Lesson 5: Rational Numbers and the Number Line

👨🏽‍🏫 Introduction

Have you ever shared a pizza equally among friends 🍕, or checked a bank balance like -₵200? These involve rational numbers — numbers that include positive and negative fractions, decimals, and whole numbers.

Rational numbers help us measure, trade, balance, and solve everyday problems. In this lesson, we’ll explore what they are, how they’re placed on a number line, and how to compare or operate with them.

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

Understand what rational numbers are
Place and compare them on a number line
Add, subtract, multiply, and divide rational numbers

🧠 Key Concepts

📌 1. What Are Rational Numbers?

A rational number is any number that can be written as a fraction:

$png$ where p and q are integers and q ≠ 0.

This includes:

Whole numbers: $png$
Fractions: $png$
Decimals: 0.25, -1.75, $png.image?\dpi{120} 0$

Note: Every integer is a rational number, but not every rational number is a whole number.

📏 2. Rational Numbers on the Number Line

You can place rational numbers like fractions and negatives on a number line just like whole numbers.

Examples:

$png$ are all rational numbers you might see on a number line.

🔎 3. Comparing and Ordering Rational Numbers

To compare rational numbers:

Convert to decimals or use a common denominator.
Compare values or use a number line.

Examples:

Example 1:
$png.image?\dpi{120} \frac{1}{2} = 0$ ,
$png.image?\dpi{120} \frac{3}{4} = 0$ ,
so $png$

Example 2:
$png.image?\dpi{120} \frac{2}{3} = 0$ ,
$png.image?\dpi{120} \frac{1}{4} = 0$ ⇒
$png$

🧮 4. Operations with Rational Numbers

➕ Addition/Subtraction

Use common denominators
Apply sign rules

Example:
$png$

✖️ Multiplication

Multiply top × top and bottom × bottom

Example:
$png$

➗ Division

Flip the second fraction and multiply

Example:
$png$

💡 Real-Life Applications

📊 Loss of ₵250 = $png$
🏃🏽‍♂️ Running 1.5 km = $png$ km
🍕 3 pizzas shared among 4 friends = $png$ pizza each

🧮 Sample Problem Walkthroughs

Compare: $png.image?\dpi{120} \frac{5}{8} = 0$ ,
$png.image?\dpi{120} \frac{3}{4} = 0$
⇒
$png$
Find: $png$ ,
so
$png$
Multiply: $png$
Divide: $png$