Lesson 3: Binary Operations

👨🏽‍🏫 Introduction

Imagine you’re playing a game with two buttons 🔘🔘 — every time you press one, something combines with something else to make a result. That’s the idea behind a binary operation — it takes two inputs and combines them using a specific rule (like +, −, ×, or even a made-up rule like “add and then multiply by 2”).

Binary operations show up in real life too — for example, calculating your total bill after a discount, combining two scores, or programming logic in computers!

By the end of this lesson, you’ll be able to:

• Understand what a binary operation is
• Apply binary operations to numbers using given rules
• Interpret and solve problems using custom symbols like ★ or ⊗

🧠 Key Concepts

📌 1. What Is a Binary Operation?

A binary operation is a rule or process that takes two inputs (usually numbers) and gives one result.

It is written using a symbol between two elements:

Here, * is the binary operation, and a and b are the inputs.

Examples of standard binary operations:

But we can also define new operations like:

If , then

📏 2. Custom Binary Operations

In WASSCE-style questions, binary operations often use made-up symbols like:

You are expected to follow the rule exactly as given. Don’t assume it behaves like normal math!

🎯 3. Important Reminders

• Always check the rule — it tells you what to do with the two numbers.
• Order matters — in many cases, .
• Brackets matter — always simplify the inside operation first if nested.

🧮 Sample Problem Walkthroughs

1. Given:
   Find
   Step 1: Follow the rule →
   Step 2: Substitute a = 3 and b = 4 →
   Answer: 11
2. Given:
   Find
   Step 1: Use rule: square a, then subtract b →
   Answer: -1
3. Given:
   Find
   Step 1: Multiply a and b →
   Step 2: Add a →
   Answer: 9

✍🏽 Practice Exercises

🅰️ A. Apply the given binary operation rules:

1. If , find
2. If , find
3. If , find

🅱️ B. Mixed Binary Operation Reasoning

1. If , find
2. Using , evaluate

✅ Answers and Explanations

• A1:
• A2:
• A3:

• B1:
• B2:

📌 Recap

• ✅ A binary operation combines two elements into one result using a rule
• ✅ Follow the rule exactly — don’t guess or assume it means regular addition or multiplication
• ✅ Use substitution, order of operations, and brackets carefully

💭 Reflection Prompt

Can you invent your own binary operation using a fun symbol (like ⚡ or 🌟)? Just choose a rule for what the operation does. For example, if you define a new operation ⚡ where:

Then

Now try your invented operation on a few numbers. Would your classmates understand the rule just by reading how it works?