Trigonometry III

Review of Basic Trigonometric Ratios

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)

KLB Secondary Mathematics Form 4, Pages 99-103

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

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

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)

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Transformations of Sine Waves
Period Changes in Trigonometric Functions

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

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

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

General Trigonometric Functions
Cosine Wave Transformations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Introduction to Trigonometric Equations

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

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

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry III

Quadratic Trigonometric Equations

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

-Solve equations like sin²x - sin x = 0
-Apply factoring techniques to trigonometric equations
-Use substitution methods for complex equations
-Find all solutions systematically

-Demonstrate substitution method (let y = sin x)
-Factor quadratic expressions in trigonometry
-Solve resulting quadratic equations
-Back-substitute to find angle solutions

Exercise books
-Manila paper
-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

Trigonometric Equations with Identities

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

KLB Secondary Mathematics Form 4, Pages 109-112

Three Dimensional Geometry

Introduction to 3D Concepts

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

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

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

Exercise books
-Cardboard boxes
-Manila paper
-Real 3D objects

KLB Secondary Mathematics Form 4, Pages 113-115

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
Lines 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
-Rulers/sticks
-3D models

KLB Secondary Mathematics Form 4, Pages 113-115

Three Dimensional Geometry

Introduction to Projections

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

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Angle Between Line and Plane - Concept

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

-Define angle between line and plane
-Understand that angle is measured with projection
-Identify the projection of line on plane
-Recognize when line is perpendicular to plane

-Demonstrate using stick against book (plane)
-Show that angle is with projection, not plane itself
-Use protractor to measure angles with projections
-Identify perpendicular lines to planes

Exercise books
-Manila paper
-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
Introduction to Plane-Plane Angles

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
-Books
-Folded paper

KLB Secondary Mathematics Form 4, Pages 115-123

Three Dimensional Geometry

Finding Angles Between Planes

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

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

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

Exercise books
-Manila paper
-Protractor
-Building examples

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
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

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

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

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

KLB Secondary Mathematics Form 4, Pages 139-143

Longitudes and Latitudes

Latitude and Longitude Differences
Introduction to Distance Calculations

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 139-143

Longitudes and Latitudes

Distance Along Great Circles

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)

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

KLB Secondary Mathematics Form 4, Pages 143-156

Longitudes and Latitudes

Shortest Distance Problems
Advanced Distance Calculations

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

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

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

Exercise books
-Manila paper
-Calculator
-Flight path examples
-Surveying examples

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

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

KLB Secondary Mathematics Form 4, Pages 156-161

Longitudes and Latitudes

Greenwich Mean Time (GMT)

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

-Understand Greenwich as reference for world time
-Calculate local times relative to GMT
-Apply GMT to solve international time problems
-Understand time zones and their practical applications

-Use Greenwich as time reference point
-Calculate local times for cities worldwide
-Apply to international business scenarios
-Discuss practical applications of GMT

Exercise books
-Manila paper
-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

Linear Programming

Introduction to Linear Programming

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

-Understand the concept of optimization in real life
-Identify decision variables in practical situations
-Recognize constraints and objective functions
-Understand applications of linear programming

-Discuss resource allocation problems in daily life
-Identify optimization scenarios in business and farming
-Introduce decision-making with limited resources
-Use simple examples from student experiences

Exercise books
-Manila paper
-Real-life examples
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 165-167

Linear Programming

Forming Linear Inequalities from Word Problems

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

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

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

Exercise books
-Manila paper
-Local business examples
-Agricultural scenarios

KLB Secondary Mathematics Form 4, Pages 165-167

Linear Programming

Types of Constraints

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

-Identify non-negativity constraints
-Understand resource constraints and their implications
-Form demand and supply constraints
-Apply constraint formation to various industries

-Practice with non-negativity constraints (x ≥ 0, y ≥ 0)
-Form material and labor constraints
-Apply to manufacturing and service industries
-Use school resource allocation examples

Exercise books
-Manila paper
-Industry examples
-School scenarios

KLB Secondary Mathematics Form 4, Pages 165-167

Linear Programming

Objective Functions
Complete Problem Formulation

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

-Define objective functions for maximization problems
-Define objective functions for minimization problems
-Understand profit, cost, and other objective measures
-Connect objective functions to real-world goals

-Form profit maximization functions
-Create cost minimization functions
-Practice with revenue and efficiency objectives
-Apply to business and production scenarios

Exercise books
-Manila paper
-Business examples
-Production scenarios
-Complete examples
-Systematic templates

KLB Secondary Mathematics Form 4, Pages 165-167

Linear Programming

Introduction to Graphical Solution Method

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

-Understand graphical representation of inequalities
-Plot constraint lines on coordinate plane
-Identify feasible and infeasible regions
-Understand boundary lines and their significance

-Plot simple inequality x + y ≤ 10 on graph
-Shade feasible regions systematically
-Distinguish between ≤ and < inequalities
-Practice with multiple examples on manila paper

Exercise books
-Manila paper
-Rulers
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 166-172

Linear Programming

Plotting Multiple Constraints

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

-Plot multiple inequalities on same graph
-Find intersection of constraint lines
-Identify feasible region bounded by multiple constraints
-Handle cases with no feasible solution

-Plot system of 3-4 constraints simultaneously
-Find intersection points of constraint lines
-Identify and shade final feasible region
-Discuss unbounded and empty feasible regions

Exercise books
-Manila paper
-Rulers
-Different colored pencils

KLB Secondary Mathematics Form 4, Pages 166-172

Linear Programming

Properties of Feasible Regions

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

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

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

Exercise books
-Manila paper
-Calculators
-Algebraic methods

KLB Secondary Mathematics Form 4, Pages 166-172

Linear Programming

Introduction to Optimization
The Corner Point Method

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

-Understand concept of optimal solution
-Recognize that optimal solution occurs at corner points
-Learn to evaluate objective function at corner points
-Compare values to find maximum or minimum

-Evaluate objective function at each corner point
-Compare values to identify optimal solution
-Practice with both maximization and minimization
-Verify optimal solution satisfies all constraints

Exercise books
-Manila paper
-Calculators
-Evaluation tables
-Evaluation templates
-Systematic approach

KLB Secondary Mathematics Form 4, Pages 172-176

Linear Programming

The Iso-Profit/Iso-Cost Line Method

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

-Understand concept of iso-profit and iso-cost lines
-Draw family of parallel objective function lines
-Use slope to find optimal point graphically
-Apply sliding line method for optimization

-Draw iso-profit lines for given objective function
-Show family of parallel lines with different values
-Find optimal point by sliding line to extreme position
-Practice with both maximization and minimization

Exercise books
-Manila paper
-Rulers
-Sliding technique

KLB Secondary Mathematics Form 4, Pages 172-176

Linear Programming

Comparing Solution Methods

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

-Compare corner point and iso-line methods
-Understand when each method is most efficient
-Verify solutions using both methods
-Choose appropriate method for different problems

-Solve same problem using both methods
-Compare efficiency and accuracy of methods
-Practice method selection based on problem type
-Verify consistency of results

Exercise books
-Manila paper
-Method comparison
-Verification examples

KLB Secondary Mathematics Form 4, Pages 172-176

Linear Programming
Differentiation

Business Applications - Production Planning
Introduction to Rate of Change

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

-Apply linear programming to production problems
-Solve manufacturing optimization problems
-Handle resource allocation in production
-Apply to Kenyan manufacturing scenarios

-Solve factory production optimization problem
-Apply to textile or food processing examples
-Use local manufacturing scenarios
-Calculate optimal production mix

Exercise books
-Manila paper
-Manufacturing examples
-Kenyan industry data
-Real-world examples
-Graph examples

KLB Secondary Mathematics Form 4, Pages 172-176

Differentiation

Average Rate of Change

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

-Calculate average rate of change between two points
-Use formula: average rate = Δy/Δx
-Apply to distance-time and other practical graphs
-Understand limitations of average rate calculations

-Calculate average speed between two time points
-Find average rate of population change
-Use coordinate points to find average rates
-Compare average rates over different intervals

Exercise books
-Manila paper
-Calculators
-Graph paper

KLB Secondary Mathematics Form 4, Pages 177-182

Differentiation

Instantaneous Rate of Change

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

-Understand concept of instantaneous rate
-Recognize instantaneous rate as limit of average rates
-Connect to tangent line gradients
-Apply to real-world motion problems

-Demonstrate instantaneous speed using car speedometer
-Show limiting process using smaller intervals
-Connect to tangent line slopes on curves
-Practice with motion and growth examples

Exercise books
-Manila paper
-Tangent demonstrations
-Motion examples

KLB Secondary Mathematics Form 4, Pages 177-182

Differentiation

Gradient of Curves at Points

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

-Find gradient of curve at specific points
-Use tangent line method for gradient estimation
-Apply limiting process to find exact gradients
-Practice with various curve types

-Draw tangent lines to curves on manila paper
-Estimate gradients using tangent slopes
-Use the limiting approach with chord sequences
-Practice with parabolas and other curves

Exercise books
-Manila paper
-Rulers
-Curve examples

KLB Secondary Mathematics Form 4, Pages 178-182

Differentiation

Introduction to Delta Notation
The Limiting Process

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

-Understand delta (Δ) notation for small changes
-Use Δx and Δy for coordinate changes
-Apply delta notation to rate calculations
-Practice reading and writing delta expressions

-Introduce delta as symbol for "change in"
-Practice writing Δx, Δy, Δt expressions
-Use delta notation in rate of change formulas
-Apply to coordinate geometry problems

Exercise books
-Manila paper
-Delta notation examples
-Symbol practice
-Limit tables
-Systematic examples

KLB Secondary Mathematics Form 4, Pages 182-184

Differentiation

Introduction to Derivatives

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

-Define derivative as limit of rate of change
-Use dy/dx notation for derivatives
-Understand derivative as gradient function
-Connect derivatives to tangent line slopes

-Introduce derivative notation dy/dx
-Show derivative as gradient of tangent
-Practice derivative concept with simple functions
-Connect to previous gradient work

Exercise books
-Manila paper
-Derivative notation
-Function examples

KLB Secondary Mathematics Form 4, Pages 182-184

Differentiation

Derivative of Linear Functions

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

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

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

Exercise books
-Manila paper
-Linear function examples
-Verification methods

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Derivative of y = x^n (Basic Powers)

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

-Find derivatives of power functions
-Apply the rule d/dx(x^n) = nx^(n-1)
-Practice with x², x³, x⁴, etc.
-Verify using first principles for simple cases

-Derive d/dx(x²) = 2x using first principles
-Apply power rule to various functions
-Practice with x³, x⁴, x⁵ examples
-Verify selected results using definition

Exercise books
-Manila paper
-Power rule examples
-First principles verification

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Derivative of Constant Functions
Derivative of Coefficient Functions

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

-Understand that derivative of constant is zero
-Apply to functions like y = 5, y = -3
-Explain geometric meaning of zero derivative
-Combine with other differentiation rules

-Show that horizontal lines have zero gradient
-Find derivatives of constant functions
-Explain why rate of change of constant is zero
-Apply to mixed functions with constants

Exercise books
-Manila paper
-Constant function graphs
-Geometric explanations
-Coefficient examples
-Rule combinations

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Derivative of Polynomial Functions

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

-Find derivatives of polynomial functions
-Apply term-by-term differentiation
-Practice with various polynomial degrees
-Verify results using first principles

-Differentiate y = x³ + 2x² - 5x + 7
-Apply rule to each term separately
-Practice with various polynomial types
-Check results using definition for simple cases

Exercise books
-Manila paper
-Polynomial examples
-Term-by-term method

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Applications to Tangent Lines

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

-Find equations of tangent lines to curves
-Use derivatives to find tangent gradients
-Apply point-slope form for tangent equations
-Solve problems involving tangent lines

-Find tangent to y = x² at point (2, 4)
-Use derivative to get gradient at specific point
-Apply y - y₁ = m(x - x₁) formula
-Practice with various curves and points

Exercise books
-Manila paper
-Tangent line examples
-Point-slope applications

KLB Secondary Mathematics Form 4, Pages 187-189

Differentiation

Applications to Normal Lines

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

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

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

Exercise books
-Manila paper
-Normal line examples
-Perpendicular concepts

KLB Secondary Mathematics Form 4, Pages 187-189

Differentiation

Introduction to Stationary Points
Types of Stationary Points

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

-Define stationary points as points where dy/dx = 0
-Identify different types of stationary points
-Understand geometric meaning of zero gradient
-Find stationary points by solving dy/dx = 0

-Show horizontal tangents at stationary points
-Find stationary points of y = x² - 4x + 3
-Identify maximum, minimum, and inflection points
-Practice finding where dy/dx = 0

Exercise books
-Manila paper
-Curve sketches
-Stationary point examples
-Sign analysis charts
-Classification examples

KLB Secondary Mathematics Form 4, Pages 189-195

Differentiation

Finding and Classifying Stationary Points

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

-Solve dy/dx = 0 to find stationary points
-Apply systematic classification method
-Use organized approach for point analysis
-Practice with polynomial functions

-Work through complete stationary point analysis
-Use systematic gradient sign testing
-Create organized solution format
-Practice with cubic and quartic functions

Exercise books
-Manila paper
-Systematic templates
-Complete examples

KLB Secondary Mathematics Form 4, Pages 189-195

Differentiation

Curve Sketching Using Derivatives

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

-Use derivatives to sketch accurate curves
-Identify key features: intercepts, stationary points
-Apply systematic curve sketching method
-Combine algebraic and graphical analysis

-Sketch y = x³ - 3x² + 2 using derivatives
-Find intercepts, stationary points, and behavior
-Use systematic curve sketching approach
-Verify sketches using derivative information

Exercise books
-Manila paper
-Curve sketching templates
-Systematic method

KLB Secondary Mathematics Form 4, Pages 195-197

Differentiation

Introduction to Kinematics Applications
Acceleration as Second Derivative

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

-Apply derivatives to displacement-time relationships
-Understand velocity as first derivative of displacement
-Find velocity functions from displacement functions
-Apply to motion problems

-Find velocity from s = t³ - 2t² + 5t
-Apply v = ds/dt to motion problems
-Practice with various displacement functions
-Connect to real-world motion scenarios

Exercise books
-Manila paper
-Motion examples
-Kinematics applications
-Second derivative examples
-Motion analysis

KLB Secondary Mathematics Form 4, Pages 197-201

Differentiation

Motion Problems and Applications

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

-Solve complete motion analysis problems
-Find displacement, velocity, acceleration relationships
-Apply to real-world motion scenarios
-Use derivatives for motion optimization

-Analyze complete motion of falling object
-Find when particle changes direction
-Calculate maximum height in projectile motion
-Apply to vehicle motion problems

Exercise books
-Manila paper
-Complete motion examples
-Real scenarios

KLB Secondary Mathematics Form 4, Pages 197-201

Differentiation

Introduction to Optimization

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

-Apply derivatives to find maximum and minimum values
-Understand optimization in real-world contexts
-Use calculus for practical optimization problems
-Connect to business and engineering applications

-Find maximum area of rectangle with fixed perimeter
-Apply calculus to profit maximization
-Use derivatives for cost minimization
-Practice with geometric optimization

Exercise books
-Manila paper
-Optimization examples
-Real applications

KLB Secondary Mathematics Form 4, Pages 201-204

Differentiation

Geometric Optimization Problems

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

-Apply calculus to geometric optimization
-Find maximum areas and minimum perimeters
-Use derivatives for shape optimization
-Apply to construction and design problems

-Find dimensions for maximum area enclosure
-Optimize container volumes and surface areas
-Apply to architectural design problems
-Practice with various geometric constraints

Exercise books
-Manila paper
-Geometric examples
-Design applications

KLB Secondary Mathematics Form 4, Pages 201-204

Differentiation

Business and Economic Applications
Advanced Optimization Problems

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

-Apply derivatives to profit and cost functions
-Find marginal cost and marginal revenue
-Use calculus for business optimization
-Apply to Kenyan business scenarios

-Find maximum profit using calculus
-Calculate marginal cost and revenue
-Apply to agricultural and manufacturing examples
-Use derivatives for business decision-making

Exercise books
-Manila paper
-Business examples
-Economic applications
-Complex examples
-Engineering applications

KLB Secondary Mathematics Form 4, Pages 201-204

Integration

Introduction to Reverse Differentiation
Basic Integration Rules - Power Functions

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

-Define integration as reverse of differentiation
-Understand the concept of antiderivative
-Recognize the relationship between gradient functions and original functions
-Apply reverse thinking to simple differentiation examples

-Q/A review on differentiation formulas and rules
-Demonstration of reverse process using simple examples
-Working backwards from derivatives to find original functions
-Discussion on why multiple functions can have same derivative
-Introduction to integration symbol ∫

Graph papers
-Differentiation charts
-Exercise books
-Function examples
Calculators
-Graph papers
-Power rule charts

KLB Secondary Mathematics Form 4, Pages 221-223

Integration

Integration of Polynomial Functions
Finding Particular Solutions
Introduction to Definite Integrals

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

-Integrate polynomial functions with multiple terms
-Apply linearity: ∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx
-Handle constant coefficients and addition/subtraction
-Solve integration problems requiring algebraic simplification

-Step-by-step integration of polynomials like 3x² + 5x - 7
-Working with coefficients and constants
-Integration of expanded expressions: (x+2)(x-3)
-Practice with mixed positive and negative terms
-Exercises from textbook Exercise 10.1

Calculators
-Algebraic worksheets
-Polynomial examples
-Exercise books
Graph papers
-Calculators
-Curve examples
-Geometric models
-Integration notation charts

KLB Secondary Mathematics Form 4, Pages 223-225

Integration

Evaluating Definite Integrals
Area Under Curves - Single Functions

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

-Apply Fundamental Theorem of Calculus
-Evaluate definite integrals using [F(x)]ₐᵇ = F(b) - F(a)
-Understand why constant of integration cancels
-Practice numerical evaluation of definite integrals

-Step-by-step evaluation process demonstration
-Multiple worked examples showing limit substitution
-Verification that constant c cancels out
-Practice with various polynomial and power functions
-Exercises from textbook Exercise 10.2

Calculators
-Step-by-step worksheets
-Exercise books
-Evaluation charts
Graph papers
-Curve sketching tools
-Colored pencils
-Calculators
-Area grids

KLB Secondary Mathematics Form 4, Pages 226-230

Integration

Areas Below X-axis and Mixed Regions
Area Between Two Curves

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

-Handle negative areas when curve is below x-axis
-Understand absolute value consideration for areas
-Calculate areas of regions crossing x-axis
-Apply integration to mixed positive/negative regions

-Demonstration of negative integrals and their meaning
-Working with curves that cross x-axis multiple times
-Finding total area vs net area
-Practice with functions like y = x³ - x
-Problem-solving with complex area calculations

Graph papers
-Calculators
-Curve examples
-Colored materials
-Exercise books
-Equation solving aids
-Colored pencils

KLB Secondary Mathematics Form 4, Pages 230-235