Base 16 Numbers and Their Representation for SS 1

Class: SS 1
Term: First Term
Week: 10
Age: 15 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Developing competency in students on the basic computer language codes
Key competencies/values: Communication; ICT and Digital Competencies; Digital Competencies
Skills:

• Coding skill

Previous Lesson: Application to computer programming
Topic: Number Base System (2): (Base 16)
Subject Matter: (base 16)

Specific Objectives

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

Cognitive Domain

• Explain the meaning and representation of hexadecimal (base 16) numbers.
• Identify the digits used in base 16.
• State the positional values of digits in base 16.

Affective Domain

• Appreciate the importance of hexadecimal numbers in computer programming.
• Show interest in learning about different number base systems.

Psychomotor Domain

• Convert numbers from base 16 to base 10.
• Convert numbers from base 10 to base 16.
• Perform addition and subtraction operations in base 16.
• Demonstrate the basic use of hexadecimal (base 16) in simple contexts.

Social Domain

• Collaborate with peers to solve problems involving number base conversions.

Reference Materials

The following resources were used in planning this lesson:

• 2025 New Revised Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• Essential Mathematics for Senior Secondary Schools by A.J.S. Oluwasanmi
• The HeadTeacher Scheme of work For The New Revised Senior Secondary Education Curriculum (SSEC)

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard and markers
• Charts showing hexadecimal digits and their decimal equivalents
• Video clips on how hexadecimal numbers are used in programming
• Calculators (optional, for checking answers)

Rationale for the Lesson

Understanding Base 16 (hexadecimal) numbers is fundamental for students pursuing computer science or related fields. This lesson provides the foundational knowledge for interpreting computer memory addresses, colour codes, and other digital data representations. It also reinforces the general concept of number base systems and positional values, which are crucial mathematical skills.

Prerequisite/Previous Knowledge

Students should have prior knowledge of number base systems (e.g., base 2, base 8, base 10), including how to convert between them and perform basic arithmetic operations in other bases.

Lesson Content/Board Summary

Number Base System (2): Base 16

Meaning of Base 16 (Hexadecimal) Numbers

Base 16, also known as hexadecimal, is a number base system that uses 16 distinct symbols to represent numbers. It is widely used in computer science and digital systems because it provides a more compact way to represent binary numbers, which are the native language of computers.

Digits Used in Base 16

The 16 symbols used in hexadecimal are the digits 0-9 and the letters A-F. Each letter represents a decimal value greater than 9:

1. 0 = 0 (decimal)
2. 1 = 1 (decimal)
3. 2 = 2 (decimal)
4. 3 = 3 (decimal)
5. 4 = 4 (decimal)
6. 5 = 5 (decimal)
7. 6 = 6 (decimal)
8. 7 = 7 (decimal)
9. 8 = 8 (decimal)
10. 9 = 9 (decimal)
11. A = 10 (decimal)
12. B = 11 (decimal)
13. C = 12 (decimal)
14. D = 13 (decimal)
15. E = 14 (decimal)
16. F = 15 (decimal)

Place Value in Base 16

In any number base system, the position of a digit determines its value. For base 16, each position represents a power of 16. Moving from right to left, the place values are (16^0, 16^1, 16^2, 16^3), and so on.

For example, in a hexadecimal number ( (D3F)_{16} ):

1. F is in the (16^0) (units) position.
2. 3 is in the (16^1) (sixteens) position.
3. D is in the (16^2) (two hundred fifty-sixes) position.

Conversion from Base 16 to Base 10

To convert a hexadecimal number to base 10, multiply each digit by its corresponding positional value (power of 16) and then sum the results.

Example 1

Question: Convert ( (A5)_{16} ) to base 10.

Solution:

Step 1: Identify the digits and their decimal equivalents.

( A = 10 ), ( 5 = 5 )

Step 2: Assign positional values (powers of 16) from right to left, starting with (16^0).

( A times 16^1 + 5 times 16^0 )

Step 3: Substitute decimal values and calculate.

( (10 times 16) + (5 times 1) )

( 160 + 5 )

( 165 )

Answer: ( (A5)_{16} = (165)_{10} )

Example 2

Question: Convert ( (2C7)_{16} ) to base 10.

Solution:

Step 1: Identify the digits and their decimal equivalents.

( 2 = 2 ), ( C = 12 ), ( 7 = 7 )

Step 2: Assign positional values.

( 2 times 16^2 + C times 16^1 + 7 times 16^0 )

Step 3: Substitute decimal values and calculate.

( (2 times 256) + (12 times 16) + (7 times 1) )

( 512 + 192 + 7 )

( 711 )

Answer: ( (2C7)_{16} = (711)_{10} )

Conversion from Base 10 to Base 16

To convert a base 10 number to hexadecimal, use the method of repeated division by 16. Record the remainders at each step, and then read the remainders from bottom to top to form the hexadecimal number. Any remainder greater than 9 should be represented by its corresponding hexadecimal letter (A-F).

Example 1

Question: Convert ( (250)_{10} ) to base 16.

Solution:

Step 1: Divide the base 10 number by 16 and note the remainder.

( 250 div 16 = 15 ) remainder ( 10 ) (which is A in hexadecimal)

Step 2: Divide the quotient from the previous step by 16.

( 15 div 16 = 0 ) remainder ( 15 ) (which is F in hexadecimal)

Step 3: Read the remainders from bottom to top.

The remainders are F and A.

Answer: ( (250)_{10} = (FA)_{16} )

Example 2

Question: Convert ( (400)_{10} ) to base 16.

Solution:

Step 1: Divide 400 by 16.

( 400 div 16 = 25 ) remainder ( 0 )

Step 2: Divide 25 by 16.

( 25 div 16 = 1 ) remainder ( 9 )

Step 3: Divide 1 by 16.

( 1 div 16 = 0 ) remainder ( 1 )

Step 4: Read the remainders from bottom to top.

The remainders are 1, 9, and 0.

Answer: ( (400)_{10} = (190)_{16} )

Arithmetic Operations in Base 16

Addition in Base 16

To add hexadecimal numbers, add the digits in each column starting from the rightmost column. If the sum in a column is 16 or more, divide by 16, write down the remainder, and carry over the quotient to the next column.

Example 1

Question: Add ( (2A)_{16} + (1B)_{16} )

Solution:

Step 1: Add the rightmost column (A + B).

( A = 10 ), ( B = 11 )

( 10 + 11 = 21 )

( 21 div 16 = 1 ) remainder ( 5 ). Write down 5, carry over 1.

Step 2: Add the next column (2 + 1 + carry 1).

( 2 + 1 + 1 = 4 ). Write down 4.

Answer: ( (2A)_{16} + (1B)_{16} = (45)_{16} )

Subtraction in Base 16

To subtract hexadecimal numbers, subtract the digits in each column starting from the rightmost column. If a digit in the top number is smaller than the corresponding digit in the bottom number, borrow 16 from the next column to the left. The borrowed 16 is added to the digit in the current column before subtraction.

Example 1

Question: Subtract ( (5C)_{16} – (2D)_{16} )

Solution:

Step 1: Subtract the rightmost column (C – D).

( C = 12 ), ( D = 13 ). Since ( 12 < 13 ), borrow 16 from the 5 in the next column.

The C becomes ( 12 + 16 = 28 ).

( 28 – 13 = 15 ) (which is F in hexadecimal). Write down F.

Step 2: Subtract the next column (5 – 2, after borrowing 1 from 5).

The 5 becomes 4 (after lending 1). So, ( 4 – 2 = 2 ). Write down 2.

Answer: ( (5C)_{16} – (2D)_{16} = (2F)_{16} )

Applications of Hexadecimal Numbers

Hexadecimal numbers are used in various fields, especially in computing:

1. Memory Addresses: Computer memory locations are often represented in hexadecimal.
2. Colour Codes: In web design and graphics, colours are specified using hexadecimal codes (e.g., #FF0000 for red).
3. MAC Addresses: Unique identifiers for network interfaces are typically displayed in hexadecimal.
4. Assembly Language: Programmers often use hexadecimal to represent binary data more concisely.

Teaching Methods/Instructional Techniques

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

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Recalling/Engaging

Teacher’s Activity: The teacher greets the students and reviews previous lessons on number base systems, asking questions about base 2 and base 8. The teacher then introduces Base 16 as another important number base, especially in computer science.

Pupils’ Activity: Pupils respond to questions about other number bases and listen attentively to the introduction of Base 16.

Learning Point: Introduction to Base 16

Step 2: Meaning and Digits of Base 16

Time: 10 minutes

Teaching Skill: Explanation/Illustration

Teacher’s Activity: The teacher explains what Base 16 (hexadecimal) means, its 16 symbols (0-9 and A-F), and their decimal equivalents. The teacher uses a chart to illustrate the symbols and their values clearly.

Pupils’ Activity: Pupils listen, observe the chart, and ask questions for clarification. They note the symbols and their decimal values.

Learning Point: Hexadecimal symbols and values

Step 3: Conversion from Base 16 to Base 10

Time: 10 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher explains the process of converting from Base 16 to Base 10, emphasising positional values. The teacher works through Example 1 and Example 2 from the Board Summary on the board, explaining each step clearly.

Pupils’ Activity: Pupils pay attention to the steps, ask questions, and attempt to solve similar problems as guided by the teacher.

Learning Point: Base 16 to Base 10 conversion

Step 4: Conversion from Base 10 to Base 16

Time: 10 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher explains the method of repeated division by 16 for converting from Base 10 to Base 16. The teacher works through Example 1 and Example 2 from the Board Summary, ensuring students understand how to represent remainders greater than 9 with letters.

Pupils’ Activity: Pupils follow the teacher’s examples, practice the conversion steps, and actively participate in solving problems.

Learning Point: Base 10 to Base 16 conversion

Step 5: Addition in Base 16

Time: 5 minutes

Teaching Skill: Explanation/Application

Teacher’s Activity: The teacher explains the rules for adding hexadecimal numbers, focusing on carrying over when the sum exceeds 15. The teacher demonstrates Example 1 from the Board Summary on the board.

Pupils’ Activity: Pupils observe the addition process, ask questions about carrying, and try to follow the steps.

Learning Point: Hexadecimal addition rules

Step 6: Subtraction in Base 16

Time: 5 minutes

Teaching Skill: Explanation/Application

Teacher’s Activity: The teacher explains the rules for subtracting hexadecimal numbers, focusing on borrowing 16 from the next column. The teacher demonstrates Example 1 from the Board Summary on the board.

Pupils’ Activity: Pupils observe the subtraction process, ask questions about borrowing, and try to follow the steps.

Learning Point: Hexadecimal subtraction rules

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 base 16?
2. Convert ( (3F)_{16} ) to base 10.
3. Convert ( (180)_{10} ) to base 16.
4. Perform the addition: ( (1A)_{16} + (2B)_{16} ).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Understanding Base 16 operations

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on Base 16 numbers, conversions, and arithmetic operations into their notebooks.

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

Learning Point: Recording lesson content

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Summarising/Reinforcement

Teacher’s Activity: The teacher briefly summarises the key concepts of Base 16 numbers, including their representation, conversion methods, and basic arithmetic. The teacher reminds students of the importance of hexadecimal in computing and encourages further practice.

Pupils’ Activity: Pupils listen to the summary and ask any final questions.

Learning Point: Consolidation of Base 16 concepts

Continuous Assessment/Further Study

Type: Homework

Instruction: Answer the following questions in your notebook:

1. List the hexadecimal digits and their decimal equivalents.
2. Convert the following hexadecimal numbers to base 10:
   1. ( (7D)_{16} )
   2. ( (1F2)_{16} )
3. Convert the following base 10 numbers to base 16:
   1. ( (195)_{10} )
   2. ( (500)_{10} )
4. Perform the following operations in base 16:
   1. ( (3E)_{16} + (1C)_{16} )
   2. ( (9A)_{16} – (4F)_{16} )
5. Research and write down two real-world applications of hexadecimal numbers in technology.

Lesson Keywords

• Base 16 – A number system that uses 16 unique symbols.
• Hexadecimal – Another name for Base 16.
• Digits (0-9, A-F) – The symbols used to represent numbers in hexadecimal.
• Positional Value – The value a digit holds based on its position in a number.
• Conversion – The process of changing a number from one base to another.

Differentiation

For struggling learners, provide extra practice with conversion charts and simpler examples. Encourage peer tutoring. For advanced learners, introduce multiplication and division in Base 16, or challenge them to convert between Base 16 and Base 2 directly.

Suggested Lesson Videos

YouTube search for “Base 16 numbers SS1 Mathematics”