binary arithmetic
4 detailed 50-minute lessons with teaching scripts, worked examples, parent guides, and assessment criteria.
Lesson Overview
Total Lessons: 4
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Tier: Foundation and Higher
Duration: 50 minutes per lesson (200 minutes total)
Exam Boards: AQA, Edexcel, OCR, Eduqas, CCEA
Learning Objectives
- Explain the key ideas of binary arithmetic
- Apply binary arithmetic to exam-style questions
- Develop independent study skills
- Build cross-topic connections
Prerequisites
- Read the revision notes for binary arithmetic first (GCSE Revise summary)
- Key vocab to pre-teach: Binary addition, Overflow, Detecting overflow
- Basic skills: reading the summary notes and answering the practice questions there
Materials & Equipment
- Exercise book, coloured pens
- Ruler
- Printed revision notes (link below)
- Internet for videos (see Resources)
Lesson 1: Introduction: binary arithmetic
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Write down everything you already know about binary arithmetic. Then check against the key terms: Binary addition, Overflow, Detecting overflow. Use a mini-whiteboard or paper.
Main Content (35 minutes)
Parent/Teacher Guide:
Before lesson: Read the script below. Pre-teach key vocab: Binary addition, Overflow, Detecting overflow.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Before lesson: Read the script below. Pre-teach key vocab: Binary addition, Overflow, Detecting overflow.
If stuck: Re-read the revision notes (link above), then break the content into smaller steps.
Extension: See the Stretch & Challenge ideas in Lesson 4.
Teaching Script (35 mins):
Mins 0-5 - Hook: "Today: binary arithmetic. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Computer Science because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
Mins 0-5 - Hook: "Today: binary arithmetic. By the end you will be able to answer exam questions on it unaided. It connects to the rest of Computer Science because the ideas here recur across the spec."
Mins 5-20 - Direct Instruction: Work through the core ideas below one at a time; after each, ask your student to explain it back in their own words.
Mins 20-30 - Guided Practice: Model the worked example together, then let your student attempt the first practice question with guidance.
Mins 30-35 - Independent Practice: 2-3 practice questions from Lesson 3 below, with immediate feedback.
First Look
Start with the revision notes summary, then attempt: Add the binary numbers 01011010 and 00110101. Show your working.
Plenary (5 minutes)
Check Out
Your student states one thing they learned and one question they still have about binary arithmetic.
Lesson 2: Core Concepts: binary arithmetic
Duration: 50 minutes
Starter Activity (5 minutes)
Review Previous Lesson
Quick recap: write 3 key points from Lesson 1 on binary arithmetic. Check them against the notes below.
Main Content (35 minutes)
Binary addition: follows the same principles as decimal addition but with only two digits (0 and 1). There are only four rules to remember.
Overflow: occurs when the result of a binary addition requires more bits than are available. If adding two 8-bit numbers produces a 9th bit (a carry out from the leftmost column), the result cannot be stored correctly in 8 bits. This is an overflow error.
Detecting overflow: If there is a carry out from the most significant bit (leftmost column), overflow has occurred. The stored result will be incorrect. In programming, you need to check for this condition.
To add more than two binary numbers,: add them two at a time. Add the first two, then add the third to that result, and so on. Check for overflow at each step.
A binary shift left: moves all bits one position to the left. A 0 fills the empty position on the right. The leftmost bit is discarded. Shifting left by 1 position multiplies the number by 2 . Shifting left by n positions multiplies by 2^n.
Warning: If a 1 bit is shifted out from the leftmost position, data is lost. This is similar to overflow. For example, shifting 10110000 left by 1 would lose the leading 1.
| Term | Meaning | Example |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 1 |
Practice (10 minutes)
Q: Add the binary numbers 01011010 and 00110101. Show your working.
Answer: Carry: 0 0 1 1 0 0 0 0 Num A: 0 1 0 1 1 0 1 0 (90) Num B: 0 0 1 1 0 1 0 1 (53) ----------------------- Result: 1 0 0 0 1 1 1 1 (143)
Plenary (5 minutes)
Explain Back
Your student teaches the key points back to you without looking. Fill any gaps immediately.
Lesson 3: Application: binary arithmetic
Duration: 50 minutes
Starter Activity (5 minutes)
Quick Recall
Recall the key terms: Binary addition, Overflow, Detecting overflow. Define each in one sentence.
Main Content (35 minutes)
Parent/Teacher Guide: Let your student attempt each question alone first, then compare with the model answer. Award method marks for correct working even if the final answer is wrong.
Q1: Add the binary numbers 01011010 and 00110101. Show your working.
Answer: Carry: 0 0 1 1 0 0 0 0 Num A: 0 1 0 1 1 0 1 0 (90) Num B: 0 0 1 1 0 1 0 1 (53) ----------------------- Result: 1 0 0 0 1 1 1 1 (143)
Q2: Add 10110011 and 01101010. Does overflow occur? Explain why.
Answer: Num A: 1 0 1 1 0 0 1 1 (179) Num B: 0 1 1 0 1 0 1 0 (106) Result would be 100101101 (285), but this needs 9 bits. With only 8 bits, we get 00101101 (45). Overflow occurs because 179 + 106 = 285, which exceeds the maximum 8-bit value of 255. The carry out from the leftmost column indicates overflow.
Q3: Perform a binary shift left by 2 on 00101100. What is the result in decimal?
Answer: Original: 00101100 (44). Shift left 1: 01011000 (88). Shift left 2: 10110000 (176). Result = 176. Check: 44 x 2^2 = 44 x 4 = 176.
Q4: Perform a binary shift right by 1 on 01101010. What is the result in decimal?
Answer: Original: 01101010 (106). Shift right 1: 00110101 (53). Result = 53. Check: 106 / 2 = 53.
Q5: What mathematical operation does a shift left by 3 perform?
Answer: A shift left by 3 multiplies the number by 2^3 = 8. For example, 00000101 (5) shifted left by 3 becomes 00101000 (40), and 5 x 8 = 40.
Plenary (5 minutes)
Error Review
Review any questions answered incorrectly. Identify whether the error was knowledge, method, or reading the question.
Lesson 4: Exam Practice: binary arithmetic
Duration: 50 minutes
Starter Activity (5 minutes)
Command Words
Review what these command words require: state (one point), describe (say what happens), explain (say why), compare (both sides), evaluate (judgement).
Main Content (35 minutes)
Extended Answer
Extended question: Full-Mark Response Perform the binary addition: 10110110 + 01101001. Show your working including any carries, and state whether overflow has occurred in an 8-bit system. [4 marks] <div class="
10110110 + 01101001 -------- 100011111 Working right to left: 0+1=1, 1+0=1, 1+0=1, 0+1=1, 1+0=1, 1+1=0 carry 1, 0+1+1=0 carry 1, 1+0+1=0 carry 1 Result: 100011111 (9 bits) Overflow has occurred because the result requires 9 bits but an 8-bit register can only hold 8 bits. The 9th bit (carry out) is lost, and the stored result would be 00011111 = 31, which is incorrect.
Exam Tips: Write binary additions in columns, working from right to left | Always show your carry bits - examiners award marks for working | Check for overflow: if there is a carry out from the MSB (most significant bit), overflow has occurred | Shift left = multiply by 2^n; shift right = integer divide by 2^n | When shifting, lost bits are discarded (not wrapped around) | A shift right on an odd number will lose the remainder
Common Errors: ✗ Forgetting to carry when binary addition column sum exceeds 1 ✓ In binary: 1+1 = 10 (write 0, carry 1). 1+1+1 = 11 (write 1, carry 1). Always carry just like decimal addition. ✗ Confusing binary shift direction — left vs right ✓ Left shift multiplies by 2 (fills with 0 on right). Right shift divides by 2 and discards the remainder (fills with 0 on left for positive numbers). ✗ Thinking overflow only occurs in very large numbers ✓ Overflow occurs whenever the result exceeds the available number of bits. Even small numbers like 11111111 + 1 = 100000000 overflow an 8-bit register. ✗ Losing bits during a right binary shift and not recording them ✓ A right shift discards the least significant b
Stretch & Challenge (Grade 8-9):
- Synoptic links: explain how binary arithmetic connects to another Computer Science topic you have studied
- Real-world: research one real-world use or example of binary arithmetic
- Critical: "What are the limitations of the models used in binary arithmetic?"
Plenary (5 minutes)
Assessment Criteria
- Got it: Confident explanation + correct worked examples
- Getting there: Main points OK, needs support with detail
- Not yet: Confused on key concepts - re-run Lesson 2
Homework & Consolidation
- Consolidation: Re-answer any Lesson 3 practice questions answered incorrectly (20 mins)
- Retrieval: Write flashcards for the key terms: Binary addition, Overflow, Detecting overflow (10 mins)
- Exam practice: One past-paper question on binary arithmetic from the board websites (15 mins)
- Extension: Explain binary arithmetic to someone else in your own words (10 mins)