Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 2nd Term
Week: 2
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Functions I.
Topic: SEQUENCES AND SERIES
Subject Matter: Geometric Progression (GP): definition, common ratio, nth term, sum of n terms, solving problems on GP, convergent and divergent geometric progressions, and sum to infinity of a convergent GP.

Specific Objectives

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

Cognitive Domain:

Define a Geometric Progression (GP).
State the formula for the nth term and the sum of the first n terms of a GP.
Distinguish between convergent and divergent geometric progressions.
Calculate the sum to infinity of a convergent geometric progression.

Affective Domain:

Appreciate the real-world applications of geometric progressions.
Develop a logical approach to solving problems involving GP.

Psychomotor Domain:

Solve problems related to the nth term and sum of n terms of a GP.
Accurately determine if a given GP is convergent or divergent.
Compute the sum to infinity for convergent GPs.

Collaborate effectively with peers to solve problems on geometric progressions.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Education Curriculum for Further Mathematics.
State Unified Scheme of Work for Further Mathematics SSS 1.
New Further Mathematics Project for Senior Secondary Schools 1.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts illustrating convergent and divergent geometric progressions.
Whiteboard and markers.
Textbook examples.

Rationale for the Lesson

This lesson helps pupils understand sequences where terms change by multiplication, which is important for understanding growth and decay in various real-world situations. It enables them to solve problems involving patterns of growth and to predict future outcomes based on these patterns.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of sequences and series, including Arithmetic Progression (AP), and basic algebraic operations.

Lesson Content/Board Summary

Geometric Progression (GP)

Definition of Geometric Progression

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Common Ratio (r)

The common ratio `r` is found by dividing any term by its preceding term. For example, `r = T2/T1 = T3/T2`.

General Term (nth term) of a GP

The formula for the nth term of a GP is given by: `Tn = ar^(n-1)`, where `a` is the first term, `r` is the common ratio, and `n` is the term number.

Sum of the First n Terms of a GP (Sn)

The sum of the first n terms of a GP is given by:

`Sn = a(r^n – 1) / (r – 1)` for `r > 1`
`Sn = a(1 – r^n) / (1 – r)` for `r < 1`

Solving Problems on Geometric Progression

The following are examples of solving problems on GP:

Example 1: Find the 5th term of the GP: 2, 6, 18, …
Solution: Here, a=2, r = 6/2 = 3. Using `Tn = ar^(n-1)`, T5 = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162.
Example 2: Find the sum of the first 4 terms of the GP: 3, 6, 12, …
Solution: Here, a=3, r = 6/3 = 2. Using `Sn = a(r^n – 1) / (r – 1)`, S4 = 3(2^4 – 1) / (2 – 1) = 3(16 – 1) / 1 = 3 * 15 = 45.

Convergent Geometric Progression

A geometric progression is convergent if the absolute value of its common ratio `|r|` is less than 1 (i.e., -1 < r < 1). For a convergent GP, as `n` approaches infinity, the terms of the sequence approach zero.

Divergent Geometric Progression

A geometric progression is divergent if the absolute value of its common ratio `|r|` is greater than or equal to 1 (i.e., `r >= 1` or `r <= -1`). For a divergent GP, as `n` approaches infinity, the terms of the sequence grow larger or oscillate without approaching a specific value.

Sum to Infinity of a Convergent GP (S∞)

The sum to infinity exists only for convergent geometric progressions (`|r| < 1`). The formula for the sum to infinity is: `S∞ = a / (1 – r)`.

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 reviews the previous lesson on Arithmetic Progression (AP) by asking questions like, “What is an AP?” and “How do you find the common difference?” The teacher then introduces Geometric Progression as another type of sequence.
Pupils’ Activity: Pupils respond to the questions and listen attentively to the introduction of the new topic.
Learning Point: Pupils recall previous knowledge and are prepared for the new topic.

Step 2: Definition and Common Ratio of GP

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher defines a Geometric Progression and explains how to find the common ratio (r). The teacher provides examples like 2, 6, 18, … and 8, 4, 2, … to illustrate the concept and calculates their common ratios.
Pupils’ Activity: Pupils listen, take notes, and identify the common ratios of given sequences.
Learning Point: Pupils understand the definition of a GP and how to calculate its common ratio.

Step 3: Formulae for GP

Time: 7 minutes
Teaching Skill: Explanation/Formula Derivation
Teacher’s Activity: The teacher presents the formulae for the nth term (`Tn = ar^(n-1)`) and the sum of the first n terms (`Sn`) of a GP, explaining when to use each version of the sum formula.
Pupils’ Activity: Pupils copy the formulae into their notebooks and ask questions for clarification.
Learning Point: Pupils learn the key formulae for calculating terms and sums in a GP.

Step 4: Solving Problems on GP

Time: 8 minutes
Teaching Skill: Problem Solving/Demonstration
Teacher’s Activity: The teacher works through examples on the board to demonstrate how to use the formulae for the nth term and sum of n terms of a GP. For instance, finding a specific term or the sum of the first few terms.
Pupils’ Activity: Pupils observe the teacher’s steps, ask questions, and attempt similar problems.
Learning Point: Pupils learn to apply the GP formulae to solve practical problems.

Step 5: Convergent and Divergent GP

Time: 7 minutes
Teaching Skill: Explanation/Classification
Teacher’s Activity: The teacher explains the conditions for a GP to be convergent (`|r| = 1`). The teacher uses the charts of convergent and divergent GPs to visually illustrate the concepts.
Pupils’ Activity: Pupils observe the charts, listen to the explanation, and identify whether given GPs are convergent or divergent.
Learning Point: Pupils can distinguish between convergent and divergent geometric progressions based on their common ratio.

Step 6: Sum to Infinity of a Convergent GP

Time: 5 minutes
Teaching Skill: Explanation/Formula Application
Teacher’s Activity: The teacher introduces the concept of sum to infinity and presents its formula (`S∞ = a / (1 – r)`) for convergent GPs. The teacher solves a simple example.
Pupils’ Activity: Pupils note the formula and work through the example with the teacher.
Learning Point: Pupils understand how to calculate the sum to infinity for convergent GPs.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define a Geometric Progression.
State the formula for the nth term of a GP.
Explain the condition for a GP to be convergent.
Calculate the sum to infinity of a GP with first term 8 and common ratio 1/2.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher briefly summarizes the key points of the lesson (definition of GP, formulae, convergent/divergent GPs, and sum to infinity). The teacher assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy the homework assignment.
Learning Point: Pupils consolidate their learning and are given tasks for practice.

Lesson Keywords

Geometric Progression (GP) – A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
Common Ratio (r) – The fixed non-zero number by which each term in a GP is multiplied to get the next term.
Convergent GP – A GP where the absolute value of the common ratio is less than 1 (`|r| < 1`), and the terms approach zero as `n` increases.
Divergent GP – A GP where the absolute value of the common ratio is greater than or equal to 1 (`|r| >= 1`), and the terms do not approach zero.
Sum to Infinity – The sum of all terms in a convergent geometric progression.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving finding the first term or common ratio given other terms or sums. For pupils needing more support, the teacher can provide additional guided examples and simplified practice problems, focusing on identifying the common ratio and applying basic formulae.

Note for teachers using this lesson plan

Ensure that pupils have a solid understanding of basic algebraic manipulation before tackling problems on geometric progression. Emphasize the conditions for convergence and divergence clearly, as this is a common area of confusion. Encourage pupils to show all their working steps when solving problems.