Matrices and Transformation

Matrices of Transformation
Identifying Common Transformation Matrices

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

KLB Secondary Mathematics Form 4, Pages 1-5

Matrices and Transformation

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

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

-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

-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
-Chalk/markers
-String
-Coloured pencils

KLB Secondary Mathematics Form 4, Pages 6-16

Matrices and Transformation

Properties of Inverse Transformations
Area Scale Factor and Determinant
Shear Transformations

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

-Calculate determinants of 2×2 matrices
-Use determinant formula for matrix inverses
-Identify when inverse matrices exist
-Apply inverse matrix formula efficiently

-Practice determinant calculations on chalkboard
-Use formula: A⁻¹ = (1/det A) × adj A
-Identify singular matrices (det = 0)
-Solve systems using inverse matrices

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

KLB Secondary Mathematics Form 4, Pages 24-26

Matrices and Transformation

Stretch Transformations
Combined Shear and Stretch Problems
Isometric and Non-isometric Transformations

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

-Define stretch transformation and scale factors
-Distinguish between one-way and two-way stretches
-Construct matrices for stretch transformations
-Apply stretch transformations to solve problems

-Demonstrate stretch using rubber bands and paper
-Practice with x-axis and y-axis invariant stretches
-Construct stretch matrices systematically
-Compare stretches with enlargements

Exercise books
-Rubber bands
-Manila paper
-Ruler
-Chalk/markers
-Paper cutouts

KLB Secondary Mathematics Form 4, Pages 28-34

Statistics II

Introduction to Advanced Statistics
Working Mean Concept
Mean Using Working Mean - Simple Data

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

-Review measures of central tendency from Form 2
-Identify limitations of simple mean calculations
-Understand need for advanced statistical methods
-Recognize patterns in large datasets

-Review mean, median, mode from previous work
-Discuss challenges with large numbers
-Examine real data from Kenya (population, rainfall)
-Q&A on statistical applications in daily life

Exercise books
-Manila paper
-Real data examples
-Chalk/markers
-Sample datasets
-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

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

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Advanced Working Mean Techniques
Introduction to Quartiles, Deciles, Percentiles
Calculating Quartiles for Ungrouped Data

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

-Apply coding techniques with working mean
-Divide by class width to simplify further
-Use transformation methods efficiently
-Solve complex grouped data problems

-Demonstrate coding method on chalkboard
-Show how dividing by class width helps
-Practice reverse calculations to get original mean
-Work with economic data from Kenya

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

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Quartiles for Grouped Data
Deciles and Percentiles Calculations
Introduction to Cumulative Frequency

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
-Ruler
-Class data

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Drawing Cumulative Frequency Curves (Ogives)
Reading Values from Ogives
Applications of Ogives

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

-Draw accurate ogives using proper scales
-Plot cumulative frequency against upper boundaries
-Create smooth curves through plotted points
-Label axes and scales correctly

-Practice plotting on large manila paper
-Use rulers for accurate scales
-Demonstrate smooth curve drawing technique
-Students create their own ogives

Exercise books
-Manila paper
-Ruler
-Pencils
-Completed ogives
-Real problem datasets

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Introduction to Measures of Dispersion
Range and Interquartile Range
Mean Absolute Deviation

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

-Define dispersion and its importance
-Understand limitations of central tendency alone
-Compare datasets with same mean but different spread
-Identify different measures of dispersion

-Compare test scores of two classes with same mean
-Show how different spreads affect interpretation
-Discuss variability in real-world data
-Introduce range as simplest measure

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

KLB Secondary Mathematics Form 4, Pages 60-65

Statistics II

Introduction to Variance
Variance Using Alternative Formula

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

-Define variance as mean of squared deviations
-Calculate variance using definition formula
-Understand why deviations are squared
-Compare variance with other dispersion measures

-Work through variance calculation step by step
-Explain squaring deviations eliminates negatives
-Calculate variance for simple datasets
-Compare with mean absolute deviation

Exercise books
-Manila paper
-Simple datasets
-Chalk/markers
-Frequency data

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II

Standard Deviation Calculations
Standard Deviation for Grouped Data
Advanced Standard Deviation Techniques

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

-Calculate standard deviation as square root of variance
-Apply standard deviation to ungrouped data
-Use standard deviation to compare datasets
-Interpret standard deviation in practical contexts

-Calculate SD for student exam scores
-Compare SD values for different subjects
-Interpret what high/low SD means
-Use SD to identify consistent performance

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

KLB Secondary Mathematics Form 4, Pages 65-70

Loci

Introduction to Loci
Basic Locus Concepts and Laws
Perpendicular Bisector Locus

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

-Define locus and understand its meaning
-Distinguish between locus of points, lines, and regions
-Identify real-world examples of loci
-Understand the concept of movement according to given laws

-Demonstrate door movement to show path traced by corner
-Use string and pencil to show circular locus
-Discuss examples: clock hands, pendulum swing
-Students trace paths of moving objects

Exercise books
-Manila paper
-String
-Chalk/markers
-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
Locus of Points at Fixed Distance from a Line

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Angle Bisector Locus
Properties and Applications of Angle Bisector
Constant Angle Locus

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

-Define angle bisector locus
-Construct angle bisectors using compass and ruler
-Prove equidistance property of angle bisector
-Apply angle bisector to find incenters

-Construct angle bisectors for various angles
-Verify equidistance from angle arms
-Find incenter of triangle using angle bisectors
-Practice with acute, obtuse, and right angles

Exercise books
-Manila paper
-Compass
-Protractor
-Ruler

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Advanced Constant Angle Constructions
Introduction to Intersecting Loci

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Intersecting Circles and Lines
Triangle Centers Using Intersecting Loci
Complex Intersecting Loci Problems

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

-Find intersections of circles with lines
-Determine intersections of two circles
-Solve problems with line and circle combinations
-Apply to geometric construction problems

-Construct intersecting circles and lines
-Find common tangents to circles
-Solve problems involving circle-line intersections
-Apply to wheel and track problems

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

KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Introduction to Loci of Inequalities
Distance Inequality Loci
Combined Inequality Loci

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

-Understand graphical representation of inequalities
-Identify regions satisfying inequality conditions
-Distinguish between boundary lines and regions
-Apply inequality loci to practical constraints

-Shade regions representing simple inequalities
-Use broken and solid lines appropriately
-Practice with distance inequalities
-Apply to real-world constraint problems

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

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
Trigonometry III

Advanced Chord Problems
Integration of All Loci Types
Review of Basic Trigonometric Ratios

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
-Rulers
-Calculators (if available)

KLB Secondary Mathematics Form 4, Pages 92-94

Trigonometry III

Deriving the Identity sin²θ + cos²θ = 1
Applications of sin²θ + cos²θ = 1

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

-Understand the derivation of fundamental identity
-Apply Pythagoras theorem to unit circle
-Use the identity to solve trigonometric equations
-Convert between sin, cos using the identity

-Demonstrate using right-angled triangle with hypotenuse 1
-Show algebraic derivation step by step
-Practice substituting values to verify identity
-Solve equations using the fundamental identity

Exercise books
-Manila paper
-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

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

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

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

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Transformations of Sine Waves
Period Changes in Trigonometric Functions
Combined Amplitude and Period 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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Phase Angles and Wave Shifts
General Trigonometric Functions
Cosine Wave Transformations

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

-Understand concept of phase angle
-Plot graphs of y = sin(x + θ) functions
-Identify horizontal shifts in wave patterns
-Apply phase differences to wave analysis

-Plot y = sin(x + 45°), y = sin(x - 30°)
-Demonstrate horizontal shifting of waves
-Compare leading and lagging waves
-Apply to electrical circuits or sound waves

Exercise books
-Manila paper
-Colored pencils
-Phase shift examples
-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
Matrices
Matrices
Matrices
Matrices
Matrices

Trigonometric Equations with Identities
Introduction and real-life applications
Order of a matrix and elements
Square matrices, row and column matrices
Addition of matrices
Subtraction of matrices

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
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
Chalk and blackboard, exercise books, number cards made from cardboard

KLB Secondary Mathematics Form 4, Pages 109-112

Matrices

Combined addition and subtraction
Scalar multiplication
Introduction to matrix multiplication
Matrix multiplication (2×2 matrices)
Matrix multiplication (larger matrices)

By the end of the lesson, the learner should be able to:
Perform multiple matrix operations
Apply order of operations in matrix calculations
Solve complex combined problems
Demonstrate systematic problem-solving

Q/A on operation order using BODMAS rules
Discussions on complex expressions using step-by-step approach
Solving multi-step problems using organized methods
Demonstrations using systematic blackboard work
Explaining operation sequencing using flowcharts

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

KLB Mathematics Book Three Pg 171-174

Matrices

Properties of matrix multiplication
Real-world matrix multiplication applications

By the end of the lesson, the learner should be able to:
Understand non-commutativity of matrix multiplication
Apply associative and distributive properties
Distinguish between pre and post multiplication
Solve problems involving multiplication properties

Q/A on multiplication properties using counterexamples
Discussions on order importance using practical examples
Solving property-based problems using verification
Demonstrations using concrete examples
Explaining distributive law using expansion

Chalk and blackboard, exercise books, cardboard for property cards
Chalk and blackboard, local price lists, exercise books

KLB Mathematics Book Three Pg 174-179

Matrices

Identity matrix
Determinant of 2×2 matrices
Inverse of 2×2 matrices - theory

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

KLB Mathematics Book Three Pg 182-183

Matrices

Inverse of 2×2 matrices - practice
Introduction to solving simultaneous equations
Solving 2×2 simultaneous equations using matrices

By the end of the lesson, the learner should be able to:
Calculate inverses of 2×2 matrices systematically
Verify inverse calculations through multiplication
Apply inverse properties correctly
Solve complex inverse problems

Q/A on inverse calculation verification methods
Discussions on accuracy checking using multiplication
Solving advanced inverse problems using practice
Demonstrations using verification procedures
Explaining checking methods using 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

KLB Mathematics Book Three Pg 185-187

Matrices

Advanced simultaneous equation problems
Matrix applications in real-world problems
Transpose of matrices

By the end of the lesson, the learner should be able to:
Solve complex simultaneous equation systems
Handle systems with no solution or infinite solutions
Interpret determinant values in solution context
Apply matrix methods to word problems

Q/A on complex systems using special cases
Discussions on solution types using geometric interpretation
Solving challenging problems using complete analysis
Demonstrations using classification methods
Explaining geometric meaning using line concepts

Chalk and blackboard, exercise books, graph paper if available
Chalk and blackboard, local business examples, exercise books
Chalk and blackboard, exercise books, paper cutouts for demonstration

KLB Mathematics Book Three Pg 188-190

Matrices
Formulae and Variations
Formulae and Variations

Matrix equation solving
Introduction to formulae
Subject of a formula - basic cases

By the end of the lesson, the learner should be able to:
Solve matrix equations systematically
Find unknown matrices in equations
Apply inverse operations to solve equations
Verify matrix equation solutions

Q/A on equation solving using algebraic analogy
Discussions on unknown determination using systematic methods
Solving matrix equations using step-by-step approach
Demonstrations using organized solution procedures
Explaining verification using checking methods

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

KLB Mathematics Book Three Pg 183-190

Formulae and Variations

Subject of a formula - intermediate cases
Subject of a formula - advanced cases

By the end of the lesson, the learner should be able to:
Make complex variables the subject of formulae
Handle formulae with fractions and powers
Apply multiple inverse operations systematically
Solve intermediate difficulty problems

Q/A on complex rearrangement using systematic approach
Discussions on fraction handling using common denominators
Solving intermediate problems using organized methods
Demonstrations using step-by-step blackboard work
Explaining systematic approaches using flowcharts

Chalk and blackboard, fraction strips made from paper, exercise books
Chalk and blackboard, squared paper patterns, exercise books

KLB Mathematics Book Three Pg 191-193

Formulae and Variations

Applications of formula manipulation
Introduction to variation
Direct variation - introduction

By the end of the lesson, the learner should be able to:
Apply formula rearrangement to practical problems
Solve real-world problems using formula manipulation
Calculate unknown quantities in various contexts
Interpret results in meaningful situations

Q/A on practical applications using local examples
Discussions on real-world formula use in farming/building
Solving application problems using formula rearrangement
Demonstrations using construction and farming scenarios
Explaining practical interpretation using community examples

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

KLB Mathematics Book Three Pg 191-193

Sequences and Series

Introduction to sequences and finding terms
General term of sequences and applications
Arithmetic sequences and nth term

By the end of the lesson, the learner should be able to:
Define sequences and identify sequence patterns
Find next terms using established patterns
Recognize different types of sequence patterns
Apply pattern recognition systematically

Q/A on number patterns from daily life
Discussions on counting patterns using classroom arrangements
Solving pattern completion problems step-by-step
Demonstrations using bead or stone arrangements
Explaining sequence terminology and pattern continuation

Chalk and blackboard, stones or beans for patterns, exercise books
Chalk and blackboard, numbered cards made from paper, exercise books
Chalk and blackboard, measuring tape or string, exercise books

KLB Mathematics Book Three Pg 207-208

Sequences and Series

Arithmetic sequence applications
Geometric sequences and nth term
Geometric sequence applications

By the end of the lesson, the learner should be able to:
Solve complex arithmetic sequence problems
Apply arithmetic sequences to real-world problems
Handle word problems involving arithmetic sequences
Model practical situations using arithmetic progressions

Q/A on practical applications using local business examples
Discussions on salary progression and savings plans
Solving real-world problems using sequence methods
Demonstrations using employment and finance scenarios
Explaining practical interpretation using meaningful contexts

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

KLB Mathematics Book Three Pg 209-210

Sequences and Series

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 series as sums of sequences
Derive the sum formula for arithmetic series
Apply the arithmetic series formula systematically
Calculate sums efficiently using the formula

Q/A on series concepts using summation examples
Discussions on sequence-to-series relationships and formula derivation
Solving arithmetic series problems using step-by-step approach
Demonstrations using cumulative sum examples
Explaining derivation logic using algebraic reasoning

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 214-215

Sequences and Series
Vectors (II)

Sequences in nature and technology
Coordinates in two dimensions

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

KLB Mathematics Book Three Pg 207-219

Vectors (II)

Coordinates in three dimensions
Column and position vectors in three dimensions
Position vectors and applications

By the end of the lesson, the learner should be able to:
Identify the coordinates of a point in three dimensions
Understand the three-dimensional coordinate system
Plot points in 3D space systematically
Apply 3D coordinates to spatial problems

Q/A on 3D coordinate understanding using room corner references
Discussions on height, length, and width measurements
Solving 3D coordinate problems using systematic approaches
Demonstrations using classroom corners and building structures
Explaining 3D visualization using physical room examples

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

KLB Mathematics Book Three Pg 222

Vectors (II)

Column vectors in terms of unit vectors i, j, k
Vector operations using unit vectors
Magnitude of a vector in three dimensions

By the end of the lesson, the learner should be able to:
Express vectors in terms of unit vectors
Convert between column and unit vector notation
Understand the standard basis vector system
Apply unit vector representation systematically

Q/A on unit vector concepts using direction examples
Discussions on component representation using organized methods
Solving unit vector problems using systematic conversion
Demonstrations using perpendicular direction examples
Explaining basis vector concepts using coordinate axes

Chalk and blackboard, direction indicators, unit vector reference charts, exercise books
Chalk and blackboard, component calculation aids, exercise books
Chalk and blackboard, 3D measurement aids, exercise books

KLB Mathematics Book Three Pg 226-228

Vectors (II)

Magnitude applications and unit vectors
Parallel vectors
Collinearity

By the end of the lesson, the learner should be able to:
Calculate the magnitude of a vector in three dimensions
Find unit vectors from given vectors
Apply magnitude concepts to practical problems
Use magnitude in vector normalization

Q/A on magnitude and unit vector relationships
Discussions on normalization and direction finding
Solving magnitude and unit vector problems
Demonstrations using direction and length separation
Explaining practical applications using navigation examples

Chalk and blackboard, direction finding aids, exercise books
Chalk and blackboard, parallel line demonstrations, exercise books
Chalk and blackboard, straight-line demonstrations, exercise books

KLB Mathematics Book Three Pg 229-230

Vectors (II)

Advanced collinearity applications
Proportional division of a line
External division of a line

By the end of the lesson, the learner should be able to:
Show that points are collinear
Apply collinearity to complex geometric problems
Integrate parallel and collinearity concepts
Solve advanced alignment problems

Q/A on advanced collinearity using complex scenarios
Discussions on geometric proof using vector methods
Solving challenging collinearity problems
Demonstrations using complex geometric constructions
Explaining advanced applications using comprehensive examples

Chalk and blackboard, complex geometric aids, exercise books
Chalk and blackboard, internal division models, exercise books
Chalk and blackboard, external division models, exercise books

KLB Mathematics Book Three Pg 232-234

Vectors (II)

Combined internal and external division
Ratio theorem

By the end of the lesson, the learner should be able to:
Divide a line internally and externally in the given ratio
Apply both division formulas systematically
Compare internal and external division results
Handle mixed division problems

Q/A on combined division using comparative methods
Discussions on division type selection using problem analysis
Solving combined division problems using systematic approaches
Demonstrations using both division types
Explaining division relationships using geometric reasoning

Chalk and blackboard, combined division models, exercise books
Chalk and blackboard, ratio theorem aids, exercise books

KLB Mathematics Book Three Pg 239

Vectors (II)

Advanced ratio theorem applications
Mid-point
Ratio theorem and midpoint integration

By the end of the lesson, the learner should be able to:
Find the position vector
Apply ratio theorem to complex scenarios
Solve multi-step ratio problems
Use ratio theorem in geometric proofs

Q/A on advanced ratio applications using complex problems
Discussions on multi-step ratio calculation
Solving challenging ratio problems using systematic methods
Demonstrations using comprehensive ratio examples
Explaining advanced applications using detailed reasoning

Chalk and blackboard, advanced ratio models, exercise books
Chalk and blackboard, midpoint demonstration aids, exercise books
Chalk and blackboard, complex problem materials, exercise books

KLB Mathematics Book Three Pg 242

Vectors (II)

Advanced ratio theorem applications
Applications of vectors in geometry
Rectangle diagonal applications

By the end of the lesson, the learner should be able to:
Use ratio theorem to find the given vectors
Apply ratio theorem to challenging problems
Handle complex geometric applications
Demonstrate comprehensive ratio mastery

Q/A on comprehensive ratio understanding using advanced problems
Discussions on complex ratio relationships
Solving advanced ratio problems using systematic methods
Demonstrations using sophisticated geometric constructions
Explaining mastery using challenging applications

Chalk and blackboard, advanced geometric aids, exercise books
Chalk and blackboard, parallelogram models, exercise books
Chalk and blackboard, rectangle models, exercise books

KLB Mathematics Book Three Pg 246-248

Vectors (II)
Binomial Expansion
Binomial Expansion

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 vectors to show geometric properties
Apply vectors to complex geometric proofs
Solve challenging geometric problems using vectors
Integrate all vector concepts in geometric contexts

Q/A on comprehensive geometric applications using vector methods
Discussions on advanced proof techniques using vectors
Solving complex geometric problems using integrated approaches
Demonstrations using sophisticated geometric constructions
Explaining advanced applications using comprehensive reasoning

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 248-250

Binomial Expansion

Pascal's triangle
Pascal's triangle applications
Pascal's triangle (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

KLB Mathematics Book Three Pg 256-257

Binomial Expansion

Pascal's triangle advanced
Applications to numerical cases

By the end of the lesson, the learner should be able to:
Use Pascal's triangle
Apply general binomial theorem concepts
Understand combination notation in expansions
Use general term formula applications

Q/A on general formula understanding using pattern analysis
Discussions on combination notation using counting principles
Solving general term problems using formula application
Demonstrations using systematic formula usage
Explaining general principles using algebraic reasoning

Chalk and blackboard, combination calculation aids, exercise books
Chalk and blackboard, simple calculation aids, exercise books

KLB Mathematics Book Three Pg 258-259

Binomial Expansion
Probability
Probability

Applications to numerical cases (continued)
Introduction
Experimental Probability

By the end of the lesson, the learner should be able to:
Use binomial expansion to solve numerical problems
Apply binomial methods to complex calculations
Handle decimal approximations using expansions
Solve practical numerical problems

Q/A on advanced numerical applications using complex scenarios
Discussions on decimal approximation using expansion techniques
Solving challenging numerical problems using systematic methods
Demonstrations using detailed calculation procedures
Explaining practical relevance using real-world examples

Chalk and blackboard, advanced calculation examples, exercise books
Chalk and blackboard, coins, dice made from cardboard, exercise books
Chalk and blackboard, coins, cardboard dice, tally charts, exercise books

KLB Mathematics Book Three Pg 259-260

Probability

Experimental Probability applications
Range of Probability Measure
Probability Space

By the end of the lesson, the learner should be able to:
Calculate the experimental probability
Apply experimental methods to various scenarios
Handle large sample experiments
Analyze experimental probability patterns

Q/A on advanced experimental techniques using extended trials
Discussions on sample size effects using comparative data
Solving complex experimental problems using systematic methods
Demonstrations using extended experimental procedures
Explaining pattern analysis using accumulated data

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

KLB Mathematics Book Three Pg 262-264

Probability

Theoretical Probability
Theoretical Probability advanced
Theoretical Probability applications

By the end of the lesson, the learner should be able to:
Calculate the probability space for the theoretical probability
Apply mathematical reasoning to find probabilities
Use equally likely outcome assumptions
Calculate theoretical probabilities systematically

Q/A on theoretical calculation using mathematical principles
Discussions on equally likely assumptions and calculations
Solving theoretical problems using systematic approaches
Demonstrations using fair dice and unbiased coin examples
Explaining mathematical probability using logical reasoning

Chalk and blackboard, fair dice and coins, probability calculation aids, exercise books
Chalk and blackboard, complex probability materials, advanced calculation aids, exercise books
Chalk and blackboard, local game examples, practical scenario materials, exercise books

KLB Mathematics Book Three Pg 266-268

Probability

Combined Events
Combined Events OR probability
Independent Events

By the end of the lesson, the learner should be able to:
Find the probability of a combined events
Understand compound events and combinations
Distinguish between different event types
Apply basic combination rules

Q/A on event combination using practical examples
Discussions on exclusive and inclusive event identification
Solving basic combined event problems using visual methods
Demonstrations using card drawing and dice rolling combinations
Explaining combination principles using Venn diagrams

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

KLB Mathematics Book Three Pg 272-273

Probability

Independent Events advanced
Independent Events applications

By the end of the lesson, the learner should be able to:
Find the probability of independent events
Distinguish between independent and dependent events
Apply conditional probability concepts
Handle complex independence scenarios

Q/A on independence verification using mathematical methods
Discussions on dependence concepts using card drawing examples
Solving dependent and independent event problems using systematic approaches
Demonstrations using replacement and non-replacement scenarios
Explaining conditional probability using practical examples

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 276-278

Probability
Compound Proportion and Rates of Work

Tree Diagrams
Tree Diagrams advanced
Compound Proportions

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

KLB Mathematics Book Three Pg 282

Compound Proportion and Rates of Work

Compound Proportions applications
Proportional Parts
Proportional Parts applications

By the end of the lesson, the learner should be able to:
Find the compound proportions
Apply compound proportions to complex problems
Handle multi-step compound proportion scenarios
Solve real-world compound proportion problems

Q/A on advanced compound proportion using complex scenarios
Discussions on multi-variable relationships using practical contexts
Solving challenging compound problems using systematic approaches
Demonstrations using construction and farming examples
Explaining practical applications using community-based scenarios

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 290-291

Compound Proportion and Rates of Work
Graphical Methods

Rates of Work
Rates of Work and Mixtures
Tables of given relations

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

KLB Mathematics Book Three Pg 294-295

Graphical Methods

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

By the end of the lesson, the learner should be able to:
Draw graphs of given relations
Plot points accurately on coordinate systems
Connect points to show relationships
Interpret graphs from given data

Q/A on graph plotting using coordinate methods
Discussions on point plotting and curve drawing
Solving graph construction problems using systematic plotting
Demonstrations using coordinate systems and curve sketching
Explaining graph interpretation using visual analysis

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 300

Graphical Methods

Graphical solution of cubic equations
Advanced cubic solutions

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

KLB Mathematics Book Three Pg 302-304

Graphical Methods

Introduction to rates of change
Average rates of change
Advanced average rates

By the end of the lesson, the learner should be able to:
Calculate the average rates of change
Understand rate of change concepts
Apply rate calculations to practical problems
Interpret rate meanings in context

Q/A on rate calculation using slope methods
Discussions on rate interpretation using practical examples
Solving basic rate problems using systematic calculation
Demonstrations using speed-time and distance examples
Explaining rate concepts using practical analogies

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

KLB Mathematics Book Three Pg 304-306

Graphical Methods

Introduction to instantaneous rates
Rate of change at an instant
Advanced instantaneous rates

By the end of the lesson, the learner should be able to:
Calculate the rate of change at an instant
Understand instantaneous rate concepts
Distinguish between average and instantaneous rates
Apply instant rate methods

Q/A on instantaneous rate concepts using limiting methods
Discussions on instant vs average rate differences
Solving basic instantaneous rate problems
Demonstrations using tangent line concepts
Explaining instantaneous rate using practical examples

Chalk and blackboard, tangent line examples, exercise books
Chalk and blackboard, detailed graph examples, exercise books
Chalk and blackboard, advanced rate examples, exercise books

KLB Mathematics Book Three Pg 310-311

Graphical Methods

Empirical graphs
Advanced empirical methods

By the end of the lesson, the learner should be able to:
Draw the empirical graphs
Understand empirical data representation
Plot experimental data systematically
Analyze empirical relationships

Q/A on empirical data plotting using experimental examples
Discussions on real data representation using practical scenarios
Solving empirical graphing problems using systematic methods
Demonstrations using experimental data examples
Explaining empirical analysis using practical interpretations

Chalk and blackboard, experimental data examples, exercise books
Chalk and blackboard, complex data examples, exercise books

KLB Mathematics Book Three Pg 315-316