Multiply two-digit by one-digit

Split the number

A two-digit number is two numbers wearing one coat. The number 36 is 30 plus 6. When you multiply 36 by 4, you can take the coat apart: multiply the 30 by 4, multiply the 6 by 4, and add the two answers. Both little multiplications use facts you already know.

This works because multiplication counts equal groups, and a group of 36 is a group of 30 sitting next to a group of 6. Four groups of 36 are the same as four groups of 30 plus four groups of 6. Nothing is lost in the split; you have only made the counting easier.

Two steps, one sum

Here is the method written as steps. First, split the two-digit number into its tens and ones. Next, multiply each part by the one-digit number. Finally, add the two products. For 4 × 36: split 36 into 30 and 6, work out 4 × 30 = 120 and 4 × 6 = 24, then add 120 + 24 = 144.

The two products are easy to hold in your head because one of them always ends in zero. The tens part gives a multiple of ten, and the ones part stays small. Adding them is the only real work.

Why the area model helps

The area model turns the calculation into a picture. Draw a rectangle whose width is the two-digit number and whose height is the one-digit number. Cut the rectangle into a tens part and a ones part. The area of each part is one of the little multiplications, and the area of the whole rectangle is the final answer.

The model also gives you a way to check whether an answer is sensible. A rectangle of 4 by 36 is a little smaller than a rectangle of 4 by 40, which has an area of 160. So an answer near 140 makes sense and an answer near 400 does not.

A short note about carrying

Some books teach a shorter written method that carries a digit from the ones place into the tens. That method is the same split-and-add idea, just written vertically. The split method explains why it works, so learn it first. When the split feels automatic, the shorter method becomes easy to read.