Learning Area : A6 : Progressions Week Learning Objectives Learning Outcomes Points to note

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1. Understand and use the concept of arithmetic progression.

Suggested Teaching and Learning Activities Use examples from real-life situations, scientific or graphing calculators and computer software to explore arithmetic progressions

1.1 Identify characteristics of arithmetic progressions.1.2 Determine whether a given sequence is an arithmetic progression.1.3 Determine by using formula: a) specific terms in arithmetic progressions; b) the number of terms in arithmetic progressions.1.4 Find: a) the sum of the first n terms of arithmetic progressions. b) the sum of a specific number of consecutive terms of arithmetic progressions. c) the value of n, given the sum of the first n terms of arithmetic progressions.1.5 Solve problems involving arithmetic progressions.

Begin with sequences to introduces arithmetic and geometric progressions.

Include examples in algebraic form

Include the use of formula T n = S n − S 1−n

Include problems involving real-life situations.

2&3 2. Understand and use the concept of geometric progression.

2.1 Identify characteristics of geometric progressions.2.2 Determine whether a given sequence is a geometric progression.2.3 Determine by using formula: a) specific terms in geometric progression, b) the number of terms in geometric progressions.2.4 Find: a) the sum of the first n terms of geometric progressions; b) the sum of a specific number of consecutive terms of geometric progressions. c) the value of n, given the sum of the first n terms of geometric progressions.

Include examples in algebraic form.

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Week Learning Objectives Learning Outcomes Points to note

3&42.5 Find: a) the sum to infinity of geometric progressions b) the first term or common ratio, given the sum to infinity of geometric progressions.

2.6 Solve problems involving geometric progressions.

Discuss :As n → ∞ , r 0→n then

.1 r

as

−=∞

∞S read as “ sum to infinity”. Include recurring decimals. Limit to2 recurring digits such as 0.333…, 0.151515 …

Exclude :a) combination of

arithmetic progressions and geometric progressions.

b) cumulative sequences such as, (1), (2,3), (4,5,6), (7,8,9,10),…

Learning Area : A7 : Linear Law Week Learning Objectives Learning Outcomes Points to note

4 1. Understand and use the concept of lines of best fit.

Suggested Teaching and learning Activities Use examples from real-life situations to introduce the concept of linear law.

1.1 Draw lines of best fit by inspection of given data.1.2 Write equation for lines of best fit..1.3 Determine values of variables from:a) lines of best fit;b) equations of lines of best fit.

Limit data to linear relation between two variables.

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Week Learning Objectives Learning Outcomes Points to note

5-6 2. Apply linear law to non-linear relations.

2.1 Reduce non-linear relations to linear form.2.2 Determine values of constants of non-linear relations given:a) lines of best fitb) data2.3 Obtain information from:a) lines of best fitb) equations of lines of best fit.

Learning Area : C2 : Integration Week Learning Objectives Learning Outcomes Points to note

8-9 1. Understand and use the concept of indefinite integral.

Suggested Teaching and learning Activities Use computer software such as Geometer’s Sketchpad to explore the concept of integration.

Determine integrals by reversing differentiation.1.2 Determine integrals of nax , where a is a constant and n is an integer, 1−≠n .

Determine integrals of algebraic expressions.Find constant of integration, c , in indefinite integrals.Determine equations of curves from functions of gradients.

1.6 Determine by substitution the integrals of the form ( )nbax + , where a and b are constants, n is an integer and 1−≠n .

Emphasise constant of integration.

∫ dxy read as “integration of y with respect to x ”

Limit integration of ∫ dxun ,

where baxu +=

10-11 2. Understand and use the concept of definite integral.

Suggested Teaching and learning Activities Use scientific or graphing calculators to explore the concept of definite integrals.

Use computer software and graphing calculators to explore areas under curves and the significance of positive and negative values of areas.

2.1 Find definite integrals of algebraic expressions.

2.2 Find areas under curves as the limit of a sum of areas.

2.3 Determine areas under curve using formula.

Include

( ) ( )∫∫ =b

a

b

a

dxxfkdxxfk

( ) ( )∫∫ −=a

b

b

a

dxxfdxxf

Derivation of formulae not required.

Limit to one curve

12-13 Use dynamics computer software to explore volumes of revolutions.

2.4 Find volume of revolutions when region bounded by a curve is rotated completely about the a) x-axis

Derivation of formulae not required.

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Week Learning Objectives Learning Outcomes Points to noteb) y-axisas the limit of a sum of volumes

2.5 Determine volumes of revolutions using formula. Limit volumes of revolution about the x-axis or y-axis

Learning Area : G2 : Vectors

Week Learning Objectives Learning Outcomes Points to note

14-17 1. Understand and use the concept of vector

Suggested Teaching and learning Activities

Use examples from real-life situations and dynamic computer software such as Geometer’s sketchpad to explore vectors.

1.1 Differentiate between and scalar quantities.

1.2 Draw and label directed line segments to represent vectors.

1.3 Determine the magnitude and direction of vectors represented by directed line segments.

1.4 Determine whether two vectors are equal.

1.5 Multiply vectors by scalar.

1.6 Determine whether two vectors are parallel.

Use notations :Vector : ,, ABa a, AB

Magnitude :ABa , ,│a│, │AB│

Zero vector : 0

Emphasize that a zero vector has a magnitude of zero.

Emphasize negative vector:

BAAB =−

Include negative scalar

Include :a) Collinear pointsb) Non-parallel non-zero

vectors.Emphasize:

If a and b are not parallel and

bkah = , then h=k=0

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Learning Area : T2 : Trigonometric Functions Week Learning Objectives Learning Outcomes Points to note

27 1. Understand the concept of positive and negative angles measured in degrees and radians.

Suggested Teaching and learning Activities • Use dynamic computer software such as Geometer’s Sketchpad to explore angles in Cartesian plane.

1.1 Represent in a Cartesian plane, angles greater than 360˚ or 2π radians for:a) positive anglesb) negative angles.

28 2. Understand and use the six trigonometric functions of any angle.

Suggested Teaching and learning Activities • Use dynamic computer software to explore trigonometric functions in degrees and radians.

• Use scientific or graphing calculators to explore trigonometric functions of any angle.

2.1 Define sine, cosine and tangent of any angle in a Cartesian plane.

2.2 Define cotangent, secant and cosecant of any angle in a Cartesian plane.

2.3 Find values of the six trigonometric functions of any angle.2.4 Solve trigonometric equations.

Use unit circle to determine the sign of trigonometric ratios.

Emphasise:Sin θ = cos (90 - θ)Cos θ = sin (90˚- θ)Tan θ = cot (90˚- θ)Cosec θ = sec (90˚- θ)Sec θ = cosec (90˚- θ)Cot θ = tan (90˚- θ)

Emphasise the use of triangles to find trigonometric ratios for special angles 30˚, 45˚ and 60˚.

Week Learning Objectives Learning Outcomes Points to note

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283. Understand and use graphs of

sine, cosine and tangent functions.

Suggested Teaching and learning Activities

• Use examples from real-life situations to introduce graphs of trigonometric functions.• Use graphing calculators and dynamic computer software such as Geometer’s Sketchpad to explore graphs of trigonometric functions.

3.1 Draw and sketch graphs of trigonometric functions: a) y = c + a sin bx, b) y = c + a cos bx, c) y = c + a tan bx, where a, b and c are constants and b>0.

3.2 Determine the number of solutions to a trigonometric equation using sketched graphs.

3.3 Solve trigonometric equations using drawn graphs.

Use angles ina) degreesb) radians, in terms of

π .

Emphasise the characteristics of sine, cosine and tangent graphs. Include trigonometric functions involving modulus.

Exclude combinations of trigonometric functions.

294. Understand and use basic

identities.

Suggested Teaching and learning Activities• Use scientific or graphing calculators and dynamic computer software such as Geometer’s Sketchpad to explore basic identities,5. Understand and use addition formulae and double-angle formulae.

Suggested Teaching and learning Activities• Use dynamic computer software such as Geometer’s sketchpad to explore addition formulae and double-angle formulae.

4.1 Prove basic identities:a) sin2 A + cos2 A = 1b) 1 + tan2 A = sec2 Ac) 1 + cot2 A = cosec2 A

4.2 Prove trigonometric identities using basic identities.

4.3 Solve trigonometric equations using basic identities.

5.1 Prove trigonometric identities using addition formulae for sin (A ± B), cos (A ± B) and tan (A ± B).

5.2 Derive double-angle formulae for sin 2A, cos 2A and tan 2A.

5.3 Prove trigonometric identities using addition formulae and/or double-angle formulae.

5.4 Solve trigonometric equations.

Basic identities are also known as Pythagorean identities.

Include learning outcomes 2.1 and 2.2.

Derivation of addition formulae not required.

Discuss half-angle formulae.

ExcludeA cosx + b sinx = c, where c ≠ 0.

Learning Area : A6 : Permutations and Combinations

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Week Learning Objectives Learning Outcomes Points to note

19 1. Understand and use the concept of permutation.

Suggested Teaching and learning Activities • Use manipulative materials to

explore multiplication rule • Use real-life situations and

computer software such as spreadsheet to explore permutations

1.1. Determine the total number of ways to perform successive events using multiplication rule.

1.2 Determine the number of permutations of n different objects.

1.3 Determine the number of permutations of n different objects taken r at a time.

1.4 Determine the number of permutations of n different objects for given conditions

1.5 Determine the number of permutations of n different objects taken r at a time for given conditions

For this topic:a) Introduce to concept by

using numerical examples.

b) Calculators should only be used after students have understood the concept

Limit to 3 events.

Exclude cases involving identical objects.

Explain the concept of permutations by listing all possible arrangements.

Include notation: a) n! = n( n -1)(n -2)…(3)(2)(1)

b) 0! = 1 n ! read as “ n factorial”.

Exclude cases involving arrangement of objects in a circle

Week Learning Objectives Learning Outcomes Points to note

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20 2. Understand and use the concept of combination.

Suggested Teaching and learning Activities Explore combinations using real- life situations and computer software

2.1. Determine the number of combinations of r objects chosen from n different objects.

2.2. Determine the number of combinations of r objects chosen from n different objects for given conditions.

Explain the concept of combinations by listing all possible selections.

Use examples to illustrate

!r

PC r

n

rn =

Learning Area : A7 : Probability Week Learning Objectives Learning Outcomes Points to note

21 1. Understand and use the concept of probability.

Suggested Teaching and learning Activities Use real-life situations to introduce probability.

Use manipulative materials, computer software, and scientific or graphing calculators to explore the concept of probability.

1.1 Describe the sample space of an experiment.

1.2 Determine the number of outcomes of an event.

1.3 Determine the probability of an event.

1.4 Determine the probability of two events:a) A or B occurringb) A and B occurring.

Use set notations.

Discuss:a) classical probability

(theoretical probability)b) subjective probabilityc) relative frequency

probability (experimental probability).

Emphasize:Only classical probability is used to solve problems.Emphasize:P(A ∪ B)= P(A) + P (B) – P(A ∩ B) Using Venn diagrams.

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Week Learning Objectives Learning Outcomes Points to note

22 2. Understand and use the concept of probability of mutually exclusive events.

Suggested Teaching and learning Activities Use manipulative materials and graphing calculators to explore the concept of probability of mutually exclusive events.

Use computer software to simulate experiments involving probability of mutually exclusive events.

2.1 Determine whether two events are mutually exclusive.

2.2 Determine the probability of two or more events that are mutually exclusive.

Include events that are mutually exclusive and exhaustive.

Limit to three mutually exclusive events.

22 3. Understand and use the concept of probability of independent events.

Suggested Teaching and learning Activities Use manipulative materials and graphing calculators to explore the concept of probability of independent events.

Use computer software to simulate experiments involving probability of independent events.

3.1 Determine whether two events are independent.

3.2 Determine the probability of two independent events.

3.3 Determine the probability of three independent events.

Include three diagrams.

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Learning Area : S4 : PROBABILITY DISTRIBUTIONS

Week Learning Objectives Learning Outcomes Points to note 23 1. Understand and use the concept

of binomial distribution.

Suggested Teaching and learning Activities Use real-life situations to introduce the concept of binomial distribution.

1.1 List all possible values of a discrete variable..1.2 Determine the probability of an event in a binomial distribution.1.3 Plot binomial distribution graphs1.4 Determine mean ,variance and standard deviation of a binomial distribution.1.5 Solve problems involving binomial distributions.

Include the characteristics of Bernoulli trials

For learning outcomes 1.2 and 1.4,derivation of formulae not required.

24 2. Understand and use the concept of normal distribution.

Suggested Teaching and learning Activities Use real-life situations and computer software such as statistical packages to explore the concept of normal distributions.

2.1 Describle continuous random variables using set notations.2.2 Find probability of z-values for standard normal distribution.2.3 Convert random variable of normal distributuins,X,to standardized variable,Z 2.4 Represent probability of an event using set notation.2.5 Determine probability of an event2.6 Solve problems involving normal distributions

Discuss characteristics of:(a) normal distribution graphs(b) standard normal distribution graphs.

Z is called standardized variable.

Integration of normal distribution to determine probability is not required.

Learning Area : AST2 – Motion Along A Straight Line

Week Learning Objectives Learning Outcomes Points to note

25 1. Understand and use the concept of displacement.

Suggested Teaching and learning Activities Use examples from real-life situations, scientific or graphing calculators and computer software to explore displacement.

Emphasise the use of the following symbols:

s= displacementv= velocitya= acceleration t = time

where s, v and a are functions of time

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Week Learning Objectives Learning Outcomes Points to note

1.1 Identify direction of displacement of a particle from fixed point.

1.2 Determine displacement of a particle from a fixed point.

1.3 Determine the total distance traveled by a particle over a time interval using graphical method.

Emphasise the difference between displacement and distance.

Discuss positive, negative and zero displacements.

Include the use of number line.

25 2. Understand and use the concept of velocity.

Suggested Teaching and learning Activities Use examples from real-life situations, scientific or graphing calculators and computer software to explore the concept of velocity.

2.1 Determine velocity function of a particle by differentiation.

2.2 Determine instantaneous velocity of a particle.

Emphasise velocity as the rate of change of displacement.Include graphs of velocity functions.

Discuss:a) uniform velocityb) zero instantaneous velocityc) positive velocityd) negative velocity

26 3. Understand and use the concept of acceleration

Suggested Teaching and learning Activities Use examples from real-life situations, scientific or graphing calculators and computer software to explore the concept of acceleration.

3.1 Determine acceleration function of a particle by differentiation.

3.2 Determine instantaneous acceleration of a particle.

3.3 Determine instantaneous velocity of a particle from acceleration function by integration.

3.4 Determine displacement of particle from acceleration function by integration.

3.5 Solve problems involving motion along a straight line.

Emphasise acceleration as the rate of change of velocity.

Discuss:

a) uniform accelerationb) zero accelerationc) positive accelerationd) negative acceleration

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Learning Area : LINEAR PROGRAMMING Week Learning Objectives Learning Outcomes Points to note

301. Understand and use the concept of graphs of linear inequalities.

Suggested Teaching and learning Activities Use examples from real-life situations, graphing calculators and dynamic computer software such as Geometer’s Sketchpad to explore linear programming.

1.1 Identify and shade the region on the graph that satisfies a linear inequality.

1.2 Find the linear inequality that defines a shaded region.

1.3 Shade region on the graph that satisfies several linear inequalities.

1.4 Find linear inequalities that define a shaded region.

Emphasise the use of solid lines and dashed lines.

Limit to regions defined by a maximum of 3 linear inequalities (not including the x-axis and y-axis)

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31-3233-3536-42

43

2. Understand and use the concept of linear programming.

Revision (Paper 1 & Paper 2)TRIAL EXAMINATION

REVISION

SPM EXAMINATION

2.1 Solve problems related to linear programming by:

a) writing linear inequalities and equations describing a situation.

b) shading the region of feasible solutions.

c) determining and drawing the objective function ax + by = k where a, b and k are constants.

d) determining graphically the optimum value of the objective function.

Optimum values refer to maximum or minimum value.

Include the use of vertices to find the optimum value.

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