Class: Junior Secondary School 3 (JSS 3 / JS3)
Term: Third Term
Week: 6
Age: 14 years
Duration: 45 minutes
Subject: Mathematics
Curriculum Theme: Mathematics
Previous Lesson: Fractions (revision): Addition and Subtraction of Fractions.
Topic: RIGHT-ANGLED TRIANGLES
Subject Matter: Use of Pythagoras rule in finding missing sides of triangles, Pythagoras triples: Finding trigonometric ratios of sine, cosine, and tangent using sides of

Specific Objectives

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

Cognitive Domain:

Define Pythagoras theorem.
Identify the hypotenuse, opposite, and adjacent sides of a right-angled triangle.
State the trigonometric ratios of sine, cosine, and tangent.

Affective Domain:

Appreciate the importance of Pythagoras theorem in solving real-life problems.
Develop an interest in using trigonometric ratios for calculations.

Psychomotor Domain:

Calculate the missing side of a right-angled triangle using Pythagoras theorem.
Determine the sine, cosine, and tangent of an acute angle in a right-angled triangle.

Collaborate with peers to solve problems involving right-angled triangles.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum
State Unified Scheme of Work
New General Mathematics for Junior Secondary Schools 3

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing different right-angled triangles.
Ruler and protractor.
Whiteboard/blackboard and markers/chalk.

Rationale for the Lesson

This lesson helps pupils understand the fundamental properties of right-angled triangles. It provides essential tools like the Pythagoras theorem and basic trigonometric ratios, which are useful for solving problems in geometry, engineering, and various real-world situations.

Prerequisite/Previous Knowledge

Pupils are expected to have basic knowledge of types of triangles, squares, and square roots from previous lessons.

Lesson Content/Board Summary

RIGHT-ANGLED TRIANGLES

Pythagoras Theorem

The Pythagoras theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Formula: c² = a² + b²

Where ‘c’ is the hypotenuse (the longest side).
‘a’ and ‘b’ are the other two sides (legs) of the triangle.

Worked Example 1: Finding the Hypotenuse

A right-angled triangle has sides of length 3 cm and 4 cm. Find the length of its hypotenuse.

Step 1: Identify the known sides and the unknown side.

a = 3 cm
b = 4 cm
c = ? (hypotenuse)

Step 2: Apply the Pythagoras theorem: c² = a² + b²

c² = 3² + 4²
c² = 9 + 16
c² = 25

Step 3: Find the square root of c².

c = √25
c = 5 cm

The length of the hypotenuse is 5 cm.

Worked Example 2: Finding a Shorter Side

A right-angled triangle has a hypotenuse of 13 cm and one side of 5 cm. Find the length of the other side.

Step 1: Identify the known sides and the unknown side.

c = 13 cm
a = 5 cm
b = ?

Step 2: Apply the Pythagoras theorem: c² = a² + b²

13² = 5² + b²
169 = 25 + b²

Step 3: Isolate b².

b² = 169 – 25
b² = 144

Step 4: Find the square root of b².

b = √144
b = 12 cm

The length of the other side is 12 cm.

Pythagorean Triples

A Pythagorean triple is a set of three positive integers a, b, and c, such that a² + b² = c². These are integer side lengths of a right-angled triangle.

The following are common Pythagorean triples:

(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)

Trigonometric Ratios (Sine, Cosine, Tangent)

Trigonometric ratios relate the angles of a right-angled triangle to the ratios of the lengths of its sides.

Consider a right-angled triangle with an acute angle θ:

Opposite (O): The side directly across from angle θ.
Adjacent (A): The side next to angle θ that is not the hypotenuse.
Hypotenuse (H): The side opposite the right angle (always the longest side).

The three basic trigonometric ratios are:

Sine (sin θ) = Opposite / Hypotenuse (SOH)
Cosine (cos θ) = Adjacent / Hypotenuse (CAH)
Tangent (tan θ) = Opposite / Adjacent (TOA)

Worked Example: Calculating Trigonometric Ratios

In a right-angled triangle, the opposite side is 8 units, the adjacent side is 6 units, and the hypotenuse is 10 units. Find sin θ, cos θ, and tan θ for the angle θ.

Step 1: Identify the lengths of the sides relative to θ.

Opposite (O) = 8
Adjacent (A) = 6
Hypotenuse (H) = 10

Step 2: Calculate the sine ratio.

sin θ = Opposite / Hypotenuse = 8 / 10 = 4/5

Step 3: Calculate the cosine ratio.

cos θ = Adjacent / Hypotenuse = 6 / 10 = 3/5

Step 4: Calculate the tangent ratio.

tan θ = Opposite / Adjacent = 8 / 6 = 4/3

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils and asks them to recall different types of triangles they know, focusing on right-angled triangles.
Pupils’ Activity: Pupils respond by mentioning various triangles and identify characteristics of a right-angled triangle.
Learning Point: Pupils recall prior knowledge of triangles, specifically right-angled triangles.

Step 2: Explanation of Right-Angled Triangles and Pythagoras Theorem

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains what a right-angled triangle is, identifies its sides (hypotenuse, opposite, adjacent), and introduces the Pythagoras theorem with its formula.
Pupils’ Activity: Pupils listen attentively, observe the diagram, and copy the definition and formula into their notebooks.
Learning Point: Pupils understand the components of a right-angled triangle and the Pythagoras theorem.

Step 3: Worked Examples on Pythagoras Theorem

Time: 10 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher solves two worked examples on the board, demonstrating how to find missing sides (hypotenuse and a shorter side) using the Pythagoras theorem.
Pupils’ Activity: Pupils observe the steps, ask questions for clarification, and copy the examples into their notes.
Learning Point: Pupils learn how to apply the Pythagoras theorem to calculate missing sides.

Step 4: Introduction to Pythagorean Triples

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher defines Pythagorean triples and provides common examples, explaining their significance.
Pupils’ Activity: Pupils listen and note down the definition and examples of Pythagorean triples.
Learning Point: Pupils understand what Pythagorean triples are.

Step 5: Explanation of Trigonometric Ratios

Time: 5 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher introduces the concepts of sine, cosine, and tangent, explaining them as ratios of sides relative to an acute angle using the acronym SOH CAH TOA.
Pupils’ Activity: Pupils pay attention, identify the sides (opposite, adjacent, hypotenuse) on diagrams, and write down the definitions of the ratios.
Learning Point: Pupils learn the definitions of sine, cosine, and tangent ratios.

Step 6: Worked Example on Trigonometric Ratios

Time: 5 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher solves a worked example on the board, showing how to calculate sine, cosine, and tangent for a given angle in a right-angled triangle.
Pupils’ Activity: Pupils follow the steps shown by the teacher and copy the example into their notebooks.
Learning Point: Pupils learn to calculate basic trigonometric ratios.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

State the Pythagoras theorem.
In a right-angled triangle with sides 6 cm and 8 cm, find the hypotenuse.
List two examples of Pythagorean triples.
Define sine, cosine, and tangent in terms of opposite, adjacent, and hypotenuse.

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the key points of the lesson and assigns homework, which includes solving problems on Pythagoras theorem and trigonometric ratios.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their understanding of the lesson and prepare for further practice.

Lesson Keywords

Pythagoras – A theorem relating the sides of a right-angled triangle.
Hypotenuse – The longest side of a right-angled triangle, opposite the right angle.
Opposite – The side across from a given acute angle in a right-angled triangle.
Adjacent – The side next to a given acute angle that is not the hypotenuse.
Sine – A trigonometric ratio (Opposite/Hypotenuse).
Cosine – A trigonometric ratio (Adjacent/Hypotenuse).
Tangent – A trigonometric ratio (Opposite/Adjacent).
Right angle – An angle measuring exactly 90 degrees.
Triples – Sets of three integers that satisfy the Pythagoras theorem.

Differentiation

For pupils who are struggling, the teacher will provide simpler problems with smaller numbers and offer one-on-one guidance. For advanced learners, the teacher will provide more complex problems involving multiple steps or real-world application scenarios.

Note for teachers using this lesson plan

Ensure that pupils clearly understand how to identify the hypotenuse, opposite, and adjacent sides relative to a specified angle. Emphasize the importance of drawing diagrams for each problem. Encourage accuracy in calculations and proper use of units.