Post on 04-Apr-2018
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7/30/2019 Yearly Plan Add Maths Form 4
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Topic/Learning Area Al : FUNCTION --- 3 weeks
1. Understand the
concept of relations.
1 Represent relations using
2 arrow diagrams
3 ordered pairs
4 graphs
5 Identify domain, co domain,
object, image and range of a
relation.
1.3 Classify a relation shownon a mapped diagram as: oneto one, many to one, one tomany or many to manyrelation.
Use pictures, role-play and
computer software to introduce the
concept of relations.
Skill : Interpretation, observe
connection between domain, codomain, object, image and range of
a 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 theconcept 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
vice versa.
• Give examples of finding images
given the object and vice versa.
(a) Given f : x → 4x – x2. Find
image 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 arrow
diagrams, ordered pairs or graphs,
e.g.
( ) x x f x x f 2,2: =→“ x x f 2: → ” is read as “function
f maps x to 2 x”.
• “ ( ) x x f 2= ” is read as “2 x is the
image of x under the function f ”.
Include examples of functions that
are not mathematically based.
Examples of functions includealgebraic (linear and quadratic),
trigonometric and absolute value.
Define and sketch absolute value
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functions.
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 theother related function.
• Use arrow diagrams or algebraic
method to determine composite
functions.
• Give examples of finding images
given 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 a
function when the composite
function is given and one other
function is also given.
Example :
Given f : x→2x – 1. find function
g if
a. The composite function fg is
given as fg : x →7 – 6x b. 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 functions
Students do not need to find ff(x)
first then substitute x=2.
4. Understand theconcept of inversefunctions.
4.1 Find the object by inverse
mapping given its image and
function.
• Limit to algebraic functions.
• Exclude inverse of composite
functions.
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4.2 Determine inverse functions
using algebra.
4.3 Determine and state the
condition for existence of an
inverse function
Additional Exercises
Use sketches of graphs to show the
relationship between a function and
its inverse.
Examples :
Given f: x 23 +→ x , find1− f
• Emphasise that the inverse of a
function is not necessarily a
function.
Topic A2 : Quadratic Equations ---3 weeks
1. Understand the
concept of quadraticequations andtheir roots.
1.1 Recognise a quadratic
equation and express it ingeneral form.
1. 2 Determine whether agiven value is the root of aquadratic equation by
6 substitution;
a) inspection.
1.3 Determine roots of quadratic equations by
trial and improvementmethod.
Use graphing calculators or
computer software such as theGeometer’s 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=++ b xa x ; aand b are numerical values.
2. Understand theconcept of
quadraticequations.
2.1 Determine the roots of aquadratic equation by
a) factorisation;
b) completing the
If x = p and x = q are the roots, thenthe quadratic equation is
( ) ( ) 0=−− q x p x , that is
( ) 02 =++− pq xq p x .
Discuss when
( ) ( ) 0=−− q x p x , hence 0=− p x
or 0=− q x .
Include cases when p = q.
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square
c) using the formula.
2.2 Form a quadratic equation
from given roots.
Involve the use of:
b
aα β + = − and a
c
=αβ
where α and β are roots of thequadratic equation
02 =++ cbxax
Skills : Mental process, trial andimprovement method
Derivation of formula for 2.1c is
not required.
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 − in
quadratic equations to:
a) find an unknown value;
b) derive a relation.
Additional Exercises
Giving quadratic equations with thefollowing conditions : 042 >− acb
042 =− acb , 042 <− acb
and ask pupils to find out the type of
roots the equation has in each case.
Using Geometer’s Sketchpad to show
the relationship between the values of
acb 42 − and the types of roots
Values: Making conclusion,connection and comparison
Explain that “no roots” means “noreal roots”.
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Topic A3 : Quadratics Functions---3 weeks
1. Understand theconcept of
quadraticfunctions and
their graphs.
1.1 Recognise quadraticfunctions
1) Use graphing calculators or
Geometer’s 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;
1 b) by tabulating values
2 based on given functions.
1) Use examples of everyday
situations to introduce graphs of
quadratic functions.
• Contextual learning
1.3 Recognise shapes of graphs of quadratic
functions.
Discuss the form of graph if
a > 0 and a < 0 for
( ) cbxax x f ++= 2
Explain the term parabola.
1.4 Relate the position of quadratic function graphswith types of roots for
Recall the type of roots if :
a) b2 – 4ac > 0
b) b2 – 4ac < 0
Relate the type of roots with
the position of the graphs.
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( ) 0= x f .c) b2 – 4ac = 0
2. Find themaximum and
minimum valuesof quadratic
functions.
2.1 Determine the maximumor minimum value of a
quadratic function bycompleting the square.
Use graphing calculators or dynamicgeometry software such as the
Geometer’s 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.
3. Sketch graphs of
quadratic functions.
3.1 Sketch quadratic function
graphs by determining the
maximum or minimum point
and two other points.
Use graphing calculators or
dynamic geometry software suchas the Geometer’s Sketchpad to
reinforce the understanding of graphs 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 of
maximum or minimum point andtwo other points on the graphs
drawn or by finding the axis of symmetry and the intersection with
the y-axis.
Determine other points by finding
the intersection with the x-axis (if it
exists).
4. Understand and usethe concept of
quadratic inequalities.
4.1 Determine the ranges of
values of x that satisfies
quadratic inequalities.
Use graphing calculators or
dynamic geometry software such as
the Geometer’s Sketchpad to
Emphasise on sketching graphs and
use of number lines when necessary.
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explore the concept of quadratic
inequalities
1. Solvesimultaneousequations in twounknowns: one
linear equationand one non-
linear equation.
1.1 Solve simultaneousequations using thesubstitution method.
Use graphing calculators or
Geometer’s 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.
1.2Solve simultaneousequations involving real-
life situations.
Additional Exercises
Use examples in real-lifesituations such as area, perimeter
and others.
Pedagogy: Contextual LearningValues : Connection between
mathematics and other subjects
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Topic G1. Coordinate Geometry---5 weeks
1. Find distance between two
points.
1.1 Find the distance between
two points ( )11, y x ,
( )22 , y x using formula
Skill : Use of formula
Use the Pythagoras’ Theorem to find
the formula for distance between two
points.
2.Understand the
concept of division of linesegments
2.1Find the midpoint of two
given points.
2.2Find the coordinates of a point that divides a lineaccording to a given ratiom : n.
Skill : Use of formula
Value : Accurate & neat work
Limit to cases where m and n are positive.
Derivation of the formula
++
++
nm
myny
nm
mxnx 2121 ,
is not required.
3 Find areas of polygons.
3.1 Find the area of a triangle based on the area of specific geometricalshapes.
3.2 Find the area of a triangle
by using formula.
13
13
21
21
2
1
y y
x x
y y
x x
3.3 Find the area of aquadrilateral usingformula.
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
21
y x y x
y x y x y x y xis not
required.
Emphasise that when the area of
polygon is zero, the given points are
collinear.
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4 Understand anduse the conceptof equation of astraight line.
4.1 Determine the x-interceptand the y-intercept of aline.
4.2 Find the gradient of a
straight line that passesthrough two points.
4.3 Find the gradient of astraight line using the x-
intercept and y-intercept
4.4 Find the equation of astraight line given:
a) gradient and one point;
b) two points;
c)x-intercept and y-
intercept.4.5Find the gradient and the
intercepts of a straight linegiven the equation.
4.6Change the equation of astraight line to the generalform
4.7Find the point of intersection of two lines.
Use dynamic geometry software suchas the Geometer’s 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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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 and parallel to a given line.
5.3 Determine whether two
straight lines are perpendicular when thegradients of both lines areknown and vice versa.
5.4 Determine the equation of a straight line that passesthrough a fixed point and
perpendicular to a givenline.
5.5 Solve problems involvingequations of straightlines.
Use examples of real-life situations to
explore parallel and perpendicular lines.
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 perpendicular lines
121
−=mm .
Derivation of 121 −=mm is not
required.
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6 Understand and
use the conceptof equation of locus involving
distance between two
points.
6.1 Find the equation of locus that satisfies the
condition if:
a)the distance of a moving point from a fixed point isconstant;
b) the ratio of the distancesof a moving point fromtwo fixed points isconstant
6.2 Solve problems involvingloci.
Additional Exercises
Use examples of real-life situations to
explore equation of locus involvingdistance between two points.Use graphic calculators and dynamic
geometry software such as theGeometer’s Sketchpad to explore the
concept of parallel and perpendicular lines.
Value : Patience, hard working
Pedagogy: contextual learningSkill : drawing relevant diagrams
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Topic T1: Circular Measures---3 weeks
1. Understand theconcept of
radian.
1.1 Convert measurements in
radians to degrees andvice versa.
Use dynamic geometry software such
as the Geometer’s 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 different
radius.
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 π.
π rad = 1800
2. Understand and use
the concept of length
of arc of a circle to
solve problems.
bulatan
2.1 Determine:
i) length of arc;
radius; and
iii) angle subtended atthe centre of a circle
based on given information.
Use examples of real-life situations toexplore circular measure.Derivation of S = j θ by use of ratio or
by deduction using definition of radian.Skill : Making conclusion or deduction, application of formula
Major and minor arc lengthsdiscussedEmphasize that the angle must be inradian.Students can also use formula
S= 2360
x jπ
°×
°when the angle
given is in degree
2.2 Find perimeter of segments of circles.2.3 Solve problemsinvolving lengths of arcs.
Solving problems with help of diagrams
Value : Accurate
Perimeter of segment
= 2j sin2
θ +jθ
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3. Understand and
use the conceptof area of sector of a circle to
solve 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 problems
involving areas of sectors.
Additional Exercises
Deriving the formula L= ½ j2 θ
Using ratio
Skill : drawing relevant diagrams ,
recognising relationship & makingconclusion
Value : Systematic & logical
Emphasize that the angle must be in
radian.Area of major sektor need to bediscussed
Students can also use formula
L=2
360
x jπ
°×
°if the angle given is
in degree.
21Area of sector =
2 j θ ,
emphasize that mustbe in radianθ
Area of segment = ( )21sin
2 j θ θ −
1. Understand anduse the concept
of indices andlaws of indicesto solve
problems.
1.1 Find the value of numbersgiven in the form of:
integer indices.
fractional indices.
1.2 Use laws of indices to findthe value of numbers inindex form that aremultiplied, divided or raised to a power.
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
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2. Understand and
use the conceptof logarithmsand laws of
logarithms tosolve problems.
2.1 Express equation in index
form to logarithm formand vice versa.
2.2 Find logarithm of anumber
Use scientific calculators to enhance
the understanding of the concept of logarithm.
Explain definition of logarithm.
N = ax; loga N = x with a > 0, a ≠ 1.
Value : systematic, abide by the laws
Pedagogy:Mastery learning
Emphasise that:
loga 1 = 0; loga a = 1.
Emphasise that:
a) logarithm of negative numbers isundefined;
b) logarithm of zero is undefined.Discuss cases where the given
number is in:a) index form
b) numerical form.
2.3 Find logarithm of
numbers by using laws of logarithms
2.4 Simplify logarithmicexpressions to thesimplest form.
Activities : Demonstration
Value : systematic and organised
Skill : recognising pattern andrelationship, application of laws
Discuss laws of logarithms
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2 Understand and
use the changeof base of logarithms to
solve problems.
3.1 Find the logarithm of a
number by changing the base of the logarithm to asuitable base.
Aktivities : Demonstration
Questions and answersPedagogy: Mastery learning, problem solving
Discuss:
ab
b
alog
1log = ,
loglog
log
ca
c
bb
a=
3.2 Solve problems involvingthe change of base andlaws of logarithms.
Aktivities : DemonstrationPedagogy: Mastery learning
, problem solving.
4. Solve equations
involvingindices and
logarithms
4.1 Solve equations
involving indices.
Aktivities : Demonstration
Pedagogy: Mastery learning
, problem solving.
Equations that involve indices andlogarithms are limited to equationswith single solution only.
Solve equations involving indices
by:a) comparison of indices and bases;b) using logarithms.
. 4.2 Solve equations involvinglogarithms.
Additional/reinforcementExercises on this topic
Values : Systematic & logicalthinking
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Topic S1: Statistics ---4 Weeks
1 Understand anduse the conceptof measures of central tendencyto solve
problems.
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 of
grouped 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
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
Discuss grouped data and ungrouped
data.
Involve uniform class intervals only.
Derivation of the median formula is
not required.
Ogive is also known as cumulativefrequency curve.
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1.9 Determine the effects on
mode, median and mean for a set of data when:
i) each data is changed uniformly;
ii) extreme valuesexist;
iii) certain data is added or
removed
1.10 Determine the most suitable
measure of central tendency
for given data.
Use Geometer’s Sketchpad to show
the effects on mode, median, meanfor a set of data when each data is
changed uniformly
Skills : Classification; observing
relationship, course and effect, able to
analise and make conclusion
Involve grouped and ungrouped data
2. Understand anduse the conceptof measures of dispersion tosolve problems.
2.1 Find the range of ungrouped data.
2.2 Find the interquartilerange of ungrouped data.
2.2 Find the range of
grouped data
Activities :1. Teacher gives real life exampleswhere values of mean, mode adnmedium are more or less the same andnot sufficient to determine theconsistency of the data and that leadto the need of finding measures of
dispersion
2.3 Find the interquartilerange of grouped datafrom the cumulativefrequency table
2.5 Determine the
Values :1. Honest2. cooperative
Determine the upper and lower quartiles by using the first principle.
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interquartile range of grouped data from an
ogive.
2.6Determine the variance of
a)ungrouped data;
b)grouped data.
2.7 Determine the standarddeviation of:
(i) ungrouped data
(ii) grouped data.
Pedagogy : Contextual learning
2.8 Determine the effects onrange, interquartile range,variance and standarddeviation for a set of datawhen:
a) each data is changeduniformly;
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.
Skills :1. Compare and contrast2. Classification3. Problem Solving4. Sorting data from small to big
Pedagogy : Contextual learning
Values : Logical thinking Emphasise that comparison betweentwo sets of data using only measures
of central tendency is not sufficient.
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Points to Note
1. Understand anduse 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
sine rule.
Use dynamic geometry software suchas the Geometer’s Sketchpad toexplore the sine rule.
Use examples of real-life situations toexplore the sine rule.
Skill : Interpretation of problemValue : Accuracy
Include obtuse-angled triangles
2. Understand and usethe concept of cosine rule tosolve problems.
2.1 Verify cosine rule.
2.2 Use cosine rule to findunknown sides or anglesof a triangle.
2.3 Solve problemsinvolving cosine rule.
2.4Solve problemsinvolving sine andcosine rules
Use dynamic geometry software suchas the Geometer’s Sketchpad toexplore the cosine rule.
Use examples of real-life situations to
explore the cosine rule.
Acticities : DemonstrationSkill : Interpretation of datas givenValue : Accuracy.
Include obtuse-angled triangles
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3. Understand and usethe formula for areas of triangles to
solve problems.
3.1 Find the areas of triangles
using the formula C ab sin2
1
or its equivalent
3.2.Solve problems
involving three-dimensional objects.
Additional Exercises
Use dynamic geometry software suchas the Geometer’s Sketchpad toexplore the concept of areas of
triangles.
Use dynamic geometry software suchas the Geometer’s Sketchpad to
explore the concept of areas of triangles.
Skills : Recognising RelationshipAnalising data
Use examples of real-life situations toexplore area of triangles.
Value : Systematic
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Topic ASS1: INDEX NUMBER---1 week
1. Understand and use
the concept of index number to
solve problems
1.1 Calculate index number.
1.2 Calculate price index.
Find Q0 or Q 1 given relevant
information.
Use examples of real-life situations to
explore index numbers.Skill : Analise, problem solving
Value : Systematic, thrifty Q0 = Quantity at base time.Q1 = Quantity at specific time.
2. Understand and use
the concept of composite index tosolve problems
2.1 Calculate composite index.
2.2 Find index number or weightage given relevantinformation.
2.3 Solve problems involving
index number and composite
index.Additional Exercises or
past year questions
Use examples of real-life situations to
explore composite index. EgComposite index of share.
Skill : Analise, problem solvingValue : SystematicUse daily life examples:e.g monthly expenditure;national budget; etc
Explain weightage and composite
index using real life examples likemonthly expenditure in bar chart or
pie chart etc
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Points to Note
2. Understand and use
the concept of firstderivative of
polynomial
functions to solve
problems.
2.1 Determine the firstderivative of the function
nax y = using formula.
2.2 Determine value of thefirst derivative of the
function nax y = for a
given value of x.
2.3Determine first derivativeof
a function involving:
a) addition, or
b) subtractionof algebraic terms.
2.4Determine the firstderivative of a product of two polynomials.
2.5 Determine the firstderivative of a quotient of
two polynomials.
2.6Determine the first
derivative of compositefunction using chain rule.
2.7Determine the gradient of
tangent at a point on acurve.
Pedagogy : Constructivism
Skills : Logical Thinking,relationship, application of rules,making inference, making deduction
Value : Logical thinking,Perserverance
Activities : Explanation anddemonstration by teacher
Limit cases in Learning Outcomes 2.7through 2.9 to rules introduced in 2.4through 2.6.
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2.8Determine the equation
of tangent at a point on acurve.
2.9 Determine the equationof normal at a point on a
curve
3. 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.
Use graphing calculators or dynamic
geometry software to explore theconcept of maximum and minimum
valuesPedagogy : Constructivism
Value : 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
4. Understand anduse the concept
of rates of change to solve
problems.
4.1 Determine rates of change for related
quantities.
Value : logical thinkingUse graphing calculators withcomputer base ranger to explore the
concept of rates of change.Skills : Interpretation of problem; Application of appropratemethod/formula
Limit problems to 3 variables only.
5. Understand and
use the concept of
small changes and
approximations to
solve problems.
5.1 Determine small changes in
quantities
5.2 Determine approximatevalues using
Skills : Interpretation of problem; Application of appropratemethod/formulaValue : Accuracy
Exclude cases involving percentagechange.
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differentiation.
6. Understand and use
the concept of
second derivative
to solve problems.
6.1 Determine the second
derivative of ( ) x f y = .
6.2 Determine whether a
turning point is maximum or
minimum point of a curve
using the second derivative
Additional Exercises
Mathematical logic
Value : systematic problem solving
Introduce2
2
dx
yd as
dx
dy
dx
d or
( ) ( )( ) x f dx
d x f ''' =
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PROJECT WORK
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.
Emphasise the use of Polya’s four-step problem solving process.
Use at least two problem solving strategies.
Emphasise reasoning and effective mathematical communication.
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