Ordering Negative Numbers
Questions designed to stretch thinking, reveal misconceptions, and spark mathematical reasoning.
Convince Me That…
Students must construct a mathematical argument for why each statement is true.
On a number line, −2 is to the right of −6. Numbers further to the right are always greater. Alternatively, −2 is only 2 steps below zero, while −6 is 6 steps below zero — so −2 is closer to zero, making it greater.
A real-life argument: −2°C is warmer than −6°C. If you owe £2 you’re better off than if you owe £6. In every context, −2 > −6.
Zero means “nothing” and −1 means “one less than nothing.” On a number line, −1 sits to the left of 0. Since every negative number is to the left of zero, −1 < 0.
Think of it as a bank balance: £0 means you have nothing, but −£1 means you actually owe money. Owing money is worse than having nothing, so −1 is less than 0. This is important because many students think 0 is the smallest possible number.
−0.5 is only half a step below zero, while −2 is two full steps below. On a number line, −0.5 is to the right of −2, so it is greater. The fact that 2 looks “bigger” than 0.5 is irrelevant — with negative numbers, being closer to zero means being greater.
A money analogy helps: owing 50p is far better than owing £2. This question combines two tricky ideas — ordering negatives and comparing integers with decimals.
The distance from −7 to −1 is 6. Half of 6 is 3. Starting at −7 and moving 3 steps to the right gives −4. Starting at −1 and moving 3 steps to the left also gives −4. So −4 is exactly in the middle.
Alternatively, the mean of −7 and −1 is \( \frac{-7 + (-1)}{2} = \frac{-8}{2} = -4 \). This extends the idea of finding a midpoint to negative numbers — the same method works, but students need confidence working with negative addition.
For positive fractions, \(\frac{1}{3}\) is a larger slice than \(\frac{1}{4}\). But with negative numbers, being further from zero (having a larger magnitude) means the value is actually smaller.
Since \(-\frac{1}{3}\) is further to the left on the number line than \(-\frac{1}{4}\), \(-\frac{1}{4}\) is the greater value. Thinking about decimals can also help here: \( -0.25 \) is closer to zero than \( -0.333… \)
Give an Example Of…
For each prompt, provide: an example, another example, one no-one else will think of, and one someone might think works but doesn’t.
Example: −3
Another: −1
Creative: −4.999 — as close to −5 as possible without reaching it. Or −0.001 — barely below zero, still negative, and much greater than −5.
Trap: −6 — a student might think that because 6 > 5, then −6 must be greater than −5. This is the classic “bigger digit means bigger number” misconception applied to negatives. In fact, −6 is less than −5.
Example: −9, −5, −3
Another: −100, −50, −1
Creative: −0.3, −0.2, −0.1 — negative decimals in ascending order. Or −1000, −999, −998 — three consecutive negative numbers.
Trap: −1, −3, −5 — the digits 1, 3, 5 are ascending, but the values −1, −3, −5 are actually descending. This is the most common error — students order the digits and forget that with negatives, the order reverses.
Example: −2.5
Another: −2.1
Creative: −2.999 — as close to −3 as possible without reaching it. Or −2.001 — barely less than −2.
Trap: 2.5 — a student might look for a number “between 2 and 3” and forget about the negative signs entirely. 2.5 is positive and nowhere near the region between −3 and −2 on the number line.
Example: 5 and −2 (distance from zero: 5 vs 2, so −2 is closer)
Another: 10 and −1
Creative: 0.001 and −0.0001 — the negative number is ten times closer to zero than the positive one. This challenges the idea that positive numbers are always “nearer” to zero.
Trap: 3 and −3 — they are the same distance from zero (both 3 units away), so the negative number isn’t closer. A student might assume that any negative number is automatically closer to zero than a positive number, and pick this pair without checking the actual distances.
Example: −5
Another: −3
Creative: −2.0001 — checking understanding of decimals within the context of negative boundaries. Or \( -6\frac{1}{2} \).
Trap: −8 — a student might look at the digits 2 and 7, forget about the signs, and think 8 lies just beyond 7 in the sequence. But −8 is less than −7, so it sits outside the required range.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
Every negative number sits to the left of zero on the number line, and every positive number sits to the right. Since left means smaller, any negative number is always less than any positive number.
Students might try extreme cases like −1000 and 0.001, but even here, −1000 < 0.001. The “size” of the digit doesn’t matter — the sign determines which side of zero you’re on.
It’s true on the positive side: 7 is further from zero than 3 and 7 > 3. But it fails on the negative side: −7 is further from zero than −3, yet −7 < −3.
This exposes the “distance from zero equals value” misconception. Distance from zero (absolute value) tells you the magnitude of a number, not its position on the number line. For negative numbers, greater distance from zero actually means a smaller value.
Zero is neither positive nor negative. It is the boundary between positive and negative numbers on the number line. Negative numbers are defined as numbers less than zero, but zero is not less than itself.
Students often group zero with the negatives because it “isn’t positive” or because it appears on the left-hand side of their number line if they start counting from 1. This misconception matters when ordering — if students think 0 is negative, they may misplace it in an ordered list.
This rule works perfectly for positive whole numbers: 100 has three digits and is larger than 9, which has one. But the rule breaks down immediately when introduced to negative numbers (and decimals).
For example, −100 has three digits, but it is less than −9. By explicitly challenging this “fragile rule” learned in primary school, students are forced to rely on the number line rather than shortcuts.
Odd One Out
Which is the odd one out? Can you make a case for each one? There’s no single right answer!
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: −1, −3, −7
Reasoning: “1 is less than 3, and 3 is less than 7. So −1 is the smallest and −7 is the largest.”
The student is ignoring the negative signs and ordering the numbers as if they were positive. This is the classic “order the digits” misconception — treating −1, −3, −7 as if they were 1, 3, 7.
For negative numbers, the order reverses: a bigger digit means a smaller value (further from zero). The correct order from smallest to largest is −7, −3, −1. On a number line, −7 is furthest to the left and −1 is closest to zero.
Answer: −4 is larger ✓
Reasoning: “−4 has a smaller number, and with negatives, smaller numbers are always bigger.”
The answer is correct — −4 is larger — but the reasoning is dangerously muddled. The student is confusing the number’s actual value with its magnitude (distance from zero) or absolute value.
4 is a smaller magnitude than 9, meaning −4 is closer to zero. Because it sits to the right of −9 on the number line, −4 is the larger number. A counterexample to the student’s rule: by their logic, 2 is a “smaller number” than 7, so 2 should be “bigger” — which is obviously wrong for positives.
Answer: −1, −8, 0, 3
Reasoning: “The negatives go first because they’re smallest. −1 comes before −8 because 1 is less than 8. Then 0, then 3.”
The student correctly identifies that negatives are less than positives and places 0 and 3 at the end. But they order the negative numbers by their digits (1 < 8), placing −1 before −8. This mixes up “digit size” with “value” for negatives.
−8 is further from zero than −1, so −8 is the smaller number. The correct order is −8, −1, 0, 3. This is a very common error when students correctly separate negatives and positives but then forget to reverse the ordering within the negatives.
Answer: −3.5 is closer to zero
Reasoning: “Decimals are really small numbers, so −3.5 must be closer to zero than −3.”
The student assumes that “decimals are small” and therefore closer to zero. This is a misconception carried over from working with positive decimals between 0 and 1. Here, 3.5 is larger than 3, so −3.5 is further from zero than −3.
On a number line, −3.5 sits to the left of −3 — it is half a step further from zero, not closer. The correct answer is −3 is closer to zero (distance of 3 vs distance of 3.5). This question exposes how misconceptions from decimals and negatives can combine.
Answer: \( -\frac{1}{3} \) is larger
Reasoning: “Because 4 is bigger than 3, a quarter is smaller than a third. So \( -\frac{1}{4} \) must be smaller than \( -\frac{1}{3} \).”
The student correctly identifies that \( \frac{1}{3} \) has a larger magnitude than \( \frac{1}{4} \). However, they forget to reverse their logic for negative numbers.
A larger negative magnitude means it is further to the left on the number line, making it smaller. Therefore, \( -\frac{1}{4} \) is actually the larger number. This reveals a “double misconception” where students have to juggle both fraction size rules and negative size rules simultaneously.