Topics · Primary Maths · Multiplication & Division
Multiplying and dividing: equal groups both ways
Multiplication is equal groups joined; division is equal groups made. They are the same picture read in two directions — which is why knowing one of them checks the other.
- 396 exam papers analysed
- appears in 86% of P2 Maths papers
- ~4 min read
01 One picture, two questions
4 × 3 is four groups of three, or three rows of four — the same 12 either way, which is why multiplication can be done in whichever order is easier.
12 ÷ 3 asks the reverse: how many threes fit into 12? — or equally, if 12 is shared into 3 equal groups, how big is each? Both are the same rectangle read from the other end.
Because they undo each other, a division can always be checked by multiplying back.
02 Multiplying bigger numbers
Long multiplication is only the distributive idea written tidily: split the number by place value, multiply each part, add the results.
- Split by place value. 24 × 6 is (20 × 6) + (4 × 6) = 120 + 24 = 144.
- By 10, 100, 1000: the digits move left. 24 × 100 = 2,400. Do not say “add two noughts” — it fails immediately on 2.4 × 100.
- Two-digit by two-digit: multiply by the ones, then by the tens, then add. The nought at the end of the second line is there because you are multiplying by tens, not by ones.
- Estimate first. 24 × 6 is about 25 × 6 = 150, so 144 is believable and 1,44 or 30 is not.
03 Dividing, and what to do with what is left
Long division works down from the largest place: how many go into the thousands, then carry what is left into the hundreds, and on down.
The remainder is where the marks actually are, because the arithmetic gives you a number and the story decides what to do with it. The same division has three different right answers depending on the question.
- Round up. 230 children, 8 per bus. 230 ÷ 8 = 28 r 6. The six still need a bus, so 29 buses.
- Round down. $230, notebooks cost $8. 28 r 6 means 28 notebooks — $6 is not enough for another.
- Use the remainder itself. How much money is left? The answer is the $6.
- Never just write “28 r 6” unless the question asked for it. Read the last sentence and answer that.
04 Order matters one way and not the other
4 × 3 and 3 × 4 are the same. Multiplication can be done in any order, and rearranging is often what makes a calculation easy: 25 × 17 × 4 is 25 × 4 × 17 = 100 × 17 = 1,700.
12 ÷ 3 and 3 ÷ 12 are not the same. Division has a fixed direction, and writing it the wrong way round is a wrong answer rather than an untidy one.
In a word problem the phrase “shared among” tells you which number is being divided: the total goes first.
05 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- productsayPROD-uktmore than oneproducts
The answer to a multiplication. The product of 4 and 3 is 12.
- quotientsayKWOH-shuhntmore than onequotients
The answer to a division. 12 ÷ 3 has a quotient of 4.
- remaindersayri-MAYN-duhmore than oneremainders
What is left over when a division does not go exactly. What to do with it depends on the question.
- factormore than onefactors
A number that divides exactly into another. 3 is a factor of 12.
- multiplesayMUL-ti-puhlmore than onemultiples
The result of multiplying by a whole number. 12 is a multiple of 3.
06 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P2
Times tables, equal groups, and sharing equally.
Practise P2 Multiplication & Division — 10 questions →86% of P2 papers
- P3
Multiplying and dividing two- and three-digit numbers, and the first remainders.
Practise P3 Multiplication & Division — 10 questions →69% of P3 papers
- P4
Long multiplication and division, factors and multiples, and deciding what a remainder means.
Practise P4 Multiplication & Division — 10 questions →66% of P4 papers
- P5
Multiplying and dividing decimals and fractions, and problems with several steps.
Practise P5 Multiplication & Division — 10 questions →51% of P5 papers
- P6
Multiplication and division inside ratio, rate, percentage and speed problems.
Practise P6 Multiplication & Division — 10 questions →56% of P6 papers
07 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“230 ÷ 8 = 28 r 6, so 28 buses.”
The six children left over still need a bus. It is 29. Whether a remainder rounds up, rounds down or is the answer itself depends on the story.
“Multiplying by 100 means adding two noughts.”
It works on whole numbers and fails on 2.4 × 100, which is 240, not 2.400. The digits move two places left — that description always works.
“The question said shared among 4, so I did 4 ÷ 20.”
The total is what gets divided: 20 ÷ 4 = 5. Division has a direction, unlike multiplication.
“A number multiplied always gets bigger.”
Not when multiplying by less than one. 20 × 0.5 is 10, and half of something is a multiplication.
“I forgot the nought on the second line of the long multiplication.”
That line is multiplying by tens, so it lands one column left. Without the nought the answer is short by a factor of ten.
Now try it
50 questions across 5 levels, marked as you go with the working shown.