Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 6
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra
Previous Lesson: Differentiation: Higher Derivatives and Implicit Functions.
Topic: BINOMIAL EXPANSION
Subject Matter: Meaning of binomial expansion (expanding powers of a binomial expression), Pascal triangle, binomial expansion of (a+b)^n where n is positive integer, negative integer or fractional value

Specific Objectives

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

Cognitive Domain:

Define binomial expansion.
State the properties of Pascal’s triangle.
Recall the binomial theorem for positive integer powers.
Recall the binomial theorem for negative and fractional powers.

Affective Domain:

Appreciate the application of binomial expansion in various mathematical problems.
Develop interest in solving problems involving binomial expressions.

Psychomotor Domain:

Construct Pascal’s triangle correctly.
Expand binomial expressions for positive integer powers.
Expand binomial expressions for negative and fractional powers up to a specified term.

Collaborate with peers to solve binomial expansion problems.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics Project 2 for Senior Secondary Schools
https://www.mathsisfun.com/algebra/binomial-theorem.html
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c929ba:binomial-theorem

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing Pascal’s triangle
Whiteboard and markers
Textbooks

Rationale for the Lesson

This lesson helps pupils understand how to expand expressions with two terms raised to a power. It provides a foundation for more advanced topics in calculus and statistics by simplifying complex algebraic expressions.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic algebra, indices, and combinations.

Lesson Content/Board Summary

BINOMIAL EXPANSION

Meaning of Binomial Expansion

A binomial expression is an algebraic expression that contains two terms, for example, (a+b), (x-y), (2x+3). Binomial expansion is the process of expanding an expression of the form (a+b)ⁿ, where ‘n’ is a positive integer, negative integer, or a fraction.

Pascal’s Triangle

Pascal’s Triangle is a triangular array of numbers where each number is the sum of the two numbers directly above it. It provides the coefficients for binomial expansions of the form (a+b)ⁿ where ‘n’ is a positive integer.

Construction of Pascal’s Triangle:

The top row (row 0) is 1.
Each subsequent row starts and ends with 1.
Every other number is the sum of the two numbers directly above it.

Example:

n=0:      1
n=1:     1 1
n=2:    1 2 1
n=3:   1 3 3 1
n=4:  1 4 6 4 1

Binomial Expansion for Positive Integer ‘n’

For a positive integer ‘n’, the expansion of (a+b)ⁿ is given by:

(a+b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁a^n-1b¹ + ⁿC₂a^n-2b² + … + ⁿC_n-1a¹b^n-1 + ⁿC_na⁰bⁿ

Where ⁿC_r = n! / (r!(n-r)!) are the binomial coefficients, which can also be obtained from Pascal’s Triangle.

Properties of expansion (a+b)ⁿ:

The number of terms in the expansion is (n+1).
The sum of the powers of ‘a’ and ‘b’ in each term is always ‘n’.
The powers of ‘a’ decrease from ‘n’ to 0, while the powers of ‘b’ increase from 0 to ‘n’.

Worked Example 1: Expand (2x – y)³

Step 1: Identify ‘a’, ‘b’, and ‘n’. Here, a = 2x, b = -y, n = 3.

Step 2: Use the coefficients from Pascal’s Triangle for n=3, which are 1, 3, 3, 1.

Step 3: Apply the binomial expansion formula:

(2x – y)³ = 1(2x)³(-y)⁰ + 3(2x)²(-y)¹ + 3(2x)¹(-y)² + 1(2x)⁰(-y)³

Step 4: Simplify each term:

= 1(8x³)(1) + 3(4x²)(-y) + 3(2x)(y²) + 1(1)(-y³)

= 8x³ – 12x²y + 6xy² – y³

Binomial Theorem for Any Real ‘n’ (Negative Integer or Fractional)

For any real number ‘n’ (negative integer or fraction), and for |x| < 1, the expansion of (1+x)ⁿ is given by:

(1+x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + …

Note: This expansion is valid only if the absolute value of x is less than 1, i.e., |x| < 1.

Worked Example 2: Expand (1 + 3x)^-2 up to the term in x³.

Step 1: Identify ‘n’ and ‘x’. Here, n = -2, and the ‘x’ in the formula is 3x.

Step 2: Apply the binomial theorem formula:

(1 + 3x)^-2 = 1 + (-2)(3x) + [-2(-2-1)/2!](3x)² + [-2(-2-1)(-2-2)/3!](3x)³ + …

Step 3: Simplify each term:

= 1 – 6x + [-2(-3)/2](9x²) + [-2(-3)(-4)/6](27x³) + …

= 1 – 6x + [6/2](9x²) + [-24/6](27x³) + …

= 1 – 6x + 3(9x²) – 4(27x³) + …

= 1 – 6x + 27x² – 108x³ + …

This expansion is valid for |3x| < 1, which means |x| < 1/3.

Worked Example 3: Find the first four terms in the expansion of (4 – x)^1/2.

Step 1: Rewrite the expression in the form (1+ax)ⁿ. Factor out 4:

(4 – x)^1/2 = [4(1 – x/4)]^1/2 = 4^1/2(1 – x/4)^1/2 = 2(1 – x/4)^1/2

Step 2: Identify ‘n’ and ‘x’ for the (1 – x/4)^1/2 part. Here, n = 1/2, and the ‘x’ in the formula is (-x/4).

Step 3: Apply the binomial theorem formula to (1 – x/4)^1/2:

(1 – x/4)^1/2 = 1 + (1/2)(-x/4) + [(1/2)(1/2-1)/2!](-x/4)² + [(1/2)(1/2-1)(1/2-2)/3!](-x/4)³ + …

Step 4: Simplify each term:

= 1 – x/8 + [(1/2)(-1/2)/2](x²/16) + [(1/2)(-1/2)(-3/2)/6](-x³/64) + …

= 1 – x/8 + [(-1/4)/2](x²/16) + [(3/8)/6](-x³/64) + …

= 1 – x/8 – (1/8)(x²/16) + (1/16)(-x³/64) + …

= 1 – x/8 – x²/128 – x³/1024 + …

Step 5: Multiply the result by the factored out constant (2):

2(1 – x/8 – x²/128 – x³/1024 + …) = 2 – x/4 – x²/64 – x³/512 + …

This expansion is valid for |-x/4| < 1, which means |x| < 4.

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 reminds them about their previous knowledge of algebraic expressions and indices. The teacher then introduces binomial expansion as a method to efficiently expand expressions like (a+b)ⁿ.
Pupils’ Activity: Pupils respond to greetings and recall previous knowledge, listening attentively to the introduction.
Learning Point: Pupils are prepared for the new topic.

Step 2: Meaning of Binomial Expansion

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains what a binomial expression is and defines binomial expansion. The teacher writes examples of binomial expressions on the board.
Pupils’ Activity: Pupils listen, take notes, and ask questions for clarification.
Learning Point: Pupils understand the basic definition of binomial expansion.

Step 3: Pascal’s Triangle

Time: 10 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher demonstrates how to construct Pascal’s triangle on the board, explaining the pattern and how it provides coefficients for positive integer powers.
Pupils’ Activity: Pupils observe, copy the construction, and participate in generating the next rows of the triangle.
Learning Point: Pupils learn to construct Pascal’s triangle and its relation to binomial coefficients.

Step 4: Binomial Expansion for Positive Integer ‘n’

Time: 10 minutes
Teaching Skill: Explanation and Worked Examples
Teacher’s Activity: The teacher explains the binomial theorem for positive integer powers, showing how to use Pascal’s triangle or combinations to find coefficients. The teacher works through Example 1 from the board summary.
Pupils’ Activity: Pupils listen, copy notes, and follow the steps of the worked example.
Learning Point: Pupils learn to expand binomial expressions with positive integer powers.

Step 5: Binomial Theorem for Any Real ‘n’

Time: 10 minutes
Teaching Skill: Explanation and Worked Examples
Teacher’s Activity: The teacher introduces the binomial theorem for negative and fractional powers, emphasizing the condition for validity (|x|<1). The teacher works through Examples 2 and 3 from the board summary.
Pupils’ Activity: Pupils pay attention to the formula, conditions, and worked examples, taking down notes.
Learning Point: Pupils learn to expand binomial expressions with negative and fractional powers.

Step 6: Further Practice

Time: 5 minutes
Teaching Skill: Practice/Supervision
Teacher’s Activity: The teacher gives pupils a short practice problem to solve individually or in pairs, providing guidance as needed. For example, “Expand (1 – 2x)^-1/2 up to the term in x³.”
Pupils’ Activity: Pupils attempt to solve the practice problem.
Learning Point: Pupils apply their understanding to solve problems.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

What is a binomial expansion?
Construct the first five rows of Pascal’s Triangle.
Expand (x + 3)⁴.
State the binomial theorem for (1+x)ⁿ where ‘n’ is a negative integer or fraction.
Expand (1 – 4x)^1/2 up to the term in x².

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
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the importance of binomial expansion and assigning homework from the textbook.
Pupils’ Activity: Pupils listen and copy the homework assignment.
Learning Point: Pupils consolidate their learning and prepare for independent practice.

Lesson Keywords

Binomial – An algebraic expression with two terms.
Expansion – The process of writing out the full form of a power of an expression.
Pascal’s Triangle – A triangular array of numbers that gives binomial coefficients.
Theorem – A mathematical statement that has been proven.
Coefficient – The numerical factor of a term in an algebraic expression.
Power – The exponent indicating how many times a base number or expression is multiplied by itself.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex problems involving approximations or applications of binomial expansion. For pupils who need more support, the teacher can provide additional guided practice with simpler expansions and provide a pre-filled Pascal’s triangle.

Note for teachers using this lesson plan

Ensure pupils understand the condition |x|<1 for the binomial theorem of general 'n'. Emphasize the step-by-step approach for solving expansion problems, especially when 'a' or 'b' are complex terms or when 'n' is negative or fractional. Encourage pupils to check their work by comparing terms.