One large product, many small products

Before this page: know the multiplication tables to 12 × 12 from the KS2 page on multiplication and division, be able to add in columns, and know what each digit is worth in integers and decimals.

347 × 26 is not a table fact. There is no row of the times tables that reaches it and no single step that produces it. What is already known is 6 × 7, 6 × 4, 2 × 3 and the rest of the small products it is built from.

Both numbers can be broken into what their digits are worth. 347 is 300 + 40 + 7, and 26 is 20 + 6. Multiplying by 26 means multiplying by 20 and by 6 and adding the two results, and the same is true of the three parts of 347. Every part of one number has to meet every part of the other, which comes to six small multiplications.

Everything on this page is those six products. What changes, from one section to the next, is how much of the working is written down — and the short form at the end is the one that writes the least, which is exactly why it is the hardest to read.

Every pair of parts

A rectangle gives each of the six products a box of its own. The parts of one number run along the top, the parts of the other down the side, and each box holds what its row and its column make together.

Not one of the six needs anything beyond a table fact and a count of zeros. 20 × 300 is 2 × 3 with three zeros after it. The grid never has to be drawn in any particular order, because the boxes do not depend on each other.

The same products, written downwards

The boxes can be written as a column of lines instead. Nothing is recalculated. Each box becomes one line, placed so that its digits stand in the columns they are worth, with the product that made it written alongside so that no line has to be taken on trust.

This is long multiplication with all of its working showing. The order is fixed now — the ones of the multiplier first, and within that the ones of the number above — so that each line reaches a little further left than the one before it. Every line can be checked against its box, and the last step is one ordinary column addition.

Gathering the lines

Six lines is a lot of writing for one calculation. They fall into two families: three came from the 6 in 26 and three from the 20. Each family can be added on its own before anything else happens.

That leaves one row for each digit of the multiplier. The six products are all still there; they have simply been added in two stages instead of one.

Why a row ends in zeros

Look at the three lines that came from the 20: 140, 800 and 6,000. Every one of them ends in a zero, because every one of them is 20 times something. Their total, 6,940, ends in a zero too, and it could not do anything else.

A third digit in the multiplier is worth a hundred, so all of its lines end in two zeros and its row does too. The pattern continues for as many digits as the lower number has: one more zero for each place.

The short form

The short form produces those two rows without ever writing the lines that make them. Instead of 1,800 and 240 and 42 going down the page, each column's product is written straight into the row and anything too large for one digit is carried into the next column. The adding happens as the row is made.

The same six products, the same two rows and the same answer. What has gone is the writing, and with it the chance to check each line against a box. Anyone who loses their way in the short form can go back to the lines, which show their working at every step.

A three-digit multiplier

Three digits below give three rows, and the third starts two places further left again because its digit is worth a hundred. Nine products in all this time, gathered into three rows.

Where it turns up

Exam questions rarely say multiply. Two situations account for most of them, and both are answered by the same rows.

Total, altogether, each, per, product and area all lead to the same arrangement. What changes is the unit attached to the answer, which the question supplies and the columns do not.

Multiply two chosen integers

Common mistakes

The arithmetic in each step is small enough to be reliable. What goes wrong is nearly always the arrangement of the rows or the handling of a carried digit.

• The missing place-holder zeroA second row without its zero is ten times too small, and a third row without its two zeros is a hundred times too small. Write the zeros first, before the row is started.
• Adding the carry before multiplyingIn 347 × 6 the ones give 42, so 4 is carried. The tens are then 6 × 4 = 24, and the carried 4 joins afterwards to make 28. It is not (4 + 4) × 6 = 48.
• Carrying between rowsEach row starts again with nothing carried into it. A digit carried in the ones row belongs to the ones row alone.
• Losing one of the productsThree digits over two is six products, and three over three is nine. Counting them is a quick check that none has been missed.

Summary

1. 347 × 26 is six small products: every part of 347 against every part of 26.
2. A grid gives each product a box. A column layout gives each one a line, placed in the columns it is worth.
3. The lines belonging to one digit of the multiplier add together to make one row.
4. A row made by the tens digit ends in one zero, and one made by the hundreds digit ends in two, because every line in it did.
5. The short form writes each row directly, carrying instead of setting the lines out, and then adds the rows in columns.