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Long multiplication and long division - Mathematics Year 6

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Introduction

Why does a written method work, and what should happen to a remainder? Multiply four-digit values by two-digit values, divide four-digit values by two-digit divisors, and turn a calculation into an answer that fits its context. Use estimates, place value and inverse checks rather than trust a neat-looking column.

Success criteria. Use formal multiplication and division with correct alignment; justify a suitable representation or shortcut for an unfamiliar problem; interpret a remainder; and communicate a complete checked answer with appropriate units.

This Year 6 course develops England's upper Key Stage 2 arithmetic. Three teaching chapters connect diagrams, an original Commons animation, formal written working and the original publisher's video. Native practice and worked assessment guidance provide an independent route. All packing and ribbon situations are fictional; online answers do not certify actual workshop measurement or production.

Make every multiplication row represent the correct place

Begin with the meaning of a product. If a fictional printer makes 4,326 sheets in each batch and completes 27 equal batches, the total is 4,326 multiplied by 27. The units are sheets. Repeated equal batches justify multiplication; a keyword alone does not decide an operation.

Partition the multiplier. Twenty-seven is twenty plus seven. Multiplying by each part and adding the results gives the same total:

$4,326 \times 27 = 4,326 \times 20 + 4,326 \times 7 = 86,520 + 30,282 = 116,802$

This distributive structure explains the two rows of long multiplication. It is not two unrelated tricks. An area or grid model uses the same decomposition; the formal layout records it more compactly.

Work the ones row. Seven times six is forty-two: record two ones and regroup four tens. Seven times two tens plus four tens is eighteen tens: record eight tens and regroup one hundred. Seven times three hundreds plus one hundred is twenty-two hundreds: record two hundreds and regroup two thousands. Seven times four thousands plus two thousands is thirty thousands. The first partial product is 30,282.

Work the tens row. The 2 in 27 represents twenty, not two. Twice 4,326 is 8,652; multiplying by twenty makes 86,520. A zero placeholder or an explicitly shifted row keeps the ones place correct. If a layout omits the written zero, the shift still has to represent it. Do not add an unshifted 8,652 as if it were the twenty-batch contribution.

Add aligned places. Read the total 116,802 as sheets, not batches. A product of a four-digit and a two-digit number can have more than four digits. Do not remove a leading digit to make the answer fit an expected number of columns.

Estimate independently. About 4,300 sheets times about 30 batches gives about 129,000 sheets. This rough scale is compatible with 116,802 but not 11,680 or 1,168,020. An estimate is a reasonableness check, not proof of every digit. For a stronger check, reverse the calculation or use a different partition.

A second exact route. Twenty-seven is thirty minus three. Thirty batches give 129,780 sheets, three batches 12,978; subtracting gives 116,802. This is a genuinely different decomposition from adding the twenty and seven contributions.

$4,326 \times 27 = 129,780 - 12,978 = 116,802$

Read the introductory diagram. Its smaller example is 57 × 23. The first row contributes 171. The shifted digits 114 represent 1,140, not one hundred and fourteen added in the ones column. Together they give 1,311. Explain that place-value shift in words as well as recognise the final digits.

Watch a second example.

Animation building the written multiplication forty-three times twenty-five. — Original animated multiplication by Miljutin. The example is 43 × 25; connect its partial rows to 43 × 5 and 43 × 20.

The animation gives 215 and 860 as contributions, totalling 1,075. Its numeric layout illustrates the method; written explanations here do not require following the animation's timing. Source language or layout choices do not alter the place values.

A zero in the multiplier. For 3,024 × 40, the zero ones contribution is zero and the tens contribution is 120,960. Compute 3,024 × 4 = 12,096, then scale that product by ten. Do not drop the tens place and report 12,096.

Choose a method. Long multiplication is reliable for a general two-digit multiplier. A mental regrouping can be efficient for forty, twenty-five or a value near thirty when you can explain it accurately. Show the structure and a check, not a complicated layout merely to make a short calculation look difficult.

Divide by two digits and preserve quotient places

Name the quantities. In 3,847 ÷ 14, the dividend is 3,847 and the divisor is 14. The quotient counts complete groups or the amount per group, depending on context. A remainder records what is left after the whole-number part. It is not a decimal tail that can be attached to the quotient.

Estimate first. Four thousand divided by about twenty gives a rough scale of two hundred, while 14 × 200 = 2,800 and 14 × 300 = 4,200 place the exact quotient between 200 and 300. The bracket check is more informative here than the coarse estimate alone.

Formal long division, first stage. Fourteen does not fit into the leading 3 thousands, so consider the first two digits, 38. Fourteen fits twice with a remainder of ten. At this stage those two quotient units occupy the hundreds column: 200 groups account for 2,800 of the original amount. The written prefix calculation compresses that place-value reasoning.

Second stage. Combine the prefix remainder ten with the next digit four to form 104. Fourteen fits seven times, giving 98 and leaving six. This quotient digit is in the tens place; seventy groups contribute 980 of the original amount.

Third stage. Bring the final digit seven into the remaining prefix to form 67. Fourteen fits four times, giving 56 and leaving eleven. The final quotient is 274 with remainder 11.

Prefix being divided	Quotient digit and its place	Prefix product	Prefix remainder
38	2 hundreds	28	10
104	7 tens	98	6
67	4 ones	56	11

Compact written record.

Read the digits with the table and place explanations. “Bring down” without explaining what the prefix represents can hide the reason the method works. Smaller layouts may look different while preserving the same quotient positions and subtractions.

Two essential checks. Multiply the quotient by the divisor and add the remainder:

$14 \times 274 + 11 = 3,836 + 11 = 3,847$

Also require the remainder to be non-negative and smaller than the positive divisor:

$0 \leq 11 < 14$

A proposed remainder of twenty-five with divisor fourteen would still contain another complete group and therefore would not be finished. Both the reconstruction equation and remainder bound matter.

An alternative by chunks. Remove 200 groups, then 70, then 4. The partial quantities are 2,800, 980 and 56. They sum to 3,836, leaving eleven. The total quotient is 200 + 70 + 4. This helps explain the formal digits; the standard algorithm is a compact record of structured grouping.

A zero quotient digit matters. For 4,248 ÷ 21, the first prefix 42 gives two hundreds. The next digit four is smaller than twenty-one, so record zero tens before considering 48. Two ones use forty-two and leave six. The result is 202 remainder 6, not 22 remainder 6.

$21 \times 202 + 6 = 4,242 + 6 = 4,248$

A practical checking question. If someone proposes 22 remainder 6, reconstructing gives 21 × 22 + 6 = 468, which is far below 4,248. The inverse check exposes the missing quotient place. A tidy-looking written record is not enough.

Khan Academy's original Introduction to dividing by 2 digits uses a smaller example to show grouping by a two-digit divisor. Identify dividend, divisor and quotient, then explain how the same structure extends to 3,847 ÷ 14. US grade labels do not define the England route. The written table and worked checks remain available without the video.

Independent practice. Divide 6,725 by 25; 5,423 by 17; 7,426 by 24; and 8,005 by 32. Show formal working, estimate or bracket each quotient, then use multiplication plus remainder to check. Keep a zero quotient digit where a place requires it. The assessment guidance provides results after your attempt.

Interpret remainders and select a method that fits

One calculation can answer different questions. The whole-number result 3,847 ÷ 14 = 274 remainder 11 does not decide by itself how a real question should be answered. State what is being grouped or shared and what must happen to the remainder. An answer may require complete groups only, enough containers for everything, or a fractional share.

Context A: complete bundles. A fictional store has 3,847 sheets and makes bundles of fourteen. It can make 274 complete bundles, with eleven sheets not in a complete bundle. The answer is not 274.11 bundles: the remainder eleven is measured in sheets, while the quotient counts bundles.

Context B: containers for every sheet. If each container holds at most fourteen sheets and every sheet must be stored, 274 containers are insufficient: eleven sheets remain outside them. Use 275 containers, with the last partly filled. This is a contextual ceiling decision, not ordinary rounding to the nearest whole number.

Context C: equal divisible shares. A fictional ribbon is 3,847 cm long and is shared equally into fourteen pieces, with no cutting loss in the model. Each piece is 274 and eleven-fourteenths centimetres long:

$\frac{3,847}{14} = 274 + \frac{11}{14}$

This value is between 274 and 275 cm. It is not 274.11 cm. A real cutting task would need a suitable measurement process, precision and loss allowance; the model's exact answer is not proof that a physical ribbon has been measured or cut.

Why eleven-fourteenths? After assigning 274 cm to each of the fourteen pieces, eleven centimetres remain to share among fourteen pieces. Each receives eleven divided by fourteen centimetres more. The divisor belongs in the fraction denominator; the remainder is not automatically a count of hundredths.

Method selection workshop. For 3,024 × 40, multiply by four and then ten. For 4,326 × 27, two partial products or the thirty-minus-three decomposition can be efficient. For 3,847 ÷ 14, formal long division is systematic; chunks can explain its places. Choose on mathematical structure, the numbers and the required explanation.

A plausibility trap. A learner multiplies 4,326 by 27 as 30,282 + 8,652 and obtains 38,934. The addition of those two numbers is correct; the first mathematical error was treating the multiplier's two tens as two ones. Correct that row to 86,520, then add again. Checking only the final addition would not repair the method.

Another trap. A learner writes 3,847 ÷ 14 = 274.11 because there are eleven left. Multiply 274.11 by fourteen and it does not reconstruct 3,847. The decimal notation represents hundredths, not the remainder's share by fourteen. Use the stated remainder form or the exact fraction form appropriate to the question.

Four multiplication practice cases. Work independently before reading the assessment answers: 2,456 × 34; 3,024 × 40; 5,078 × 26; 9,806 × 17. Include an estimate and explain at least one regrouping in each formal method. Use a different partition to check one answer.

Context transfer. A fictional workshop has 7,426 plain cards and uses twenty-four per complete pack. Determine the complete packs and spare cards. Then determine how many trays of maximum capacity twenty-four are needed to hold every card. Use the same division and explain why the two whole-number answers differ.

Remainder comparison. For 8,005 ÷ 32, the whole-number quotient is 250 with five left. If dividing a continuous quantity into thirty-two equal shares, the extra share is five thirty-seconds of the relevant unit. If counting full packs of thirty-two discrete items, five items remain. Always connect the unit to the quantity being described.

A complete solution record. Write what the question asks; identify equal groups or sharing; estimate or bracket the result; choose and carry out a method; reconstruct with an inverse check; then interpret the result with units and the context rule. A correct bare quotient is not a full answer to a question about enough containers.

Teacher assessment. Look for four-digit by two-digit formal calculations, place-preserving partial products and quotient digits, a valid remainder bound and contextual choices. If a learner can execute a method but not explain a remainder, return to the bundles/containers/ribbon comparison rather than prescribe more identical columns.

Exit challenge. Create two questions using the same dividend and divisor that need different contextual answers. Give formal working once, explain each interpretation and show how a multiplication check supports the arithmetic while the wording determines what to report. An oral explanation with annotated working is an equally valid route to showing reasoning.

Interactive Tasks

Quiz: Test Your Knowledge

What does the two in the multiplier twenty-seven represent? (Twenty, so its partial product must be placed in the tens position) (!Two ones with no change of place) (!Two hundred in every multiplication) (!A zero contribution that can always be discarded)

What are the correct partial products for four thousand three hundred and twenty-six times twenty-seven? (Eighty-six thousand five hundred and twenty plus thirty thousand two hundred and eighty-two) (!Eight thousand six hundred and fifty-two plus thirty thousand two hundred and eighty-two) (!Eighty-six thousand five hundred and twenty plus three thousand and twenty-eight) (!Four thousand three hundred and twenty-six plus twenty-seven)

What does the shifted row labelled one-one-four represent in the introductory diagram? (One thousand one hundred and forty) (!One hundred and fourteen added in the ones column) (!Eleven and four tenths) (!One hundred and forty-one)

What is three thousand and twenty-four multiplied by forty? (One hundred and twenty thousand nine hundred and sixty) (!Twelve thousand and ninety-six) (!Three thousand and sixty-four) (!One million two hundred and nine thousand six hundred)

Which statement correctly describes three thousand eight hundred and forty-seven divided by fourteen? (The quotient is two hundred and seventy-four with remainder eleven) (!The quotient is twenty-seven with remainder forty-seven) (!The exact answer is two hundred and seventy-four point eleven) (!The remainder can be fourteen or greater in a completed whole-number result)

Which check is necessary for a completed positive-divisor remainder calculation? (Divisor times quotient plus remainder reconstructs the dividend, and the remainder is smaller than the divisor) (!The quotient and remainder must contain the same digits) (!The remainder must always equal the divisor) (!Only the final addition needs checking, regardless of grouping)

Why is a zero tens digit required in four thousand two hundred and forty-eight divided by twenty-one? (After the hundreds step the next prefix is smaller than the divisor, but the tens place must remain present) (!Zero always means the entire quotient is zero) (!Every division must have exactly three non-zero quotient digits) (!The remainder replaces the tens digit)

How many capacity-fourteen containers are needed for all three thousand eight hundred and forty-seven sheets? (Two hundred and seventy-five containers) (!Two hundred and seventy-four containers with eleven sheets unstored) (!Two hundred and seventy-four point eleven containers) (!Exactly fourteen containers regardless of quantity)

What does the remaining eleven mean when a ribbon length is shared equally among fourteen pieces? (Each piece gets eleven-fourteenths of a centimetre beyond the whole-number share) (!Each piece automatically gets eleven hundredths of a centimetre) (!The original total has eleven centimetres added fourteen times) (!The quotient must be rounded down even when exact equal lengths are requested)

Which strategy best supports an unfamiliar context problem? (Use a justified method, an independent arithmetic check and the context’s rule for the remainder) (!Always append the remainder as decimal digits) (!Choose an operation only because one keyword appears) (!Use the same whole-number answer for complete packs and enough containers)

Memory Game

Dividend	The amount being divided
Divisor	The number by which the dividend is divided
Quotient	The result or whole-number part of division
Remainder	The amount left after complete whole-number grouping
Partial product	The contribution from one part of a multiplication
Estimate	An approximate check on scale or size
Inverse check	A reverse operation used to check a result

Drag and Drop

Tens partial product	The twenty-batch contribution is eighty-six thousand five hundred and twenty
Quotient placeholder	Zero tens preserve the two-hundreds-and-two result
Complete-group interpretation	Count only full bundles and state the leftover sheets
Enough-container interpretation	Add a partly filled container when items still remain
Divisible-share interpretation	Share the remainder by the divisor as a fraction of the unit

Crossword Puzzle

DIVIDEND	The amount to be divided
DIVISOR	The number dividing the amount
QUOTIENT	The division result or its whole-number part
REMAINDER	What is left after complete grouping
PRODUCT	The result of multiplication
ESTIMATE	An approximate reasonableness check

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Read partial rows: Explain the two partial products in 4,326 × 27 and in the introductory 57 × 23 diagram. State how the shifted row keeps its place value.
Four formal products: Complete 2,456 × 34; 3,024 × 40; 5,078 × 26; 9,806 × 17. Show alignment, estimates and a different decomposition for one check.
Four formal divisions: Complete 6,725 ÷ 25; 5,423 ÷ 17; 7,426 ÷ 24; 8,005 ÷ 32. Give quotient and remainder, formal working, a reconstruction and the remainder bound.
Zero quotient place: Explain 4,248 ÷ 21 using prefix stages and the zero tens digit. Check the correct result and show why twenty-two remainder six is wrong.

Standard

Same arithmetic different questions: Use 3,847 ÷ 14 for complete fourteen-sheet bundles, capacity-fourteen containers holding everything, and fourteen equal ribbon-length shares. Interpret units and remainders separately.
Find the first error: Analyse the incorrect product formed from 30,282 + 8,652. Explain why correct addition can follow an earlier place-value error and repair the original method.
Compare division methods: Solve 3,847 ÷ 14 formally and by chunks of two hundred, seventy and four groups. Explain how the methods represent the same structure and verify the remainder.
Workshop transfer: Solve both versions of the 7,426-card problem with packs or trays of twenty-four. Include estimates, formal work, a reconstruction and justified contextual whole-number answers.

Advanced

Remainder as a fraction: Explain why the exact equal-share ribbon length uses eleven-fourteenths beyond the whole-number quotient, rather than point-eleven. Check by multiplying the share by fourteen.
Method efficiency: Compare two justified methods for 4,326 × 27 and for 3,024 × 40. Explain when the formal layout or a structural shortcut is clearer, without claiming one method is best for all numbers.
Create a paired problem: Invent two questions with a four-digit dividend and two-digit divisor that require complete groups versus enough containers. Show one arithmetic solution and two clearly different interpretations.
Diagnose and improve: Review a partner’s formal working for place alignment, zero quotient digits, remainder bound, inverse check and units. Identify the first incorrect step, correct it and explain why the revised answer fits the context.

Learning Check

Four formal multiplications: Calculate the four independent products with checks.
Four formal divisions: Calculate the four independent divisions and verify quotient/remainder conditions.
Place-value explanation: Explain the formal product and the zero-containing quotient.
Context interpretations: Interpret 3,847 divided by fourteen in the three stated contexts.
New contextual transfer: Answer the card-pack and tray questions with a checked conclusion.
Choose and communicate: Show an unfamiliar solution with method choice, units and error correction.

Worked checks and assessment guidance

In the stated order: 2,456 × 34 = 83,504; 3,024 × 40 = 120,960; 5,078 × 26 = 132,028; 9,806 × 17 = 166,702. Require correct place alignment and an estimate. A structural check may use a different partition; a plausible estimate alone does not verify every digit.
In the stated order: 6,725 ÷ 25 = 269 remainder 0; 5,423 ÷ 17 = 319 remainder 0; 7,426 ÷ 24 = 309 remainder 10; 8,005 ÷ 32 = 250 remainder 5. Reconstruct each dividend with divisor × quotient + remainder. The remainder must be non-negative and smaller than the positive divisor.
4,326 × 27 gives 30,282 from seven and 86,520 from twenty, totalling 116,802. The smaller diagram’s shifted 114 represents 1,140, making 57 × 23 = 1,311. For 4,248 ÷ 21, quotient digits two hundreds, zero tens and two ones give 202 remainder 6. Omitting the zero changes the value and fails reconstruction.
There are 274 complete bundles and eleven sheets left. Holding every sheet in capacity-fourteen containers requires 275 containers. An exact fourteen-way ribbon share is 274 and eleven-fourteenths centimetres per piece. It is not 274.11. Arithmetic checks the grouping; the question determines which interpretation to report.
7,426 cards give 309 complete packs of twenty-four with ten cards left, because 24 × 309 + 10 = 7,426. To hold every card in trays with maximum capacity twenty-four requires 310 trays. State cards, packs and trays distinctly. The last tray is partly filled; this is not rounding to the nearest whole quotient.
Require a four-digit/two-digit calculation, justified formal or structural method, estimate or bracket, exact inverse check and context conclusion. A wrong tens row or missing zero quotient place must be repaired at its first incorrect step. For remainder claims, check both reconstruction and bounds rather than append digits as a decimal.

Learning evidence

Keep four multiplications, four divisions with estimates and exact inverse checks, a zero-quotient explanation and at least two contrasting remainder contexts. Record method choice and correct units. Annotated working, spoken explanation or written solutions are suitable; no actual cutting, packing or production competence is certified by online arithmetic.

OER and Wikipedia

Checked on 11 November 2026. Department for Education: mathematics programmes of study, Year 6 addition/subtraction/multiplication/division, informs formal methods through four-digit by two-digit calculations, estimation, inverse checking and contextual remainder interpretation. This course is part of the wider programme, not a complete arithmetic qualification.

The printer, bundles, containers, workshop cards and ribbon situations, numerical work, formal layouts and tasks are original fictional teaching material. Prefix steps are explained with their place value. An exact length model does not establish actual instrument precision, cutting loss or observed practical competence.

Embedded original: Introduction to dividing by 2 digits, Khan Academy. Its smaller example supports the divisor method before transfer to the larger course cases. The video is embedded from its publisher rather than copied or described as Commons-licensed; written teaching and checks are independent.

Unchanged Commons still image and original animation are credited below. The 57 × 23 diagram uses a shifted tens row whose omitted final zero is explained; the 43 × 25 animation is connected to its two correct partial products. No dependence on colour alone or animation timing is required.

Long multiplication.png, Phlsph7, CC BY-SA 4.0. Unchanged.

Poser-une-multiplication.gif, Miljutin, CC BY 3.0. Unchanged.