Arrays and the Area Model
Open an egg box, look at a muffin tray, or glance at a bar of chocolate: the pieces sit in neat
rows and columns, and you can tell how many there are without counting each one. That tidy
rectangle is the shortcut this page is about.
An array arranges things in neat rows and columns — a tidy
rectangle, with the same number in every row and the same number in every column. The moment
things line up like that, you can read off a
multiplication at a
glance: count the rows, count the columns, and multiply.
\text{rows} \times \text{columns} = \text{total}
So an array with a rows of b dots holds
a \times b dots in all — no counting one by one.
Here is a tray with 3 rows of 4 cookies:
You don't have to count "1, 2, 3, …, 12". You see 3 rows of 4 and write
3 \times 4 = 12 \text{ cookies}.
Turn the array — the order doesn't matter
Tip the same tray on its side and the cookies become 4 rows of 3. Not a single
cookie was added or eaten, so the total is still 12:
3 \times 4 = 4 \times 3 = 12.
Count the rectangle by rows and you get a rows of
b, which is a \times b. Count the very same
rectangle by columns and you get b columns of
a, which is b \times a. It is one rectangle,
so the total cannot change:
a \times b = b \times a
That is a picture of why the order of a multiplication never matters — turn the array on
its side and every square is still there.
See it built
Watch an a \times b rectangle fill in, square by square. First the
rows, then the columns — and the total is just rows times columns. Step through it.
Drawing a product as the rows-by-columns rectangle is called the area model: the
product is the area of a rectangle that is a units tall and
b units wide, measured in little unit squares.
\text{area} = \text{length} \times \text{width} = \text{product}
Because the squares fill the whole rectangle exactly, "how many squares" and "what is the area"
are the same question — and both equal a \times b.
Arrays are everywhere once you look. A box of 2 rows of 5 oranges is
2 \times 5 = 10 oranges:
Egg cartons, muffin trays, chocolate bars, window panes, the squares on a chessboard — every one
is an array waiting to be read as a multiplication.
The clever bit: split a big rectangle into easy pieces
The area model is the trick that lets us multiply bigger numbers in our heads. Take
4 \times 13. A 13-times-table is hard to
remember — but a 13-wide rectangle can be cut into a
friendly 10-wide piece and a 3-wide piece:
4 \times 13 = 4 \times 10 + 4 \times 3 = 40 + 12 = 52.
Cutting the rectangle does not change its area — the two pieces hold exactly the squares the whole
rectangle did, so the parts must add back to the whole. Each piece is an easy fact
(4 \times 10 and 4 \times 3), and we add
them. That is the whole idea behind the long-multiplication you'll meet later.
Picture 4 rows of 13 cookies. Slide a tray
divider after the 10th cookie in every row. The left tray is
4 \times 10 = 40 cookies; the right tray is
4 \times 3 = 12 cookies. Nothing fell on the floor, so the two trays
together are still 40 + 12 = 52 — the same as before you split them.
You may split a rectangle any way you like; the pieces always add back to the whole.
Worked examples
-
An array. A garden has 5 rows of
6 tomato plants. That is 5 \times 6 = 30
plants — and turning it round, 6 columns of 5
gives the same 30.
-
An area. A rug is 3 m long and
4 m wide. Its area is 3 \times 4 = 12 square
metres — twelve 1\,\text{m} \times 1\,\text{m} tiles would cover it.
-
A split. 6 \times 12 = 6 \times 10 + 6 \times 2 = 60 + 12 = 72.
The hard product becomes two easy ones that add to the whole.
Two array traps to avoid:
-
Count rows AND columns — not one edge twice. The total is
rows \times columns. Counting the dots along one side and squaring it,
or counting the same edge twice, gives the wrong answer. For a 3 \times 4
array the answer is 12, not 3 + 4 = 7 (that's
the distance round, not the amount inside) and not 4 \times 4.
-
The split parts must add back to the whole. When you cut
4 \times 13 into 4 \times 10 and
4 \times 3, the second piece is 4 \times 3 — the
same number of rows, only fewer columns. A common slip is to change the rows too;
keep the rows fixed and only the widths add: 10 + 3 = 13.
Make your own array
Here is a fresh rectangle every time. The squares are shaded in two colours to show one way of
splitting it: the left block plus the right block add back to the whole product.
Press Refresh for a new array.
See it explained
Khan Academy builds multiplication up from area models: