Lesson 2: Operations of Arithmetic

New Concept

The fundamental operations of arithmetic are addition, subtraction, multiplication, and division. Understanding their properties is key to solving all types of math problems.

What’s next

Now, let’s begin our journey by exploring the specific vocabulary and rules for each operation, starting with worked examples and practice problems.

Property

Commutative Property of Addition: $a + b = b + a$. Commutative Property of Multiplication: $a \cdot b = b \cdot a$.

Examples

Adding numbers in any order: $21 + 7 = 28$ is the same as $7 + 21 = 28$.
Multiplying factors in any order: $9 \cdot 4 = 36$ gives the same product as $4 \cdot 9 = 36$.
Be careful, because $12 - 5$ is not the same as $5 - 12$!

Explanation

Order doesn't matter for adding or multiplying! Think of it like a commute—the distance is the same from home to school as school to home. This trick lets you rearrange numbers to make math easier, but remember it does not work for subtraction or division, where order is king!

Property

Associative Property of Addition: $(a + b) + c = a + (b + c)$. Associative Property of Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.

Examples

To solve $(25 \cdot 17) \cdot 4$, regroup to get $17 \cdot (25 \cdot 4) = 17 \cdot 100 = 1700$.
Adding $(58 + 25) + 75$ is much easier if you calculate it as $58 + (25 + 75) = 58 + 100 = 158$.
This property helps turn tricky calculations into simple steps you can do in your head!

Explanation

When you are adding or multiplying a long list of numbers, you can regroup them to create 'friendly' pairs. It's like choosing which friends to talk to first in a group—it doesn’t change the overall group! This is a great strategy for simplifying problems before you start calculating.

Property

'Three times five' can be written as $3 \times 5$, $3 \cdot 5$, $3(5)$, or $(3)(5)$. 'Six divided by two' can be written as $6 \div 2$,$\frac{6}{2}$.

Examples

The expression $8(5)$ means the exact same thing as $8 \cdot 5$; both are equal to 40.
A problem asking you to solve $\frac{45}{9}$ is just another way of writing $45 \div 9$, which equals 5.

Explanation

Math has different ways of saying the same thing, just like we can say 'hello' or 'hi!' Knowing all the symbols for multiplying and dividing is like being fluent in the language of math. This helps you understand problems no matter how they are written, whether in a textbook or on a test.