Yearly Plan Add Maths f5

8/14/2019 Yearly Plan Add Maths f5

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Ministry ofEducation

Malaysia

Integrated Curriculum for Secondary Schools

YEARLY PLAN

ADDITIONAL MATHEMATICSFORM 5


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A6LEARNINGAREA:

Form 5LEARNING OBJECTIVES

Pupils will be taught toSUGGESTED TEACHING AND

LEARNING ACTIVITIES

LEARNING OUTCOMES

Pupils will be able toPOINTS TO NOTE

1 Understand and use

the concept ofarithmetic

progression.

Use examples from real-life

situations, scientific orgraphing

calculators and computer

software to explore arithmetic

progressions.

(i) Identify characteristics of

arithmetic progressions.

(ii) Determine whether a given

sequence is an arithmetic

progression.

(iii) Determine by using formula:

a) specific terms in arithmetic

progressions,

b) the number of terms in


(iv) Find:

a) the sum of the first n terms of

arithmetic progressions,

b) the sum of a specific number

of consecutive terms of


c) the value ofn, given the sumof the first n terms of


(v) Solve problems involving


Begin with sequences to

introduce arithmetic and

geometric progressions.

Include examples in

algebraic form.

Include the use ofthe

formula Tn

= Sn

Sn

1

Include problems

involving real-life

situations.

1

WEEK

POINTS TO NOTE


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A6LEARNINGAREA:



LEARNING ACTIVITIES

LEARNING OUTCOMES


2 Understand and use the

concept ofgeometric

progression.


situations, scientific orgraphing

calculators; and computer

software to explore geometric

progressions.

(i) Identify characteristics of


(ii) Determine whether a given

sequence is a geometric

progression.

Include examples in

algebraic form.

(iii) Determine by using formula:

a) specific terms in

geometric progressions,

b) the number of terms in


(iv) Find:

a) the sum of the first n terms

of geometric progressions,

b) the sum of a specific number

of consecutive terms ofgeometric progressions,

c) the value ofn, given the sum

of the first n terms of


2

WEEK

POINTS TO NOTE


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1 r

A6LEARNINGAREA:



LEARNING ACTIVITIES

LEARNING OUTCOMES


(v) Find:

a) the sum to infinity of


b) the first term orcommon

ratio, given the sum to infinity

of geometric progressions.

(vi) Solve problems involving


Discuss:

As n , rn 0

thenS =a

S read as sum toinfinity.

Include recurring decimals.

Limit to 2.recu

.r.ring digits

such as 0.3, 0.15,

Exclude:

a) combination of

arithmetic progressions

and geometric

progressions,

b) cumulative

sequences such as,

(1), (2, 3), (4, 5, 6),

(7, 8, 9, 10),

3

WEEK

POINTS TO NOTE


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LEARNING OBJECTIVES


LEARNING ACTIVITIES

LEARNING OUTCOMES

Pupils will be able toPOINTS TO NOTE WEEK


concept of lines of best fit.

2 Apply linear law to non-

linearrelations.


situations to introduce theconcept of linear law.

Use graphing calculators or

computer software such as

Geometers Sketchpad to

explore lines of best fit.

(i) Draw lines of best fitby

inspection of given data.

(ii) Write equations for lines ofbest

fit.

(iii) Determine values ofvariables

from:

a) lines of best fit,

b) equations of lines of best fit.

(i) Reduce non-linear relations to

linearform.

(ii) Determine values ofconstants

of non-linear relations given:


b) data.

(iii) Obtain information from:


b) equations of lines of best fit.

Limit data to linear

relations between two

variables.

A7LEARNING AREA:

Form 5

4


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LEARNING OBJECTIVES


LEARNING ACTIVITIES

LEARNING OUTCOMES


1 Understand and use the Use computer software such as

concept of indefinite Geometers Sketchpad to

integral.explore the concept ofintegration.

(i) Determine integrals by reversing

differentiation.

(ii) Determine integrals of axn, where

a is a constant and n is an integer,

n 1.

(iii) Determine integrals ofalgebraic

expressions.

(iv) Find constants of integration, c, in

indefinite integrals.

(v) Determine equations ofcurves

from functions ofgradients.

(vi) Determine by substitution the

integrals of expressions ofthe

form (ax + b)n, where a and b areconstants, n is an integerand

n 1.

Emphasise constant of

integration.

ydx read as integrationofy with respect tox

Limit integration of

un dx,where u = ax + b.

C2 Form 5

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LEARNING OBJECTIVES


LEARNING ACTIVITIES

LEARNING OUTCOMES


2 Understand and use the Use scientific orgraphing

concept of definite integral. calculators to explore the

concept of definite integrals.

Use computer software and

graphing calculator to explore

areas under curves and the

significance of positive andnegative values ofareas.

Use dynamic computer

software to explore volumes of

revolutions.

(i) Find definite integrals of

algebraic expressions.

(ii) Find areas under curves as the

limit of a sum ofareas.

(iii) Determine areas undercurvesusing formula.

(iv) Find volumes of revolutions when

region bounded by a curve is

rotated completely about the

a) x-axis,

b) y-axis

as the limit of a sum ofvolumes.

(v) Determine volumes ofrevolutions

using formula.

Includeb b

akf(x)dx = kaf(x)dxb a

af(x)dx =

bf(x)dx

Derivation of formulae not

required.

Limit to one curve.

Derivation of formulae not

required.

Limit volumes of

revolution about the

x-axis ory-axis.

C2 Form 5

6


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LEARNING OBJECTIVES

Pupils will be taught to

SUGGESTED TEACHING AND

LEARNING ACTIVITIES

LEARNING OUTCOMES



concept ofvector.


situations and dynamic computer

software such as Geometers

Sketchpad to explore vectors.

(i) Differentiate between vectorand

scalarquantities.

(ii) Draw and label directed line

segments to represent vectors.

(iii) Determine the magnitude and

direction of vectors represented

by directed line segments.

(iv) Determine whether two vectors

are equal.

(v) Multiply vectors by scalars.

Use notations:

Vector:~

a,AB, a , AB.

Magnitude:

|a|, |AB|, |a|, |AB|.~

Zero vector:~0

Emphasise that a zero

vector has a magnitude of

zero.

Emphasise negative

vector:

AB=BA

Include negative scalar.

G2LEARNINGAREA:

Form 5

7


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LEARNING OBJECTIVES


LEARNING ACTIVITIES

LEARNING OUTCOMES



concept of addition and

subtraction ofvectors.

Use real-life situations and

manipulative materials to

explore addition and

subtraction ofvectors.

(vi) Determine whether two vectors

areparallel.

(i) Determine the resultant vectorof

two parallel vectors.

(ii) Determine the resultant vectorof

two non-parallel vectors using:

a) triangle law,

b) parallelogram law.

(iii) Determine the resultant vectorof

three or more vectors using the

polygon law.

Include:

a) collinearpoints

b) non-parallel

non-zero vectors.

Emphasise:

If~

a and~

b are not

parallel and ha = kb, then

h = k= 0.

G2LEARNINGAREA:

Form 5

~ ~

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LEARNING OBJECTIVES



LEARNING ACTIVITIES

LEARNING OUTCOMES



vectors in the Cartesian

plane.

Use computer software to

explore vectors in the Cartesian

plane.

(iv) Subtract two vectors which are:

a) Parallel,

b) non-parallel.

(v) Represent a vector as a

combination of othervectors.

(vi) Solve problems involving

addition and subtraction of

vectors.

(i) Express vectors in the form:

a) x~

i +yj

~

xb) .

y

Emphasise:

a~~b = ~a + (~b)

Relate unit vector i andj~ ~

to Cartesian coordinates.

Emphasise:

1Vector i = and

~ 0

0Vectorj =

~ 1

G2LEARNINGAREA:

Form 5

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LEARNING OBJECTIVES



LEARNING ACTIVITIES

LEARNING OUTCOMES


(ii) Determine magnitudes ofvectors.

(iii) Determine unit vectors in given

directions.

(iv) Add two or more vectors.

(v) Subtract two vectors.

(vi) Multiply vectors by scalars.

(vii) Perform combined operations onvectors.

(viii) Solve problems involving vectors.

For learning outcomes 3.2

to 3.7, all vectors are given

in the form

xxi +yj or .

~ ~ y

Limit combined operations

to addition, subtraction

and multiplication of

vectors by scalars.

G2LEARNINGAREA:

Form 5

10


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T2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES

Form 5Pupils will be taught to

LEARNING ACTIVITIES Pupils will be able toPOINTS TO NOTE

1 Understand the concept

of positive and negative

angles measured in degrees

and radians.


six trigonometric functions

of any angle.



Sketchpad to explore angles in

Cartesian plane.


software to explore

trigonometric functions indegrees and radians.

Use scientific orgraphing

(i) Represent in a Cartesian plane,

angles greater than 360o or 2

radians for:

a) positive angles,

b) negativeangles.

(i) Define sine, cosine and tangent of

any angle in a Cartesian plane.

(ii) Define cotangent, secant and

cosecant of any angle in a

Cartesian plane.

Use unit circle to

determine the sign of

trigonometric ratios.

Emphasise:

sin = cos (90o)

o

calculators to explore

trigonometric functions ofany(iii) Find values of the six

trigonometric functions ofany

cos = sin (90

tan = cot (90o)

)

oangle.

angle.cosec = sec (90 )

(iv) Solve trigonometric equations.

sec = cosec (90o)

cot = tan (90o)

Emphasise the use of

triangles to find

trigonometric ratios for

special angles 30o, 45

oand

60o.

11

WEEK

POINTS TO NOTE


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graphs of sine, cosine and

tangent functions.


situations to introduce graphs

of trigonometric functions.

Use graphing calculators and

dynamic computer software

such as Geometers Sketchpad

to explore graphs of

trigonometric functions.

(i) Draw and sketch graphs of

trigonometric functions:

a) y = c + a sin bx,

b) y = c + a cos bx,

c) y = c + a tan bx

where a, b and c are constants

and b > 0.

(ii) Determine the numberof

solutions to a trigonometric

equation using sketched graphs.

Use angles in

a) degrees

b) radians, in terms of.

Emphasise the

characteristics ofsine,

cosine and tangent graphs.

Include trigonometric

functions involving

modulus.

Exclude combinations of

trigonometric functions.

(iii) Solve trigonometric equations

using drawn graphs.

12

WEEK

POINTS TO NOTE


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basic identities.

Use scientific orgraphing

calculators and dynamic


Geometers Sketchpad to

explore basic identities.

(i) Prove basic identities:

a) sin2A + cos

2A = 1,

b) 1 + tan2A = sec

2A,

c) 1 + cot2A = cosec

2A.

(ii) Prove trigonometric identities

using basic identities.

(iii) Solve trigonometric equationsusing basic identities.

Basic identities are also

known as Pythagorean

identities.

Include learning outcomes

2.1 and 2.2.


addition formulae and

double-angle formulae.



Sketchpad to explore addition

formulae and double-angle

formulae.

(i) Prove trigonometric identities

using addition formulae for

sin (A B), cos (A B) andtan (A B).(ii) Derive double-angle formulae for

sin 2A, cos 2A and tan 2A.

(iii) Prove trigonometric identities

using addition formulae and/or

double-angle formulae.

(iv) Solve trigonometric equations.

Derivation ofaddition

formulae not required.

Discuss half-angle

formulae.

Excludea cosx + b sinx = c,

where c 0.

13

WEEK

POINTS TO NOTE


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LEARNING OBJECTIVESPupils will be taught to

SUGGESTED TEACHING ANDLEARNING ACTIVITIES

LEARNING OUTCOMESPupils will be able to

POINTS TO NOTE WEEK


concept ofpermutation.

Use manipulative materials to

explore multiplication rule.



spreadsheet to explore

permutations.

(i) Determine the total numberof

ways to perform successive

events using multiplication rule.


permutations ofn different

objects.

For this topic:

a) Introduce the concept

by using numerical

examples.

b) Calculators should only

be used afterstudents

have understood the

concept.

Limit to 3 events.

Exclude cases involving

identical objects.

Explain the concept of

permutations by listing all

possible arrangements.

Include notations:

a) n! = n(n 1)(n 2)(3)(2)(1)

b) 0! =1

n! read as n factorial.

S2 Form 5

14


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POINTS TO NOTE WEEK


concept ofcombination.

Explore combinations using

real-life situations and

computersoftware.

(iii) Determine the numberof

permutations ofn different objects

taken rat a time.

(iv) Determine the numberof

permutations ofn different objects

for given conditions.

(v) Determine the numberof

permutations ofn different objectstaken rat a time forgiven

conditions.

(i) Determine the numberof

combinations ofrobjects chosen

from n different objects.


combinations robjects chosen

from n different objects forgivenconditions.

Exclude cases involving

arrangement of objects in

a circle.

Explain the concept of

combinations by listing all

possible selections.

Use examples to illustratenPn

C =r

r

r!

S2 Form 5

15


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POINTS TO NOTE WEEK


concept ofprobability.

Use real-life situations to

introduceprobability.

Use manipulative materials,

computer software, and

scientific orgraphing

calculators to explore the

concept ofprobability.

(i) Describe the sample space ofan

experiment.


outcomes of an event.

(iii) Determine the probability ofan

event.

(iv) Determine the probability oftwo

events:

a) A orB occurring,

b) A andB occurring.

Use set notations.

Discuss:

a) classical probability

(theoreticalprobability)

b) subjectiveprobability

c) relative frequency

probability

(experimental

probability).

Emphasise:

Only classical probability

is used to solve problems.

Emphasise:

P(A B) = P(A) + P(B)P(A B)using Venn diagrams.

S3 Form 5

16


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POINTS TO NOTE WEEK


concept of probability of

mutually exclusive events.


concept of probability of

independent events.

Use manipulative materials and

graphing calculators to explore

the concept of probability of

mutually exclusive events.


simulate experiments involving

probability ofmutually

exclusive events.

Use manipulative materials and

graphing calculators to explore

the concept of probability of

independent events.


simulate experiments involving

probability ofindependent

events.

(i) Determine whether two events are

mutually exclusive.

(ii) Determine the probability oftwo

or more events that are mutually

exclusive.

(i) Determine whether two events are

independent.

(ii) Determine the probability oftwo

independent events.

(iii) Determine the probability ofthree

independent events.

Include events that are

mutually exclusive and

exhaustive.

Limit to three mutually

exclusive events.

Include tree diagrams.

S3 Form 5

17


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POINTS TO NOTE WEEK


concept ofbinomial

distribution.

Use real-life situations to

introduce the concept of

binomial distribution.

Use graphing calculators and

computer software to explore

binomial distribution.

(i) List all possible values ofa

discrete random variable.

(ii) Determine the probability ofan

event in a binomial distribution.

(iii) Plot binomial distribution graphs.

(iv) Determine mean, variance and

standard deviation of abinomial

distribution.

(v) Solve problems involving

binomial distributions.

Include the characteristics

of Bernoulli trials.

For learning outcomes 1.2

and 1.4, derivation of

formulae not required.

S4 Form 5

18


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POINTS TO NOTE WEEK


concept ofnormal

distribution.



statistical packages to explore

the concept ofnormal

distributions.

(i) Describe continuous random

variables using set notations.

(ii) Find probability ofz-values for

standard normal distribution.

(iii) Convert random variable of

normal distributions, X, to

standardised variable, Z.

(iv) Represent probability ofan

event using set notation.

(v) Determine probability of an event.(vi) Solve problems involving

normal distributions.

Discuss characteristics

of:a) normal distribution

graphs

b) standardnormal

distributiongraphs

.

Zis called standardised

variable.

Integration ofnormal

distribution function to

determine probability is

not required.

S4 Form 5

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AST2LEARNING OBJECTIVES SUGGESTED TEACHING AND LEARNING OUTCOMES




the concept of

displacement.

Use real-life examples, graphing

calculators and computer


Sketchpad to explore

displacement.

(i) Identify direction ofdisplacement

of a particle from a fixed point.

(ii) Determine displacement ofa

particle from a fixedpoint.

(iii) Determine the total distance

travelled by a particle over a time

interval using graphical method.

Emphasise the use ofthe

following symbols:

s = displacement

v = velocity

a = acceleration

t= time

where s, v and a are

functions oftime.

Emphasise the difference

between displacement and

distance.

Discuss positive, negative

and zero displacements.

Include the use ofnumber

line.

20

WEEK

POINTS TO NOTE


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concept ofvelocity.Use real-life examples, graphing

calculators and dynamic


Geometers Sketchpad to explore

the concept ofvelocity.

(i) Determine velocity function ofa

particle by differentiation.

(ii) Determine instantaneous velocity

of aparticle.

(iii) Determine displacement ofa

particle from velocity functionby

integration.

Emphasise velocity as the

rate of change of

displacement.

Include graphs ofvelocity

functions.

Discuss:

a) uniform velocity b) zeroinstantaneous

velocity

c) positive velocity

d) negative velocity.

21

WEEK

POINTS TO NOTE


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concept ofacceleration.Use real-life examples and


Geometers Sketchpad to explore

the concept ofacceleration.

(i) Determine acceleration function

of a particle by differentiation.

(ii) Determine instantaneous

acceleration of aparticle.

(iii) Determine instantaneous velocityof a particle from acceleration

function by integration.

(iv) Determine displacement ofa

particle from acceleration

function by integration.

Emphasise acceleration as

the rate of change of

velocity.

Discuss:

a) uniform

acceleration

b) zeroacceleration

c) positive

acceleration

d) negative

acceleration.

(v) Solve problems involving motion

along a straight line.

22

WEEK

POINTS TO NOTE


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POINTS TO NOTE WEEK


concept of graphs of linear

inequalities.

Use real-life examples,

graphing calculators and

dynamic computersoftware

such as Geometers Sketchpad

to explore linearprogramming.

(i) Identify and shade the region on

the graph that satisfies a linear

inequality.

(ii) Find the linear inequality that

defines a shaded region.

(iii) Shade region on the graph that

satisfies several linear

inequalities.

(iv) Find linear inequalities that define

a shaded region.

Emphasise the use ofsolid

lines and dashed lines.

Limit to regions defined

by a maximum of 3 linear

inequalities (not including

thex-axis andy-axis).

ASS2LEARNING AREA:

Form 5

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LEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to

POINTS TO NOTE WEEK


concept oflinear

programming.

(i) Solve problems related to linear

programming by:

a) writing linear inequalities and

equations describing a

situation,

b) shading the region offeasible

solutions,

c) determining and drawing theobjective function ax + by = k

where a, b and kare

constants,

d) determining graphically the

optimum value ofthe

objective function.

Optimum values referto

maximum orminimum

values.

Include the use ofvertices

to find the optimum value.

ASS2LEARNING AREA:

Form 5

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LEARNING ACTIVITIESLEARNING OUTCOMESPupils will be able to

POINTS TO NOTE WEEK

ASS2LEARNING AREA:

Form 5

1 Carry out project work. Use scientific calculators,

graphing calculators orcomputer software to carry outproject work.

Pupils are allowed to carry out

project work in groupsbut

written reports must be done

individually.

Pupils should be given the

opportunities to give oral

presentations of theirproject

work.

(i) Define the problem/situation tobe

studied.

(ii) State relevant conjectures.

(iii) Use problem-solving strategies to

solveproblems.

(iv) Interpret and discuss results.

(v) Draw conclusions and/orgeneralisations based on critical

evaluation ofresults.

(vi) Present systematic and

comprehensive written reports.

Emphasise the use of

Polyas four-step problem-

solvingprocess.

Use at least two problem-

solving strategies.

Emphasise reasoning and

effective mathematical

communication.

25

Yearly Plan Add Maths f5

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