Matrices and Transformation

Matrices of Transformation
Identifying Common Transformation Matrices
Finding the Matrix of a Transformation
Using the Unit Square Method
Successive Transformations
Matrix Multiplication for Combined Transformations
Single Matrix for Successive Transformations

By the end of the lesson, the learner should be able to:

-Define transformation and identify types
-Recognize that matrices can represent transformations
-Apply 2×2 matrices to position vectors
-Relate matrix operations to geometric transformations

-Determine the matrix representing a given transformation
-Use coordinate geometry to find transformation matrices
-Apply algebraic methods to find matrix elements
-Verify transformation matrices using test points

-Review transformation concepts from Form 2
-Demonstrate matrix multiplication using position vectors
-Plot objects and images on coordinate plane
-Practice identifying transformations from images

-Work through algebraic method of finding matrices
-Use simultaneous equations to solve for matrix elements
-Practice with different types of transformations
-Verify results by applying matrix to test objects

Exercise books
-Manila paper
-Ruler
-Pencils
-String
Exercise books
-Manila paper
-Ruler
-Chalk/markers
-String
-Coloured pencils

KLB Secondary Mathematics Form 4, Pages 1-5
KLB Secondary Mathematics Form 4, Pages 6-16

Matrices and Transformation

Inverse of a Transformation
Properties of Inverse Transformations
Area Scale Factor and Determinant

By the end of the lesson, the learner should be able to:

-Define inverse transformation conceptually
-Find inverse matrices using algebraic methods
-Apply inverse transformations to return objects to original position
-Verify inverse relationships using matrix multiplication

-Demonstrate inverse transformations geometrically
-Practice finding inverse matrices algebraically
-Verify that A × A⁻¹ = I
-Apply inverse transformations to solve problems

Exercise books
-Manila paper
-Ruler
-Chalk/markers
det A

KLB Secondary Mathematics Form 4, Pages 24-26

Matrices and Transformation

Shear Transformations
Stretch Transformations

By the end of the lesson, the learner should be able to:

-Define shear transformation and its properties
-Identify invariant lines in shear transformations
-Construct matrices for shear transformations
-Apply shear transformations to geometric objects

-Demonstrate shear using cardboard models
-Identify x-axis and y-axis invariant shears
-Practice constructing shear matrices
-Apply shears to triangles and rectangles

Exercise books
-Cardboard pieces
-Manila paper
-Ruler
-Rubber bands

KLB Secondary Mathematics Form 4, Pages 28-34

Matrices and Transformation
Statistics II

Combined Shear and Stretch Problems
Isometric and Non-isometric Transformations
Introduction to Advanced Statistics

By the end of the lesson, the learner should be able to:

-Apply shear and stretch transformations in combination
-Solve complex transformation problems
-Identify transformation types from matrices
-Calculate areas under shear and stretch transformations

-Work through complex transformation sequences
-Practice identifying transformation types
-Calculate area changes under different transformations
-Solve real-world applications

Exercise books
-Manila paper
-Ruler
-Chalk/markers
-Paper cutouts
-Real data examples

KLB Secondary Mathematics Form 4, Pages 28-34

Statistics II

Working Mean Concept
Mean Using Working Mean - Simple Data

By the end of the lesson, the learner should be able to:

-Define working mean (assumed mean)
-Explain why working mean simplifies calculations
-Identify appropriate working mean values
-Apply working mean to reduce calculation errors

-Demonstrate calculation difficulties with large numbers
-Show how working mean simplifies arithmetic
-Practice selecting suitable working means
-Compare results with and without working mean

Exercise books
-Manila paper
-Sample datasets
-Chalk/markers
-Student data

KLB Secondary Mathematics Form 4, Pages 39-42

Statistics II

Mean Using Working Mean - Frequency Tables
Mean for Grouped Data Using Working Mean
Advanced Working Mean Techniques

By the end of the lesson, the learner should be able to:

-Calculate mean using working mean for frequency data
-Apply working mean to discrete frequency distributions
-Use the formula with frequencies correctly
-Solve real-world problems with frequency data

-Demonstrate with family size data from local community
-Practice calculating fx and fd systematically
-Work through examples step-by-step
-Students practice with their own collected data

Exercise books
-Manila paper
-Community data
-Chalk/markers
-Real datasets
-Economic data

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Introduction to Quartiles, Deciles, Percentiles
Calculating Quartiles for Ungrouped Data

By the end of the lesson, the learner should be able to:

-Define quartiles, deciles, and percentiles
-Understand how they divide data into parts
-Explain the relationship between these measures
-Identify their importance in data analysis

-Use physical demonstration with student heights
-Arrange 20 students by height to show quartiles
-Explain percentile ranks in exam results
-Discuss applications in grading systems

Exercise books
-Manila paper
-Student height data
-Measuring tape
-Test score data
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Quartiles for Grouped Data
Deciles and Percentiles Calculations

By the end of the lesson, the learner should be able to:

-Calculate quartiles using interpolation formula
-Identify quartile classes correctly
-Apply the formula: Q = L + [(n/4 - CF)/f] × h
-Solve problems with continuous grouped data

-Work through detailed examples on chalkboard
-Practice identifying quartile positions
-Use cumulative frequency systematically
-Apply to real examination grade data

Exercise books
-Manila paper
-Grade data
-Chalk/markers
-Performance data

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Introduction to Cumulative Frequency
Drawing Cumulative Frequency Curves (Ogives)
Reading Values from Ogives

By the end of the lesson, the learner should be able to:

-Construct cumulative frequency tables
-Understand "less than" cumulative frequencies
-Plot cumulative frequency against class boundaries
-Identify the characteristic S-shape of ogives

-Create cumulative frequency table with class data
-Plot points on manila paper grid
-Join points to form smooth curve
-Discuss properties of ogive curves

Exercise books
-Manila paper
-Ruler
-Class data
-Pencils
-Completed ogives

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Applications of Ogives
Introduction to Measures of Dispersion
Range and Interquartile Range
Mean Absolute Deviation
Introduction to Variance

By the end of the lesson, the learner should be able to:

-Use ogives to solve real-world problems
-Find number of values above/below certain points
-Calculate percentage of data in given ranges
-Compare different datasets using ogives

-Calculate range for different datasets
-Find interquartile range (Q3 - Q1)
-Calculate quartile deviation (semi-interquartile range)
-Compare advantages and limitations of each measure

-Solve problems about pass rates in examinations
-Find how many students scored above average
-Calculate percentages for different grade ranges
-Use agricultural production data for analysis

-Calculate range for student heights in class
-Find IQR for the same data
-Discuss effect of outliers on range
-Compare IQR stability with range

Exercise books
-Manila paper
-Real problem datasets
-Ruler
-Comparative datasets
-Chalk/markers
Exercise books
-Manila paper
-Student data
-Measuring tape
-Test score data
-Chalk/markers
-Simple datasets

KLB Secondary Mathematics Form 4, Pages 52-60
KLB Secondary Mathematics Form 4, Pages 60-65

Statistics II

Variance Using Alternative Formula
Standard Deviation Calculations

By the end of the lesson, the learner should be able to:

-Apply the formula: σ² = (Σx²/n) - x̄²
-Use alternative variance formula efficiently
-Compare computational methods
-Solve variance problems for frequency data

-Demonstrate both variance formulas
-Show computational advantages of alternative formula
-Practice with frequency tables
-Students choose efficient method

Exercise books
-Manila paper
-Frequency data
-Chalk/markers
-Exam score data

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II
Loci

Standard Deviation for Grouped Data
Advanced Standard Deviation Techniques
Introduction to Loci

By the end of the lesson, the learner should be able to:

-Calculate standard deviation for frequency distributions
-Use working mean with grouped data for SD
-Apply coding techniques to simplify calculations
-Solve complex grouped data problems

-Work with agricultural yield data from local farms
-Use coding method to simplify calculations
-Calculate SD step by step for grouped data
-Compare variability in different crops

Exercise books
-Manila paper
-Agricultural data
-Chalk/markers
-Transformation examples
-String

KLB Secondary Mathematics Form 4, Pages 65-70

Loci

Basic Locus Concepts and Laws
Perpendicular Bisector Locus

By the end of the lesson, the learner should be able to:

-Understand that loci follow specific laws or conditions
-Identify the laws governing different types of movement
-Distinguish between 2D and 3D loci
-Apply locus concepts to simple problems

-Physical demonstrations with moving objects
-Students track movement of classroom door
-Identify laws governing pendulum movement
-Practice stating locus laws clearly

Exercise books
-Manila paper
-String
-Real objects
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 73-75

Loci

Properties and Applications of Perpendicular Bisector
Locus of Points at Fixed Distance from a Point

By the end of the lesson, the learner should be able to:

-Understand perpendicular bisector in 3D space
-Apply perpendicular bisector to find circumcenters
-Solve practical problems using perpendicular bisector
-Use perpendicular bisector in triangle constructions

-Find circumcenter of triangle using perpendicular bisectors
-Solve water pipe problems (equidistant from two points)
-Apply to real-world location problems
-Practice with various triangle types

Exercise books
-Manila paper
-Compass
-Ruler
-String

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Locus of Points at Fixed Distance from a Line
Angle Bisector Locus
Properties and Applications of Angle Bisector

By the end of the lesson, the learner should be able to:

-Define locus of points at fixed distance from straight line
-Construct parallel lines at given distances
-Understand cylindrical surface in 3D
-Apply to practical problems like road margins

-Construct parallel lines using ruler and set square
-Mark points at equal distances from given line
-Discuss road design, river banks, field boundaries
-Practice with various distances and orientations

Exercise books
-Manila paper
-Ruler
-Set square
-Compass
-Protractor

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Constant Angle Locus
Advanced Constant Angle Constructions
Introduction to Intersecting Loci
Intersecting Circles and Lines
Triangle Centers Using Intersecting Loci

By the end of the lesson, the learner should be able to:

-Understand constant angle locus concept
-Construct constant angle loci using arc method
-Apply circle theorems to constant angle problems
-Solve problems involving angles in semicircles

-Understand concept of intersecting loci
-Identify points satisfying multiple conditions
-Find intersection points of two loci
-Apply intersecting loci to solve practical problems

-Demonstrate constant angle using protractor
-Construct arc passing through two points
-Use angles in semicircle property
-Practice with different angle measures

-Demonstrate intersection of two circles
-Find points equidistant from two points AND at fixed distance from third point
-Solve simple two-condition problems
-Practice identifying intersection points

Exercise books
-Manila paper
-Compass
-Protractor
Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 75-82
KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Complex Intersecting Loci Problems
Introduction to Loci of Inequalities

By the end of the lesson, the learner should be able to:

-Solve problems with three or more conditions
-Find regions satisfying multiple constraints
-Apply intersecting loci to optimization problems
-Use systematic approach to complex problems

-Solve treasure hunt type problems
-Find optimal locations for facilities
-Apply to surveying and engineering problems
-Practice systematic problem-solving approach

Exercise books
-Manila paper
-Compass
-Real-world scenarios
-Ruler
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Distance Inequality Loci
Combined Inequality Loci

By the end of the lesson, the learner should be able to:

-Represent distance inequalities graphically
-Solve problems with "less than" and "greater than" distances
-Find regions satisfying distance constraints
-Apply to safety zone problems

-Shade regions inside and outside circles
-Solve exclusion zone problems
-Apply to communication range problems
-Practice with multiple distance constraints

Exercise books
-Manila paper
-Compass
-Colored pencils
-Ruler

KLB Secondary Mathematics Form 4, Pages 89-92

Loci

Advanced Inequality Applications
Introduction to Loci Involving Chords
Chord-Based Constructions

By the end of the lesson, the learner should be able to:

-Apply inequality loci to linear programming introduction
-Solve real-world optimization problems
-Find maximum and minimum values in regions
-Use graphical methods for decision making

-Solve simple linear programming problems
-Find optimal points in feasible regions
-Apply to business and farming scenarios
-Practice identifying corner points

Exercise books
-Manila paper
-Ruler
-Real problem data
-Compass

KLB Secondary Mathematics Form 4, Pages 89-92

Loci

Advanced Chord Problems
Integration of All Loci Types

By the end of the lesson, the learner should be able to:

-Solve complex problems involving multiple chords
-Apply power of point theorem
-Find loci related to chord properties
-Use chords in circle geometry proofs

-Apply intersecting chords theorem
-Solve problems with chord-secant relationships
-Find loci of points with equal power
-Practice with tangent-chord angles

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 92-94

Trigonometry III

Review of Basic Trigonometric Ratios
Deriving the Identity sin²θ + cos²θ = 1
Applications of sin²θ + cos²θ = 1

By the end of the lesson, the learner should be able to:

-Recall sin, cos, tan from right-angled triangles
-Apply Pythagoras theorem with trigonometry
-Use basic trigonometric ratios to solve problems
-Establish relationship between trigonometric ratios

-Review right-angled triangle ratios from Form 2
-Practice calculating unknown sides and angles
-Work through examples using SOH-CAH-TOA
-Solve simple practical problems

Exercise books
-Manila paper
-Rulers
-Calculators (if available)
-Unit circle diagrams
-Calculators
-Trigonometric tables
-Real-world examples

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Additional Trigonometric Identities
Introduction to Waves
Sine and Cosine Waves
Transformations of Sine Waves

By the end of the lesson, the learner should be able to:

-Derive and apply tan θ = sin θ/cos θ
-Use reciprocal ratios (sec, cosec, cot)
-Apply multiple identities in problem solving
-Verify trigonometric identities algebraically

-Plot graphs of y = sin x and y = cos x
-Identify amplitude and period of basic functions
-Compare sine and cosine wave patterns
-Read values from trigonometric graphs

-Demonstrate relationship between tan, sin, cos
-Introduce reciprocal ratios with examples
-Practice identity verification techniques
-Solve composite identity problems

-Plot sin x and cos x on same axes using manila paper
-Mark key points (0°, 90°, 180°, 270°, 360°)
-Measure and compare wave characteristics
-Practice reading values from completed graphs

Exercise books
-Manila paper
-Identity reference sheet
-Calculators
-String/rope
-Wave diagrams
Exercise books
-Manila paper
-Rulers
-Graph paper (if available)
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 99-103
KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Period Changes in Trigonometric Functions
Combined Amplitude and Period Transformations
Phase Angles and Wave Shifts

By the end of the lesson, the learner should be able to:

-Understand effect of coefficient on period
-Plot graphs of y = sin(bx) for different values of b
-Calculate periods of transformed functions
-Apply period changes to cyclical phenomena

-Plot y = sin(2x), y = sin(x/2) on manila paper
-Compare periods with y = sin x
-Calculate period using formula 360°/b
-Apply to frequency and musical pitch examples

Exercise books
-Manila paper
-Rulers
-Period calculation charts
-Transformation examples
-Colored pencils
-Phase shift examples

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

General Trigonometric Functions
Cosine Wave Transformations

By the end of the lesson, the learner should be able to:

-Work with y = a sin(bx + c) functions
-Identify amplitude, period, and phase angle
-Plot complex trigonometric functions
-Solve problems involving all transformations

-Plot y = 2 sin(3x + 60°) step by step
-Identify all transformation parameters
-Practice reading values from complex waves
-Apply to real-world periodic phenomena

Exercise books
-Manila paper
-Rulers
-Complex function examples
-Temperature data

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Introduction to Trigonometric Equations
Solving Basic Trigonometric Equations
Quadratic Trigonometric Equations

By the end of the lesson, the learner should be able to:

-Understand concept of trigonometric equations
-Identify that trig equations have multiple solutions
-Solve simple equations like sin x = 0.5
-Find all solutions in given ranges

-Demonstrate using unit circle or graphs
-Show why sin x = 0.5 has multiple solutions
-Practice finding principal values
-Use graphs to identify all solutions in range

Exercise books
-Manila paper
-Unit circle diagrams
-Trigonometric tables
-Calculators
-Solution worksheets
-Factoring techniques
-Substitution examples

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III

Equations Involving Multiple Angles
Using Graphs to Solve Trigonometric Equations

By the end of the lesson, the learner should be able to:

-Solve equations like sin(2x) = 0.5
-Handle double and triple angle cases
-Find solutions for compound angle equations
-Apply to periodic motion problems

-Work through sin(2x) = 0.5 systematically
-Show relationship between 2x solutions and x solutions
-Practice with cos(3x) and tan(x/2) equations
-Apply to pendulum and rotation problems

Exercise books
-Manila paper
-Multiple angle examples
-Real applications
-Rulers
-Graphing examples

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III
Three Dimensional Geometry
Three Dimensional Geometry

Trigonometric Equations with Identities
Introduction to 3D Concepts
Properties of Common Solids

By the end of the lesson, the learner should be able to:

-Use trigonometric identities to solve equations
-Apply sin²θ + cos²θ = 1 in equation solving
-Convert between different trigonometric functions
-Solve equations using multiple identities

-Solve equations using fundamental identity
-Convert tan equations to sin/cos form
-Practice identity-based equation solving
-Work through complex multi-step problems

Exercise books
-Manila paper
-Identity reference sheets
-Complex examples
-Cardboard boxes
-Real 3D objects
-Cardboard
-Scissors
-Tape/glue

KLB Secondary Mathematics Form 4, Pages 109-112

Three Dimensional Geometry

Understanding Planes in 3D Space
Lines in 3D Space
Introduction to Projections
Angle Between Line and Plane - Concept

By the end of the lesson, the learner should be able to:

-Define planes and their properties in 3D
-Identify parallel and intersecting planes
-Understand that planes extend infinitely
-Recognize planes formed by faces of solids

-Understand concept of projection in 3D geometry
-Find projections of points onto planes
-Identify foot of perpendicular from point to plane
-Apply projection concept to shadow problems

-Use books/boards to represent planes
-Demonstrate parallel planes using multiple books
-Show intersecting planes using book corners
-Identify planes in classroom architecture

-Use light source to create shadows (projections)
-Drop perpendiculars from corners to floor
-Identify projections in architectural drawings
-Practice finding feet of perpendiculars

Exercise books
-Manila paper
-Books/boards
-Classroom examples
-Rulers/sticks
-3D models
Exercise books
-Manila paper
-Light source
-3D models
-Protractor
-Rulers/sticks

KLB Secondary Mathematics Form 4, Pages 113-115
KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Calculating Angles Between Lines and Planes
Advanced Line-Plane Angle Problems
Introduction to Plane-Plane Angles

By the end of the lesson, the learner should be able to:

-Calculate angles using right-angled triangles
-Apply trigonometry to 3D angle problems
-Use Pythagoras theorem in 3D contexts
-Solve problems involving cuboids and pyramids

-Work through step-by-step calculations
-Use trigonometric ratios in 3D problems
-Practice with cuboid diagonal problems
-Apply to pyramid and cone angle calculations

Exercise books
-Manila paper
-Calculators
-3D problem diagrams
-Real scenarios
-Problem sets
-Books
-Folded paper

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Finding Angles Between Planes
Complex Plane-Plane Angle Problems

By the end of the lesson, the learner should be able to:

-Construct perpendiculars to find plane angles
-Apply trigonometry to calculate dihedral angles
-Use right-angled triangles in plane intersection
-Solve angle problems in prisms and pyramids

-Work through construction method step-by-step
-Practice finding intersection lines first
-Calculate angles in triangular prisms
-Apply to roof and building angle problems

Exercise books
-Manila paper
-Protractor
-Building examples
-Complex 3D models
-Architecture examples

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Practical Applications of Plane Angles
Understanding Skew Lines
Angle Between Skew Lines

By the end of the lesson, the learner should be able to:

-Apply plane angles to real-world problems
-Solve engineering and construction problems
-Calculate angles in roof structures
-Use in navigation and surveying contexts

-Calculate roof pitch angles
-Solve bridge construction angle problems
-Apply to mining and tunnel excavation
-Use in aerial navigation problems

Exercise books
-Manila paper
-Real engineering data
-Construction examples
-Rulers
-Building frameworks
-Translation examples

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Advanced Skew Line Problems
Distance Calculations in 3D

By the end of the lesson, the learner should be able to:

-Solve complex skew line angle calculations
-Apply to engineering and architectural problems
-Use systematic approach for difficult problems
-Combine with other 3D geometric concepts

-Work through power line and cable problems
-Solve bridge and tower construction angles
-Practice with space frame structures
-Apply to antenna and communication tower problems

Exercise books
-Manila paper
-Engineering examples
-Structure diagrams
-Distance calculation charts
-3D coordinate examples

KLB Secondary Mathematics Form 4, Pages 128-135

Three Dimensional Geometry

Volume and Surface Area Applications
Coordinate Geometry in 3D

By the end of the lesson, the learner should be able to:

-Connect 3D geometry to volume calculations
-Apply angle calculations to surface area problems
-Use 3D relationships in optimization
-Solve practical volume and area problems

-Calculate slant heights using 3D angles
-Find surface areas of pyramids using angles
-Apply to packaging and container problems
-Use in architectural space planning

Exercise books
-Manila paper
-Volume formulas
-Real containers
-3D coordinate grid
-Room corner reference

KLB Secondary Mathematics Form 4, Pages 115-135

Three Dimensional Geometry
Longitudes and Latitudes
Longitudes and Latitudes

Integration with Trigonometry
Introduction to Earth as a Sphere
Great and Small Circles
Understanding Latitude
Properties of Latitude Lines

By the end of the lesson, the learner should be able to:

-Apply trigonometry extensively to 3D problems
-Use multiple trigonometric ratios in solutions
-Combine trigonometry with 3D geometric reasoning
-Solve complex problems requiring trig and geometry

-Define latitude and its measurement
-Identify equator as 0° latitude reference
-Understand North and South latitude designations
-Recognize that latitude ranges from 0° to 90°

-Work through problems requiring sin, cos, tan
-Use trigonometric identities in 3D contexts
-Practice angle calculations in pyramids
-Apply to navigation and astronomy problems

-Mark latitude lines on globe using tape
-Show equator as reference line (0°)
-Demonstrate measurement from equator to poles
-Practice identifying latitude positions

Exercise books
-Manila paper
-Trigonometric tables
-Astronomy examples
-Globe/spherical ball
-Chalk/markers
-Globe
-String
Exercise books
-Globe
-Tape/string
-Protractor
-Calculator
-Manila paper

KLB Secondary Mathematics Form 4, Pages 115-135
KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Understanding Longitude
Properties of Longitude Lines
Position of Places on Earth

By the end of the lesson, the learner should be able to:

-Define longitude and its measurement
-Identify Greenwich Meridian as 0° longitude reference
-Understand East and West longitude designations
-Recognize that longitude ranges from 0° to 180°

-Mark longitude lines on globe using string
-Show Greenwich Meridian as reference line
-Demonstrate measurement East and West from Greenwich
-Practice identifying longitude positions

Exercise books
-Globe
-String
-World map
-Manila paper
-Kenya map

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Latitude and Longitude Differences
Introduction to Distance Calculations

By the end of the lesson, the learner should be able to:

-Calculate latitude differences between two points
-Calculate longitude differences between two points
-Understand angular differences on same and opposite sides
-Apply difference calculations to navigation problems

-Calculate difference between Nairobi and Cairo
-Practice with points on same and opposite sides
-Work through systematic calculation methods
-Apply to real navigation scenarios

Exercise books
-Manila paper
-Calculator
-Navigation examples
-Globe
-Conversion charts

KLB Secondary Mathematics Form 4, Pages 139-143

Longitudes and Latitudes

Distance Along Great Circles
Distance Along Small Circles (Parallels)

By the end of the lesson, the learner should be able to:

-Calculate distances along meridians (longitude lines)
-Calculate distances along equator
-Apply formula: distance = angle × 60 nm
-Convert distances between nautical miles and kilometers

-Calculate distance from Nairobi to Cairo (same longitude)
-Find distance between two points on equator
-Practice conversion between units
-Apply to real geographical examples

Exercise books
-Manila paper
-Calculator
-Real examples
-African city examples

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Shortest Distance Problems
Advanced Distance Calculations
Introduction to Time and Longitude

By the end of the lesson, the learner should be able to:

-Understand that shortest distance is along great circle
-Compare great circle and parallel distances
-Calculate shortest distances between any two points
-Apply to navigation and flight path problems

-Compare distances: parallel vs great circle routes
-Calculate shortest distance between London and New York
-Apply to aircraft flight planning
-Discuss practical navigation implications

Exercise books
-Manila paper
-Calculator
-Flight path examples
-Surveying examples
-Globe
-Light source
-Time zone examples

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Local Time Calculations
Greenwich Mean Time (GMT)

By the end of the lesson, the learner should be able to:

-Calculate local time differences between places
-Understand that places east are ahead in time
-Apply rule: 4 minutes per degree of longitude
-Solve time problems involving East-West positions

-Calculate time difference between Nairobi and London
-Practice with cities at various longitudes
-Apply East-ahead, West-behind rule consistently
-Work through systematic time calculation method

Exercise books
-Manila paper
-World time examples
-Calculator
-World map
-Time zone charts

KLB Secondary Mathematics Form 4, Pages 156-161

Longitudes and Latitudes

Complex Time Problems
Speed Calculations

By the end of the lesson, the learner should be able to:

-Solve time problems involving date changes
-Handle calculations crossing International Date Line
-Apply to travel and communication scenarios
-Calculate arrival times for international flights

-Work through International Date Line problems
-Calculate flight arrival times across time zones
-Apply to international communication timing
-Practice with business meeting scheduling

Exercise books
-Manila paper
-International examples
-Travel scenarios
-Calculator
-Navigation examples

KLB Secondary Mathematics Form 4, Pages 156-161