Matrices and Transformation

Matrices of Transformation

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

KLB Secondary Mathematics Form 4, Pages 1-5

Matrices and Transformation

Identifying Common Transformation Matrices
Finding the Matrix of a Transformation

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

-Identify matrices for reflection, rotation, enlargement
-Describe transformations represented by given matrices
-Apply identity matrix and understand its effect
-Distinguish between different types of transformations

-Use unit square drawn on paper to identify transformations
-Practice with specific matrices like (0 1; 1 0), (-1 0; 0 1)
-Draw objects and images under various transformations
-Q&A on transformation properties

Exercise books
-Manila paper
-Ruler
-String
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 1-5

Matrices and 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:

-Use unit square to find transformation matrices
-Read matrix elements directly from unit square images
-Apply unit square method to various transformations
-Compare unit square method with algebraic method

-Demonstrate unit square method systematically
-Practice reading transformation matrices from diagrams
-Apply method to reflections, rotations, enlargements
-Compare efficiency of different methods

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

KLB Secondary Mathematics Form 4, Pages 6-16

Matrices and Transformation

Single Matrix for Successive Transformations
Inverse of a Transformation
Properties of Inverse Transformations

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

KLB Secondary Mathematics Form 4, Pages 21-24

Matrices and Transformation

Area Scale Factor and Determinant

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

-Establish relationship between area scale factor and determinant
-Calculate area scale factors for transformations
-Apply determinant to find area changes
-Solve problems involving area transformations

-Measure areas of objects and images using grid paper
-Calculate determinants and compare with area ratios
-Practice with various transformation types
-Verify the relationship: ASF =

det A

Matrices and Transformation

Shear 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

KLB Secondary Mathematics Form 4, Pages 28-34

Matrices and Transformation

Stretch 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

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

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

KLB Secondary Mathematics Form 4, Pages 39-42

Statistics II

Working Mean Concept

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 39-42

Statistics II

Mean Using Working Mean - Simple Data
Mean Using Working Mean - Frequency Tables

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

-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

-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
-Student data
-Chalk/markers
-Community data

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Mean for Grouped Data Using Working Mean

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

-Calculate mean for grouped continuous data
-Select appropriate working mean for grouped data
-Use midpoints of class intervals correctly
-Apply working mean formula to grouped data

-Use height/weight data of students in class
-Practice finding midpoints of class intervals
-Work through complex calculations step by step
-Students practice with agricultural production data

Exercise books
-Manila paper
-Real datasets
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Advanced Working Mean Techniques

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

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

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

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Quartiles for Grouped Data

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

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Deciles and Percentiles Calculations

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

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Introduction to Cumulative Frequency
Drawing Cumulative Frequency Curves (Ogives)

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Reading Values from Ogives

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

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Applications of Ogives

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

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

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

Exercise books
-Manila paper
-Real problem datasets
-Ruler

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Introduction to Measures of Dispersion

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

KLB Secondary Mathematics Form 4, Pages 60-65

Statistics II

Range and Interquartile Range
Mean Absolute Deviation

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 60-65

Statistics II

Introduction to Variance

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

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II

Variance Using Alternative Formula

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

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II

Standard Deviation Calculations
Standard Deviation for Grouped Data

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

KLB Secondary Mathematics Form 4, Pages 65-70

Statistics II

Advanced Standard Deviation Techniques

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

KLB Secondary Mathematics Form 4, Pages 65-70

Loci

Introduction to Loci

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

KLB Secondary Mathematics Form 4, Pages 73-75

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

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Locus of Points at Fixed Distance from a Point

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

-Define circle as locus of points at fixed distance from center
-Construct circles with given radius using compass
-Understand sphere as 3D locus from fixed point
-Solve problems involving circular loci

-Construct circles of different radii
-Demonstrate with string of fixed length
-Discuss radar coverage, radio signal range
-Students create circles with various measurements

Exercise books
-Manila paper
-Compass
-String

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Locus of Points at Fixed Distance from a Line
Angle Bisector Locus

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

Properties and Applications of Angle Bisector

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

-Understand relationship between angle bisectors in triangles
-Apply angle bisector theorem
-Solve problems involving inscribed circles
-Use angle bisectors in geometric constructions

-Construct inscribed circle using angle bisectors
-Apply angle bisector theorem to solve problems
-Find external angle bisectors
-Solve practical surveying problems

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Constant Angle Locus

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

Advanced Constant Angle Constructions

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

KLB Secondary Mathematics Form 4, Pages 75-82

Loci

Introduction to Intersecting Loci
Intersecting Circles and Lines

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

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

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Triangle Centers Using Intersecting Loci

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

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

-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

KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Complex Intersecting Loci Problems

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 83-89

Loci

Introduction to Loci of Inequalities
Distance 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

Combined Inequality Loci

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

-Solve problems with multiple inequality constraints
-Find intersection regions of inequality loci
-Apply to optimization and feasibility problems
-Use systematic shading techniques

-Find feasible regions for multiple constraints
-Solve planning problems with restrictions
-Apply to resource allocation scenarios
-Practice systematic region identification

Exercise books
-Manila paper
-Ruler
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 89-92

Loci

Advanced Inequality Applications

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

KLB Secondary Mathematics Form 4, Pages 89-92

Loci

Introduction to Loci Involving Chords
Chord-Based Constructions

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

-Review chord properties in circles
-Understand perpendicular bisector of chords
-Apply chord theorems to loci problems
-Construct equal chords in circles

-Review chord bisector theorem
-Construct chords of given lengths
-Find centers using chord properties
-Practice with chord intersection theorems

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 92-94

Loci

Advanced Chord Problems

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

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

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

Exercise books
-Manila paper
-Compass
-Ruler

KLB Secondary Mathematics Form 4, Pages 92-94

Loci

Integration of All Loci Types

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

KLB Secondary Mathematics Form 4, Pages 73-94

Trigonometry III

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

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

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

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

Exercise books
-Manila paper
-Rulers
-Calculators (if available)
-Unit circle diagrams
-Calculators

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Applications of sin²θ + cos²θ = 1

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

-Solve problems using the fundamental identity
-Find missing trigonometric ratios given one ratio
-Apply identity to simplify trigonometric expressions
-Use identity in geometric problem solving

-Work through examples finding cos when sin is given
-Practice simplifying complex trigonometric expressions
-Solve problems involving unknown angles
-Apply to real-world navigation problems

Exercise books
-Manila paper
-Trigonometric tables
-Real-world examples

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Additional Trigonometric Identities

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

KLB Secondary Mathematics Form 4, Pages 99-103

Trigonometry III

Introduction to Waves

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

-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)

Exercise books
-Manila paper
-String/rope
-Wave diagrams

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Sine and Cosine Waves
Transformations of Sine Waves

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

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

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

Exercise books
-Manila paper
-Rulers
-Graph paper (if available)
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Period Changes in Trigonometric Functions

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

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

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

Exercise books
-Manila paper
-Rulers
-Period calculation charts

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Combined Amplitude and Period Transformations

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Phase Angles and Wave Shifts
General Trigonometric Functions

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Cosine Wave Transformations

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

-Apply transformations to cosine functions
-Plot y = a cos(bx + c) functions
-Compare cosine and sine transformations
-Use cosine functions in modeling

-Plot various cosine transformations on manila paper
-Compare with equivalent sine transformations
-Practice identifying cosine wave parameters
-Model temperature variations using cosine

Exercise books
-Manila paper
-Rulers
-Temperature data

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Introduction to 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

KLB Secondary Mathematics Form 4, Pages 109-112

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

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

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III

Using Graphs to Solve Trigonometric Equations

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

-Solve equations graphically using intersections
-Plot trigonometric functions on same axes
-Find intersection points as equation solutions
-Verify algebraic solutions graphically

-Plot y = sin x and y = 0.5 on same axes
-Identify intersection points as solutions
-Use graphical method for complex equations
-Compare graphical and algebraic solutions

Exercise books
-Manila paper
-Rulers
-Graphing examples

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III
Three Dimensional Geometry

Trigonometric Equations with Identities
Introduction to 3D Concepts

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

KLB Secondary Mathematics Form 4, Pages 109-112

Three Dimensional Geometry

Properties of Common Solids

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

-Identify properties of cubes, cuboids, pyramids
-Count faces, edges, vertices systematically
-Apply Euler's formula (V - E + F = 2)
-Classify solids by their geometric properties

-Make models using cardboard and tape
-Create table of properties for different solids
-Verify Euler's formula with physical models
-Compare prisms and pyramids systematically

Exercise books
-Cardboard
-Scissors
-Tape/glue

KLB Secondary Mathematics Form 4, Pages 113-115

Three Dimensional Geometry

Understanding Planes in 3D Space

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

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

Exercise books
-Manila paper
-Books/boards
-Classroom examples

KLB Secondary Mathematics Form 4, Pages 113-115

Three Dimensional Geometry

Lines in 3D Space

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

-Understand different types of lines in 3D
-Identify parallel, intersecting, and skew lines
-Recognize that skew lines don't intersect and aren't parallel
-Find examples of different line relationships

-Use rulers/sticks to demonstrate line relationships
-Show parallel lines using parallel rulers
-Demonstrate skew lines using classroom edges
-Practice identifying line relationships in models

Exercise books
-Rulers/sticks
-3D models
-Manila paper

KLB Secondary Mathematics Form 4, Pages 113-115

Three Dimensional Geometry

Introduction to Projections
Angle Between Line and Plane - Concept

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

-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 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
-Light source
-3D models
-Protractor
-Rulers/sticks

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Calculating Angles Between Lines and Planes

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

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Advanced Line-Plane Angle Problems

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

-Solve complex angle problems systematically
-Apply coordinate geometry methods where helpful
-Use multiple right-angled triangles in solutions
-Verify answers using different approaches

-Practice with tent and roof angle problems
-Solve ladder against wall problems in 3D
-Work through architectural angle calculations
-Use real-world engineering applications

Exercise books
-Manila paper
-Real scenarios
-Problem sets

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Introduction to Plane-Plane Angles
Finding Angles Between Planes

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

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Complex Plane-Plane Angle Problems

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

-Solve advanced dihedral angle problems
-Apply to frustums and compound solids
-Use systematic approach for complex shapes
-Verify solutions using geometric properties

-Work with frustum of pyramid problems
-Solve wedge and compound shape angles
-Practice with architectural applications
-Use geometric reasoning to check answers

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

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Practical Applications of Plane Angles

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

KLB Secondary Mathematics Form 4, Pages 123-128

Three Dimensional Geometry

Understanding Skew Lines
Angle Between Skew Lines

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

-Define skew lines and their properties
-Distinguish skew lines from parallel/intersecting lines
-Identify skew lines in 3D models
-Understand that skew lines exist only in 3D

-Use classroom edges to show skew lines
-Demonstrate with two rulers in space
-Identify skew lines in building frameworks
-Practice recognition in various 3D shapes

Exercise books
-Manila paper
-Rulers
-Building frameworks
-Translation examples

KLB Secondary Mathematics Form 4, Pages 128-135

Three Dimensional Geometry

Advanced Skew Line Problems

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

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

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

Exercise books
-Manila paper
-Engineering examples
-Structure diagrams

KLB Secondary Mathematics Form 4, Pages 128-135

Three Dimensional Geometry

Distance Calculations in 3D

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

KLB Secondary Mathematics Form 4, Pages 115-135

Three Dimensional Geometry

Volume and Surface Area Applications
Coordinate Geometry in 3D

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 115-135

Three Dimensional Geometry

Integration with Trigonometry

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

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

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

Exercise books
-Manila paper
-Trigonometric tables
-Astronomy examples

KLB Secondary Mathematics Form 4, Pages 115-135

Longitudes and Latitudes

Introduction to Earth as a Sphere

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

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Great and Small Circles

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

-Define great circles and small circles on a sphere
-Identify properties of great and small circles
-Understand that great circles divide sphere into hemispheres
-Recognize examples of great and small circles on Earth

-Demonstrate great circles using globe and string
-Show that great circles pass through center
-Compare radii of great and small circles
-Identify equator as the largest circle

Exercise books
-Globe
-String
-Manila paper

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Understanding Latitude
Properties of Latitude Lines

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

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

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

Exercise books
-Globe
-Tape/string
-Protractor
-Calculator
-Manila paper

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Understanding Longitude

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

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

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

Exercise books
-Globe
-String
-World map

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Properties of Longitude Lines

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

-Understand that longitude lines are great circles
-Recognize that all longitude lines pass through poles
-Understand that longitude lines converge at poles
-Identify that opposite longitudes differ by 180°

-Show longitude lines converging at poles
-Demonstrate that longitude lines are great circles
-Find opposite longitude positions
-Compare longitude and latitude line properties

Exercise books
-Globe
-String
-Manila paper

KLB Secondary Mathematics Form 4, Pages 136-139

Longitudes and Latitudes

Position of Places on Earth
Latitude and Longitude Differences

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

KLB Secondary Mathematics Form 4, Pages 139-143

Longitudes and Latitudes

Introduction to Distance Calculations

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

-Understand relationship between angles and distances
-Learn that 1° on great circle = 60 nautical miles
-Define nautical mile and its relationship to kilometers
-Apply basic distance formulas for great circles

-Demonstrate angle-distance relationship using globe
-Show that 1' (minute) = 1 nautical mile
-Convert between nautical miles and kilometers
-Practice basic distance calculations

Exercise books
-Globe
-Calculator
-Conversion charts

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Distance Along Great Circles

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

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

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

Exercise books
-Manila paper
-Calculator
-Real examples

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Distance Along Small Circles (Parallels)
Shortest Distance Problems

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

-Understand that parallel distances use different formula
-Apply formula: distance = longitude difference × 60 × cos(latitude)
-Calculate radius of latitude circles
-Solve problems involving parallel of latitude distances

-Derive formula using trigonometry
-Calculate distance between Mombasa and Lagos
-Show why latitude affects distance calculations
-Practice with various latitude examples

Exercise books
-Manila paper
-Calculator
-African city examples
-Flight path examples

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Advanced Distance Calculations

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

-Solve complex distance problems with multiple steps
-Calculate distances involving multiple coordinate differences
-Apply to surveying and mapping problems
-Use systematic approaches for difficult calculations

-Work through complex multi-step distance problems
-Apply to surveying land boundaries
-Calculate perimeters of geographical regions
-Practice with examination-style problems

Exercise books
-Manila paper
-Calculator
-Surveying examples

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Introduction to Time and Longitude

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

KLB Secondary Mathematics Form 4, Pages 156-161

Longitudes and Latitudes

Local Time Calculations
Greenwich Mean Time (GMT)

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 156-161

Longitudes and Latitudes

Complex Time Problems
Speed Calculations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 156-161