Commutativity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Commutativity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A property where swapping the order of two operands does not change the result: a \ \star\ b = b\ \star\ a.

3 + 5 = 5 + 3 and 3 \times 5 = 5 \times 3. Swapping the order doesn't change the answer.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Commutative operations can have their inputs swapped freely.

Common stuck point: Subtraction and division are NOT commutative: 5 - 3 \neq 3 - 5.

Sense of Study hint: Try computing both orders (e.g., 3 + 5 and 5 + 3) and check if the answers match, then test subtraction to see it fails.

Worked Examples

Example 1

easy
Show that \(6 + 9 = 9 + 6\). Does the order of adding matter?

Solution

  1. 1
    Calculate \(6 + 9\): count on from 9 to 15. \(6 + 9 = 15\).
  2. 2
    Calculate \(9 + 6\): count on from 9 to 15. \(9 + 6 = 15\).
  3. 3
    Both equal 15, so \(6 + 9 = 9 + 6\).
  4. 4
    The order does NOT matter for addition.

Answer

Both equal 15; order does not matter
The commutative property of addition states \(a + b = b + a\). You get the same sum no matter which number you start with.

Example 2

medium
Verify that \(4 \times 7 = 7 \times 4\) using a rectangular array. Explain what changes and what stays the same.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
If \(13 + 27 = 40\), what is \(27 + 13\)? Explain without calculating.

Example 2

medium
Does commutativity work for subtraction? Check with \(10 - 3\) and \(3 - 10\).
← Back to Commutativity Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionmultiplication