Number Bases, Review and Base Conversion for SS 3

Class: SS 3
Term: First Term
Week: 11
Age: 14 years
Duration: 45 minutes
Subject: Computer Science
Topic: OVERVIEW OF NUMBER BASES
Subject Matter: Review of number bases: decimal, hexadecimal: Conversion in number bases
Previous Lesson: BASIC Programming and Rectangle Calculations

Specific Objectives

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

Cognitive Domain

• Define number bases.
• List the digits used in binary, octal, decimal, and hexadecimal number systems.
• Explain the process of converting numbers from base 10 to other bases.
• Explain the process of converting numbers from other bases to base 10.
• Explain the process of converting between binary and hexadecimal bases.

Psychomotor Domain

• Convert given decimal numbers to binary and hexadecimal.
• Convert given binary and hexadecimal numbers to decimal.
• Convert given binary numbers to hexadecimal and vice versa.

Affective Domain

• Appreciate the relevance of different number bases in computer operations.
• Demonstrate accuracy and carefulness when performing number base conversions.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• A suitable Computer Science textbook for SS 3
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard and markers
• Projector (if available)
• Charts showing different number bases and their digits
• Charts illustrating number base conversion steps
• Calculators (for verification, not primary calculation)

Rationale for the Lesson

This lesson is important as number bases are fundamental to computer science and digital systems. Understanding how computers represent and process data requires knowledge of different number bases and their conversions. This skill is essential for students pursuing further studies or careers in technology, enabling them to comprehend low-level computer operations.

Prerequisite/Previous Knowledge

Students should have a basic understanding of counting, place values in the decimal system, and an introductory knowledge of binary numbers from previous classes.

Lesson Content/Board Summary

OVERVIEW OF NUMBER BASES

Introduction to Number Bases

A number base, also known as a radix, defines the number of unique digits (including zero) used to represent numbers in a positional numeral system. The most common number base is base 10 (decimal), which uses ten digits (0-9). Computers, however, primarily use base 2 (binary).

Common Number Bases and Their Digits

Different number bases use a specific set of digits. Each digit’s position determines its value, which is a power of the base.

1. Binary (Base 2): Uses two digits: 0, 1.
2. Octal (Base 8): Uses eight digits: 0, 1, 2, 3, 4, 5, 6, 7.
3. Decimal (Base 10): Uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
4. Hexadecimal (Base 16): Uses sixteen digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Here, A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15.

Conversion of Number Bases

Conversion from Base 10 (Decimal) to Other Bases

To convert a decimal number to another base (e.g., binary, octal, hexadecimal), the division method is used. The decimal number is repeatedly divided by the target base, and the remainders are collected in reverse order.

Decimal to Binary Conversion

Divide the decimal number by 2 repeatedly and record the remainders. The binary equivalent is formed by reading the remainders from bottom to top.

Example 1

Question: Convert (8_{10}) to binary.

Solution:

Step 1: Divide the number by the target base (2).

(8 div 2 = 4) remainder (0)

(4 div 2 = 2) remainder (0)

(2 div 2 = 1) remainder (0)

(1 div 2 = 0) remainder (1)

Step 2: Collect the remainders from bottom to top.

Answer: (1000_2)

Example 2

Question: Convert (25_{10}) to hexadecimal.

Solution:

Step 1: Divide the number by the target base (16).

(25 div 16 = 1) remainder (9)

(1 div 16 = 0) remainder (1)

Step 2: Collect the remainders from bottom to top.

Answer: (19_{16})

Conversion from Other Bases to Base 10 (Decimal)

To convert a number from any base to decimal, multiply each digit by its positional value (the base raised to the power of its position) and then sum the results. Positions are counted from right to left, starting from 0.

Binary to Decimal Conversion

Multiply each binary digit by (2^n), where (n) is the digit’s position (starting from 0 on the right), and sum the products.

Example 1

Question: Convert (1000_2) to decimal.

Solution:

Step 1: Assign positional values to each digit.

(1 times 2^3 + 0 times 2^2 + 0 times 2^1 + 0 times 2^0)

Step 2: Calculate the powers and multiply.

(1 times 8 + 0 times 4 + 0 times 2 + 0 times 1)

(8 + 0 + 0 + 0)

Step 3: Sum the results.

(8)

Answer: (8_{10})

Hexadecimal to Decimal Conversion

Multiply each hexadecimal digit (converting A-F to their decimal equivalents) by (16^n), where (n) is the digit’s position, and sum the products.

Example 2

Question: Convert (19_{16}) to decimal.

Solution:

Step 1: Assign positional values to each digit.

(1 times 16^1 + 9 times 16^0)

Step 2: Calculate the powers and multiply.

(1 times 16 + 9 times 1)

(16 + 9)

Step 3: Sum the results.

(25)

Answer: (25_{10})

Conversion Between Non-Decimal Bases (Binary and Hexadecimal)

Conversion between binary and hexadecimal is straightforward because 16 is a power of 2 ((2^4 = 16)). This means four binary digits can represent one hexadecimal digit.

Binary to Hexadecimal Conversion

Group the binary digits into sets of four, starting from the right. If the leftmost group has fewer than four digits, pad it with leading zeros. Then, convert each group of four binary digits into its corresponding hexadecimal digit.

Example 1

Question: Convert (1101011_2) to hexadecimal.

Solution:

Step 1: Group binary digits in fours from right to left, padding with zeros if necessary.

(0110 1011_2)

Step 2: Convert each group to its hexadecimal equivalent.

(0110_2 = 6_{16})

(1011_2 = B_{16}) (since (1011_2 = 8+2+1 = 11_{10}), which is B in hexadecimal)

Answer: (6B_{16})

Hexadecimal to Binary Conversion

Convert each hexadecimal digit into its four-bit binary equivalent. Combine the binary groups to form the complete binary number.

Example 2

Question: Convert (6B_{16}) to binary.

Solution:

Step 1: Convert each hexadecimal digit to its 4-bit binary equivalent.

(6_{16} = 0110_2)

(B_{16} = 1011_2)

Step 2: Combine the binary groups.

(01101011_2)

Answer: (1101011_2) (leading zero can be omitted)

Teaching Methods/Instructional Techniques

Discussion, Explanation, Demonstration, Guided Practice, Question and Answer, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Activating Prior Knowledge

Teacher’s Activity: The teacher greets the students and asks them to recall what they know about numbers and counting. The teacher then introduces the topic of number bases by asking how numbers are represented in computers.

Pupils’ Activity: Students respond to the questions and share their previous knowledge about numbers and computer representation.

Learning Point: Introduction to number bases

Step 2: Review of Number Bases and Their Digits

Time: 7 minutes

Teaching Skill: Explanation/Listing

Teacher’s Activity: The teacher explains what a number base is and reviews common number bases: binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16). The teacher lists the digits used in each base, emphasizing that hexadecimal uses letters A-F for values 10-15.

Pupils’ Activity: Students listen attentively, ask questions, and note down the different number bases and their digits.

Learning Point: Digits of number bases

Step 3: Conversion from Base 10 (Decimal) to Other Bases

Time: 8 minutes

Teaching Skill: Demonstration/Explanation

Teacher’s Activity: The teacher explains the division method for converting decimal numbers to other bases. Using the whiteboard, the teacher demonstrates the conversion of (8_{10}) to binary, following the example from the curriculum.

Pupils’ Activity: Students observe the demonstration, follow the steps, and ask clarifying questions.

Learning Point: Decimal to binary conversion

Step 4: Guided Practice (Decimal to Other Bases)

Time: 7 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher provides another decimal number, e.g., (25_{10}), and guides students to convert it to hexadecimal on their own or in pairs, providing support as needed.

Pupils’ Activity: Students attempt to convert (25_{10}) to hexadecimal, working individually or with partners, and seek assistance from the teacher.

Learning Point: Decimal to hexadecimal conversion

Step 5: Conversion from Other Bases to Base 10 (Decimal)

Time: 7 minutes

Teaching Skill: Demonstration/Explanation

Teacher’s Activity: The teacher explains the multiplication method for converting numbers from other bases to decimal. The teacher demonstrates converting (1000_2) to decimal and (19_{16}) to decimal, showing the positional value calculation.

Pupils’ Activity: Students pay attention to the new conversion method and examples, noting the differences from the previous method.

Learning Point: Other bases to decimal conversion

Step 6: Conversion Between Non-Decimal Bases (Binary to Hexadecimal)

Time: 6 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains how to convert between binary and hexadecimal using the grouping of four bits. The teacher demonstrates converting (1101011_2) to hexadecimal and (6B_{16}) to binary.

Pupils’ Activity: Students observe the direct conversion method between binary and hexadecimal, understanding the 4-bit grouping concept.

Learning Point: Binary and hexadecimal conversion

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. What are the digits used in the hexadecimal number system?
2. Explain the steps to convert a decimal number to binary.
3. Convert (12_{10}) to binary.
4. Convert (1011_2) to decimal.
5. Convert (1A_{16}) to decimal.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Number base conversion assessment

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on number bases and conversion methods into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Number base concepts recorded

Step 9: Conclusion

Time: 1 minute

Teaching Skill: Summarizing

Teacher’s Activity: The teacher briefly summarizes the importance of number bases in computing and reiterates the main conversion methods discussed. The teacher encourages students to practice more examples.

Pupils’ Activity: Students listen to the summary and prepare for the next lesson.

Learning Point: Lesson recap

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following conversion problems in your notebooks:

1. Convert (27_{10}) to binary.
2. Convert (45_{10}) to hexadecimal.
3. Convert (1101_2) to decimal.
4. Convert (2F_{16}) to decimal.
5. Convert (111010_2) to hexadecimal.

Lesson Keywords

• Binary – Base 2 number system using digits 0 and 1.
• Octal – Base 8 number system using digits 0-7.
• Decimal – Base 10 number system using digits 0-9.
• Hexadecimal – Base 16 number system using digits 0-9 and A-F.
• Base – The number of unique digits in a number system.
• Conversion – The process of changing a number from one base to another.
• Digits – The symbols used to represent numbers in a number system.

Differentiation

For weaker learners, provide simpler conversion exercises with smaller numbers and offer one-on-one guidance. Encourage peer tutoring. For faster learners, provide more complex conversion problems, including those involving octal, or ask them to research the applications of different number bases in computer memory addressing or colour codes.

Suggested Lesson Videos

For further understanding, search on YouTube for:

• Number Bases Conversion Computer Science SS3
• Decimal Binary Hexadecimal Conversion Explained

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have a clear understanding of all conversion methods and can demonstrate them accurately. Prepare charts or slides for visual presentation of number bases and their digits. During the lesson, follow the sequence of introducing each base, explaining its digits, and then demonstrating conversion methods step-by-step. Pay close attention to common errors students make, such as incorrect division, misplacing remainders, or errors in positional values. Allow ample time for guided practice and individual attempts. Students should copy the Board Summary notes in Step 8 to reinforce their learning. Regularly check for understanding through questions and quick exercises. Provide additional support to students struggling with the concepts and challenge advanced learners with more complex problems or applications.