Every digit has a value

A digit's value depends on its column. Reading left to right, each column is worth ten times the one to its right — and to the right of the decimal point the columns get ten times smaller.

Watch out: the 0 in 306 is a place-holder — it keeps the 3 in the hundreds column. Drop it and 306 becomes 36, a totally different number.

What's it worth?

Which is bigger?

To order numbers, compare them column by column, starting from the left. We write comparisons with inequality symbols:

Watch out with decimals: more digits does not mean bigger! 0.7 is greater than 0.65, because 7 tenths beats 6 tenths — compare the tenths column first.

Negatives: the further left on the number line, the smaller. So -5 < -2 (−5 is colder / lower than −2).

Pick the bigger number

Tap the greater number in each pair. Compare column by column!

Building blocks of number

• A factor divides into a number with no remainder. Factors of 12: 1, 2, 3, 4, 6, 12.
• A multiple is in a number's times-table. Multiples of 4: 4, 8, 12, 16, …
• A prime has exactly two factors — 1 and itself (2, 3, 5, 7, 11, …).

Watch out: 1 is not prime (it has only one factor). 2 is the only even prime — every other even number is divisible by 2.

Spot the prime

Prime or not prime?

Tap a number, then tap the box it belongs in.

Splitting into primes

Every whole number can be written as a product of primes. A factor tree breaks it down step by step until every branch ends on a prime:

Tip: circle each prime as you reach it so you don't forget it. The order you split doesn't matter — you always get the same primes.

Highest common factor & lowest common multiple

The HCF is the biggest number that divides into both. The LCM is the smallest number that is a multiple of both. A Venn diagram of prime factors makes it easy:

Your turn — HCF

Your turn — LCM

Is 0.65 bigger than 0.7? It has more digits.

Can taking away make a number bigger?

Adding, subtracting & the sign rules

Add, subtract, multiply and divide work with integers, decimals and fractions. The trickiest part at KS3 is negative numbers:

• Subtracting a negative adds: 5 − (−3) = 5 + 3 = 8.
• Two same signs multiply/divide to a +: (−4) × (−2) = +8.
• Two different signs multiply/divide to a −: (−6) ÷ 2 = −3.

Your turn — negatives

BIDMAS — do things in the right order

When a calculation has several operations, do them in this order:

Watch out: reading strictly left to right gives the wrong answer. 3 + 4 × 2 = 11, not 14, because you multiply before you add.

Your turn — BIDMAS

Squares, cubes & roots

A power (or index) tells you how many times to multiply a number by itself. A root undoes a power — they are inverse operations.

• 2³ ("2 cubed") = 2 × 2 × 2 = 8, and the cube root ∛8 = 2.
• Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100…

Watch out: 5² is not 5 × 2 = 10. The little 2 means "multiply by itself", so 5 × 5 = 25.

Your turn — powers

Writing huge (and tiny) numbers

Standard form writes a number as A × 10ⁿ, where A is between 1 and 10. It's a neat way to handle very big or very small numbers.

• Big numbers → positive power: 50 000 = 5 × 10⁴.
• Small numbers → negative power: 0.004 = 4 × 10⁻³.

Watch out: the first part must be at least 1 but under 10. So 32 × 10² is not standard form — it should be written 3.2 × 10³.

Into standard form

Can taking away make a number bigger?

20% off £45 or 45% off £20: which saves more?

Three ways to write a part

Fractions, decimals and percentages are the same idea written differently. You should be able to swap between them:

Terminating decimals (like 0.5, 0.75, 0.2) stop. Each one is a fraction over 10, 100, 1000…: 0.75 = 75/100 = ¾.

Finding a percentage of an amount

"Percent" means "out of 100". To find a percentage of an amount, turn it into a fraction or decimal and multiply:

Handy building blocks: 50% = ÷2, 25% = ÷4, 10% = ÷10, 1% = ÷100. Build any percentage from these.

Your turn — percentages

Rounding, estimating & error bounds

To round, look at the next digit: 5 or more rounds up, 4 or less rounds down.

• 47 to the nearest 10 → 50 (the 7 rounds up).
• 3.14159 to 2 decimal places → 3.14 (the next digit, 1, rounds down).
• To 1 significant figure, 68 → 70.

Estimate by rounding every number to 1 significant figure first: 19 × 31 ≈ 20 × 30 = 600.

Error bounds: a length given as 12 cm to the nearest cm could really be anywhere from 11.5 cm up to (but not including) 12.5 cm — half a unit either side.

Rounding check

20% off £45 or 45% off £20: which saves more?

The big ideas to remember

You've covered the key ideas in the KS3 Number strand. Press Finish to see your score.