Mathematics - Senior Secondary 1 - Modular arithmetic

Modular arithmetic

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TERM: 1ST TERM

WEEK: 3

Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Modular Arithmetic
Focus: Revision of integer operations (addition, division, multiplication, subtraction), introduction to modular arithmetic, and operations in modular arithmetic.

SPECIFIC OBJECTIVES:

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

  1. Revise the addition, subtraction, multiplication, and division of integers.
  2. Define modular arithmetic and explain its significance.
  3. Perform addition, subtraction, multiplication, and division in modular arithmetic.
  4. Apply modular arithmetic to real-life situations (e.g., shift duties, menstrual charts, market days).

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Practice exercises
  • Discussions
  • Real-life connections
  • Use of charts and visual aids

 

INSTRUCTIONAL MATERIALS:

  • Modular arithmetic charts
  • Samples of shift duty charts
  • Menstrual cycle charts
  • Whiteboard and markers

 

PERIOD 1 & 2: Introduction to Modular Arithmetic

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Revision of Integer Operations

The teacher revises basic operations on integers: addition, subtraction, multiplication, and division. Demonstrates each with examples on the board.

Students answer questions about each operation and practice solving problems on the board.

Step 2: Defining Modular Arithmetic

Teacher explains the concept of modular arithmetic using the notation "a ≡ b (mod m)" to mean that "a and b leave the same remainder when divided by m." Example: 8 ≡ 2 (mod 3).

Students listen and take notes on the definition and examples provided.

Step 3: Connection to Real Life

Teacher connects modular arithmetic to real-life applications, like calculating shift duties or tracking menstrual cycles (e.g., using mod 7 to represent a weekly cycle).

Students engage in a discussion on how modular arithmetic is used in their everyday lives.

NOTE ON BOARD:
Modular Arithmetic:

  • a ≡ b (mod m): "a and b give the same remainder when divided by m."
  • Example: 8 ≡ 2 (mod 3), because when 8 is divided by 3, the remainder is 2.

EVALUATION (5 exercises):

  1. What does a ≡ b (mod m) mean?
  2. If 10 ≡ 4 (mod 6), what is the remainder when 10 is divided by 6?
  3. Provide an example where modular arithmetic can be used in daily life.
  4. What is the modulus in the equation 12 ≡ 5 (mod 7)?
  5. True or False: 14 ≡ 3 (mod 5). Justify your answer.

CLASSWORK (5 questions):

  1. 15 ≡ ? (mod 6)
  2. 19 ≡ ? (mod 4)
  3. 22 ≡ ? (mod 7)
  4. Calculate 17 ≡ ? (mod 3).
  5. Is 18 ≡ 0 (mod 9)? Explain why or why not.

ASSIGNMENT (5 tasks):

  1. What is modular arithmetic used for in tracking time?
  2. Convert the following: 21 ≡ ? (mod 5).
  3. Research how modular arithmetic is used in encryption.
  4. What is the equivalent of 35 mod 8?
  5. Discuss the use of modular arithmetic in creating shift duty schedules.

 

PERIOD 3 & 4: Operations in Modular Arithmetic (Addition, Subtraction, Multiplication, Division)

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Modular Addition

Teacher demonstrates how to add two numbers in modular arithmetic. Example: 8 + 7 ≡ 2 (mod 5). Students then work through similar problems.

Students follow along and solve similar modular addition problems.

Step 2: Modular Subtraction

Teacher demonstrates how to subtract two numbers in modular arithmetic. Example: 9 - 5 ≡ 4 (mod 6). Students then practice modular subtraction problems.

Students solve modular subtraction problems individually.

Step 3: Modular Multiplication

Teacher explains modular multiplication using examples like 4 × 3 ≡ 12 (mod 5), and how to reduce the result to a smaller number within the modulus.

Students complete modular multiplication exercises in pairs.

Step 4: Modular Division

Teacher introduces division in modular arithmetic using multiplicative inverses. Example: Solve 6 ÷ 2 (mod 5). Explains how to find the inverse and perform the division.

Students practice modular division with teacher support.

NOTE ON BOARD:
Addition in Modular Arithmetic:

  • (a + b) mod m = [(a mod m) + (b mod m)] mod m
  • Example: (8 + 7) mod 5 = (15 mod 5) = 0

Subtraction in Modular Arithmetic:

  • (a - b) mod m = [(a mod m) - (b mod m)] mod m
  • Example: (9 - 5) mod 6 = (4 mod 6) = 4

Multiplication in Modular Arithmetic:

  • (a × b) mod m = [(a mod m) × (b mod m)] mod m
  • Example: (4 × 3) mod 5 = (12 mod 5) = 2

Division in Modular Arithmetic:

  • To divide by a number, find the multiplicative inverse of the divisor.

EVALUATION (5 exercises):

  1. Compute (7 + 4) mod 5.
  2. Compute (15 - 9) mod 6.
  3. Compute (3 × 5) mod 4.
  4. Compute 6 ÷ 2 (mod 5).
  5. Compute (20 ÷ 4) mod 7.

CLASSWORK (5 questions):

  1. (8 + 6) mod 5 = ?
  2. (12 - 7) mod 5 = ?
  3. (5 × 3) mod 7 = ?
  4. (18 ÷ 3) mod 4 = ?
  5. (21 + 15) mod 9 = ?

ASSIGNMENT (5 tasks):

  1. Solve (14 + 9) mod 6.
  2. Solve (25 - 14) mod 7.
  3. Solve (6 × 4) mod 8.
  4. Solve 18 ÷ 3 (mod 5).
  5. Research how modular arithmetic is used in creating secure passwords.

 

PERIOD 5: Application of Modular Arithmetic in Real-Life Situations

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1: Shift Duty Example

Teacher explains how modular arithmetic helps to plan shift duties, e.g., with a 7-day workweek (mod 7), the days of the week cycle. Demonstrates with a sample shift chart.

Students practice using modular arithmetic to create their own shift duty schedules.

Step 2: Menstrual Cycle Example

Teacher explains how the menstrual cycle follows a similar modular pattern (e.g., mod 28 for a 28-day cycle). Students analyze and practice with charts.

Students use modular arithmetic to track and predict dates on a menstrual cycle chart.

Step 3: Market Days Example

Teacher demonstrates the use of modular arithmetic to calculate market days (e.g., mod 4 for a 4-day market rotation).

Students practice with real-life examples of market day rotations.

NOTE ON BOARD:
Applications of Modular Arithmetic:

  • Shift duty charts: Mod 7
  • Menstrual cycles: Mod 28
  • Market day schedules: Mod 4

EVALUATION (5 exercises):

  1. How is modular arithmetic used in shift scheduling?
  2. How can modular arithmetic predict menstrual cycle days?
  3. Give an example of how modular arithmetic can be applied to market day schedules.
  4. True or False: Modular arithmetic is only used in mathematics, not in real life.
  5. Explain how modular arithmetic helps to avoid overlapping shifts.

CLASSWORK (5 questions):

  1. Apply modular arithmetic to create a 5-day shift duty schedule.
  2. Predict the 5th day of the menstrual cycle starting on Day 3.
  3. Calculate the 3rd market day in a 6-day cycle, starting from Day 1.
  4. Use modular arithmetic to solve a shift duty schedule problem.
  5. Create a calendar for 4 market days using mod 4.

ASSIGNMENT (5 tasks):

  1. Apply modular arithmetic to design a 7-day rotating shift schedule for workers.
  2. Calculate the 10th day of a menstrual cycle starting from Day 5 (mod 28).
  3. Create a 4-day market day schedule using mod 4.
  4. Use modular arithmetic to solve a real-life example from work.

Discuss why modular arithmetic is important in day-to-day scheduling tasks.