Stacking multiplications 7 - rectangles

Here is a stacked multiplication. First, we multiply, digit by digit, and then we finish by adding several numbers. But why do we need to add? We’re doing multiplication here! To answer this, we need to remember that a multiplication, like 12 times 34, can be seen as a rectangle: 12 steps in one direction and 34 steps in another direction.

This creates a rectangle with 12 times 34 small squares. When we count all the squares, we find that there are 408, because 12 times 34 is 408. Let’s look more closely at the rectangle. This side, 12, can also be seen as 10 plus 2. Let’s draw a slightly thicker line here.

Now we see the large rectangle divided into two smaller rectangles. And do you remember? Each rectangle is a multiplication. So, the large rectangle is split up into two smaller rectangles. And the large multiplication is split up into two smaller multiplications: This one, 2 times 34.

This one, 10 times 34. The number of squares in the large rectangle is still the same, 408. This means that 2 times 34, plus 10 times 34, equals 408. Let's put this aside, return to the stack and do the calculation again. Let’s start here: 2 times 4.

This is 8. And then 2 times 3. This is 6. The number written below the bar here, 68, is the exact number of small squares in this rectangle: 68. Then we go to the next calculation of the stack.

1 times 4 is 4. 1 times 3 is 3. But keep in mind that this 1 has a tens place value! So, this is 34 tens, 340. And this is the exact number of squares in this rectangle.

340. And then we add it up. 68 plus 340 is 408. Just like the numbers in the large rectangle to the right. So, when we stack a multiplication, we split it into smaller parts.

Then we add them. The same way this large rectangle, one large multiplication, can be split into two smaller rectangles, two smaller multiplications, that we finish by adding together. This explains why we sometimes use addition, even though we are really calculating multiplication.