Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 1st Term
Week: 6
Age: 15 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: General Mathematics
Previous Lesson: Logarithm.
Topic: LOGARITHMS (I)
Subject Matter: Deducing logarithm from indices and standard form., Definition of logarithm., Graph of y=10^x using x=0.1, 0.2, … and reading values., Finding logarithm and antilogarithm of numbers greater than 1.

Specific Objectives

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

Cognitive Domain:

Deduce the relationship between indices and logarithms.
Define logarithm.
Find values of expressions like log_a N.
Determine the logarithm of numbers greater than 1.
Determine the antilogarithm of numbers greater than 1.

Affective Domain:

Appreciate the importance of logarithms in simplifying complex calculations.
Participate actively in classroom discussions and activities.

Psychomotor Domain:

Plot the graph of y=10^x using given values.
Read values accurately from the graph of y=10^x.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum (Mathematics)
State Unified Scheme of Work (Mathematics)
New General Mathematics for Senior Secondary Schools Book 1 by J.B. Channon, A. McLeish Smith, H.C. Head
Online resources for logarithm tables and graph examples: https://www.mathsisfun.com/algebra/logarithms.html

Instructional Materials

The teacher will teach this lesson with the aid of:

Indices/logarithms chart
Definition chart of logarithm
Graph board with a prepared graph of y=10^x
Graph books or graph papers
Logarithm tables

Rationale for the Lesson

This lesson helps pupils understand logarithms as an inverse operation to indices. Understanding logarithms enables pupils to simplify calculations involving large numbers and powers, which is useful in various scientific and engineering fields.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of indices, standard form of numbers, and basic algebraic operations.

Lesson Content/Board Summary

LOGARITHMS (I)

1. Relationship between Indices and Logarithms

Logarithm is the inverse operation of exponentiation (indices).

If an equation is expressed in index form as a^x = N, it can be written in logarithmic form as x = log_a N.

The following are key terms in this relationship:

Base (a): The number being raised to a power.
Exponent/Index (x): The power to which the base is raised.
Number (N): The result of the exponentiation.

Examples:

Since 2^3 = 8, then 3 = log_2 8.
Since 10^2 = 100, then 2 = log_10 100.
Since 5^2 = 25, then 2 = log_5 25.

2. Definition of Logarithm

The logarithm of a number N to a given base ‘a’ is the power to which the base must be raised to obtain the number N.

Mathematically, if a^x = N, then log_a N = x. For this definition, the base ‘a’ must be a positive number and not equal to 1 (a > 0, a ≠ 1), and the number N must be positive (N > 0).

3. Graph of y = 10^x

The graph of y = 10^x can be plotted by selecting various values for x (e.g., 0.1, 0.2, 0.3, 0.4, …) and calculating the corresponding values for y.

Steps to plot the graph:

Create a table of values for x and y=10^x.
Choose appropriate scales for the x and y axes.
Plot the calculated points on a graph paper.
Draw a smooth curve connecting the plotted points.

This graph can be used to find the logarithm (x-value) of a number (y-value) to base 10, and vice versa.

4. Finding Logarithm of Numbers Greater Than 1

To find the logarithm of a number greater than 1 (e.g., log_10 N), logarithm tables or scientific calculators are typically used.

A logarithm to base 10 consists of two parts:

Characteristic: The integral part of the logarithm. It is determined by the position of the decimal point in the original number. For numbers greater than 1, the characteristic is one less than the number of digits before the decimal point.
Mantissa: The decimal part of the logarithm. It is found from logarithm tables using the significant figures of the number. The mantissa is always positive.

Steps to find the logarithm of a number greater than 1:

Determine the characteristic: Count the number of digits before the decimal point and subtract 1.
Find the mantissa: Use the logarithm table to find the mantissa corresponding to the significant figures of the number.
Combine the characteristic and mantissa to write the full logarithm.

Example: To find log_10 345.6

Characteristic: There are 3 digits before the decimal point (3, 4, 5), so the characteristic is 3 – 1 = 2.
Mantissa: Look up 34 in the log table, then under 5, then difference under 6.

5. Finding Antilogarithm of Numbers Greater Than 1

Antilogarithm is the inverse process of finding a logarithm. If log_10 N = x, then N = antilog x (or N = 10^x).

Steps to find the antilogarithm of a number:

Use the decimal part (mantissa) of the given logarithm to find the corresponding number from the antilogarithm table.
Use the integral part (characteristic) of the given logarithm to place the decimal point in the number found from the table. If the characteristic is ‘n’, there will be ‘n+1’ digits before the decimal point in the antilogarithm.

Example: To find antilog 2.5386

Mantissa (.5386) is used to find the digits from the antilog table.
Characteristic (2) means there will be 2+1 = 3 digits before the decimal point in the final answer.

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction/Recalling Prior Knowledge
Teacher’s Activity: The teacher greets the pupils and reviews the previous lesson on indices, asking questions like, “What is an index?” and “How do we write 2 x 2 x 2 x 2 in index form?”. The teacher then introduces logarithms as the reverse of indices.
Pupils’ Activity: Pupils respond to questions about indices and listen attentively to the introduction of logarithms.
Learning Point: Pupils recall knowledge of indices, which forms a foundation for understanding logarithms.

Step 2: Deducing Logarithm from Indices

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher writes examples of index forms on the board (e.g., 2^3 = 8, 10^2 = 100) and guides pupils to express them in logarithmic form (e.g., log_2 8 = 3, log_10 100 = 2). The teacher explains the relationship between the base, index, and number in both forms.
Pupils’ Activity: Pupils observe the examples, ask questions, and practice converting simple index forms to logarithmic forms.
Learning Point: Pupils understand that logarithms are the inverse of indices.

Step 3: Definition of Logarithm

Time: 5 minutes
Teaching Skill: Definition/Explanation
Teacher’s Activity: The teacher formally defines logarithm using the definition chart, explaining the conditions for the base and the number (a > 0, a ≠ 1, N > 0). The teacher provides more examples and asks pupils to state the definition in their own words.
Pupils’ Activity: Pupils write down the definition and examples, and some state the definition orally.
Learning Point: Pupils learn the formal definition of logarithm and its conditions.

Step 4: Graph of y = 10^x

Time: 10 minutes
Teaching Skill: Demonstration/Guidance
Teacher’s Activity: The teacher demonstrates how to plot the graph of y = 10^x on the graph board, using specific values for x (e.g., 0.1, 0.2, 0.3) to find corresponding y values. The teacher shows how to read values from the graph to find logarithms or antilogarithms.
Pupils’ Activity: Pupils observe the plotting process, ask questions, and practice reading values from the graph.
Learning Point: Pupils learn how to plot and use the graph of y=10^x to find values.

Step 5: Finding Logarithm of Numbers Greater Than 1

Time: 8 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains the two parts of a logarithm: characteristic and mantissa. Using examples, the teacher demonstrates how to determine the characteristic for numbers greater than 1 and how to use logarithm tables to find the mantissa.
Pupils’ Activity: Pupils pay attention, copy examples, and practice finding characteristics and using logarithm tables under the teacher’s guidance.
Learning Point: Pupils learn to find the logarithm of numbers greater than 1 using characteristics and mantissa from tables.

Step 6: Finding Antilogarithm of Numbers Greater Than 1

Time: 5 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains that finding antilogarithm is the reverse of finding logarithm. The teacher demonstrates how to use the antilogarithm table with the mantissa and how to use the characteristic to place the decimal point correctly.
Pupils’ Activity: Pupils observe the examples and attempt to find antilogarithms from the table.
Learning Point: Pupils learn to find the antilogarithm of numbers greater than 1.

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 relationship between indices and logarithms using an example.
Define logarithm.
Convert 3^4 = 81 to logarithmic form.
Mention the two parts of a common logarithm.
Explain how to determine the characteristic of a number greater than 1.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 3 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definition of logarithm, its relationship with indices, and the process of finding logarithms and antilogarithms. The teacher assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils consolidate their understanding of the topic and get practice for reinforcement.

Lesson Keywords

Logarithm – The power to which a base must be raised to produce a given number.
Index – The power or exponent to which a number is raised.
Base – The number that is raised to a power in an exponential expression, or the number to which a logarithm is taken.
Characteristic – The integral part of a common logarithm.
Mantissa – The decimal part of a common logarithm, always positive.
Antilogarithm – The number whose logarithm is a given number.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex examples or challenge problems involving logarithm properties. For pupils needing extra support, the teacher will offer one-on-one guidance, provide simpler examples, or encourage peer tutoring. Group work can be used to foster collaborative learning and allow pupils to explain concepts to each other.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in indices before introducing logarithms. Emphasize the inverse relationship between the two. Provide ample practice with logarithm tables, explaining clearly how to read both the logarithm and antilogarithm tables. Encourage pupils to use graph paper for plotting the graph of y=10^x to ensure accuracy.