Lesson 2 of 10 · Grades 3 to 4
Arrays and the commutative property
An array arranges objects in neat rows and columns. Count the rows, count the columns, multiply, and you have the total. Arrays also show something surprising: turning the array around gives a different multiplication with the same answer.
Rows and columns
An array is a neat arrangement of objects in rows and columns. Every row holds the same number of objects, and every column holds the same number too. A tray of cupcakes, a page of stickers, the windows on a building and a grid of dots can all be arrays.
An array lets you see a multiplication instead of adding it. Count the rows, count the number in one row, and multiply. An array with 3 rows of 8 has 3 × 8 dots. You could just as well count the columns: 8 columns of 3, which is 8 × 3. Both counts describe the same dots, and both give 24.
Turn the array around
Imagine you turn the whole array a quarter turn, like turning a page in a book. The dots do not move away or appear; they simply line up the other way. What were 3 rows of 8 are now 8 rows of 3. The shape has changed, but the number of dots has not.
This is a picture proof of something important: 3 × 8 = 8 × 3. The order of the factors does not change the product. You can always swap the two numbers you multiply and get the same answer. Mathematicians call this the commutative property. Long word, simple idea: factors can travel.
Why the turn-around rule helps
The rule cuts the number of facts you need to remember. If you know 3 × 8, you also know 8 × 3 without any extra work. When a larger times table fact looks hard, look at it the other way round. Some children find 2 × 9 easier to picture as 9 groups of 2, even though 2 groups of 9 is equally correct.
The turn-around rule works for any two whole numbers, no matter how big. It even works when one of the numbers is 1: 1 × 7 and 7 × 1 both equal 7. Multiplication is flexible, and you are allowed to choose the direction that suits you.
Useful tip
When you meet a difficult multiplication, try both orders and pick the one you can picture. 12 × 3 can be read as 12 groups of 3, which is easy to double and double again: 3, 6, 12, 24, 36. That is the same work as 3 groups of 12, just with steps you may already know.
Worked example
A sheet of stickers is arranged in 6 rows with 7 stickers in each row. How many stickers are on the sheet? Then write the turn-around fact.
Step 1: an array multiplies rows by columns. There are 6 rows and 7 columns, so the multiplication is 6 × 7.
Step 2: work out the product. 6 × 7 = 42.
Step 3: turn the sheet a quarter turn. Now there are 7 rows of 6, which is 7 × 6.
Step 4: 7 × 6 also equals 42. Turning the array did not add or remove a single sticker.
Answer: 42 stickers. The turn-around fact is 7 × 6 = 42.
Remember
- An array has equal rows and equal columns.
- Rows × columns gives the total number of objects.
- Turning the array around swaps the factors.
- The commutative property says a × b = b × a.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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An array has 3 rows with 8 tiles in each row. How many tiles are there?
Show answer
24 tiles. 3 × 8 = 24.
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You turn that same array a quarter turn. Now it has 8 rows. Write the multiplication and solve it.
Show answer
8 × 3 = 24. The tiles did not change; only the way you read the array changed.
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An array has 4 rows and 6 columns. How many dots does it hold?
Show answer
24 dots. 4 × 6 = 24.
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Does 9 × 2 give the same answer as 2 × 9?
Show answer
Yes. Both give 18. Swapping the factors does not change the product.
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An array has 5 rows and 5 columns. How many dots are in it?
Show answer
25 dots. 5 × 5 = 25.
Puzzle · Tile mosaic
Puzzle time
A mosaic has 3 rows with 8 tiles in each row. Every tile is the same size. How many tiles are in the mosaic?
Show a hint
Multiply the number of rows by the number of tiles in one row.
Show the answer
Answer: 24. 3 × 8 = 24 tiles. Turning the mosaic around would count 8 rows of 3, which is still 24.