Number Base System and Conversion from any base to base 10 for SS 1

Class: SS 1
Term: First Term
Week: 1
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: Simple Equations: Change of Subject and Substitution
Topic: Number Base System (1)
Subject Matter: Meaning of Number Base System, Conversion from any base to base 10, Conversion of fractions in any base to base 10

Specific Objectives

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

Cognitive Domain

• define Number Base System;
• convert whole numbers from any base to base 10;
• convert fractions in any base to base 10.

Affective Domain

• demonstrate responsibility in solving number base problems;
• collaborate effectively in group activities to convert number bases.

Psychomotor Domain

• accurately perform calculations for converting numbers from any base to base 10;
• apply the conversion procedure to practical examples.

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

Rationale for the Lesson

Understanding number base systems is fundamental to computer science and digital technology, as computers operate using binary (base 2). This lesson provides students with the foundational knowledge and skills to convert numbers between different bases, particularly to base 10, which is essential for further mathematical and computational studies. It helps students appreciate the structure of numbers beyond the familiar decimal system.

Prerequisite/Previous Knowledge

Students should have a basic understanding of place values in the decimal (base 10) system and be familiar with basic arithmetic operations (addition, subtraction, multiplication).

Lesson Content/Board Summary

Number Base System (1)

Meaning of Number Base System

A number base system is a method of representing quantities using a fixed set of digits and place values based on powers of the base. The base of a number system indicates the number of unique digits, including zero, used to represent numbers in that system. For example, the decimal system (base 10) uses ten digits (0-9), while the binary system (base 2) uses two digits (0, 1).

Positional Values in Number Bases

In any number base system, the position of each digit determines its value. Each position represents a power of the base. For a number (d_n d_{n-1} … d_1 d_0 . d_{-1} d_{-2} … d_{-m}) in base (b), its value in base 10 is given by:

[ (d_n times b^n) + (d_{n-1} times b^{n-1}) + … + (d_1 times b^1) + (d_0 times b^0) + (d_{-1} times b^{-1}) + (d_{-2} times b^{-2}) + … + (d_{-m} times b^{-m}) ]

Where:

1. (d_i) represents a digit in the number.
2. (b) represents the base of the number system.
3. (i) represents the position of the digit, starting from 0 for the digit immediately to the left of the decimal point, increasing to the left, and decreasing (negative) to the right.

Conversion from Any Base to Base 10 (Whole Numbers)

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

Example 1

Question: Convert (1101_2) to base 10.

Solution:

Step 1: Identify the digits and their positional values.

The number is (1101_2). The base is 2.

Digits from left to right are 1, 1, 0, 1.

Positional values (powers of 2) from right to left are (2^0, 2^1, 2^2, 2^3).

Step 2: Multiply each digit by its positional value.

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

Step 3: Calculate the powers and perform multiplication.

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

( 8 + 4 + 0 + 1 )

Step 4: Sum the results.

( 13 )

Answer: (1101_2 = 13_{10})

Example 2

Question: Convert (243_5) to base 10.

Solution:

Step 1: Identify the digits and their positional values.

The number is (243_5). The base is 5.

Digits from left to right are 2, 4, 3.

Positional values (powers of 5) from right to left are (5^0, 5^1, 5^2).

Step 2: Multiply each digit by its positional value.

( (2 times 5^2) + (4 times 5^1) + (3 times 5^0) )

Step 3: Calculate the powers and perform multiplication.

( (2 times 25) + (4 times 5) + (3 times 1) )

( 50 + 20 + 3 )

Step 4: Sum the results.

( 73 )

Answer: (243_5 = 73_{10})

Conversion of Fractions in Any Base to Base 10

To convert a fractional number from any base to base 10, multiply each digit after the radix point (decimal point) by its corresponding negative positional value (negative power of the base) and then sum the results. For mixed numbers, convert the whole number part and the fractional part separately, then add them.

Example 1

Question: Convert (110.11_2) to base 10.

Solution:

Step 1: Separate the whole number and fractional parts.

Whole number part: (110_2)

Fractional part: (.11_2)

Step 2: Convert the whole number part to base 10.

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

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

( 4 + 2 + 0 = 6_{10} )

Step 3: Convert the fractional part to base 10.

Digits after the radix point are 1, 1.

Positional values (negative powers of 2) are (2^{-1}, 2^{-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} )

Step 4: Add the converted whole number and fractional parts.

( 6 + 0.75 = 6.75_{10} )

Answer: (110.11_2 = 6.75_{10})

Example 2

Question: Convert (11021.101_3) to base 10.

Solution:

Step 1: Separate the whole number and fractional parts.

Whole number part: (11021_3)

Fractional part: (.101_3)

Step 2: Convert the whole number part to base 10.

( (1 times 3^4) + (1 times 3^3) + (0 times 3^2) + (2 times 3^1) + (1 times 3^0) )

( (1 times 81) + (1 times 27) + (0 times 9) + (2 times 3) + (1 times 1) )

( 81 + 27 + 0 + 6 + 1 = 115_{10} )

Step 3: Convert the fractional part to base 10.

Digits after the radix point are 1, 0, 1.

Positional values (negative powers of 3) are (3^{-1}, 3^{-2}, 3^{-3}).

( (1 times 3^{-1}) + (0 times 3^{-2}) + (1 times 3^{-3}) )

( (1 times frac{1}{3}) + (0 times frac{1}{9}) + (1 times frac{1}{27}) )

( frac{1}{3} + 0 + frac{1}{27} )

( frac{9}{27} + frac{1}{27} = frac{10}{27} approx 0.37037_{10} )

Step 4: Add the converted whole number and fractional parts.

( 115 + 0.37037… approx 115.37037_{10} )

Answer: (11021.101_3 approx 115.37037_{10})

Teaching Methods/Instructional Techniques

Discussion, Explanation, Demonstration, Guided Practice, Question and Answer, Group Work, Video Presentation

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Explaining/Questioning

Teacher’s Activity: The teacher introduces the topic by asking students about the number system they commonly use (base 10) and if they know of other number systems. The teacher explains that different number systems exist, especially in computing.

Pupils’ Activity: Pupils respond to questions and listen attentively, sharing their prior knowledge about number systems.

Learning Point: Introduction to bases

Step 2: Meaning of Number Base System

Time: 10 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher defines a number base system, explaining that it uses a fixed set of digits and place values based on powers of the base. The teacher uses examples like base 10 (decimal) and base 2 (binary) to illustrate the concept and explains positional values.

Pupils’ Activity: Pupils listen, ask questions for clarification, and copy the definition into their notebooks.

Learning Point: Meaning of number base

Step 3: Positional Values in Number Bases

Time: 10 minutes

Teaching Skill: Demonstration/Illustration

Teacher’s Activity: The teacher demonstrates how to determine the positional value of each digit in a number for different bases, emphasising that each position represents a power of the base. The teacher writes out the general formula for expansion.

Pupils’ Activity: Pupils observe the examples, participate in identifying positional values, and copy the formula.

Learning Point: Positional values explained

Step 4: Conversion from Any Base to Base 10 (Whole Numbers)

Time: 10 minutes

Teaching Skill: Demonstration/Guided Practice

Teacher’s Activity: The teacher demonstrates the step-by-step process of converting whole numbers from any base to base 10 using the expansion method. The teacher works through Example 1 and 2 from the Board Summary, guiding students through each step.

Pupils’ Activity: Pupils follow the teacher’s demonstration, ask questions, and attempt to solve similar problems individually or in pairs.

Learning Point: Whole number conversion

Step 5: Practice on Whole Number Conversion

Time: 5 minutes

Teaching Skill: Problem Solving/Collaboration

Teacher’s Activity: The teacher provides additional practice questions for whole number conversion, encouraging students to work in groups using number base conversion charts if available. The teacher monitors and provides support.

Pupils’ Activity: Students work in groups or individually to solve the practice problems, discussing their methods and solutions.

Learning Point: Practice whole number conversion

Step 6: Conversion of Fractions in Any Base to Base 10

Time: 5 minutes

Teaching Skill: Demonstration/Explanation

Teacher’s Activity: The teacher explains how to convert fractional parts of numbers from any base to base 10, using negative powers of the base. The teacher demonstrates Example 1 and 2 from the Board Summary, explaining the process for mixed numbers.

Pupils’ Activity: Pupils observe the demonstration, noting the difference in positional values for fractional parts, and ask clarifying questions.

Learning Point: Fractional number 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 is a Number Base System?
2. Convert (1011_2) to base 10.
3. Convert (32_4) to base 10.
4. Convert (10.1_2) to base 10.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Number base conversion

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, including definitions, formulas, and worked examples for whole and fractional number base conversions, 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: Summarising

Teacher’s Activity: The teacher summarises the key points of the lesson, reiterating the definition of a number base system and the method for converting numbers (whole and fractional) from any base to base 10. The teacher encourages students to practise more examples.

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

Learning Point: Reinforcing base conversion

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your mathematics notebook:

1. Define a Number Base System in your own words.
2. Convert the following numbers to base 10:
   1. (10110_2)
   2. (402_5)
   3. (123_8)
3. Convert the following fractional numbers to base 10:
   1. (11.01_2)
   2. (21.12_3)
4. Watch the video clip on number base conversion again and summarise the main steps.

Lesson Keywords

• Number Base System – A system for representing numbers using a specific set of digits and positional values based on a chosen base.
• Base 10 (Decimal) – The standard number system using ten digits (0-9).
• Base 2 (Binary) – A number system using two digits (0, 1), fundamental to computers.
• Positional Value – The value of a digit determined by its position in a number, representing a power of the base.
• Radix Point – The point (like a decimal point) that separates the whole number part from the fractional part in any base system.

Differentiation

For students who grasp the concept quickly, provide more complex conversion problems involving larger numbers or different bases (e.g., base 16 – hexadecimal). For students who are struggling, provide additional guided practice with simpler binary and base 3 conversions, using physical number base charts to visualise positional values. Pair stronger students with weaker ones for peer tutoring during group activities.

Suggested Lesson Videos

YouTube search for Number Base System Conversion SS1 Mathematics
How to Convert Numbers from Any Base to Base 10 | Number Base System