Common Mistakes Kids Make With the Commutative Property (and How To Fix Them)

Math rules can sometimes feel confusing, especially when they involve abstract ideas. One concept that often puzzles young learners is the commutative property, which is the idea that numbers can be added in any order and still give the same result. 

When you help children spot commutative patterns, they can solve problems more efficiently. This also helps build their confidence when solving multi-step tasks and word problems.

What Is the Commutative Property of Addition?

The commutative property says that changing the order of numbers that you are adding or multiplying doesn’t change the sum or product. For example, whether you have “two groups of four” or “four groups of two,” you still have eight in total.

Let’s try it on a number line jump. Begin at zero and do two jumps of five for addition and see which number you land on. After that, begin at zero again and do five jumps of two. Notice that you land in the same place.

The same is true when you work with arrays for multiplication. Notice that the rows × columns can be flipped to be the columns × rows without changing the total.

Common Mistakes Kids Make (and How To Fix Them)

Learners often misapply the commutative property by using it with subtraction/division, confusing it with grouping, or overlooking patterns. You can help them avoid these errors through clear visuals, hands-on practice, and consistent reinforcement. 

Here are some of the common mistakes you should be aware of:

Thinking It Works for Subtraction or Division

Mistake: Some learners assume that 5 − 3 is equal to 3 − 5, or that 12 ÷ 3 equals 3 ÷ 12.

Fix: Clarify that addition and multiplication are the only operations that follow the commutative property rule. You can also model the principle using counters by attempting to remove three from five versus removing five from three to have a visual that shows different results.

Confusing Order (Commutative) With Grouping (Associative)

Mistake: Students change parentheses but think they’re changing order, e.g., (2 + 3) + 4 vs. 2 + (3 + 4).

Fix: Show the difference between order and grouping. You can do that using counters.

• Demonstrate the commutative property by swapping the position of items. Show the equation: 1 + 2 + 3 = 6. Move the 1 to the end, like so: 2 + 3 + 1. Explain that nothing is grouped, and you're just swapping the order so the total stays the same.
  
  
• Move to associative, where you group first. Use this example: 5 + 3 - 2, but this time circle the first two (5 + 3) - 2, then circle the last two: 5 + (3 - 2). You’re regrouping, not moving numbers.

Not Recognizing Patterns Across Representations

Mistake: Kids don’t notice that reverse-order sums or products match.

Fix: Reinforce patterns using visuals like number lines, arrays, or counters. Here are a few exercises to try:

• With Objects: Put 3 red and 2 blue counters on the table. Count them: 5. Now swap their places (2 blue, 3 red). Count again, and explain that you switched places, but the total stayed the same.
• On a Number Line: Start at 0. Jump 3, then 2, and you land on 5. Now jump 2, then 3, and you still land on 5.
• On Paper: Write 3 + 2 and 2 + 3. Circle both answers and explain that both make 5.

Skipping Concrete, Real-Life Practice

Mistake: Having students learn only from hypothetical problems on a page, making the idea feel abstract.

Fix: Apply the rule to snacks, building blocks, or classroom supplies. Use this commutative property of addition example: “We have 3 snack packs of 4 crackers. Do 4 packs of 3 crackers change the total?” Tying the rule to real objects helps anchor understanding, allowing  children to transfer the idea more smoothly to problems on paper.

Fun Activities To Reinforce the Commutative Property

It will help young learners to grasp the concepts mentioned earlier when you use fun games and activities. Here are a few ideas you can try:

Swap and Solve

Write a set of addition and multiplication facts, then have learners swap the numbers and check that the total or product matches. Ask “What changed? What stayed the same?”

Array Flip Challenge

Build arrays (e.g., 2 × 6) with tiles or stickers, then rotate them to 6 × 2. Record both equations, and write a sentence describing why the total didn’t change.

Story Problems

Create two short stories that differ only in order. For instance, you can use a story that mentions 5 red marbles and 3 blue. You can then invert the order and use 3 blue and 5 red in the story. Make the students solve both story problems and explain why the answers in both stories are the same.

Interactive Practice

Using Prodigy Math, you can assign practice skills focused on the properties of addition and let students explore them through digital gameplay. 

Tips for Parents and Teachers

• Focus on Hands-On, Visual Learning First: Start with counters, number lines, and arrays before moving to symbols. That concrete-to-representational progression helps learners internalize the rule.
• Say It Both Ways: Encourage kids to read each problem forward and in reverse: “seven plus two” and “two plus seven.” Hearing both versions keeps the emphasis on order.
• Keep It Short and Playful: Use brief, high-frequency practice so the idea stays fresh without causing fatigue.
• Praise Understanding: Use positive feedback such as “You showed why the sums match!” to highlight the importance of reasoning as much as the correct answer.

Turn Learning Into an Everyday Skill

Understanding the commutative property of addition helps kids solve addition and multiplication problems more easily. Keep practicing hands-on, visual, and connected to real-life objects so the pattern becomes second nature.

Bring math concepts to life using playful, grade-aligned challenges when you explore Prodigy Math. We provide an engaging way to strengthen understanding while kids have fun exploring patterns and math concepts.