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Advanced long multiplication
Up to 4 digits by 2 digits, with zero placeholders and accurate carrying.
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Long multiplication lets you multiply big numbers, like 3,527 × 46, without a calculator. It is built from things you already know: your times tables, multiplying by 10 and column addition.
In this lesson you will learn:
- short multiplication by a one-digit number, and how to carry
- why long multiplication works
- long multiplication step by step, with the zero placeholder
- how to spot the mistakes that cost marks
- how to estimate to check your answer
1. Short multiplication: the building block
Key idea
Multiply each digit by the one-digit number, starting with the ones. Write the ones digit of each
answer and carry the tens into the next column.
Example: 3,476 × 6
TThThHTO3476×6208562243
- Ones: 6 × 6 = 36. Write 6, carry 3.
- Tens: 7 × 6 = 42, add the carried 3: 45. Write 5, carry 4.
- Hundreds: 4 × 6 = 24, add 4: 28. Write 8, carry 2.
- Thousands: 3 × 6 = 18, add 2: 20. Write 0, carry 2.
- Nothing is left to multiply, so the carried 2 goes straight into the answer.
Answer20,856
Watch out
Multiply first, then add the carry. In the tens, 7 × 6 + 3 = 45. Adding first,
(7 + 3) × 6 = 60, is wrong.
The multiplication workshop: short multiplication
How it works
- Tap the digit that goes in the glowing box. Work from the ones, on the right.
- Get two right with no slips and the workshop moves you up a level. Level 5 has 4-digit numbers.
2. Why long multiplication works
Key idea
To multiply by a 2-digit number, split it into tens and ones.
243 × 26 is 243 × 20 plus 243 × 6.
Split 243 too, and every part is easy:
| × | 200 | 40 | 3 | Row total |
|---|---|---|---|---|
| 20 | 4,000 | 800 | 60 | 4,860 |
| 6 | 1,200 | 240 | 18 | 1,458 |
- The × 6 parts add up to 1,458. That is the first row of long multiplication.
- The × 20 parts add up to 4,860. That is the second row.
- 1,458 + 4,860 = 6,318. So 243 × 26 = 6,318.
Where the zero comes from
× 20 is the same as × 2, then × 10. Multiplying by 10 moves every digit one place to the
left, and a 0 fills the ones column. That 0 is the zero placeholder.
The splitter: change the numbers and watch the parts
Top number
Multiply by
The boxes show the parts. They are not drawn to scale.
Things to try
- Tap a row of boxes to see which row of long multiplication it makes.
- Make the multiplier 10, 20, 30. What happens to the × ones row?
3. Long multiplication step by step
Key idea
Row 1: multiply by the ones digit. Row 2: write a 0 in the ones column,
then multiply by the tens digit. Last: add the two rows.
Example: 243 × 26
ThHTO243×261458+48606318
- Row 1: 243 × 6 = 1,458.
- Row 2: you are multiplying by 2 tens (20), so write a 0 in the ones column first. Then 243 × 2 = 486, written next to it: 4,860.
- Add the rows: 1,458 + 4,860 = 6,318.
Answer6,318
A bigger one: 3,527 × 46
HThTThThHTO3527×4621162+141080162242
- Row 1: 3,527 × 6 = 21,162.
- Row 2: write the 0, then 3,527 × 4 = 14,108: so 141,080.
- Add the rows: 21,162 + 141,080 = 162,242.
Answer162,242
Tidy carrying
Write each carry small. When row 1 is finished, cross its carries out before you start row 2, so you
never add them twice.
The multiplication workshop: long multiplication
How it works
- Fill row 1, then row 2 (it starts with the zero placeholder), then add the rows.
- Carries show small above the top number while you work on a row.
- Two right with no slips moves you up a level. Level 5 is 4 digits by 2 digits.
4. Spot the mistake
Key idea
Most lost marks come from four slips: leaving out the zero placeholder, forgetting a
carry, adding a carry before multiplying, and adding the two rows wrongly.
The most common one: no zero placeholder
243×261458+4861944
- Row 2 should be 243 × 20 = 4,860. Without its 0, it became 486, which is only 243 × 2.
- So the answer, 1,944, is far too small. The right answer is 6,318.
Be the teacher: is this working right?
5. Estimate to check
Key idea
Round each number to make an easy sum, then multiply. Your answer should be close to the
estimate.
Example: is 6,318 a sensible answer to 243 × 26?
- 243 is about 200. 26 is about 30.
- 200 × 30 = 6,000.
- 6,318 is close to 6,000, so it is sensible. An answer like 1,944 is far too small: something went wrong.
- Round to the nearest 10, 100 or 1000, whichever makes the numbers easy. Rounding is Lesson 1.3.
- An estimate tells you the size of the answer. It does not replace working it out.
Sensible or silly? Use an estimate to find the right answer
6. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Use rough paper and set each one out in columns. Move up when you are getting most of them right.
The mistakes that cost marks
- Leaving out the zero placeholder. 243 × 26 worked out as 1,458 + 486 = 1,944. row 2 is × 20, so start it with a 0: 4,860. Then 1,458 + 4,860 = 6,318.
- Adding the carry before multiplying. In 3,476 × 6, working out the tens as (7 + 3) × 6 = 60. multiply first, then add the carry: 7 × 6 = 42, and 42 + 3 = 45.
- Forgetting to carry. 3,476 × 6 written as 18,426, with every carry left out. write each carry small under the next column and add it in: 20,856.
- Using row 1’s carries again in row 2. The old carries get added to the new row by mistake. cross out row 1’s carries before you start row 2.
- Adding the two rows wrongly. The rows are added in columns, with carrying, just like Lesson 2.1. line up the ones, add from the right, and carry when a column makes 10 or more.
- Not checking the size. An answer of 1,944 for 243 × 26 should ring alarm bells. estimate: 200 × 30 = 6,000, so the answer should be about 6,000.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 2.2.
Download
Print these and work through them away from the screen.
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