YEARLY Plan Add Maths f4

8/14/2019 YEARLY Plan Add Maths f4

1/29

22000066


2/29

Ministry ofEducation

Malaysia

Integrated Curriculum for Secondary Schools

YEARLY PLAN

ADDITIONAL MATHEMATICSFORM 4


3/29

A1LEARNING AREA:

Form 4LEARNING OBJECTIVES

Pupils will be taught to

SUGGESTED TEACHING

AND LEARNING ACTIVITIES

LEARNING OUTCOMES

Pupils will be able toPOINTS TO NOTE WEEK

1 Understand the conceptofrelations.

2 Understand the concept

offunctions.

Use pictures, role-play andcomputer software to introduce

the concept ofrelations.

(i) Represent relations using:a) arrow diagrams,

b) orderedpairs,

c) graphs.

(ii) Identify domain, object,

image and range ofa

relation.

(iii) Classify a relation shown on

a mapped diagram as:

one-to-one, many-to-one,

one-to-many ormany-to-many relation.

(i) Recognise functions as a

special relation.

(ii) Express functions using

function notation.

(iii) Determine domain, object,

image and range ofa

function.

Discuss the idea of set andintroduce set notation.

Represent functions using arrow

diagrams, ordered pairs or

graphs, e.g.

f :x 2x

f(x)= 2x

f :x 2x is read as

function fmaps x to 2x.

f(x)= 2x is read as 2x is the

image ofx under the function f.

Include examples offunctions

that are not mathematically

based.

1


4/29

A1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES



of composite functions.


of inverse functions.

Use graphing calculators orcomputer software to explore

the image offunctions.

Use arrow diagrams or

algebraic method to determine

composite functions.

Use sketches of graphs to show

the relationship between a

function and its inverse.

(iv) Determine the image ofafunction given the object

and vice versa.

(i) Determine composition of

two functions.

(ii) Determine the image of

composite functions given

the object and vice versa.

(iii) Determine one ofthefunctions in a given

composite function given

the other related function.

(i) Find the object by inversemapping given its image

and function.

(ii) Determine inverse functions

using algebra.

(iii) Determine and state the

condition for existence ofan

inverse function.

Examples of functions includealgebraic (linear and quadratic),

trigonometric and absolute value.

Define and sketch absolute valuefunctions.

Involve algebraic functions only.

Images of composite functions

include a range of values. (Limit

to linear composite functions).

Limit to algebraic functions.

Exclude inverse ofcomposite

functions.

Emphasise that inverse of a

function is not necessarily a

function.

2


5/29

A2 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand the conceptof quadratic equations and

theirroots.


of quadratic equations.

Use graphing calculators orcomputer software such as the

Geometers Sketchpad and

spreadsheet to explore the

concept of quadratic equations.

(i) Recognise a quadraticequation and express it ingeneral form.

(ii) Determine whether a given

value is the root ofa

quadratic equation by:

a) substitution,

b) inspection.

(iii) Determine roots of

quadratic equations by trial

and improvement method.

(i) Determine the roots ofa

quadratic equation by:

a) factorisation,

b) completing thesquare,

c) using the formula.

.

Questions for 1.2(b) are given in

the form of; a and b are

numerical values.

Discuss when

(xp)(xq) = 0,

hence xp = 0 or xq = 0.

Include cases when p = q.

Derivation of formula for 2.1c isnot required.

3


6/29



SUGGESTED TEACHING


LEARNING OUTCOMES


3 Understand and use the

conditions forquadraticequations to have

a) two different roots;

b) two equal roots;

c) no roots.

(ii) Form a quadratic equationfrom given roots.

(i) Determine types of roots of

quadratic equations from

the value of b2

4ac .

(ii) Solve problems involving

b2 4ac in quadratic

equations to:

a) find an unknown value,

b) derive arelation.

Ifx =p and x = q are the roots,then the quadratic equation is(xp)(xq) = 0, that is

x2 (p + q)x +pq = 0.

Involve the use of:

+=b

and =c

a awhere andare roots ofthe

quadratic equation

ax2

+ bx + c = 0

b2

4ac > 0

b2

4ac = 0

b2 4ac < 0

Explain that no roots means

no real roots.

4


7/29



SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand the conceptof quadratic functions and

theirgraphs.

2 Find the maximum and

minimum values ofquadratic functions.

Use graphing calculators orcomputer software such as

Geometers Sketchpad to

explore the graphs ofquadratic

functions.

Use examples ofeveryday

situations to introduce graphs

of quadratic functions.

Use graphing calculators or

dynamic geometry software

such as Geometers Sketchpad

to explore the graphs of

quadratic functions.

(i) Recognise quadraticfunctions.

(ii) Plot quadratic function

graphs:

a) based on given

tabulated values,

b) by tabulatingvalues

based on given

functions.(iii) Recognise shapes ofgraphs

of quadratic functions.

(iv) Relate the position of

quadratic function graphs

with types of roots for

f(x)= 0 .

(i) Determine the maximum or

minimum value ofa

quadratic function by

completing the square.

Discuss cases where

a > 0 and a < 0 for

f(x) = ax2

+ bx + c

5


8/29



SUGGESTED TEACHING


LEARNING OUTCOMES


3 Sketch graphs ofquadratic functions.


concept ofquadratic

inequalities.

Use graphing calculators ordynamic geometry software

such as the Geometers

Sketchpad to reinforce the

understanding of graphs of

quadratic functions.



such as the GeometersSketchpad to explore the

concept ofquadratic

inequalities.

(i) Sketch quadratic functiongraphs by determining the

maximum orminimum

point and two otherpoints.

(i) Determine the ranges of

values ofx that satisfies

quadratic inequalities.

Emphasise the marking ofmaximum or minimum point and

two other points on the graphs

drawn or by finding the axis of

symmetry and the intersection

with they-axis.

Determine other pointsby

finding the intersection with the

x-axis (if it exists).

Emphasise on sketching graphs

and use of number lines when

necessary.

6


9/29



SUGGESTED TEACHING


LEARNING OUTCOMES


1 Solve simultaneousequations in two unknowns:

one linear equation and onenon-linearequation.



Sketchpad to explore the

concept ofsimultaneous

equations.

Use examples in real-life

situations such as area,

perimeter and others.

(i) Solve simultaneousequations using thesubstitution method.

(ii) Solve simultaneous

equations involving real-life

situations.

Limit non-linear equations up tosecond degree only.

7


10/29



SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand and use theconcept of indices and laws

of indices to solveproblems.


concept of logarithms andlaws of logarithms to solveproblems.

Use examples ofreal-lifesituations to introduce theconcept ofindices.

Use computer software such as

the spreadsheet to enhance the

understanding ofindices.

Use scientific calculators to

enhance the understanding ofthe concept oflogarithms.

(i) Find the values ofnumbersgiven in the form of:

a) integerindices,

b) fractionalindices.

(ii) Use laws of indices to find

the values of numbers in

index form that are

multiplied, divided orraised

to apower.

(iii) Use laws of indices to

simplify algebraic

expressions.

(i) Express equation in index

form to logarithm form and

vice versa.

(ii) Find logarithm of a number.

Discuss zero index and negativeindices.

Explain definition oflogarithm.

N= ax; loga N=x with a > 0,

a 1.

Emphasise that:

loga 1 = 0; loga a = 1.

Emphasise that:

a) logarithm of negative numbers

is undefined;

b) logarithm of zero isundefined.

Discuss cases where the given

number is in:

a) index form,

b) numerical form.

8


11/29



SUGGESTED TEACHING


LEARNING OUTCOMES



change of base oflogarithms to solveproblems.

4 Solve equations

involving indices andlogarithms.

(iii) Find logarithm ofnumbersby using laws oflogarithms.

(iv) Simplify logarithmic

expressions to the simplest

form.

(i) Find the logarithm ofa

number by changing the

base of the logarithm to a

suitable base.


the change of base and laws

oflogarithms.

(i) Solve equations involving

indices.

(ii) Solve equations involvinglogarithms.

Discuss laws oflogarithms.

Discuss:

1loga b =

logb a

Equations that involve indices

and logarithms are limited to

equations with single solution

only.

Solve equations involving

indices by:

a) comparison of indices and

bases,

b) using logarithms.

9


12/29

G1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES


1 Find distance betweentwopoints.


of division of line segments.

3 Find areas ofpolygons.

Use examples ofreal-lifesituations to find the distance between twopoints.

Use dynamic geometry

software such as the


explore the concept of areas of

polygons.

Use1 x1 x2 x3 x4

2 y1 y2 y3 y4

for substitution ofcoordinates

into the formula.

(i) Find the distance betweentwo points using formula.

(i) Find the midpoint oftwo

givenpoints.

(ii) Find the coordinates ofa

point that divides a lineaccording to a given ratio

m : n.

(i) Find the area of a triangle

based on the area ofspecific

geometrical shapes.

(ii) Find the area of a triangle

by using formula.

(iii) Find the area ofa

quadrilateral by using

formula.

Use the Pythagoras Theorem tofind the formula fordistance between twopoints.

Limit to cases where m and n arepositive.

Derivation of the formula

nx1 + mx2 ny1 + my2 , m + n m + n

is not required.

Limit to numerical values.

Emphasise the relationship

between the sign of the value for

area obtained with the orderof

the vertices used.

Derivation of the formula:

x1y1 + x2y3 + x3y1 x2y1 1 2

x y x y

3 2 1 3

is not required.

Emphasise that when the area of

polygon is zero, the givenpoints

are collinear.

10


13/29

G1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES


4 Understand and use theconcept of equation ofa

straight line.

Use dynamic geometrysoftware such as theGeometers Sketchpad to

explore the concept ofequation

of a straight line.

(i) Determine thex-interceptand they-intercept of a line.

(ii) Find the gradient ofa

straight line thatpasses

through twopoints.

(iii) Find the gradient ofa

straight line using thex-intercept andy-intercept.

(iv) Find the equation ofa

straight line given:

a) gradient and onepoint,

b) points,

c) x-intercept and

y-intercept.

(v) Find the gradient and the

intercepts of a straight line

given the equation.

(vi) Change the equation ofastraight line to the general

form.

(vii) Find the point of

intersection of two lines.

Answers for learning outcomes

4.4(a) and 4.4(b) must be stated

in the simplest form.

Involve changing the equation

into gradient and intercept form.

11


14/29

G1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES


5 Understand and use theconcept of parallel and

perpendicular lines.

Use examples ofreal-lifesituations to explore paralleland perpendicular lines.

Use graphic calculator and



to explore the concept of

parallel andperpendicular

lines.

(i) Determine whether twostraight lines areparallel

when the gradients ofboth

lines are known and vice

versa.

(ii) Find the equation ofa


through a fixed point and

parallel to a given line.

(iii) Determine whether two

straight lines are

perpendicular when the

gradients of both lines are

known and vice versa.

(iv) Determine the equation ofa


through a fixed point and

perpendicular to a given

line.

(v) Solve problems involving

equations of straight lines.

Emphasise that for parallel lines:

m1

= m2.

Emphasise that forperpendicular

lines

m1m

2= 1.

Derivation of m1m2 = 1 is not

required.

12


15/29

G1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES


6 Understand and use theconcept of equation oflocus

involving distance between

twopoints.

Use examples ofreal-lifesituations to explore equation

of locus involving distance

between twopoints.

Use graphing calculators and



Sketchpad to explore the

concept of parallel and

perpendicular lines.

(i) Find the equation of locusthat satisfies the condition

if:

a) the distance ofa

moving point from a

fixed point is constant,

b) the ratio of thedistances

of a moving point from

two fixed points is

constant.


loci.

13


16/29

S1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand and use theconcept of measures of

central tendency to solveproblems.

Use scientific calculators,graphing calculators and

spreadsheets to explore

measures of central tendency.

Pupils collect data from real-

life situations to investigate

measures of central tendency.

(i) Calculate the mean ofungrouped data.

(ii) Determine the mode of

ungrouped data.

(iii) Determine the median of

ungrouped data.

(iv) Determine the modal class

of grouped data fromfrequency distribution

tables.

(v) Find the mode from

histograms.

(vi) Calculate the mean of

grouped data.

(vii) Calculate the median ofgrouped data from

cumulative frequency

distribution tables.

(viii) Estimate the median of

grouped data from an ogive.

Discuss grouped data andungrouped data.

Involve uniform class intervals

only.

Derivation of the median

formula is not required.

Ogive is also known as

cumulative frequency curve.

14


17/29



SUGGESTED TEACHING


LEARNING OUTCOMES



concept of measures of

dispersion to solve

problems.

(ix) Determine the effects onmode, median and mean fora set of data when:

a) each data is changeduniformly,

b) extreme valuesexist,

c) certain data is added or

removed.

Involve grouped and ungrouped

data

Determine the upper and lower

quartiles by using the first

principle.

(x) Determine the most suitable

measure of central tendency

for given data.

(i) Find the range ofungrouped

data.

(ii) Find the interquartile range

of ungrouped data.

(iii) Find the range ofgroupeddata.

(iv) Find the interquartile range

of grouped data from the

cumulative frequency table.

15


18/29



SUGGESTED TEACHING


LEARNING OUTCOMES


(v) Determine the interquartilerange of grouped data froman ogive.

(vi) Determine the variance of:

a) ungrouped data,

b) groupeddata.

(vii) Determine the standard

deviation of:

a) ungrouped data,

b) groupeddata.

(viii) Determine the effects onrange, interquartile range,

variance and standard

deviation for a set ofdata

when:

a) each data is changed

uniformly,

b) extreme valuesexist,

c) certain data is added or

removed.

(ix) Compare measures of

central tendency anddispersion between two sets

ofdata.

Emphasise that comparison

between two sets of data usingonly measures ofcentral

tendency is not sufficient.

16


19/29

T1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand the conceptofradian.


concept of length of arc ofacircle to solveproblems.

Use dynamic geometrysoftware such as the


explore the concept ofcircular

measure.

Use examples ofreal-life

situations to explore circular

measure.

(i) Convert measurements inradians to degrees and viceversa.

(i) Determine:

a) length ofarc,

b) radius,

c) angle subtended at thecentre of a circle

based on given information.

(ii) Find perimeter ofsegments

ofcircles.

(iii) Solve problems involving

lengths ofarcs.

Discuss the definition ofoneradian.

rad is the abbreviation of

radian.

Include measurements in radians

expressed in terms of.

17


20/29

T1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


3 Understand and use theconcept of area of sector of

a circle to solve problems.

(i) Determine the:a) area ofsector,

b) radius,

c) angle subtended at the

centre of a circle

based on given information.

(ii) Find the area of segments of

circles.

(iii) Solve problems involvingareas ofsectors.

18


21/29

C1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand and use theconcept of gradients of

curve and differentiation.


concept of first derivative ofpolynomial functions tosolveproblems.



to explore the concept of

differentiation.

(i) Determine the value ofafunction when its variableapproaches a certain value.

(ii) Find the gradient of a chord

joining two points on a

curve.

(iii) Find the first derivative ofa

function y = f(x), as thegradient of tangent to its

graph.

(iv) Find the first derivative of

polynomials using the first

principle.

(v) Deduce the formula forfirst

derivative of the function

y = f(x

)by induction.

(i) Determine the first

derivative of the function

y = axn

using formula.

Idea of limit to a function can beillustrated using graphs.

The concept of first derivative of

a function is explained as a

tangent to a curve and canbe

illustrated using graphs.

Limit to y = axn ;

a, n are constants, n = 1, 2, 3Notation off'(x) is equivalentdyto when y = f(x),

dx

f'(x)read as fprime ofx.

19


22/29



SUGGESTED TEACHING


LEARNING OUTCOMES


(ii) Determine value of the firstderivative of the function

y = axn for a given value of

x.

(iii) Determine first derivative of

a function involving:

a) addition, or

b) subtraction

of algebraic terms.

(iv) Determine the first

derivative of a product oftwopolynomials.

(v) Determine the first

derivative of a quotient of

twopolynomials.

(vi) Determine the first

derivative ofcomposite

function using chain rule.

(vii) Determine the gradient of

tangent at a point on a

curve.

(viii) Determine the equation of

tangent at a point on acurve.

Limit cases in Learning

Outcomes 2.7 through 2.9 to

rules introduced in 2.4 through

2.6.

20


23/29



SUGGESTED TEACHING


LEARNING OUTCOMES



concept of maximum andminimum values to solveproblems.


concept of rates ofchange

to solveproblems.


concept of small changes

and approximations to solveproblems.


dynamic geometry software to

explore the concept ofmaximum and minimum values.

Use graphing calculators with

computer base ranger to

explore the concept of rates of

change.

(ix) Determine the equation ofnormal at a point on acurve.

(i) Determine coordinates of

turning points of a curve.

(ii) Determine whethera

turning point is a maximum

or a minimum point.

(iii) Solve problems involvingmaximum orminimum

values.

(i) Determine rates ofchange

for related quantities.

(i) Determine small changes in

quantities.

(ii) Determine approximate

values using differentiation.

Emphasise the use offirst

derivative to determine the

turningpoints.

Exclude points ofinflexion.

Limit problems to two variablesonly.

Limit problems to 3 variables

only.

Exclude cases involving

percentage change.

21


24/29



SUGGESTED TEACHING


LEARNING OUTCOMES


6 Understand and use theconcept ofsecondderivative to solve

problems.

(i) Determine the second

derivative ofy = f(x).

(ii) Determine whethera

turning point is maximum

or minimum point ofa

curve using the second

derivative.

2

Introduced y

asd dy

ordx

2dx dx

f(x) =d

(f(x))dx

22


25/29

AST1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand and use theconcept of sine rule to solveproblems.


concept of cosine rule tosolveproblems.



explore the sine rule.


situations to explore the sine

rule.

Use dynamic geometry

software such as the


explore the cosine rule.


situations to explore the cosine

rule.

(i) Verify sine rule.

(ii) Use sine rule to find

unknown sides or angles of

a triangle.

(iii) Find the unknown sides and

angles of a triangle

involving ambiguous case.

(iv) Solve problems involving

the sine rule.

(i) Verify cosine rule.

(ii) Use cosine rule to find

unknown sides or angles of

a triangle.


cosine rule.

(iv) Solve problems involving

sine and cosine rules.

Include obtuse-angled triangles.

Include obtuse-angled triangles

23


26/29

AST1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


3 Understand and use theformula for areas oftriangles to solve problems.



explore the concept of areas of

triangles.


situations to explore areas of

triangles.

(i) Find the areas oftrianglesusing the formula1

ab sin Cor its equivalent.2


three-dimensional objects.

24


27/29

ASS1 Form 4LEARNING OBJECTIVES


SUGGESTED TEACHING


LEARNING OUTCOMES


1 Understand and use theconcept of index number tosolveproblems.


concept of composite indexto solveproblems.

Use examples ofreal-lifesituations to explore index

numbers.


situations to explore composite

index.

(i) Calculate index number.

(ii) Calculate price index.

(iii) Find Q0 orQ 1 given

relevant information.

(i) Calculate composite index.

(ii) Find index numberor

weightage given relevant

information.


index numberand

composite index.

Explain index number.

Q0 = Quantity at base time.

Q1 = Quantity at specific time.

Explain weightage andcomposite index.

25


28/29

PW1LEARNING AREA:



SUGGESTED TEACHING


LEARNING OUTCOMES


1 Carry out project work. Use scientific calculators,graphing calculators or

computer software to carry out

project work.

Pupils are allowed to carry out

project work in groups but

written reports must be done

individually.

Pupils should be givenopportunity to give oral

presentation of theirproject

work.

(i) Define the problem/situation tobe

studied.

(ii) State relevant conjectures.

(iii) Useproblem-solving

strategies to solve problems.

(iv) Interpret and discuss results.

(v) Draw conclusions and/or

make generalizations based

on critical evaluation of

results.

(vi) Present systematic and

comprehensive written

reports.

Emphasize the use ofPolyasfour-step problem-solving

process.

Use at least two problem-solving

strategies.

Emphasize reasoning and

effective mathematical

communication.

26


29/29

YEARLY Plan Add Maths f4

Documents

Transcript of YEARLY Plan Add Maths f4