Further Mathematics - Senior Secondary 1 - Binary operations II

Binary operations II

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TERM: 3RD TERM

WEEK 9
Class: Senior Secondary School 1
Age: 15 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Binary Operations II
Focus: Laws of Complementation, Identity Elements, Inverse of an Element
SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Understand the laws of complementation as in sets.
  2. Recognize and apply the identity elements in binary operations.
  3. Understand the inverse of an element in binary operations.
  4. Solve problems using the laws of binary operations.

INSTRUCTIONAL TECHNIQUES:
• Explanation with examples
• Guided demonstration
• Class discussion
• Problem-solving exercises
• Visual aids (charts displaying laws of binary operations)

INSTRUCTIONAL MATERIALS:
• Whiteboard and markers
• Charts illustrating the laws of binary operations
• Worksheets for practice
• Flashcards with binary operations examples

PERIOD 1 & 2: Introduction to Binary Operations Laws
PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of binary operations and briefly explains how they relate to sets. Defines the complement, identity, and inverse elements in the context of binary operations.

Students listen attentively and ask clarifying questions.

Step 2 - Law of Complementation

Explains the law of complementation using set theory analogies. For example, in binary operations, the complement of an element is its "opposite." Demonstrates with examples, e.g., 1' = 0 and 0' = 1.

Students observe and take notes on the law of complementation.

Step 3 - Identity Element

Explains the concept of an identity element (1 for AND, 0 for OR) in binary operations and provides examples. For example, 1 AND x = x and 0 OR x = x.

Students follow along and work through examples.

Step 4 - Inverse of an Element

Introduces the inverse of an element, where x AND x' = 0 or x OR x' = 1. Demonstrates with examples, e.g., 1 AND 0 = 0 or 1 OR 0 = 1.

Students participate in examples and practice with the teacher.

NOTE ON BOARD:

    
  • Complement Law: x' = opposite of x
  • Identity Element: x AND 1 = x; x OR 0 = x
  • Inverse Element: x AND x' = 0; x OR x' = 1

EVALUATION (5 exercises):

  1. What is the complement of 0 in binary operations?
  2. What is the identity element for OR in binary operations?
  3. What is the inverse of 1 in binary operations?
  4. Give an example where 1 AND x = x.
  5. Solve: What is the result of x AND x' for any value of x?

CLASSWORK (5 questions):

  1. What is the complement of 1?
  2. What is the identity element for AND?
  3. What is the inverse of 0?
  4. Simplify: 1 OR 0.
  5. Simplify: 0 AND 1.

ASSIGNMENT (5 tasks):

  1. List all identity elements for binary operations.
  2. Write out the complement of 1 and 0.
  3. Solve: What is the result of 0 OR 0?
  4. Solve: What is the result of 1 AND 1?
  5. Explain the concept of an identity element with a real-life example.

PERIOD 3 & 4: Problem-Solving and Practice
PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Review and Example Solving

Reviews the laws of complementation, identity elements, and inverse elements with examples on the board. Provides further problems to practice in class.

Students ask questions for clarification and solve problems independently.

Step 2 - Guided Practice

Assigns binary operations problems where students apply the laws of complementation, identity elements, and inverses to solve. Encourages collaboration in pairs.

Students solve problems in pairs and ask for teacher guidance when needed.

Step 3 - Independent Practice

Provides a series of problems for independent practice. Encourages students to apply the laws correctly in different scenarios.

Students complete the problems independently and submit their answers for review.

NOTE ON BOARD:

    
  • Complementation and Identity Law Problems
  • Inverse Element Application
  • Example problems to practice with step-by-step solutions

EVALUATION (5 exercises):

  1. Solve: 1 AND 0 = ?
  2. Solve: 0 OR 1 = ?
  3. Solve: 1 OR 1 = ?
  4. Solve: x AND x' = ?
  5. Solve: x OR x' = ?

CLASSWORK (5 questions):

  1. What is the complement of 1?
  2. What is the identity element for AND?
  3. What is the inverse of 0 in binary operations?
  4. Solve: 1 AND 0.
  5. Solve: 0 OR 1.

ASSIGNMENT (5 tasks):

  1. Explain the inverse law in binary operations with an example.
  2. Solve: 1 AND 1.
  3. Solve: 0 OR 0.
  4. Find the complement of 1 in binary.

Create a real-life scenario where binary operations laws could apply.