37756909 yearly-plan-add-maths-form-4-edit-kuching-1

Week

NoLearning Objectives

Pupils will be taught to.....

Learning Outcomes Pupils will be able to…

No of Periods

Suggested Teaching & Learning activities/Learning Skills/Values

Points to Note

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

First Term

1 1. Understand the concept of relations.

1.1 Represent relations using

1 arrow diagrams

2 ordered pairs

3 graphs

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.

1

1

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

Skill : Interpretation, observe connection between domain, co domain, object, image and range of a relation.

Discuss the idea of set and introduce set notation.

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 vice versa.

1

1

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.

“ ” is read as “function f maps x to 2x”.

“ ”is read

as “2x is the image of x under the function f”.

Include examples of functions that are not mathematically based.Examples of functions include

1

Yearly Plan – Additional Mathematics Form 4

WeekNo

Learning ObjectivesPupils will be taught to.....


No of Periods


Points to Note

algebraic (linear and quadratic), trigonometric and absolute value. Define and sketch absolute value functions.

2 3. Understand the concept of composite functions.

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 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

2

WeekNo



No of Periods


Points to Note

3 4. Understand the concept of inverse functions.

4.1 Find the object by inverse mapping given its image and 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 , find

Limit to algebraic functions. Exclude inverse of composite

functions.

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

Topic A2 : Quadratic Equations ---3 weeks

41. Understand the

concept of quadratic equations and their roots.

1.1 Recognise a quadratic equation and express it in general form.

1. 2 Determine whether a given value is the root of a quadratic equation by

4 substitution;

a) inspection.

1.3 Determine roots of quadratic equations by trial and improvement method.

1

1

Use graphing calculators or computer software such as the Geometer’s Sketchpad and spreadsheet to explore the concept of quadratic equations

Values : Logical thinkingSkills : seeing connection, using trial and error methods

Questions for 1..2(b) are given in the form of ; a and b are numerical values.

3

WeekNo



No of Periods


Points to Note

5

2. Understand the concept of quadratic equations.

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 equation from given roots.

1

1

2

If x = p and x = q are the roots, then the quadratic equation is

, that is

.Involve the use of:

and

where α and β are roots of the quadratic equation

Skills : Mental process, trial and error

Discuss when , hence

or . Include cases when p = q.

Derivation of formula for 2.1c is not required.

6

3. Understand and use the conditions for quadratic equations to have

a) two different roots;b) two equal roots;c) no roots.punca berbeza;

3.1 Determine types of roots of quadratic equations from the value of .

3.2 Solve problems involving in quadratic

equations to:

a) find an unknown value;

b) derive a relation.


2

2

2

Giving quadratic equations with the following conditions :

, and ask pupils to find out the type of roots the equation has in each case.

Values: Making conclusion, connection and comparison

Explain that “no roots” means “no real roots”.

4

WeekNo



No of Periods


Points to Note

Topic A3 : Quadratics Functions---3 weeks

71. Understand the

concept of quadratic functions and their graphs.

1.1 Recognise quadratic functions

1 1) Use graphing calculators or computer software such as Geometer’s Sketchpad to explore the graphs of quadratic functions.

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

* pedagogy : ConstructivismSkills : making comparison

& making conclusion

1.2 Plot quadratic function graphs:

a)based on given tabulated values;

1 b) 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 of graphs of quadratic functions.

1

Discuss the form of graph if a > 0 and a < 0 for

Explain the term parabola.

81.4 Relate the position of

quadratic function graphs with types of roots for

.

2 Recall the type of roots if :

a)b2 – 4ac > 0b) b2 – 4ac < 0c) b2 – 4ac = 0

Relate the type of roots with the position of the graphs.

5

WeekNo



No of Periods


Points to Note

2. Find the maximum and minimum values of quadratic functions.

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

2

Use graphing calculators or dynamic geometry software such as the Geometer’s Sketchpad to explore the graphs 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 or dynamic geometry software such as 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 and two 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 use the concept of quadratic inequalities.

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

2Use graphing calculators or dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of quadratic inequalities.

Emphasise on sketching graphs and use of number lines when necessary.

6

WeekNo



No of Periods


Points to Note

Topic A4: SIMULTANEOUS EQUATIONS---2 weeks

101. Solve

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

1.1 Solve simultaneous equations using the substitution method.

4 Use graphing calculators or dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of simultaneous equations.Value: systematicSkills: interpretation of mathematical problem

Limit non-linear equations up to second degree only.

111.2Solve simultaneous

equations involving real-life situations.


2

2

Use examples in real-life situations such as area, perimeter and others.

Pedagogy: Contextual Learning Values : Connection between mathematics and other subjects

Topic G1. Coordinate Geometry---5 weeks

121. Find distance

between two points.

1.1 Find the distance between two points , using formula

1 Skill : Use of formula

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

7

WeekNo



No of Periods


Points to Note

2.Understand the concept of division of line segments

2.1Find the midpoint of two given points.

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

1

2

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.

13 3 Find areas of polygons.

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

3.2 Find the area of a triangle by using formula.

3.3 Find the area of a quadrilateral 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:

is not required.Emphasise that when the area of polygon is zero, the given points are collinear.

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

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

1

1

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

Skills : drawing relevant diagrams, using formula, recognising relationship, compare and contrast.

8

WeekNo



No of Periods


Points to Note

14 straight line using the x-intercept and y-intercept

4.4 Find the equation of a straight line given:

a) gradient and one point;

b) two points;

c) x-intercept and y-intercept.

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

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

4.7Find the point of intersection of two lines.

2

1

1

Values : Neat & systematicPedagogy: contextual learning

Finding point of intersection of two lines by solving simultaneous equations

Answers for learning outcomes 4.4(a) and 4.4(b) must be stated in the simplest form.Involve changing the equation into

gradient and intercept form

155. Understand and

use the concept of parallel and perpendicular lines.

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

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

1

1

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

Skill: Use of formula; making comparison

Emphasise that for parallel lines:.

9

WeekNo



No of Periods


Points to Note

16

5.3 Determine whether two straight lines are perpendicular when the gradients of both lines are known 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.

1

1

2

Students to be exposed to SPM exam type of questions.

Values : hard work, cooperative

Pedagogy : Mastery learning

Emphasise that for perpendicular lines

.

Derivation of is not required.

6 Understand and use the concept of 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 is constant;

b) the ratio of the distances of a moving point from two fixed points is constant

6.2 Solve problems involving loci.

1

1

2

Use examples of real-life situations to explore equation of locus involving distance between two points.Use graphic calculators and dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of parallel and perpendicular lines.

Value : Patience, hard workingPedagogy: contextual learningSkill : drawing relevant diagrams

10

WeekNo



No of Periods


Points to Note

Topic T1: Circular Measures---3 weeks

171. Understand the

concept of radian.

1.1 Convert measurements in

radians to degrees and vice versa.

1 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 learningValue : Accurate, making conclusion.

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

ii) radius; and

iii) angle subtended at the centre of a circle

based on given information.

2 Use examples of real-life situations to explore 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 lengths discussed

Emphasize that the angle must be in radian.Students can also use formula

S= when the angle

given is in degree

18

2.2 Find perimeter of segments of circles.

Solve problems involving lengths of arcs.

1

1

11

WeekNo



No of Periods


Points to Note

19

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

3.1 Determine the:

a) area of sector;

b)radius; and

c)angle subtended at the centre of a circle

based on given information.

3.2 Find the area of segments of circles.

3.3 Solve problems involving areas of sectors.

2

2

2

Deriving the formula L= ½ j2 θUsing ratioSkill : drawing relevant diagrams , recognising relationship & making conclusion Value : Systematic & logical

Emphasize that the angle must be in radian.Area of major sektor need to be discussedStudents can also use formula

L= if the angle given

is in degree.

Topic A5 : INDICES AND LOGARITHMS---4 weeksSecond Term

1 1. Understand and

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

1.1 Find the value of numbers given in the form of:

i. integer indices.

ii. fractional indices.

1.2 Use laws of indices to find the value of numbers in index form that are multiplied, divided or raised to 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 : ConstructivismSkill : making inference, use of lawsValue : systematic, logical thinking

Discuss zero index and negative indices.

12

WeekNo



No of Periods


Points to Note

1.3 Use laws of indices to simplify algebraic expressions

1

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

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

2.2 Find logarithm of a number

1 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

is undefined;b) logarithm of zero is undefined.Discuss cases where the given number is in:a) index formb) numerical form.

22.3 Find logarithm of

numbers by using laws of logarithms

2.4 Simplify logarithmic expressions to the simplest form.

2

1

Activities : DemonstrationValue : systematic and organised

Skill : recognising pattern and relationship, application of laws

Discuss laws of logarithms

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

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

1 Aktivities : DemonstrationPedagogy: Mastery learning, problem solving

Discuss:

33.2 Solve problems involving

the change of base and laws of logarithms.

2Aktivities : DemonstrationPedagogy: Mastery learning, problem solving.

13

WeekNo



No of Periods


Points to Note

4. Solve equations involving indices and logarithms

4.1 Solve equations involving indices.

2 Aktivities : Demonstration

Pedagogy: Mastery learning

, problem solving.

Equations that involve indices and logarithms are limited to equations with single solution only. Solve equations involving indices by: a) comparison of indices and

bases;b) using logarithms.

4. 4.2 Solve equations involving

logarithms.

Additional/reinforcement Exercises on this topic

2

2

Values : Systematic & logical thinking

Topic S1: Statistics ---4 Weeks

51 Understand and

use the concept of measures of central tendency to 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

2

1

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 compound2). Marks for Add maths in the class.

Values : Cooperative; honest , logical thinkingSkill : classification, making conclusion

Pedagogy : 1. Contextual learning

Discuss grouped data and ungrouped data.

Involve uniform class intervals only.

Derivation of the median formula is not required.

14

WeekNo



No of Periods


Points to Note

6

1.8 Estimate the median of grouped data from an ogive

1.9 Determine the effects on mode, median and mean for a set of data when:

i) each data is changed uniformly;

ii) extreme values exist;

iii) certain data is added or removed

1.10 Determine the most suitable

measure of central tendency for given data.

1

2

1

2. Constructivism3. Multiple intelligence

Skills : Classification; observing relationship, course and effect, able to analise and make conclusion

Ogive is also known as cumulative frequency curve.

Involve grouped and ungrouped data

72. Understand and

use the concept of measures of dispersion to solve problems.

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

1 Activities : 1. Teacher gives real life examples where values of mean, mode adn medium are more or less the same and not sufficient to determine the consistency of the data and that lead to the need of finding measures of dispersion

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

1Values :1. Honest2. cooperative

Determine the upper and lower quartiles by using the first principle.

15

WeekNo



No of Periods


Points to Note

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

2.6Determine the variance of

a)ungrouped data;

b)grouped data.2.7 Determine the standard

deviation of:

ii) ungrouped data

grouped data.

1

2

Pedagogy : Contextual learning

82.8 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;

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 contrast2. Classification3. Problem Solving4. Sorting data from small to big

Pedagogy : Contextual learning

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

16

WeekNo



No of Periods


Points to Note

Topic AST1: SOLUTION OF TRIANGLES---2 weeks

91. Understand and

use the concept of sine rule to solve 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.

1

1

1

1

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

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

Skill : Interpretation of problemValue : Accuracy

Include obtuse-angled triangles

10

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

2.1 Verify cosine rule.2.2 Use cosine rule to find unknown sides or angles of a triangle.2.3 Solve problems involving cosine rule.2.4Solve problems

involving sine and cosine rules

1

1

2

Use dynamic geometry software such as the Geometer’s Sketchpad to explore 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

11 3. Understand and use the formula for areas of triangles to

3.1 Find the areas of triangles 1 Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of areas of

17

WeekNo



No of Periods


Points to Note

solve problems.using the formula

or its equivalent

3.2.Solve problems involving three- dimensional objects.


2

1

triangles.

Use dynamic geometry software such as the Geometer’s Sketchpad to explore the concept of areas of triangles.Skills : Recognising RelationshipAnalising data Use examples of real-life situations to explore area of triangles.

Value : Systematic

Topic ASS1: INDEX NUMBER---1 week

121. 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.

1

1

Use examples of real-life situations to explore index numbers.Skill : Analise, problem solvingValue : Systematic Q0 = Quantity at base time.

Q1 = Quantity at specific time.

13

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

2.1 Calculate composite index.2.2 Find index number or weightage given relevant information.

2.3 Solve problems involving index number and composite index.

Additional Exercises or past year questions

2

2

2

Use examples of real-life situations to explore composite index. Eg Composite index of share.

Skill : Analise, problem solvingValue : Systematic

Explain weightage and composite index.

18

WeekNo



No of Periods


Points to Note

Topic K1 : Differentiation---5 Weeks

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

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

1.2Find the gradient of a chord joining two points on a curve.

1.3Find the first derivative of a function , as the gradient of tangent to its graph.

1.3 Find the first derivative of polynomials using the first principles.

1.4 Deduce the formula for first derivative of the function

by induction.

1

1

2

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

Skills : Logical Thinking, relationship, application of rules, making inference, making deduction

Pedagogy : Constructivism

Activities : Explanation & demonstration

Idea of limit to a function can be illustrated using graphs.

The concept of first derivative of a function is explained as a tangent to a curve can be illustrated using graphs.

Limit to ;

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

Notation of is equivalent to

when ,

read as “f prime x”.

15

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

2.1 Determine the first derivative of the function

using formula.2.2 Determine value of the

first derivative of the function for a given value of x.

1

1

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

19

WeekNo



No of Periods


Points to Note

&

16

2.3Determine first derivative of a function involving:

a) addition, or

b) subtraction of algebraic terms.

2.4Determine the first derivative of a product of two polynomials.

2.5 Determine the first derivative of a quotient of two polynomials.

2.6Determine the first derivative of composite function using chain rule.

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

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

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

1

1

1

1

1

1

Activities : Explanation and demonstration by teacher

Limit cases in Learning Outcomes 2.7 through 2.9 to rules introduced in 2.4 through 2.6.

20

WeekNo



No of Periods


Points to Note

173. Understand and

use the concept of maximum and minimum values to solve problems.

3.1 Determine coordinates of turning points of a curve.

3.2 Determine whether a turning point is a maximum or a minimum point.

3.3 Solve problems involving maximum or minimum values.

2

1

Use graphing calculators or dynamic geometry software to explore the concept of maximum and minimum values Pedagogy : Constructivism

Skills : Interpretation of problem; Application of approprate method/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 and use the concept of rates of change to solve problems.

4.1 Determine rates of change for related quantities.

1 Use graphing calculators with computer base ranger to explore the concept of rates of change.Skills : Interpretation of problem; Application of approprate method/formula

Limit problems to 3 variables only.

18

5. Understand and

use the concept of small changes and approximations to solve problems.

5.1 Determine small changes in quantities

5.2 Determine approximate values using differentiation.

1 Skills : Interpretation of problem; Application of approprate method/formula

Exclude cases involving percentage change.

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

6.1 Determine the second derivative of .

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

1

1

Mathematical logic

Value : systematic problem solving

Introduce as or

21

37756909 yearly-plan-add-maths-form-4-edit-kuching-1

Documents

image of x

x p x q

function f maps x

roots of aif x

image of functions

concept of functions

examples of functions

algebraic functions