Indices and Their Laws for SS 1

Class: SS 1
Term: First Term
Week: 3
Age: 15 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Applying laws of indices to solve practical daily problem
Key competencies/values: Critical Thinking; Respect; Responsibility
Skills:

• Applying laws of indices to solve practical daily problem

Previous Lesson: Number Base Conversion and Arithmetic Operations
Topic: Laws Of Indices
Subject Matter: Meaning of Laws of Indices

Specific Objectives

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

Cognitive Domain

• Define indices and the laws of indices.
• State the conditions for applying each law of indices.
• Identify the correct law to use for different indices problems.

Affective Domain

• Appreciate the importance of indices in simplifying mathematical expressions.
• Demonstrate responsibility in applying the laws of indices accurately.
• Show critical thinking when solving problems involving indices.

Psychomotor Domain

• Generate the laws of indices from examples of repeated multiplication.
• Use the laws of indices to solve various problems involving multiplication, division, and powers of indices.
• Apply laws of indices to solve practical daily problems.

Social Domain

• Collaborate with peers to verify the laws of indices using numerical examples.

Reference Materials

The following resources were used in planning this lesson:

• 2025 New Revised Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 1
• The HeadTeacher Scheme of work For The New Revised Senior Secondary Education Curriculum (SSEC)

Instructional Materials

The teacher will teach this lesson with the aid of:

• Repeated multiplication chart.
• Charts displaying the laws of indices.
• Whiteboard and markers.
• Textbooks and exercise books.

Rationale for the Lesson

This lesson is important because indices provide a concise way to represent repeated multiplication, simplifying complex mathematical expressions. Understanding the laws of indices is foundational for advanced topics in algebra, calculus, and various scientific fields. It also equips students with critical thinking skills applicable to solving real-world problems involving exponential growth or decay.

Prerequisite/Previous Knowledge

Pupils/students should have a basic understanding of multiplication, division, and powers of numbers from their Junior Secondary education.

Lesson Content/Board Summary

Laws of Indices

Meaning of Indices

Indices (singular: index) are a way of writing repeated multiplication in a compact form. An index (also called a power or exponent) indicates how many times a base number is multiplied by itself.

For example, in (2^3):

1. (2) is the base.
2. (3) is the index (or power/exponent).
3. (2^3) means (2 times 2 times 2), which equals (8).

Laws of Indices

The laws of indices are rules that simplify expressions involving powers. These laws apply when the base numbers are the same.

Law 1: Multiplication Law

When multiplying numbers with the same base, add their indices.

(a^x times a^y = a^{x+y})

Condition: The bases must be the same.

Example 1

Question: Simplify (2^3 times 2^4).

Solution:

Step 1: Write the formula.

(a^x times a^y = a^{x+y})

Step 2: Substitute the values.

(2^3 times 2^4 = 2^{3+4})

Step 3: Simplify and write the answer.

(2^{3+4} = 2^7)

Answer: (2^7)

Example 2

Question: Simplify (m^5 times m^2).

Solution:

Step 1: Write the formula.

(a^x times a^y = a^{x+y})

Step 2: Substitute the values.

(m^5 times m^2 = m^{5+2})

Step 3: Simplify and write the answer.

(m^{5+2} = m^7)

Answer: (m^7)

Law 2: Division Law

When dividing numbers with the same base, subtract the index of the divisor from the index of the dividend.

(a^x div a^y = a^{x-y})

Conditions: The bases must be the same, and (a neq 0).

Example 1

Question: Simplify (3^5 div 3^2).

Solution:

Step 1: Write the formula.

(a^x div a^y = a^{x-y})

Step 2: Substitute the values.

(3^5 div 3^2 = 3^{5-2})

Step 3: Simplify and write the answer.

(3^{5-2} = 3^3)

Answer: (3^3)

Example 2

Question: Simplify (frac{p^7}{p^3}).

Solution:

Step 1: Write the formula.

(a^x div a^y = a^{x-y})

Step 2: Substitute the values.

(frac{p^7}{p^3} = p^{7-3})

Step 3: Simplify and write the answer.

(p^{7-3} = p^4)

Answer: (p^4)

Law 3: Power Law (Power of a Power)

When raising a power to another power, multiply the indices.

((a^x)^y = a^{xy})

Condition: Applicable to any non-zero base.

Example 1

Question: Simplify ((2^3)^2).

Solution:

Step 1: Write the formula.

((a^x)^y = a^{xy})

Step 2: Substitute the values.

((2^3)^2 = 2^{3 times 2})

Step 3: Simplify and write the answer.

(2^{3 times 2} = 2^6)

Answer: (2^6)

Example 2

Question: Simplify ((y^4)^3).

Solution:

Step 1: Write the formula.

((a^x)^y = a^{xy})

Step 2: Substitute the values.

((y^4)^3 = y^{4 times 3})

Step 3: Simplify and write the answer.

(y^{4 times 3} = y^{12})

Answer: (y^{12})

Law 4: Zero Index Law

Any non-zero number raised to the power of zero is equal to 1.

(a^0 = 1)

Condition: (a neq 0).

Example 1

Question: Simplify (5^0).

Solution:

Step 1: Apply the Zero Index Law.

(5^0 = 1)

Answer: (1)

Example 2

Question: Simplify ((xy)^0).

Solution:

Step 1: Apply the Zero Index Law.

((xy)^0 = 1)

Answer: (1)

Law 5: Negative Index Law

A number raised to a negative index is the reciprocal of the number raised to the positive index.

(a^{-x} = frac{1}{a^x})

Condition: (a neq 0).

Example 1

Question: Simplify (2^{-3}).

Solution:

Step 1: Apply the Negative Index Law.

(2^{-3} = frac{1}{2^3})

Step 2: Simplify the denominator.

(frac{1}{2^3} = frac{1}{2 times 2 times 2} = frac{1}{8})

Answer: (frac{1}{8})

Example 2

Question: Simplify (x^{-5}).

Solution:

Step 1: Apply the Negative Index Law.

(x^{-5} = frac{1}{x^5})

Answer: (frac{1}{x^5})

Practical Application of Indices: Population Growth

Indices can be used to model growth patterns, such as population growth over generations.

Consider a family where each child gives birth to 3 children only:

1. 1st Generation: Starting with 1 child. This can be represented as (3^0 = 1).
2. 2nd Generation: Each of the 1 child gives birth to 3, so (1 times 3 = 3) children. This is (3^1).
3. 3rd Generation: Each of the 3 children gives birth to 3, so (3 times 3 = 9) children. This is (3^2).
4. 4th Generation: Each of the 9 children gives birth to 3, so (9 times 3 = 27) children. This is (3^3).

This pattern shows that the number of children in a given generation can be expressed using indices, demonstrating exponential growth.

Teaching Methods/Instructional Techniques

Discussion, Explanation, Demonstration, Guided Practice, Question and Answer, Problem Solving

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Recalling/Engaging

Teacher’s Activity: The teacher greets the students and asks them to recall what they know about repeated multiplication, e.g., (2 times 2 times 2). The teacher then introduces the concept of indices as a shorter way to write such expressions.

Pupils’ Activity: Pupils respond to questions about repeated multiplication and listen to the introduction of indices.

Learning Point: Introduction to indices

Step 2: Meaning of Indices

Time: 10 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher explains the meaning of indices, defining the base and the exponent using the repeated multiplication chart. The teacher provides examples like (5^2) and (x^4).

Pupils’ Activity: Pupils listen, ask questions, and copy the definition and examples into their notebooks.

Learning Point: Understanding index notation

Step 3: Multiplication Law of Indices

Time: 10 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher introduces the multiplication law (a^x times a^y = a^{x+y}), explaining its derivation through examples like (2^2 times 2^3 = (2 times 2) times (2 times 2 times 2) = 2^5). The teacher then guides students through worked examples from the Board Summary.

Pupils’ Activity: Pupils observe, participate in the derivation, and solve practice problems with teacher guidance.

Learning Point: Applying multiplication law

Step 4: Division Law of Indices

Time: 10 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher explains the division law (a^x div a^y = a^{x-y}), demonstrating with examples like (3^5 div 3^2). The teacher highlights the condition that the base must be non-zero and guides students through more examples.

Pupils’ Activity: Pupils listen, ask clarifying questions, and solve division problems involving indices.

Learning Point: Applying division law

Step 5: Power Law of Indices

Time: 8 minutes

Teaching Skill: Demonstration/Application

Teacher’s Activity: The teacher introduces the power law ((a^x)^y = a^{xy}), using examples like ((2^3)^2 = 2^6). The teacher ensures students understand that the indices are multiplied in this case and guides them through examples.

Pupils’ Activity: Pupils observe the examples and practice applying the power law.

Learning Point: Applying power law

Step 6: Zero and Negative Index Laws & Practical Application

Time: 7 minutes

Teaching Skill: Explanation/Connection

Teacher’s Activity: The teacher introduces the zero index law (a^0 = 1) and the negative index law (a^{-x} = frac{1}{a^x}) with simple examples. The teacher then guides students to apply indices to the family generation problem, showing how powers represent generational growth.

Pupils’ Activity: Pupils learn the new laws, ask questions, and discuss the practical application of indices.

Learning Point: Zero and negative indices

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

1. What is an index?
2. State the multiplication law of indices.
3. Simplify (5^3 times 5^2).
4. Simplify (x^7 div x^4).
5. Simplify ((3^2)^3).
6. What is the value of (100^0)?
7. Express (4^{-2}) in positive index form.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Understanding indices laws

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on the meaning and laws of indices into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson content

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Summarising

Teacher’s Activity: The teacher summarises the key laws of indices discussed and reiterates their importance in simplifying mathematical expressions and solving real-world problems. The teacher encourages students to practice regularly.

Pupils’ Activity: Pupils listen attentively and ask any final questions.

Learning Point: Reinforcing indices laws

Continuous Assessment/Further Study

Type: Homework/Practice Exercise

Instruction: Solve the following problems using the appropriate laws of indices:

1. Simplify (7^6 times 7^{-2}).
2. Simplify (frac{a^8}{a^3}).
3. Simplify ((p^5)^4).
4. Evaluate ( (2x)^0 ).
5. Express (y^{-7}) with a positive index.
6. In a bacterial culture, the number of bacteria doubles every hour. If you start with 1 bacterium, how many bacteria will there be after 5 hours? Express your answer using indices.

Lesson Keywords

• Index – A number that shows how many times a base number is multiplied by itself.
• Base – The number that is multiplied by itself in an index form.
• Exponent – Another term for index or power.
• Power – The value of the index.
• Reciprocal – The multiplicative inverse of a number (e.g., the reciprocal of (a) is (frac{1}{a})).

Differentiation

Support: Provide simpler numerical examples and step-by-step guidance for students struggling with the concept. Use flashcards for index laws.

Extension: Challenge advanced students with problems involving multiple laws in one expression or introduce fractional indices for self-study.

Suggested Lesson Videos

Search YouTube for “laws of indices SS1 mathematics”