may go on to see that top and bottom can be further divided by 3: 4/9. This will be the answer. In effect 24/54 has been divided by 2 and 3, or 2 x 3 = 6.
3. 'Set out work well.' This plea occurs in reports year after year. Setting out work well secures marks in two ways - first, by reducing error commission - fewer mistakes are made when small steps are written out systematically; second, and even more importantly, by showing knowledge of method (whether a correct answer is obtained in the end or not). Valuable marks are awarded for this.
4. The limits of Algebra. Many students can solve or find (by rearranging and dividing) what 'x' equals in the following form of equation:
3x + 4 = 22 (3x = 22 - 4 = 18; x = 18/3 = 6)
But relatively few candidates at Foundation level know what to do when an equation of this type contains brackets:
3(x+2) = 18
Solving requires only one more step - multiplying out the brackets: 3(x +2) = 3 times x + 3 times 2 = 3x +6.
So then 3x +6 = 18 (3x = 18 - 6 = 12; x = 12/3 = 4)
Being able to multiply out those brackets combined with setting out work well will bring bountiful marks.
5. And don't forget (3 short ones at random to finish).
* when finding the area of a triangle to divide by 2
* when marking points on a scatter diagram to use a 'small neat cross'
* when writing 3 figure bearings to show 3 figures - for example, not 75 degrees but 075 degrees