What you'll learn
Multiplication is repeated addition of equal groups, and division is the opposite: it either shares an amount into equal groups or finds how many equal groups fit inside an amount. If 6 crates each hold 24 mangoes, multiplication tells you there are 6 × 24 = 144 mangoes altogether, and division tells you that 144 mangoes packed 24 to a crate need 144 ÷ 24 = 6 crates. This topic covers times-table facts, multiplying and dividing by 10, 100 and 1,000, formal long multiplication, formal long division with and without remainders, deciding what a remainder means in a real situation, and choosing the right operation in a word problem. Number is the heaviest strand on the SEA Mathematics paper — 19 of the 40 items — and multiplication and division sit underneath almost all of the rest of it, including money, area and averages.
Key terms and definitions
Product — the answer to a multiplication. The product of 7 and 8 is 56.
Factor — a number being multiplied. In 7 × 8 = 56, both 7 and 8 are factors of 56.
Quotient — the answer to a division. The quotient of 56 ÷ 8 is 7.
Dividend — the number being divided (the 56 in 56 ÷ 8).
Divisor — the number you are dividing by (the 8 in 56 ÷ 8).
Remainder — what is left over when a division does not work out exactly. 25 ÷ 4 = 6 remainder 1.
Multiple — the result of multiplying a number by a whole number. The multiples of 6 are 6, 12, 18, 24, 30 and so on.
Inverse operation — the operation that undoes another. Division undoes multiplication, so 144 ÷ 6 = 24 checks that 6 × 24 = 144.
Partial product — one line of a long multiplication before the lines are added together.
Core concepts
Multiplication is about equal groups
Every multiplication question is really the same question: how many altogether, when the groups are all the same size? Three shelves of 8 books is 3 × 8 = 24 books. If the groups are not equal, multiplication is the wrong tool and the amounts must be added separately.
Multiplication can be done in either order. 8 × 3 gives the same answer as 3 × 8. This is worth using deliberately: 3 × 40 is easier to think about than 40 × 3 for some children, and 2 × 17 is far easier than 17 × 2. Choosing the friendlier order is not cheating, it is good technique.
Division is sharing or grouping
Division answers two different-sounding questions with the same calculation.
Sharing: 30 sweets shared equally among 5 children. Each child gets 30 ÷ 5 = 6.
Grouping: 30 sweets put into bags of 5. That makes 30 ÷ 5 = 6 bags.
Both are 30 ÷ 5. Recognising that these are the same sum is a large part of the topic, because SEA word problems use both wordings.
Unlike multiplication, division cannot be turned around. 30 ÷ 5 is 6, but 5 ÷ 30 is not. The larger number nearly always goes first at this level, and if a child writes it the other way round the answer will be obviously wrong.
Multiplying and dividing by 10, 100 and 1,000
When you multiply by 10, every digit moves one place to the left and becomes worth ten times as much. Multiplying by 100 moves them two places, and by 1,000 three places.
| Calculation | Answer | Digits move |
|---|---|---|
| 46 × 10 | 460 | one place left |
| 46 × 100 | 4,600 | two places left |
| 307 × 1,000 | 307,000 | three places left |
| 5,200 ÷ 10 | 520 | one place right |
| 5,200 ÷ 100 | 52 | two places right |
It is much safer to say "the digits move" than "add a zero", because "add a zero" stops working the moment decimals appear later on. But for whole numbers at SEA level, the zeros are a reliable shortcut.
This helps with bigger sums too. 30 × 40 is 3 × 4 = 12, then two zeros back on, giving 1,200.
Formal long multiplication
For a three-digit number times a two-digit number, split the second number into its tens and its ones, multiply by each, and add the two lines.
236
× 24
-----
944 (236 × 4)
4720 (236 × 20)
-----
5664
The zero at the end of the second line is not decoration. The second line is 236 multiplied by twenty, not by two, and the zero is what records that. Leaving it out is the commonest error in the whole topic and it always gives an answer far too small.
Formal long division
Divide from the left, one digit at a time, carrying anything left over to the next digit.
For 875 ÷ 7:
- 8 ÷ 7 = 1, remainder 1. Write 1 above the 8 and carry the 1.
- The carried 1 makes the next digit 17. 17 ÷ 7 = 2, remainder 3. Write 2 and carry the 3.
- The carried 3 makes the last digit 35. 35 ÷ 7 = 5, remainder 0. Write 5.
The answer is 125. Check it by multiplying back: 7 × 125 = 875. ✔
For 4,872 ÷ 6:
- 4 ÷ 6 = 0, so take the first two digits together: 48 ÷ 6 = 8.
- 7 ÷ 6 = 1, remainder 1.
- The carried 1 makes 12. 12 ÷ 6 = 2.
The answer is 812, and 6 × 812 = 4,872. ✔
What to do with a remainder
The remainder is where SEA questions catch people out, because the answer depends on the situation, not on the arithmetic.
| Situation | Sum | What the remainder means |
|---|---|---|
| 25 pupils, 4 to a table | 25 ÷ 4 = 6 r 1 | Need 7 tables — the last pupil still needs one |
| $25 shared among 4 | 25 ÷ 4 = 6 r 1 | $6 each, $1 left over |
| 25 m of ribbon cut into 4 m pieces | 25 ÷ 4 = 6 r 1 | Only 6 whole pieces can be cut |
Same numbers, three different correct answers. Ask "does the leftover bit still need something, or is it thrown away?"
Estimating before you calculate
Rounding both numbers to something friendly gives a rough answer that tells you whether the real one is sensible. For 236 × 24, round to 240 × 25 = 6,000. The exact answer, 5,664, is close, so it is believable. An answer of 566 or 56,640 would have been caught instantly.
Useful shortcuts
Splitting (the distributive idea). 7 × 23 = (7 × 20) + (7 × 3) = 140 + 21 = 161.
Doubling and halving. 25 × 16 = 50 × 8 = 100 × 4 = 400. Doubling one factor while halving the other leaves the product unchanged.
Times tables to 12. There is no substitute. A child who is unsure of 7 × 8 will be slow on every long multiplication and every long division for the rest of the paper.
Worked examples
Example 1: Long multiplication in context
A wholesaler packs oranges into crates. Each crate holds 236 oranges. How many oranges are in 24 crates?
Estimate first: 240 × 25 = 6,000, so expect something a little under 6,000.
236
× 24
-----
944
4720
-----
5664
The first line is 236 × 4 = 944. The second line is 236 × 20 = 4,720. Adding them gives 5,664 oranges, which sits neatly under the estimate of 6,000.
Example 2: A remainder that must be rounded up
250 pupils are going on a school trip. Each maxi taxi seats 24 pupils. How many maxi taxis are needed?
250 ÷ 24: twenty-four tens is 240, and 250 − 240 = 10. So 250 ÷ 24 = 10 remainder 10.
Ten maxi taxis carry 240 pupils. That leaves 10 pupils standing on the pavement, and they still have to get to the trip, so one more taxi is needed.
The answer is 11 maxi taxis.
Writing "10 remainder 10" here scores nothing, because a remainder is not a number of taxis.
Example 3: A two-step multiply-then-divide problem
A school buys 8 boxes of exercise books. Each box holds 45 books. The books are shared equally among 15 classes. How many books does each class get?
Step one, find the total: 8 × 45. Split it — 8 × 40 = 320 and 8 × 5 = 40, so 320 + 40 = 360 books.
Step two, share them: 360 ÷ 15. Fifteen tens is 150, fifteen twenties is 300, and 360 − 300 = 60, which is another four fifteens. So 20 + 4 = 24.
Each class gets 24 books.
Check by multiplying back: 15 × 24 = 360. ✔
Common mistakes and how to avoid them
Leaving out the placeholder zero in long multiplication. Writing 472 instead of 4,720 on the tens line. Fix: before starting the second line, write the zero down first, then multiply.
Columns not lined up. Ones under ones, tens under tens. On squared paper this looks after itself; on plain paper, rule the columns. A misaligned addition at the end ruins a perfectly good multiplication.
Getting division the wrong way round. Writing 5 ÷ 30 when the question meant 30 ÷ 5. Fix: ask "am I sharing out the big number or the small one?" You cannot share 5 sweets among 30 children and give them one each.
Leaving the answer as "remainder 3". Fix: reread the question and decide whether to round up, round down, or state the leftover.
Stopping after the first step of a two-step problem. Fix: underline the final question before doing any arithmetic, and check your answer against it.
Forgetting to carry, or carrying and then forgetting to add the carry in. Fix: write the carried digit small above the next column, and cross it out once it is used.
Trusting a shaky times table. Fix: whenever a table fact is uncertain, build it from one you know. If 7 × 8 is unsure, use 7 × 4 = 28 and double it to 56.
How parents can help at home
You do not need to remember any of the methods yourself to be genuinely useful here.
Drill the tables in short bursts. Two minutes in the car is better than twenty minutes at a table. Go out of order — 6 × 7, then 9 × 4, then 8 × 8 — because chanting a table in order hides the gaps.
Use the shop. "Six tins at $9 each — how much?" "This pack of 12 costs $60, so what is one?" A parent asking real questions in a supermarket is doing exactly what the SEA paper does, which is dress arithmetic up as a situation.
When your child is stuck, say "draw the columns". Almost every multiplication or division error at this level is a layout error, not a thinking error. Getting the digits into ruled columns fixes most of them without you saying anything more.
Ask for the check, not the answer. "How do you know?" invites them to multiply back after dividing. A child who habitually checks 15 × 24 = 360 after working out 360 ÷ 15 will catch their own mistakes in the exam, where nobody is there to catch them.
Talk about the leftovers. When a packet does not divide evenly, ask what happens to the extra. This one habit turns remainder questions from a trap into an easy mark.
Exam technique for multiplication and division
The SEA Mathematics paper carries 100 marks against 60 for ELA and 40 for ELA Writing, so it is the single most valuable paper of the three. It has 40 items worth 75 marks, and with Number contributing 19 of those items, multiplication and division are worth working at.
Estimate before you calculate on any long multiplication. It catches the answers that are ten times too big or too small.
Show the columns even in a multiple-choice item — the working space is there to be used.
Read the last sentence of every word problem twice. Many items ask for the number of taxis, the number of full boxes, or the money left over, not the raw quotient.
In multiple-choice items, the wrong options are built from the standard errors — the missing placeholder zero, the un-rounded remainder, the answer to step one only. If your answer matches an option exactly but you rushed the working, check it again rather than feeling reassured.
Quick revision summary
- Multiplication counts equal groups; division shares them out or measures how many fit.
- Multiplication can be done in any order; division cannot.
- Multiplying by 10, 100 or 1,000 moves the digits one, two or three places to the left.
- In long multiplication, the tens line must end in a placeholder zero.
- In long division, work from the left and carry the remainder to the next digit.
- Check a division by multiplying back, and a multiplication by estimating first.
- A remainder can mean round up, round down, or "left over" — the situation decides.
- Split awkward multiplications: 7 × 23 = 140 + 21 = 161.
- Doubling one factor and halving the other leaves the product unchanged.
- Underline the final question in a two-step problem before you start.