add math t5 2013

YEARLY TEACHING PLAN FOR ADDITIONAL MATHEMATICS FORM 5

WEEK/S LEARNING OBJECTIVESStudents will be taught to…

LEARNING OUTCOMESStudents will be able to…

SUGGESTED TEACHING AND LEARNING ACTIVITIES

VALUES AND POINTS TO NOTE

TEACHING Aids / CCTS

ALGEBRAIC COMPONENT

A6: PROGRESSION

1. Understand and use the concept of arithmetic progression.

Level 11.1 Identify characteristics

of arithmetic progressions.

1.2 Determine whether given sequence is an arithmetic progression.

Level 21.3 Determine by using

formula:a) specific terms in arithmetic

progressions;b) the number of terms in

arithmetic progressions.

Use example from real-life situations, scientific or graphing calculator software to explore arithmetic progressions.

Begin with sequences to introduce arithmetical and geometrical progressions.

Include examples in algebraic form.

Patience and diligence.

Include the use of the formula

1−−= nnn SST

Careful, systematic and rational.

Coloured blocks, blackboard, text book, chards, scientific calculator, work sheet, list of formulae.

Identifying patterns, conceptualizing and characterizing

Evaluating, analyzing and working backwards

1






2. Understand and use the concept of geometric progression.

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.

Level 31.5 Solve problems involving

arithmetic progressions.

Level 12.1 Identify characteristics of

geometric progressions.

2.2 Determine whether a given sequence is a geometric progression.

Use example from real-life situations, scientific or graphing calculator software to explore geometric progressions.

Include problems involving real-life situations.

Rational, careful, independent and co-operation

Include examples in algebraic form.

Independent and co-operation

Problem solving and interpreting

OHP/CD/Worksheet

Difference or similarities and identifying patterns

2






Level 22.3 Determine by using formula:

a) specific terms in geometric progressions,

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.

Level 32.5 Find:

a) the sum to infinity of geometric progressions.

b) the first term or common ratio, given the sum to infinity of geometric progressions.

Careful and systematic

Discuss:As ,∞→n

0→nr then

r

aS

−=∞ 1

nS read as “sum to infinity”.

Include recurring decimals. Limit to 2 recurring digits such as 0.3, 0.15, .…

Evaluating and analyzing

Estimating, finding all possible solution and predicting

3






2.6 Solve problems involving geometric progressions.

Exclude: a) combination of arithmetic progressions and geometric progressions.b) Cumulative sequences such as, (1), (2, 3), (4, 5, 6), (7, 8, 9, 10), …

Rational and systematic

ALGEBRAIC COMPONENT

A7: LINEAR LAW

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

2. Apply linear law to non-

Level 11.4 Draw lines of best fit by

inspection of given data.

Level 21.5 Write equations for

lines of best fit.1.6 Determine values of

variables form:a) lines of best fitb) equations of lines of

best fit.

Level 3

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

Use graphing calculators or computer software such as Geometer’s Sketchpad to explore lines of best fit.

Respect, Fairness,careful

Limit data to linear relations between two variables.

Aids:Teaching Courseware

Geometer’s Sketchpad

Graphing Calculator

CCTS:Identifying Relationship

Drawing Graph

4






linear relations. 2.1 Reduce non-linear relations to linear form.

2.2 Determine value of constants of non-linear relations given:a) lines of best fitb) data

2.3 Obtain information from:c) lines of best fit

b) equations of lines of best fit.

Problem Solving

CALCULUS COMPONENT

C2: INTEGRATION

1. Understand and use the concept of indefinite integral.

Level 11.1 Determine integrals by reversing

differentiation.

1.2 Determine integrals of axn , where a is a constant and n is an integer, n ≠ - 1.

1.3 Determine integrals of algebraic expressions.

1.4 Find constants of integration, c, in indefinite integrals.

Level 2

Use computer software such as Geometer’s Sketchpad to explore the concept of integration.

Systematic

Confidence

Patience

Careful

AID :Colourful White Board Maker,

Text Book,

Chart,

OHP

5






1.5 Determine equations of curves from functions of gradients.

1.6 Determine by substitution the integrals of expressions of the form (ax + b)n , where a and b are constants, n is an integer and

n ≠ -1.

2. Understand and use the concept of definite integral.

Level 22.1 Find definite integrals of algebraic expressions.

Level 32.2 Find areas under curves as the limit of a sum of areas.

2.3 Determine areas under curves using formula.

2.4 Find volumes of revolutions when region bounded by a curve is rotated completely about the

a) x- axisb) y- axisas the limit of a sum of volumes.

2.5 Determine volumes of revolutions using formula.

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.

Use dynamic computer software to explore volumes of revolutions.

Systematic

Confidence

Patience

Careful

CCTS :Identifying relationship,

Interpreting,

Making Conclusion,

Generating Idea,

6






GEOMETRIC COMPONENT

G2: VECTORS

1. Understand and use the concept of vectors.

Level 11.1 Differentiate between vector and

scalar quantities .1.2 Draw and label directed line

segments to represent vectors.

Level 21.3 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 scalars.1.6 Determine whether two vectors

are parallel.

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

Systematic and cooperative.

Use notations:

Vector: ~a , a, AB.

Magnitude:

~a , AB , |a|, |

AB|

Zero vector: ~0

Emphasise that a zero vector has a magnitude of zero.Emphasise negative vector: - AB = BA

Include a) collinear points b) non-parallel

non-zero vectors.

Teaching courseware,GSP , text books and workbooks.

7






2. Understand and use the concept of addition and subtraction of vectors .

3. Understand and use vectors in the Cartesian plane

Level 12.1 Determine the resultant vector of

two parallel vectors.

Level 22.2 Determine the resultant vector of

two non-parallel vectors using :a) triangle lawb) parallelogram law

2.3 Determine the resultant vector of three or more vectors using the polygon law .

Level 32.4 Subtract two vectors which are :

a) parallelb) non-parallel.

2.5 Represent vectors as a combination of other vectors.

2.6 Solve problems involving addition and subtraction of vectors.

Level 13.1 Express vectors in the forma) jyix +

Use real-life situations and manipulative material to explore addition and subtraction of vectors..

Use computer software to explore vectors in the Cartesian plane.

Patience, rational, careful, systematic and diligence.

Emphasise:)(

~~~~baba −+=−

Relate unit vector i andj to Cartesian

coordinates.Emphasis:

GSP

Teaching courseware

Scientific CalculatorTextbookWorksheetDifference and SimilaritiesProblem SolvingClassifying

GSP

Teaching courseware

Scientific Calculator

8






b)

y

x

3.2 Determine magnitudes of vectors

Level 23.3 Determine unit vectors in given

directions.3.4 Add two or more vectors.3.5 Subtract two vectors.3.6 Multiply vectors by scalars.

Level 33.7 Perform combined operations on

vectors.3.8 Solve problem involving vectors.

For learning outcomes 3.2 to 3.7 all vectors are given in the form

jyix + or

y

x

Limit combined operations to addition, subtraction and multiplication of vectors by scalars.

Vectors i =

0

1

and

Vectors j =

1

0

Rational

Systematic

Co-operation

Textbook

Worksheet

Difference and Similarities

Problem Solving

Classifying

9






TRIGONOMETRIC COMPONENT

T2: TRIGONOMETRIC FUNCTIONS

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

2. Understand and use the six trigonometric functions and of any angles.

Level 11.1 Represent in a Cartesian plane

angles greater than 360° or 2π radians for: a) Positive anglesb) Negative angles

Level 12.1 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.

Level 22.3 Find values of six trigonometric functions of any angle.

2.4 Solve trigonometric equations.

Introduce the four quadrants in the Cartesian plane.Teacher explain clockwise and anticlockwise direction to draw negative angles and positive angles.

Use dynamic computer software such as a Geometer Sketchpad to explore angles in Cartesian plane.

Use dynamic computer software to explore trigonometric functions in degrees and radians

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

Careful, rational and hardworking.

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°.

Use angles ina) degreesb)radians, in term

Identifying relationship, Drawing diagram,

Using of ICT, Teaching Courseware, Projector And Geometer’s Sketchpad. Making Inference, Finding all possible solution.

10






3. Understand and use graphs of sine, cosine and tangent functions.

Level 23.1 Draw and sketch graphs of trigonometric functions. a. y = c + a sin bx, b. y = c + a cos bx c. y = c + a tan bxwhere 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 equation using drawn graphs.

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.

Discuss minimum and maximum value of sin and cos functions.

Discuss properties of graphs of trigonometric functions:

a. y = c + a sin bx, b. y = c + a cos bx c. y = c + a tan bxwhere a, b, and c are constants and b> 0.

of π.

Emphasise the characteristics of sine, cosine and tangent graphs.

Include trigonometric functions involving modulus.

Exclude combinations of trigonometric functions.

STATISTICS COMPONENT

S2: PERMUTATIONS AND

11






COMBINATIONS

1. Understand and use the concept of permutations.

2. Understand and use the concept of Combinations.

Level 11.1 Determine the total number of

ways to perform successive events using multiplication rule.

1.2 Determine the number of permutations of n different objects.

Level 21.3 Determine the number of

permutations of n different objects taken r at a time

Level 31.4 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.

Level 22.1 To determine the number of

combination of r objects chosen from n different objects.

Level 32.2 To determine the number of

combinations of r objects for given condition.

Use manipulative materials to explore multiplication rule. If A event occurs in r ways, and B event occurs in s ways. The total number of ways for event A and B to occur successively is r x s ways

To expand the concept of permutations by listing down all the possible arrangements notation.

n! = n(n-1)(n-2)(3)(2)(1) 0! = 1

Use real-life situations and computer software such as spreadsheet to explore permutations

To expand the concept of combination of r objects from n objects by listing down all the possible selections.

Use examples to illustrate

nCr = !r

Prn

Rational and careful

Systematic

Careful

Scientific calculator

CCTS

Arranging in orderIdentifying patternsAnalyzing

Problem solving

Difference or similarities

Scientific calculator

12







S3 : PROBABILITY

1. Understand and use the concept of probability.

Level 11.1 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.

Use real-life situations to introduce probability.

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

Use set notations.

Discuss:a) classical

probability (theoretical probability)

b) subjective probability

c) relative frequency probability (experimental probability).

Emphasise:Only classical probability is used to solve problems.

Computer Software

Reference books

CCTS:- Relate- Applying concept-Problem solving

13






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

3. Understand and use the concept of probability of

Level 21.4 Determine the probability of two

events:d) A or B occurring;e) A and B occurring.

Level 22.1 Determine whether two events

are mutually exclusive.

Level 32.2 Determine the probability of two

or more events that are mutually exclusive.

Level 33.1 Determine whether two events

are independent

Use manipulative materials and graphing calculator to explore the concept of probability of mutually exclusive events.

Use computer software to stimulate experiment involving probability of mutually exclusive events.

Use manipulative materials and graphing calculators to explore

Emphasise:P(A U B) = P(A) + P(B) – P (A ∩ B) using Venn diagrams.

-Cooperation- Patient- Accuracy

Include events that are mutually exclusive and exhaustive.

Limit to three mutually exclusive events.


Include tree diagrams.

Reference books

Courseware

CCTS:- Relate- Find value- Identify pattern- Predicting- Classifying

Computer software

14






independent events. 3.2 Determine the probability of two independent events

3.3 Determine the probability of three independent events

the concept of probability of independent events.

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


Reference books

CCTS:- Relate- Applying concept- Problem solving


S4 : PROBABILITY DISTRIBUTIONS

Understand and use the concept of binomial distribution .

Level 11.1 List all possible values of a

discrete random variable1.2 Determine the probability of an

event in a binomial distribution.

Level 21.3 Plot binomial distribution graphs

1.4 Determine mean , variance and standard deviation of a binomial distribution.

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

Use graphing calculators and computer software to explore binomial distribution.

Find out the difference or similarities of

P( x ≥ a ) = 1 – P ( x < a )

Identifying relationship between

FairnessRationalHonestyIndependent

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

Include the characteristics of Bernoulli trialsCoinDiceMarble

15






2. Understand and use the concept of normal distributions

Level 31.5 Solve problems involving binomial distributions.

Level 12.1 Describe continuous random

variables using set notations.2.2 Find probability of z-values for

Standard normal distributions.

Level 22.3 Convert random variable of normal distributions, X, to standardised variable, Z.

2.4 Represent probability of an eventusing set notation.

2.5 Determine probability of an event

2.6 Solve problem involving normaldistributions.

p and q

p = 1 – q

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

Co-operation,Fairness,Careful,Hardworking,Rational.

Discuss characteristics of :

a) normal distribution graphs.

b) standard normal distribution graphs.

Z is called standardised variable.

Integration of normal distribution function to determine probability is not required.

Blackboard, text book, scientific calculator, work sheet, list of formulae. Teaching courseware,Interpreting, Identifying relations, Making Inference, Using of ICT, Problem solving, Mathematical communication, Statistical table, Evaluating.

16






SCIENCE AND TECHNOLOGY PACKAGE

AST2: MOTION ALONG A STRAIGHT LINE

1 Understand and use the concept of displacement.

Level 11.1 Identify direction of

displacement of a particle from a fixed point.

Level 21.2 Determine displacement of a

particle from a fixed point.

Level 31.3 Determine the total distance

traveled by a particle over a time interval using graphical method.

Use real-life examples, graphing calculators and computer software such as Geometer’s Sketchpad to explore displacement.

Emphasize the use of the following symbols:s = displacement,v = velocity,a = accelerationt = timewhere s, v and a are functions of time.

Emphasise the differences between displacement and distance.

Discuss positive, negative and zero displacements.

Include the use of number line.

Careful, systematic, rational, confidence.

Coloured pens, blackboard, text book, charts, scientific calculator, worksheet, list of formulae.

Interpreting, Identifying relations, Making Inference, translating,Identifying patterns, comparing and contrasting, drawing diagram, working backwards.

17






2 Understand and use the concept of velocity

3 Understand and use the concept of acceleration

Level 22.1 Determine velocity function of a

particle by differentiation .2.2 Determine instantaneous velocity

of a particle.

Level 32.3 Determine displacement of a

particle from velocity function by integration.

Level 23.1 Determine acceleration function

of a particle by differentiation.

Level 33.2 Determine instantaneous

acceleration of a particle.3.3 Determine instantaneous velocity

of a particle from acceleration function by integration.

Use real-life examples, graphing calculators and computer software such as Geometer’s Sketchpad to explore the concept of velocity.

Use real-life examples, graphing calculators and computer software such as Geometer’s Sketchpad to explore acceleration.

Emphasise velocity as the rate of change of displacement.Include graphs of velocity functions.Discussa)uniform velocity.b)Zero instantaneous velocity.c)positive velocityd)negative velocity.

Careful, systematic, rational, confidence.

Emphasise acceleration as the rate of change of velocity.Discuss: a)uniform accelerationb)zero acceleration c)positive acceleration

Using of ICT, Problem solving, Mathematical communication

18






3.4 Determine displacement of a particle from acceleration function by integration.

3.5 Solve problems involving motion along a straight line.

.

d)negative .acceleration.

Careful, systematic, rational, confidence

SOCIAL SCIENCE PACKAGE

ASS2 : LINEAR PROGRAMMING

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

Level 11.1 Identify and shade the region on

the graph that satisfies a linear inequality.

1.2 Find the linear inequality that defines a shaded region.

Level 21.3 Shade region on the graph that

satisfies several linear inequalities.

1.4 Find linear inequalities that define a shaded region.

Use example from real-life situations, scientific or graphing calculator software.

Emphasize 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)

Co-operation, Rational, Systematic, Hardworking

Whiteboard, text book, colored marker pens, color pencil, graph paper, graph chart, scientific calculator, dynamic computer software, GSP, worksheet.

Identifying relationships

19






2. Understand and use the concept of linear programming.

Level 32.1 Solve problems related to linear

programming by :

a) Writing linear inequalities and equation describing a situation.

b) Shading the region of feasible solution.

c) Determining and drawing the objective function

kbyax =+ where ba, and k are constants.

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

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

Constructivism

Cooperative learning.

Optimum values refer to maximum or minimum values

Include the use of vertices to find the optimum valueCo-operation, diligence, courage, cleanliness, systematic,confidence, patience, careful and hardworking

and patterns.Characterizing and translating.

Whiteboard, text book, colored marker pens, color pencil, set square , rulers, graph paper, graph chart, scientific calculator, graphic calculator, dynamic computer software, GSP, worksheet.

Identifying relationship and patterns.Analyzing, drawing diagram and interpreting.Problem

20






solving and making conclusions.

21

add math t5 2013

Documents

arithmetic progressions

geometrical progressions

learning outcomes students

s n s n

learning activities

suggested teaching

value of n

lines of best fit b