Arrays and the Order Property

Look at a carton of eggs, a chocolate bar or the windows on a building. The items are often arranged in neat rows and columns. This arrangement is called an array, and it is one of the best ways to see multiplication. In this lesson you will build arrays and learn a powerful shortcut called the order property.

What is an array?

An array is a set of objects arranged in equal rows and equal columns.

• Rows go across, like a line of people standing side by side.
• Columns go up and down, like the columns that hold up a building.

Reading an array

Here is an array of stars with 3 rows and 4 columns:

We can describe it as 3 rows of 4, which is 3 × 4 = 12.

The order property

Turn the array on its side. Now there are 4 rows of 3. Did the number of stars change? No! There are still 12 stars.

3 × 4 = 12 and 4 × 3 = 12

This is called the order property of multiplication (in maths books it is also called the commutative property). It means you can multiply two numbers in any order and get the same product.

Why this is a superpower

The order property cuts the number of facts you need to learn almost in half! If you know 2 × 9 = 18, you also know 9 × 2 = 18. If you know 5 × 7 = 35, you also know 7 × 5 = 35.

Making arrays

You can draw arrays with dots or build them with counters, coins or blocks. Arrays help you check your answers and solve problems you have not memorised yet.

Example

Kiran is planting a vegetable garden. He plants 6 rows of carrots with 5 carrots in each row. How many carrots does he plant?

Kiran draws an array with 6 rows and 5 dots in each row. He skip counts by 5s down the rows: 5, 10, 15, 20, 25, 30. So 6 × 5 = 30 carrots.

Splitting an array

Big arrays can be split into smaller, easier arrays. For 7 × 4, split the 7 rows into 5 rows and 2 rows. Then 5 × 4 = 20 and 2 × 4 = 8. Add them: 20 + 8 = 28. You will use this splitting trick again for harder tables.

Key takeaways

• An array shows equal rows and columns.
• An array with 3 rows of 4 shows 3 × 4.
• The order property: 3 × 4 = 4 × 3.
• Big arrays can be split into smaller ones to make facts easier.

Practice

1. Draw an array for 2 × 6. What is the product?
2. If 4 × 7 = 28, what is 7 × 4?
3. Split 8 × 3 into 5 × 3 and 3 × 3. What is the product?

Answers: 1 is 12 (2 rows of 6 dots). 2 is 28, because of the order property. 3 is 15 + 9 = 24.