Number Base Conversion and Arithmetic Operations for SS 1

Class: SS 1
Term: First Term
Week: 2
Age: 15 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Focal competence: Converting from one base to another for computer programming
Key competencies/values: Collaboration; Responsibility
Skills:

• converting from one base to another

Previous Lesson: Graph of y = 10x, Finding Logarithms and Antilogarithms
Topic: Number Base System (1): Conversion Of Number From One Base To Another Base
Subject Matter: Conversion of number from one base to another base, Addition, Subtraction, Multiplication and Division of number bases

Specific Objectives

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

Cognitive Domain

• convert numbers from one base to another base;
• solve number base problems involving addition;
• solve number base problems involving subtraction;
• solve number base problems involving multiplication;
• solve number base problems involving division.

Affective Domain

• appreciate the importance of accuracy in number base conversions and operations;
• demonstrate responsibility in checking their solutions;
• collaborate effectively with peers to solve number base problems.

Psychomotor Domain

• demonstrate the step-by-step process of converting numbers between different bases;
• accurately perform arithmetic operations on numbers in various bases.

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
• New General Mathematics for Senior Secondary Schools 1
• 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:

• Number base conversion charts
• Whiteboard and markers
• Short video clips on number base conversion and operations. e.g. youtube.com/watch?v=jpK0Q0PN21
• Calculators (optional, for checking answers)

Rationale for the Lesson

Understanding number base systems is fundamental to computer science and digital electronics. This lesson provides students with the essential skills to convert numbers between different bases and perform arithmetic operations, which are crucial for further studies in technology and problem-solving in various contexts.

Prerequisite/Previous Knowledge

Students should have a basic understanding of place values in the decimal system and elementary arithmetic operations.

Lesson Content/Board Summary

Number Base System (1): Conversion Of Number From One Base To Another Base

Introduction to Number Base Systems

A number base system is a system for representing quantities using a fixed set of digits and place values based on powers of the base. The most common number base system is base 10 (decimal), which uses ten digits (0-9). Other common bases include base 2 (binary), base 8 (octal), and base 16 (hexadecimal).

The place value of each digit in a number depends on its position and the base of the number system. For example, in base 10, the number 234 means (2 times 10^2 + 3 times 10^1 + 4 times 10^0).

Conversion of Numbers from One Base to Another

Conversion from Base 10 to Other Bases

To convert a number from base 10 to another base, repeatedly divide the base 10 number by the target base and record the remainders. The conversion is read from the last remainder upwards.

Example 1: Convert (25_{10}) to base 2.

Solution:

1. (25 div 2 = 12) remainder (1)
2. (12 div 2 = 6) remainder (0)
3. (6 div 2 = 3) remainder (0)
4. (3 div 2 = 1) remainder (1)
5. (1 div 2 = 0) remainder (1)

Reading the remainders upwards: (11001_2). So, (25_{10} = 11001_2).

Example 2: Convert (150_{10}) to base 8.

Solution:

1. (150 div 8 = 18) remainder (6)
2. (18 div 8 = 2) remainder (2)
3. (2 div 8 = 0) remainder (2)

Reading the remainders upwards: (226_8). So, (150_{10} = 226_8).

Conversion from Other Bases to Base 10

To convert a number from another base to base 10, multiply each digit by its corresponding place value (power of the base) and sum the results.

Example 1: Convert (1101_2) to base 10.

Solution:

(1101_2 = (1 times 2^3) + (1 times 2^2) + (0 times 2^1) + (1 times 2^0))

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

(= 8 + 4 + 0 + 1 = 13_{10})

So, (1101_2 = 13_{10}).

Example 2: Convert (435_6) to base 10.

Solution:

(435_6 = (4 times 6^2) + (3 times 6^1) + (5 times 6^0))

(= (4 times 36) + (3 times 6) + (5 times 1))

(= 144 + 18 + 5 = 167_{10})

So, (435_6 = 167_{10}).

Conversion between Other Bases (e.g., Base 2 to Base 8)

To convert a number from one non-base 10 to another non-base 10, first convert the number to base 10, then convert the base 10 result to the target base.

Example: Convert (1101_2) to base 5.

Solution:

Step 1: Convert (1101_2) to base 10.

(1101_2 = (1 times 2^3) + (1 times 2^2) + (0 times 2^1) + (1 times 2^0))

(= 8 + 4 + 0 + 1 = 13_{10})

Step 2: Convert (13_{10}) to base 5.

1. (13 div 5 = 2) remainder (3)
2. (2 div 5 = 0) remainder (2)

Reading remainders upwards: (23_5). So, (1101_2 = 23_5).

Conversion of Fractions in Number Bases

To convert a fraction from another base to base 10, multiply each digit after the decimal point by negative powers of the base and sum the results.

Example 1: Convert (0.11_2) to base 10.

Solution:

(0.11_2 = (1 times 2^{-1}) + (1 times 2^{-2}))

(= (1 times frac{1}{2}) + (1 times frac{1}{4}))

(= 0.5 + 0.25 = 0.75_{10})

So, (0.11_2 = 0.75_{10}).

To convert a number with both integer and fractional parts, convert each part separately and then combine them.

Example 2: Convert (110.11_2) to base 10.

Solution:

Integer part: (110_2 = (1 times 2^2) + (1 times 2^1) + (0 times 2^0) = 4 + 2 + 0 = 6_{10})

Fractional part: (0.11_2 = (1 times 2^{-1}) + (1 times 2^{-2}) = 0.5 + 0.25 = 0.75_{10})

Combining: (6 + 0.75 = 6.75_{10}). So, (110.11_2 = 6.75_{10}).

Arithmetic Operations in Number Bases

Arithmetic operations (addition, subtraction, multiplication, division) in other bases follow similar principles to base 10, but carrying and borrowing are done with respect to the given base.

Addition of Number Bases

Add digits column by column. If the sum in a column is equal to or greater than the base, divide the sum by the base, write down the remainder, and carry the quotient to the next column.

Example 1: Add (110_2 + 11_2).

Solution:

  110_2
+  11_2
-----

1. Rightmost column: (0 + 1 = 1). Write (1).
2. Middle column: (1 + 1 = 2). (2 div 2 = 1) remainder (0). Write (0), carry (1).
3. Leftmost column: (1) (from (110_2)) (+ 1) (carry) ( = 2). (2 div 2 = 1) remainder (0). Write (0), carry (1).
4. Next column (empty): Write the carry (1).

  110_2
+  11_2
-----
 1001_2

So, (110_2 + 11_2 = 1001_2).

Example 2: Add (23_5 + 14_5).

Solution:

  23_5
+ 14_5
-----

1. Rightmost column: (3 + 4 = 7). (7 div 5 = 1) remainder (2). Write (2), carry (1).
2. Leftmost column: (2 + 1 + 1) (carry) ( = 4). Write (4).

  23_5
+ 14_5
-----
  42_5

So, (23_5 + 14_5 = 42_5).

Subtraction of Number Bases

Subtract digits column by column. If a digit is smaller than the one being subtracted from it, borrow from the next column. When borrowing, the value borrowed is equal to the base.

Example 1: Subtract (110_2 – 11_2).

Solution:

  110_2
-  11_2
-----

1. Rightmost column: (0 – 1). Cannot subtract. Borrow from the next column. The (1) in the middle column becomes (0), and the (0) becomes (0 + 2 = 2). Now, (2 – 1 = 1). Write (1).
2. Middle column: (0 – 1). Cannot subtract. Borrow from the next column. The (1) in the leftmost column becomes (0), and the (0) becomes (0 + 2 = 2). Now, (2 – 1 = 1). Write (1).
3. Leftmost column: (0). Write (0).

  110_2
-  11_2
-----
   11_2

So, (110_2 – 11_2 = 11_2).

Example 2: Subtract (42_5 – 14_5).

Solution:

  42_5
- 14_5
-----

1. Rightmost column: (2 – 4). Cannot subtract. Borrow from the next column. The (4) in the leftmost column becomes (3), and the (2) becomes (2 + 5 = 7). Now, (7 – 4 = 3). Write (3).
2. Leftmost column: (3 – 1 = 2). Write (2).

  42_5
- 14_5
-----
  23_5

So, (42_5 – 14_5 = 23_5).

Multiplication of Number Bases

Multiply digits similar to base 10 multiplication. When a product is greater than or equal to the base, convert it to the base by dividing and carrying the quotient.

Example 1: Multiply (11_2 times 10_2).

Solution:

  11_2
x 10_2
-----
  00  (11_2 x 0)
+110  (11_2 x 1, shifted)
-----
 110_2

So, (11_2 times 10_2 = 110_2).

Example 2: Multiply (12_3 times 2_3).

Solution:

  12_3
x  2_3
-----

1. (2 times 2 = 4). (4 div 3 = 1) remainder (1). Write (1), carry (1).
2. (2 times 1 = 2). Add the carry: (2 + 1 = 3). (3 div 3 = 1) remainder (0). Write (0), carry (1).
3. Write the carry (1).

  12_3
x  2_3
-----
 101_3

So, (12_3 times 2_3 = 101_3).

Division of Number Bases

Division in other bases can be complex. One common method is to convert both numbers to base 10, perform the division, and then convert the result back to the desired base if necessary.

Example: Divide (110_2 div 10_2).

Solution:

Step 1: Convert to base 10.

(110_2 = (1 times 2^2) + (1 times 2^1) + (0 times 2^0) = 4 + 2 + 0 = 6_{10})

(10_2 = (1 times 2^1) + (0 times 2^0) = 2 + 0 = 2_{10})

Step 2: Perform division in base 10.

(6_{10} div 2_{10} = 3_{10})

Step 3: Convert the result back to base 2.

1. (3 div 2 = 1) remainder (1)
2. (1 div 2 = 0) remainder (1)

Reading remainders upwards: (11_2). So, (110_2 div 10_2 = 11_2).

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Pair Work, Group Work, Video Presentation.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Questioning/Recall

Teacher’s Activity: The teacher asks students about their understanding of place values in everyday numbers (base 10) and introduces the idea that numbers can be represented in other bases. “What does the ‘3’ in 345 mean? What about the ‘4’?”

Pupils’ Activity: Pupils respond, explaining place values in base 10.

Learning Point: Place value concept

Step 2: Conversion from Base 10 to Other Bases

Time: 10 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains and demonstrates the method for converting numbers from base 10 to other bases (e.g., base 2, base 8) using repeated division and remainders. The teacher uses a number base conversion chart to illustrate.

Pupils’ Activity: Pupils observe, ask questions, and follow the demonstration, using the conversion charts.

Learning Point: Base 10 to other base conversion

Step 3: Conversion from Other Bases to Base 10

Time: 10 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher explains and demonstrates how to convert numbers from other bases (e.g., base 2, base 5) to base 10 using expansion by powers of the base. Students are guided to convert (110_2) to base 10.

Pupils’ Activity: Pupils practice converting numbers from other bases to base 10 with teacher guidance.

Learning Point: Other base to Base 10 conversion

Step 4: Conversion of Fractions and between Other Bases

Time: 10 minutes

Teaching Skill: Demonstration/Problem Solving

Teacher’s Activity: The teacher guides students to convert fractions in any base to base 10 (e.g., (110.11_2) to base 10) and also demonstrates conversion between two non-base 10 systems (e.g., base 2 to base 5) by first converting to base 10. A short video clip on conversion is shown.

Pupils’ Activity: Pupils watch the video, ask questions, and attempt conversion problems involving fractions and different bases in groups.

Learning Point: Fractional and inter-base conversion

Step 5: Arithmetic Operations: Addition and Subtraction

Time: 8 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher explains and demonstrates addition and subtraction of numbers in different bases, highlighting the concept of carrying and borrowing relative to the base. Students are given problems to solve in pairs.

Pupils’ Activity: Pupils work in pairs to solve addition and subtraction problems in various bases.

Learning Point: Base number addition and subtraction

Step 6: Arithmetic Operations: Multiplication and Division

Time: 7 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher explains and demonstrates multiplication and division of numbers in different bases, emphasising conversion to base 10 for division. Another short video clip on arithmetic operations is shown.

Pupils’ Activity: Pupils watch the video and attempt multiplication and division problems individually or in groups.

Learning Point: Base number multiplication and division

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. Convert (37_{10}) to base 2.
2. Convert (1011_2) to base 10.
3. Add (101_2 + 11_2).
4. Subtract (1101_2 – 101_2).

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Number base conversion and 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 number base conversion and arithmetic operations into their notebooks.

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

Learning Point: Recording lesson notes

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Consolidation

Teacher’s Activity: The teacher summarises the key concepts of number base conversion and arithmetic operations, reiterating their importance in various fields, especially computing. The teacher encourages students to practice more problems.

Pupils’ Activity: Pupils listen attentively and ask any final clarifying questions.

Learning Point: Reinforcement of key concepts

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebooks:

1. Convert (58_{10}) to base 3.
2. Convert (213_4) to base 10.
3. Convert (10110_2) to base 8.
4. Add (111_2 + 101_2).
5. Subtract (32_5 – 14_5).
6. Multiply (21_3 times 12_3).

Lesson Keywords

• Number Base – A system for representing numbers using a specific set of digits and a fixed base value.
• Base 10 (Decimal) – The standard number system using ten digits (0-9).
• Base 2 (Binary) – A number system using two digits (0 and 1), fundamental to computers.
• Base 8 (Octal) – A number system using eight digits (0-7).
• Base 16 (Hexadecimal) – A number system using sixteen symbols (0-9 and A-F).
• Conversion – The process of changing a number from one base system to another.
• Place Value – The value of a digit based on its position in a number.

Differentiation

Support: Provide simpler conversion problems and extra practice sheets for students struggling with the concept. Encourage peer tutoring and use number line visual aids for carrying/borrowing in different bases.

Extension: Challenge advanced students with conversions involving larger numbers, more complex fractional conversions, or operations in base 16 (hexadecimal). Encourage them to research real-world applications of different number bases.

Suggested Lesson Videos

Search on YouTube for: “Number Base Conversion and Arithmetic Operations SS1 Mathematics”