Mathematics 1

11.1.0 MATHEMATICS I

11.1.01 INTRODUCTION TO THE MODULE

This module unit is designed to equip the trainee with the relevant mathematical knowledge, skills, techniques and attitudes necessary to enhance better understanding of the respective trade area.

11.1.02 GENERAL OBJECTIVES

At the end of the module unit, the trainee should be able to:

1. Use mathematical concepts and techniques in solving problems related to respective trade area
2. Organise, draw simple deductions and conclusions from a given data
3. Interpret graphical representation of functions relevant to the respective trade area

11.1.03 MODULE UNIT SUMMARY AND TIME ALLOCATION

CODE	SUB-MODULE UNIT	CONTENT	TIME(HOURS)
11.1.1	NUMBER SYSTEMS	Types of numbers: natural integers, rational, irrational Place value, rounding off, significant figures Factors and prime numbers, GCF and LCM Powers, fractions and decimals	3
11.1.2	INDICES AND LOGARITHMS	Powers Laws of indices Indices operations/logarithms Laws of logarithms Operations Base e Change of base Logarithmic equations Scientific calculator	3
11.1.3	ALGEBRA	Algebraic expressions Operations of algebraic expressions Factorization of algebraic expressions Quadratic expressions Solution of equations linear quadratic cubic and polynomial Partial fractions Simultaneous equations - linear with 2 unknowns - linear with 3 unknowns - linear and quadratic Transposition of formulae Evaluation of polynomials Division of polynomials, remainder and factor theorem	6
11.1.4	GEOMETRY AND SCALE DRAWING	Scale Drawing basic figures plane Drawing of solids	4
11.1.5	SEQUENCE AND SERIES	Sequences, arithmetic and geometric progressions Series Solution of problems related to simple and compound interest Convergent series	6
11.1.6	GRAPHS	Linear Quadratic Exponential Solution of equations - linear - quadratic Linear and quadratic Tangents Determination of laws	4
11.1.7	TRIGONOMETRY	Angles - radian measure - minutes and seconds Trigonometric ratios and their reciprocals Angles of elevation and depression Sine rule Cosine rule Solution of triangles Graphs of trigonometric functions Compound angle formula Derivation of double angle formula Basic trigonometric equations - formula - half angle formula - tangent rule - factor formula - further trigonometric equations - parametric equations - Heron’s formula	6
11.1.8	LATITUDES AND LONGITUDES	Latitudes and longitudes The equator and the Greenwich meridian Distance between two points along small and great circles Time between longitude Speed	4
11.1.9	COORDINATE GEOMETRY AND LOCI	Polar equations Conversion of Cartesian to polar and vice versa Graphs of polar equations Definitions of locus in relation points, lines, planes, ellipses, parabola, hyperbola	5
11.1.10	MENSURATION	Units of measurements Perimeter and areas of regular figures Volume of regular solids Surface areas of regular solids Area of irregular figures Area and volumes using Pappus theorem	6
11.1.11	PERMUTATIONS, COMBINATIONS AND BINOMIAL EXPANSION	Definition of terms – permutation and combination factorial notation Solving problems involving permutations and combinations Binomial expansion: - pascal triangle - binomial theorem - power series using binomial theorem - roots of numbers by binomial theorem - apply binomial theorem to approximations	5
11.1.12	VECTORS	Vectors and scalar in two and three dimensions operations on vectors: Addition and subtraction Position vectors Resolution of vectors Scalar product	6
11.1.13	PROBABILITY I	Definitions Laws of probability Mutually exclusive, independent events, conditional probability Tree diagram, sample point, Venn diagram	4
11.1.4	STATISTICS I	Data arrangement Representation of data Measures of central tendency Measures of dispersion	6
	TOTAL		66

NUMBER SYSTEM

THEORY

Specific Objectives

By the end of the sub-module unit, the trainee should be able to:

1. identify types of numbers
2. carry out arithmetic operations on numbers
3. state the place value of digit in a number
4. round off numbers
5. state accuracy to given significant figures
6. express numbers as product of prime factors
7. find the GCD and LCM of a set of numbers
8. perform operations on fractions and decimals

Content

• Types of numbers: natural, integers, rational, irrational, decimals, fractions on numbers.
• Arithmetic operations on numbers
• Arithmetic operations on numbers
• Place value of digits
• Rounding off
• Prime factors
• GCD and LCM Fractional and decimals

Competence

Ability to:

1. performs arithmetic operations and numbers accurately

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Assessment Methods

• Oral tests
• Written
• Number games
• Quizzes

INDICES AND LOGARITHMS

THEORY

Specific Objectives

By the end of the sub-module unit, the trainee should be able to:

1. define an index of a number
2. state the laws of indices
3. perform indicial operations
4. write numbers in standard form
5. state laws of logarithms
6. perform logarithmic operations
7. Evaluate natural logarithms
8. convert numbers from one
9. solve logarithmic equations base to another
10. use calculator/tables

Content

• Definition of an index of a number
• Laws of indices
• Indicial operations
• Numbers in standard form
• Laws of logarithms
• Logarithmic operations
• Natural logarithms
• Logarithmic equations
• Calculator/tables

Competence

Ability to:

1. use logarithms to solve mathematical problems

Suggested Learning Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

ALGEBRA

THEORY

Specific Objectives

By the end of the sub-module unit, the trainee should be able to:

1. write statements in algebraic form.
2. simplify algebraic expressions
3. factorise algebraic expressions
4. change the subject of a formula
5. divide polynomials
6. solve equations
7. obtain partial fractions from composite fractions
8. solve simultaneous equation
9. evaluate polynomials
10. apply the remainder factor

Content

• Statement in algebraic form
• Simplification of algebraic expressions
• Factorisation of algebraic expression
• Transposition of formulae
• Division of polynomials
• Solution of equations

- linear - quadratic - cubic

• Partial fractions
• Simultaneous equations

- linear in 2 unknown - linear in 3 unknown - linear and quadratic

• Evaluation of polynomials
• Remainder and factor theorems

Competence

Ability to:

1. perform algebraic operations accurately

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Assessment Methods

• Oral tests
• Written
• Number games
• Quizzes

GEOMETRY AND SCALE DRAWING

THEORY

Specific Objectives

By the end of the sub-module unit, the trainee should be able to:

1. conversion of scales
2. draw plane figures to a given scale
3. state the properties of different types of triangles
4. solve problems involving angle properties of a circle
5. draw regular solids

Content

• Scale
• Scale drawing
• Types of triangles
• Angle properties of a circle
• Simple solids: cube, cuboids, cylinder, cones, pyramids

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Assessment Methods

• Oral tests
• Written
• Number games
• Quizzes

SEQUENCE AND SERIES

THEORY

Specific Objectives

By the end of the sub-module unit, the trainee should be able to:

1. distinguish between a sequence and a series
2. solve problems related to series
3. use series to calculate simple and compound interest
4. determine series convergence

Content

• A sequence and a series
• Series

- arithmetic progression - geometric progression

• Simple and compound interest
• Convergent series

GRAPHS

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. identify linear functions
2. plot graphs of linear
3. identify quadratic functions
4. plot graphs of quadratic functions
5. identify exponential functions
6. plot graphs of exponential functions
7. solve quadratic equation by graph
8. solve simultaneous equations – two linear with two unknowns
9. solve simultaneous equations – one linear one quadratic
10. determine linear laws
11. linearize non-linear laws
12. determine non-linear laws

Content

• Linear functions
• Straight line graph
• Quadratic functions
• Graphs of quadratic functions
• Exponential functions of the forms:

- y=abx - y=axb - y=aex

• Graphs of exponential functions
• Graphical solution of simultaneous quadratic equations
• Graphical solution of equations – two linear with two unknowns
• Solve simultaneous equations – one linear one quadratic
• Determine linear laws
• Linearization
• Determination of non-linear laws

Competences

Ability to:

1. Plot graphs of functions
2. Solve equations by suing graphs
3. Determine laws

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Assessment Methods

• Oral tests
• Written
• Number games
• Quizzes

TRIGONOMETRY

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. convert angles from one measurement to another
2. define the trigonometric ratios and their reciprocals
3. draw graphs of trigonometric functions
4. solve problems related to trigonometric ratios
5. calculate angles of elevation and depression
6. derive the sine and cosine rule
7. solve problems using the cosine and sine rule
8. drive compound angle formula
9. use the compound angle formula in solving problems
10. deduce the double angle formula
11. solve problems using the double angle formula
12. derive the trigonometric identities

Content

• Conversion of angles

- radian measure - minutes and seconds

• Trigonometric ratios and their reciprocals
• Graphs of trigonometric functions
• Equations involving trigonometric ratios
• Angles of elevation and depression
• Sine and Cosine rule
• Use of sine and cosine rules
• The compound angle formula
• Using the compound angle formula
• Double angle formula Using the double angle formula
• Trigonometric identities

- t-formulae - factor formulae - half-angle formulae - tangent rule

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Assessment Methods

• Oral tests
• Written
• Number games
• Quizzes

LATITUDES AND LONGITUDES

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. differentiate between latitudes and longitudes
2. identify the Equator and the Greenwich Meridian
3. determine the distance between two points along the small and great circles
4. calculate time between longitudes
5. calculate speed

Content

• Latitudes and longitudes
• The equator and the Greenwich Meridian
• Distance between two points along small and great circles
• Time between longitudes
• Speed

Suggested Leaning Activities

• Discussion
• Question and answer
• Demonstration

Suggested Learning Resources

• Sphere

COORDINATE GEOMETRY AND LOCI

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. define polar equations
2. convert polar equations to Cartesian equations and vice versa
3. plot graphs of polar equations
4. define the locus of a point
5. determine the locus of points in relation to other points to other points, lines, planes, eclipses, parabola and hyper bola

Content

• Definition of polar equations
• Conversion of polar equations to Cartesian equations and vice versa
• Graphs of polar equation
• Definition of a locus of a point
• Locus of points

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Demonstration
• Question and answer
• Group work
• Discussion
• Working out problems

Suggested Learning Resources

• Charts
• Graphs
• Plotting paper
• Geometrical instruments ns

MENSURATION=

THEORY

SPECIFIC OBJECTIVES

By the end of the sub-module, the trainee should be able to:

1. state different units of measurements
2. calculate perimeters and areas of regular figures
3. determine volumes of regular solids
4. calculate surface areas of regular solids
5. calculate areas of irregular figures
6. determine area and volume of solids using Pappus theorem

Content

• Units of measurement
• Perimeter and area of regular figures
• Volume of regular solids
• Surface area of regular solids
• Area of irregular figures

- Trapezoidal rule - Simpson’s rule

• Pappus theorem

Competence

Ability to:

1. Solve op problems in combinations and binomial exercises

Suggested Learning/Teaching Resources

• Charts
• Number line
• Factor tree diagrams
• Bells
• Alarm
• Flickering lights

Suggested Learning Teaching Activities

• Measuring
• Drawing and sketching figures
• Measuring lengths of models

Suggested Assessment Methods

• Written tests
• Assignments

Suggested Learning Resources

• Charts of solids

PERMUTATIONS, COMBINATIONS AND BINOMIAL EXPANSION

SPECIFIC OBJECTIVES

By the end of the sub-module, the trainee should be able to:

1. define permutation and combinations
2. simplify problem using factorial notation
3. determine binomial coefficients
4. deduce the binomial theorem
5. obtain binomial series using binomial expansion
6. approximate roots and errors using binomial theorem

Content

• Definitions of permutation and combination
• Factorial notation
• Binomial expansion

- coefficients using - Pascal’s triangle - binomial theorem

• Deduction of binomial theorem from Pascal’s triangle
• Power series using binomial theorem
• Approximations, roots and errors using binomial theorem

Suggested Learning Resources

• Charts
• Audio/visual materials

VECTORS

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. define a vector
2. represent a vector by a directed straight line
3. add vectors
4. subtract vectors
5. write a vector in terms of components
6. multiply a vector with a scalar
7. write a vector in terms of component unit vectors
8. set up the coordinate system for representing vectors
9. obtain the direction cosines of a vector
10. calculate the scalar product

Content

• Definition of scalar and vector quantity
• Vector representation
• Addition of vectors
• Subtraction of vectors
• Components of a vector
• Scalar multiplication
• Components of a vector in terms of unit vectors
• Vectors in space
• Direction cosines
• Scalar/dot product of two vectors

PROBABILITY I

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

• define the term probability
• state and apply the laws of probability
• distinguish between mutually exclusive, dependent and independent events
• compute conditional probabilities
• use tree diagram to solve probability problems
• draw probability space of a given case
• pick a sample point from probability space
• draw a Venn diagram for a given situation

Content

• Definition of probability
• Laws of probability
• Mutually exclusive, dependent and independent events
• Conditional probabilities
• Probability tree diagrams
• Probability space
• Sample point
• Venn diagram

Suggested Learning Resources

• Charts
• Audio/visual materials
• Balls
• Beads
• Cards
• Coins

STATISTICS I

THEORY

Specific Objectives

By the end of the sub-module, the trainee should be able to:

1. distinguish between discrete and continuous data
2. construct frequency distribution tables
3. construct cumulative frequency table
4. determine class boundaries, class intervals, central values
5. represent data graphically
6. determine measures of central tendency
7. determine measures of dispersion

Content

• Types of data
• Frequency table grouped/ungrouped data
• Cumulative frequency table
• Class boundaries, class intervals, central values
• Histogram, frequency polygons, bar groups, pie charts, pictograms
• Mean, mode, median
• Range, variance and standard deviation