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

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

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

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

KLB Secondary Mathematics Form 4, Pages 1-5

Matrices and Transformation

Single Matrix for Successive Transformations
Inverse of a Transformation
Properties of Inverse Transformations
Area Scale Factor and Determinant
Shear Transformations
Stretch Transformations
Combined Shear and Stretch Problems

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

-Find single matrix equivalent to successive transformations
-Apply commutativity properties in matrix multiplication
-Determine order of operations in transformations
-Solve complex transformation problems efficiently

-Demonstrate equivalence of successive and single matrices
-Practice finding single equivalent matrices
-Compare geometric and algebraic approaches
-Solve real-world transformation problems

Exercise books
-Manila paper
-Ruler
-Chalk/markers
det A
-Cardboard pieces
-Rubber bands

KLB Secondary Mathematics Form 4, Pages 21-24

Matrices and Transformation
Statistics II
Statistics II
Statistics II
Statistics II
Statistics II
Statistics II

Isometric and Non-isometric Transformations
Introduction to Advanced Statistics
Working Mean Concept
Mean Using Working Mean - Simple Data
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:

-Distinguish between isometric and non-isometric transformations
-Classify transformations based on shape and size preservation
-Identify isometric transformations from matrices
-Apply classification to solve problems

-Compare congruent and non-congruent images using cutouts
-Classify transformations systematically
-Practice identification from matrices
-Discuss real-world applications of each type

Exercise books
-Paper cutouts
-Manila paper
-Ruler
-Real data examples
-Chalk/markers
-Sample datasets
-Student data
-Community data
-Real datasets
-Economic data

KLB Secondary Mathematics Form 4, Pages 35-38

Statistics II

Introduction to Quartiles, Deciles, Percentiles
Calculating Quartiles for Ungrouped Data
Quartiles for Grouped Data
Deciles and Percentiles Calculations
Introduction to Cumulative Frequency
Drawing Cumulative Frequency Curves (Ogives)

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
-Grade data
-Performance data
-Ruler
-Class data
-Pencils

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Reading Values from Ogives
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:

-Read median from cumulative frequency curve
-Find quartiles using ogive
-Estimate any percentile from the curve
-Interpret readings in real-world context

-Demonstrate reading techniques on large ogive
-Practice finding median position (n/2)
-Read quartile positions systematically
-Students practice reading their own curves

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

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II
Loci
Loci
Loci

Variance Using Alternative Formula
Standard Deviation Calculations
Standard Deviation for Grouped Data
Advanced Standard Deviation Techniques
Introduction to Loci
Basic Locus Concepts and Laws
Perpendicular Bisector Locus

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
-Agricultural data
-Transformation examples
-String
-Real objects
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 65-70

Loci

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

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
-Set square
-Protractor

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Advanced Constant Angle Constructions
Introduction to Intersecting Loci
Intersecting Circles and Lines
Triangle Centers Using Intersecting Loci
Complex Intersecting Loci Problems
Introduction to Loci of Inequalities

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

-Construct constant angle loci for various angles
-Find centers of constant angle arcs
-Solve complex constant angle problems
-Apply to geometric theorem proving

-Find centers for 60°, 90°, 120° angle loci
-Construct major and minor arcs
-Solve problems involving multiple angle constraints
-Verify constructions using measurement

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

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

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
-Real problem data

KLB Secondary Mathematics Form 4, Pages 89-92

Loci
Trigonometry III
Trigonometry III
Trigonometry III
Trigonometry III
Trigonometry III
Trigonometry III

Integration of All Loci Types
Review of Basic Trigonometric Ratios
Deriving the Identity sin²θ + cos²θ = 1
Applications of sin²θ + cos²θ = 1
Additional Trigonometric Identities
Introduction to Waves
Sine and Cosine Waves

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

-Combine different types of loci in single problems
-Solve comprehensive loci challenges
-Apply multiple loci concepts simultaneously
-Use loci in geometric investigations

-Solve multi-step loci problems
-Combine circle, line, and angle loci
-Apply to real-world complex scenarios
-Practice systematic problem-solving

Exercise books
-Manila paper
-Compass
-Ruler
-Rulers
-Calculators (if available)
-Unit circle diagrams
-Calculators
-Trigonometric tables
-Real-world examples
-Identity reference sheet
-String/rope
-Wave diagrams
-Graph paper (if available)

KLB Secondary Mathematics Form 4, Pages 73-94

Trigonometry III

Transformations of Sine Waves
Period Changes in Trigonometric Functions
Combined Amplitude and Period Transformations
Phase Angles and Wave Shifts
General Trigonometric Functions
Cosine Wave Transformations

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

-Understand effect of coefficient on amplitude
-Plot graphs of y = k sin x for different values of k
-Compare transformed waves with basic sine wave
-Apply amplitude changes to real situations

-Plot y = 2 sin x, y = 3 sin x on manila paper
-Compare amplitudes with y = sin x
-Demonstrate stretching effect of coefficient
-Apply to sound volume or signal strength examples

Exercise books
-Manila paper
-Colored pencils
-Rulers
-Period calculation charts
-Transformation examples
-Phase shift examples
-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
Equations Involving Multiple Angles
Using Graphs to Solve Trigonometric Equations
Trigonometric Equations with Identities

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
-Multiple angle examples
-Real applications
-Rulers
-Graphing examples
-Identity reference sheets
-Complex examples

KLB Secondary Mathematics Form 4, Pages 109-112

Three Dimensional Geometry

Introduction to 3D Concepts
Properties of Common Solids
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:

-Distinguish between 1D, 2D, and 3D objects
-Identify vertices, edges, and faces of 3D solids
-Understand concepts of points, lines, and planes in space
-Recognize real-world 3D objects and their properties

-Use classroom objects to demonstrate dimensions
-Count vertices, edges, faces of cardboard models
-Identify 3D shapes in school environment
-Discuss difference between area and volume

Exercise books
-Cardboard boxes
-Manila paper
-Real 3D objects
-Cardboard
-Scissors
-Tape/glue
-Books/boards
-Classroom examples
-Rulers/sticks
-3D models
-Light source
-Protractor

KLB Secondary Mathematics Form 4, Pages 113-115

Three Dimensional Geometry

Calculating Angles Between Lines and Planes
Advanced Line-Plane Angle Problems
Introduction to Plane-Plane Angles
Finding Angles Between Planes
Complex Plane-Plane Angle Problems
Practical Applications of Plane Angles
Understanding Skew Lines

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
-Protractor
-Building examples
-Complex 3D models
-Architecture examples
-Real engineering data
-Construction examples
-Rulers
-Building frameworks

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Angle Between Skew Lines
Advanced Skew Line Problems
Distance Calculations in 3D
Volume and Surface Area Applications
Coordinate Geometry in 3D
Integration with Trigonometry

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

-Understand how to find angle between skew lines
-Apply translation method for skew line angles
-Use parallel line properties in 3D
-Calculate angles by creating intersecting lines

-Demonstrate translation method using rulers
-Translate one line to intersect the other
-Practice with cuboid edge problems
-Apply to framework and structure problems

Exercise books
-Manila paper
-Rulers
-Translation examples
-Engineering examples
-Structure diagrams
-Distance calculation charts
-3D coordinate examples
-Volume formulas
-Real containers
-3D coordinate grid
-Room corner reference
-Trigonometric tables
-Astronomy examples

KLB Secondary Mathematics Form 4, Pages 128-135

Longitudes and Latitudes

Introduction to Earth as a Sphere
Great and Small Circles
Understanding Latitude
Properties of Latitude Lines
Understanding Longitude
Properties of Longitude Lines

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

-Understand Earth as a sphere for mathematical purposes
-Identify poles, equator, and axis of rotation
-Recognize Earth's dimensions and basic structure
-Connect Earth's rotation to day-night cycle

-Use globe or spherical ball to demonstrate Earth
-Identify North Pole, South Pole, and equator
-Discuss Earth's rotation and its effects
-Show axis of rotation through poles

Exercise books
-Globe/spherical ball
-Manila paper
-Chalk/markers
-Globe
-String
-Tape/string
-Protractor
-Calculator
-World map

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Position of Places on Earth
Latitude and Longitude Differences
Introduction to Distance Calculations
Distance Along Great Circles
Distance Along Small Circles (Parallels)
Shortest Distance Problems
Advanced Distance Calculations

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

-Express position using latitude and longitude coordinates
-Use correct notation for positions (e.g., 1°S, 37°E)
-Identify positions of major Kenyan cities
-Locate places given their coordinates

-Find positions of Nairobi, Mombasa, Kisumu on globe
-Practice writing coordinates in correct format
-Locate cities worldwide using coordinates
-Use maps to verify coordinate positions

Exercise books
-Globe
-World map
-Kenya map
-Manila paper
-Calculator
-Navigation examples
-Conversion charts
-Real examples
-African city examples
-Flight path examples
-Surveying examples

KLB Secondary Mathematics Form 4, Pages 139-143

Longitudes and Latitudes
Linear Programming

Introduction to Time and Longitude
Local Time Calculations
Greenwich Mean Time (GMT)
Complex Time Problems
Speed Calculations
Introduction to Linear Programming

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

-Understand relationship between longitude and time
-Learn that Earth rotates 360° in 24 hours
-Calculate that 15° longitude = 1 hour time difference
-Understand concept of local time

-Demonstrate Earth's rotation using globe
-Show how sun position determines local time
-Calculate time differences for various longitudes
-Apply to understanding sunrise/sunset times

Exercise books
-Globe
-Light source
-Time zone examples
-Manila paper
-World time examples
-Calculator
-World map
-Time zone charts
-International examples
-Travel scenarios
-Navigation examples
-Real-life examples
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 156-161

Linear Programming

Forming Linear Inequalities from Word Problems
Types of Constraints
Objective Functions
Complete Problem Formulation
Introduction to Graphical Solution Method
Plotting Multiple Constraints

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

-Translate real-world constraints into mathematical inequalities
-Identify decision variables in word problems
-Form inequalities from resource limitations
-Use correct mathematical notation for constraints

-Work through farmer's crop planning problem
-Practice translating budget constraints into inequalities
-Form inequalities from production capacity limits
-Use Kenyan business examples for relevance

Exercise books
-Manila paper
-Local business examples
-Agricultural scenarios
-Industry examples
-School scenarios
-Business examples
-Production scenarios
-Complete examples
-Systematic templates
-Rulers
-Colored pencils
-Different colored pencils

KLB Secondary Mathematics Form 4, Pages 165-167

Linear Programming

Properties of Feasible Regions
Introduction to Optimization
The Corner Point Method
The Iso-Profit/Iso-Cost Line Method
Comparing Solution Methods
Business Applications - Production Planning

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

-Understand that feasible region is convex
-Identify corner points (vertices) of feasible region
-Understand significance of corner points
-Calculate coordinates of corner points

-Identify all corner points of feasible region
-Calculate intersection points algebraically
-Verify corner points satisfy all constraints
-Understand why corner points are important

Exercise books
-Manila paper
-Calculators
-Algebraic methods
-Evaluation tables
-Evaluation templates
-Systematic approach
-Rulers
-Sliding technique
-Method comparison
-Verification examples
-Manufacturing examples
-Kenyan industry data

KLB Secondary Mathematics Form 4, Pages 166-172

Differentiation

Introduction to Rate of Change
Average Rate of Change
Instantaneous Rate of Change
Gradient of Curves at Points
Introduction to Delta Notation
The Limiting Process
Introduction to Derivatives

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

-Understand concept of rate of change in daily life
-Distinguish between average and instantaneous rates
-Identify examples of changing quantities
-Connect rate of change to gradient concepts

-Discuss speed as rate of change of distance
-Examine population growth rates
-Analyze temperature change throughout the day
-Connect to gradients of lines from coordinate geometry

Exercise books
-Manila paper
-Real-world examples
-Graph examples
-Calculators
-Graph paper
-Tangent demonstrations
-Motion examples
-Rulers
-Curve examples
-Delta notation examples
-Symbol practice
-Limit tables
-Systematic examples
-Derivative notation
-Function examples

KLB Secondary Mathematics Form 4, Pages 177-182

Differentiation

Derivative of Linear Functions
Derivative of y = x^n (Basic Powers)
Derivative of Constant Functions
Derivative of Coefficient Functions
Derivative of Polynomial Functions
Applications to Tangent Lines

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

-Find derivatives of linear functions y = mx + c
-Understand that derivative of linear function is constant
-Apply to straight line gradient problems
-Verify using limiting process

-Find derivative of y = 3x + 2 using definition
-Show that derivative equals the gradient
-Practice with various linear functions
-Verify results using first principles

Exercise books
-Manila paper
-Linear function examples
-Verification methods
-Power rule examples
-First principles verification
-Constant function graphs
-Geometric explanations
-Coefficient examples
-Rule combinations
-Polynomial examples
-Term-by-term method
-Tangent line examples
-Point-slope applications

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Applications to Normal Lines
Introduction to Stationary Points
Types of Stationary Points
Finding and Classifying Stationary Points
Curve Sketching Using Derivatives
Introduction to Kinematics Applications

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

-Find equations of normal lines to curves
-Use negative reciprocal of tangent gradient
-Apply to perpendicular line problems
-Practice with normal line calculations

-Find normal to y = x² at point (2, 4)
-Use negative reciprocal relationship
-Apply perpendicular line concepts
-Practice normal line equation finding

Exercise books
-Manila paper
-Normal line examples
-Perpendicular concepts
-Curve sketches
-Stationary point examples
-Sign analysis charts
-Classification examples
-Systematic templates
-Complete examples
-Curve sketching templates
-Systematic method
-Motion examples
-Kinematics applications

KLB Secondary Mathematics Form 4, Pages 187-189

Differentiation
Matrices and Transformations
Matrices and Transformations

Acceleration as Second Derivative
Motion Problems and Applications
Introduction to Optimization
Geometric Optimization Problems
Business and Economic Applications
Advanced Optimization Problems
Transformation on a Cartesian plane
Basic Transformation Matrices

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

-Understand acceleration as derivative of velocity
-Apply a = dv/dt = d²s/dt² notation
-Find acceleration functions from displacement
-Apply to motion analysis problems

-Find acceleration from velocity functions
-Use second derivative notation
-Apply to projectile motion problems
-Practice with particle motion scenarios

Exercise books
-Manila paper
-Second derivative examples
-Motion analysis
-Complete motion examples
-Real scenarios
-Optimization examples
-Real applications
-Geometric examples
-Design applications
-Business examples
-Economic applications
-Complex examples
-Engineering applications
Square boards
-Peg boards
-Graph papers
-Mirrors
-Rulers
-Protractors
-Calculators

KLB Secondary Mathematics Form 4, Pages 197-201

Matrices and Transformations

Identification of transformation matrix
Two Successive Transformations
Complex Successive Transformations
Single matrix of transformation for successive transformations
Matrix Multiplication Properties
Identity Matrix and Transformation
Inverse of a matrix
Determinant and Area Scale Factor
Area scale factor and determinant relationship

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

-Determine transformation matrix from object and image coordinates
-Identify type of transformation from given matrix
-Use algebraic methods to find unknown matrices
-Classify transformations based on matrix properties

-Worked examples finding matrices from coordinate pairs
-Analysis of matrix elements to identify transformation type
-Solving simultaneous equations to find matrix elements
-Practice with various transformation identification problems
-Discussion on matrix patterns for each transformation

Graph papers
-Calculators
-Exercise books
-Matrix examples
Square boards
-Peg boards
-Graph papers
-Colored pencils
-Rulers
Calculators
-Matrix multiplication charts
-Matrix worksheets
-Formula sheets
-Solve problems involving area changes under transformations
-Area calculation tools

KLB Secondary Mathematics Form 4, Pages 6-16

Matrices and Transformations
Integration
Integration
Integration
Integration
Integration
Integration
Integration
Integration
Integration
Quadratic Expressions and Equations

Shear Transformation
Stretch Transformation and Review
Introduction to Reverse Differentiation
Basic Integration Rules - Power Functions
Integration of Polynomial Functions
Finding Particular Solutions
Introduction to Definite Integrals
Evaluating Definite Integrals
Area Under Curves - Single Functions
Areas Below X-axis and Mixed Regions
Area Between Two Curves
Factorisation of quadratic expressions

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

-Define shear transformation and its properties
-Find matrices for shear parallel to x-axis and y-axis
-Calculate images under shear transformations
-Understand that shear preserves area but changes shape

-Physical demonstration using flexible materials
-Derivation of shear transformation matrices
-Drawing effects of shear on rectangles and parallelograms
-Verification that area is preserved under shear
-Practice exercises Ex 1.6

Square boards
-Flexible materials
-Graph papers
-Rulers
-Calculators
Graph papers
-Elastic materials
-Comparison charts
-Review materials
-Differentiation charts
-Exercise books
-Function examples
Calculators
-Power rule charts
-Algebraic worksheets
-Polynomial examples
-Curve examples
-Geometric models
-Integration notation charts
-Step-by-step worksheets
-Evaluation charts
-Curve sketching tools
-Colored pencils
-Area grids
-Colored materials
-Equation solving aids
Calculators, charts showing factorization patterns

KLB Secondary Mathematics Form 4, Pages 10-13, 28-34

Quadratic Expressions and Equations

Factorisation of quadratic expressions
Completing squares
Completing squares
Solving quadratic expressions by completing square
Solving quadratic expressions by factorization
The quadratic formula
The quadratic formula
Formation of quadratic equations
Graphs of quadratic functions
Graphs of quadratic functions

By the end of the lesson, the learner should be able to:
Factorize quadratic expressions using different methods
Identify common factors in expressions
Apply grouping method to factorize

Q/A on previous lesson concepts
Discussions on advanced factorization
Solving complex factorization problems
Demonstrations of grouping methods
Explaining various factorization techniques

Calculators, factorization method charts
Calculators, perfect square charts
Calculators, vertex form examples
Calculators, equation solving guides
Calculators, method selection charts
Calculators, formula derivation charts
Calculators, discriminant interpretation guides
Calculators, word problem templates
Graph papers, calculators, plotting guides
Graph papers, calculators, rulers

KLB Mathematics Book Three Pg 1-2

Quadratic Expressions and Equations
Approximations and Errors
Approximations and Errors
Approximations and Errors

Graphical solutions of quadratic equation
Graphical solutions of simultaneous equations
Computing using calculators
Computing using calculators
Approximation

By the end of the lesson, the learner should be able to:
Draw graphs of quadratic functions
Solve quadratic equations using the graphs
Find roots as x-intercepts

Q/A on graph-equation relationships
Discussions on graphical solutions
Solving equations graphically
Demonstrations of root finding
Explaining intersection concepts

Graph papers, calculators, rulers
Graph papers, calculators, estimation guides
Graph papers, calculators, intersection analysis guides
Calculators, operation guides
Calculators, verification worksheets
Calculators, rounding charts

KLB Mathematics Book Three Pg 15-17

Approximations and Errors

Estimation
Accuracy and errors
Percentage error
Rounding off error and truncation error
Propagation of errors
Propagation of errors

By the end of the lesson, the learner should be able to:
Approximate values by truncation
Estimate values using appropriate methods
Compare estimation techniques

Q/A on estimation strategies
Discussions on truncation vs rounding
Solving estimation problems
Demonstrations of truncation methods
Explaining when to use different techniques

Calculators, estimation guides
Calculators, error calculation sheets
Calculators, percentage error worksheets
Calculators, error comparison charts
Calculators, error propagation guides
Calculators, verification worksheets

KLB Mathematics Book Three Pg 30

Approximations and Errors
Trigonometry (II)
Trigonometry (II)

Propagation of errors in multiplication
Propagation of errors in division
Propagation of errors in division
Word problems
The unit circle
The unit circle

By the end of the lesson, the learner should be able to:
Find the propagation of errors in multiplication
Calculate relative errors in products
Apply multiplication error rules

Q/A on multiplication error concepts
Discussions on product error calculation
Solving multiplication error problems
Demonstrations of relative error computation
Explaining multiplication error principles

Calculators, multiplication error guides
Calculators, method comparison charts
Calculators, division error worksheets
Calculators, verification guides
Calculators, word problem sets, comprehensive review sheets
Calculators, protractors, rulers, pair of compasses

KLB Mathematics Book Three Pg 36-37

Trigonometry (II)

Trigonometric ratios of angles greater than 90°
Trigonometric ratios of negative angles
Trigonometric ratios of angles greater than 360°
Use of mathematical tables
Use of mathematical tables

By the end of the lesson, the learner should be able to:
Find the trigonometric values of angles
Calculate trigonometric ratios for obtuse angles
Apply reference angle concepts

Q/A on basic trigonometric ratios
Discussions on angle extensions
Solving obtuse angle problems
Demonstrations of reference angles
Explaining quadrant relationships

Calculators, protractors, rulers, pair of compasses
Calculators, quadrant charts
Geoboards, graph books, calculators
Mathematical tables, calculators

KLB Mathematics Book Three Pg 44-45

Trigonometry (II)

Use of calculators
Radian measure
Simple trigonometric graphs
Graphs of cosines
Graphs of tan
The sine rule

By the end of the lesson, the learner should be able to:
Use calculators to find sine, cosine and tan
Apply calculator functions for trigonometry
Verify calculator accuracy

Q/A on calculator trigonometric functions
Discussions on calculator modes
Solving problems using calculators
Demonstrations of function keys
Explaining degree vs radian modes

Calculators, function guides
Calculators, conversion charts
Calculators, graph papers, plotting guides
Calculators, triangle worksheets

KLB Mathematics Book Three Pg 56-58

Trigonometry (II)
Surds
Surds
Surds
Surds

Cosine rule
Problem solving
Rational and irrational numbers
Order of surds and simplification
Simplification of surds practice
Addition of surds

By the end of the lesson, the learner should be able to:
State the cosine rule
Apply cosine rule to find solution of triangles
Choose appropriate rule for triangle solving

Q/A on cosine rule concepts
Discussions on rule selection
Solving complex triangle problems
Demonstrations of cosine rule
Explaining when to use each rule

Calculators, triangle worksheets
Calculators, comprehensive problem sets, real-world examples
Calculators, number classification charts
Calculators, surd order examples
Calculators, factor trees, simplification worksheets
Calculators, addition rule charts

KLB Mathematics Book Three Pg 71-75

Surds
Further Logarithms
Further Logarithms

Subtraction of surds
Multiplication of surds
Division of surds
Rationalizing the denominator
Advanced rationalization techniques
Introduction
Laws of logarithms

By the end of the lesson, the learner should be able to:
Subtract surds with like terms
Apply subtraction rules to surds
Simplify surd subtraction expressions

Q/A on subtraction principles
Discussions on surd subtraction methods
Solving subtraction problems
Demonstrations of systematic approaches
Explaining subtraction verification

Calculators, subtraction worksheets
Calculators, multiplication rule guides
Calculators, division worksheets
Calculators, rationalization guides
Calculators, advanced technique sheets
Calculators, logarithm definition charts
Calculators, logarithm law charts

KLB Mathematics Book Three Pg 80

Further Logarithms

Laws of logarithms
Logarithmic equations and expressions
Logarithmic equations and expressions
Further computation using logarithms
Further computation using logarithms

By the end of the lesson, the learner should be able to:
Use laws of logarithms to solve problems
Apply advanced logarithmic laws
Combine multiple laws in calculations

Q/A on law mastery and applications
Discussions on power and root laws
Solving complex law-based problems
Demonstrations of combined law usage
Explaining advanced law techniques

Calculators, advanced law worksheets
Calculators, challenging problem sets
Calculators, equation-solving guides
Calculators, advanced equation worksheets
Calculators, computation worksheets
Calculators, intermediate problem sets

KLB Mathematics Book Three Pg 90-93

Further Logarithms
Commercial Arithmetic
Commercial Arithmetic
Commercial Arithmetic

Further computation using logarithms
Problem solving
Problem solving
Simple interest
Simple interest
Compound interest

By the end of the lesson, the learner should be able to:
Solve problems involving logarithms
Master advanced logarithmic computations
Apply logarithms to complex mathematical scenarios

Q/A on advanced computational mastery
Discussions on complex calculation strategies
Solving advanced computation problems
Demonstrations of sophisticated methods
Explaining optimal computational approaches

Calculators, advanced computation guides
Calculators, comprehensive problem sets
Calculators, real-world application examples
Calculators, simple interest charts
Calculators, real-world problem sets
Calculators, compound interest tables

KLB Mathematics Book Three Pg 95-96

Commercial Arithmetic
Circles: Chords and Tangents

Compound interest
Appreciation
Depreciation
Hire purchase
Hire purchase
Income tax and P.A.Y.E
Length of an arc

By the end of the lesson, the learner should be able to:
Calculate the compound interest
Solve advanced compound interest problems
Compare simple and compound interest

Q/A on advanced compounding scenarios
Discussions on investment comparisons
Solving complex compound problems
Demonstrations of comparison methods
Explaining investment decisions

Calculators, comparison worksheets
Calculators, appreciation examples
Calculators, depreciation charts
Calculators, hire purchase examples
Calculators, complex hire purchase worksheets
Income tax tables, calculators
Geometrical set, calculators

KLB Mathematics Book Three Pg 102-107

Circles: Chords and Tangents

Length of an arc
Chords
Parallel chords
Equal chords
Intersecting chords
Intersecting chords

By the end of the lesson, the learner should be able to:
Calculate the length of an arc
Solve complex arc length problems
Apply arc concepts to real situations

Q/A on advanced arc applications
Discussions on practical arc measurements
Solving complex arc problems
Demonstrations of real-world applications
Explaining engineering and design uses

Geometrical set, calculators

KLB Mathematics Book Three Pg 124-125

Circles: Chords and Tangents

Chord properties
Tangent to a circle
Tangent to a circle
Properties of tangents to a circle from an external point
Tangent properties
Tangents to two circles

By the end of the lesson, the learner should be able to:
Solve comprehensive chord problems
Integrate all chord concepts
Apply chord knowledge systematically

Q/A on comprehensive chord understanding
Discussions on integrated problem-solving
Solving mixed chord problems
Demonstrations of systematic approaches
Explaining complete chord mastery

Geometrical set, calculators

KLB Mathematics Book Three Pg 126-139

Circles: Chords and Tangents

Tangents to two circles
Contact of circles
Contact of circles
Circle contact
Angle in alternate segment
Angle in alternate segment

By the end of the lesson, the learner should be able to:
Calculate the tangents of transverse common tangents
Find transverse tangent properties
Compare direct and transverse tangents

Q/A on transverse tangent concepts
Discussions on tangent type differences
Solving transverse tangent problems
Demonstrations of comparison methods
Explaining tangent classifications

Geometrical set, calculators

KLB Mathematics Book Three Pg 150-151

Circles: Chords and Tangents
Matrices
Matrices
Matrices
Matrices

Circumscribed circle
Escribed circles
Centroid
Orthocenter
Circle and triangle relationships
Introduction and real-life applications
Order of a matrix and elements
Square matrices, row and column matrices
Addition of matrices

By the end of the lesson, the learner should be able to:
Construct circumscribed circles
Find circumscribed circle properties
Apply circumscription concepts

Q/A on circumscription concepts
Discussions on circumscribed circle construction
Solving circumscription problems
Demonstrations of construction techniques
Explaining circumscription applications

Geometrical set, calculators
Old newspapers with league tables, chalk and blackboard, exercise books
Chalk and blackboard, ruled exercise books, class register
Paper cutouts, chalk and blackboard, counters or bottle tops
Counters or stones, chalk and blackboard, exercise books

KLB Mathematics Book Three Pg 165

Matrices

Subtraction of matrices
Combined addition and subtraction
Scalar multiplication
Introduction to matrix multiplication
Matrix multiplication (2×2 matrices)
Matrix multiplication (larger matrices)
Properties of matrix multiplication
Real-world matrix multiplication applications

By the end of the lesson, the learner should be able to:
Subtract matrices of the same order
Apply matrix subtraction rules correctly
Understand order requirements for subtraction
Solve complex matrix subtraction problems

Q/A on matrix subtraction using simple numbers
Discussions on element-wise subtraction using examples
Solving subtraction problems on blackboard
Demonstrations using number line concepts
Explaining sign changes using practical examples

Chalk and blackboard, exercise books, number cards made from cardboard
Chalk and blackboard, exercise books, locally made operation cards
Beans or stones for grouping, chalk and blackboard, exercise books
Chalk and blackboard, rulers for tracing, exercise books
Chalk and blackboard, exercise books, homemade grid templates
Chalk and blackboard, large sheets of paper for working, exercise books
Chalk and blackboard, exercise books, cardboard for property cards
Chalk and blackboard, local price lists, exercise books

KLB Mathematics Book Three Pg 170-171

Matrices

Identity matrix
Determinant of 2×2 matrices
Inverse of 2×2 matrices - theory
Inverse of 2×2 matrices - practice
Introduction to solving simultaneous equations
Solving 2×2 simultaneous equations using matrices
Advanced simultaneous equation problems

By the end of the lesson, the learner should be able to:
Define and identify identity matrices
Understand identity matrix properties
Apply identity matrices in multiplication
Recognize the multiplicative identity role

Q/A on identity concepts using number 1 analogy
Discussions on multiplicative identity using examples
Solving identity problems using pattern recognition
Demonstrations using multiplication by 1 concept
Explaining diagonal properties using visual patterns

Chalk and blackboard, exercise books, pattern cards made from paper
Chalk and blackboard, exercise books, crossed sticks for demonstration
Chalk and blackboard, exercise books, fraction examples
Chalk and blackboard, exercise books, scrap paper for verification
Chalk and blackboard, exercise books, equation examples from previous topics
Chalk and blackboard, exercise books, previous elimination method examples
Chalk and blackboard, exercise books, graph paper if available

KLB Mathematics Book Three Pg 182-183

Matrices
Formulae and Variations
Formulae and Variations
Formulae and Variations

Matrix applications in real-world problems
Transpose of matrices
Matrix equation solving
Introduction to formulae
Subject of a formula - basic cases
Subject of a formula - intermediate cases

By the end of the lesson, the learner should be able to:
Apply matrix operations to practical scenarios
Solve business, engineering, and scientific problems
Model real situations using matrices
Interpret matrix solutions in context

Q/A on practical applications using local examples
Discussions on modeling using familiar situations
Solving comprehensive problems using matrix tools
Demonstrations using community-based scenarios
Explaining solution interpretation using meaningful contexts

Chalk and blackboard, local business examples, exercise books
Chalk and blackboard, exercise books, paper cutouts for demonstration
Chalk and blackboard, exercise books, algebra reference examples
Chalk and blackboard, measuring tape or string, exercise books
Chalk and blackboard, simple balance (stones and stick), exercise books
Chalk and blackboard, fraction strips made from paper, exercise books

KLB Mathematics Book Three Pg 168-190

Formulae and Variations
Sequences and Series
Sequences and Series

Subject of a formula - advanced cases
Applications of formula manipulation
Introduction to variation
Direct variation - introduction
Introduction to sequences and finding terms
General term of sequences and applications

By the end of the lesson, the learner should be able to:
Make variables subject in complex formulae
Handle square roots and quadratic expressions
Apply advanced algebraic manipulation
Solve challenging subject transformation problems

Q/A on advanced manipulation using careful steps
Discussions on square root handling using examples
Solving complex problems using systematic approach
Demonstrations using detailed blackboard work
Explaining quadratic handling using factoring

Chalk and blackboard, squared paper patterns, exercise books
Chalk and blackboard, local measurement tools, exercise books
Chalk and blackboard, local price lists from markets, exercise books
Chalk and blackboard, beans or stones for counting, exercise books
Chalk and blackboard, stones or beans for patterns, exercise books
Chalk and blackboard, numbered cards made from paper, exercise books

KLB Mathematics Book Three Pg 191-193

Sequences and Series

Arithmetic sequences and nth term
Arithmetic sequence applications
Geometric sequences and nth term
Geometric sequence applications
Arithmetic series and sum formula
Geometric series and applications
Mixed problems and advanced applications

By the end of the lesson, the learner should be able to:
Define arithmetic sequences and common differences
Calculate common differences correctly
Derive and apply the nth term formula
Solve problems using arithmetic sequence concepts

Q/A on arithmetic patterns using step-by-step examples
Discussions on constant difference patterns and formula derivation
Solving arithmetic sequence problems systematically
Demonstrations using equal-step progressions
Explaining formula structure using algebraic reasoning

Chalk and blackboard, measuring tape or string, exercise books
Chalk and blackboard, local employment/savings examples, exercise books
Chalk and blackboard, objects for doubling demonstrations, exercise books
Chalk and blackboard, population/growth data examples, exercise books
Chalk and blackboard, counting materials for summation, exercise books
Chalk and blackboard, convergence demonstration materials, exercise books
Chalk and blackboard, mixed problem collections, exercise books

KLB Mathematics Book Three Pg 209-210

Sequences and Series
Vectors (II)
Vectors (II)
Vectors (II)
Vectors (II)
Vectors (II)

Sequences in nature and technology
Coordinates in two dimensions
Coordinates in three dimensions
Column and position vectors in three dimensions
Position vectors and applications
Column vectors in terms of unit vectors i, j, k

By the end of the lesson, the learner should be able to:
Identify mathematical patterns in natural phenomena
Analyze sequences in biological and technological contexts
Apply sequence concepts to environmental problems
Appreciate mathematics in the natural and modern world

Q/A on natural and technological patterns using examples
Discussions on biological sequences and digital applications
Solving nature and technology-based problems
Demonstrations using natural pattern examples
Explaining mathematical beauty using real phenomena

Chalk and blackboard, natural and technology examples, exercise books
Chalk and blackboard, squared paper or grid drawn on ground, exercise books
Chalk and blackboard, 3D models made from sticks and clay, exercise books
Chalk and blackboard, movement demonstration space, exercise books
Chalk and blackboard, origin marking systems, exercise books
Chalk and blackboard, direction indicators, unit vector reference charts, exercise books

KLB Mathematics Book Three Pg 207-219

Vectors (II)

Vector operations using unit vectors
Magnitude of a vector in three dimensions
Magnitude applications and unit vectors
Parallel vectors
Collinearity
Advanced collinearity applications

By the end of the lesson, the learner should be able to:
Express vectors in terms of unit vectors
Perform vector addition using unit vector notation
Calculate vector subtraction with i, j, k components
Apply scalar multiplication to unit vectors

Q/A on vector operations using component-wise calculation
Discussions on systematic operation methods
Solving vector operation problems using organized approaches
Demonstrations using component separation and combination
Explaining operation logic using algebraic reasoning

Chalk and blackboard, component calculation aids, exercise books
Chalk and blackboard, 3D measurement aids, exercise books
Chalk and blackboard, direction finding aids, exercise books
Chalk and blackboard, parallel line demonstrations, exercise books
Chalk and blackboard, straight-line demonstrations, exercise books
Chalk and blackboard, complex geometric aids, exercise books

KLB Mathematics Book Three Pg 226-228

Vectors (II)

Proportional division of a line
External division of a line
Combined internal and external division
Ratio theorem
Advanced ratio theorem applications
Mid-point

By the end of the lesson, the learner should be able to:
Divide a line internally in the given ratio
Apply the internal division formula
Calculate division points using vector methods
Understand proportional division concepts

Q/A on internal division using systematic formula application
Discussions on ratio division using proportional methods
Solving internal division problems using organized approaches
Demonstrations using internal point construction examples
Explaining internal division using geometric visualization

Chalk and blackboard, internal division models, exercise books
Chalk and blackboard, external division models, exercise books
Chalk and blackboard, combined division models, exercise books
Chalk and blackboard, ratio theorem aids, exercise books
Chalk and blackboard, advanced ratio models, exercise books
Chalk and blackboard, midpoint demonstration aids, exercise books

KLB Mathematics Book Three Pg 237-238

Vectors (II)
Binomial Expansion
Binomial Expansion

Ratio theorem and midpoint integration
Advanced ratio theorem applications
Applications of vectors in geometry
Rectangle diagonal applications
Advanced geometric applications
Binomial expansions up to power four
Binomial expansions up to power four (continued)

By the end of the lesson, the learner should be able to:
Use ratio theorem to find the given vectors
Apply midpoint and ratio concepts together
Solve complex ratio and midpoint problems
Integrate division and midpoint methods

Q/A on integrated problem-solving using combined methods
Discussions on complex scenario analysis using systematic approaches
Solving challenging problems using integrated techniques
Demonstrations using comprehensive geometric examples
Explaining integration using logical problem-solving

Chalk and blackboard, complex problem materials, exercise books
Chalk and blackboard, advanced geometric aids, exercise books
Chalk and blackboard, parallelogram models, exercise books
Chalk and blackboard, rectangle models, exercise books
Chalk and blackboard, advanced geometric models, exercise books
Chalk and blackboard, rectangular cutouts from paper, exercise books
Chalk and blackboard, squared paper for geometric models, exercise books

KLB Mathematics Book Three Pg 244-245

Binomial Expansion

Pascal's triangle
Pascal's triangle applications
Pascal's triangle (continued)
Pascal's triangle advanced
Applications to numerical cases
Applications to numerical cases (continued)

By the end of the lesson, the learner should be able to:
Use Pascal's triangle
Construct Pascal's triangle systematically
Apply triangle coefficients for binomial expansions
Recognize number patterns in the triangle

Q/A on triangle construction using addition patterns
Discussions on coefficient relationships using triangle analysis
Solving triangle construction and application problems
Demonstrations using visual triangle building
Explaining pattern connections using systematic observation

Chalk and blackboard, triangular patterns drawn/cut from paper, exercise books
Chalk and blackboard, Pascal's triangle reference charts, exercise books
Chalk and blackboard, advanced triangle patterns, exercise books
Chalk and blackboard, combination calculation aids, exercise books
Chalk and blackboard, simple calculation aids, exercise books
Chalk and blackboard, advanced calculation examples, exercise books

KLB Mathematics Book Three Pg 256-257

Probability

Introduction
Experimental Probability
Experimental Probability applications
Range of Probability Measure
Probability Space
Theoretical Probability

By the end of the lesson, the learner should be able to:
Calculate the experimental probability
Understand probability concepts in daily life
Distinguish between certain and uncertain events
Recognize probability situations

Q/A on uncertain events from daily life experiences
Discussions on weather prediction and game outcomes
Analyzing chance events using coin tossing and dice rolling
Demonstrations using simple probability experiments
Explaining probability language using familiar examples

Chalk and blackboard, coins, dice made from cardboard, exercise books
Chalk and blackboard, coins, cardboard dice, tally charts, exercise books
Chalk and blackboard, extended experimental materials, data recording sheets, exercise books
Chalk and blackboard, number line drawings, probability scale charts, exercise books
Chalk and blackboard, playing cards (locally made), spinners from cardboard, exercise books
Chalk and blackboard, fair dice and coins, probability calculation aids, exercise books

KLB Mathematics Book Three Pg 262-264

Probability

Theoretical Probability advanced
Theoretical Probability applications
Combined Events
Combined Events OR probability
Independent Events
Independent Events advanced
Independent Events applications

By the end of the lesson, the learner should be able to:
Calculate the probability space for the theoretical probability
Apply theoretical probability to complex problems
Handle multiple outcome scenarios
Solve advanced theoretical problems

Q/A on advanced theoretical applications using complex scenarios
Discussions on multiple outcome analysis using systematic methods
Solving challenging theoretical problems using organized approaches
Demonstrations using complex probability setups
Explaining advanced theoretical concepts using detailed reasoning

Chalk and blackboard, complex probability materials, advanced calculation aids, exercise books
Chalk and blackboard, local game examples, practical scenario materials, exercise books
Chalk and blackboard, playing cards, multiple dice, Venn diagram drawings, exercise books
Chalk and blackboard, Venn diagram materials, card examples, exercise books
Chalk and blackboard, multiple coins and dice, independence demonstration materials, exercise books
Chalk and blackboard, playing cards for replacement scenarios, multiple experimental setups, exercise books
Chalk and blackboard, complex experimental materials, advanced calculation aids, exercise books

KLB Mathematics Book Three Pg 268-270

Probability
Compound Proportion and Rates of Work
Compound Proportion and Rates of Work
Compound Proportion and Rates of Work
Compound Proportion and Rates of Work

Tree Diagrams
Tree Diagrams advanced
Compound Proportions
Compound Proportions applications
Proportional Parts
Proportional Parts applications

By the end of the lesson, the learner should be able to:
Draw tree diagrams to show the probability space
Construct tree diagrams systematically
Represent sequential events using trees
Apply tree diagram methods

Q/A on tree construction using step-by-step methods
Discussions on sequential event representation
Solving basic tree diagram problems using systematic drawing
Demonstrations using branching examples and visual organization
Explaining tree structure using logical branching principles

Chalk and blackboard, tree diagram templates, branching materials, exercise books
Chalk and blackboard, complex tree examples, detailed calculation aids, exercise books
Chalk and blackboard, local business examples, calculators if available, exercise books
Chalk and blackboard, construction/farming examples, exercise books
Chalk and blackboard, sharing demonstration materials, exercise books
Chalk and blackboard, business partnership examples, exercise books

KLB Mathematics Book Three Pg 282

Compound Proportion and Rates of Work
Graphical Methods
Graphical Methods
Graphical Methods
Graphical Methods

Rates of Work
Rates of Work and Mixtures
Tables of given relations
Graphs of given relations
Tables and graphs integration
Introduction to cubic equations

By the end of the lesson, the learner should be able to:
Calculate the rate of work
Understand work rate relationships
Apply time-work-efficiency concepts
Solve basic rate of work problems

Q/A on work rate calculation using practical examples
Discussions on efficiency and time relationships using work scenarios
Solving basic rate of work problems using systematic methods
Demonstrations using construction and labor examples
Explaining work rate concepts using practical work situations

Chalk and blackboard, work scenario examples, exercise books
Chalk and blackboard, mixture demonstration materials, exercise books
Chalk and blackboard, ruled paper for tables, exercise books
Chalk and blackboard, graph paper or grids, rulers, exercise books
Chalk and blackboard, graph paper, data examples, exercise books
Chalk and blackboard, cubic function examples, exercise books

KLB Mathematics Book Three Pg 294-295

Graphical Methods

Graphical solution of cubic equations
Advanced cubic solutions
Introduction to rates of change
Average rates of change
Advanced average rates
Introduction to instantaneous rates
Rate of change at an instant
Advanced instantaneous rates
Empirical graphs
Advanced empirical methods

By the end of the lesson, the learner should be able to:
Draw graphs of cubic equations
Plot cubic curves accurately
Use graphs to solve cubic equations
Find roots using graphical methods

Q/A on cubic curve plotting using systematic point plotting
Discussions on curve characteristics and root finding
Solving cubic graphing problems using careful plotting
Demonstrations using cubic curve construction
Explaining root identification using graph analysis

Chalk and blackboard, graph paper, cubic equation examples, exercise books
Chalk and blackboard, advanced graph examples, exercise books
Chalk and blackboard, rate calculation examples, exercise books
Chalk and blackboard, graph paper, rate examples, exercise books
Chalk and blackboard, advanced rate scenarios, exercise books
Chalk and blackboard, tangent line examples, exercise books
Chalk and blackboard, detailed graph examples, exercise books
Chalk and blackboard, advanced rate examples, exercise books
Chalk and blackboard, experimental data examples, exercise books
Chalk and blackboard, complex data examples, exercise books

KLB Mathematics Book Three Pg 302-304