Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Addition of numbers in base 2 numerals.
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Addition of numbers in base 2 numerals.
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 1st Term
Week: 7
Theme: Basic Operations
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Students should be able to add two or three 3-digit binary numbers.
the new carry-over.
Leftmost column (fours place): 1 (carry) + 1 + 1 + 0 = 1 (with a carry-over of 1). Write down 1. ``` 1 1 1 (carries) 1 1 12 1 0 12 + 0 1 12 ------- 1 1 12 ``` Since there's a carry-over of 1 and no further columns, bring down the carry-over 1. ``` 1 1 1 1 1 12 1 0 12 + 0 1 12 ------- 1 1 1 12 ``` Therefore, 1112 + 1012 + 0112 =
1
1
1
1
2. Verification:** 1112 = 710 1012 = 510 0112 = 310 710 + 510 + 310 = 1510 11112 = 1×23 + 1×22 + 1×21 + 1×20 = 8 + 4 + 2 + 1 =
1
5
1
0. The answer is correct. This section outlines the essential concepts and rules for performing addition in base 2. 2.
1. The Binary Number System (Base 2) The binary system is a positional numeral system with a radix, or base, of
2. It uses only two distinct symbols: 0 (zero) and 1 (one). Each digit in a binary number is referred to as a "bit." This system is fundamental to computer science as digital electronic circuits represent information using two stable states, typically represented as 0 and 1. 2.
2. Rules for Binary Addition The process of adding binary numbers is similar to adding decimal numbers, with the key difference being that "carrying over" occurs when the sum of bits in a column equals or exceeds the base (2). The basic rules for adding two binary digits are: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 with a carry-over of 1 (This means 102 which is equivalent to 210) When adding three binary digits, an additional rule applies: 1 + 1 + 1 = 1 with a carry-over of 1 (This means 112 which is equivalent to 310) 2.
3. Step-by-Step Worked Examples Example 1: Adding two 3-digit binary numbers Add 1012 and
1
1
0
2. Step 1: Arrange the numbers vertically, aligning by place value. ``` 1 0 12 + 1 1 02 ------- ``` Step 2: Start addition from the rightmost column (least significant bit).
Rightmost column (units place): 1 + 0 =
1. Write down 1. ``` 1 0 12 + 1 1 02 ------- 12 ``` Step 3: Move to the next column to the left.
Middle column (twos place): 0 + 1 =
1. Write down 1. ``` 1 0 12 + 1 1 02 ------- 1 12 ``` Step 4: Move to the leftmost column.
Leftmost column (fours place): 1 + 1 = 0, with a carry-over of 1 to the next (non-existent) column. Write down 0. ``` 1 1 0 12 + 1 1 02 ------- 0 1 12 ``` Since there's a carry-over of 1 and no further columns, bring down the carry-over 1. ``` 1 1 0 12 + 1 1 02 ------- 1 0 1 12 ``` Therefore, 1012 + 1102 =
1
0
1
1
2. Verification (optional, but good for understanding): 1012 = 1×22 + 0×21 + 1×20 = 4 + 0 + 1 = 510 1102 = 1×22 + 1×21 + 0×20 = 4 + 2 + 0 = 610 510 + 610 = 1110 10112 = 1×23 + 0×22 + 1×21 + 1×20 = 8 + 0 + 2 + 1 =
1
1
1
0. The answer is correct.
Example 2: Adding three 3-digit binary numbers Add 1112, 1012, and
0
1
1
2. Step 1: Arrange the numbers vertically, aligning by place value. ``` 1 1 12 1 0 12 + 0 1 12 ------- ``` Step 2: Start addition from the rightmost column (units place).
Rightmost column: 1 + 1 + 1 = 1 (with a carry-over of 1). Write down 1. ``` 1 (carry to next column) 1 1 12 1 0 12 + 0 1 12 ------- 12 ``` Step 3: Move to the next column to the left, including the carry-over.
Middle column (twos place): 1 (carry) + 1 + 0 + 1 = 1 (with a carry-over of 1). Write down 1. ``` 1 1 (carries) 1 1 12 1 0 12 + 0 1 12 ------- 1 12 ``` Step 4: Move to the leftmost column, including the new carry-over.
Leftmost column (fours place): 1 (carry) + 1 + 1 + 0 = 1 (with a carry-over of 1). Write down 1. ``` 1 1 1 (carries) 1 1 12 1 0 12 + 0 1 12 ------- 1 1 12 ``` * Since there's a carry-over of 1 and no further columns, bring down the carry-over 1. ``` 1 1 1 1 1 12 1 0 12 + 0 1 12 ------- 1 1 1 12 ``` Therefore, 1112 + 1012 + 0112 =
1
1
1
1
2. Verification: 1112 3.
1. Teacher Activities: Introduction (5 minutes): Begin by reviewing the concept of base 10 addition, particularly carrying over. Ask students to solve a simple base 10 addition problem (e.g., 37 + 45) to reactivate prior knowledge. Briefly reintroduce the binary system (base 2) and its digits (0, 1), emphasizing its importance in digital technology. Explanation of Binary Addition Rules (10 minutes): Present the basic binary addition rules (0+0, 0+1, 1+0, 1+1, 1+1+1) clearly on the board. Explain the concept of "carrying over" in base 2 (e.g., 1+1 = 102, where 0 is written down and 1 is carried). Relate it to carrying over 10 in base
1
0. Demonstration with Worked Examples (15 minutes): Work through "Example 1" (adding two 3-digit binary numbers) step-by-step on the board, vocalizing each step and carry. Encourage questions. Work through "Example 2" (adding three 3-digit binary numbers) step-by-step, again vocalizing and emphasizing the handling of double carries (e.g., 1+1+1). Use different coloured markers for carries to enhance clarity.
Guided Practice (15 minutes): Provide 2-3 new practice problems on the board. Ask students to attempt them individually or in pairs. Circulate around the classroom, providing immediate feedback and correcting misconceptions. Select students to demonstrate their solutions on the board, ensuring they explain their steps.
Consolidation and Summary (5 minutes): Reiterate the key rules of binary addition and the importance of careful alignment and handling of carries. Briefly connect the topic back to its real-world relevance in computers and digital devices. 3.
2. Student Activities: Actively participate in the review of base 10 addition. Listen attentively to explanations of binary addition rules and examples, taking notes. Ask clarifying questions when concepts are unclear. Attempt guided practice problems individually or with peers. Present solutions to problems on the board, explaining their reasoning. Engage in class discussions. The following practice questions are designed to reinforce the addition of two or three 3-digit binary numbers.
Question 1: Add 1102 and
1
0
1
2. Solution 1: ``` 1 1 02 + 1 0 12 ``` Units column: 0 + 1 = 1 Twos column: 1 + 0 = 1 Fours column: 1 + 1 = 0, carry 1 Carried 1: Bring down the carry. ``` 1 1 1 02 + 1 0 12 1 0 1 12 ``` Answer: 10112
Commentary: This problem practices the basic 1+1 carry operation in a two-number addition.
Question 2: Find the sum of 1002 and
1
1
1
2. Solution 2: ``` 1 0 02 + 1 1 12 ``` Units column: 0 + 1 = 1 Twos column: 0 + 1 = 1 Fours column: 1 + 1 = 0, carry 1 Carried 1: Bring down the carry. ``` 1 1 0 02 + 1 1 12 1 0 1 12 ``` Answer: 10112
Commentary: This problem reinforces carrying over, even with zeros in the original numbers.
Question 3: Add 1102, 0112, and
1
0
1
2. Solution 3: ``` 1 1 02 0 1 12 + 1 0 12 ``` Units column: 0 + 1 + 1 = 102 (0, carry 1)
Twos column (with carry): 1 (carry) + 1 + 1 + 0 = 112 (1, carry 1)
Fours column (with carry): 1 (carry) + 1 + 0 + 1 = 112 (1, carry 1)
Carried 1: Bring down the carry. ``` 1 1 1 1 1 02 0 1 12 + 1 0 12 1 1 1 02 ``` Answer: 11102
Commentary: This problem is more challenging, involving multiple carries and the 1+1+1 rule. Careful tracking of carries is essential. ---
Understanding binary addition has several practical implications and connections within the Nigerian context: Digital Communication and Mobile Phones: Every Nigerian using a mobile phone, from making calls to sending SMS messages or browsing the internet, is interacting with systems that fundamentally operate using binary arithmetic. When a phone adds two numbers, calculates data usage, or processes an image, it performs binary addition at its core. This knowledge helps demystify the "black box" of technology that is integral to daily life, especially with the prevalence of mobile money and e-commerce across Nigeria.
Computer Systems and Data Processing: Whether it's a computer used in a government office for record-keeping, a student's laptop for assignments, or the servers processing transactions in Nigerian banks, all digital computations (addition, subtraction, logic operations) are executed using binary numbers. Learning binary addition provides a foundational understanding of how these machines process information, which is critical for anyone considering a career in IT, engineering, or even just being a digitally literate citizen in Nigeria.
Basic Programming and Logic Circuits: For students interested in computer programming or electronics, understanding binary addition is a gateway. Simple logical operations in programming languages or the design of basic electronic circuits (e.g., adders) directly depend on these binary principles. This concept forms an essential prerequisite for higher-level studies in STEM fields that are vital for Nigeria's technological advancement and economic diversification.