add maths form 4
TRANSCRIPT
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7/27/2019 Add Maths Form 4
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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
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7/27/2019 Add Maths Form 4
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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
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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.
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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.
Additional Exercises
2
2
2
Giving quadratic equations with the
following conditions : 042 > acb04
2 = acb , 042
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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
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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.
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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.
Additional Exercises
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.
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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.
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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
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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
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points. b) the ratio of the distancesof a moving point from
two fixed points isconstant
6.2 Solve problems involving
loci.
Additional Exercises
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
2. Understand and use
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.
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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.
Additional Exercises
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
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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.
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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=
33.2 Solve problems involving
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
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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
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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.
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(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
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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.
Additional Exercises
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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Learning Outcomes
Pupils will be able to
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activities/Learning Skills/Values
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
Pedagogy : Constructivism
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.
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2. Understand and use
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
Pedagogy : Constructivism
Skills : Logical Thinking,relationship, application of rules,making inference, making deductionValue : Logical thinking,Perserverance
Activities : Explanation anddemonstration by teacher
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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
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.
3.3 Solve problems involving
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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Learning Outcomes
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Suggested Teaching & Learning
activities/Learning Skills/Values
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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