MATHEMATICS LESSON NOTE

CLASS: SS 1
TERM: First Term
WEEK: 4
TOPIC: Concept of Modular Arithmetic and Cyclic Events

Instructional Materials

• Wall charts
• Online resources
• Pictures
• Related audio-visual materials
• Mathematics textbooks
• A chart showing modular arithmetic
• Samples of duty shifts
• Menstrual chart

Reference Materials

• Scheme of work
• Online information
• Textbooks
• Workbooks
• Education curriculum

Previous Knowledge

Students have previous knowledge of:

• Conversion from Base Ten to other bases
• Conversion from one base to another

Behavioral Objectives

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

1. Define modular arithmetic.
2. Perform basic operations (addition, subtraction, multiplication, and division) in modular arithmetic.
3. Apply modular arithmetic to real-life situations.

CONTENT

SUB-TOPIC 1: Definition of Modular Arithmetic

Modular arithmetic is a system of arithmetic where numbers “wrap around” after reaching a certain value called the modulus. Unlike standard arithmetic, which follows an infinite sequence, modular arithmetic cycles through a set of numbers repeatedly.

For example, on a clock, after 12, the numbers reset to 1 instead of continuing to 13, 14, and so on. This is an example of modular arithmetic with a modulus of 12.

Interactive Questions and Answers

1. What is a modulus?

• The modulus is the value at which numbers “wrap around” in modular arithmetic.

2. What is standard arithmetic?

• Standard arithmetic follows a fixed number sequence without wrapping around.

3. How do numbers “wrap around” in modular arithmetic?

• Numbers reset to zero or another starting point once they reach the modulus.

4. What is the difference between modular arithmetic and standard arithmetic?

• Modular arithmetic repeats after reaching a modulus, while standard arithmetic continues infinitely.

5. What happens if two numbers in modular arithmetic are not relatively prime to the modulus?

• Their results may collide and get reduced modulo the modulus.

SUB-TOPIC 2: Operations in Modular Arithmetic

Addition in Modular Arithmetic

Addition in modular arithmetic follows these steps:

1. Add the given numbers.
2. Divide the sum by the modulus.
3. The remainder is the final answer.

Examples:

1. $3+7$ (mod 10) = 10. Since 10 mod 10 = 0, the answer is 0.
2. $5+3$ (mod 7) = 8. Since 8 mod 7 = 1, the answer is 1.
3. $2+1$ (mod 3) = 3. Since 3 mod 3 = 0, the answer is 0.
4. $4+3$ (mod 5) = 7. Since 7 mod 5 = 2, the answer is 2.
5. $7+5$ (mod 9) = 12. Since 12 mod 9 = 3, the answer is 3.

Subtraction in Modular Arithmetic

Subtraction in modular arithmetic follows similar steps:

1. Subtract the numbers.
2. If the result is negative, add the modulus.
3. Find the remainder when divided by the modulus.

Examples:

1. $7-3$ (mod 10) = 4. Since 4 mod 10 = 4, the answer is 4.
2. $5-3$ (mod 7) = 2. Since 2 mod 7 = 2, the answer is 2.
3. $2-1$ (mod 3) = 1. Since 1 mod 3 = 1, the answer is 1.
4. $4-3$ (mod 5) = 1. Since 1 mod 5 = 1, the answer is 1.
5. $7-5$ (mod 9) = 2. Since 2 mod 9 = 2, the answer is 2.

Multiplication in Modular Arithmetic

For multiplication in modular arithmetic:

1. Multiply the numbers.
2. Find the remainder when divided by the modulus.

Examples:

1. $3\times 7$ (mod 10) = 21. Since 21 mod 10 = 1, the answer is 1.
2. $5\times 3$ (mod 7) = 15. Since 15 mod 7 = 1, the answer is 1.
3. $2\times 1$ (mod 3) = 2. Since 2 mod 3 = 2, the answer is 2.
4. $4\times 3$ (mod 5) = 12. Since 12 mod 5 = 2, the answer is 2.
5. $7\times 5$ (mod 9) = 35. Since 35 mod 9 = 8, the answer is 8.

Division in Modular Arithmetic

Division in modular arithmetic requires finding the modular inverse. The modular inverse of a number $a$ modulo $m$ is the number $b$ such that:
$a\times b\equiv 1(modm)$.

Examples:

1. $7÷3$ (mod 10) = 1
2. $5÷3$ (mod 7) = 2
3. $2÷1$ (mod 3) = 2
4. $4÷3$ (mod 5) = 1
5. $7÷5$ (mod 9) = 4

SUB-TOPIC 3: Application of Modular Arithmetic in Daily Life

Modular arithmetic is widely used in:

• Timekeeping: The 12-hour and 24-hour clocks operate on mod 12 and mod 24, respectively.
• Cryptography: Used in encryption and security algorithms.
• Banking Systems: Used for generating and verifying account numbers.
• Scheduling: Helps in rotating work shifts and class timetables.
• Menstrual Cycle Calculations: Used in predicting periods based on a cycle length.

Evaluation Questions

1. Reduce 72 to its simplest form in:
   a. Modulo 3
   b. Modulo 4
   c. Modulo 5
   d. Modulo 6
   e. Modulo 7
2. Find the following sums in modulo 5:
   a. 3 + 9
   b. 65 + 32
   c. 41 + 52
   d. 8 + 17
3. Solve the following in modulo 4:
   a. $2\times 2$
   b. $5\times 7$
   c. $6\times 73$

Conclusion

The teacher will:

• Mark students’ work and provide corrections.
• Summarize the lesson.
• Explain the importance of modular arithmetic in daily life.