3-Digit × 2-Digit Written Multiplication

Thirty-four times twenty-six is the area of a rectangle thirty-four long and twenty-six wide. Split thirty-four into thirty and four, and twenty-six into twenty and six, and the rectangle splits into four smaller ones whose areas are easy: thirty times twenty is six hundred, thirty times six is one hundred and eighty, four times twenty is eighty, four times six is twenty-four. Add them: eight hundred and eighty-four. That is the area model. It turns one multiplication you cannot do in your head into four that you can, and it shows you exactly where each part of the answer comes from.

With a three-digit number there are six parts instead of four, and nothing else changes. Two hundred and thirty-four times twenty-six: split into two hundred, thirty, four across the top and twenty, six down the side. The top row, everything times twenty, gives four thousand, six hundred, eighty: four thousand six hundred and eighty. The bottom row, everything times six, gives twelve hundred, one hundred and eighty, twenty-four: one thousand four hundred and four. Add the rows: six thousand and eighty-four. Look at those two row totals — four thousand six hundred and eighty is two hundred and thirty-four times twenty, and one thousand four hundred and four is two hundred and thirty-four times six. The written method is going to be exactly those two rows.

Now the same multiplication in columns. Row one: two hundred and thirty-four times the six. Six fours are twenty-four, write four carry two; six threes are eighteen plus two is twenty, write zero carry two; six twos are twelve plus two is fourteen. One thousand four hundred and four. Row two: two hundred and thirty-four times the twenty. Because it is twenty and not two, the row ends in a zero — write that zero down first — then two hundred and thirty-four times two, which is four hundred and sixty-eight, in front of it: four thousand six hundred and eighty. Add the rows: six thousand and eighty-four. Those are the two row totals from the area model, so you already know they are right.

Multiplication mistakes are mostly the wrong size: a dropped zero makes the answer far too small, an extra one makes it far too big. So the estimate here is about size, not closeness. Two hundred and eighteen times thirty-four: round to two hundred times thirty, six thousand. The exact answer, seven thousand four hundred and twelve, is a four-digit number in the same region, so it passes. One thousand five hundred and twenty-six — someone forgot the zero on the tens row — is a quarter of the estimate and fails. Sixty-six thousand — an extra zero — is ten times too big and fails. Round to the front digit, multiply, and check that your answer is the same size.

On a test the multiplication is often wrapped in words, and the words are the hard part. Multiply the smallest four-digit whole number by the largest two-digit whole number: the smallest four-digit number is one thousand, the largest two-digit number is ninety-nine, and one thousand times ninety-nine is ninety-nine thousand. A crate holds thirty-six bottles, how many in fourteen crates: thirty-six times fourteen. Twenty-eight rows of thirty-six seats: twenty-eight times thirty-six. And when the product is given with a factor missing — a number times twenty-four is six thousand and seventy-two — divide. Quiz and exercises now.