Written Multiplication and Division Methods Explained

Reliable written multiplication and division methods keep place values aligned, record every regrouping and make each step checkable. For Year 5 multiplication, write the digits in columns, multiply from the ones and regroup into the next column; for a two-digit multiplier, calculate two place-value products and add them. For short division, divide from the highest place value, regroup any remainder into the next column, and interpret a final remainder in its context.

The layout is not a string of rules to memorise blindly. It records what happens to ones, tens, hundreds and thousands. A child who can explain why a carried digit changes the next column, why the second multiplication row represents tens, or why a division remainder is regrouped is more likely to spot an error and repair it.

Contents

What Year 5 pupils are expected to do

Before using a formal written method

Worked example: short multiplication

Worked example: long multiplication

Worked example: short division

What to do with a remainder

A parent practice sequence

Common mistakes and precise fixes

Adjustment cues

Frequently asked questions

Next step

What Year 5 pupils are expected to do

England’s Year 5 mathematics programme says pupils should multiply numbers of up to four digits by a one- or two-digit number using a formal written method, including long multiplication for a two-digit multiplier. It also specifies short division of numbers up to four digits by a one-digit number, with remainders interpreted appropriately for the context.[1]

That defines the scope here: formal calculation, place-value layout, regrouping and checks. It does not replace secure multiplication-table knowledge, and it does not teach how to choose an operation from a word problem. Those are connected skills, but a child can choose multiplication correctly and still need help with the written calculation.

The EEF’s Key Stages 2 and 3 guidance recommends connecting facts, procedures and concepts, teaching pupils to understand procedures, and building conceptual and procedural knowledge together.[2] In practice, that means asking “What does this digit represent?” alongside “What do you write next?” Physical counters or place-value equipment can help reveal the structure, but the EEF stresses that a representation needs a clear purpose and should support eventual independence.[2]

Before using a formal written method

Start with three quick checks.

Can the child read the place values? In 2,347, the 3 means three hundreds, not three ones.

Can they recall or derive the needed facts? Slow recall is manageable, but guessing products makes the written method unstable.

Can they estimate the rough size? For 2,347 × 6, an estimate based on 2,300 × 6 shows that the answer should be near 14,000, not 1,400 or 140,000.

Use squared paper if columns drift. Say place-value names during early practice: “five hundreds”, “four tens”, “six ones”. Then shorten the language as the structure becomes secure.

Worked example: short multiplication

Calculate 2,347 × 6.

Write 2,347 in one row with 6 beneath the ones digit. Work from right to left because a product may create a new group for the next place.

```text 2347 × 6 ------- 14082 ```

Here is the reasoning behind every recorded digit:

Ones: 6 × 7 ones = 42 ones. Write 2 in the ones column and regroup 40 as 4 tens.

Tens: 6 × 4 tens = 24 tens. Add the 4 regrouped tens to make 28 tens. Write 8 in the tens column and regroup 20 tens as 2 hundreds.

Hundreds: 6 × 3 hundreds = 18 hundreds. Add the 2 regrouped hundreds to make 20 hundreds. Write 0 in the hundreds column and regroup 20 hundreds as 2 thousands.

Thousands: 6 × 2 thousands = 12 thousands. Add the 2 regrouped thousands to make 14 thousands.

So 2,347 × 6 = 14,082.

Check the scale: 2,300 × 6 = 13,800, so 14,082 is plausible. Check exactly with a different decomposition: 2,347 × 3 = 7,041, and doubling 7,041 gives 14,082.

A zero in the answer is not an empty column. It states that the final number contains zero hundreds after regrouping. Omitting it would change 14,082 into 1,482.

Worked example: long multiplication

Calculate 326 × 24.

The multiplier 24 means 4 ones and 2 tens. Calculate one partial product for each part.

```text 326 × 24 ------- 1304 6520 ------- 7824 ```

First calculate 326 × 4 = 1,304. This is the ones row.

Next calculate 326 × 20 = 6,520. The 2 in 24 represents 2 tens, not 2 ones, so this row is ten times the value of 326 × 2. Its first place is therefore the tens column. A placeholder zero makes that value visible.

Finally add the partial products:

1,304 + 6,520 = 7,824

Therefore 326 × 24 = 7,824.

Use distributivity as an independent check:

326 × (20 + 4) = (326 × 20) + (326 × 4)

6,520 + 1,304 = 7,824

If a child writes 652 rather than 6,520 in the second row, do not say only “remember the zero”. Ask what the 2 in 24 is worth. The repair is place-value understanding: 20 groups of 326 must be ten times 2 groups of 326.

Worked example: short division

Calculate 3,276 ÷ 6.

Division begins at the highest place value because each remainder must be regrouped into the next smaller unit.

```text 546 6 ) 3276 ```

Read the process by place value:

Thousands and hundreds: 3 thousands cannot be shared into 6 whole groups, so regroup them as 30 hundreds. With the existing 2 hundreds, there are 32 hundreds. Six fits into 32 five times, making 5 hundreds, with 2 hundreds remaining.

Tens: Regroup the remaining 2 hundreds as 20 tens. Add the existing 7 tens to make 27 tens. Six fits four times, making 4 tens, with 3 tens remaining.

Ones: Regroup 3 tens as 30 ones. Add the existing 6 ones to make 36 ones. Six fits exactly six times.

The quotient is 546, so 3,276 ÷ 6 = 546.

Check with the inverse operation: 546 × 6 = 3,276. Also estimate: 3,300 ÷ 6 = 550, so 546 is a sensible size.

The small regrouped values in a short-division layout are not extra quotient digits. They record the amount that could not yet be divided into whole groups and has been exchanged into the next place value.

What to do with a remainder

Calculate 1,258 ÷ 4.

Short division gives 314 remainder 2, because:

4 × 314 = 1,256, leaving 2.

That remainder does not have one automatic final form. Its meaning depends on the question:

If 1,258 counters are placed into complete bags of 4, there are 314 full bags and 2 counters left.

If 1,258 people need cars holding 4 people, 315 cars are needed; the final partly filled car still counts.

As an exact numerical result, the remainder can be written as 2/4 = 1/2, giving 314½.

The curriculum explicitly expects remainders to be interpreted appropriately.[1] Before changing a remainder, ask: What is being counted, and can the answer be partial?

A parent practice sequence

1. Establish meaning without the full algorithm

Use place-value counters, bundled sticks or a quick partition. Show that 247 × 3 means three groups of 200, 40 and 7. For division, share a small amount into equal groups and exchange one ten for ten ones when needed. Keep the representation purposeful rather than decorative.

2. Model one example aloud

Write one correctly aligned example and narrate the place values. Say, “42 ones becomes 4 tens and 2 ones,” not only “write 2, carry 4”. For long multiplication, name the two partial products before adding them.

3. Complete one together

Let the child direct each step while you write. Ask short questions: “Which column are we in?”, “What is this digit worth?” and “Where does the remainder go?” Do not answer your own question immediately.

4. Fade the support

Give a near example with one prompt removed. Then let the child complete a fresh calculation independently. The EEF advises that manipulatives and representations act as a temporary scaffold towards independent mathematics, not a permanent extra procedure.[2]

5. Check by another route

Use estimation first, then an inverse or decomposition. A check should be sufficiently different from the original process to catch the likely error.

6. Respond to the error you actually see

The EEF recommends using assessment to identify the specifics of what a pupil does and does not know, then adapting support.[2] Separate fact errors, place-value errors, layout errors and remainder-interpretation errors instead of assigning a page of mixed repetition.

Common mistakes and precise fixes

| Mistake | What it may mean | Precise response | | --- | --- | --- | | Columns drift left or right | Place value is not visible in the layout | Return to squared paper; label ones, tens and hundreds temporarily | | A regrouped amount is forgotten | The child is treating it as a mark rather than exchanged value | Say the full exchange aloud and circle the regrouped digit | | The long-multiplication tens row is too small | The tens digit was treated as ones | Ask for the value of the multiplier digit; rewrite 24 as 20 + 4 | | Partial products are correct but the total is wrong | The multiplication is secure; the final addition is not | Check only the addition rather than restarting the whole method | | A short-division remainder disappears | Regrouping is being copied without meaning | Exchange the remainder into the next place with counters | | The inverse check repeats the same wrong idea | The check is not independent | Use magnitude or decomposition before recalculating | | Every remainder is rounded up | Context has been ignored | Ask whether a partial group is allowed and what the answer counts |

Adjustment cues

Step back to concrete place value when:

the child cannot say what a carried or regrouped digit represents;

zeros are routinely omitted;

division remainders are moved between columns at random;

a two-digit multiplier is treated as two unrelated one-digit calculations.

Keep the method but reduce load when:

the layout is understood but multiplication facts interrupt every step;

one dense page causes more errors than two spaced examples;

the child succeeds with squared paper but not plain paper;

explanation is accurate but writing speed is slow.

Use smaller digits, provide a multiplication-square reference if appropriate, or pre-draw columns. The goal is to isolate the difficulty without changing the mathematics.

Increase independence when:

the child explains each regrouping without a script;

estimates identify answers that are ten times too large or small;

they choose an inverse or decomposition check themselves;

they interpret a remainder from the wording rather than by habit.

Then vary the numbers, include internal zeros, or ask the child to diagnose a deliberately incorrect worked example.

Frequently asked questions

Should a child always use a written method?

No. Mental calculation may be more efficient for numbers such as 300 × 6 or 840 ÷ 7. The written method is useful when it makes a multi-step calculation reliable and inspectable.

Is “carry the 4” wrong language?

It is incomplete language. It can become shorthand once the child understands that 4 represents four regrouped tens, hundreds or thousands in that particular column.

Why does short division start on the left?

It starts with the greatest place value so any amount that cannot yet form a whole group can be exchanged into the next smaller place value.

Does Year 5 include long division?

The Year 5 programme specifies short division by a one-digit divisor. Formal long division by a two-digit divisor appears in the Year 6 programme, so this guide does not teach it as a Year 5 method.[1]

How many examples should we do at once?

Use enough to expose the pattern without allowing attention to collapse. One modelled example, one shared example and one independent example can be more informative than a long sheet. Adjust from the child’s explanations and error pattern.

What is the best check for multiplication or division?

Combine a rough magnitude check with a mathematically independent check. Estimate first; then use division to check multiplication, multiplication to check division, or split the calculation in a different way.

Next step

Choose one method that currently feels fragile, model its place-value meaning, complete one shared calculation and finish with one independent checked example. Try a Maths lesson.

Sources

[1] Department for Education, National curriculum in England: mathematics programmes of study: https://www.gov.uk/government/publications/national-curriculum-in-england-mathematics-programmes-of-study

[2] Education Endowment Foundation, Improving Mathematics in Key Stages 2 and 3: https://educationendowmentfoundation.org.uk/education-evidence/guidance-reports/maths-ks-2-3