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Written multiplication and division strategies - Mathematics Year 5

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Introduction

A print workshop needs 24 packs of 3,246 cards. A club has 2,738 objects to pack six per box. Which operation fits each situation, and what should happen to a remainder?

Connect place-value reasoning with written multiplication and short division. Explain partial products, exchanges and zeroes in a quotient, then interpret an answer in its context. The workshop and club figures are invented teaching examples.

Your outcome: accurate working for four larger multiplications and four divisions, an independent check, an explanation of a first error and two different interpretations of a remainder. Use a calculator only after your written reasoning when checking your answers.

Start with equal groups and place value

Multiplication can describe equal groups, arrays or scaling. Division can describe sharing into a known number of groups or finding how many equal groups fit. Identify what the quantities mean before choosing an operation; a single keyword is not a reliable method.

If four packs each contain eleven counters, the total is $4 \times 11 = 44$ . If forty-four counters are shared equally among four packs, each pack receives eleven. If packs hold eleven each, forty-four counters fill four packs. The same fact can answer different questions, but the units attached to the answer change.

The numbers being multiplied are factors; the result is the product.

Place-value partitioning: split 23 into 20 and 3. Three groups of 23 contain three groups of 20 and three groups of 3:

$23 \times 3 = (20 \times 3) + (3 \times 3) = 60 + 9 = 69.$

This is the distributive relationship. A grid or partial-product record makes it visible; a compact written algorithm records the same values more briefly. A written method should preserve meaning rather than become a sequence of unexplained marks.

For whole numbers, multiplying by ten makes each digit represent ten times its previous value. For example, 42 becomes 420. Dividing 4,200 by 100 gives 42. Explain the change in value; do not rely on 'add or remove zeroes' as a universal rule for all numbers.

Try it: represent 32 times 4 as tens and ones, then calculate. Explain why 30 times 4 and 2 times 4 must be added. The result is 128. For 128 divided by 4, the same partition can support an inverse check.

Estimate first: if each of 24 packs contains about 3,200 cards, you expect tens of thousands of cards, not a few thousand. Estimates are useful for scale; they do not prove every digit of an exact result.

Multiply by one digit with explained exchanges

Work through $1,836 \times 7$ . The ones, tens, hundreds and thousands have different values, even when you record a carried amount with a single small digit.

Seven times six ones is 42 ones. Record two ones and exchange forty ones for four tens.
Seven times three tens is 21 tens. Add the four exchanged tens to make 25 tens. Record five tens and carry two hundreds.
Seven times eight hundreds is 56 hundreds. Add the two carried hundreds to make 58 hundreds. Record eight hundreds and carry five thousands.
Seven times one thousand is seven thousands. Add the five carried thousands to make twelve thousands.

$1,836 \times 7 = 12,852.$

The carried four in the first step means four tens, not four ones. The later carried five means five thousands. Recording the place name makes the algorithm explainable.

Partition	Partial product
1,000 times 7	7,000
800 times 7	5,600
30 times 7	210
6 times 7	42
Sum	12,852

The partitioned check uses the same mathematical relationship but a different organisation, helping you spot a carrying mistake. A rough estimate of 1,800 times 7 gives 12,600, which supports the answer's scale.

Diagnose the first error: a pupil writes two ones after 7 times 6, then calculates the tens as only 7 times 3. The first error is forgetting the four tens exchanged from the ones column. Correct that step before carrying on; simply replacing the final answer does not explain the problem.

Practice: calculate 2,408 times 6 and 6,025 times 7. The answers are 14,448 and 42,175. Explain how a zero in the original number still occupies a place and why the algorithm cannot skip the position.

Check: a carried digit should be added at the appropriate later step once, not ignored and not added twice.

Connect two-digit multiplication with partial products

For the print workshop, calculate $3,246 \times 24$ . Twenty-four is twenty plus four. Make one partial product for each part, then combine them.

$3,246 \times 24 = (3,246 \times 20) + (3,246 \times 4) .$

The four-ones row is 12,984. The two-tens row is 64,920, because 3,246 times 2 is 6,492 and multiplying by twenty makes that ten times as large. The zero in the ones position records a row of tens; it is not a decorative extra character.

$\begin{matrix} 3246 \\ \times 24 \\ 12984 \\ 64920 \\ 77904 \end{matrix}$

The workshop needs 77,904 cards. Use quantity labels when interpreting the result. The two partial rows can be added in either order if their place values remain aligned.

Viewing task: the original Khan Academy video demonstrates multiplying two-digit numbers. It supplies a smaller foundation for the four-digit by two-digit example here. Identify the ones row, tens row and final addition; then explain why the same structure extends to 3,246 times 24. Its US grade label is not the England year allocation used for this course.

An area-style grid: partition 24 into 20 and 4 and 3,246 into 3,000, 200, 40 and 6. Every cell is one product; the sum of all cells is the total.

Multiplier part	3,000	200	40	6	Row total
20	60,000	4,000	800	120	64,920
4	12,000	800	160	24	12,984

Error analysis: adding 12,984 and 6,492 would treat the tens digit as two ones. It produces 19,476, which is a result for multiplication by six rather than twenty-four. The first error is in the value of the second multiplier part.

Practice: calculate 2,408 times 36, 5,072 times 48 and 6,025 times 17. Before checking the answers, explain the meaning of each row. The products are 86,688, 243,456 and 102,425. For the first, compare 72,240 plus 14,448 with your compact method.

Check: a rough calculation of 3,200 times 25 gives 80,000, close to 77,904. This helps reject 19,476, but an inverse or a complete partial-product check is needed to test the exact answer.

Use short division without losing a zero or a remainder

In $2,738 \div 6$ , sharing into six equal groups can be followed place by place. Two thousands cannot give a whole thousand to each group, so exchange them into hundreds. There are now 27 hundreds to share.

Twenty-seven hundreds divided by six gives four hundreds to each group, with three hundreds remaining.
Exchange the remaining three hundreds into thirty tens and include the three tens already present: 33 tens. Each group receives five tens, with three tens remaining.
Exchange those three tens into thirty ones and include the eight ones: 38 ones. Each group receives six ones, with two ones left.

The number being divided is the dividend; six is the divisor. The whole-number part of the answer is the quotient, and the leftover amount is the remainder.

The result is 456 remainder 2. The remainder must be non-negative and smaller than the divisor; a remainder of eight would mean at least one more group of six could be formed.

$2,738 = 6 \times 456 + 2.$

This equation is an exact inverse check. It also explains the units: six groups of 456 account for 2,736 of the original objects, leaving two.

Zero in the quotient: calculate 4,092 divided by 4. Four thousands give one thousand to each group. There are zero hundreds to share, so the quotient needs a zero in the hundreds place. Nine tens give two tens each with one ten remaining; twelve ones then give three ones each.

$4,092 \div 4 = 1,023.$

Writing 123 would lose the empty hundreds place and would fail the check: four times 123 is only 492. A zero in the quotient preserves place value.

Practice: divide 5,075 by 5, 6,484 by 7 and 8,037 by 4. The results are 1,015; 926 remainder 2; and 2,009 remainder 1. Check each using divisor times quotient plus remainder. These include empty places and exchanges, so record positions carefully.

Try a second route: 2,738 can be partitioned as 2,400 plus 300 plus 36 plus 2. Dividing the divisible parts by six gives 400 plus 50 plus 6, with two remaining. This agrees with the short-division result and makes its place-value structure explicit.

Check: do not append a remainder as though it were a decimal digit. A remainder of two when dividing by six represents two sixths of one shared unit, not two tenths.

Let the context decide what the remainder means

The same calculation can lead to different practical answers. Start from $2,738 \div 6 = 456 remainder 2$ , then ask what the divisor, quotient and remainder represent.

Invented situation	Appropriate answer	Reason
Pack 2,738 counters in boxes holding at most six each; every counter must be packed.	457 boxes	456 full boxes hold 2,736 counters; the two remaining counters require another box.
Sell only complete packs of six counters from 2,738 counters.	456 full packs and two counters left	The last two counters do not form another complete sale pack.
Share a 2,738 cm length equally into six mathematically equal lengths, ignoring cutting loss in this model.	456 and one third cm each	The final two centimetres are shared into six, giving two sixths, or one third, of a centimetre each.

The mathematical remainder is not a command to round up every time. Whole containers may require an extra one; complete groups may require keeping only the full-group count; a divisible quantity may be shared further as a fraction. State the modelling assumptions, such as ignoring cutting loss in the length example.

$456 + \frac{2}{6} = 456 + \frac{1}{3} .$

A planning check: 457 boxes have capacity for 2,742 counters, leaving four unused spaces. This is consistent with a final box containing two of its six possible counters. Counting spare spaces is a different contextual check on the rounding-up decision.

Choose an efficient method: 48 times 25 can be found as 48 times 100 divided by 4, giving 1,200. That may be quicker than long multiplication, while still needing an explanation. Use a written method when its clear record helps reliability; do not assume the longest method is always the best.

Your independent set: show full working for 3,246 times 24; 2,408 times 36; 5,072 times 48; and 6,025 times 17. Then divide 2,738 by 6; 4,092 by 4; 6,484 by 7; and 8,037 by 4. The earlier chapters contain the answer key, so complete your first attempt before consulting it.

For two calculations, add an estimate or inverse check. For two contexts using the same remainder, explain why the practical answer differs. Finally, choose one plausible incorrect solution and identify its first wrong step.

Worked review: 'I carried a four' is incomplete. 'I exchanged forty ones for four tens and included those tens in the next place' explains the method. 'I rounded up because there was a remainder' is incomplete. 'Every counter must fit, so the final two require one more box' explains the decision.

Transfer: when a new problem includes a remainder, describe what can and cannot be split before choosing a final answer. Keep the unit and the purpose visible throughout your reasoning.

Interactive Tasks

Quiz: Test Your Knowledge

In 3,246 times 24, what is the partial product for the two tens? (64,920) (!6,492) (!12,984) (!77,904)

After multiplying six ones by seven, what does the carried four represent? (Four tens exchanged from forty ones) (!Four extra ones that can be ignored) (!Four hundreds from the original number) (!Four separate answers to add at the end)

Which result preserves place value in 4,092 divided by 4? (1,023) (!123) (!1,032) (!10,023)

Which equation checks 2,738 divided by 6 as 456 remainder 2? (2,738 equals 6 times 456 plus 2) (!2,738 equals 6 times 456 minus 2) (!2,738 equals 456 plus 6 plus 2) (!2,738 equals 6 times 458)

Why is 455 remainder 8 not a finished whole-number division answer for division by six? (Eight remaining objects still contain another full group of six) (!A remainder must always be zero) (!The quotient must always be rounded up) (!Eight is smaller than the divisor six)

Every one of 2,738 counters must fit in boxes holding at most six. How many boxes are needed? (457 boxes) (!456 boxes) (!458 boxes) (!2,738 boxes)

Only complete packs of six can be sold from 2,738 counters. What is available? (456 full packs with two counters left) (!457 full packs with no counters left) (!456 full packs with six counters left) (!458 full packs with two counters left)

In the length-sharing model, what does the remainder two mean when sharing 2,738 cm equally among six? (Each length receives another two sixths of a centimetre) (!Each length receives another two tenths of a centimetre) (!Two centimetres are added to every share) (!The length must always be rounded up to a whole metre)

What is 5,072 times 48? (243,456) (!60,864) (!24,345) (!202,880)

What can a rough estimate establish? (Whether the scale is plausible, while an exact check is still needed) (!That every final digit is necessarily correct) (!That no working needs to be recorded) (!That a remainder always means an extra container)

Memory Game

Factor	A number being multiplied
Product	Result of multiplication
Dividend	Number being divided
Divisor	Number by which the dividend is divided
Quotient	Result of division, with a whole-number part in a remainder answer
Remainder	Amount left after forming as many whole equal groups as possible

Drag and Drop

Two-tens row	Multiply the other factor by twenty
A zero in the quotient	Preserve an empty place-value position
All objects must fit	Use another container when a non-zero remainder remains
Only complete packs count	Keep the whole-group count and report what is left
A divisible shared length	Share the remaining amount as a fraction of a unit

Crossword Puzzle

PRODUCT	Result of multiplying factors
FACTOR	A number being multiplied
DIVISOR	Number by which another is divided
QUOTIENT	Result of division
REMAINDER	Amount left after whole groups have been formed
EXCHANGE	Replace one larger place-value unit with equivalent smaller units

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

Represent equal groups with place value: Use four groups of eleven and 32 times 4 to connect objects, a partitioned calculation and an inverse division. Label the meaning and units of each number.
Explain a carried amount: Work through 1,836 times 7 and explain each exchanged amount in ones, tens, hundreds or thousands. Do not describe every small carried digit as ones.
Build the two partial-product rows: Partition 24 into twenty and four, then calculate each row for 3,246 times 24. Explain why the tens row has no ones.
Preserve a zero in short division: Calculate 4,092 divided by 4 and explain the zero in 1,023. Use multiplication to show why 123 cannot be correct.

Standard

Complete four larger multiplications: Show full working for 3,246 times 24; 2,408 times 36; 5,072 times 48; and 6,025 times 17. Check at least one with a grid or a different partition.
Complete four short divisions: Divide 2,738 by 6; 4,092 by 4; 6,484 by 7; and 8,037 by 4. Record place-value positions and verify each with divisor times quotient plus remainder.
Interpret one remainder in three contexts: Use 2,738 divided by 6 for boxes, complete sale packs and equal lengths. Give a unit-labelled practical answer and a reason for each.
Find the first multiplication error: A learner adds 12,984 and 6,492 for 3,246 times 24. Identify the first incorrect interpretation, repair the row and check the total.

Advanced

Explain a written method through distributivity: Connect the full area-style grid with the two compact rows for 3,246 times 24. Explain why every part is included once and why the order of adding rows does not change the total.
Compare efficient multiplication strategies: Calculate 48 times 25 using a written method and using multiplication by 100 followed by division by 4. Explain the relationship and when each record is useful.
Evaluate a misleading remainder answer: Analyse 455 remainder 8 for 2,738 divided by 6. Show why the equation can balance while the grouping is unfinished, then give the canonical whole-number quotient and remainder.
Model capacity and spare spaces: Explain why 457 six-counter boxes give four spare spaces for 2,738 counters. Contrast this with selling full packs, and design a new example where the same numerical remainder has different practical interpretations.

Learning Check

Calculate four written products: Solve 3,246 times 24; 2,408 times 36; 5,072 times 48; and 6,025 times 17 with explained rows.
Calculate and check four divisions: Divide 2,738 by 6; 4,092 by 4; 6,484 by 7; and 8,037 by 4, then use inverse checks.
Explain exchange and place value: Explain the first carried four in 1,836 times 7 and the hundreds-place zero in 4,092 divided by 4.
Interpret a remainder in different models: Use the 2,738 divided by 6 result for containers, complete packs and equal lengths, including assumptions and units.
Critique a method and select a check: Correct a two-tens-row error or an overlarge remainder, and distinguish a scale estimate from an exact check.

Worked checks and assessment guidance

The products are 77,904; 86,688; 243,456; and 102,425. The main example combines 64,920 and 12,984. Check that a tens row represents multiplication by twenty, thirty or forty as appropriate, rather than by two, three or four alone.
The results are 456 remainder 2; 1,023; 926 remainder 2; and 2,009 remainder 1. Each can be checked as dividend equals divisor times quotient plus remainder. Remainders are smaller than their divisors, and zeroes in the quotient preserve the correct places.
Seven times six ones gives forty-two ones; forty ones are exchanged into four tens, which join the next step. In the division, the zero hundreds must remain between the one thousand and the later tens and ones. The resulting quotient is 1,023, not 123.
All counters packed requires 457 boxes, with two counters in the final box and four unused spaces. Complete sale packs give 456 packs with two counters left. Equal lengths give 456 and one third cm per length when cutting loss is ignored. The context, rather than the mere presence of a remainder, determines the practical answer.
For 3,246 times 24, replacing the tens row with 6,492 loses a factor of ten; it should be 64,920. For division by six, a remainder eight contains another full group and must be regrouped. An estimate such as about 80,000 supports scale, while a complete partial-product or inverse calculation checks the exact result.

Learning evidence

Knowledge: Explain factor, product, dividend, divisor, quotient, remainder, exchange and partial product.

Skills: Multiply up to four-digit whole numbers by one- or two-digit numbers, use short division by a positive one-digit divisor, preserve place-value positions and interpret remainders through context.

Products: Keep four long multiplications, four divisions, a grid connection, an inverse check, a first-error explanation and two different practical remainder interpretations.

Assessment: Correct final answers matter, but explanations must show the value of carried amounts and partial rows. Accept a different valid method that is clear and checked. A calculator result alone does not show understanding of the written algorithm.

Transfer: Select a method from a new problem's structure, identify units and constraints, and explain whether a leftover amount forms an extra container, remains outside complete groups or can be shared further.

OER and Wikipedia

Department for Education: Mathematics, Year 5 multiplication and division. The course connects formal written methods, mental strategies, place value and contextual interpretations. The workshop, packing and length examples, numerical working and tasks are original teaching material.

Images, unmodified: Dienes Material and ZR100, four tens and four ones, Rhetos, CC0. The materials illustrate arithmetic representations, without making a diagnostic or therapeutic claim about learners. The second photo represents the total 44, not four groups already arranged as eleven each.

Video: Multiplying: 2 digit numbers, Khan Academy, original publisher's YouTube upload. Its smaller multiplication example supports the larger original worked example here; the US grade label does not define the England Year 5 route. The video is embedded rather than downloaded or reproduced.