Properties of Operations Cheat Sheet

A printable reference covering commutative, associative, distributive, identity, zero, inverse, and order of operations properties for grades 3-8.

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Properties of operations are rules that explain how numbers behave when you add, subtract, multiply, and divide. This cheat sheet helps students recognize patterns, simplify expressions, and choose efficient mental math strategies. These properties are especially useful when working with multi-step problems, variables, and algebraic expressions. Understanding them makes arithmetic and algebra feel more organized and predictable. The most important ideas include changing order, grouping numbers, distributing multiplication, and using special numbers like $0$ and $1$ . Addition and multiplication follow commutative and associative properties, but subtraction and division usually do not. The distributive property connects multiplication with addition or subtraction, such as $a(b+c)=ab+ac$ . Identity, zero, and inverse properties help students simplify expressions quickly and accurately.

Key Facts

The commutative property of addition says $a+b=b+a$ .
The commutative property of multiplication says $ab=ba$ .
The associative property of addition says $(a+b)+c=a+(b+c)$ .
The associative property of multiplication says $(ab)c=a(bc)$ .
The distributive property says $a(b+c)=ab+ac$ and $a(b-c)=ab-ac$ .
The additive identity property says $a+0=a$ .
The multiplicative identity property says $a\cdot 1=a$ .
The zero property of multiplication says $a\cdot 0=0$ .

Vocabulary

Commutative Property: A rule that says changing the order of numbers does not change the sum or product.
Associative Property: A rule that says changing the grouping of numbers does not change the sum or product.
Distributive Property: A rule that multiplies a number by each term inside parentheses, such as $a(b+c)=ab+ac$ .
Identity Element: A number that leaves another number unchanged, such as $0$ for addition or $1$ for multiplication.
Inverse Operation: An operation that undoes another operation, such as addition and subtraction or multiplication and division.
Order of Operations: A set of rules for simplifying expressions in the correct order, usually parentheses, exponents, multiplication and division, then addition and subtraction.

Common Mistakes to Avoid

Using the commutative property with subtraction is wrong because $a-b$ is usually not equal to $b-a$ .
Using the commutative property with division is wrong because $a\div b$ is usually not equal to $b\div a$ .
Forgetting to distribute to every term is wrong because $a(b+c)$ means $a\cdot b+a\cdot c$ , not just $ab+c$ .
Changing grouping in mixed operations is wrong when the operation changes, because $(a+b)c$ is not the same as $a+bc$ .
Confusing identity numbers is wrong because $a+1$ changes $a$ , while $a\cdot 1=a$ , and $a+0=a$ .

Practice Questions

1 Use a property to rewrite $7+12=12+7$ and name the property.
2 Simplify $4(6+9)$ using the distributive property.
3 Which property is shown by $(3\cdot 5)\cdot 2=3\cdot(5\cdot 2)$ ?
4 Explain why $8-3$ cannot be rewritten as $3-8$ using the commutative property.

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