KS3 · Maths

Ordering and comparing numbers

Which is bigger: 4.032 or 4.32? Lots of people pick 4.032 because it looks longer. It's the smaller one, and soon you'll know which digit proves it.

Where do they really sit?

Five decimals, one line

Drag each number to where you think it sits between 4 and 4.8. Commit to your guesses before you check. If one of them surprises you, that's the whole point.

Place 4.032Place 4.3Place 4.7Place 4.07Place 4.32

Predict, then check

Don't count the digits. Think about how you would prove your answer.

Which is greater: 8.04781009 or 8.047809?

Words into symbols

Which symbol does each phrase need?

Pick a phrase, then pick the symbol it needs. Every time, ask: is the boundary number itself allowed?

Still to sort

< less than (0)

Below the boundary. The boundary itself is not included.

Where the line is: Differs from ≤ only at the boundary value.

> greater than (0)

Above the boundary. The boundary itself is not included.

Where the line is: Differs from ≥ only at the boundary value.

≤ less than or equal to (0)

Up to the boundary, and the boundary counts.

≥ greater than or equal to (0)

The boundary and everything above it.

5 of 5 still to sort.

Watch out: < and ≤ only disagree about one number: the boundary. That one number is the whole question.

Careful with units

Which is longer: 0.4 m or 40 mm?

Two lengths: 0.4 m and 40 mm.

Which is closest to what you think right now?

40 mm is longer. 40 is a much bigger number than 0.4.You can't compare them. One is in metres and the other is in millimetres.0.4 m is longer. In millimetres it is 400 mm.They're the same. Both are just a 4.

How sure are you?

I'm sureI think soI'm guessing

40 mm is longer. 40 is a much bigger number than 0.4.

This is a common misconception

That's the trap. When the units are different, the number on its own doesn't tell you the size. A millimetre is tiny, a thousandth of a metre. Convert first: 1 m = 1000 mm, so 0.4 m = 400 mm. Now 400 mm > 40 mm, so 0.4 m is the longer length. The greater number is not always the greater measurement.

You can't compare them. One is in metres and the other is in millimetres.

This is a common misconception

You can: both are lengths, so they measure the same kind of thing. They just need to be in the same unit first. 0.4 m = 400 mm, and 400 mm > 40 mm. What really can't be compared is measures of different TYPES. Lengths go only with lengths, masses only with masses, and capacities only with capacities.

0.4 m is longer. In millimetres it is 400 mm.

This matches the evidence

Exactly. Putting both in the same unit is what makes the comparison fair: 0.4 m = 400 mm, and 400 > 40, so 0.4 m > 40 mm. You could go the other way too: 40 mm = 4 cm = 0.04 m, and 0.4 m > 0.04 m.

They're the same. Both are just a 4.

This is a common misconception

The digit is the same, but its place value isn't. Convert properly and the 4 lands in different columns: 0.4 m = 400 mm, so that 4 is worth 400 mm, while the other is worth 40 mm. Same digit, different column, different size.

Before comparing two measurements, ask two questions. Same type? Then same unit. Only then compare the numbers.

Convert, then compare

Your turn to supply the steps

Which is heavier: 0.456 kg or 465 g?

The units are different, so write both in grams first.465 g is already in grams, so only one number needs converting.
missing step

Which line is step 2?

1 kg = 100 g, so multiply by 100: 0.456 kg = 45.6 g1 kg = 1000 g, so multiply by 1000: 0.456 kg = 456 gGrams are smaller, so divide by 1000: 0.456 kg = 0.000456 g

Mixed units

Four journeys, shortest to longest

Put these journeys in order of length

1 · Shortest4 · Longest

2.34 km
432 000 mm
324 000 cm
423 m

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Longer doesn't mean bigger. Line up the columns and let the first difference decide.

What you need to know

Place value decides size. In 4.32 the 3 is worth 3 tenths; in 4.032 it is worth only 3 hundredths.
Place value works for big numbers too: 0.4 million has a 0 in the millions column and a 4 in the hundred thousands column, so 0.4 million = 400 000, which is greater than 40 000.
Equal values can be written in different ways: 9 = 9.00, 43 hundreds = 4300, and 47 thousandths = 4.7 hundredths.
Five symbols record a comparison: =, <, >, ≤ and ≥.

The big picture

To compare two numbers, line up their place values and read from the left: the first column where the digits differ decides, however many digits each number has. Record the result with =, <, >, ≤ or ≥, and let the words in a question tell you whether the boundary value is included. For measurements, check they are the same type, convert them to the same unit, and only then compare.

Key points

1Longer isn't larger. Line up the decimal points and compare from the left: the first column that differs decides.

2< means less than and > means greater than. Swap the two numbers and the sign swaps too.

3≤ and ≥ include the boundary value; < and > leave it out. 'At least' is ≥, 'more than' is >, 'maximum' is ≤ and 'under' is <.

4Only measures of the same type can be compared: lengths with lengths, masses with masses, capacities with capacities.

5kilo means 1000 times, centi means a hundredth and milli means a thousandth. A metric conversion keeps the same digits in the same order.

6To order a list of measures, write them all in one unit, put them in order, then write the answer with < signs.

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Worked example

Problem

Put these capacities in ascending order: 0.33 l, 303 ml, 0.03 l, 30 cl.

⚠ Watch out

Deciding by length. 4.032 looks bigger than 4.32, and 1.09 looks bigger than 1.4, because of the extra digits. Line up the points and check the tenths: 0 against 3, then 0 against 4. So 4.032 < 4.32 and 1.09 < 1.4.

🧠

Memory hook

Line up, left to right, first difference wins. And remember the speed limit: at 30 mph, exactly 30 is fine but 30.001 isn't, so it's speed ≤ 30, never speed < 31.

✓

Check yourself

In your head: which is longer, 985 000 m or 958 km? And 0.098 m or 99 mm? Write each answer with < or >, then the other way round.

Flashcards

(13)

What do < and > mean?

< means less than (4.07 < 4.7); > means greater than (0.13 > 0.098). Each time, read the left-hand value first.

Which two symbols include the boundary value?

≤, less than or equal to, and ≥, greater than or equal to.

How do you compare two decimals?

Line up the decimal points, then compare column by column from the left. The first column where the digits differ decides.

Is the number with more digits always bigger?

No. 4.32 > 4.032 and 1.09 < 1.4. Sometimes the longer number is bigger, but only because of the first column that differs, never because of its length.

Write 0.13 > 0.098 the other way round.

0.098 < 0.13. When the numbers swap sides, the sign turns round too.

Which symbol does 'at least' need?

≥, because the boundary counts. 'At least 12 minutes' means time ≥ 12.

Which symbol does 'more than' need?

>, because the boundary doesn't count. 'More than 4000' leaves out 4000 itself.

What do kilo, centi and milli mean?

kilo means 1000 times, centi means a hundredth and milli means a thousandth. So 1 km = 1000 m, 1 m = 100 cm and 1 litre = 1000 ml.

What must be true before two measurements can be compared?

They must be the same type of measure (length with length, mass with mass, capacity with capacity) and written in the same unit.

Is 406 cm the same length as 4.6 m?

No. 4.6 m = 460 cm = 4600 mm. 406 cm has a 0 between the 4 and the 6, so its digits are not in the same order.

Which unit suits an ant that is 3 mm long?

Millimetres. In metres it would be 0.003 m, but 3 mm is a whole number in a unit sized for small lengths.

What is 0.4 million as an ordinary number?

400 000: a 0 in the millions column and a 4 in the hundred thousands column.

What does ascending order mean?

Smallest first, going up to the largest.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Five decimals, one line

Which symbol does each phrase need?

Which is longer: 0.4 m or 40 mm?

Your turn to supply the steps

Four journeys, shortest to longest

More KS3 Maths topics