Arrays and the Commutative Property

Step 1: Let's Learn

Read it, or press Listen and follow the words.

Watch the idea first — 51 seconds. Then read on, and try it yourself in the next step.

An array of 3 rows of 4 turned sideways is 4 rows of 3. Nothing was added or taken away.

So the order does not matter.

Commutative property means you can multiply in either order and get the same product.

Why this is worth knowing

It halves how many facts you have to learn. Knowing 8 × 3 gives you 3 × 8 free.

Same product, different story

3 plates of 4 buns and 4 plates of 3 buns both give 12, but they are not the same picture.

Step 2: Try It Yourself

Tap and try it out.

Swap the rows and columns round. Watch the product stay put.

4 + 4 + 4 = 12

3 × 4 = 12

3 rows of 4 makes 12.

Rows

Columns

The same total, laid out every way it goes. What will not fit sits outside.

3 × 4 = 12. 3 and 4 are a factor pair of 12.

Pairs you have found

3 × 4

How many squares

How many rows

Step 3: In Real Life

At the cinema

Cinema seats are an array. A room with 8 rows of 10 seats holds 8 × 10 = 80 people. Turn the plan sideways and it is 10 rows of 8. Same seats, same 80, whichever way you count them.

Step 4: Watch an Example

One step at a time.

Watch Leo Halve His Work

Leo is learning his facts and is asked for 2 × 9.

Step 1
He does not know 2 × 9 by heart yet.

Step 5: Your Turn

Practice makes it stick.

The Garden

Problem 1 of 2

A garden has 5 rows of 7 plants. How many plants? What is 7 × 5?

plants

Two Ways

Problem 2 of 2

You know 6 × 4 = 24. What is 4 × 6?

Either Order

1 of 4

You know 8 × 2 = 16. What is 2 × 8?

2 of 4

5 × 3 = ?

3 of 4

10 × 4 = ?

4 of 4

Does 7 × 6 give the same product as 6 × 7?

Build the Array

Build it, then check.

Build 3 rows of 4.

Set the rows and the columns.

1 = 1

1 row of 1 makes 1.

Rows

In each row

Step 6: Quick Check

Show what you know.

Question 1 of 2

You know 9 × 2 = 18. What is 2 × 9?

Question 2 of 2

Why is the commutative property useful?

What You Learned

Multiplication can be done in either order.
3 × 4 and 4 × 3 both give 12.
That halves how many facts you have to learn.

For the grown-up

Multiplication is equal groups before it is a fact to recall. Three fours is three groups of four, and division is the same picture asked from the other end.

Watch for multiplication and division being learned as separate subjects. They are one relationship, and a child who sees that halves what there is to remember.

When they are stuck on a division, ask the multiplication instead: "what times 4 gives 24?" Nearly every child can answer that when they cannot answer 24 divided by 4.

Why turning the array proves it — An array with 3 rows of 6 has 18 counters. Turn the page a quarter turn and the same array now reads as 6 rows of 3 — still 18 counters, because rotating a picture does not remove anything. That single observation proves 3 × 6 = 6 × 3 without calculating either one.

It halves how much you must learn — The times tables up to 10 contain 100 facts, but because order does not matter, nearly half are repeats. Learn 7 × 3 and you have 3 × 7 for free. The 100 facts are really about 55 distinct ones, which is a much less daunting number.

Division does not work this way — 12 ÷ 3 = 4, but 3 ÷ 12 is not 4. Multiplication can be flipped and division cannot, in exactly the way addition can be flipped and subtraction cannot. When a property holds it is worth naming, and it is just as worth knowing where it stops.

Turn the array on its side — An array with 3 rows of 5. Count it: 5 + 5 + 5 = 15 Now turn the same array a quarter turn. It is 5 rows of 3. Count it: 3 + 3 + 3 + 3 + 3 = 15 3 × 5 = 15 and 5 × 3 = 15 Nothing was added or removed. Only which way you looked at it.