Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Subtraction of numbers in base 2 numerals.
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Subtraction of numbers in base 2 numerals.
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 1st Term
Week: 8
Theme: Basic Operations
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This lesson introduces students to the fundamental operation of subtraction within the binary number system (base 2). Understanding binary subtraction is crucial for grasping how digital devices, such as computers, smartphones, and various electronic gadgets prevalent in Nigeria (e.g., POS terminals, ATMs, digital payment systems), perform arithmetic calculations. This foundational knowledge supports the understanding of computer architecture and programming logic, which are increasingly relevant skills for economic participation in Nigeria's digital economy.
1 = 0$. Write down
0. Result: $001_2$ or simply $1_2$.
Example 4: 3-digit subtraction with multiple borrows Subtract $011_2$ from $100_2$. $100_2 - 011_2$ $1\ 0\ 0_2$ - $0\ 1\ 1_2$ --------- $0\ 0\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. We need to borrow. The middle column has '0', so we must borrow from the leftmost column. Borrow '1' from the leftmost column (the '1' becomes '0'). The middle '0' becomes '10' (binary). Now, from this '10', we borrow '1' for the rightmost column. So, the middle '10' becomes '1' (binary). The rightmost '0' becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 2 (Middle column): This column was originally '0', received '10' (binary) from the leftmost column, and then lent '1' to the rightmost column. So, it is now '1'. Now, $1_2 - 1_2 = 0_2$. Write down
0. Step 3 (Leftmost column): The '1' in this column became '0' due to initial borrowing. We now have $0 - 0 = 0$. Write down
0. Result: $001_2$ or simply $1_2$. (Self-check using base 10: $100_2 = 4_{10}$, $011_2 = 3_{10}$. $4_{10} - 3_{10} = 1_{10}$. The result $001_2$ is $1_{10}$, so it is correct.)* This topic focuses on subtracting numbers expressed in base 2 (binary).
The binary system uses only two digits: 0 and
1. Subtraction in base 2 follows similar principles to subtraction in base 10, but with specific rules for borrowing.
Recall of Basic Binary Arithmetic: Digits: The only digits available are 0 and
1. Place Value: Positions represent powers of 2 (e.g., ... $2^3$, $2^2$, $2^1$, $2^0$). Rules for Binary Subtraction (Column by Column): The basic rules for subtracting individual binary digits are: 1. $0 - 0 = 0$ 2. $1 - 0 = 1$ 3. $1 - 1 = 0$ 4. $0 - 1 = ?$ (This scenario requires borrowing).
Understanding Borrowing in Base 2: When a '0' in a column needs to subtract a '1', it cannot do so directly. It must "borrow" from the digit in the column immediately to its left. If the digit to the left is '1', it changes to '0', and the borrowed '1' is carried over to the current column. When '1' is borrowed from the left, it represents '2' in base 10 for the current column. In base 2, this borrowed '1' (which is $1 \times 2^1$) becomes '10' (binary for two) in the column where it was borrowed.
Therefore, if a '0' needs to subtract '1' after borrowing: The '0' becomes '10' (binary). Then, $10_2 - 1_2 = 1_2$. (Because $2_{10} - 1_{10} = 1_{10}$, and $1_{10}$ is $1_2$). Step-by-Step Procedure for Binary Subtraction:
1. Align the numbers vertically by their place values.
2. Start subtracting from the rightmost column (least significant bit).
3. Apply the basic subtraction rules (0-0, 1-0, 1-1).
4. If $0 - 1$ occurs, borrow '1' from the next column to the left. The digit from which '1' was borrowed becomes '0' (if it was '1') or needs to borrow itself (if it was '0'). The '0' in the current column becomes '10' (binary). Then, perform $10_2 - 1_2 = 1_2$.
5. Continue this process for all columns until the subtraction is complete. Worked
Examples: Example 1: Simple 2-digit subtraction (no borrowing) Subtract $10_2$ from $11_2$. $11_2 - 10_2$ $1\ 1_2$ - $1\ 0_2$ ------ $0\ 1_2$ Step 1 (Rightmost column): $1 - 0 = 1$. Write down
1. Step 2 (Leftmost column): $1 - 1 = 0$. Write down
0. Result: $01_2$ or simply $1_2$.
Example 2: 2-digit subtraction with one borrow Subtract $01_2$ from $10_2$. $10_2 - 01_2$ $1\ 0_2$ - $0\ 1_2$ ------ $0\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. This requires borrowing. Borrow '1' from the left column (the '1' becomes '0'). The '0' in the current column becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 2 (Leftmost column): The '1' in this column became '0' due to borrowing. We now have $0 - 0 = 0$. Write down
0. Result: $01_2$ or simply $1_2$.
Example 3: 3-digit subtraction with one borrow Subtract $101_2$ from $110_2$. $110_2 - 101_2$ $1\ 1\ 0_2$ - $1\ 0\ 1_2$ --------- $0\ 0\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. Borrow '1' from the middle column (the '1' becomes '0'). The '0' in the current column becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 2 (Middle column): The '1' in this column became '0' due to borrowing. We now have $0 - 0 = 0$. Write down
0. Step 3 (Leftmost column): We have $1 - 1 = 0$. Write down
0. Result: $001_2$ or simply $1_2$.
Example 4: 3-digit subtraction with multiple borrows Subtract $011_2$ from $100_2$. $100_2 - 011_2$ $1\ 0\ 0_2$ - $0\ 1\ 1_2$ --------- $0\ 0\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. We need to borrow. The middle column has '0', so we must borrow from the leftmost column. Borrow '1' from the leftmost column (the '1' becomes '0'). The middle '0' becomes '10' (binary). * Now, from this '10', we borrow '1' for the Teacher Activities: Introduction & Review (5 minutes): Begin by reviewing binary numbers and basic binary addition from the previous lesson. Briefly explain the importance of binary arithmetic in digital technology relevant to Nigerian students. Introduction to Binary Subtraction Rules (10 minutes): Present the four basic rules for binary subtraction (0-0, 1-0, 1-1, 0-1). Focus intensely on explaining the "borrowing" concept for $0-1$. Use a visual aid on the board to demonstrate how '1' borrowed from the left becomes '10' in the current column. Draw parallels with base 10 borrowing to make it relatable (e.g., borrowing 1 from tens place makes units place 10 higher). Demonstration of Worked Examples (15 minutes): Work through Example 1 and Example 2 (simple 2-digit cases) step-by-step on the board, verbalizing each action and rule application. Progress to Example 3 and Example 4 (3-digit cases, including multiple borrows), clearly illustrating the borrowing process. Encourage questions and clarify any misconceptions immediately.
Guided Practice (10 minutes): Present 2-3 guided practice problems (from the "Guided Practice" section) on the board or projector. Instruct students to attempt them individually or in pairs. Circulate around the classroom, providing support, checking understanding, and offering constructive feedback. Select a few students to present their solutions and explanations on the board. Addressing Misconceptions & Reinforcement (5 minutes): Reiterate the critical step of borrowing ('1' becoming '10' in base 2). Emphasize writing down intermediate steps when borrowing to avoid errors.
Independent Practice Setup (5 minutes): Assign questions from the "Independent Practice" section for students to complete individually. Explain expectations for neatness and showing working.
Student Activities: Actively participate in the review of binary numbers and addition. Pay close attention during the explanation of binary subtraction rules and borrowing. Take comprehensive notes on the rules and worked examples provided by the teacher. Ask clarifying questions when concepts are unclear. Attempt guided practice problems as instructed, working individually or collaboratively with peers. Present solutions to guided practice problems on the board if called upon. Engage in independent practice, solving the assigned problems individually, ensuring all steps are shown. The following problems are designed to build student confidence, starting with simpler cases and progressing to 3-digit numbers with multiple borrows.
Question 1: Subtract $1_2$ from $11_2$. $1\ 1_2$ $0\ 1_2$ Solution 1: $1\ 1_2$ $0\ 1_2$ $1\ 0_2$ Step 1 (Rightmost column): $1 - 1 = 0$. Write down
0. Step 2 (Leftmost column): $1 - 0 = 1$. Write down
1. Commentary: This is a straightforward subtraction with no borrowing required, ideal for starting.
Question 2: Subtract $1_2$ from $10_2$. $1\ 0_2$ $0\ 1_2$ Solution 2: $\cancel{1}\ 0_2$ (The 1 changes to 0 after borrowing) $0\ 1_2$ $0\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the rightmost column becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 2 (Leftmost column): The '1' became '0'. We now have $0 - 0 = 0$. Write down
0. Commentary: This introduces the core concept of borrowing, where '1' borrowed from the left transforms the current '0' into '10' (base 2), which is equivalent to '2' in base
1
0. Question 3: Subtract $101_2$ from $111_2$. $1\ 1\ 1_2$ $1\ 0\ 1_2$ Solution 3: $1\ 1\ 1_2$ $1\ 0\ 1_2$ $0\ 1\ 0_2$ Step 1 (Rightmost column): $1 - 1 = 0$. Write down
0. Step 2 (Middle column): $1 - 0 = 1$. Write down
1. Step 3 (Leftmost column): $1 - 1 = 0$. Write down
0. Commentary: A 3-digit subtraction problem without any borrowing, reinforcing the direct application of rules.
Question 4: Subtract $011_2$ from $110_2$. $1\ 1\ 0_2$ $0\ 1\ 1_2$ Solution 4: $1\ \cancel{1}\ \cancel{0}_2$ (The middle 1 becomes 0; the rightmost 0 becomes 10) $0\ 1\ 1_2$ $0\ 1\ 1_2$ Step 1 (Rightmost column): We have $0 - 1$. Borrow '1' from the middle column. The '1' in the middle column becomes '0'. The '0' in the rightmost column becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 2 (Middle column): The '1' became '0' due to borrowing. We now have $0 - 1$. This requires another borrow. Borrow '1' from the leftmost column. The '1' in the leftmost column becomes '0'. The '0' in the middle column becomes '10' (binary). Now, $10_2 - 1_2 = 1_2$. Write down
1. Step 3 (Leftmost column): The '1' became '0' due to borrowing. We now have $0 - 0 = 0$. Write down
0. Result: $011_2$.
Commentary: This problem demonstrates a 3-digit subtraction involving multiple consecutive borrows, which is often challenging for students. Emphasize tracking the changes in digits after each borrow.
Example 1: Simple 2-digit subtraction (no borrowing)
Subtract $10_2$ from $11_2$.
$11_2 - 10_2$
$1\ 1_2$
$1\ 0_2$
$0\ 1_2$
Step 1 (Rightmost column): $1 - 0 = 1$. Write down
1. Step 2 (Leftmost column): $1 - 1 = 0$. Write down
0. Result: $01_2$ or simply $1_2$.
Example 2: 2-digit subtraction with one borrow
Subtract $01_2$ from $10_2$.
$10_2 - 01_2$
Digital Computing and Electronics in Nigeria: Binary subtraction is a fundamental operation within Central Processing Units (CPUs) of computers. Every time a subtraction is performed on a computer (e.g., deducting stock from inventory in a Nigerian business, calculating change at a POS terminal, or performing calculations in an Excel sheet), it is ultimately executed using binary arithmetic operations like subtraction. This understanding helps demystify the "black box" of technology, making students more informed users and potential innovators.
Network Protocols and Data Transmission: In digital communication, data is transmitted as binary signals. Subtraction algorithms are used in error detection and correction codes (e.g., checksums) to verify data integrity. For example, when transferring money via mobile banking apps or making online purchases (common activities across Nigeria), binary subtraction might be part of the underlying algorithms that ensure the correct amount is debited or credited and that the data transmitted is accurate.
Basic Robotics and Automation: As Nigeria increasingly embraces technological advancements, including robotics and automation in manufacturing and agriculture, the control systems for these machines rely heavily on binary logic and arithmetic. Subtraction could be used in programming a robot to adjust its position, reduce a count, or manage resources, making this an essential foundational skill for students interested in engineering and technology-driven agriculture.