Trigonometry III

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

-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

-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

-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
-Rulers
-Calculators (if available)
-Unit circle diagrams
-Calculators
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
Sine and Cosine 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
-Rulers
-Graph paper (if available)

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Transformations of Sine Waves

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

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
Phase Angles and Wave Shifts

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

General Trigonometric Functions
Cosine Wave Transformations

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

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

-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

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

KLB Secondary Mathematics Form 4, Pages 103-109

Trigonometry III

Introduction to Trigonometric Equations
Solving Basic Trigonometric Equations

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

-Understand concept of trigonometric equations
-Identify that trig equations have multiple solutions
-Solve simple equations like sin x = 0.5
-Find all solutions in given ranges

-Demonstrate using unit circle or graphs
-Show why sin x = 0.5 has multiple solutions
-Practice finding principal values
-Use graphs to identify all solutions in range

Exercise books
-Manila paper
-Unit circle diagrams
-Trigonometric tables
-Calculators
-Solution worksheets

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

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
Trigonometric Equations with Identities

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
-Identity reference sheets
-Complex examples

KLB Secondary Mathematics Form 4, Pages 109-112

Trigonometry (II)

The unit circle

By the end of the lesson, the learner should be able to:
Draw the unit circle
Identify coordinates on the unit circle
Understand the unit circle concept

Q/A on basic circle properties
Discussions on unit circle construction
Solving problems using unit circle
Demonstrations of circle drawing
Explaining unit circle applications

Calculators, protractors, rulers, pair of compasses

KLB Mathematics Book Three Pg 41-42

Trigonometry (II)

The unit circle
Trigonometric ratios of angles greater than 90°

By the end of the lesson, the learner should be able to:
Solve problems using the unit circle
Apply unit circle to find trigonometric values
Use unit circle for angle measurement
Find the trigonometric values of angles
Calculate trigonometric ratios for obtuse angles
Apply reference angle concepts

Q/A on unit circle mastery
Discussions on practical applications
Solving trigonometric problems
Demonstrations of value finding
Explaining angle relationships
Q/A on basic trigonometric ratios
Discussions on angle extensions
Solving obtuse angle problems
Demonstrations of reference angles
Explaining quadrant relationships

Calculators, protractors, rulers, pair of compasses
Calculators, protractors, rulers, pair of compasses
Calculators, quadrant charts

KLB Mathematics Book Three Pg 43-44
KLB Mathematics Book Three Pg 44-45

Trigonometry (II)

Trigonometric ratios of negative angles

By the end of the lesson, the learner should be able to:
Find the trigonometric values of negative angles
Apply negative angle identities
Solve problems involving negative angles

Q/A on negative angle concepts
Discussions on angle direction
Solving negative angle problems
Demonstrations of identity applications
Explaining clockwise rotations

Geoboards, graph books, calculators

KLB Mathematics Book Three Pg 48-49

Trigonometry (II)

Trigonometric ratios of angles greater than 360°

By the end of the lesson, the learner should be able to:
Find the trigonometric values of angles greater than 360°
Apply coterminal angle concepts
Reduce angles to standard position

Q/A on angle reduction concepts
Discussions on coterminal angles
Solving extended angle problems
Demonstrations of angle reduction
Explaining periodic properties

Geoboards, graph books, calculators

KLB Mathematics Book Three Pg 49-51

Trigonometry (II)

Use of mathematical tables

By the end of the lesson, the learner should be able to:
Use mathematical tables to find sine and cosine
Read trigonometric tables accurately
Apply table interpolation methods

Q/A on table reading skills
Discussions on table structure
Solving problems using tables
Demonstrations of interpolation
Explaining table accuracy

Mathematical tables, calculators

KLB Mathematics Book Three Pg 51-55

Trigonometry (II)

Use of calculators

By the end of the lesson, the learner should be able to:
Use calculators to find sine, cosine and tan
Apply calculator functions for trigonometry
Verify calculator accuracy

Q/A on calculator trigonometric functions
Discussions on calculator modes
Solving problems using calculators
Demonstrations of function keys
Explaining degree vs radian modes

Calculators, function guides

KLB Mathematics Book Three Pg 56-58

Trigonometry (II)

Radian measure

By the end of the lesson, the learner should be able to:
Convert degrees to radians and vice versa
Apply radian measure in calculations
Understand radian-degree relationships

Q/A on angle measurement systems
Discussions on radian concepts
Solving conversion problems
Demonstrations of conversion methods
Explaining radian applications

Calculators, conversion charts

KLB Mathematics Book Three Pg 58-61

Trigonometry (II)

Simple trigonometric graphs
Graphs of cosines
Graphs of tan

By the end of the lesson, the learner should be able to:
Draw tables for sine of values
Plot graphs of sine functions
Identify sine graph properties
Draw tables for tan of values
Plot graphs of tan functions
Identify asymptotes and discontinuities

Q/A on coordinate graphing
Discussions on periodic functions
Solving graphing problems
Demonstrations of sine plotting
Explaining graph characteristics
Q/A on tangent behavior
Discussions on function domains
Solving tangent graphing problems
Demonstrations of asymptote identification
Explaining discontinuous functions

Calculators, graph papers, plotting guides

KLB Mathematics Book Three Pg 62-63
KLB Mathematics Book Three Pg 64-65

Trigonometry (II)

The sine rule

By the end of the lesson, the learner should be able to:
State the sine rule
Apply sine rule to find solution of triangles
Solve triangles using sine rule

Q/A on triangle properties
Discussions on sine rule applications
Solving triangle problems
Demonstrations of rule application
Explaining ambiguous case

Calculators, triangle worksheets

KLB Mathematics Book Three Pg 65-70

Trigonometry (II)

Cosine rule
Problem solving

By the end of the lesson, the learner should be able to:
State the cosine rule
Apply cosine rule to find solution of triangles
Choose appropriate rule for triangle solving

Q/A on cosine rule concepts
Discussions on rule selection
Solving complex triangle problems
Demonstrations of cosine rule
Explaining when to use each rule

Calculators, triangle worksheets
Calculators, comprehensive problem sets, real-world examples

KLB Mathematics Book Three Pg 71-75

Matrices and Transformation

Matrices of Transformation
Identifying Common Transformation Matrices

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

-Define transformation and identify types
-Recognize that matrices can represent transformations
-Apply 2×2 matrices to position vectors
-Relate matrix operations to geometric transformations

-Review transformation concepts from Form 2
-Demonstrate matrix multiplication using position vectors
-Plot objects and images on coordinate plane
-Practice identifying transformations from images

Exercise books
-Manila paper
-Ruler
-Pencils
-String

KLB Secondary Mathematics Form 4, Pages 1-5

Matrices and Transformation

Finding the Matrix of a Transformation
Using the Unit Square Method
Successive Transformations

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

-Determine the matrix representing a given transformation
-Use coordinate geometry to find transformation matrices
-Apply algebraic methods to find matrix elements
-Verify transformation matrices using test points

-Work through algebraic method of finding matrices
-Use simultaneous equations to solve for matrix elements
-Practice with different types of transformations
-Verify results by applying matrix to test objects

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

KLB Secondary Mathematics Form 4, Pages 6-16

Matrices and Transformation

Matrix Multiplication for Combined Transformations
Single Matrix for Successive Transformations
Inverse of a Transformation

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

-Multiply 2×2 matrices to find combined transformations
-Apply matrix multiplication rules correctly
-Verify combined transformations geometrically
-Solve problems involving multiple transformations

-Practice matrix multiplication systematically on chalkboard
-Verify results by applying to test objects
-Work through complex transformation sequences
-Check computations step by step

Exercise books
-Manila paper
-Ruler
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 16-24

Matrices and Transformation

Properties of Inverse Transformations
Area Scale Factor and Determinant
Shear Transformations

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

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

-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

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

-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 =

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

KLB Secondary Mathematics Form 4, Pages 24-26

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

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

KLB Secondary Mathematics Form 4, Pages 28-34

Matrices and Transformation
Statistics II

Isometric and Non-isometric Transformations
Introduction to Advanced Statistics

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

-Distinguish between isometric and non-isometric transformations
-Classify transformations based on shape and size preservation
-Identify isometric transformations from matrices
-Apply classification to solve problems

-Compare congruent and non-congruent images using cutouts
-Classify transformations systematically
-Practice identification from matrices
-Discuss real-world applications of each type

Exercise books
-Paper cutouts
-Manila paper
-Ruler
-Real data examples
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 35-38

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

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

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Mean Using Working Mean - Frequency Tables
Mean for Grouped Data Using Working Mean
Advanced Working Mean Techniques

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

-Calculate mean using working mean for frequency data
-Apply working mean to discrete frequency distributions
-Use the formula with frequencies correctly
-Solve real-world problems with frequency data

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

-Demonstrate with family size data from local community
-Practice calculating fx and fd systematically
-Work through examples step-by-step
-Students practice with their own collected data

-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
-Community data
-Chalk/markers
-Real datasets
Exercise books
-Manila paper
-Economic data
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 42-48

Statistics II

Introduction to Quartiles, Deciles, Percentiles

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

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Calculating Quartiles for Ungrouped Data
Quartiles for Grouped Data

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

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

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

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

KLB Secondary Mathematics Form 4, Pages 49-52

Statistics II

Deciles and Percentiles Calculations

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

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

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Drawing Cumulative Frequency Curves (Ogives)
Reading Values from Ogives

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

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

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

Exercise books
-Manila paper
-Ruler
-Pencils
-Completed ogives

KLB Secondary Mathematics Form 4, Pages 52-60

Statistics II

Applications of Ogives
Introduction to Measures of Dispersion

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

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

-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

-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
-Real problem datasets
-Ruler
Exercise books
-Manila paper
-Comparative datasets
-Chalk/markers

KLB Secondary Mathematics Form 4, Pages 52-60
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