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Turnarounds: Commutative Property of Multiplication

This lesson explains turnarounds, the idea that multiplication facts can be flipped, such as 3 times 4 and 4 times 3, and still give the same product. Students see this with arrays and equal groups, build a table of turnaround pairs, and practice writing and checking turnaround facts.

What is a turnaround in multiplication?

A "turnaround" is a pair of multiplication facts that use the same two numbers but in reversed order, such as \( 3 \times 4 \) and \( 4 \times 3 \). Both facts always give the same product. This idea has a formal name in math, the commutative property of multiplication, but many students first meet it as a simple, friendly rule: you can flip the order of the numbers you are multiplying and the answer will not change.

In symbols, for any two numbers \( a \) and \( b \):

\( a \times b = b \times a \)

So \( 5 \times 2 = 10 \) and \( 2 \times 5 = 10 \) are turnarounds of each other. Once you know one fact in a turnaround pair, you already know the other, which is one of the most useful shortcuts for learning multiplication facts.

Why turnarounds work: the array model

The easiest way to see why turnarounds work is to build an array, an arrangement of objects in equal rows and columns, and then rotate it. Think back to how you built groups and arrays to model multiplication. An array with 3 rows of 4 dots has the same total number of dots as an array with 4 rows of 3 dots, because turning the array on its side does not add or remove any dots.

Rotating an array from 3 rows of 4 into 4 rows of 3 keeps the same total, showing why \( 3 \times 4 \) and \( 4 \times 3 \) are turnarounds.

Writing turnaround facts as statements

Once you can picture the array, it helps to write the pair as two multiplication statements side by side, so the pattern is easy to check:

\( 2 \times 6 = 12 \) and \( 6 \times 2 = 12 \)
\( 4 \times 5 = 20 \) and \( 5 \times 4 = 20 \)
\( 7 \times 3 = 21 \) and \( 3 \times 7 = 21 \)

You can also confirm a turnaround pair by skip counting. Skip counting by 3s four times lands on 12, and skip counting by 4s three times also lands on 12, matching the array picture above.

Using turnarounds to learn facts faster

Because \( a \times b \) always equals \( b \times a \), every turnaround pair is really just one fact to memorize instead of two. For example, once you know \( 6 \times 8 = 48 \), you automatically know \( 8 \times 6 = 48 \) as well. This roughly cuts the multiplication facts you need to practice in half, which is why turnarounds are such a useful tool when you are still building up your multiplication table.

Worked example

Problem: A garden has 6 rows of tomato plants with 9 plants in each row. Without recounting, how many plants would there be if the garden instead had 9 rows with 6 plants in each row?

Solution: The two arrangements are turnarounds of each other, \( 6 \times 9 \) and \( 9 \times 6 \). Since \( 6 \times 9 = 54 \), the turnaround \( 9 \times 6 \) must also equal \( 54 \). The number of plants stays the same, only the shape of the array changes.

What turnarounds do not apply to

Turnarounds work for multiplication and for addition, because \( a + b = b + a \) as well. They do not work for subtraction or division. For example, \( 10 - 4 \) is not the same as \( 4 - 10 \), and \( 8 \div 2 \) is not the same as \( 2 \div 8 \). Keeping this distinction in mind helps avoid a common mixup once students get comfortable flipping numbers in multiplication.

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