What is the area model in multiplication?

A rectangle split up by place value. For 46 × 23, split 46 into 40 and 6 and 23 into 20 and 3, find the four smaller areas (800, 120, 120 and 18), then add them to get 1058.

Is long multiplication the same as the standard algorithm?

Yes. In the US it is usually called the standard algorithm, and in England it is called long multiplication when you multiply by a two-digit number. Both mean the column method with one row for each digit of the bottom number.

Why do you put a zero in long multiplication?

Because the second row multiplies by the tens digit, which is worth ten times as much. In 46 × 23 the second row is 46 × 20 = 920, and the 0 holds the ones place.

How do you multiply a 3 digit number by a 2 digit number?

The same way as 2 digit by 2 digit: one row for the ones digit, one row starting with a 0 for the tens digit, then add. For 331 × 34, the rows are 1324 and 9930, and 1324 + 9930 = 11,254.

What grade learns 2 digit by 2 digit multiplication?

In the US Common Core, grade 4, usually with area models (4.NBT.B.5), then the standard algorithm in grade 5 (5.NBT.B.5). In England, long multiplication by a two-digit number begins in Year 5.

Is there a free multi-digit multiplication worksheet with answers?

Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

Grades 4 and 5 (Years 5 and 6 in England)

Multi-digit multiplication: long multiplication and the area model

Here is how to multiply bigger numbers two ways: the area model, which shows why it works, and long multiplication, which is quick on paper. The worked example is 2 digit by 2 digit, and the free practice set below goes up to four digits by two. Print a fresh worksheet with an answer key when you are ready.

The method, step by step

Write the bigger number on top and line up the ones.
Multiply the top number by the ones digit of the bottom number and write the result as the first row.
Start the second row with a 0 in the ones place, because you are now multiplying by tens.
Multiply the top number by the tens digit and write the result after the 0.
Add the two rows to get the answer, then check it with an estimate.

CCSS 4.NBT.B.5CCSS 5.NBT.B.5England: Y4 Multiplication and divisionEngland: Y5 Multiplication and divisionEngland: Y6 Multiplication and division

Worked example

46 × 23 = ?

Multiply by the ones digit: 46 × 3 = 138.
Multiply by the tens digit: 46 × 20 = 920. Write a 0 first, then 46 × 2 = 92.
Add the rows: 138 + 920 = 1058.
Check with an area model: 40 × 20 = 800, 40 × 3 = 120, 6 × 20 = 120 and 6 × 3 = 18.
800 + 120 + 120 + 18 = 1058, the same answer.
Estimate: 46 is about 50 and 23 is about 20, and 50 × 20 = 1000, so 1058 is sensible.

Practice now

Question 1 of 100 correct

8131 × 17 = ?

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Why multi-digit multiplication works

Every method splits the numbers by place value and multiplies the parts. 23 is 20 + 3, so 46 × 23 is 46 × 20 plus 46 × 3. This is the distributive property, and children have usually met it already in times tables: 7 × 12 = 7 × 10 + 7 × 2 = 70 + 14 = 84.

Area model multiplication turns this into a picture. Draw a rectangle 46 long and 23 wide. Split the long side into 40 and 6 and the short side into 20 and 3, which cuts the rectangle into four smaller ones: 40 × 20 = 800, 6 × 20 = 120, 40 × 3 = 120 and 6 × 3 = 18. Their total, 1058, is the area of the whole rectangle, which is 46 × 23. Every partial product is a piece you can see, so none gets forgotten.

Long multiplication, step by step

Long multiplication, called the standard algorithm in the US, is the area model with the pieces combined. The first row, 46 × 3 = 138, holds the 120 and 18 pieces. The second row, 46 × 20 = 920, holds the 800 and 120 pieces. That is also why the second row starts with a 0: you are multiplying by 2 tens, not by 2.

Each row has its own carrying. For 46 × 3: 6 × 3 = 18, so write 8 and carry 1. Then 4 × 3 = 12, plus the carried 1 makes 13, so the row reads 138. Cross out those carries before you start the next row so they do not get mixed up with the new ones.

From 2 digit by 2 digit to bigger numbers

The rows grow longer but the steps stay the same. These examples get a little harder each time.

Two digits by one: 62 × 7 = 60 × 7 + 2 × 7 = 420 + 14 = 434.
2 digit by 2 digit: 35 × 12. 35 × 2 = 70 and 35 × 10 = 350, so 70 + 350 = 420.
Carrying in both rows: 78 × 64. 78 × 4 = 312 and 78 × 60 = 4680, so 312 + 4680 = 4992.
Three digits by two: 331 × 34. 331 × 4 = 1324 and 331 × 30 = 9930, so 1324 + 9930 = 11,254.
Four digits by two: 2575 × 46. 2575 × 6 = 15,450 and 2575 × 40 = 103,000, so 15,450 + 103,000 = 118,450.

An estimate catches the biggest slips. 331 × 34 is about 300 × 30 = 9000, so 11,254 is the right size. If you forget the 0 and add 331 × 3 = 993 instead of 9930, you get 2317, which the estimate shows is far too small.

How it is taught in school

In the US, Common Core grade 4 multiplies a number of up to four digits by a one-digit number and multiplies two two-digit numbers, using place value strategies and explaining them with equations, rectangular arrays or area models (4.NBT.B.5). Grade 5 multiplies multi-digit whole numbers fluently using the standard algorithm (5.NBT.B.5). So the standards bring in the area model first and the standard algorithm a year later.

In England, Year 4 pupils multiply two-digit and three-digit numbers by a one-digit number using a formal written layout. Year 5 multiply numbers up to four digits by a one- or two-digit number, including long multiplication for two-digit numbers, and Year 6 use long multiplication for numbers up to four digits by a two-digit number. The grid method you may see in English primary schools is the area model drawn as a table.

How parents can help

Check times tables first. Long multiplication is mostly single-digit facts, so if 7 × 8 needs a long pause, practise the tables alongside.
Draw the area model together, even if your child already knows long multiplication. It shows where every number comes from.
Use squared paper so the digits and rows stay lined up.
Ask for an estimate before every answer, rounding each number to the nearest ten or hundred.
Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.

Common mistakes

Forgetting the 0 on the second row: 46 × 23 worked as 138 + 92 = 230. The second row is 46 × 20, not 46 × 2.
Multiplying only the matching digits: 4 × 2 and 6 × 3, giving 818 for 46 × 23. This misses the two middle pieces of the area model, 40 × 3 and 6 × 20.
Adding a carried digit before multiplying instead of after. In 46 × 3 it is 4 × 3 + 1 = 13, not (4 + 1) × 3 = 15.
Leaving the first row's carries in place, so they get added into the second row by mistake. Cross them out first.
Getting every multiplication right and then slipping in the final addition. Line the rows up carefully and check each column.

Questions people ask

What is the area model in multiplication?: A rectangle split up by place value. For 46 × 23, split 46 into 40 and 6 and 23 into 20 and 3, find the four smaller areas (800, 120, 120 and 18), then add them to get 1058.
Is long multiplication the same as the standard algorithm?: Yes. In the US it is usually called the standard algorithm, and in England it is called long multiplication when you multiply by a two-digit number. Both mean the column method with one row for each digit of the bottom number.
Why do you put a zero in long multiplication?: Because the second row multiplies by the tens digit, which is worth ten times as much. In 46 × 23 the second row is 46 × 20 = 920, and the 0 holds the ones place.
How do you multiply a 3 digit number by a 2 digit number?: The same way as 2 digit by 2 digit: one row for the ones digit, one row starting with a 0 for the tens digit, then add. For 331 × 34, the rows are 1324 and 9930, and 1324 + 9930 = 11,254.
What grade learns 2 digit by 2 digit multiplication?: In the US Common Core, grade 4, usually with area models (4.NBT.B.5), then the standard algorithm in grade 5 (5.NBT.B.5). In England, long multiplication by a two-digit number begins in Year 5.
Is there a free multi-digit multiplication worksheet with answers?: Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

Updated November 1, 2026.

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