N1: Ordering Numbers
Order positive and negative integers, decimals and fractions; use the symbols =, ≠, <, >, ≤, ≥
📋 Key Concepts
Definition: Ordering numbers means arranging them from smallest to largest (ascending) or largest to smallest (descending).
Comparison Symbols
| Symbol | Meaning | Example |
|---|---|---|
| = | Equal to | 5 = 5 |
| ≠ | Not equal to | 3 ≠ 7 |
| < | Less than | 2 < 8 |
| > | Greater than | 9 > 4 |
| ≤ | Less than or equal to | x ≤ 10 |
| ≥ | Greater than or equal to | x ≥ 0 |
Number Line
Negative numbers are smaller than positive numbers. The further left on a number line, the smaller the value.
... -3, -2, -1, 0, 1, 2, 3 ...
(left = smaller, right = larger)
(left = smaller, right = larger)
📝 Ordering Integers
Rule: Negative numbers are always smaller than positive numbers. The larger the negative number (ignoring the minus sign), the smaller its value.
Example 1
Order from smallest to largest: -5, 3, -1, 0, 2
Solution: -5, -1, 0, 2, 3
Explanation: -5 is furthest left on the number line, so it is the smallest.
Example 2
Which is smaller: -7 or -3?
Solution: -7 is smaller than -3
Explanation: -7 is further left on the number line.
📝 Ordering Decimals
Method: Compare digits from left to right, starting with the highest place value. Use place value columns if needed.
Example 3
Order from smallest to largest: 0.45, 0.5, 0.398, 0.42
Solution: 0.398, 0.42, 0.45, 0.5
Tip: Add zeros to make them the same length: 0.450, 0.500, 0.398, 0.420
📝 Ordering Fractions
Method 1: Convert all fractions to have the same denominator (common denominator)
Method 2: Convert all fractions to decimals
Method 2: Convert all fractions to decimals
Example 4
Order from smallest to largest: 3/4, 2/5, 1/2
Solution (using common denominator):
Common denominator = 20
3/4 = 15/20, 2/5 = 8/20, 1/2 = 10/20
Order: 2/5, 1/2, 3/4
Example 5
Order from smallest to largest: 2/3, 5/8, 7/10
Solution (using decimals):
2/3 ≈ 0.667, 5/8 = 0.625, 7/10 = 0.7
Order: 5/8, 2/3, 7/10
📝 Mixed Numbers
Tip: Convert mixed numbers to improper fractions or decimals for easier comparison.
Example 6
Order from smallest to largest: 2 1/4, 2.5, 2 3/8
Solution:
2 1/4 = 2.25, 2.5 = 2.5, 2 3/8 = 2.375
Order: 2 1/4, 2 3/8, 2.5
🧮 Worked Examples — Exam Style
Ordering questions look easy and are where careless marks die. Two habits fix it: convert everything to the same form, and remember that with negatives, the bigger-looking number is smaller.
Write in order, smallest first: −3.5, −3.05, 3.05, −3
Negatives first (smallest = most negative): −3.5 < −3.05 < −3, then positives last.
Answer: −3.5, −3.05, −3, 3.05
B1 all four correct. Trap: −3.05 looks smaller than −3.5 but sits closer to zero, so it is LARGER.
Write in order of size, smallest first: 7/8, 0.9, 4/5, 0.85
Convert all to decimals: 7/8 = 0.875, 0.9, 4/5 = 0.8, 0.85
Order: 0.8 < 0.85 < 0.875 < 0.9
Answer: 4/5, 0.85, 7/8, 0.9
M1 for converting fractions to decimals (or all to fractions with a common denominator), A1 for the correct order.
Exam wording to expect: "Write … in order of size, starting with the smallest" — nearly always 1 mark, no working shown in the mark scheme but you may still use the margin.
❓ Practice Questions
Q1: Order from smallest to largest: -8, 4, -2, 0, 6
Q2: Order from smallest to largest: 0.7, 0.08, 0.71, 0.8
Q3: Order from smallest to largest: 1/2, 2/5, 3/10
Q4: Which symbol makes this correct? -5 ___ -2
Q5: Order from smallest to largest: 1.5, 1 1/3, 1.25, 1 3/4
✅ Answers
- -8, -2, 0, 4, 6
- 0.08, 0.7, 0.71, 0.8
- 3/10, 2/5, 1/2 (same as: 0.3, 0.4, 0.5)
- -5 < -2 (negative 5 is smaller)
- 1.25, 1 1/3, 1.5, 1 3/4
🧮 Worked Examples — Harder Exam Style
Write these in order of size, smallest first: −0.4, −0.45, 0.4, −0.04
All the negatives come first, ordered by distance from zero: −0.45 (furthest) then −0.4, then −0.04.
Positives last: 0.4
Answer: −0.45, −0.4, −0.04, 0.4
B1 for all four in order. The −0.04 vs −0.4 pair is the designed trap: −0.04 is closer to zero, so it is LARGER than −0.4.
📋 Model Answers — Full Working
Every mark in GCSE maths comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. Most negative is smallest: −8, −2, 0, 4, 6
Q2. Compare place by place: 0.08 (8 hundredths) < 0.7 < 0.71 < 0.8 → 0.08, 0.7, 0.71, 0.8
Q3. Convert to /10: 1/2 = 5/10, 2/5 = 4/10, 3/10 → 3/10, 2/5, 1/2
Q4. −5 is further from zero on the negative side, so −5 < −2
Q5. Convert to decimals: 1.5, 1.333…, 1.25, 1.75 → 1.25, 1⅓, 1.5, 1¾
Q6. −1.5, −1, −0.5, 0
Q7. 2/3 = 0.666… (recurring), which is greater than 0.66, so 2/3 is larger. Write 0.666… or 0.6̇ to justify — a bare "2/3" may not earn the mark.
📌 Key Facts to Memorise
- On a number line, LEFT = smaller — always
- With negatives: −8 < −2 (the further from zero, the smaller)
- Compare decimals place by place: 0.08 < 0.7 (8 hundredths < 7 tenths)
- Mixed forms: convert everything to decimals (or a common denominator) first
- Useful conversions: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125
🗺️ In the Exam
Ordering is a 1–2 mark question that opens many Paper 1 papers. It also feeds inequality number lines and estimating. Getting it wrong is expensive relative to the effort.
Links to: inequalities (A22), decimals-fractions (N10), negative numbers (N2).
⚡ Targeted Drills
Thirty-second checks — cover the answers, work each one, then mark yourself. If any takes longer than a minute, revisit the matching section above.
- Order: −6, −6.5, 6, −0.6 → −6.5, −6, −0.6, 6
- Fill in the symbol: −3 __ −7 → > (−3 is closer to zero)
- Order: 3/4, 0.72, 7/10, 0.075 → 0.075, 7/10, 0.72, 3/4
- Which is smaller: −0.09 or −0.1? → −0.1 (further from zero)
- Convert then order: 2/5, 0.45, 9/20 → 0.4, 0.45, 0.45 = 9/20 — 2/5 < 0.45 = 9/20
🧭 Board Focus — OCR
Spec J560. Three papers of 100 marks each: Paper 1 non-calculator, Papers 2 and 3 calculator, at Foundation/Higher tiers. OCR marks problem-solving explicitly, and its papers tend to front-load multi-step worded contexts — practise extracting the mathematics from prose before calculating.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always check if you need ascending (smallest to largest) or descending (largest to smallest)
- For decimals, add zeros to make all numbers the same length
- For fractions, convert to decimals if the denominators are hard to compare
- Draw a number line for negative numbers if you get confused
- Remember: -10 is smaller than -1
🧠 Problem-Solving Strategies
Problem-Solving
When ordering a mix of fractions, decimals and negatives: convert everything to the same format first. Decimals are usually easiest for comparison. For negatives, remember the number line — further left means smaller. Always re-read the question to check if ascending or descending order is needed.
Multi-Step Problem
Three students scored 5/8, 0.62 and 61% in a test. Put their scores in order from lowest to highest.
Solution: Convert all to decimals: 5/8 = 0.625, 0.62 = 0.62, 61% = 0.61. So 0.61, 0.62, 0.625. Order: 61%, 0.62, 5/8.
⚠️ Common Errors
Watch Out!
1. Wrong: -7 is bigger than -3 Correct: -7 is smaller than -3 (it is further left on the number line)
2. Wrong: 0.45 is bigger than 0.5 because 45 > 5 Correct: 0.5 = 0.50, so 0.45 < 0.50 — add zeros to compare
3. Wrong: 3/5 is bigger than 2/3 because 3 > 2 and 5 > 3 Correct: Convert to common denominator: 3/5 = 9/15, 2/3 = 10/15, so 2/3 is bigger
✍️ 6-Mark Exam Question
Extended Answer
6 marks: A race has three runners with finish times of 9.87 s, 9 4/5 s and 9.79 s. (a) Convert all times to the same format. (b) State who finished first, second and third. (c) What is the time difference between first and last place?
(a) 9 4/5 = 9.8 s. Times in decimals: 9.87 s, 9.8 s, 9.79 s.
(b) Ascending order: 9.79 s (1st), 9.8 s (2nd), 9.87 s (3rd).
(c) Difference = 9.87 − 9.79 = 0.08 s.
Mark scheme: 2 marks for correct conversions, 2 marks for correct ordering, 2 marks for correct difference with working
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A scientist measures three substances. Substance A has a temperature of −18°C, Substance B has −23°C, and Substance C has −15°C.
(a) Which substance has the lowest temperature?
(b) A fourth substance, D, has a temperature between A and C. What integer values could D's temperature be?
(c) The scientist says "Substance B is the coldest so it must have the largest temperature value." Is this correct? Explain your reasoning.
Answers: (a) Substance B (−23°C is furthest left on the number line). (b) D must be between −18 and −15, so integer values are −17 or −16. (c) No — −23 is the smallest value numerically, not the largest. The word "largest" refers to magnitude but in temperature, −23°C is the lowest/coldest, not the largest.
📝 Exam Questions by Topic
🎬 Video Resources
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