Rancangan Tahunan Baharu 954 - 3 Terms

YEARLY PLANNING FOR MATHEMATICS T 954 (PRAU 1 2013)FIRST TERM: ALGEBRA AND GEOMETRYWeekTopicTeaching

PeriodLearning OutcomeActivities

1 Functions28Candidates should be able to:

1(6 periods)1.1 Functions6(a) state the domain and range of a function, and find composite functions;

(b) determine whether a function is one-to-one, and find the inverse of a one-to-one function;

(c) sketch the graphs of simple functions, including piecewise-defined functions;

Plot graphs with computers, hand-phones & calculators

1(2 periods) ( 2(6 periods)1.2 Polynomial and

rational functions8(d) use the factor theorem and the remainder theorem;

(e) solve polynomial and rational equations and inequalities;

(f) solve equations and inequalities involving modulus signs in simple cases;

(g) decompose a rational expression into partial fractions in cases where the denominator has two distinct linear factors, or a linear factor and a prime quadratic factor;

2(2 periods) ( 3(4 periods)1.3 Exponential and

logarithmic functions6(h) relate exponential and logarithmic functions, algebraically and graphically;

(i) use the properties of exponents and logarithms;

(j) solve equations and inequalities involving exponential or logarithmic expressions;

Search of e and its use.

WeekTopicTeaching


3(4 periods) ( 4(4 periods)1.4 Trigonometric functions8(k) relate the periodicity and symmetries of the sine, cosine and tangent functions to their graphs, and identify the inverse sine, inverse cosine and inverse tangent functions and their graphs;

(l) use basic trigonometric identities and the formulae for sin (A B), cos (A B) and tan (A B), including sin 2A, cos 2A and tan 2A;(m) express a sin + b cos in the forms r sin ( ) and r cos ( );

(n) find the solutions, within specified intervals, of trigonometric equations and inequalities.

Use some software on trigonometric identities

WeekTopicTeaching


2 Sequences and Series18Candidates should be able to:

4(4 periods)2.1 Sequences4(a) use an explicit formula and a recursive formula for a sequence;

(b) find the limit of a convergent sequence;

5(8 periods)2.2 Series8(c) use the formulae for the nth term and for the sum of the first n terms of an arithmetic series and of a geometric series;

(d) identify the condition for the convergence of a geometric series, and use the formula for the sum of a convergent geometric series;

(e) use the method of differences to find the nth partial sum of a series, and deduce the sum of the series in the case when it is convergent;

History of some interesting series

6(6 periods)2.3 Binomial expansions6(f) expand (a + b)n , where n Z+ ;

(g) expand (1 + x) n , where nQ, and identify the condition | x | < 1 for the validity of this expansion;

(h) use binomial expansions in approximations.

WeekTopicTeaching


6 Vectors20Candidates should be able to:

6

(2 periods)

7

(6 periods)6.1 Vectors in two and three dimensions8(a) use unit vectors and position vectors;

(b) perform scalar multiplication, addition and subtraction of vectors;

(c) find the scalar product of two vectors, and determine the angle between two vectors;

(d) find the vector product of two vectors, and determine the area a parallelogram and of a triangle;

Some applications of dot product in applied science.

7

(2 periods),

8

(8 periods)

9

(2 periods)

6.2 Vector geometry12(e) find and use the vector and cartesian equations of lines;

(f) find and use the vector and cartesian equations of planes;

(g) calculate the angle between two lines, between a line and a plane, and between two planes;

(h) find the point of intersection of two lines, and of a line and a plane;

(i) find the line of intersection of two planes.

Some interesting 3D problem solved by vectors.

9

(2 periods)

Formative test 2Test on functions, sequence & series and vectors.

WeekTopicTeaching


Coursework12Candidates should be able to:

9

(2 periods)

Briefing on coursework.2(a) plan to carry out Assignment A,

(b) raise possible problems faced,

(c) revise the resultant velocity if necessary.

Teacher gives briefing and guideline.

9(2 periods)

10

(5 periods)

Coursework7(a) carry out assignment,

(b) refer to relevance sources related to the assignment,

(c) seek for advice and reasonable aids related to the assignment,(d) seek for advice and reasonable aids related to the assignment,

Students carry out assignment A. Teacher as adviser, observer, facilitators

10

(3 period)

Passing up coursework3(a) complete assignment reportTeacher assesses assignment report and conduct viva.

WeekTopicTeaching


3 Matrices16Candidates should be able to:

11(8 periods) 12(2 periods)3.1 Matrices10(a) identify null, identity, diagonal, triangular and symmetric matrices;

(b) use the conditions for the equality of two matrices;

(c) perform scalar multiplication, addition, subtraction and multiplication of matrices with at most three rows and three columns;

(d) use the properties of matrix operations;

(e) find the inverse of a non-singular matrix using elementary row operations;

(f) evaluate the determinant of a matrix;

(g) use the properties of determinants;

12

(6 periods)3.2 Systems of linear equations6(h) reduce an augmented matrix to row-echelon form, and determine whether a system of linear equations has a unique solution, infinitely many solution or no solutions;

(i) apply the Gaussian elimination to solve a system of linear equations;

(j) find the unique solution of a system of linear equations using the inverse of a matrix.

WeekTopicTeaching


4 Complex Numbers12Candidates should be able to:

13

(8 periods)

14

(4 periods)4 Complex Numbers12(a) identify the real and imaginary parts of a complex number;

(b) use the conditions for the equality of two complex numbers;

(c) find the modulus and argument of a complex number in cartesian form and express the complex number in polar form;

(d) represent a complex number geometrically by means of an Argand diagram;

(e) find the complex roots of a polynomial equation with real coefficients;

(f) perform elementary operations on two complex numbers expressed in cartesian form;

(g) perform multiplication and division of two complex numbers expressed in polar form;

(h) use de Moivres theorem to find the powers and roots of a complex number.

WeekTopicTeaching


5 Analytic Geometry14Candidates should be able to:

14

(4 periods),15

(8 periods)

16(2 periods)5 Analytic Geometry14(a) transform a given equation of a conic into the standard form;

(b) find the vertex, focus and directrix of a parabola;

(c) find the vertices, centre and foci of an ellipse;

(d) find the vertices, centre, foci and asymptotes of a hyperbola;

(e) find the equations of parabolas, ellipses and hyperbolas satisfying prescribed conditions (excluding eccentricity);

(f) sketch conics;

(g) find the cartesian equation of a conic defined by parametric equations;

(h) use the parametric equations of conics.

16

(6 periods)

Trial examination 6

YEARLY PLANNING FOR MATHEMATICS T 954 (PRAU 2 2013)

SECOND TERM: CALCULUSWeekTopicTeaching


7 Limits and Continuity12Candidates should be able to:

1(6 periods)7.1 Limits6(a) determine the existence and values of the left-hand limit, right-hand limit and limit of a function;

(b) use the properties of limits;

1(2 periods) 2(4 periods) 7.2 Continuity6(c) determine the continuity of a function at a point and on an interval;

(d) use the intermediate value theorem.

WeekTopicTeaching


8 Differentiation28Candidates should be able to:

2(4 periods) 3(8 periods) 8.1 Derivatives12(a) identify the derivative of a function as a limit;

(b) find the derivatives of xn (n ), ex, ln x, sin x, cos x, tan x, sin1x, cos1x, tan1x, with constant multiples, sums, differences, products, quotients and composites;

(c) perform implicit differentiation;

(d) find the first derivatives of functions defined parametrically;

4(8 periods) 5(8 periods) 8.2 Applications of differentiation16(e) determine where a function is increasing, decreasing, concave upward and concave downward;

(f) determine the stationary points, extremum points and points of inflexion;

(g) sketch the graphs of functions, including asymptotes parallel to the coordinate axes;

(h) find the equations of tangents and normals to curves, including parametric curves;

(i) solve problems concerning rates of change, including related rates;

(j) solve optimisation problems.

WeekTopicTeaching


9 Integration28Candidates should be able to:

6(8 periods) 7(6 periods)9.1 Indefinite integrals14(a) identify integration as the reverse of differentiation;

(b) integrate xn (n ), ex, sin x, cos x, sec2x, with constant multiples, sums and differences;

(c) integrate rational functions by means of decomposition into partial fractions;

(d) use trigonometric identities to facilitate the integration of trigonometric functions;

(e) use algebraic and trigonometric substitutions to find integrals;

(f) perform integration by parts;

7(2 periods), 8(8 periods)

9(4 periods)9.2 Definite integrals14(g) identify a definite integral as the area under a curve;

(h) use the properties of definite integrals;

(i) evaluate definite integrals;

(j) calculate the area of a region bounded by a curve (including a parametric curve) and lines parallel to the coordinate axes or between two curves;

(k) calculate volumes of solids of revolution about one of the coordinate axes.

WeekTopicTeaching


10 Differential Equations14Candidates should be able to:

9(4 periods),

10(8 periods) 11(2 periods)10 Differential Equations14(a) find the general solution of a first order differential equation with separable variables;

(b) find the general solution of a first order linear differential equation by means of an integrating factor;

(c) transform, by a given substitution, a first order differential equation into one with separable variables or one which is linear;

(d) use a boundary condition to find a particular solution;

(e) solve problems, related to science and technology, that can be modelled by differential equations.

11

(2 periods)

Formative test 2Test on limits & continuity, differentiation, integration and differential equations.

WeekTopicTeaching



11

(2 periods)

Briefing on coursework.2(a) plan to carry out Assignment B,


(c) revise the related topics if necessary.


11

(2 periods)

12

(5 periods)



(c) seek for advice and reasonable aids related to the assignment,

(d) seek for advice and reasonable aids related to the assignment,

Students carry out assignment B. Teacher as adviser, observer, facilitators

12

(3 period)


WeekTopicTeaching


11 Maclaurin Series12Candidates should be able to:

13

(8 periods)

14

(4 periods)11 Maclaurin Series12(a) find the Maclaurin series for a function and the interval of convergence;

(b) use standard series to find the series expansions of the sums, differences, products, quotients and composites of functions;

(c) perform differentiation and integration of a power series;

(d) use series expansions to find the limit of a function.

12 Numerical Methods14Candidates should be able to:

14

(4 periods) 15

(6 periods)12.1 Numerical solution of equations10(a) locate a root of an equation approximately by means of graphical considerations and by searching for a sign change;

(b) use an iterative formula of the form xn+1 f (xn ) = to find a root of an equation to a prescribed degree of accuracy;

(c) identify an iteration which converges or diverges;

(d) use the Newton-Raphson method;

15(2 periods)

16

(2 periods)

12.2 Numerical integration4(e) use the trapezium rule;

(f) use sketch graphs to determine whether the trapezium rule gives an over-estimate or an under-estimate in simple cases.

16

(6 periods)

Trial examination 6

YEARLY PLANNING FOR MATHEMATICS T 954 (PRAU 3 2013)

THIRD TERM: STATISTICSWeekTopicTeaching


13 Data Description14Candidates should be able to:

1(8 periods) 2(6 periods)13 Data Description14(a) identify discrete, continuous, ungrouped and grouped data;

(b) construct and interpret stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency curves;

(c) state the mode and range of ungrouped data; (d) determine the median and interquartile range of ungrouped and grouped data;

(e) calculate the mean and standard deviation of ungrouped and grouped data, from raw data and from given totals such as

(f) select and use the appropriate measures of central tendency and measures of dispersion;

(g) calculate the Pearson coefficient of skewness;

(h) describe the shape of a data distribution.

WeekTopicTeaching


14 Probability14Candidates should be able to:

2(2 periods),3(8 periods)

4(4 periods) 14 Probability14(a) apply the addition principle and the multiplication principle;

(b) use the formulae for combinations and permutations in simple cases;

(c) identify a sample space, and calculate the probability of an event;

(d) identify complementary, exhaustive and mutually exclusive events;

(e) use the formula P(A B) = P(A) + P(B) P(A B);

(f) calculate conditional probabilities, and identify independent events;

(g) use the formulae P(A B) = P(A) P(B|A) = P(B) P(A|B);

(h) use the rule of total probability.

WeekTopicTeaching


15 Probability Distributions26Candidates should be able to:

4(4 periods) 5(2 periods)15.1 Discrete random variables6(a) identify discrete random variables;

(b) construct a probability distribution table for a discrete random variable;

(c) use the probability function and cumulative distribution function of a discrete random variable;

(d) calculate the mean and variance of a discrete random variable;

5(6 periods)15.2 Continuous random variables6(e) identify continuous random variables;

(f) relate the probability density function and cumulative distribution function of a continuous random variable;

(g) use the probability density function and cumulative distribution function of a continuous random variable;

(h) calculate the mean and variance of a continuous random variable;

6(4 periods)15.3 Binomial distribution4(i) use the probability function of a binomial distribution, and find its mean and variance;

(j) use the binomial distribution as a model for solving problems related to science and technology;

WeekTopicTeaching


6(4 periods)15.4 Poisson distribution4(k) use the probability function of a Poisson distribution, and identify its mean and variance;

(l) use the Poisson distribution as a model for solving problems related to science and technology;

7(6 periods)15.5 Normal distribution6(m) identify the general features of a normal distribution, in relation to its mean and standard deviation;

(n) standardise a normal random variable and use the normal distribution tables;

(o) use the normal distribution as a model for solving problems related to science and technology;

(p) use the normal distribution, with continuity correction, as an approximation to the binomial distribution, where appropriate.

7

(2 periods)

Formative test 2Test on data description, probability and probability distributions.

WeekTopicTeaching


16 Sampling and Estimation26Candidates should be able to:

8(8 periods)

9(6 periods)16.1 Sampling14(a) distinguish between a population and a sample, and between a parameter and a statistic;

(b) identify a random sample;

(c) identify the sampling distribution of a statistic;

(d) determine the mean and standard deviation of the sample mean;

(e) use the result that X has a normal distribution if X has a normal distribution;

(f) use the central limit theorem;

(g) determine the mean and standard deviation of the sample proportion;

(h) use the approximate normality of the sample proportion for a sufficiently large sample size;

9(2 periods),10

(8 periods)

11(2 periods)16.2 Estimation12(i) calculate unbiased estimates for the population mean and population variance;

(j) calculate an unbiased estimate for the population proportion;

(k) determine and interpret a confidence interval for the population mean based on a sample from a normally distributed population with known variance;

(l) determine and interpret a confidence interval for the population mean based on a large sample;

(m) find the sample size for the estimation of population mean;(n) determine and interpret a confidence interval for the population proportion based on a large sample;

(o) find the sample size for the estimation of population proportion.

WeekTopicTeaching


17 Hypothesis Testing14Candidates should be able to:

11(6 periods) 12(8 periods)17 Hypothesis Testing14(a) explain the meaning of a null hypothesis and an alternative hypothesis;

(b) explain the meaning of the significance level of a test;

(c) carry out a hypothesis test concerning the population mean for a normally distributed population with known variance;

(d) carry out a hypothesis test concerning the population mean in the case where a large sample is used;

(e) carry out a hypothesis test concerning the population proportion by direct evaluation of binomial probabilities;

(f) carry out a hypothesis test concerning the population proportion using a normal approximation.

18 Chi-squared Tests14Candidates should be able to:

13(8 periods) 14(6 periods)18 Chi-squared Tests14(a) identify the shape, as well as the mean and variance, of a chi-squared distribution with a given number of degrees of freedom;

(b) use the chi-squared distribution tables;

(c) identify the chi-squared statistic;

(d) use the result that classes with small expected frequencies should be combined in a chi-squared test;

(e) carry out goodness-of-fit tests to fit prescribed probabilities and probability distributions with known parameters;

(f) carry out tests of independence in contingency tables (excluding Yates correction).

WeekTopicTeaching



14

(2 periods)

Briefing on coursework.2(a) plan to carry out Assignment C,


(c) revise the related topics if necessary.


15

(7 periods)



(c) seek for advice and reasonable aids related to the assignment,

(d) seek for advice and reasonable aids related to the assignment,

Students carry out assignment C. Teacher as adviser, observer, facilitators

15

(1 period)

16

(2 periods)


16

(6 periods)

Trial examination 6

SCHEME OF WORK

MATHEMATICS T

PRAU 1

YEAR 2013

SCHEME OF WORK

MATHEMATICS T

PRAU 2

YEAR 2013

SCHEME OF WORK

MATHEMATICS T

PRAU 3

YEAR 2013

Rancangan Tahunan Baharu 954 - 3 Terms

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