Ever sorted contacts on your phone, compared shop prices, or tracked daily temperatures? These everyday tasks all involve ordering numbers. In maths, this means putting them in sequence from smallest to largest (ascending) or largest to smallest (descending).
This guide will help you get to grips with ordering all sorts of numbers and unlock the meaning of important inequality symbols like $<$, $>$, $\le$, and $\ge$.
Understanding the Core Concepts
The Mighty Number Line
The number line is a visual ruler for all numbers. The key rule is: as you move right, numbers get larger. As you move left, numbers get smaller.
This is especially useful for negative numbers. For example, -5°C is colder (smaller) than -2°C, and on the number line, -5 is to the left of -2.
How to Order Different Number Types
Ordering Decimals
The secret is to compare them place value by place value from the left. To make it easier, pad the numbers with zeros so they have the same number of decimal places.
Example: Order 0.5, 0.25, 0.3
| Original | Padded |
|---|---|
| 0.25 | 0.25 |
| 0.3 | 0.30 |
| 0.5 | 0.50 |
Comparing the padded numbers, the correct ascending order is 0.25, 0.3, 0.5.
Ordering Fractions
The most reliable way is to find a common denominator. This means making the bottom number of all the fractions the same.
Example: Order $\frac{1}{2}, \frac{3}{4}, \frac{2}{3}$
- The lowest common multiple of 2, 4, and 3 is 12.
- Convert the fractions: $\frac{1}{2} = \frac{6}{12}$, $\frac{3}{4} = \frac{9}{12}$, $\frac{2}{3} = \frac{8}{12}$.
- Compare the numerators: 6, 8, 9.
The correct ascending order is $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$.
Using Inequality Symbols
These symbols tell us the relationship between two numbers. A great memory tip is to think of the symbol as a crocodile’s mouth that always wants to eat the bigger number!
| Symbol | Meaning | Example |
|---|---|---|
| $<$ | Less than | $3 < 5$ |
| $>$ | Greater than | $7 > 4$ |
| $\le$ | Less than or equal to | $x \le 10$ |
| $\ge$ | Greater than or equal to | $y \ge -2$ |
Tutor Insights
🤔 Common Misunderstandings
- Negative Numbers: Thinking -10 is greater than -5. Always use a number line or think about temperature to remember it’s the other way around!
- Fractions: Forgetting to find a common denominator before comparing.
- Decimals: Not comparing place value correctly (e.g., thinking 0.5 is smaller than 0.25).
📝 Common Exam Mistakes
- Mixing up ascending/descending. Always double-check what the question is asking for.
- Not converting to a common format when ordering a mixed list of numbers.
- Failing to show working, especially when finding a common denominator for fractions.
Practice Questions
- Order the following integers in ascending order: 7, -1, 0, -6, 3.
- Place the correct inequality symbol ($<$, $>$, or $=$) between: a) -8 ___ -2, b) 0.5 ___ 0.50.
- Order the following decimals in descending order: 0.8, 0.08, 0.88, 0.808.
- Order the following numbers in ascending order: -0.25, $\frac{1}{5}$, -1, 0.3.
- Which is smaller, $\frac{7}{10}$ or 0.75? Show your working.
Show Answers
- -6, -1, 0, 3, 7
- a) -8 < -2, b) 0.5 = 0.50
- 0.88, 0.808, 0.8, 0.08
- -1, -0.25, $\frac{1}{5}$, 0.3 (Tip: Convert $\frac{1}{5}$ to 0.2 to compare)
- $\frac{7}{10}$ is smaller. (Working: $\frac{7}{10} = 0.7$. Comparing 0.70 and 0.75, 0.70 is smaller).
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