advanced math.pdf

THE UNITED REPUBLIC OF TANZANIA

MINISTRY OF EDUCATION AND VOCATIONAL TRAINING

MATHEMATICS ACADEMIC SYLLABUS FOR DIPLOMA IN SECONDARY

EDUCATION

2009

i

Ministry of Education and Vocational Training 2009

All rights reserved. No part of this publication may be reproduced, reported, stored in

any retrieval system or transmitted in any form or by any means of electronic,

mechanical, photocopying, recording or otherwise without the prior permission of the

copyright owner.

Designed and prepared by:

Tanzania Institute of Education

P.O. Box 35094

Dar -es -Salaam

Tanzania

ii

DECLARATION

The Secondary Education Diploma course is a two year course which has been designed

to prepare professional teachers who will teach in ordinary level secondary schools. A

student teacher shall be recognized as a teacher when he/she successfully completes a

secondary education diploma course for two years within which he/she successfully

performed teaching practice.

This document is hereby declared as the Mathematics Academic Syllabus for Diploma

in Secondary Education course of 2009.

iii

TABLE OF CONTENTS

DECLARATION ................................................................................................................ ii

Introduction......................................................................................................................... v

Subject description.............................................................................................................. v

Rationale for review of the subject syllabus ....................................................................... v

Subject goals ....................................................................................................................... v

Subject Competences to be developed............................................................................... vi

Subject objectives .............................................................................................................. vi

Organization of the syllabus .............................................................................................. vi

Assessment of the subject .................................................................................................. vi

1.0 SIMILARITY AND CONGRUENCE ..................................................................... 1

1.1 Similar Figures.......................................................................................................... 1

1.2 Congruency of Triangles........................................................................................... 2

2.0 LOGIC ...................................................................................................................... 3

2.1 Statements ................................................................................................................. 3

2.2 Logical Connectives.................................................................................................. 3

2.3 Laws of Algebra of Propositions .............................................................................. 4

2.4 Validity of Arguments ............................................................................................... 5

2.5 Electrical Network ..................................................................................................... 6

3.0 CALCULATING DEVICES ..................................................................................... 7

3.1 Scientific Calculations ............................................................................................... 7

3.2 Computer Packages.................................................................................................... 7

4.0 COORDINATE GEOMETRY II............................................................................... 9

4.1 Conic section.............................................................................................................. 9

4.2 The parabola............................................................................................................... 9

4.3 The ellipse................................................................................................................ 10

4.4 The hyperbola .......................................................................................................... 11

5.0 LINEAR PROGRAMMING ................................................................................... 12

5.1 Graphical solution.................................................................................................... 12

5.2 Application of linear programming ......................................................................... 12

6.0 PROBABILITY ....................................................................................................... 14

6.1 Fundamental Principles of Counting (FPC)............................................................. 14

6.2 Permutations ............................................................................................................ 14

6.3 Combinations ........................................................................................................... 15

6.4 Probability Axioms and Theorems .......................................................................... 16

6.5 Conditional Probability............................................................................................ 17

7.0 ALGEBRA............................................................................................................... 18

7.1 Series........................................................................................................................ 18

7.2 Roots of polynomial function .................................................................................. 18

8.0 TRIGONOMETRY.................................................................................................. 20

8.1 Compound Angle Formulae..................................................................................... 20

8.2 Double Angle Formulae........................................................................................... 20

8.3 Factor Formulae ....................................................................................................... 21

9.0 DIFFERENTIATION .............................................................................................. 23

9.1 Differentiation of a Function ................................................................................... 23

iv

9.2 Applications of Differentiation ............................................................................. 23

10.0 INTEGRATION ................................................................................................... 25

10.1 Integration of a Function....................................................................................... 25

10.2 Applications of Integration ................................................................................... 25

11.0 HYPERBOLIC FUNCTIONS.............................................................................. 27

11.1 Definition of Hyperbolic Functions ...................................................................... 27

11.2 Derivative of Hyperbolic Functions...................................................................... 27

11.3 Integration of Hyperbolic Functions..................................................................... 28

12.0 VECTORS ............................................................................................................ 30

12.1 Vector representation ............................................................................................ 30

12.2 Dot product ........................................................................................................... 30

12.3 Cross Product ........................................................................................................ 31

Reference .......................................................................................................................... 32

v

Introduction

This mathematics academic syllabus for Diploma in secondary Education is a

consolidated version of the 2007 syllabus. It has been improved to develop student

teachers academic ability to handle the ordinary level mathematics syllabus as well as

being able to continue with further education.

Subject description

The mathematics academic syllabus is a two years course. It will be implemented in the

teachers college and is estimated to take 128 hours. It is designed for student teacher

specializing in Mathematics. It aims at preparing teachers who have academic

competence for effective teaching of Basic and Additional Mathematics in the ordinary

level secondary schools.

Rationale for review of the subject syllabus

Mathematics is poorly performed subject at al levels of education in the country.

However its significance in all walks of life leaves much to be desired. Generally the

contents among other things include the mastery of what and how to teach

mathematics. This academic syllabus has been reviewed to addresses the what aspects

where student teachers are prepared to develop sound academic competences in ordinary

level secondary and also to be able to pursue further studies mathematics

Subject goals

The goal of teaching mathematics to prepare student teachers with mathematical

knowledge, skills and attitudes in solving mathematical problems; in which they will be

able to:-

(a) master the ordinary level secondary mathematics ready for teaching

(b) enhance the advanced level secondary mathematics ready for further studies.

vi

Subject Competences to be developed

By the end of the course, the student teacher should have the ability to:

(a) Interpret and solve Mathematics problems

(b) Apply mathematics knowledge and skills in other related fields

(c) Communicate precisely in a mathematical language

(d) Apply the mathematical knowledge and skills for further studies.

Subject objectives

By the end of the course, the student teacher should be able to:

(a) Understand basic principles and theories in Mathematics

(b) Understand Mathematical concepts and their relation in other fields

(c) Use Mathematics knowledge and skills in solving problems in daily life

(d) Promote capabilities for studying mathematics in higher education.

Organization of the syllabus

This syllabus has been organized into topics and sub-topics. Each sub-topic consists of:

estimated time, specific objectives, proposed teaching and learning strategies, resources

and assessment procedures. You are strongly advised to use the proposed strategies

resources and assessment in order to achieve the required objectives. A reading list is

included at the end of the syllabus. This list is not exhaustive.

Assessment of the subject

The subject will be assessed as one examination paper. Continuous assessment will

contribute 25% for academic and 25 % for pedagogy. The final examination conducted

by NECTA will contribute 50% making a total mark of 100%. The Block Teaching

Practice (BTP) will be treated as an independent assessment item which carries 100%.

The following table illustrate the continuous assessment and final examination for

mathematics academic syllabus:

No Assessment type Frequency Weight

1 Test 4 5%

2 Portfolio 2 5%

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3 Project 1 5%

4 Terminal exams 3 10%

Continuous 25%

Final exam ACK 50%

TOTAL 100%

1

1.0 SIMILARITY AND CONGRUENCE

1.1 Similar Figures

Estimated Time: 4hours

Specific Objectives

By the end of this sub-topic, the student teacher should be able to:

a) Construct different polygons using paper cuttings and geo - board;

b) Identify similar polygons;

c) Prove similarity theorems using triangles; and

d) Solve problems using similarity theorems of triangles.

e) Apply similarity theorems in daily life.

Teaching and Learning Strategies

a) Demonstration on the construction of polygons;

b) Group work and discussion.

c) Demonstration of the proof of similarity theorems;

d) Individual exercise on solving problems involving similarity theorems

of triangles and real life experiences.

Teaching and Learning Resources

Manila sheets, Ruler, Pair of scissors, Razor blade, Flip charts and Marker pens.

Assessment Procedures

a) Observation schedule/checklist on drawing and cutting out the polygons.

b) Tests.

c) Written exercise

2

1.2 Congruency of Triangles

Estimated Time: 4 hours

Specific Objectives


a) Determine the conditions for congruence of triangles,

b) Prove congruence of triangles, and

c) Apply theorems on congruence of triangle to solve problems.

d) Identify triangular objects which are congruent in real life.


a) Discussion and presentation.

b) Demonstration on the proof of congruence of triangles.

c) Individual exercises.

d) Group work.


Manila sheets, Flip charts, Ruler, scissors, razor blades and Marker pens.


a) Quiz, written exercise.

b) Portfolio.

c) Anecdotal record

3

2.0 LOGIC

2.1 Statements


Specific Objectives

By the end of this sub-topic, the student teacher should be able to::

a) Identify mathematical statements and non mathematical statements;

b) Distinguish between simple and compound statements;

c) Analyse the difference between the converse and contrapositive of a

statement.


a) Brainstorming.

b) Group discussion and presentation.

c) Role play on generating logical statements which are used by

Lawyers.


Mixed examples of mathematical statements and non mathematical statements.


a) Quizzes and oral tests

b) Portfolios

c) Written report

2.2 Logical Connectives


Specific Objectives


a) Represent a compound sentence in symbolic form

4

b) Construct truth tables on compound statements of up to three

statements

c) Distinguish among tautologies, contradictions and equivalent

statements using truth tables.

d) Draw the meaning from the conclusion of the truth tables.


a) Questions and answers.

b) Demonstration on construction of truth tables of compound

statements.

c) Discussion and presentation.


Chart showing compound, sentences symbols and their meaning


a) Tests

b) Group assignments

c) Written exercise

2.3 Laws of Algebra of Propositions Estimated Time: 4 hours

Specific Objectives


a) Use the laws of algebra to simplify propositions;

b) Use the laws of algebra of propositions to prove equivalence of

statements; and

c) Apply laws of algebra in real life.


5

Small group discussions and presentations demonstration.


Prior knowledge of the laws of algebra.


a) Portfolio

b) Observations schedule

c) Written exercise

2.4 Validity of Arguments


Specific Objectives


a) Write down an argument in symbolic form;

b) Determine the validity of an argument.

c) Reflect on the significance of the valid argument.


a) Brainstorming

b) Interactive Lecture

c) Questions and answer.

d) Individual reflections.


Various arguments symbols and their meaning.


a) Observation schedule.

b) Peer assessment.

c) Written exercise

6

2.5 Electrical Network


Specific Objectives


a) Describe ways to represent compound statement into electrical network

and vice versa;

b) Represent compound statement in electrical network and vice-versa; and

c) Simplify electrical network.


a) Questions and answers

b) Discussion and presentation.

c) Demonstration of a simple electrical network by using electrical circuits

constructed in parallel and in series.


Various compound statement, electrical circuits.


a) Individual assignments.

b) Practical report

c) Observation checklist

7

3.0 CALCULATING DEVICES

3.1 Scientific Calculations

Estimated Time 3 hours

Specific Objectives


a) Identify features of scientific calculator the significance of using

scientific calculators in working out mathematics problems.

b) Use scientific calculators to perform mathematical computations.


(i) Brainstorming and participatory lecture.

(ii) Group practicals and activities.

Teaching and learning resources

Scientific calculation and guiding manuals.

Assessment Procedure

(i) Quiz and test.

(ii) Observation schedule.

3.2 Computer Packages

Estimated time: 4 hours

Specific objectives

By the end of this sub-topic, the student teacher should be able to::-

a) Identify the appropriate computer package for solving mathematics

problems

b) Describe the importance of different computer package.

c) Use computer package to solve related mathematical problems.

8


a) Think-pair-share.

b) Group discussion and presentation.

c) Demonstration.


Computers, Mathematical computer packages such as maple, matlab,

mathematical device SPSS.


a) Quiz.

b) Observation schedule.

c) Written exercise

9

4.0 COORDINATE GEOMETRY II

4.1 Conic section


Specific objectives


a) Describe the concept of conic section

b) Locate conic sections in a cone.


a) Group discussion and gallery walk.

b) Demonstration and practicals.


Cone, graph papers, rubber bend, geometrical board.


a) Practical drawing

b) Quiz.

c) Observation schedule.

4.2 The parabola


Specific Objectives


a) Derive an equation of a parabola.

b) Sketch the graph of a parabola.

c) Relate the concept of parabola with real life.

10


a) Demonstration.

b) Practical activities in small groups.


Graph paper, geoboard, rubber bends.

Assessment Procedure

a) Observation check list.

b) Individual exercises

c) Group reflection report.

4.3 The ellipse


Specific Objectives


a) Derive an equation of an ellipse.

b) Sketch the graph of an ellipse.

c) Relate the concept of ellipse with real life.


a) Demonstration.

b) Practical activities in small graphs.

c) Internet or library search on concept of ellipse with real life.


Graph papers, geoboard, rabbler bends.

Assessment procedures

a) Observation check list

b) Individual exercises.


11

d) Written Essay

4.4 The hyperbola


Specific Objectives


a) Derive an equation of hyperbola.

b) Sketch the graph of hyperbola.

c) Reflect on the significance of hyperbola in daily life.


a) Demonstration

b) Practical activities in small groups.


Graph papers, geoboard, rabber bends.

Assessment Procedures.

a) Observation checklist



12

5.0 LINEAR PROGRAMMING

5.1 Graphical solution


Specific Objectives


a) Solve linear programming problems graphically.

b) Find the minimum and maximum values using the objective function.

c) Analyse the meaning of minimum and maximum values.

Teaching and learning Strategies

a) Individual exercises.

b) Questions and answers.

c) Group discussion and presentation.


Geometrical instruments, graph papers, graph board and in rubber bands


a) Individual exercise.

b) Tests.

c) Group report.

5.2 Application of linear programming


Specific Objectives:


a) Transform transportation problems into a mathematical model

b) Solve transportation problems graphically.

c) Judge the essence of transportation problem in linear programming.

13


a) Group discussions and presentation.

b) Group practicals.

c) Think-pair-shore.


Geometrical instruments, graph papers, graph board and rubber bends.


a) Test.

b) Group report

c) Individual reflection.

14

6.0 PROBABILITY

6.1 Fundamental Principles of Counting (FPC)

Estimated Time: 3 hours.

Specific Objectives


a) Describe the fundamental principles of counting; and

b) Apply FPC to solve problems.

c) Identify the link between FPC and life problems.


a) KWL.

b) Group experiment.



Physical objects, mathematical set of instruments


a) Self assessment.

b) Group report

c) Quiz and written exercise.

6.2 Permutations


Specific Objectives


a) Explain the concept of permutations;

b) Describe different application of permutations

c) Apply permutations to solve problems.

15


a) Brainstorming.

b) Pair experimentation.





a) Self assessment.

b) Peer assessment.

c) Test.

6.3 Combinations


Specific Objectives


a) Explain the concept of combinations

b) Differentiate combinations from permutations

c) Apply combinations to solve problems.



b) Group experiments.




16


a) Group written report.

b) Written exercise

6.4 Probability Axioms and Theorems


Specific Objectives


a) Describe probability axioms; and

b) Use probability axioms to prove theorems.


i) Questions and answer to establish the axioms, 0 P(E) 1, P (E) =

P(E1) + P(E2) + P(E3) + .. + P(En) = 1 and P(E) + P(E1) = 1

ii) Demonstration on the proofs of the theorems

P(A1 B1) = P(A B) 1

P [ ])B)' (A =P(A1 B1)

P(AUB) = P(A) +P(B) P(A B)

P(A) P(B) P(A) P(B)

iii) Individual activities


Physical objects, mathematical set of instruments, prior knowledge on set.


a) Written tests


c) Self assessment.

d) Individual exercises

17

6.5 Conditional Probability


Specific objectives


a) Explain the meaning of conditional probability.

b) Apply conditional probability to solve problems.

c) Relate the concept of conditional probability to real life experiences.


a) Brainstorming.

b) Demonstration on the calculation of the conditional probability.

c) Min project to relate conditional probability with life.


Die, Coins, Physical objects and Flash cards


a) Quiz.

b) Observation checklist.

c) Min project report.

18

7.0 ALGEBRA

7.1 Series


Specific objectives


a) Use sigma notation in writing series.

b) Find the sum of the first 1st squares and cubic number.

Teaching and learning strategies

a) Demonstration and group discussion.

b) Questions and answers in finding sum of the first sequence and cubic

numbers.

Teaching and learning resources

Number cards, colored chalks and number patterns.

Assessment procedure

a) Group reflection

b) Tests.

c) Written exercise

7.2 Roots of polynomial function


Specific objectives


a) Find roots of a polynomial function

b) Identify the relationship between roots and coefficient of cubic

equations.

c) Write equations from known roots.

19

Teaching and learning Strategies

a) Demonstration and pair discussion.

b) Individual exercise.


Colored chalks, number patterns.

Assessment procedure

a) Tests

b) Peer assessment

c) Written exercises

20

8.0 TRIGONOMETRY

8.1 Compound Angle Formulae


Specific Objectives


a) Derive the compound angle formulae; and

b) Simplify trigonometric expressions in solving compound angles.



b) Pair reflection.


Trigonometric tables and scientific calculators




c) Oral reflection.

8.2 Double Angle Formulae

Estimated Time: 4 hours.

Specific Objectives


a) Deduce the double angle formulae from compound angle formulae; and

b) Simplify trigonometric equations using double angle formulae.

21


a) Discussion and presentation.


c) Pair reflection.


Trigonometric tables and scientific calculators.




c) Oral reflection.

8.3 Factor Formulae


Specific objectives


a) Derive the factor formulae by using the knowledge of compound angle

formulae.

b) Simplify expressions involving factor formulae; and

c) Solve trigonometric equations involving factor formulae.


a) Question and answer.



Trigonometric tables and scientific calculators.


i) Observation schedule on derivation of factor formulae.

22

ii) Tests.

iii) Written exercises.

23

9.0 DIFFERENTIATION

9.1 Differentiation of a Function


Specific Objectives


a) Differentiate polynomial functions;

b) Differentiate trigonometric functions and their inverses;

c) Differentiate exponential functions; and

d) Differentiate logarithmic function.


a) Groups discussion and jig-saw.


c) Group Reflection.


Graph papers, Graph board and rubber bands


a) Observation checklist.

b) Portfolio.

c) Reflection report

d) Written exercises

9.2 Applications of Differentiation


Specific Objectives


24

a) Find maximum and minimum values of a function;

b) Determine rates and change in velocity and acceleration and

c) Solve real life problems involving maximum and minimum values.


a) Pair discussion and presentation.

b) Out of class practicals.

c) Peer teaching.

d) Individual exercises.


Graph papers, Graph board


a) Pair assessment.

b) Observation schedule and practical report.

c) Group reflection.

25

10.0 INTEGRATION

10.1 Integration of a Function


Specific Objectives


a) Integrate polynomial functions

b) Integrate trigonometric functions

c) Integrate exponential functions

d) Integrate logarithmic functions.


a) Group discussion jig-saw and presentation.




Graph papers, Graph board and rubber bands.



b) Portfolio based on jig-saw presentation.

c) Reflection report.

d) Written exercise

10.2 Applications of Integration


Specific objectives


26

a) Calculate area under the curve;

b) Find area of a sector;

c) Calculate the volume of solids of revolutions and

d) Calculate the length of an arc.


a) Library search.

b) Out of class practicals.



Graph papers, Graph board and rubber bands.


a) Written notes;

b) Portfolio, practical report and reflection.

c) Tests.

27

11.0 HYPERBOLIC FUNCTIONS

11.1 Definition of Hyperbolic Functions


Specific Objectives

By the end of this sub-topic, the student-teacher should be able to:

a) Explain the concept of hyperbolic functions;

b) Draw graphs of hyperbolic functions; and

c) Solve problems involving hyperbolic functions.


a) KWL.

b) Activities drawing.



Graph papers and scientific calculators.


a) Self assessment.


c) Reflection report.

11.2 Derivative of Hyperbolic Functions


Specific Objectives


a) Differentiate hyperbolic functions;

b) Differentiate inverses of hyperbolic functions; and

c) Apply derivatives of hyperbolic functions to solve problems.

d) Relate differentiation of hyperbolic function with daily life.

28


a) Pair discussion.


c) Reflection.


Graph papers, scientific calculators, computer programmers such as maple drive

4 etc.


a) Tests.

b) Portfolio.

c) Written Group assignments.

11.3 Integration of Hyperbolic Functions


Specific Objectives


a) Integrate hyperbolic functions;

b) Apply hyperbolic functions to integrate other related functions.

c) Relate integration of hyperbolic functions with daily life.


a) Demonstration.


c) Reflection.


Graph papers, Scientific calculators, computers using programmers such as

maple, drive 4 etc.

29


a) Tests.


c) Anecdotal record.

30

12.0 VECTORS

12.1 Vector representation


Specific objectives


a) Represent a vector in 3 dimensional space

b) Find unit vector in a give direction in a 3 dimensional space

c) Reflect on the practical experiences in real life connected to 3

dimensional space.


a) Guide practicals activities on presenting a vector in 3 dimensional

spaces.

b) Small group discussion and presentation.

Teaching and Leaning Resources

dimensional model, graph paper geometrical instruments, scientific a

inculcator and computer programmes.


a) Observation checklist

b) Group report.

c) Written exercises

12.2 Dot product


Specific Objectives


a) Find projection of a vector onto another vector.

b) Prove cosine rule using dot product of vectors.

31


a) Guided group discussion and presentation.

b) Questions and answer.


Dimensional model, geometrical instruments, graph papers, tourch and

pointer.


a) Individual exercise

b) Tests.

12.3 Cross Product


Specific Objectives


a) Calculate the areas of a triangle and a parallelogram using cross

product of vector

b) Prove sine rule using cross product of two vector.


a) Guided group discussion and presentation.



Dimensional model geometrical instruments and graph papers.


a) Group assignment.

b) Tests.

c) Anecdotal record.

32

Reference

Clarke, L.H (1977): Pure Mathematics at Advanced Level, Heinemann Educational

Books Ltd, Great Britain.

Fobes, M.P and Smyth R.B. (1963): Calculus and Analytic Geometry Vol 1, Prentice

- Hall Inc, London.

Ministry of Education and Culture. (1997): Advanced Level Mathematics Syllabus

Form V-VI, Dar es Salaam, Tanzania.

Ministry of Education and Culture (2005): Additional Mathematics Syllabus for

Secondary Schools Form I-IV, Dar es salaam, Tanzania.

Ministry of Education and Culture (1997): Basic Mathematics Syllabus for

Secondary Schools, Form V-VI, Dar es Salaam, Tanzania.

Shayo,L.K (1989): Advanced Level mathematics Vol 1, University Press, Dar es

Salaam, Tanzania.

Setek,W.M. (1989): Fundamental of Mathematics 5th

Education, Macmillan

Publishing Company, New York.

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