# 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
• Group work
• Discussion
• Working out problems

• 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
• Group work
• Discussion
• Working out problems

• 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
• Group work
• Discussion
• Working out problems

• 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
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
• 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
• Group work
• Discussion
• Working out problems

• 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
• Group work
• Discussion
• Working out problems

• 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
• Demonstration

• 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
• 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
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
• 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
• 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