An array of 3 rows of 4 turned sideways is 4 rows of 3. Nothing was added or taken away.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Ravi builds 2 rows of 6 stars. Noor builds 6 rows of 2. Who has more stars?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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.
4 + 4 + 4 = 12
3 × 4 = 12
3 rows of 4 makes 12.
3 × 4 = 12. 3 and 4 are a factor pair of 12.
Pairs you have found
- 3 × 4
Step 3: In Real Life
At the cinema
Cinema seats are an array. A room has 8 rows of 10 seats. It 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 Ana Turn an Array
Ana builds 4 rows of 8 stars. Then she turns it on its side.
- Step 1
Ana builds 4 rows of 8. Each row is a group of 8.
Watch Leo Halve His Work
Leo is asked 10 × 2. Leo starts, and you finish.
- Step 1
Leo reads 10 × 2 as ten twos. That is a long add.
Step 5: Your Turn
Practice makes it stick.
The Bookshop
Problem 1 of 12
Priya buys 5 books at $9 each. She pays $45. Leo buys 9 books at $5 each. How much does Leo pay?
dollars
The Parking Lot
Problem 2 of 12
A parking lot has 6 rows of 7 cars. Seen from the side, it is 7 rows of 6. How many cars fill the lot?
cars
Either Order
Turn the fact round. The product stays put.
Problem 3 of 12
Ana knows 7 × 8 = 56. What is 8 × 7?
Problem 4 of 12
Turn this array on its side. What is it now?
Problem 5 of 12
Which has the same product as 4 × 10?
Find the Mistake
One line on each page is a slip. Find it, then fix it.
Problem 6 of 12
Maya knows 9 × 3 = 27. She uses it for 3 × 9. Which line is a slip?
Maya's answer for: 3 × 9
- 19 × 3 = 27
- 2Turn it round: 3 × 9
- Answer3 × 9 = 72
Problem 7 of 12
What should Maya's product for 3 × 9 be?
Problem 8 of 12
Dev counts an array, then turns it and counts again. Which line is a slip?
Dev's count for 5 rows of 7, then turned
- 15 rows of 7 is 35
- 2Turned: 7 rows of 5
- Answer7 rows of 5 is 40
Problem 9 of 12
What should Dev's second count be?
Same Product?
Same product, or not? Look for what is the same.
Problem 10 of 12
Sam has 6 bags of 9 shells. Ana has 9 bags of 6 shells. What is the same?
Problem 11 of 12
Which sentences give the number of stars?
Turn the Array
Build it, then check.
Problem 12 of 12
Priya built 6 rows of 8. Build it turned on its side.
1 = 1
1 × 1 = 1
1 row of 1 makes 1.
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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Dev learns three facts: 2 × 7, 3 × 8 and 4 × 9. How many facts does he get free by turning them?
facts
Stretch 2 of 2
Sam turns his array on its side. It looks exactly the same. It has 3 rows. How many in each row?
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.
The slip this lesson targets is thinking a turned fact has a turned answer: 9 × 3 = 27, so 3 × 9 = 72. Have them turn something real, an egg box or a muffin tray, and count it both ways; the count that stays put is the proof. Dev's slip is the same idea on an array: a turned array cannot hold a different number. The opposite trap follows a day later. Having just learned that order does not matter, a child says 6 bags of 9 and 9 bags of 6 are the same picture. Ask "how many bags?" to bring the story back: same product, different story.
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.