Add Maths Form 4

7/27/2019 Add Maths Form 4

1/23

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Learning Objectives

Pupils will be taught

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Learning Outcomes

Pupils will be able to

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Suggested Teaching & Learning

activities/Learning Skills/Values

Points to Note

Topic/Learning Area Al : FUNCTION --- 3 weeks

First Term

1 1. Understand theconcept of relations.

1 Represent relations using

2 arrow diagrams

3 ordered pairs

4 graphs

5 Identify domain, co domain,

object, image and range of arelation.

1.3 Classify a relation shownon a mapped diagram as: oneto one, many to one, one tomany or many to manyrelation.

1

1

Use pictures, role-play and

computer software to introduce the

concept of relations.

Skill : Interpretation, observe

connection between domain, codomain, object, image and range ofa relation.

Use of daily life examples

Values : systematic

Discuss the idea of set and introduce

set notation.

Emphasis :

(a) f(x) as image

(b) x as object

2. Understand the

concept of functions.

2.1 Recognise functions as a

special relation..

2.2 Express functions using

function notation.

2.3 Determine domain, object,

image and range of a

function.

2.4 Determine the image of a

function given the object and

1

1

Give examples of finding imagesgiven the object and vice versa.

(a) Given f : x 4x x2. Findimage of 5.

(b) Given function h : x 3x 12. Find object with image =

0.

Use graphing calculators and

computer software to explore the

image of functions.

Represent functions using arrowdiagrams, ordered pairs or graphs,

e.g.

( ) xxfxxf 2,2: = xxf 2: is read as function

fmapsx to 2x.

( ) xxf 2= is read as 2x isthe image ofx under the function

f.

Include examples of functions that

are not mathematically based.

Examples of functions include

1

Yearly Plan Additional Mathematics Form 4


2/23

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Points to Note

vice versa. algebraic (linear and quadratic),trigonometric and absolute value.

Define and sketch absolute value

functions.

2 3. Understand theconcept of

compositefunctions.

3.1 Determine composition of

two functions.

3.2 Determine the image of

composite functions given

the object and vice versa

3.3 Determine one of the

functions in a given

composite function given the

other related function.

1

1

2

Use arrow diagrams or algebraicmethod to determine composite

functions.

Give examples of finding imagesgiven the object and vice versa for

composite functions

For example :Given f : x 3x 4. Find(a) ff(2),

(b) range of value of x if ff(x) > 8.

Give examples for finding afunction when the composite

function is given and one other

function is also given.

Example :

Given f : x 2x 1. find functiong ifa. The composite function fg is

given as fg : x 7 6xb. composite function gf is given as

gf : x 5/2x.

Involve algebraic functions only.

Images of composite functions

include a range of values. (Limit to

linear composite functions).

Define composite functionsStudents do not need to find ff(x)

first then substitute x=2.

3 4. Understand theconcept of inverse

4.1 Find the object by inverse

mapping given its image and

Limit to algebraic functions. Exclude inverse of composite

2


3/23

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Points to Note

functions. function.

4.2 Determine inverse functions

using algebra.

4.3 Determine and state the

condition for existence of an

inverse function

Additional Exercises

1

1

1

1

Use sketches of graphs to show the

relationship between a function and

its inverse.

Examples :

Given f: x 23 + x , find1f

functions.

Emphasise that the inverse of afunction is not necessarily a

function.

Topic A2 : Quadratic Equations ---3 weeks

41. Understand the

concept ofquadraticequations andtheir roots.

1.1 Recognise a quadraticequation and express it ingeneral form.

1. 2 Determine whether agiven value is the root of a

quadratic equation by6 substitution;

a) inspection.

1.3 Determine roots ofquadratic equations bytrial and improvementmethod.

1

1

Use graphing calculators orcomputer software such as theGeometers Sketchpad andspreadsheet to explore the conceptof quadratic equations

Values : Logical thinkingSkills : seeing connection, usingtrial and improvement method.

Questions for 1..2(b) are given in

the form of ( ) ( ) 0=++ bxax ; aand b are numerical values.

3


4/23

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Points to Note

5

2. Understand the

concept ofquadraticequations.

2.1 Determine the roots of a

quadratic equation by

a) factorisation;

b) completing the

square

c) using the formula.

2.2 Form a quadratic equationfrom given roots.

1

1

2

Ifx =p andx = q are the roots, then

the quadratic equation is( ) ( ) 0= qxpx , that is

( ) 02 =++ pqxqpx .Involve the use of:

b

a + = and

a

c=

where andare roots of thequadratic equation

02 =++ cbxax

Skills : Mental process, trial andimprovement method

Discuss when

( ) ( ) 0= qxpx , hence0=px or 0=qx .

Include cases whenp = q.

Derivation of formula for 2.1c isnot required.

6

3. Understand anduse the conditionsfor quadraticequations to have

a) two different roots;

b) two equal roots;c) no roots.

a)dua punca berbeza;

3.1 Determine types of roots of

quadratic equations from the

value of acb 42 .

3.2 Solve problems

involving acb 42

inquadratic equations to:a) find an unknown value;

b) derive a relation.


2

2

2

Giving quadratic equations with the

following conditions : 042 > acb04

2 = acb , 042


5/23

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Topic A3 : Quadratics Functions---3 weeks

71. Understand the

concept of

quadraticfunctions andtheir graphs.

1.1 Recognise quadraticfunctions

1 1) Use graphing calculators or

Geometers Sketchpad to explore the

graphs of quadratic functions.

a) f(x) = ax2 + bx + c

b) f(x) = ax2 + bx

c) f(x) = ax2 + c

pedagogy : Constructivism

Skills : making comparison

& making conclusion

1.2 Plot quadratic functiongraphs:

a)based on giventabulated

values;

1b) by tabulating values

2 based on given functions.

2

1) Use examples of everyday

situations to introduce graphs of

quadratic functions.

Contextual learning

1.3 Recognise shapes ofgraphs of quadratic

functions.1

Discuss the form of graph if

a > 0 and a < 0 for

( ) cbxaxxf ++= 2

Explain the term parabola.

81.4 Relate the position of

quadratic function graphs 2Recall the type of roots if :

a)b2 4ac > 0 Relate the type of roots with

5


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with types of roots for

( ) 0=xf .

b) b2 4ac < 0

c) b2 4ac = 0

the position of the graphs.

2. Find themaximum andminimum valuesof quadraticfunctions.

2.1 Determine the maximumor minimum value of aquadratic function bycompleting the square.

2

Use graphing calculators or dynamicgeometry software such as theGeometers Sketchpad to explore thegraphs of quadratic functions

Skills : mental process ,interpretation

Students be reminded of the steps

involved in completing square and

how to deduce maximum or

minimum value from the function

and also the corresponding values of

x.

9 3. Sketch graphs of

quadratic functions.

3.1 Sketch quadratic function

graphs by determining the

maximum or minimum point

and two other points.

2 Use graphing calculators ordynamic geometry software suchas the Geometers Sketchpad toreinforce the understanding ofgraphs of quadratic functions.

Steps to sketch quadratic graphs:

a) Determining the form or

b) finding maximum or minimum

point and axis of symmetry.

c) finding the intercept with x-axis

and y-axis.

d) plot all points

e) write the equation of the axis of

symmetry

Emphasise the marking ofmaximum or minimum point andtwo other points on the graphsdrawn or by finding the axis ofsymmetry and the intersection with

they-axis.Determine other points by finding

the intersection with thex-axis (if it

exists).

4. Understand and use

the concept of

4.1 Determine the ranges of

values ofx that satisfies 2

Use graphing calculators or

dynamic geometry software such as

Emphasise on sketching graphs and

use of number lines when necessary.

6


7/23

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Points to Note

quadratic inequalities. quadratic inequalities. the Geometers Sketchpad to

explore the concept of quadratic

inequalities.

Topic A4: SIMULTANEOUS EQUATIONS---2 weeks

101. Solve

simultaneous

equations in twounknowns: one

linear equation

and one non-linear equation.

1.1 Solve simultaneousequations using the

substitution method.

4 Use graphing calculators or

Geometers Sketchpad to explore the

concept of simultaneous equations.

Value: systematic

Skills: interpretation of mathematical

problem

Revise through solving simultaneous

linear equations before entering into

second degree equations.

Limit non-linear equations up to

second degree only.

111.2Solve simultaneous

equations involving real-

life situations.


2

2

Use examples in real-lifesituations such as area, perimeter

and others.

Pedagogy: Contextual LearningValues : Connection between

mathematics and other subjects

Topic G1. Coordinate Geometry---5 weeks

121. Find distance

between twopoints.

1.1 Find the distance between

two points ( )11 , yx ,

( )22 , yx using formula1 Skill : Use of formula

Use the Pythagoras Theorem to find

the formula for distance between two

points.

7


8/23

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2.Understand theconcept ofdivision of linesegments

2.1Find the midpoint of twogiven points.

2.2Find the coordinates of a

point that divides a lineaccording to a given ratio

m : n.

1

2

Skill : Use of formula

Value : Accurate & neat work

Limit to cases where m and n arepositive.Derivation of the formula

++

++

nm

myny

nm

mxnx 2121 ,

is not required.

13 3 Find areas of

polygons.

3.1 Find the area of a trianglebased on the area of

specific geometricalshapes.

3.2 Find the area of a triangle

by using formula.

13

13

21

21

2

1

yy

xx

yy

xx

3.3 Find the area of aquadrilateral using

formula.

1

1

Values : Systematic & neat

Skills : use of formula , recognise

relationship and patterns

Limit to numerical values.Emphasise the relationship between

the sign of the value for area

obtained with the order of the

vertices used.

Derivation of the formula:

++

3123

12133211

2

1

yxyx

yxyxyxyxis not

required.

Emphasise that when the area of

polygon is zero, the given points are

collinear.

8


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14

4 Understand and

use the conceptof equation of astraight line.

4.1 Determine thex-intercept

and they-intercept of a line.

4.2 Find the gradient of astraight line that passes

through two points.

4.3 Find the gradient of astraight line using thex-intercept andy-intercept

4.4 Find the equation of astraight line given:

a) gradient and one point;

b) two points;

c) x-intercept andy-intercept.

4.5Find the gradient and theintercepts of a straight linegiven the equation.

4.6Change the equation of astraight line to the generalform

4.7Find the point ofintersection of two lines.

1

1

1

1

1

Use dynamic geometry software such

as the Geometers Sketchpad toexplore the concept of equation of astraight line.

Skills : drawing relevant diagrams,

using formula, recognisingrelationship, compare and contrast.

Values : Neat & systematic

Pedagogy: contextual learning

Finding point of intersection of twolines by solving simultaneousequations

Answers for learning outcomes4.4(a) and 4.4(b) must be stated inthe simplest form.

Involve changing the equation intogradient and intercept form

9


10/23

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15

5. Understand and

use the conceptof parallel and

perpendicular

lines.

5.1 Determine whether two

straight lines are parallelwhen the gradients of

both lines are known and

vice versa.

5.2 Find the equation of astraight line that passes

through a fixed point andparallel to a given line.

5.3 Determine whether twostraight lines are

perpendicular when thegradients of both lines areknown and vice versa.

5.4 Determine the equation ofa straight line that passesthrough a fixed point and

perpendicular to a givenline.

5.5 Solve problems involvingequations of straight

lines.

1

1

2

Use examples of real-life situations to

explore parallel and perpendicularlines.

Skill: Use of formula; makingcomparison

Students to be exposed to SPMexam type of questions.

Values : hard work, cooperative

Pedagogy : Mastery learning

Emphasise that for parallel lines:

21 mm = .

Emphasise that for perpendicularlines

121 =mm .Derivation of 121 =mm is notrequired.

6 Understand anduse the conceptof equation oflocus involvingdistance

between two

6.1 Find the equation oflocus that satisfies thecondition if:

a)the distance of a moving

point from a fixed point isconstant;

1 Use examples of real-life situations toexplore equation of locus involvingdistance between two points.Use graphic calculators and dynamicgeometry software such as theGeometers Sketchpad to explore the

10


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16

points. b) the ratio of the distancesof a moving point from

two fixed points isconstant

6.2 Solve problems involving

loci.


2

2

concept of parallel and perpendicularlines.

Value : Patience, hard workingPedagogy: contextual learning

Skill : drawing relevant diagrams

Topic T1: Circular Measures---3 weeks

17

1. Understand the

concept ofradian.

1.1 Convert measurements in

radians to degrees and

vice versa.

1 Use dynamic geometry software such

as the Geometers Sketchpad to

explore the concept of circular

measure.

Students measure angle subtended at

the centre by an arc length equal the

length of radius. Repeat with differentradius.

Skill : contextual learning

Value : Accurate, making conclusion.

Discuss the definition of one radian.

rad is the abbreviation of radian.

Include measurements in radians

expressed in terms of .

rad = 1800


the concept of length

of arc of a circle to

solve problems.

2.1 Determine:

i) length of arc;

ii) radius; and

2 Use examples of real-life situations toexplore circular measure.Derivation of S = j by use of ratio or

by deduction using definition of

Major and minor arc lengthsdiscussedEmphasize that the angle must be inradian.

11


12/23

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bulatan iii) angle subtended atthe centre of a circle

based on given information.

radian.Skill : Making conclusion or

deduction, application of formula

Students can also use formula

S= 2360

xj

when the angle

given is in degree

2.2 Find perimeter of

segments of circles.2.3 Solve problems

involving lengths of arcs.

1 Solving problems with help of

diagrams

Value : Accurate

Perimeter of segment

= 2j sin2

+j

18

19

3. Understand and

use the conceptof area of sector

of a circle tosolve problems

3.1 Determine the:

a) area of sector;

b)radius; and

c)angle subtended at thecentre of a circle

based on given information.

3.2 Find the area of segmentsof circles.

3.3 Solve problemsinvolving areas of sectors.


2

2

2

2

Deriving the formula L= j2

Using ratio

Skill : drawing relevant diagrams ,recognising relationship & making

conclusionValue : Systematic & logical

Emphasize that the angle must be in

radian.Area of major sektor need to be

discussedStudents can also use formula

L=2

360

xj

if the angle given is

in degree.

21Area of sector =2

j ,

emphasize that mustbe in radian

Area of segment = ( )21

sin2

j

Topic A5 : INDICES AND LOGARITHMS---4 weeks

Second Term

12


13/23

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1

1. Understand and

use the conceptof indices andlaws of indices

to solveproblems.

1.1 Find the value of numbersgiven in the form of:

i. integer indices.

ii. fractionalindices.

1.2 Use laws of indices to findthe value of numbers inindex form that are

multiplied, divided or raisedto a power.

1

1

Use examples of real-life situations to

introduce the concept of indices.

Use computer software such as the

spreadsheet to enhance the

understanding of indices.

Pedagogy : Constructivism

Skill : making inference, use of laws

Value : systematic, logical thinking

Discuss zero index and negative

indices.

Can show the following

0 m ma a

=

1

m

m

a= =

1.3 Use laws of indices tosimplify algebraicexpressions

1

2. Understand anduse the conceptof logarithmsand laws of

logarithms tosolve problems.

2.1 Express equation in indexform to logarithm formand vice versa.

2.2 Find logarithm of anumber

1 Use scientific calculators to enhance

the understanding of the concept of

logarithm.

Explain definition of logarithm.

N= ax; logaN=x with a > 0, a 1.

Value : systematic, abide by the laws

Pedagogy:Mastery learning

Emphasise that:loga 1 = 0; logaa = 1.

Emphasise that:

a) logarithm of negative numbers isundefined;

b) logarithm of zero is undefined.Discuss cases where the givennumber is in:a) index form

b) numerical form.

13


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2

2.3 Find logarithm of

numbers by using laws oflogarithms

2.4 Simplify logarithmic

expressions to thesimplest form.

2

1

Activities : Demonstration

Value : systematic and organised

Skill : recognising pattern and

relationship, application of laws

Discuss laws of logarithms

3 Understand anduse the change

of base oflogarithms to

solve problems.

3.1 Find the logarithm of anumber by changing the

base of the logarithm to asuitable base.

1 Aktivities : DemonstrationQuestions and answers

Pedagogy: Mastery learning, problem solving

Discuss:

ab

b

alog

1log = ,

loglog

log

ca

c

bb

a=


the change of base andlaws of logarithms.

2

Aktivities : Demonstration

Pedagogy: Mastery learning

, problem solving.

4. Solve equationsinvolvingindices andlogarithms

4.1 Solve equationsinvolving indices.

2 Aktivities : Demonstration

Pedagogy: Mastery learning

, problem solving.

Equations that involve indices andlogarithms are limited to equationswith single solution only.

Solve equations involving indicesby:a) comparison of indices and bases;b) using logarithms.

4. 4.2 Solve equations involving

logarithms.

Additional/reinforcementExercises on this topic

2

2

Values : Systematic & logicalthinking

Topic S1: Statistics ---4 Weeks

14


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5

6

1 Understand and

use the conceptof measures ofcentral tendency

to solveproblems.

1.1 Calculate the mean of

ungrouped data.1.2 Determine the mode of

ungrouped data.

1.3 Determine the median of

ungrouped data

1.4Determine the modal class of

grouped data from frequency

distribution tables.1.5 Find the mode from

histograms.

1.6 Calculate the mean ofgrouped data

1.7 Calculate the median of

grouped data from

cumulative frequency

distribution tables.

1.8 Estimate the median of

grouped data from an ogive

1.9 Determine the effects on

mode, median and mean fora set of data when:

i) each data is changed uniformly;

ii) extreme valuesexist;

iii) certain data is added or

removed

1

2

1

1

2

Use scientific calculators, graphing

calculators and spreadsheets to

explore measures of central tendency.

Students collect data from real-life

situations to investigate measures of

central tendency.

Eg. 1) Length of leaves in school

compound

2). Marks for Add maths in the class.

Values : Cooperative; honest , logical

thinking

Skill : classification, making

conclusion

Pedagogy :

1. Contextual learning

2. Constructivism

3. Multiple intelligence

Use Geometers Sketchpad to showthe effects on mode, median, mean

for a set of data when each data is

changed uniformly

Skills : Classification; observing

relationship, course and effect, able to

analise and make conclusion

Discuss grouped data and ungrouped

data.

Involve uniform class intervals only.

Derivation of the median formula is

not required.

Ogive is also known as cumulative

frequency curve.

Involve grouped and ungrouped data

15


16/23

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Points to Note

1.10 Determine the most suitable

measure of central tendencyfor given data.

1

72. Understand and

use the conceptof measures ofdispersion to

solve problems.

2.1 Find the range ofungrouped data.

2.2 Find the interquartile

range of ungrouped data.

2.2 Find the range of groupeddata

1 Activities :1. Teacher gives real life exampleswhere values of mean, mode adnmedium are more or less the same and

not sufficient to determine theconsistency of the data and that lead

to the need of finding measures ofdispersion

2.3 Find the interquartile range

of grouped data from thecumulative frequency table

2.5 Determine theinterquartile range ofgrouped data from anogive.

2.6Determine the variance of

a)ungrouped data;

b)grouped data.

2.7 Determine the standarddeviation of:

(i) ungrouped data

1

1

2

Values :

1. Honest2. cooperative

Pedagogy : Contextual learning

Determine the upper and lower

quartiles by using the first principle.

16


17/23

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Points to Note

(ii) grouped data.

82.8 Determine the effects onrange, interquartile range,

variance and standarddeviation for a set of data

when:

a) each data is changed

uniformly;

b) extreme values exist;

c) certain data is added or

removed.2.9 Compare measures of

central tendency and

dispersion between two sets

of data.

2

2

Skills :1. Compare and contrast

2. Classification3. Problem Solving

4. Sorting data from small to big

Pedagogy : Contextual learning

Values : Logical thinking Emphasise that comparison between

two sets of data using only measuresof central tendency is not sufficient.

Topic AST1: SOLUTION OF TRIANGLES---3 weeks

91. Understand and

use the conceptof sine rule tosolve problems.

1.1Verify sine rule.

1.2Use sine rule to find

unknown sides or angles of a

triangle.

1.3Find the unknown sides and

angles of a triangle involving

ambiguous case

1.4Solve problems involving the

1

1

1

Use dynamic geometry software suchas the Geometers Sketchpad toexplore the sine rule.

Use examples of real-life situations toexplore the sine rule.

Skill : Interpretation of problem

Value : Accuracy

Include obtuse-angled triangles

17


18/23

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Points to Note

sine rule. 1

10

2. Understand and usethe concept ofcosine rule tosolve problems.

2.1 Verify cosine rule.2.2 Use cosine rule to find

unknown sides or anglesof a triangle.

2.3 Solve problemsinvolving cosine rule.

2.4Solve problemsinvolving sine and

cosine rules

1

1

2

Use dynamic geometry software suchas the Geometers Sketchpad toexplore the cosine rule.

Use examples of real-life situations toexplore the cosine rule.

Acticities : Demonstration

Skill : Interpretation of datas givenValue : Accuracy.

Include obtuse-angled triangles

11 3. Understand and usethe formula forareas of triangles to

solve problems.

3.1 Find the areas of triangles

using the formula

Cab sin2

1or its equivalent

3.2.Solve problemsinvolving three-dimensional objects.


1

2

1

Use dynamic geometry software such

as the Geometers Sketchpad toexplore the concept of areas of

triangles.

Use dynamic geometry software suchas the Geometers Sketchpad toexplore the concept of areas oftriangles.Skills : Recognising Relationship

Analising dataUse examples of real-life situations toexplore area of triangles.

Value : Systematic

Topic ASS1: INDEX NUMBER---1 week

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20/23

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Points to Note

a function ( )xfy = , asthe gradient of tangent to

its graph.

1.3 Find the first derivative of

polynomials using the firstprinciples.

1.4 Deduce the formula for firstderivative of the function

( )xfy = by induction.

2


Activities : Explanation &demonstration

Values : accuracy, systematic,tolerance , patient

a, n are constants, n = 1, 2, 3.

Notation of ( )xf' is equivalent to

dx

dywhen ( )xfy = ,

( )xf' read as fprimex.

15

16


the concept of firstderivative of

polynomial

functions to solve

problems.

2.1 Determine the first

derivative of the function

naxy = using formula.2.2 Determine value of the

first derivative of the

function naxy = for a

given value ofx.

2.3Determine first derivativeofa function involving:

a) addition, or

b) subtractionof algebraic terms.

2.4Determine the firstderivative of a product oftwo polynomials.

2.5 Determine the firstderivative of a quotient of

two polynomials.

1

1

1

1

1

1


Skills : Logical Thinking,relationship, application of rules,making inference, making deductionValue : Logical thinking,Perserverance

Activities : Explanation anddemonstration by teacher

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Points to Note

2.6Determine the firstderivative of composite

function using chain rule.

2.7Determine the gradient of

tangent at a point on acurve.

2.8Determine the equationof tangent at a point on a

curve.

2.9 Determine the equationof normal at a point on a

curve

1

1

Limit cases in Learning Outcomes 2.7through 2.9 to rules introduced in 2.4

through 2.6.

173. Understand and

use the conceptof maximum

and minimumvalues to solve

problems.

3.1 Determine coordinates of

turning points of a curve.

3.2 Determine whether a

turning point is a maximumor a minimum point.


maximum or minimum

values.

2

1

Use graphing calculators or dynamicgeometry software to explore theconcept of maximum and minimum

valuesPedagogy : ConstructivismValue : rational

Skills : Interpretation of problem; Application of appropratemethod/formula

Emphasise the use of first derivative

to determine the turning points.

Limit problems to two variables

only.Exclude points of inflexion.

Limit problems to two variables only

Value : logical thinking

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Week

No

Learning Objectives


to.....

Learning Outcomes


No of

Periods



Points to Note

Carry out project work In carrying out the project work

1.1Define the problem/situation

to be studied.

1.2 State relevant conjectures

1.3 Use problem solving strategies

to solve problems

1.4 Interpret and discuss results.

1.5 Draw conclusions and/or

make generalisations based

on critical evaluation of

results.

1.6 Present systematic and

comprehensive written reports.

Use scientific calculators, graphing calculators or

computer software to carry out project work.

Students are allowed to carry out project work in

groups but written reports must be done

individually.

Students should be given opportunity to give oral

presentation of their project work.

Emphasise the use of Polyas four-

step problem solving process.

Use at least two problem solving

strategies.

Emphasise reasoning and effective

mathematical communication.

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Documents

Add Maths Form 4