This document provides guidance for mathematics teachers to improve student performance in Additional Mathematics for the SPM 2009 exam. It identifies common weaknesses and mistakes by student category (very weak to excellent). Suggestions are given to rectify issues for different topics in Paper 1 and Paper 2, such as functions, quadratic equations, vectors, and integration. For weaker students, the focus is on getting partial marks. For stronger students, emphasis is placed on careless mistakes. Teachers are advised to provide targeted practice addressing specific weaknesses.

1. KURSUS PEMANTAPAN PENGAJARAN DAN PEMBELAJARAN
UNTUK GURU-GURU MATEMATIK TAMBAHAN
SBP ZON TENGAH
2009
PENINGKATAN PRESTASI
MATEMATIK TAMBAHAN
SPM 2009
Mengenalpasti kelemahan dan kesilapan lazim
pelajar dan cadangan cara mengatasinya
Mengikut kategori pelajar:
• Pelajar Lemah (Markah 0 – 29)
• Pelajar Sederhana (Markah 30 – 50)
• Pelajar Target A (Markah 51 – 69)
• Pelajar Cemerlang (Markah 70 – 100)
GENERAL COMMENTS
For Very Weak Students :
Organise a class where teacher emphasises on how to get at least one mark for every
question (sub questions) either in Paper 1 or Paper 2.
For Intermediate students:
Emphasise on topics where questions are certain to come out.
For Excellent Students
Do a lot of papers in SPM format to reduce careless mistakes.
Emphasise on the steps / details required to get full marks, especially in
paper 2 such as round off errors.
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2. PAPER 1
Topic Weakness / Common Mistakes Rectification / Reminder
Functions Weak Students (0 – 29)
(a) Unable to determine the type of
function especially when the relation
is not given as arrow diagram.
(b) Unable to distinguish between object
and image eg: f(x) = 9
(c) Make mistakes when question asks to
find the object if the image is given.
Average Students (30 – 50)
(d) Unable to find inverse function.
Excellent Students (70 – 100)
(e) Did not write the condition for f(x)
involving fractions.
Suggestion
(a) Ask student to convert to arrow diagram.
(b) Emphasised on object and image.
Draw an arrow diagram
Front  object Back  image
(d) Suggestion:
Explained inverse function as the inverse of the
operation for simple function as ax + b .
f (x) = x ax + b
Thus
f -1
(x)=
a
bx −
x
(e) For functions involving fractions, the
condition for f(x) MUST be written:
for f(x) =
x
2
, x ≠ 0
Quadratic
Equations
Weak Students (0 – 29)
(a) Difficulty in converting to general
form.
Unable to identify the constants a, b
and c when the equation is not
written in general form.
Example : 3x2
– kx + 2k = 1 – 5x
Suggestion:
i. Make the RHS = 0  3x2
– kx + 2k – 1 + 5x = 0
ii. Draw 3 boxes for each constant as such:
x2
+ x + = 0
Fill in any Fill in any Fill in any
constants constants constants
with x2
with x without x or x2
[ a ] [ b ] [ c ]
Quadratic
Functions
Targeted A Students (51 – 69)
(a) could not relate the information
given in equation of completing the
square form with the graph given.
(b) Problems in inequalities  Unable
to determine the range.
(a) Do more exercise on sketching. Give emphasis on
Max / min point and the axis of symmetry.
(b) Show at least two ways of solving:
- using graph
- using number line
- using tables
Students may choose the one that he/she
understands the most.
2
× a + b
– b÷ a
x 9f

3. S7
S2
S5
S4
Topic Weakness / Common Mistakes Rectification / Reminder
Coordinate
Geometry
Weak Students (0 – 29)
(a) Weak in perpendicular gradient
Average Students (30 – 50)
(b) did not use midpoint to find the
fourth coordinate of a parallelogram or
rhombus.
(a) Must relate to  m1 m2 = –1 or
to inverse the values and change the sign.
(b) Remind students that the diagonals of rhombus,
parallelogram, rectangle and square share the
same midpoint.
Linear Law Average Students (30 – 50)
(a) Problem in finding c, example,
3
25
514
=
−
−
=m
Students tend to do: Y = mX + c
xy = 3 x2
+ c
subst. (2, 5) to x and y. (2)(5) = 3(2)2
+c
Suggestion:
• Write down: Y = mX + c
• Find m and c BEFORE substituting the y and
the x-axis given from the graph.
OR
• Use calculator to find the y-intercept and the
gradient.
Progression Targeted A Students (51 – 69)
(a) Weak in finding, example,
(i) S3 – 7
(ii) Tn when Sn is given.
Suggestion
(a) Explain using diagram : example
(i) 1 2 3 4 5 6 7 S3 – 7 = S7 – S2
(Sa – b = Sb – Sa-1)
(ii) 1 2 3 4 5 T5 = S5 – S4
(Tn = Sn – Sn-1)
(b) Use only 2 data to show it is a GP or
an AP.
(b) Remind students that they must use at least 3 data
to show whether it’s a GP or an AP.
Circular
Measures
Weak Students (0 – 29)
(a) Unable to change degree to radian
and vice versa.
(b) Use angle in degree in the formula.
Average Students (30 – 50)
(c) When finding θ using s=rθ or
A=r2
θ  students tend to multiply the
value of θ with
180
π
as if to change it to
rad.
(d) Difficulty finding the area of
segment
 formula not given.
(a) Emphasise on the use of calculator to change
radian to degree.
(c) Emphasise that θ is in radian.
(d) Remind students of the formulae not given.
** For Area of segment = r2
( θ - sin θ)
(since θ is in radian, to get sin θ  use calculator
to mark it as rad, thus, no need to change to degree
Example: If r = 3 and θ = 1.25 radian
0.5 × 3 x2
( 1.25 - sin 1.25 shift Ans 2 ) =
3
(2, 5)
(5, 14)
xy
x
Mode
AC
Reg Lin
Enter points
2 5 M+
5 14
Shift 2 Press right arrow twice A =
,
, M+
y-intercept
Shift 2 Press right arrow twice B = gradient

4. Topic Weakness / Common Mistakes Rectification / Reminder
On screen  0.5x32
(1.25 – sin 1.25r
)
Vectors Weak Students (0 – 29)
(a) Unable to write unit vector correctly
(b) for parallel vectors : a is parallel to b
 students tend to take a = b .
(a) Emphasise on phythagoras theorem to find the
magnitude.
Show the formula and how to substitute.
Suggestion
(b) Use the ratio of each vector.
Example: If p = 3i + 4j and q = mi + 8j are
parallel find m.
Solutions:
8
43
=
m
[similar
triangle]
Differentiatio
n
Weak Students (0 – 29)
(a) No attempt to answer
Targeted A Students (51 – 69)
(b) Confuse between rate of change and
small changes.
(a) Teach how to get at least one mark.
Suggestion
(b)• Observe the unit given  ms-1
 rate of change
• List out questions involving the subtopics and
ask students to identify the exact subtopic.
Triogonometric
Functions
Targeted A Students (51 – 69)
(a) Do not give all the values of θ
especially involving 0°.
Suggestion
(a)(i) Use the R Q R A steps.
- Get the Reference angle,
- Determine the Quadrant
- find new Range (if neccessary)
- Get All angles
(ii) For 0° 
Normal
Distribution
Weak Students (0 – 29)
(a) No attempt to answer.
Excellent Students (70 – 100)
(b) Difficulty answering question
that gave the probability and ask to find
Z or X.
(a) Emphasise on the use of calculator to find the
probability.
(b) Remind students of the two types of normal
distribution questions – Type 1 and Type 2.
Type 1: Given: X Z P
Type 2: Given: P Z X
Permutation
&
Combination
(a) Confuse between using P and C. (a) Try both methods.
Statistics Weak Students (0 – 29)
(a) Unable to use the formula given
correctly.
(a) Ask students to copy the required formula of the
statistics, before answering the question.
4
0 90 18
0
27
0
360
sin 0 1 0 -1 0
cos 1 0 -1 0 1
tan 0 ∞ 0 ∞ 0
formula Use Calculator
Use Tables formula

5. Topic Weakness / Common Mistakes Rectification / Reminder
PAPER 2
Topic Weakness / Common Mistakes Rectification / Reminder
Simultaneous
Equations
Weak Students (0 – 29)
(a) Unable to complete the solutions.
Average Students (30 – 50)
& Targeted A Students(30 – 69)
(b) Forget to find the other unknown
(c) Give answer not as instructed by the
questions, eg, give to three decimal places
(a) Emphasise on the steps where marks are
given.
- identify the linear equation
- get one unknown in terms of the other.
- Substitute into the non-linear
- simplify to get the general form
(RHS = 0)
- factorise or use formula
- Get the values of the unknowns
- get the values of the other unknowns.
Suggestion
Drill students to
(b) write their final answer as:
when x1 = ? , y2 = ?
x2 = ? , y2 = ?
(c) highlight the instructions for the answers.
Quadratic
Equations
Weak Students (0 – 29)
a) Weak in factorization
- Without calc: Unable to factorise correctly.
- With calc:  did not show the factors
 write Quadratic Equation
that cannot be factorise in the
factor form: (x – 0.234)
(a) Ask students to use the calculator
- After getting the solutions, press the
fraction button: [Shift  d/c]
• if it changes to fraction  it can be
factorise, thus, show the factors.
• if no changes  cannot be factorise,
thus, use the formula
Progression (a) In GP, problems finding r if there are two
equations to be solve simultaneously.
(b) In problem solving questions, confuse
when to use T, S or S∞.
(a) Remind students to divide the equations.
In this way fewer calculations are
required.
(b) A lot of practice is needed for problem
solving.
Suggestion: S∞ is used if the question does
not stated the value for n.
Statistics Weak Students (0 – 29)
(a) Histogram: - Draw with incorrect axis
- Forget how to get the mode
- Unable to read the value.
(a) Drilling
5

6. Topic Weakness / Common Mistakes Rectification / Reminder
(b) Unable to use the formula for median
correctly wrong median class.
 use the F from the median class
(b) Drilling
Coordinate
Geometry
Average Students (30 – 50)
(a) Find Area, forget,
i. to close the points: A B C A
ii. the constant  included in the formula.
(b) Problems to find the equations of line.
Targeted A Students (51 – 69)
(c) Unable to find the equation of locus.
Average Students (30 – 50)
& Targeted A Students (51 – 69)
(d) Locus: Make mistake when giving the ratio
of the distance.
(e) Locus: When solving, it is required to
square both sides to get rid of the square
roots  students forget to square the
constants too.
(b) Drill students with finding equations of
parallel line and perpendicular lines.
- use formula y = mx + c
or y – y1 = m(x – x1)
(c) Remind students to apply the distance
formula whenever the word ‘locus’ is seen.
(d) Convert the ratio into fraction, eg,
AB : AC = 2 : 5 
5
2
=
AC
AB
or 5AB = 2AC
Normal
Distribution
Excellent Students (70 – 100)
(a) Difficulty solving problems when the
probability is given. Example
P(Z > k) = 0.7, find k.
Suggestion:
(a) (i) sketch the graph
(ii) remember these shortcuts.
Integrations Average Students (30 – 50)
(a) When given the gradient function :
Students make mistake such as:
(i) use equation of the straight line to find
the equation of the curve.
(ii) differentiate the gradient function to
get the gradient of tangent.
(a) Emphasise on
(i) the term gradient function
(ii) y
∫ dxdx
dy
dx
dy
6
inequality
Value of
probability
Value of k
> > 0.5 –
> < 0.5 +
< > 0.5 +
< < 0.5 –

7. Topic Weakness / Common Mistakes Rectification / Reminder
Linear Law Weak Students (0 – 29)
(a) Do not show tables for the new axes.
(b) Best-fit line does not touch the y-axis.
Average Students (30 – 50)
& Targeted A Students (51 – 69)
(c) Mistakes when finding the value of x given
y or v.v.  do not use the axes given.
(d) Values in tables are not given to at least 2
decimal places.
(e) Do not use points on the graph to find the
gradient  students tend to take any 2
points from the table without checking
whether the points are on the line or not.
When doing revision, emphasise on where
the marks are given.
For weaker students, make sure that they
are able to do part(a) of the questions.
Trigonometric
Functions
Targeted A Students (51 – 69)
(a) Unable to get the equation of the required
straight line to solve the equation given.
(a) Give more practice.
- Use the method of simultaneous equation
- Separate trigo expression from the non
trigo expression.
Linear
Programming
Weak Students (0 – 29)
(a) Unable to interpret the problem given into
mathematical inequality.
(b) Weak in drawing y = mx.
Average Students (30 – 50)
(c) Unable to get the correct shaded region 
Example for x > 2y  students shade the
upper region.
(a) Emphasise on the terms: at least, at most,
not more than, not less than, exceeds by,
twice, three times, …
(b) Remind students to read from the
direction of positive y.
x > 2y can be written as 2y < x
Read from positive-y, thus, it is the lower
region.
** Two ways to find the max / min value
It is recommended to use the points of the
corners of R, and substitute into the
optimum equation.
The End
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