Lesson Notes By Weeks and Term v3 - Junior Secondary 1
Addition of numbers in base 2 numerals.
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Addition of numbers in base 2 numerals.
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: General Mathematics
Class: Junior Secondary 1
Term: 3rd Term
Week: 6
Theme: Basic Operations
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Students should be able to Add two or three 3-digits binary numbers.
This section outlines the core principles and steps required for adding numbers in base
2. A.
Understanding Base 2 (Binary)
Numbers: The binary number system uses only two digits: 0 and
1. Each position in a binary number represents a power of 2, starting from 2^0 (which is 1) from the rightmost digit.
Example: The binary number `1101_2` can be converted to base 10 as follows: `1 2^3 + 1 2^2 + 0 2^1 + 1 2^0` `= 1 8 + 1 4 + 0 2 + 1 1` `= 8 + 4 + 0 + 1 = 13_10`
B. Basic Rules of Binary Addition: Binary addition follows specific rules, similar to decimal addition but limited to the digits 0 and
1. The concept of "carrying over" is vital. 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 10_2 (This means 0 with a carry-over of 1 to the next left position). In base 10, this is 2. 1 + 1 + 1 = 11_2 (This means 1 with a carry-over of 1 to the next left position). In base 10, this is
3. This rule is especially important when adding three binary numbers or when multiple carries occur.
C. Step-by-Step Addition Process: To add binary numbers, arrange them vertically, aligning the digits by their place values, just like in decimal addition. Begin addition from the rightmost column (least significant digit). Worked
Examples: Example 1: Adding two 3-digit binary numbers (without a carry beyond the initial digits) Calculate `101_2 + 010_2` Step 1: Arrange vertically. ``` 1 0 1_2 + 0 1 0_2 ``` Step 2: Add the rightmost column (units column). `1 + 0 = 1` ``` 1 0 1_2 + 0 1 0_2 1 ``` Step 3: Add the middle column. `0 + 1 = 1` ``` 1 0 1_2 + 0 1 0_2 1 1 ``` Step 4: Add the leftmost column. `1 + 0 = 1` ``` 1 0 1_2 + 0 1 0_2 1 1 1_2 ``` Result: `101_2 + 010_2 = 111_2` Example 2: Adding two 3-digit binary numbers (with carrying) Calculate `110_2 + 101_2` Step 1: Arrange vertically. ``` 1 1 0_2 + 1 0 1_2 ``` Step 2: Add the rightmost column. `0 + 1 = 1` ``` 1 1 0_2 + 1 0 1_2 1 ``` Step 3: Add the middle column. `1 + 0 = 1` ``` 1 1 0_2 + 1 0 1_2 1 1 ``` Step 4: Add the leftmost column. `1 + 1 = 10_2`. Write down 0, carry over 1 to the next column (which is effectively a new column to the left). ``` 1 1 1 0_2 + 1 0 1_2 1 0 1 1_2 ``` Result: `110_2 + 101_2 = 1011_2` Example 3: Adding three 3-digit binary numbers (with multiple carrying) Calculate `101_2 + 110_2 + 011_2` Step 1: Arrange vertically. ``` 1 0 1_2 1 1 0_2 + 0 1 1_2 ``` Step 2: Add the rightmost column. `1 + 0 + 1 = 10_2`. Write down 0, carry over 1. ``` 1 (carry) 1 0 1_2 1 1 0_2 + 0 1 1_2 0 ``` Step 3: Add the middle column, including the carry. `0 + 1 + 1 + 1(carry) = 1 + 1 = 10_2`. Write down 0, carry over 1. ``` 1 1 (carries) 1 0 1_2 1 1 0_2 + 0 1 1_2 0 0 ``` Step 4: Add the leftmost column, including the carry. `1 + 1 + 0 + 1(carry) = 10_2 + 1 = 11_2`. Write down 1, carry over 1. ``` 1 1 1 (carries) 1 0 1_2 1 1 0_2 + 0 1 1_2 1 1 0 0_2 ``` Result: `101_2 + 110_2 + 011_2 = 1100_2` --- This section outlines practical activities for both the teacher and students to facilitate understanding and mastery of binary addition.
A. Teacher Activities: Introduction (5 minutes): Begins by briefly revising the concept of number bases, particularly base 10 and base
2. Asks students for examples of situations where numbers other than base 10 might be used (e.g., computers).
States the lesson topic: "Addition of numbers in base 2 numerals." Clearly states the learning objectives for the lesson. Concept Explanation and Demonstration (15 minutes): Explains the rules of binary addition (0+0=0, 0+1=1, 1+0=1, 1+1=10_2, 1+1+1=11_2) using a whiteboard or chart. Demonstrates the addition process with two clear, step-by-step examples on the board, starting with two 3-digit numbers, involving carrying. Uses the examples from Section
2. Emphasises the concept of "carry-over" and how it differs from base 10 carrying. Demonstrates an example of adding three 3-digit binary numbers, highlighting the 1+1+1=11_2 rule. Interactive Practice and Questioning (10 minutes): Provides a simple binary addition problem (e.g., two 2-digit numbers) and invites individual students to come to the board to solve it, guiding them through each step. Asks probing questions to check understanding: "Why did we write 0 and carry 1 here?", "What does '10 base 2' mean in base 10?" Encourages class participation in verifying the steps.
Group Work Instruction (5 minutes): Divides the class into small groups (e.g., 4-5 students per group). Distributes practice questions (from Guided Practice) to each group. Instructs groups to work collaboratively, discussing each step and arriving at a common solution. Moves around the classroom, observing group discussions, providing hints, and clarifying misconceptions.
Review and Consolidation (5 minutes): Selects a few groups to present their solutions to the class. Corrects any errors identified during presentations and reiterates key rules. Summarises the main points of binary addition.
B. Student Activities: Active Listening and Participation: Listens attentively to the teacher's explanation and demonstration. Asks questions when concepts are unclear. Volunteers to solve problems on the board when called upon.
Collaborative Learning (Group Work): Works effectively in assigned groups to solve practice problems. Discusses strategies and helps group members understand the concepts. Presents group solutions to the class.
Note-Taking: Copies important rules and worked examples from the board into their notebooks.
Independent Practice: Completes independent practice exercises assigned by the teacher to reinforce understanding. --- These questions are designed to be worked through with teacher guidance or in collaborative groups, reinforcing the concepts learned.
Question 1: Add the binary numbers: `111_2 + 001_2` Solution 1: ``` 1 (carry) 1 1 1_2 + 0 0 1_2 1 0 0 0_2 ``` Rightmost column: `1 + 1 = 10_2`. Write down 0, carry over
1. Middle column: `1 + 0 + 1(carry) = 1 + 1 = 10_2`. Write down 0, carry over
1. Leftmost column: `1 + 0 + 1(carry) = 1 + 1 = 10_2`. Write down 0, carry over
1. New column (left of leftmost): Write down the final carry of
1. Result: `1000_2`
Commentary: This example demonstrates how carrying propagates across multiple columns, resulting in a larger binary number.
Question 2: Find the sum of `101_2` and `111_2`.
Solution 2: ``` 1 1 (carries) 1 0 1_2 + 1 1 1_2 1 1 0 0_2 ``` Rightmost column: `1 + 1 = 10_2`. Write down 0, carry over
1. Middle column: `0 + 1 + 1(carry) = 1 + 1 = 10_2`. Write down 0, carry over
1. Leftmost column: `1 + 1 + 1(carry) = 10_2 + 1 = 11_2`. Write down 1, carry over
1. New column (left of leftmost): Write down the final carry of
1. Result: `1100_2`
Commentary: This problem further solidifies the carrying concept, including a scenario where the sum of a column plus a carry results in `11_2`.
Question 3: Add the three binary numbers: `100_2 + 011_2 + 111_2` Solution 3: ``` 1 1 1 (carries) 1 0 0_2 0 1 1_2 + 1 1 1_2 1 1 1 0_2 ``` Rightmost column: `0 + 1 + 1 = 10_2`. Write down 0, carry over
1. Middle column: `0 + 1 + 1 + 1(carry) = 1 + 1 + 1 = 11_2`. Write down 1, carry over
1. Leftmost column: `1 + 0 + 1 + 1(carry) = 1 + 1 + 1 = 11_2`. Write down 1, carry over
1. New column (left of leftmost): Write down the final carry of
1. Result: `1110_2`
Commentary: This example explicitly applies the `1 + 1 + 1 = 11_2` rule in the middle and leftmost columns, demonstrating addition of three binary numbers with multiple carries. --- Strategies to cater to the diverse learning needs within a typical Nigerian classroom.
A. Differentiation for Struggling Learners (Remediation): Visual Aids and Manipulatives: Use physical objects like pebbles, bottle caps, or drawing dots to represent 0s and 1s. Students can physically move these to simulate addition and carrying (e.g., two pebbles in a column "carry over" as one pebble in the next column).
Repetitive Practice with Small Numbers: Start with adding two 2-digit binary numbers, then gradually move to 3-digit numbers. Provide ample repetitive practice with step-by-step guidance.
Peer Tutoring: Pair struggling learners with more capable classmates for one-on-one support during practice sessions.
Simplified Worksheets: Provide worksheets that break down addition into individual column steps, prompting them to write down the carry-over explicitly. Re-explain the "Carry-Over" Concept: Relate the binary carry-over (`1+1=10_2`) explicitly to base 10 carry-over (e.g., `5+7=12_10` where 2 is written, and 1 (representing 10) is carried over). Emphasise that in binary, the '1' carried over represents 2, which is the base.
B. Extension/Enrichment for High-Achieving Learners: Adding Longer Binary Numbers: Challenge them to add binary numbers with more than three digits (e.g., four or five digits), or even four 3-digit binary numbers.
Binary Subtraction: Introduce the concept of binary subtraction (using complements or direct method) as a natural extension.
Conversion and Verification: Ask them to convert the binary numbers to base 10, perform the addition in base 10, and then convert the base 10 sum back to binary to verify their binary addition result.
Example: Add `101_2 + 110_2`. (Result `1011_2`).
Convert: `101_2 = 5_10`, `110_2 = 6_10`. `5_10 + 6_10 = 11_10`. Convert `11_10` to binary: `1011_2`.
Real-world Problem Solving: Present problems that require converting decimal numbers to binary, adding them, and then converting back (e.g., "A computer processes two values, 7 and
9. Show how it would add them in binary.").
Example 1: Adding two 3-digit binary numbers (without a carry beyond the initial digits)
Calculate `101_2 + 010_2`
Step 1: Arrange vertically.
```
1 0 1_2
+ 0 1 0_2
-------
```
Step 2: Add the rightmost column (units column).
`1 + 0 = 1`
```
1 0 1_2
+ 0 1 0_2
-------
1
```
Step 3: Add the middle column.
`0 + 1 = 1`
```
1 0 1_2
+ 0 1 0_2
-------
1 1
```
Step 4: Add the leftmost column.
`1 + 0 = 1`
```
1 0 1_2
+ 0 1 0_2
-------
1 1 1_2
```
Result: `101_2 + 010_2 = 111_2`
Example 2: Adding two 3-digit binary numbers (with carrying)
Calculate `110_2 + 101_2`
Step 1: Arrange vertically.
```
1 1 0_2
+ 1 0 1_2
-------
```
Step 2: Add the rightmost column.
`0 + 1 = 1`
```
1 1 0_2
+ 1 0 1_2
-------
1
```
Step 3: Add the middle column.
`1 + 0 = 1`
```
1 1 0_2
+ 1 0 1_2
-------
1 1
```
Step 4: Add the leftmost column.
`1 + 1 = 10_2`. Write down 0, carry over 1 to the next column (which is effectively a new column to the left).
```
¹
1 1 0_2
+ 1 0 1_2
-------
1 0 1 1_2
```
Result: `110_2 + 101_2 = 1011_2`
Example 3: Adding three 3-digit binary numbers (with multiple carrying)
Calculate `101_2 + 110_2 + 011_2`
Step 1: Arrange vertically.
```
1 0 1_2
1 1 0_2
+ 0 1 1_2
-------
```
Step 2: Add the rightmost column.
`1 + 0 + 1 = 10_2`. Write down 0, carry over 1.
```
¹ (carry)
1 0 1_2
1 1 0_2
+ 0 1 1_2
-------
0
```
Step 3: Add the middle column, including the carry.
`0 + 1 + 1 + ¹(carry) = 1 + 1 = 10_2`. Write down 0, carry over 1.
```
¹ ¹ (carries)
1 0 1_2
1 1 0_2
+ 0 1 1_2
-------
0 0
```
Step 4: Add the leftmost column, including the carry.
`1 + 1 + 0 + ¹(carry) = 10_2 + 1 = 11_2`. Write down 1, carry over 1.
```
¹ ¹ ¹ (carries)
1 0 1_2
1 1 0_2
+ 0 1 1_2
-------
1 1 0 0_2
```
Result: `101_2 + 110_2 + 011_2 = 1100_2`
Teaching and Learning Activities
Binary addition, and binary numbers in general, are fundamental to almost every aspect of modern digital technology. Connecting this to Nigerian contexts helps students see the relevance of abstract mathematical concepts.
Computer and Mobile Phone Operations: Every single piece of information processed by a computer, tablet, or mobile phone (from sending a WhatsApp message, browsing the internet, to calculating a spreadsheet) is represented and manipulated using binary digits (bits). Binary addition is the core arithmetic operation performed by the Central Processing Unit (CPU) for all calculations. For example, when a user in Lagos adds two numbers in a calculator app, the device internally converts these numbers to binary, performs binary addition, and then converts the result back to base 10 for display.
Digital Communication and Networking: Telecommunication networks (like MTN, Glo, Airtel in Nigeria) transmit voice, video, and data as binary signals. When multiple data streams merge (e.g., during a conference call or network routing), the underlying processes involve binary arithmetic, including addition, to manage and direct the data packets efficiently. Understanding binary helps appreciate how data travels across the country and globally. Financial Transactions (ATMs and Online Banking): When a Nigerian makes a deposit or withdrawal at an ATM, or performs an online bank transfer, the amounts are internally handled as binary numbers. The addition and subtraction of these amounts are binary operations. For example, depositing N5,000 to an account that had N10,000 involves the system performing `Balance (binary) + Deposit (binary)` to update the account, ensuring accuracy and security in financial dealings. ---