AQA · GCSE · Mathematics
Course version: 8300 · Version from Sept 2015
Structure and calculation: order positive and negative integers, decimals and fractions use the symbols =, ≠, <, >, ≤, ≥
Ordering means placing numbers on a number line from least to greatest.
- 4 key terms
Study notes
From the checked teaching notes for this specification point.
The idea
Ordering means placing numbers on a number line from least to greatest. For integers, remember that negative numbers are less than zero and less than any positive number: −7 < −2 < 0 < 3. For decimals, compare place value column by column: 0.3 < 0.25 is false because 0.30 > 0.25. For fractions, rewrite with a common denominator or convert to decimals: 3/4 = 0.75, so 2/3 < 3/4. The symbols =, ≠, <, >, ≤, ≥ let you state exact relationships. For example, −5 ≤ −5 is true because ≤ includes equality, and 4 ≠ 5 shows inequality. When ordering a mixed list of positive and negative integers, decimals and fractions, convert all to decimals or a common form first, then compare.
What earns the marks
- Correctly orders positive and negative integers, including placing negative numbers to the left of zero and positive numbers to the right.
- Compares decimals by aligning place value, recognising that 0.30 is greater than 0.25.
- Compares fractions by finding a common denominator or converting to decimals, e.g. 2/3 < 3/4.
- Uses =, ≠, <, >, ≤, ≥ correctly, including the inclusive meaning of ≤ and ≥.
- Orders a mixed set of positive and negative integers, decimals and fractions by converting to a common form.
Common mistakes and how to put them right
Thinking −7 > −2 because 7 > 2.
Put it right: On a number line, −7 is further left, so −7 < −2.
Believing 0.3 < 0.25 because 3 < 25.
Put it right: Align decimal places, 0.30 > 0.25.
Writing 1/4 > 1/3 because 4 > 3 in the denominators.
Put it right: Convert to decimals, 0.25 < 0.333...
Using < when ≤ is required for equal values, e.g. writing 5 < 5 instead of 5 ≤ 5.
Exam tips
- Convert all numbers to decimals to two or three places before ordering a mixed list.
- Check the inequality direction by reading it as 'less than' from left to right.
- For fractions, cross-multiply to compare quickly: a/b < c/d if ad < bc (with positive denominators).
Try the lesson
The start of the Structure and calculation: order positive and negative integers, decimals and fractions use the symbols =, ≠, <, >, ≤, ≥ lesson, made from checked teaching notes for this specification point.
Have a go first
Which symbol belongs in the box to make this statement true? −7 □ −2
Listen
Ordering means placing numbers on a number line from least to greatest. For integers, remember that negative numbers are less than zero and less than any positive number: −7 < −2 < 0 < 3. For decimals, compare place value column by column: 0.3 0.25. For fractions, rewrite with a common denominator or convert to decimals: 3/4 = 0.75, so 2/3 < 3/4. The symbols =, ≠, , ≤, ≥ let you state exact relationships. For example, −5 ≤ −5 is true because ≤ includes equality, and 4 ≠ 5 shows inequality. When ordering a mixed list of positive and negative integers, decimals and fractions, convert all to decimals or a common form first, then compare. A strong answer earns marks for these points. Correctly orders positive and negative integers, including placing negative numbers to the left of zero and positive numbers to the right. Compares decimals by aligning place value, recognising that 0.30 is greater than 0.25. Compares fractions by finding a common denominator or converting to decimals, e.g. 2/3 < 3/4. Uses =, ≠, , ≤, ≥ correctly, including the inclusive meaning of ≤ and ≥. A common slip: Thinking −7 > −2 because 7 > 2. Instead, on a number line, −7 is further left, so −7 < −2. Exam tip: Convert all numbers to decimals to two or three places before ordering a mixed list.
See it
Put it in order
Put these steps in the order they happen.
- 1. Convert all to a common form
- 2. Compare and place from least to greatest
- 3. State relationships using =, ≠, <, >, ≤, ≥
- 4. Order a mixed set of numbers
Key words
Say what each word means before you turn the card.
Number line and ordering
Try answering aloud or in your head before turning the card.
Place value in decimals
Try answering aloud or in your head before turning the card.
Equivalent fractions and comparison
Try answering aloud or in your head before turning the card.
Inequality symbols and their meanings
Try answering aloud or in your head before turning the card.
See every definition
- Number line and ordering
- Placing numbers on a number line from least to greatest, with negative numbers to the left of zero and positive numbers to the right.
- Place value in decimals
- Comparing decimals column by column by aligning place value, so that 0.30 is greater than 0.25.
- Equivalent fractions and comparison
- Rewriting fractions with a common denominator or converting them to decimals in order to compare them.
- Inequality symbols and their meanings
- The symbols =, ≠, <, >, ≤, ≥, which let you state exact relationships, where ≤ and ≥ include equality.
Label the diagram
Some boxes have lost their names. Pick the right name for each number.
1Box 1
2Box 2
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