Teknik Peningkatan Prestasi

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 StudentsDo 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.

PAPER 1

1

Topic Weakness / Common Mistakes Rectification / ReminderFunctions 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)= x

(e) For functions involving fractions, the condition for f(x) MUST be written:

for f(x) = , 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 anyconstants constants constantswith 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

S7

S2

S5

S4

Topic Weakness / Common Mistakes Rectification / ReminderCoordinate Geometry


(a) Weak in perpendicular gradient


(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,

Students tend to do: Y = mX + c x y = 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.


(c) When finding using s=r or A=r2 students tend to multiply the

value of with 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 is in radian.

(d) Remind students of the formulae not given.

** For Area of segment = r2 ( - sin ) (since is in radian, to get sin use calculator to mark it as rad, thus, no need to change to degree Example: If r = 3 and = 1.25 radian

0.5 3 x2 ( 1.25 - sin 1.25 shift Ans 2 ) =

3

(2, 5)

(5, 14)

x y

xMode

AC

Reg LinEnter points

2 5 M+

5 14

Shift 2 Press right arrow twice A =

,

, M+

y-intercept

Shift 2 Press right arrow twice B = gradient

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: [similar

triangle]Differentiatio

nWeak 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.

TriogonometricFunctions

Targeted A Students (51 – 69) (a) Do not give all the values of 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 90 180 270 360sin 0 1 0 -1 0cos 1 0 -1 0 1tan 0 0 0

formula Use Calculator

Use Tables formula

PAPER 2

Topic Weakness / Common Mistakes Rectification / Reminder

Simultaneous Equations


(a) Unable to complete the solutions.


& 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.

SuggestionDrill students to (b) write their final answer as: when x1 = ? , y2 = ? x2 = ? , y2 = ?(c) highlight the instructions for the answers.

Quadratic Equations


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.

(b) Unable to use the formula for median correctly wrong median class.

(a) Drilling

(b) Drilling

5

Topic Weakness / Common Mistakes Rectification / Reminder use the F from the median class

Coordinate Geometry


(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.


& 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

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

6

inequalityValue of

probabilityValue of k

> > 0.5 – > < 0.5 + < > 0.5 + < < 0.5 –

Topic Weakness / Common Mistakes Rectification / ReminderLinear Law Weak Students (0 – 29)

(a) Do not show tables for the new axes.(b) Best-fit line does not touch the y-axis.


& 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.

TrigonometricFunctions

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.


(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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Technology

Teknik Peningkatan Prestasi