Long multiplication (age 10+)

For a two-digit multiplier, use short multiplication twice. Start with its ones digit and record the result. Next multiply by its tens value. Write this result beneath the first, with a zero in the ones column.

Each result is a partial product. Add them to find the complete product. This works because the multiplier has been split into its tens and ones.

Check the zero in your tens row. Without it, you have multiplied by the tens digit rather than its value, making the result too small. Round both factors before calculating to estimate what a reasonable answer might be.

A lattice organises the same products used in column multiplication. Draw a grid with a diagonal from bottom left to top right in each box. Write one factor's digits left to right above the grid and the other's top to bottom down its right side. Multiply the digits meeting at each box. Put each product's tens digit above its diagonal and its ones digit below, including a zero when needed.

Start at the bottom-right corner and add along diagonals. Write each total's ones digit at the edge. Carry any tens into the next diagonal towards the top left. Read the answer down the left edge, then across the bottom, ignoring a leading zero.

Both methods combine digit-pair products by place value. The grid helps keep digits aligned. Some learners prefer it; with familiar multiplication facts, columns can be quicker.

Place value tells you what each digit is really worth, and that makes multiplication much easier. For 2,463 times 37, think of 37 as 30 plus 7. Multiplying by 7 gives one partial product, and multiplying by 30 gives the other. Because 30 is ten times 3, that second row always ends in zero. The distributive property is the rule behind this split: a times (b plus c) equals a times b plus a times c. So 2,463 times 37 equals 2,463 times 7 plus 2,463 times 30. Once you see the split, long multiplication is just two easy multiplications and one addition.

The standard algorithm is the neat column layout grown-ups use for big multiplications. You work digit by digit from right to left, carrying tens into the next column as you go. Each row of the layout is lined up under the digit you multiplied by, which keeps ones, tens and hundreds in the right columns. With three- and four-digit numbers there is more carrying to track, so go slowly and write every carry clearly. A quick estimate tells you roughly how big the answer should be. If your answer is ten times too big or too small, a missing or extra zero is usually the cause.

Whole-number multiplication combines equal groups. With four groups of 25, repeated addition gives 25 + 25 + 25 + 25 = 100. Multiplication records this as 4 × 25 = 100.

Factors are the numbers being multiplied; their result is the product. Familiar multiplication facts help with larger calculations.

Split larger factors into smaller parts. To multiply by 30, multiply by 3, then by 10. For a whole number, multiplying by 10 places a zero after its digits. Record partial products in aligned columns and add them. This also works when multiplying a four-digit number by a two-digit number.