Lesson 2: Properties of Operations

New Concept

Arithmetic operations follow fundamental rules called properties. These rules describe how we can reorder, group, or use special numbers like 0 and 1.

What’s next

This card is your foundation. Next, you'll see worked examples of each property and justify the steps used to simplify expressions.

Property

$a + b = b + a$ and $a \times b = b \times a$

Examples

$7 + 11 = 11 + 7$, since both equal 18.
$4 \times 8 = 8 \times 4$, since both equal 32.
$12 - 5 = 5 - 12$, because $7$ is not the same as $-7$.

Explanation

Think of it like getting dressed! Putting on your left sock then right sock is the same as right then left. This property lets you swap the order of numbers when you add or multiply without changing the result. This trick does not work for subtraction or division!

Property

$(a + b) + c = a + (b + c)$ and $(a \times b) \times c = a \times (b \times c)$

Examples

$(3 + 5) + 6 = 3 + (5 + 6)$ is true, as $8 + 6 = 3 + 11$, and both equal 14.
Use this to simplify: $4 \times (25 \times 9) = (4 \times 25) \times 9 = 100 \times 9 = 900$.
$(16 \div 4) \div 2 \neq 16 \div (4 \div 2)$, because $2$ is not the same as $8$.

Explanation

Who you 'associate' with first doesn't change the final group! When you are only adding or only multiplying, you can regroup numbers using parentheses to make problems much easier to solve in your head. It’s all about smart teamwork!

Property

$a + 0 = a$ and $a \times 1 = a$

Examples

$42 + 0 = 42$. Adding zero does not change the number at all.
$153 \times 1 = 153$. Multiplying by one keeps it perfectly the same.
The missing number in $18 \times ? = 18$ must be the multiplicative identity, which is 1.

Explanation

Think of zero and one as special mirrors for numbers. Adding zero or multiplying by one always reflects the original number, keeping its 'identity' exactly the same. Zero is the additive identity, and one is the multiplicative identity. They are the superheroes of keeping things the same!