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

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

Single Matrix for Successive Transformations
Inverse of a Transformation
Properties of Inverse Transformations
Area Scale Factor and Determinant

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

-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

KLB Secondary Mathematics Form 4, Pages 21-24

Matrices and Transformation

Shear Transformations
Stretch Transformations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 28-34

Matrices and Transformation

Combined Shear and Stretch Problems
Isometric and Non-isometric Transformations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 28-34

Statistics II

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

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

-Calculate mean using working mean for ungrouped data
-Apply the formula: mean = working mean + mean of deviations
-Verify results using direct calculation method
-Solve problems with whole numbers

-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

-Work through step-by-step examples on chalkboard
-Practice with student marks and heights data
-Verify answers using traditional method
-Individual practice with guided support

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

KLB Secondary Mathematics Form 4, Pages 39-42
KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Advanced Working Mean Techniques
Introduction to Quartiles, Deciles, Percentiles

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

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Calculating Quartiles for Ungrouped Data
Quartiles for Grouped Data

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

-Find lower quartile, median, upper quartile for raw data
-Apply the position formulas correctly
-Arrange data in ascending order systematically
-Interpret quartile values in context

-Practice with test scores from the class
-Arrange data systematically on chalkboard
-Calculate Q1, Q2, Q3 step by step
-Students work with their own datasets

Exercise books
-Manila paper
-Test score data
-Chalk/markers
-Grade data

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Deciles and Percentiles Calculations
Introduction to Cumulative Frequency

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

-Calculate specific deciles and percentiles
-Apply interpolation formulas for deciles/percentiles
-Interpret decile and percentile positions
-Use these measures for comparative analysis

-Calculate specific percentiles for class test scores
-Find deciles for sports performance data
-Compare students' positions using percentiles
-Practice with national examination statistics

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

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

KLB Secondary Mathematics Form 4, Pages 60-65

Statistics II

Mean Absolute Deviation
Introduction to Variance
Variance Using Alternative Formula
Standard Deviation Calculations
Standard Deviation for Grouped Data

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

-Calculate mean absolute deviation
-Use absolute values correctly in calculations
-Understand concept of average distance from mean
-Apply MAD to compare variability in datasets

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

-Calculate MAD for class test scores
-Practice with absolute value calculations
-Compare MAD values for different subjects
-Interpret MAD in context of data spread

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

Exercise books
-Manila paper
-Test score data
-Chalk/markers
-Simple datasets
Exercise books
-Manila paper
-Frequency data
-Chalk/markers
-Exam score data
-Agricultural data

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II
Loci

Advanced Standard Deviation Techniques
Introduction to Loci

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

-Apply transformation properties of standard deviation
-Use coding with class width division
-Solve problems with multiple transformations
-Verify results using different methods

-Demonstrate coding transformations
-Show how SD changes with data transformations
-Practice reverse calculations
-Verify using alternative methods

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

KLB Secondary Mathematics Form 4, Pages 65-70

Loci

Basic Locus Concepts and Laws
Perpendicular Bisector Locus

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 73-75

Loci

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

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

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

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

Exercise books
-Manila paper
-Compass
-Ruler
-String

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

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

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Constant Angle Locus
Advanced Constant Angle Constructions

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

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

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

Exercise books
-Manila paper
-Compass
-Protractor

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Introduction to Intersecting 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:

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

-Find circumcenter using perpendicular bisector intersections
-Locate incenter using angle bisector intersections
-Determine centroid and orthocenter
-Apply triangle centers to solve problems

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

-Construct all four triangle centers
-Compare properties of different triangle centers
-Use triangle centers in geometric proofs
-Solve problems involving triangle center properties

Exercise books
-Manila paper
-Compass
-Ruler
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

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

Chord-Based Constructions
Advanced Chord Problems

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

-Construct circles through three points using chords
-Find loci of chord midpoints
-Solve problems with intersecting chords
-Apply chord properties to geometric constructions

-Construct circles using three non-collinear points
-Find locus of midpoints of parallel chords
-Solve chord intersection problems
-Practice with chord-tangent relationships

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 92-94

Loci
Trigonometry III

Integration of All Loci Types
Review of Basic Trigonometric Ratios

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)

KLB Secondary Mathematics Form 4, Pages 73-94

Trigonometry III

Deriving the Identity sin²θ + cos²θ = 1
Applications of sin²θ + cos²θ = 1
Additional Trigonometric Identities

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
-Identity reference sheet

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Introduction to Waves
Sine and Cosine Waves
Transformations of Sine Waves
Period Changes in Trigonometric Functions

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

-Define amplitude and period of waves
-Understand wave characteristics and properties
-Identify amplitude and period from graphs
-Connect waves to trigonometric functions

-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

-Use physical demonstrations with string/rope
-Draw simple wave patterns on manila paper
-Measure amplitude and period from wave diagrams
-Discuss real-world wave examples (sound, light)

-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
-String/rope
-Wave diagrams
-Rulers
-Graph paper (if available)
Exercise books
-Manila paper
-Colored pencils
-Rulers
-Period calculation charts

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Combined Amplitude and Period Transformations
Phase Angles and Wave Shifts

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

-Plot graphs of y = a sin(bx) functions
-Identify both amplitude and period changes
-Solve problems with multiple transformations
-Apply to complex wave phenomena

-Plot y = 2 sin(3x), y = 3 sin(x/2) on manila paper
-Calculate both amplitude and period for each function
-Compare multiple transformed waves
-Apply to radio waves or tidal patterns

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

General Trigonometric Functions
Cosine Wave Transformations
Introduction to Trigonometric Equations

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

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

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

Exercise books
-Manila paper
-Rulers
-Complex function examples
-Temperature data
-Unit circle diagrams
-Trigonometric tables

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Solving Basic Trigonometric Equations
Quadratic Trigonometric Equations

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

-Solve equations of form sin x = k, cos x = k
-Find all solutions in specified ranges
-Use symmetry properties of trigonometric functions
-Apply inverse trigonometric functions

-Work through sin x = 0.6 step by step
-Find all solutions between 0° and 360°
-Use calculator to find inverse trigonometric values
-Practice with multiple basic equations

Exercise books
-Manila paper
-Calculators
-Solution worksheets
-Factoring techniques
-Substitution examples

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III

Equations Involving Multiple Angles
Using Graphs to Solve Trigonometric Equations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III
Three Dimensional Geometry
Three Dimensional Geometry

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

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 109-112

Three Dimensional Geometry

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

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

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

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

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

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

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

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

Three Dimensional Geometry

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

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

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Introduction to Plane-Plane Angles
Finding Angles Between Planes
Complex Plane-Plane Angle Problems

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

-Define angle between two planes
-Understand concept of dihedral angles
-Identify line of intersection of two planes
-Find perpendiculars to intersection line

-Use two books to demonstrate intersecting planes
-Show how planes meet along an edge
-Identify dihedral angles in classroom
-Demonstrate using folded paper

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

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Practical Applications of Plane Angles
Understanding Skew Lines

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Angle Between Skew Lines
Advanced Skew Line Problems

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

KLB Secondary Mathematics Form 4, Pages 128-135

Three Dimensional Geometry

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:

-Calculate distances between points in 3D
-Find shortest distances between lines and planes
-Apply 3D Pythagoras theorem
-Use distance formula in coordinate geometry

-Calculate space diagonals in cuboids
-Find distances from points to planes
-Apply 3D distance formula systematically
-Solve minimum distance problems

Exercise books
-Manila paper
-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 115-135