Probability

Introduction
Experimental Probability

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

In groups, learners are guided to:
Q/A on uncertain events from daily life experiences
Discussions on weather prediction and game outcomes
Analyzing chance events using coin tossing and dice rolling
Demonstrations using simple probability experiments
Explaining probability language using familiar examples

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

KLB Mathematics Book Three Pg 262-264

Probability

Experimental Probability applications
Range of Probability Measure

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

In groups, learners are guided to:
Q/A on advanced experimental techniques using extended trials
Discussions on sample size effects using comparative data
Solving complex experimental problems using systematic methods
Demonstrations using extended experimental procedures
Explaining pattern analysis using accumulated data

Chalk and blackboard, extended experimental materials, data recording sheets, exercise books
Chalk and blackboard, number line drawings, probability scale charts, exercise books

KLB Mathematics Book Three Pg 262-264

Probability

Probability Space
Theoretical Probability

By the end of the lesson, the learner should be able to:
Calculate the probability space for the theoretical probability
Define sample space systematically
List all possible outcomes
Apply sample space concepts

In groups, learners are guided to:
Q/A on outcome listing using systematic enumeration
Discussions on complete outcome identification
Solving sample space problems using organized listing
Demonstrations using dice, cards, and spinner examples
Explaining probability calculation using outcome counting

Chalk and blackboard, playing cards (locally made), spinners from cardboard, exercise books
Chalk and blackboard, fair dice and coins, probability calculation aids, exercise books

KLB Mathematics Book Three Pg 266-267

Probability

Theoretical Probability advanced
Theoretical Probability applications

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

In groups, learners are guided to:
Q/A on advanced theoretical applications using complex scenarios
Discussions on multiple outcome analysis using systematic methods
Solving challenging theoretical problems using organized approaches
Demonstrations using complex probability setups
Explaining advanced theoretical concepts using detailed reasoning

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

KLB Mathematics Book Three Pg 268-270

Probability

Combined Events
Combined Events OR probability

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

In groups, learners are guided to:
Q/A on event combination using practical examples
Discussions on exclusive and inclusive event identification
Solving basic combined event problems using visual methods
Demonstrations using card drawing and dice rolling combinations
Explaining combination principles using Venn diagrams

Chalk and blackboard, playing cards, multiple dice, Venn diagram drawings, exercise books
Chalk and blackboard, Venn diagram materials, card examples, exercise books

KLB Mathematics Book Three Pg 272-273

Probability

Independent Events
Independent Events advanced

By the end of the lesson, the learner should be able to:
Find the probability of independent events
Apply multiplication rule for independent events
Calculate "A and B" probabilities
Understand independence concepts

In groups, learners are guided to:
Q/A on multiplication rule using independent event examples
Discussions on independence identification and verification
Solving AND probability problems using systematic calculation
Demonstrations using multiple coin tosses and dice combinations
Explaining multiplication rule using logical reasoning

Chalk and blackboard, multiple coins and dice, independence demonstration materials, exercise books
Chalk and blackboard, playing cards for replacement scenarios, multiple experimental setups, exercise books

KLB Mathematics Book Three Pg 274-275

Probability

Independent Events applications
Tree Diagrams

By the end of the lesson, the learner should be able to:
Find the probability of independent events
Apply independence to practical problems
Solve complex multi-event scenarios
Integrate independence with other concepts

In groups, learners are guided to:
Q/A on complex event analysis using systematic problem-solving
Discussions on rule selection and application strategies
Solving advanced combined problems using integrated approaches
Demonstrations using complex experimental scenarios
Explaining strategic problem-solving using logical analysis

Chalk and blackboard, complex experimental materials, advanced calculation aids, exercise books
Chalk and blackboard, tree diagram templates, branching materials, exercise books

KLB Mathematics Book Three Pg 278-280

Probability
Compound Proportion and Rates of Work

Tree Diagrams advanced
Compound Proportions

By the end of the lesson, the learner should be able to:
Use tree diagrams to find probability
Apply trees to multi-stage problems
Handle complex sequential events
Calculate final probabilities using trees

In groups, learners are guided to:
Q/A on complex tree application using multi-stage examples
Discussions on replacement scenario handling
Solving complex tree problems using systematic calculation
Demonstrations using detailed tree constructions
Explaining systematic probability calculation using tree methods

Chalk and blackboard, complex tree examples, detailed calculation aids, exercise books
Chalk and blackboard, local business examples, calculators if available, exercise books

KLB Mathematics Book Three Pg 283-285

Compound Proportion and Rates of Work

Compound Proportions applications
Proportional Parts

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

In groups, learners are guided to:
Q/A on advanced compound proportion using complex scenarios
Discussions on multi-variable relationships using practical contexts
Solving challenging compound problems using systematic approaches
Demonstrations using construction and farming examples
Explaining practical applications using community-based scenarios

Chalk and blackboard, construction/farming examples, exercise books
Chalk and blackboard, sharing demonstration materials, exercise books

KLB Mathematics Book Three Pg 290-291

Compound Proportion and Rates of Work

Proportional Parts applications
Rates of Work

By the end of the lesson, the learner should be able to:
Calculate the proportional parts
Apply proportional parts to complex sharing scenarios
Handle business partnership profit sharing
Solve advanced proportional distribution problems

In groups, learners are guided to:
Q/A on complex proportional sharing using business examples
Discussions on partnership profit distribution using practical scenarios
Solving advanced proportional problems using systematic methods
Demonstrations using business partnership and investment examples
Explaining practical applications using meaningful contexts

Chalk and blackboard, business partnership examples, exercise books
Chalk and blackboard, work scenario examples, exercise books

KLB Mathematics Book Three Pg 291-293

Compound Proportion and Rates of Work
Graphical Methods

Rates of Work and Mixtures
Tables of given relations

By the end of the lesson, the learner should be able to:
Calculate the rate of work
Apply work rates to complex scenarios
Handle mixture problems and combinations
Solve advanced rate and mixture problems

In groups, learners are guided to:
Q/A on advanced work rates using complex scenarios
Discussions on mixture problems using practical examples
Solving challenging rate and mixture problems using systematic approaches
Demonstrations using cooking, construction, and manufacturing examples
Explaining mixture concepts using practical applications

Chalk and blackboard, mixture demonstration materials, exercise books
Chalk and blackboard, ruled paper for tables, exercise books

KLB Mathematics Book Three Pg 295-296

Graphical Methods

Graphs of given relations
Tables and graphs integration

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

In groups, learners are guided to:
Q/A on graph plotting using coordinate methods
Discussions on point plotting and curve drawing
Solving graph construction problems using systematic plotting
Demonstrations using coordinate systems and curve sketching
Explaining graph interpretation using visual analysis

Chalk and blackboard, graph paper or grids, rulers, exercise books
Chalk and blackboard, graph paper, data examples, exercise books

KLB Mathematics Book Three Pg 300

Graphical Methods

Introduction to cubic equations
Graphical solution of cubic equations

By the end of the lesson, the learner should be able to:
Draw tables of cubic functions
Understand cubic equation characteristics
Prepare cubic function data systematically
Recognize cubic curve patterns

In groups, learners are guided to:
Q/A on cubic function evaluation using systematic calculation
Discussions on cubic equation properties using mathematical analysis
Solving cubic table preparation using organized methods
Demonstrations using cubic function examples
Explaining cubic characteristics using pattern recognition

Chalk and blackboard, cubic function examples, exercise books
Chalk and blackboard, graph paper, cubic equation examples, exercise books

KLB Mathematics Book Three Pg 301

Graphical Methods

Advanced cubic solutions
Introduction to rates of change

By the end of the lesson, the learner should be able to:
Draw graphs of cubic equations
Apply graphical methods to complex cubic problems
Handle multiple root scenarios
Verify solutions using graphical analysis

In groups, learners are guided to:
Q/A on advanced cubic graphing using complex examples
Discussions on multiple root identification using graph analysis
Solving challenging cubic problems using systematic methods
Demonstrations using detailed cubic constructions
Explaining verification methods using graphical checking

Chalk and blackboard, advanced graph examples, exercise books
Chalk and blackboard, rate calculation examples, exercise books

KLB Mathematics Book Three Pg 302-304

Graphical Methods

Average rates of change
Advanced average rates

By the end of the lesson, the learner should be able to:
Calculate the average rates of change
Apply average rate methods to various functions
Use graphical methods for rate calculation
Solve practical rate problems

In groups, learners are guided to:
Q/A on average rate calculation using graphical methods
Discussions on rate applications using real-world scenarios
Solving average rate problems using systematic approaches
Demonstrations using graph-based rate calculation
Explaining practical applications using meaningful contexts

Chalk and blackboard, graph paper, rate examples, exercise books
Chalk and blackboard, advanced rate scenarios, exercise books

KLB Mathematics Book Three Pg 304-306

Graphical Methods

Introduction to instantaneous rates
Rate of change at an instant

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

In groups, learners are guided to:
Q/A on instantaneous rate concepts using limiting methods
Discussions on instant vs average rate differences
Solving basic instantaneous rate problems
Demonstrations using tangent line concepts
Explaining instantaneous rate using practical examples

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

KLB Mathematics Book Three Pg 310-311

Graphical Methods

Advanced instantaneous rates
Empirical graphs

By the end of the lesson, the learner should be able to:
Calculate the rate of change at an instant
Handle complex instantaneous rate scenarios
Apply instant rates to advanced problems
Integrate instantaneous concepts with applications

In groups, learners are guided to:
Q/A on advanced instantaneous applications using complex examples
Discussions on sophisticated rate problems using detailed analysis
Solving challenging instantaneous problems using systematic methods
Demonstrations using comprehensive rate constructions
Explaining advanced applications using detailed reasoning

Chalk and blackboard, advanced rate examples, exercise books
Chalk and blackboard, experimental data examples, exercise books

KLB Mathematics Book Three Pg 310-315

Graphical Methods
Differentiation

Advanced empirical methods
Introduction to Rate of Change

By the end of the lesson, the learner should be able to:
Draw the empirical graphs
Apply empirical methods to complex data
Handle large datasets and trends
Interpret empirical results meaningfully

In groups, learners are guided to:
Q/A on advanced empirical techniques using complex datasets
Discussions on trend analysis using systematic methods
Solving challenging empirical problems using organized approaches
Demonstrations using comprehensive data analysis
Explaining advanced interpretations using detailed reasoning

Chalk and blackboard, complex data examples, exercise books
Exercise books
-Manila paper
-Real-world examples
-Graph examples

KLB Mathematics Book Three Pg 315-321

Differentiation

Average Rate of Change
Instantaneous 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

In groups, learners are guided to:
-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
-Tangent demonstrations
-Motion examples

KLB Secondary Mathematics Form 4, Pages 177-182

Differentiation

Gradient of Curves at Points
Introduction to Delta Notation

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

In groups, learners are guided to:
-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
-Delta notation examples
-Symbol practice

KLB Secondary Mathematics Form 4, Pages 178-182

Differentiation

The Limiting Process
Introduction to Derivatives

By the end of the lesson, the learner should be able to:
-Understand concept of limit in differentiation
-Apply "as Δx approaches zero" reasoning
-Use limiting process to find exact derivatives
-Practice systematic limiting calculations

In groups, learners are guided to:
-Demonstrate limiting process with numerical examples
-Show chord approaching tangent as Δx → 0
-Calculate limits using table of values
-Practice systematic limit evaluation

Exercise books
-Manila paper
-Limit tables
-Systematic examples
-Derivative notation
-Function examples

KLB Secondary Mathematics Form 4, Pages 182-184

Differentiation

Derivative of Linear Functions
Derivative of y = x^n (Basic Powers)

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

In groups, learners are guided to:
-Find derivative of y = 3x + 2 using definition
-Show that derivative equals the gradient
-Practice with various linear functions
-Verify results using first principles

Exercise books
-Manila paper
-Linear function examples
-Verification methods
-Power rule examples
-First principles verification

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

In groups, learners are guided to:
-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
Applications to Tangent Lines

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

In groups, learners are guided to:
-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
-Tangent line examples
-Point-slope applications

KLB Secondary Mathematics Form 4, Pages 184-188

Differentiation

Applications to Normal Lines
Introduction to Stationary Points

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

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

Exercise books
-Manila paper
-Normal line examples
-Perpendicular concepts
-Curve sketches
-Stationary point examples

KLB Secondary Mathematics Form 4, Pages 187-189

Differentiation

Types of Stationary Points
Finding and Classifying Stationary Points

By the end of the lesson, the learner should be able to:
-Distinguish between maximum and minimum points
-Identify points of inflection
-Use first derivative test for classification
-Apply gradient analysis around stationary points

In groups, learners are guided to:
-Analyze gradient changes around stationary points
-Use sign analysis of dy/dx
-Classify stationary points by gradient behavior
-Practice with various function types

Exercise books
-Manila paper
-Sign analysis charts
-Classification examples
-Systematic templates
-Complete examples

KLB Secondary Mathematics Form 4, Pages 189-195

Differentiation

Curve Sketching Using Derivatives
Introduction to Kinematics Applications

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

In groups, learners are guided to:
-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
-Motion examples
-Kinematics applications

KLB Secondary Mathematics Form 4, Pages 195-197

Differentiation

Acceleration as Second Derivative
Motion Problems and Applications

By the end of the lesson, the learner should be able to:
-Understand acceleration as derivative of velocity
-Apply a = dv/dt = d²s/dt² notation
-Find acceleration functions from displacement
-Apply to motion analysis problems

In groups, learners are guided to:
-Find acceleration from velocity functions
-Use second derivative notation
-Apply to projectile motion problems
-Practice with particle motion scenarios

Exercise books
-Manila paper
-Second derivative examples
-Motion analysis
-Complete motion examples
-Real scenarios

KLB Secondary Mathematics Form 4, Pages 197-201

Differentiation

Introduction to Optimization
Geometric Optimization Problems

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

In groups, learners are guided to:
-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
-Geometric examples
-Design applications

KLB Secondary Mathematics Form 4, Pages 201-204

Differentiation
Area Approximation

Business and Economic Applications
Advanced Optimization Problems
Area Approximation - Introduction to area approximation

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

In groups, learners are guided to:
-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
Wall map of Kenya, traced outlines, real leaves, chalkboard

KLB Secondary Mathematics Form 4, Pages 201-204

Area Approximation

Area Approximation - Tracing and overlaying on a square grid
Area Approximation - Counting full and partial squares
Area Approximation - Applying scale to find actual area
Area Approximation - Subdividing irregular regions into known shapes

By the end of the lesson, the learner should be able to:
- trace an irregular outline onto tracing paper
- overlay the tracing onto a 1 cm square grid
- distinguish between fully and partially enclosed squares

- Practical activity: learners trace an irregular shape onto tracing paper and overlay on graph paper
- Teacher demonstrates correct alignment
- Learners shade full and partial squares in different colours

Tracing paper, graph paper, pencils, coloured pencils, rulers
Traced outlines, graph paper, calculators, manila paper
Topographical map sample, graph paper, calculators, chalkboard
Manila paper outlines, rulers, set squares, calculators

KLB Sec. Maths Form 4, pg. 206

Area Approximation

Area Approximation - Deriving the trapezium rule
Area Approximation - Applying the trapezium rule to irregular shapes
Area Approximation - Estimating area under a curve using the trapezium rule
Area Approximation - Deriving and applying the mid-ordinate rule

By the end of the lesson, the learner should be able to:
- recall the formula for area of a trapezium
- derive the trapezium rule A = (h/2)[y₀ + yₙ + 2(y₁ + … + yₙ₋₁)]
- identify ordinates and the strip width h

In groups, learners are guided to:
- Review of trapezium area formula
- Step-by-step derivation by dividing a region into strips and summing areas
- Guided drawing of strips under a sample curve

Graph paper, manila paper, rulers, chalkboard
Graph paper, rulers, calculators, worksheets
Graph paper, calculators, worksheets, chalkboard

KLB Sec. Maths Form 4, pg. 210

Area Approximation
Integration
Integration

Area Approximation - Comparison of methods and consolidation
Introduction to Reverse Differentiation
Basic Integration Rules - Power Functions

By the end of the lesson, the learner should be able to:
- apply all three approximation methods to the same region
- identify sources of error and compare accuracy
- solve mixed-method problems including real-life applications

In groups, learners are guided to:
- Whole-class problem-solving using all three methods on one region
- Group work to tabulate and compare estimates
- End-of-topic short test (10 minutes) covering all three methods

Graph paper, calculators, comparison tables, test handout
Graph papers
-Differentiation charts
-Exercise books
-Function examples
Calculators
-Graph papers
-Power rule charts

KLB Sec. Maths Form 4, pg. 219

Integration

Integration of Polynomial Functions
Finding Particular Solutions
Introduction to Definite Integrals
Evaluating 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

In groups, learners are guided to:
-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
-Step-by-step worksheets
-Evaluation charts

KLB Secondary Mathematics Form 4, Pages 223-225

Integration

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

By the end of the lesson, the learner should be able to:
-Understand integration as area calculation tool
-Calculate area between curve and x-axis
-Handle regions bounded by curves and vertical lines
-Apply definite integrals to find exact areas

In groups, learners are guided to:
-Geometric demonstration of area under curves
-Drawing and shading regions on graph paper
-Working examples: area under y = x², y = 2x + 3, etc.
-Comparison with approximation methods from Chapter 9
-Practice finding areas of various regions

Graph papers
-Curve sketching tools
-Colored pencils
-Calculators
-Area grids
-Curve examples
-Colored materials
-Exercise books
-Equation solving aids

KLB Secondary Mathematics Form 4, Pages 230-233