Multiplication with arrays & the area model
Theory
An array shows a product as rows \(\times\) columns. An area model splits a factor by place value into smaller rectangles — the partial products — which you add for the total.
An array sets objects in equal rows. The number of objects is rows \(\times\) columns, so \(4\) rows of \(6\) is \(4 \times 6 = 24\).
An area model draws a product as a rectangle. Split the long side by place value into a tens part and a ones part.
Each smaller rectangle is a partial product. Work out each one, then add them for the total.
Turning an array on its side swaps the rows and columns but keeps the same product.
An area model for \(7 \times 32\) has two partial products that add to the total.
| Part | Multiply | Value |
|---|---|---|
| Tens | \(7 \times 30\) | \(210\) |
| Ones | \(7 \times 2\) | \(14\) |
| Total | \(210 + 14\) | \(224\) |
How to use an area model
- Split by place value. Break the long side into a tens part and a ones part.
- Multiply each part. Work out both smaller rectangles.
- Add the partial products. Join the two parts for the total.
Rows times columns.
| \(4 \times 6\) | \(=\) | \(24\) |
Split \(32\) into \(30 + 2\).
| \(7 \times 30\) | \(=\) | \(210\) |
| \(7 \times 2\) | \(=\) | \(14\) |
| \(210 + 14\) | \(=\) | \(224\) |
Split \(15\) into \(10 + 5\).
| \(8 \times 10\) | \(=\) | \(80\) |
| \(8 \times 5\) | \(=\) | \(40\) |
| \(80 + 40\) | \(=\) | \(120\) |
There are \(6\) chairs in each row.
| \(30 \div 5\) | \(=\) | \(6\) |
Common pitfalls
Frequently asked questions
What is an array in multiplication?
Objects set out in equal rows. The total is rows \(\times\) columns, so \(4\) rows of \(6\) is \(4 \times 6 = 24\).
How does an area model work?
Draw the product as a rectangle and split one side by place value. Multiply each smaller rectangle, then add the partial products.
What is a partial product?
The result of one of the smaller rectangles. For \(6 \times 24\) the partial products are \(6 \times 20 = 120\) and \(6 \times 4 = 24\).
Why split the number by place value?
Because the tens and ones are easy to multiply on their own. \(24 = 20 + 4\), so \(6 \times 24 = 120 + 24 = 144\).
Does turning an array change the answer?
No. A \(4 \times 6\) array turned on its side becomes \(6 \times 4\); both hold \(24\) objects.