In math, there are certain rules and patterns we learn early on that become the foundation for everything else we do.
Just like we memorize the multiplication table, learn how to add and subtract, or understand what place value means, some rules quietly guide us through math problems, even when we don’t realize it.
One of those important rules is the commutative property.
In this guide, we’ll explain what the commutative property really means, show you how it works through simple examples, clear up common questions, and give you a fun quiz to test what you’ve learned.
If we start from the meaning of the word commute, we know it means to move or switch places, like when someone commutes to work or school.
In math, the commutative property uses this same idea: it shows that we can switch the order of numbers in an operation and still get the same result.
The commutative property is one of the four basic number properties that help us understand how numbers behave and how to solve problems more easily.
The other three are:
Associative Property: Changing how numbers are grouped doesn’t change the result.
Distributive Property: Multiplying a number by a group of numbers added together is the same as multiplying each part separately and adding.
Identity Property: Adding 0 or multiplying by 1 keeps the number the same.
The commutative property of addition means that you can add numbers in any order, and the total will stay the same.
We can write this property using a simple equation:
a + b = b + a
Let’s see how this looks with actual numbers:
3 + 8 = 8 + 3 = 11
5 + 12 = 12 + 5 = 17
5 + 9 = 9 + 5 = 14
The commutative property of multiplication tells us that changing the order of two numbers in a multiplication problem does not change the product.
In math, we can express this with an equation:
ab = ba
Let’s see what that means using real numbers:
6 × 4 = 4 × 6 = 24
2 × 15 = 15 × 2 = 30
7 × 1 = 1 × 7 = 7
Knowing a rule is one thing… seeing it in action is where the fun begins!
Let’s test the commutative property in different situations to prove just how consistent it really is.
Does the commutative property still work when we use negative numbers?
Let's do a little experiment and see:
8 + (–20) = –20 + 8
We’ll simplify both sides step by step.
Left side:
8 + (–20) = –12
Right side:
–20 + 8 = –12
Both sides equal –12, so yes, the commutative property still works, even when we’re adding negative numbers.
Of course, the same would apply if we worked with multiplication.
Does the commutative property still work when we have more than two numbers?
Let’s test it out!
Try this multiplication problem:
2 × 3 × 4
Now let’s change the order of the numbers:
4 × 2 × 3
Let’s solve both and see what happens:
2 × 3 = 6, and 6 × 4 = 24
4 × 2 = 8, and 8 × 3 = 24
We still get 24 both times!
That means the commutative property works even when we’re multiplying three numbers, not just two. We can switch the order of the numbers, and the answer stays the same.
Ready for a quick challenge? Let’s see how well you really understand the commutative property with this mini quiz.
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The commutative property of numbers is one of the most straightforward rules in math. However, that doesn’t mean it can’t cause a few head-scratchers for students who are learning it for the first time.
Here are some of the most common questions about commutative property, including the ones we get at Mathnasium, along with answers to clear up any confusion!
Most students first learn about the commutative property in early elementary school, usually around 2nd or 3rd grade, as they begin working with basic addition and multiplication.
At Mathnasium, we help them see the logic behind the rule—that changing the order of numbers doesn’t change the result—so it truly makes sense.
No, because in subtraction and division, the order matters. Changing the order of the numbers gives you a different answer, which means the commutative property doesn’t apply.
Let’s look at subtraction:
5 – 3 = 2
3 – 5 = –2
The answers are not the same, so subtraction is not commutative.
Now try division:
10 ÷ 2 = 5
2 ÷ 10 = 0.2
Again, the results are different, so division doesn’t follow the commutative property either.
Yes! In addition, 0 can switch places with any number.
5 + 0 = 0 + 5 = 5
In multiplication, it still works too:
0 × 4 = 4 × 0 = 0
So zero follows the commutative property—just remember the answer might still be zero!
The commutative property shows us that we can change the order of numbers in addition or multiplication, and the answer will stay the same.
For example:
2 + 3 = 3 + 2
4 × 5 = 5 × 4
The associative property shows us that we can change how numbers are grouped in addition or multiplication, and the answer will still stay the same.
For example:
(2 + 3) + 4 = 2 + (3 + 4)
(2 × 3) × 5 = 2 × (3 × 5)
These two properties are like a math dream team. The commutative property lets us change the order of numbers. The associative property lets us change how we group them.
When we use both, we can solve problems faster and make mental math easier, especially when working with three or more numbers.
Yes, as long as you're only using addition or multiplication, you’ll still get the same result, no matter the order.
That’s the beauty of the commutative and associative properties. For example, 2 + 3 + 4 is the same as 4 + 2 + 3, and (2 × 3) × 4 is the same as 2 × (3 × 4). These properties help us see that math can be flexible and reliable.
But here’s where you want to be a little strategic: sometimes, rearranging numbers can make the math a whole lot easier to do in your head.
Take 4 × 9 × 5. You could go left to right and do 4 × 9 = 36, then 36 × 5 = 180.
But if you rearrange it to 4 × 5 × 9, you get 20 × 9 = 180—same answer, much faster and simpler!
So while the result doesn’t change, the order you choose can save time and boost confidence. That’s why we teach students to use these properties with purpose, turning every problem into an opportunity to think clearly and solve smart.
Yes! That’s a real-world example of the commutative property of multiplication.
Whether it’s 3 × 5 or 5 × 3, you still have 15 pencils. At Mathnasium, we love using everyday situations like this to help students truly understand how math works all around them.
Mathnasium is a math-only learning center dedicated to helping students of all skill levels unlock their math potential.
Using the Mathnasium Method™, a proprietary teaching approach that combines personalized learning plans with proven techniques, Mathnasium focuses on building solid math foundations, expanding mathematical thinking, and even transforming how students think and feel about math.
Mathnasium’s specially trained tutors work with elementary school students to help them learn and master any math topic, including the commutative property of numbers.
At Mathnasium, we assess each student’s current skills and consider their unique academic needs to develop personalized learning plans that will put them on the best path toward math mastery.
Whether your student is looking to catch up, keep up, or get ahead in math class, find a Mathnasium Learning Center near you, schedule an assessment, and enroll them today!
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