Groups and Arrays in Multiplication
Two powerful visual tools for understanding multiplication are equal groups and arrays. Both help you see why multiplication works the way it does.
Equal Groups
When objects are arranged into groups of the same size, you can multiply to find the total.
| Groups | Size of each group | Multiplication | Total |
|---|---|---|---|
| 4 | 5 | 4 × 5 | 20 |
| 6 | 3 | 6 × 3 | 18 |
| 2 | 9 | 2 × 9 | 18 |
Arrays
An array is a rectangular arrangement of objects organised into rows and columns.
Arrays reveal the commutative property: rotating the array (swapping rows and columns) gives the same total.
Arrays and Area
An array of unit squares IS a rectangle's area. If you fill a rectangle with rows and columns of 1-unit squares, counting those squares (rows × columns) gives exactly the same number as multiplying the rectangle's length by its width. This is why multiplication and area use the very same formula.
Look at a 4-row by 6-column array, but now drawn as a solid grid instead of separate dots. Each little square has an area of 1 square unit, so the whole grid's area is simply the number of squares inside it.
A rectangle 5 units wide and 3 units tall works exactly the same way: fill it with 1-unit squares and you'll find 5 × 3 = 15 square units inside it, arranged in 3 rows of 5. This is precisely the rectangle area formula used on the Area page — Area = length × width — and it's the same rule used later for multiplying algebraic expressions like x(x + 3), where the two side lengths of a rectangle are x and (x + 3).
Key Takeaways
- Equal groups and arrays are visual models of multiplication.
- Rows × columns = total items in the array.
- Rotating an array shows that a × b = b × a.
- Arrays connect directly to the concept of area.
Practice: Count the Array
An array of dots is shown below. Count the total using rows × columns. If you get it wrong (or leave it blank), we'll count it together.