2009 F5 Add Maths Yearly Plan

YEARLY TEACHING PLAN ADDITIONAL MATHEMATICS FORM 5

Minggu Tempoh 1 5/1 – 9/1 2 12/1 – 16/1

Tajuk A6 Progressions

Arithmetic Progressions

A6 Progressions

Geometric Progressions

3

19/1 – 23/1

A6 Progressions

4 5 6 7 8 9

A7 Linear Law A7 Linear Law A7 Linear Law C2 Integration C2 Integration C2 Integration

10 11 12 13

2/2 – 6/2 9/2 – 13/2 16/2 – 20/2 23/2 – 27/2 2/3 – 6/3 9/3 – 13/3 14/3 – 22/3 23/3 – 27/3 30/3 – 3/4 6/4 – 10/4 13/4 – 17/4

14

20/4 – 24/4

T2 Trigonometric Functions

15

27/4 – 1/5

T2 Trigonometric Functions

16 17 18 19

4/5 – 8/5 11/5 – 15/5 18/5 – 22/5 25/5 – 29/5

G2 Vectors G2 Vectors T2 Trigonometric Functions T2 Trigonometric Functions

Catatan MSY PANITIA 1 , 17/1 SEK GANTI

Arithmetic & Geometric Progressions Cuti Tahun Baru Cina Lines of best fit Non-linear relation Non-linear relation UJIAN SELARAS 1 Concept Integrals Area / Volume CUTI PERTENGAHAN PENGGAL PERTAMA concept KELAS INTENSIF PMR / SPM Mag. and dir Vector / equal and parallel UJIAN SELARAS 2 Concept and 6 trigo functions 11/4 SEK GANTI graphs MSY PANITIA 2, 15/4 M’KA BDR SJR Basic identities / Double-angle formulae / Solve trigo equations Basic identities / Double-angle formulae 1/5 CT HR PKJ / Solve trigo equations PEPERIKSAAN PERTENGAHAN TAHUN PEPERIKSAAN PERTENGAHAN TAHUN PEPERIKSAAN PERTENGAHAN TAHUN

Pembetulan Kertas CUTI PERTENGAHAN TAHUN

SEMINAR ROAD 2 SUCCESS

1

Minggu 20 21 22 23 24 25 26 27 28 29

Tempoh 15/6 – 19/6 22/6 – 26/6 29/6 – 3/7 6/7 – 10/7 13/7 – 17/7 20/7 – 24/7 27/7 – 31/7 3/8 – 7/8 10/8 – 14/8 17/8 – 21/8

Tajuk A6 Permutations and Combinations permutations A6 Permutations and Combinations Combinations A7 Probability Probability of mutually S4 Probability Distributions Binomial distribution S4 Probability Distributions Normal distribution S4 Probability Distributions Normal distribution ASS2 Linear Programming Linear inequality ASS2 Linear Programming Graph ASS2 Linear Programming Graph Ujian Selaras 3 CUTI PERTENGAHAN PENGGAL KEDUA

22/8 – 30/8

30 31 32 33 34 35 36 37 38 39 40

31/8 – 4/9 7/9 – 11/9 14/9 – 18/9

Revision Revision Revision

21/9 – 25/9 28/9 – 2/10 5/10 – 9/10 12/10 – 16/10 19/10 – 23/10 26/10 – 30/10 2/11 – 6/11 9/11 – 13/11 16/11 – 20/11

CUTI HARI RAYA AIDIFITRI

Catatan 27 SEK GANTI MSY PANITIA 3

17/8 PELANCARAN BLN PATRIOTIK BENGKEL PANITIA

MSY PANITIA 4 18/9 MAJLIS P’NT BLN PAT

Revision Percubaan SPM Percubaan SPM

PMR

SPM Model Paper

19/10 CUTI P’ISTIWA

SPM Model Paper SPM Model Paper SPM Model Paper SPM Model Paper

CUTI AKHIR TAHUN 2

Learning Area : A6 : Progressions Week

Learning Objectives 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

Week

Learning Objectives 2. Understand and use the concept of geometric progression.

Learning Outcomes 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. Learning Outcomes 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.

Points to note Begin with sequences to introduces arithmetic and geometric progressions. Include examples in algebraic form

Include the use of formula T n = S n − S n −1 Include problems involving reallife situations.

Points to note Include examples in algebraic form.

3

Week

Learning Objectives

Learning Outcomes 2.5 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.

Points to note Discuss : As n → ∞ , r n → 0 then a s∞ = . 1− r 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),…

4

Learning Area : A7 : Linear Law Week

Learning Objectives 1. Understand and use the concept of lines of best fit. Suggested Teaching and learning Activities

Learning Outcomes 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.

Points to note Limit data to linear relation between two variables.

Use examples from real-life situations to introduce the concept of linear law.

Week

Learning Objectives 2. Apply linear law to non-linear relations.

Learning Outcomes 2.1 Reduce non-linear relations to linear form. 2.2 Determine values of constants of non-linear relations given: a) lines of best fit b) data 2.3 Obtain information from: a) lines of best fit b) equations of lines of best fit.

Points to note

5

Learning Area : C2 : Integration Week

Learning Objectives 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.

Week

Learning Outcomes Points to note Emphasise constant of 1.1 Determine integrals by reversing differentiation. 1.2 Determine integrals of ax n , where a is a constant and n is an integration. ∫ y dx read as “integration of integer, n ≠ − 1 . y with respect to x ” 1.3 Determine integrals of algebraic expressions. 1.4 Find constant of integration, c , in indefinite integrals. 1.5 Determine equations of curves from functions of gradients. 1.6 Determine by substitution the integrals of the form ( ax + b ) n , n Limit integration of ∫ u dx , where a and b are constants, n is an integer and n ≠ − 1 . where u = ax + b

Learning Objectives Learning Outcomes 2. Understand and use the 2.1 Find definite integrals of algebraic expressions. concept of definite integral.

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

b

b

∫ k f ( x ) dx = k ∫ f ( x ) dx a

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

Points to note Include

a

b

a

a

b

∫ f ( x ) dx = − ∫ f ( x ) dx 2.2 Find areas under curves as the limit of a sum of areas.

Derivation required.

2.3 Determine areas under curve using formula.

Limit to one curve

2.4 Find volume of revolutions when region bounded by a curve is Derivation required. rotated completely about the a) x-axis b) y-axis as the limit of a sum of volumes

of

of

formulae

formulae

not

not

6

Week

Learning Objectives

Learning Outcomes 2.5 Determine volumes of revolutions using formula.

Points to note Limit volumes of revolution about the x-axis or y-axis

Learning Area : G2 : Vectors Week

Learning Objectives 1. Understand and use the concept of vector

Learning Outcomes 1.1 Differentiate between and scalar quantities.

Vector :

1.2 Draw and label directed line segments to represent vectors. Suggested Teaching and learning Activities Use examples from real-life situations and dynamic computer software such as Geometer’s sketchpad to explore vectors.

Points to note Use notations :

a, AB, a, AB

Magnitude :

a , AB ,│a│, │AB│ 1.3 Determine the magnitude and direction of vectors represented by directed line segments.

1.4 Determine whether two vectors are equal.

Zero vector :

0

Emphasize that a zero vector has a magnitude of zero. Emphasize negative vector:

− AB = BA

1.5 Multiply vectors by scalar.

Include negative scalar

1.6 Determine whether two vectors are parallel.

Include : a) Collinear points b) Non-parallel non-zero vectors. Emphasize: If a and b are not parallel and

h a = k b , then h=k=0

7

Learning Area : T2 : Trigonometric Functions Week

Learning Objectives Learning Outcomes 1. Understand the concept of 1.1 Represent in a Cartesian plane, angles greater than 360˚ or 2 π positive and negative angles radians for: measured in degrees and radians. a) positive angles b) negative angles. Suggested Teaching and learning Activities

Points to note

• Use dynamic computer software such as Geometer’s Sketchpad to explore angles in Cartesian plane.

2. Understand and use the six 2.1 Define sine, cosine and tangent of any angle in a Cartesian trigonometric functions of any plane. angle. 2.2 Define cotangent, secant and cosecant of any angle in a Cartesian plane. Suggested Teaching and 2.3 Find values of the six trigonometric functions of any angle. learning Activities 2.4 Solve trigonometric equations. • Use dynamic computer software to explore trigonometric functions in degrees and radians. • Use scientific or graphing calculators to explore trigonometric functions of any angle.

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 8

triangles to find trigonometric ratios for special angles 30˚, 45˚ and 60˚.

Week

Learning Objectives

Learning Outcomes

3. Understand and use graphs of 3.1 Draw and sketch graphs of trigonometric functions: sine, cosine and tangent a) y = c + a sin bx, functions. b) y = c + a cos bx, c) y = c + a tan bx, where a, b and c are constants and b>0. Suggested Teaching and learning Activities 3.2 Determine the number of solutions to a trigonometric equation • Use examples from real-life using sketched graphs. 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.

Use angles in a) degrees b) radians, in terms of π . Emphasise the characteristics of sine, cosine and tangent graphs. Include trigonometric functions involving modulus.

3.3 Solve trigonometric equations using drawn graphs. Exclude combinations trigonometric functions.

4. Understand and use basic 4.1 Prove basic identities: identities. a) sin2 A + cos2 A = 1 b) 1 + tan2 A = sec2 A c) 1 + cot2 A = cosec2 A Suggested Teaching and learning Activities • Use scientific or graphing calculators 4.2 Prove trigonometric identities using basic identities. and dynamic computer software such as Geometer’s Sketchpad to explore basic identities,

Points to note

of

Basic identities are also known as Pythagorean identities. Include learning outcomes 2.1 and 2.2.

4.3 Solve trigonometric equations using basic identities.

5. Understand and use addition formulae and double-angle 5.1 Prove trigonometric identities using addition formulae for sin Derivation (A ± B), cos (A ± B) and tan (A ± B).

of

addition 9

formulae.

formulae not required. 5.2 Derive double-angle formulae for sin 2A, cos 2A and tan 2A.

Suggested Teaching learning Activities

and

• Use dynamic computer software such as Geometer’s sketchpad to explore addition formulae and doubleangle formulae.

Discuss half-angle formulae. 5.3 Prove trigonometric identities using addition formulae and/or double-angle formulae. Exclude A cosx + b sinx = c, where 5.4 Solve trigonometric equations. c ≠ 0.

Learning Area : A6 : Permutations and Combinations Week

Learning Objectives 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

Learning Outcomes

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 1.4

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

Points to note 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)

1.5

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

b) 0! = 1 n ! read as “ n factorial”. Exclude cases involving arrangement of objects in a circle

10

Week

Learning Objectives 2. Understand and use the concept of combination.

Learning Outcomes 2.1.

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

Suggested Teaching and learning 2.2. Activities Explore combinations using reallife situations and computer software

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

Points to note Explain the concept of combinations by listing all possible selections. Use examples to illustrate n P n Cr = r r!

Learning Area : A7 : Probability Week

Learning Objectives 1. Understand and use the concept of probability.

Learning Outcomes 1.1 Describe the sample space of an experiment. 1.2 Determine the number of outcomes of an event.

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.

1.3 Determine the probability of an event. 1.4 Determine the probability of two events: a) A or B occurring b) A and B occurring.

Points to note Use set notations. Discuss: a) classical probability (theoretical probability) b) subjective probability c) 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.

11

Week

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

Learning Outcomes 2.1 Determine whether two events are mutually exclusive.

Points to note Include events that are mutually exclusive and exhaustive.

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

Limit to three mutually exclusive events.

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

3.1 Determine whether two events are independent.

Include three diagrams.

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

3.3 Determine the probability of three independent 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.

3.2 Determine the probability of two independent events.

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

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Week

Learning Objectives

Learning Outcomes

Points to note

Learning Area : S4 : PROBABILITY DISTRIBUTIONS

Week 1-2

Learning Objectives

Learning Outcomes

Points to note

1. Understand and use 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 graphs

Include the characteristics of Bernoulli trials

Suggested Teaching and

For learning outcomes 1.2 and 1.4,derivation of formulae not

13

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

3-4

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.

1.4 Determine mean ,variance and standard deviation of a binomial distribution. 1.5 Solve problems involving binomial distributions.

required.

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 event 2.6 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 1. Understand and use the concept of displacement.

Learning Outcomes

Points to note Emphasise the use of the following symbols:

14

Week 1-2

Learning Objectives

Learning Outcomes

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

s= displacement v= velocity a= acceleration t = time 1.1 Identify direction of displacement of a particle from fixed point.

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. 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.

Points to note

where s, v and a are functions of time

1.2 Determine displacement of a particle from a fixed point.

Emphasise the difference between displacement and distance.

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

Discuss positive, negative and zero displacements.

2.1 Determine velocity function of a particle by differentiation.

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

2.2 Determine instantaneous velocity of a particle.

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.

Discuss: a) uniform velocity b) zero instantaneous velocity c) positive velocity d) negative velocity Emphasise acceleration as the rate of change of velocity. Discuss: a) uniform acceleration b) zero acceleration c) positive acceleration d) negative acceleration

3.5 Solve problems involving motion along a straight line.

Learning Area : LINEAR PROGRAMMING

Week

Learning Objectives

Learning Outcomes

Points to note 15

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

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

Suggested Teaching and learning Activities

1.2 Find the linear inequality that defines a shaded region.

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

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

2. Understand and use the concept of linear programming.

2.1 Solve problems related to linear programming by:

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)

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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