Multiples and factors: why pupils mix them up

A pupil who knows the times tables can get every calculation right and still accept the sentence “6 is a multiple of 24”. Both words describe the same equation seen from two sides, and that is where the confusion starts.

One equation, two sentences

Always start from a written multiplication: 24 = 6 × 4. It gives two sentences: 24 is a multiple of 6, and 6 is a factor of 24. Have pupils say both aloud from the same equation, with different numbers each time. When the sentence rests on a visible multiplication, the vocabulary no longer depends on memory alone.

One reference point helps hesitant pupils: apart from the case of 0, the multiple is the larger of the two numbers, or equal to the other.

Arranging tiles to see a factor

A factor is easy to see with objects. Hand out 24 tiles and ask pupils to arrange them in equal rows. With 4 rows of 6, nothing is left: 4 and 6 are factors of 24. With 5 rows of 4, 4 tiles are left over: 5 is not a factor. A failed try gives information, so it should not be treated as a mistake.

The arrangement also shows why factors come in pairs. Each rectangle gives two of them: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Search in order, 1, 2, 3…, and stop when the pairs meet.

Three traps to teach directly

• The last digit. 26 ends in 6, but 26 = (6 × 4) + 2, so it is not a multiple of 6. Only 2, 5 and 10 can be recognized from the units digit.
• The cases of 0 and 1. 0 is a multiple of every number, and 1 is a factor of every number. Pupils forget these quickly because they rarely meet them.
• The lowest common multiple. The product of 6 and 8 is 48, which is a common multiple. The lowest one, apart from 0, is 24. Writing both lists side by side stops pupils from multiplying by reflex.

Two families of divisibility rules

For 2 and 5, look at the units digit. For 3 and 9, add up the digits. Pupils soon mix the two methods: a number ending in 3 looks divisible by 3. Have them check 143: 1 + 4 + 3 = 8, which is not a multiple of 3, so 143 is not divisible by 3. Conversely, 27 ends in neither 3 nor 9, yet it is divisible by both.

A short session

Five to ten minutes are enough for a focused activity: a wall of multiples at the start of the lesson, a number to arrange in rectangles, then a few sentences to judge true or false. Each time, ask for the equation that justifies the answer. In the game below, each mistake shows that equation or the remainder, and the results separate vocabulary mix-ups from times-table errors.

The Tile WorkshopMultiples, factors and divisibility with mosaic tiles to arrange in rectangles.See the game