1. Hints and Tips to Improve Performance
We will now look at some of the common errors made in GCSE.
It will probably come as relief to find out that your students aren’t the only ones who do these things !
Learn From the Mistakes !
GCSEMathsA-Spr2005

2. Common Errors at B/C Borderline
Misreading Standard Form from a calculator.
A very common error with weaker pupils (below grade B) is to write the display from a calculator down as the answer.
Although they are likely to pick up method marks, they will certainly lose accuracy marks if they do this.
It must be emphasised that the x10 needs to be inserted between the number and the power.
1.9 x is OK as the 0 is not wrong.
GCSEMathsA-WP-Spr2005

3. Common Errors at B/C Borderline (cont)
Limits
A piece of wood is measured as 80 cm to the nearest 10 cm. What are the upper and lower limits of the actual length of the piece of wood ?
Lower limit = 75
Upper limit = 85
Most students are happy with the lower limit but have a lot of difficulty with the upper limit.
84.5, 84.9, would all be common errors.
They seem to be happy with 84.9r, but very uneasy with 85.
However, should they be required to calculate with these values, as they can be at grade B - What is the maximum possible weight of 6 such books, and will certainly be at grade A/A*, then it is easier to enter 85 into a calculator than 84.9….
(This is a good topic to share strategies with colleagues on).
Strictly speaking
75 ≤ length < 85.
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4. Common Errors at B/C Borderline (cont)
Expanding brackets
(x - 4)2 = x2 - 16
(x - 4)2 = (x - 4)(x - 4)
= x2 - 4x - 4x + 16
= x2 - 8x + 18
-16
The squaring of a bracket occurs in number, algebra, surds and causes problems because students just square each term.
You should emphasise that they need to write down the bracket twice and then use whatever method they have for expanding two brackets (FOIL, box method, funny face etc.)
If they get this far, other errors are -16 for -42, -4x - 4x = + 8x (because 2 minuese make a plus), and although the +18 was initially a typo it is let in to demonstrate another common phenomenon, the mis copy. These are very common, possibly due to poor lighting, difficulty reading calculator displays, inability to read their own writing and/or vanity in not wearing glasses.
If, as in this case the answer is seen and it is not contradictory to the answer as a whole, this is called a ‘slip’ and ignored.
+8x
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5. Common Errors at C/D Borderline
Not testing ‘halfway’ value in T & I
The table shows two attempts
to find the solution to
x3 - 2x = 50.
Find the value of x to 1
decimal place.
Guess
x3-2x
Comment 3 21
Too low 4 56
Too high
3.8
47.27
3.9
51.52
3.85
49.37
Many pointless marks are lost because students (which presumebly means teachers as well) do not know the QCA rule that the mid-point (or some other valid 2dp value) must be tested to guarantee that the idp value is nearest to the required root, (even if it is obvious).
GCSEMathsA-WP-Spr2005

6. Common Errors at D/E Borderline
Conversion factors
8km ≈ 5 miles
1 litre ≈ 1.75 pints
1 kg ≈ 2.2 lbs
In spec 3f4ha. These need to be known.
4.5 litres ≈ 1 gallon
30 cm ≈ 1 foot
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7. Common Errors at C/D Borderline
Probability
P(black)=6 out of 12
P(black)= 6/12 = 1/2
P(black)= evens x
Words first instance penalised.
Ratio always penalised.
Fractions, Decimals or percentages acceptable. √
P(black)= 0.5
P(black)= 6 to 12
P(black)= 50%
P(black)= 6:12
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8. Common Errors at A*/A Borderline
Cosine Rule
x2 = x7x9 x cos 42
x2 = …..
cos A = a2 - b2 - c2 + 2bc
A very large number (30-40% of candidates) mis calculate the cosine rule.
They press the = sign after 2 x 7 x 9 and then multiply by cos 42.
Another common error is when they need to use the cos rule to calculate the angle.
Because we do not give this on the formula sheet, they try to rearrange the formula.
Always a disaster. The -2bc totally throws them.
The best strategy is to learn the formula, or at least rearrange after numbers are substituted but this also does not prove successful.
cos A = b2 + c2 - a bc
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9. Common Errors at A*/A Borderline (cont)
Fractional and negative indices
– = reciprocal
This is an A* question usually worth 2 marks. We normally award 0, sometimes 2 but rarely 1 mark.
One problem is that the answer is often given as a negative.
The problem is that that it is very difficult to show intermediate working.
One approach is to deal with it in three parts:
First use the denominator to find that root. Fifth root 32 = 2
Then use the numerator as a power. 22 = 4
Then use the negative as a reciprocal. Reciprocal 4 = 1/4. 1 4
= 2
= 22 = 4
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10. Common Errors at A*/A Borderline (cont)
Algebraic Fractions 1
Solve
= 1
x x + 1
3(3x+1) -5(x-1) = (x-1)(3x+1) M2
9x+3 -5x+5= 3x2-2x -1 A1
3x2 - 6x -9 = 0
x2 - 2x - 3 = 0 M1
This is an A* question.
There are many common errors, usually in combining the algebraic fractions and knowing what to do with the denominator and the Right hand side.
If you can encourage students to go straight to the first line as shown (note use of brackets), then they score 2 method marks straight away.
These are for dealing with the algebraic fraction (note that we do not expect an expansion for the method mark, in fact it is to be discouraged as ‘doing two things at once is a problem’). The second is for taking the denominator of the LHS and cross multiplying to the RHS.
A problem at this stage is the ‘invisible bracket’. Students should be encouraged to use brackets. After all 2 marks for actually doing ‘no work’.
If students get to this point they have a good chance of making further progress. Although the minus sign in front of 5 is problematical.
Note that the quadratic thus formed will always factorise.
(x + 1)(x - 3) = 0
x = -1 or 3. A1
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11. Common Errors at A*/A Borderline (cont)
Algebraic Fractions 2 2 3
Simplify
This is an A* question. Probably the easiest A* algebra you could get.
Unfortunately not many students appreciate this.
The common errors are to cancel the x2 terms, and a factor of 3 etc…
If these questions do occur they will always factorise and furthermore there will be one factor common to the top and the bottom because we are testing if candiadates can manipulate algebraic fractions.
However, once the right answers has been achieved any further work will be penalised as this is contradictory (normally further work is ignored).
– 1 for fw
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12. Common Errors at A*/A Borderline (cont)
Given curve y = x2 - 2x - 1
Using the graph find solutions of
x2 - x - 4 = 0
Graphs
y = x2 - 2x - 1
original -
0 = x2 - x - 4
new
This is, judging by performance over the years, the hardest topic.
Normally it is worth 3 marks but the modal mark is 0. Any marks are rare.
There are difficulties with showing working and without a valid method the accuracy cannot be awarded.
One method, which we rarely see is to subtract the equation we are trying to solve (the new one) from the equation given (the original one). The resulting answer is the graph to be drawn.
There are difficulties with this when students cannot subtract involving minuses but at least a discernible method can be seen and if say the answer was -3x + 3 (only one error) we could give, M1, A0, A1 providing the graph was drawn correctly and the intersection points were given as the solution.
y = x + 3
Draw y = -x + 3
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13. Common Errors at A/B Borderline
Difference of two squares
Factorise x2 - 9
(x - 3)(x - 3) or (x - 9)(x + 9)
x2 - 9 = (x - 3)(x + 3)
Grade B
The DOTS is a topic in the intermediate spec. Unfortunately not many students (including FM students) recognise it.
At the basic level it is grade B.
If there is a further aspect, such as taking out a common factor before using DOTS it becomes grade A,
4x = 4(x - 2)(x + 2)
Grade A
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14. Drawing graphs Plot points to start with Equal gradients = parallel.
Gradients positive or negative?
PERPENDICULARS
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15. Exam mistakes Calculator in degree mode READ THE QUESTION
Do not round until the end
Watch out for negatives
SHOW WORKING
Do what you can first
UNITS!!!!!!
DO NOT LEAVE ANYTHING!!!!!
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16. ALGEBRA Don’t rearrange a fraction until its alone
Do not solve a quadratic until its equal to zero
Watch out for negatives
Cross multiplying good but fraction must be alone to do this
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17. General Vectors-watch out for direction
Standard form 1 number in front of decimal point
Look for patterns
Reciprocal- one over
Fractions!
Travel graphs
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18. General Diagrams are NOT drawn to scale You can’t add surds!!
HENCE means you must use a previous part!!
GCSEMathsA-WP-Spr2005