Ordering Decimals
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.
0.4 is the same as 0.40 when written with two decimal places. Now we are comparing 40 hundredths with 38 hundredths. Since 40 hundredths > 38 hundredths, 0.4 > 0.38.
Alternatively, both sit between 0.3 and 0.5 on a number line. 0.4 is at the halfway mark, while 0.38 is just before it — so 0.4 is further to the right.
0.5 means 5 tenths. 0.50 means 50 hundredths, which simplifies to 5 tenths. 0.500 means 500 thousandths, which also simplifies to 5 tenths.
Adding trailing zeros after the last non-zero decimal digit doesn’t change the value — it’s like saying “I have 5 tenths, zero hundredths, and zero thousandths.” The extra zeros add precision to the measurement but not to the value.
0.1 means 1 tenth, which is the same as 10 hundredths. 0.09 means 0 tenths and 9 hundredths — just 9 hundredths. Since 9 hundredths < 10 hundredths, 0.09 < 0.1.
Think of a metre stick. 0.1 of a metre is 10 centimetres. 0.09 of a metre is 9 centimetres. You need ten hundredths to make one tenth, and you only have nine.
On a number line, 0.09 sits just to the left of the 0.1 mark. This is a case where a number with more digits is actually smaller.
Between 0.1 and 0.2, we can find 0.11, 0.12, 0.13, … 0.19 — that’s 9 decimals with two decimal places.
But between 0.1 and 0.11, there are 0.101, 0.102, … 0.109. Between 0.1 and 0.101, there are 0.1001, 0.1002, … and so on. Each time we zoom in, we find more decimals. This process never ends — we can always add another decimal place to find numbers we haven’t listed yet.
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: 0.305
Another: 0.301
Creative: 0.30999 — pushing as close to 0.31 as possible without reaching it. Or 0.300001 — barely above 0.3.
Trap: 0.35 — a student might think this is between 0.3 and 0.31 because 35 looks like it’s between 3 and 31, but 0.35 is actually greater than 0.31.
Example: 0.912
Another: 0.999
Creative: 0.901 — only just greater. Or 1.000 — it has three decimal places and is greater than 0.9, but might spark debate about whether it counts!
Trap: 0.123 — it has three decimal places but is much less than 0.9. A student might think “123 is bigger than 9” and assume 0.123 > 0.9. This is the classic “longer is larger” misconception.
Example: 0.5 and 0.50
Another: 0.7 and 0.70
Creative: 0.25 and 0.250 — or even 1.0 and 1.00. Or take it further: 0.100000 and 0.1.
Trap: 0.5 and 0.05 — these look similar (same digits!) but 0.5 is ten times larger than 0.05. The position of the 5 relative to the decimal point makes all the difference.
Example: −0.5
Another: −0.75
Creative: −0.999 — as close to −1 as possible without reaching it. Or −0.301 — barely less than −0.3.
Trap: −0.2 — a student might think that because 0.2 is less than 0.3, then −0.2 must be less than −0.3. But with negatives the order reverses: −0.2 is actually greater than −0.3 (it’s closer to zero), so it doesn’t satisfy the condition.
Example: 0.995
Another: 0.991
Creative: 0.9999 — infinitely approaching 1.
Trap: 0.100 — students confusing the next “tick” on their mental number line and accidentally starting again with 100 thousandths.
Always, Sometimes, Never
Is the statement always true, sometimes true, or never true? Students should justify their decision with examples.
It’s true that 0.125 (three d.p.) is smaller than 0.5 (one d.p.). But 0.912 (three d.p.) is larger than 0.5 (one d.p.).
The number of decimal places tells you nothing about the size of a number. What matters is the value of each digit, determined by its position.
0.5 = 0.50 = 0.500. Adding a trailing zero after the last decimal digit never changes the value.
This catches students who apply the whole number rule — where adding a zero to the end of 5 gives 50, which is different. With decimals, the zero goes into a new, smaller place-value column and contributes nothing.
This works when decimals have the same number of decimal places: comparing 0.35 and 0.41 by treating them as 35 and 41 gives the right order (35 < 41, so 0.35 < 0.41).
But it fails when decimals have different numbers of decimal places: comparing 0.5 and 0.12, treating the digits as 5 and 12 suggests 0.5 < 0.12, which is wrong. This is one of the most common misconceptions in ordering decimals.
Between any two different decimals, you can always find another. One reliable method: find the number halfway between them.
Between 0.3 and 0.4 lies 0.35. Between 0.3 and 0.31 lies 0.305. Between 0.3 and 0.301 lies 0.3005. This process never ends. Unlike whole numbers (where there’s nothing between 3 and 4), decimals are dense — there are no gaps.
Removing the zero from 0.05 gives 0.5 (gets bigger). Removing a zero from 5.02 to make it 5.2 (gets bigger).
However, removing the zero from 0.50 gives 0.5 (stays the same). This forces you to think purely about place value rather than string length.
Odd One Out
Which is the odd one out? The challenge: make a valid mathematical case for each one being the odd one out.
Explain the Mistake
Each example contains a deliberate error targeting a common misconception. Can you find where and why the reasoning goes wrong?
Answer: 0.5, 0.12, 0.308
Reasoning: “5 is less than 12, and 12 is less than 308. So 0.5 is the smallest and 0.308 is the largest.”
The student is treating the digits after the decimal point as whole numbers and comparing 5, 12, and 308. This is the classic “longer is larger” misconception.
The correct approach is to compare place values. Looking at the tenths column: 0.5, 0.12, 0.308. Since 1 < 3 < 5, the correct order from smallest to largest is 0.12, 0.308, 0.5.
Answer: 0.7 is larger ✔
Reasoning: “0.7 only has one decimal place and 0.25 has two. Shorter decimals are always bigger because the digits are worth more.”
The answer is correct — 0.7 is larger — but the reasoning is dangerously wrong. The student has invented a rule: “shorter decimals are always bigger.” This is the mirror image of “longer is larger” and will fail just as badly.
A counterexample breaks the rule immediately: 0.3 has one decimal place and 0.85 has two, but 0.85 > 0.3. The correct reasoning is to compare place values: 0.7 has 7 tenths while 0.25 has only 2 tenths, so 0.7 is larger. The number of decimal places is irrelevant.
Create Cognitive Conflict: “If shorter decimals are always bigger, then 0.1 must be bigger than 0.99. Does that make sense if we think about money? Is 10 pence more than 99 pence?”
Answer: 0.5, 0.50, 0.500
Reasoning: “5 is less than 50, and 50 is less than 500. So 0.5 is smallest and 0.500 is largest.”
The student is treating the digits after the decimal point as whole numbers (5, 50, 500) and ordering those.
In fact, all three numbers are equal. 0.5 = 0.50 = 0.500. Each one represents exactly 5 tenths. The trailing zeros add no value — they would all sit at exactly the same point on a number line. There is no “smallest to largest” here.
Answer: 0.8, 0.18, 0.081
Reasoning: “Decimals with fewer digits after the point are smaller. 0.8 has one decimal place, 0.18 has two, and 0.081 has three. So that’s my order.”
The student assumes that “fewer decimal places = smaller number”. This is the mirror image of the “longer is larger” misconception — sometimes students overcorrect and think that shorter decimals must be smaller.
The correct approach is to compare the tenths digit: 0.081 has 0 tenths, 0.18 has 1 tenth, 0.8 has 8 tenths. The correct order is 0.081, 0.18, 0.8 — exactly the reverse of what the student wrote.