Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: 1st Term
Week: 3
Age: 15 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Previous Lesson: Logarithms.
Topic: MODULAR ARITHMETIC
Subject Matter: Revision of addition, division, multiplication and subtraction of integers; Concept of modular arithmetic; Addition, subtraction and multiplication operations in modular arithmetic; Application to real life situations.

Specific Objectives

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

Cognitive Domain:

Define modular arithmetic.
Perform addition, subtraction, and multiplication operations in modular arithmetic.

Affective Domain:

Appreciate the concept and usefulness of modular arithmetic.
Value the application of modular arithmetic in daily life situations.

Psychomotor Domain:

Solve problems involving addition, subtraction, and multiplication in modular arithmetic.

Apply modular arithmetic concepts to solve real-life problems such as shift duties and market days.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Mathematics (Senior Secondary Education).
State Unified Scheme of Work for Senior Secondary School 1 Mathematics.
New General Mathematics for Senior Secondary Schools 1.

Instructional Materials

The teacher will teach this lesson with the aid of:

Modular arithmetic charts.
Samples of shift duty charts.
A menstrual chart.
Clock face diagrams.

Rationale for the Lesson

This lesson helps pupils understand a different way of doing arithmetic, which is important for solving problems involving cycles, like time, days of the week, or recurring events. It enables pupils to see how mathematical concepts are relevant to many aspects of their daily lives.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic mathematical operations (addition, subtraction, multiplication, division) involving integers.

Lesson Content/Board Summary

MODULAR ARITHMETIC

Revision of Basic Operations on Integers

Integers are whole numbers, including negative numbers, positive numbers, and zero. Basic operations include addition, subtraction, multiplication, and division.

The following are examples of operations on integers:

Addition: -5 + 3 = -2
Subtraction: 7 – (-2) = 9
Multiplication: -4 × 6 = -24
Division: 10 ÷ -2 = -5

Concept of Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers “wrap around” after reaching a certain value—the modulus. It is sometimes called “clock arithmetic” because it works like the 12-hour clock, where numbers cycle after 12.

If two integers ‘a’ and ‘b’ have the same remainder when divided by a positive integer ‘n’, we say that ‘a’ is congruent to ‘b’ modulo ‘n’. This is written as a ≡ b (mod n).

For example, 17 ≡ 5 (mod 12) because 17 divided by 12 gives a remainder of 5.

Addition in Modular Arithmetic

To add two numbers in modular arithmetic, first add them as usual, then find the remainder when the sum is divided by the modulus.

Example: 7 + 8 (mod 12)

7 + 8 = 15

15 ÷ 12 = 1 remainder 3

So, 7 + 8 ≡ 3 (mod 12)

Subtraction in Modular Arithmetic

To subtract two numbers in modular arithmetic, subtract them as usual, then find the remainder when the result is divided by the modulus. If the result is negative, add multiples of the modulus until a positive remainder is obtained.

Example: 5 – 9 (mod 12)

5 – 9 = -4

-4 + 12 = 8

So, 5 – 9 ≡ 8 (mod 12)

Multiplication in Modular Arithmetic

To multiply two numbers in modular arithmetic, multiply them as usual, then find the remainder when the product is divided by the modulus.

Example: 4 × 5 (mod 7)

4 × 5 = 20

20 ÷ 7 = 2 remainder 6

So, 4 × 5 ≡ 6 (mod 7)

Applications of Modular Arithmetic

Modular arithmetic has several real-life applications:

Time: Clock arithmetic (e.g., 5 hours after 10 o’clock is 3 o’clock, which is 10 + 5 = 15, 15 ≡ 3 (mod 12)).
Days of the Week: Calculating future or past days.
Shift Duties: Scheduling rotating work shifts.
Menstrual Charts: Predicting cycles.
Computer Science: Cryptography and error detection codes.

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 basic operations on integers. The teacher then uses a clock to illustrate how time cycles, asking “If it’s 10 o’clock now, what time will it be in 5 hours?” leading to the idea of numbers wrapping around.
Pupils’ Activity: Pupils respond to the questions and observe the clock demonstration.
Learning Point: Pupils are reminded of integer operations and introduced to the concept of cyclic numbers.

Step 2: Revision of Basic Operations on Integers

Time: 8 minutes
Teaching Skill: Review/Recall
Teacher’s Activity: The teacher writes various examples of addition, subtraction, and multiplication of integers on the board and guides pupils to solve them, reinforcing their prior knowledge.
Pupils’ Activity: Pupils solve the problems on the board and in their notebooks, recalling how to perform operations on integers.
Learning Point: Pupils refresh their understanding of integer operations.

Step 3: Concept of Modular Arithmetic

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher formally introduces modular arithmetic, defining it with examples like the clock. The teacher explains the notation ‘a ≡ b (mod n)’ and provides simple examples.
Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and write down the definition and examples.
Learning Point: Pupils understand the definition and basic concept of modular arithmetic.

Step 4: Addition in Modular Arithmetic

Time: 5 minutes
Teaching Skill: Demonstration/Practice
Teacher’s Activity: The teacher demonstrates how to perform addition in modular arithmetic using examples, such as 7 + 8 (mod 12), and guides pupils through similar problems.
Pupils’ Activity: Pupils observe the demonstration and solve practice problems on addition in modular arithmetic.
Learning Point: Pupils learn to perform addition in modular arithmetic.

Step 5: Subtraction in Modular Arithmetic

Time: 5 minutes
Teaching Skill: Demonstration/Practice
Teacher’s Activity: The teacher explains and demonstrates subtraction in modular arithmetic, including cases where the result is negative, showing how to adjust it to a positive remainder within the modulus. Example: 5 – 9 (mod 12).
Pupils’ Activity: Pupils follow the teacher’s examples and practice subtracting numbers in modular arithmetic.
Learning Point: Pupils learn to perform subtraction in modular arithmetic.

Step 6: Multiplication in Modular Arithmetic

Time: 5 minutes
Teaching Skill: Demonstration/Practice
Teacher’s Activity: The teacher demonstrates multiplication in modular arithmetic, such as 4 × 5 (mod 7), and provides practice exercises for the pupils.
Pupils’ Activity: Pupils watch the examples and practice multiplying numbers in modular arithmetic.
Learning Point: Pupils learn to perform multiplication in modular arithmetic.

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 modular arithmetic.
Evaluate 10 + 7 (mod 12).
Calculate 3 – 8 (mod 10).
Find the value of 5 × 6 (mod 8).

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/Consolidation
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the definition of modular arithmetic and its operations. The teacher assigns homework: “List two more real-life applications of modular arithmetic besides those discussed in class.”
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their understanding and are encouraged to think further about the topic.

Lesson Keywords

Modular Arithmetic – A system of arithmetic for integers, where numbers “wrap around” after reaching a certain value (the modulus).
Modulus – The number by which other numbers are divided, and whose remainder is the result in modular arithmetic.
Congruent – Two integers are congruent modulo ‘n’ if they have the same remainder when divided by ‘n’.
Remainder – The amount left over after a division.

Differentiation

For pupils who grasp the concepts quickly, the teacher will provide more complex problems involving multiple operations or larger moduli. For pupils needing more support, the teacher will offer one-on-one guidance, simplify examples, and provide additional step-by-step practice problems with smaller numbers and visual aids like number lines or clock faces.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of integer operations before introducing modular arithmetic. Use practical examples like clocks and days of the week to make the concept relatable. Encourage active participation and group work for problem-solving. Pay close attention to negative results in subtraction and guide pupils on how to convert them into positive remainders within the given modulus.