Mathematics - Senior Secondary 1 - Number Bases (II)

Step

Teacher’s Activity

Student’s Activity

Step 1 - Revision

Briefly revises the concept of binary and base 7 numbers and their digits.

Students recall the digits used in binary and base 7.

Step 2 - Binary Addition

Demonstrates binary addition with simple examples, emphasizing carrying over when the sum of digits in a place value column is 2 (1+1=10 in binary). Example: 101₂+11₂.

Students observe the process of binary addition and take notes on the carry-over rule.

Step 3 - Binary Subtraction

Demonstrates binary subtraction with simple examples, emphasizing borrowing when a digit is smaller than the digit being subtracted. Example: 110₂−11₂.

Students observe the process of binary subtraction and take notes on the borrowing rule.

Step 4 - Base 7 Addition & Subtraction

Introduces addition and subtraction in base 7, emphasizing carrying over when the sum is 7 or more, and borrowing when needed (borrowing 7). Examples: 34₇+15₇ and 52₇−26₇.

Students observe the processes of base 7 addition and subtraction, noting the carry-over and borrowing rules for base 7.

Step 5 - Guided Practice

Provides worksheets with addition and subtraction problems in binary and base 7 for students to solve in pairs. Teacher circulates to assist.

Students work in pairs to solve the given arithmetic problems in binary and base 7.

NOTE ON BOARD

Binary Addition: 1+0=1, 0+1=1, 0+0=0, 1+1=10 (carry 1). Binary Subtraction: 1-0=1, 0-0=0, 1-1=0, 0-1=borrow 1 (becomes 10_2 which is 2 in decimal). Base 7 Addition/Subtraction: Similar to base 10 but carry/borrow occurs at 7. Examples provided.

Students copy the rules and examples into their notebooks.

Step

Teacher’s Activity

Student’s Activity

Step 1 - Binary Multiplication

Demonstrates binary multiplication with simple whole numbers. Emphasizes the similarity to decimal multiplication, but with only 0 and 1. Example: 101₂×11₂.

Students observe the process of binary multiplication and note the steps.

Step 2 - Binary Division

Demonstrates simple binary division. Example: 110₂÷10₂.

Students observe the process of binary division and note the steps.

Step 3 - Binary Decimal Fractions

Explains the concept of binary decimal fractions, where digits to the right of the binary point represent negative powers of 2 (e.g., 0.12=1×2−1=0.510).

Students understand the place values after the binary point.

Step 4 - Conversion: Binary Decimal to Base 10

Demonstrates how to convert a binary decimal fraction to base 10 by multiplying each digit after the binary point by its corresponding negative power of 2 and summing the results. Example: 0.11²=(1×2⁻¹)+(1×2⁻²)=0.5+0.25=0.75₁₀. Extends to numbers with a whole number part: 10.1₂=(1×2¹)+(0×2⁰)+(1×2⁻¹)=2+0+0.5=2.5_10.

Students observe the conversion process and take notes on how to handle the fractional part.

Step 5 - Guided Practice

Provides worksheets with binary multiplication, division, and conversion of binary decimal fractions to base 10 for students to practice. Teacher assists as needed.

Students practice the operations and conversions individually or in pairs.

NOTE ON BOARD

Binary Multiplication: Similar to decimal, but multiply by 0 or 1. Binary Division: Similar to decimal long division. Binary Decimal Fractions: 0.d1d2...2=(d1×2⁻¹)+(d2×2⁻²)+... Examples shown for multiplication, division, and decimal conversion.

Students copy the rules and examples into their notebooks.

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Briefly explains that computers use the binary system to represent all information (data and instructions) because electronic circuits have two states: ON (represented by 1) and OFF (represented by 0).

Students listen and understand the fundamental reason for binary in computers.

Step 2 - Data Representation

Gives simple examples of how different types of data can be represented in binary, such as numbers, letters (using ASCII or similar codes), and simple instructions. Focuses on the concept rather than detailed encoding. For example, the decimal number 5 is 101₂, a simple instruction could be a sequence of 0s and 1s that the computer understands.

Students grasp the idea that everything in a computer is ultimately represented by 0s and 1s.

Step 3 - Basic Logic Gates

Introduces the concept of basic logic gates (AND, OR, NOT) as fundamental building blocks of computer circuits that operate on binary inputs to produce binary outputs. A simple diagram of each gate and its truth table (showing the output for different binary inputs) can be shown using the projector.

Students are introduced to the basic logic gates and understand that these gates perform operations on binary values.

Step 4 - Simple Program Example (Conceptual)

Describes a very simple hypothetical scenario where binary numbers and logic could be used in a program (without actual coding). For example, a sensor reads a value (represented in binary). If the value is above a certain binary threshold, the program performs an action (another binary output). This illustrates the flow of binary information and decision-making.

Students understand the basic idea of how binary numbers and logic can be used to make decisions in a computer program.

Step 5 - Discussion

Facilitates a brief discussion on other potential applications of binary in technology they might know or imagine.

Students participate in the discussion, sharing their thoughts and ideas.

NOTE ON BOARD

Computers use binary (0 and 1) because their circuits have two states (ON/OFF). Data (numbers, letters, instructions) is represented in binary. Basic logic gates (AND, OR, NOT) operate on binary values. Simple programs use binary to process information and make decisions based on binary logic.

Students copy the key concepts about the application of binary in computers.