Learning Objectives

Suggested Teaching and Learning Activities

Learning Outcomes Points To Note Vocabulary

A1. FUNCTIONSStudents will be taught to :1. Understand the concept of relation .

2. Understand the concept of functions

Use pictures , role-play and computer software to introduce the concept of relations .

Students will be able to :

1.1 Represent a relation using(a) arrow diagrams .(b) ordered pairs .(c) graph .

1.2 Identify domain , co-domain , object , image and range of a relation .

1.3 Classify a relation shown on a mapped diagram as : one to one , many to one , one to many or many to many relation .

2.1 Recognise function as a special relation .

2.2 Express function using function notation .2.3 Determine domain , object image and range of a function .

Discuss the idea of set and introduce set notation .

Represent function using arrow diagrams , ordered pairs of graphs e.g , f : x 2x f(x) = 2x" f : x 2x " is read as " function f maps x to 2x "Include examples of functions that are not mathematically based .Examples of function include algebraic ( linear and quadratic ) , trigonometric and

function relation object image range domain co domain map ordered pair arrow diagram

3. Understand the concept of composite functions

4. Understand the concept of inverse function .

. Use graphing calculate and computer software to explore the image of a function .

. Use arrow diagrams or algebraic method to determine composite functions .

. Use sketches of

2.4 Determine image of a function given the object and vice versa .

3.1 Determine composition of two functions .

3.2 Determine image of composite functions given the other related function .

4.1 Find object by inverse mapping given its image and function .

4.2 Determine inverse function using algebra .

absolute value .Define and sketch absolute value function .

Involve algebraic functions only .

Image of composite function include a range of values ( Limit to linear composite functions )

Limit to algebraic function .Exclude inverse of composite functions .

Emphasise that inverse of a function is not necessarily a function .

composite function

inverse

mapping .

graph to show the relationship between a function and its inverse .

4.3 Determine and state the condition for existence of an inverse function .

A 2 : Quadratic Equations .

1. Understand the concept of quadratic equation and its roots .

2. Understand the concept of quadratic equations .

. Use graphing calculator or computer software such as the Geometer's Sketchpad and spreadsheet to explore the concept of quadratic equations .

1.1 Recognize quadratic equation and express it in general form .

1.2 Determine whether a given value is the root of a quadratic equation by (a) substitution .(b) inspection .

1.3 Determine roots of a quadratic equation by trial and improvement method .

2.1 Determine the roots of a quadratic equation by (a) factorisation .(b) completing the square .(c) using the formula .

Questions for 1.2(b) are given in the form ( x+a)( x+b ) = 0 , a,b are numerical values .

Discuss when ( x-p )( x-q ) = 0hence x-p = 0 or x -q = 0 . Include case when p = q .Derivation of formula for 2.1c is not required .

quadratic equation

general form

root

substitutioninspectiontrial and improvement method .

factorisation

completing the square .

3 Understand and use the conditions for quadratic equations to have(a) two different roots .(b)two equal roots .(c)no roots .

2.2 Form a quadratic equation from given roots .

3.1 Determine types of roots of quadratic equations from the value of b 2 - 4ac .

If x=p and x=q are the roots , then the quadratic equation is ( x-p )( x-q ) = 0 , that isx2 - (p+q)x + pq = 0Involve the use of :

+ = and

=

where and are roots of the quadratic equation ax 2 + bx + c = 0 .

b2 - 4ac > 0b2 - 4ac = 0b2 - 4ac < 0Explain that " no roots " means " no real roots "

discriminant real roots .

A 3 Quadratic

Functions

1. Understand the concept of quadratic functions and their graphs .

2. Find maximum and minimum values of quadratic functions .

3. Sketch graphs of quadratic functions .

. Use graphing calculator or computer software such as Geometer's Sketchpad to explore the graphs of quadratic functions .. Use examples of everyday situation to introduce graph of quadratic functions .

. Use graphing calculator or dynamic geometry software such as the Geometer's Sketchpad to explore the graphs of quadratic functions .

. Use graphing calculator or dynamic geometry software

1.1 Recognise quadratic functions .

1.2 Plot quadratic function graphs (a) based on given tabulated values .(b) by tabulating values based on given functions .

1.3 Recognise shapes of graphs of quadratic functions .

1.4 Relate the position of quadratic function graphs with types of roots for f(x) = 0

2.1 Determine the maximum or minimum value of quadratic function by completing the square .

3.1 Sketch quadratic function graphs by determine the maximum or minimum point and two other points .

Discuss cases where a>0 and a<0 for f(x) = ax2+bx+c=0

Emphasise the marking of maximum or minimum point and two other points on the graphs draw or by finding the axis of symmetry and the intersection with the y-axis .

quadratic function

tabulated values

axis of symmetry

parabola

maximum point

completing the square .

axis of symmetry .

sketchintersectionvertical line quadratic inequalityrangenumber line

4 Understand and use the concept of quadratic inequalities .

such as the Geometer's Sketchpad to reinforce the understanding of graphs of quadratic functions .

. Use graphing calculator or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of quadratic inequalities .

4.1 Determine the range of values of x that satisfies quadratic inequalities .

Determine other points by finding the intersection with the x-axis ( if it exists )

Emphasise on sketching graphs and use number lines when necessary .

A 4 Simultaneous Equation .

1 Solve simultaneous equations in two unknowns : one linear equation and one non-linear equation .

. Use graphing calculator or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of

1.1 Solve simultaneous equations using the substitution method .

Limit non-linear equation up to second degree only .

simultaneous equations

intersetion

substitutions method .

simultaneous equation .. Use examples in real-lift situations such as area , perimeter and others .

1.2 Solve simultaneous equation involving real-life situations .

A 5 Indices and Logarithms .

1. Understand and use the concept of indices and laws of indices to solve problems .

2. Understand and use the concept of logarithms and laws of logarithms to salve problems .

. Use examples of real-life situations to introduce the concept if indices .

. Use computer software such as the spreadsheet to enhance the understanding of indices .

. Use scientific calculator to enhance the understanding of the concept of logarithm .

1.1 Find the value of numbers given in the form of :(a) integer indices .(b) fractional indices .

1.2 Use laws of indices to find the values of numbers in index form that are multiplied , divided or raised or raised to power .

1.3 Use laws of indices to simplify algebraic expressions .

2.1 Express equation in index form to logarithm form and vice versa .

2.2 Find logarithm of a number .

Discuss zero index and negative indices .

Explain definition of logarithm .N = ax : log a N = xwith a > 0 , a 1Emphasise that : log a 1 = 0 : log a a = 1

Emphasise that :(a) logarithm of negative numbers in undefined :(b) logarithm of zero is undefined .

baseinteger indicesfractional indices .index form .raised to a powerlaw of indices

index formlogarithm formlogarithmundefined

3. Understand and use the change of the change of base of logarithm to solve problems .

4. Solve equations involving indices and logarithm .

2.3 Find logarithm of numbers by using laws of logarithms.2.4 Simplify logarithm expressions to the simplest form .

3.1 Find the logarithm of a number by changing the base of the logarithm to a suitable base .

3.2 Solve problem involving the change of base and laws of logarithm .

4.1 Solve equation involving indices .

4.2 Solve equations involving logarithms .

Discuss cases where the given number is in(a) index form .(b) numerical form .

Discuss laws of logarithm .

Discuss :

log a b =

Equation that involve indices and logarithm are limited to equations with single solution only .Solve equations involving indices by ;(a)comparison of indices and base .(b) using logarithms .

G 1 Coordinate Geometry .1 Find distance between two points

. Use examples of real-life situations to find the distance between two points .

1.1 Find distance between two points using formula .

Use Pythagoras theorem to find the formula for distance between two points .

distance midpoint coordinate ratio .

2 Understand the concept of division of a line segment .

3 Find area of polygons .

4 Understand and use the concept of equation of a straight line

. Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of area of polygons .

. Use

for substitution of coordinates into the formula .

. Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of equation of a straight line .

2.1 Find midpoint of two given points .

2.2 Find coordinates of a point that divides a line according to a given ratio m : n

3.1 Find area of a triangle based on the area of specific geometrical shapes .

3.2 Find area of a triangle y using formula .

4.1 Determine the x-intercept and the y-intercept of a line .

4.2 Find the gradient of a straight line that passes through two points .

4.3 Find the gradient of a straight line using the x-intercept and y-intercept .

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

(

)is not required .

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 :

(x1y2 + x2y3 + x3y1 -

x2y1 - x3y2 - x1y3 )is not required .

Emphasise that when the area of polygon is zero , the given point are collinear .

areapolygongeometrical shapequadrilateralvertexverticesclockwiseanticlockwisemoduluscollinear

x-intercepty-interceptgradient .

5 Understand and use the concept of parallel and perpendicular line .

6 Understand and

. Use examples of real-life situation to explore parallel and perpendicular lines .

. Use graphic calculator and dynamic geometry software such as Geometer's Sketchpad to explore the concept of parallel and perpendicular lines .

4.4 Find the equation of a straight line given ;(a)gradient and one point .(b)two points(c)x-intercept and y-intercept .

4.5 Find the gradient and the intercepts of a straight line given the equation .

4.6 Change the equation of a straight line to the general form .

4.7 Find the point of intersection of two line .

5.1 Determine whether two straight lines are parallel when gradients of both lines are know and vice versa .

5.2 Find the equation of a straight line that passes through a fixed point and parallel to a given line .

5.3 Determine whether two straight line are perpendicular when gradients of both are know and vice versa .

5.4 Determine the equation of a straight line that passes through a fixed point and perpendicular to a given line .

5.5 Solve problems involving equations of straight lines .

6.1 Find the equation of locus that satisfies

Answer 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 .

Emphasise that for parallel lines :m 1 = m 2 .

Emphasise that for perpendicular line m 1 m 2 = - 1 .

Derivation of m 1 m 2 = - 1 is not required .

straight linegeneral form intersectiongradient form intercept form

parallelperpendicular

equation of locus .moving point loci .

use the concept of equation of locus involving distance between two points .

. Use examples of real-life situation to explore equation of locus involving distance between two points .

. Use graphic calculator and dynamic software such as the Geometer's Sketchpad to explore the concept of parallel and perpendicular lines .

the condition if :(a)the distance of a moving point from a fixed point is constant .(b)the ratio of the distances of a moving point from two fixed points is constant .

6.2 Solve problems involving loci .

S 1 Statistics

1 Understand and use the concept of measure of central tendency to solve problems .

. 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

1.1 Calculate mean of ungrouped data .

1.2 Determine mode of ungrouped data .

1.3 Determine median of ungrouped data .

1.4 Determine modal class of grouped data from the frequency distribution table .

1.5 Find mode from histogram .

1.6 Calculate mean of grouped data .

1.7 calculate median of grouped data from the cumulative frequency distribution table

1.8 Estimate median of grouped data from an ogive .

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

measure of central tendency meanmodemedianungrouped datafrequency distribution tablemodal classuniform classintervalhistogram midpointcumulative frequencydistribution table

2 Understand and use the concept of measure of disperaion to solve problem .

1.9 Determine the effects on mode , median and mean for a set of data when(a)each data is change uniformly .(b)extreme values exist .(c)certain data is added or removed .

1.10 Determine the most suitable measure of central tendency for given data .

2.1 Find the range of ungrouped data

2.2 Find the interquartile range of ungrouped data .

2.3 Find the range of grouped data

2.4 Find the interquartile range of grouped data from the cumulative frequency table .

2.5 Determine the interquartile range of grouped data from an ogive .

2.6 Determine the variance of(a)ungrouped data .(b)grouped data .

2.7 Determine standard deviation of(a)ungrouped data .(b)grouped data .

2.8 Determine the effects on range . interquartile range , variance and standard deviation for a set of data when (a)each data is changed uniformly .(b)extreme values exist .

curve .

Involving grouped and ungrouped data .

Determine upper and lower quartiles by using the first principle .

ogiverangeinterquartilemeasures of dispersionextreme value lower boundary

standard deviationclass intervalupper quartilelower quartile

variance

(c)certain data is added or removed .

2.9 Compare the measures of central tendency and dispersion between two sets of data .

Emphasise that comparison between two sets of data using only measures of central tendency is not sufficient .

T 1 Circular Measures

1 Understand and concept of radian .

2 Understand and use the concept of arc of a circle to solve problem

3 Understand and use the concept of area of sector of a circle to solve problem

. Use dynamic geometry software such as Geometer's Sketchpad to explore the concept of circular measure .

. Use examples of real-life situations to explore circular measure .

1.1 Convert measurements in radians to degrees and vise versa .

2.1 Determine(a)length of arc (b)radius and(c)angle subtended at the centre of a circle based on given information .

2.2 Find perimeter of segments of circles .

2.3 Solve problem involving lengths of arcs .3.1 Determine(a)area of sector(b)radius and(c)angle subtended at the centre of a circle based on given information .

Discuss the definition of one radian ." rad " is the abbreviation of radian .Include measurements in radians expressed in terms of

radian degree

length of arcangle subtended

circleperimetersegment

areasector

3.2 Find are of segments of circle .

3.3 Solve problems involving area of sectors .

C 1 Differentiation

1 Understand and use the concept of gradients of curve and differentiation .

2 Understand and use the concept of first derivative of polynomial function to solve problems

. Use graphing calculator or dynamic geometry software such as Geometer's Sketchpad to explore the concept of differentiation .

1.1 Determine value of a function when its variable approaches a certain velue .

1.2 Find gradient of a chord joining two point on a curve .

1.3 Find the first derivative of a function y = f(x) as gradient of tangent to its graph .

1.4 Find the first derivative for polynomial using first principles .

1.5 Deduce the formula for first derivative of function y = f(x) by induction .

2.1 Determine first derivative of the function y = ax n using formula .

2.2 Determine value of the first derivative of the function y = ax n for given value of x .

2.3 Determine first derivative of a function involving ,(a)addition , or(b)subtraction

Idea of limit to a function can be illustrated using graph .Concept of first derivative of a function is explained as a tangent to a curve can be illustrated using graphs .Limit to y = ax n , a , n are constant , n = 1,2,3.Notation of f '(x) is

equivalent to when y

= f ' (x) .f ' (x) read as " f prime x "

limittangentfirst derivativegradientinductioncurvefixed point

3 Understand and use the concept of maximum and minimum value to solve problems .

4 Understand and use the concept of rate of change to solve problems .

. Use graphing calculator or dynamic geometry software to explore the concept of maximum and minimum value .

. Use graphing calculator with computer base range to explore the concept of rate of change .

of algebraic terms .

2.4 Determine first derivative of a product of two polynomials .

2.5 Determine first derivative of a quotient of two polynomials .

2.6 Determine first derivative of composite function using chain rule .

2.7 Determine gradient of tangent at a point on a curve .

2.8 Determine equation of tangent at a point on a curve .

2.9 Determine equation of normal at a point on a curve .

3.1 Determine coordinates of turning points of a curve .

3.2 Determine whether a tuning points a maximum or minimum point.

3.3 Solve problem involving maximum or minimum values .

4.1 Determine rates change for related quantities .

Limit cases in learning outcomes 2.7 - 2.9 to rule introduced in 2.4 - 2.6

Emphasise the use of first derivative to determine turning points .Exclude points of inflexion .

Limit problems to two variables only .

Limit problem to 3 variables only .

productquotientcomposite functionchain rulenormal

turning pointminimum pointmaximum point

rates of change

5 Understand and use te concept of small changes and approximations to solve problems .

6 Understand and use the concept of second derivative to solve problems .

5.1 Determine small change in quantities .

5.2 Determine approximate values using differentiation .

6.1 Determine second derivative of function y = f (x) .

6.2 Determine whether a turning point is maximum point of a curve using the second derivative .

Exclude cases involving percentage change .

Introduce as

or f ''(x) =

( f '(x) )

approximation .

second derivative

AST 1 Solution of Triangle

1 Understand and use the concept of sine rule to solve problems .

2 Understand and use the concept of cosine rule to solve problems .

. Use dynamic geometry software such as the Geometer's Sketchpad to explore the sine rule .

. Use example of real-life situations to explore the sine rule .

. Use dynamic geometry software such as Geometer's sketchpad explore the cosine rule .. Use examples of real-life situation to explore the cosine rule .

1.1 Verify sine rule .

1.2 Use sine rule to find unknown sides or angle of a triangle .

1.3 Find unknown sides and angles of triangle in an ambiguous case .

1.4 Solve problem involving the sine rule .

2.1 Verify cosine rule .

2.2 Use cosine rule to find unknown sides or angles of triangle .

2.3 Solve problem involving cosine rule .

2.4 Solve problems involving sine and cosine rules .

Include obtuse-angledtriangles .

Include obtuse-angled triangles .

sine ruleacute angled triangleambiguous

cosine rule

3 Understand and use the formula for area of triangles to solve problems .

. Use dynamic geometry software such as the Geometer's Sketchpad to explore the concept of area of triangles .

. Use examples of real-life situation to explore area of triangle .

3.1 Find area of triangles using formula

ab sin C or its equivalent

3.2 Solve problem involving three-dimensional objects .

three-dimensional .

ASS1 Index Number

1 Understand and use the concept of index number to solve problems .

2 Understand and use the concept of composite index to solve problems .

. Use examples of real-life situation to explore index number

. Use example of real-life situations to explore composite index .

1.1 Calculate index number .

1.2 Calculate price index .

1.3 Find Q 0 or Q 1 given relevant information .

2.1 Calculate composite index .

2.2 Find index number or weightage given relevant information .

2.3 Solve problem involving index number and composite index .

Explain index number.

Q 0 = Quantity at base time

Q 1 + Quantity at specific time .

Explain weightage and composite index .

index numberprice indexquantity at base time .quantity at specific time .

composite indexweightage

PROJECT WORK

1 Carry out project work .

Guidelines in carrying out project :1.When using heuristic methods of

1.1 Define the problem / situation to be studied .

conjectureheuristicsystematic

problem solving or making conjectures or both , students can do the following ,(a)explain simple cases followed by complex cases .(b)State and test conjectures .(c)check result obtained , whenever necessary using different methods .(d)draw conclusions supported by mathematical reasoning .(e)make generalisations with adequate justification .2 Students should be given opportunity to present their research findings in the classroom .

3 Students are encouraged to answer question regarding their research findings

1.2 Use heuristic methods to solve problem / state conjectures and test them .

1.3 Make generalisations based on the results / conclusions .

1.4 Present clear and systematic written reports .

Use at least two heuristic methods .

conclusiongeneralisationmathematicalreasoningjustificationadequate