Basic Arithmetic Operations and Order of Operations

TL;DR

Arithmetic operations are the fundamental ways we combine numbers like addition, subtraction, multiplication, and division. The "order of operations" is a set of rules that tells you which calculation to do first to get the correct answer. Mastering these ensures you solve math problems consistently and accurately.

1. The Mental Model

Think of arithmetic as building with LEGOs: you have basic bricks (numbers) and tools (operations) to put them together. The order of operations is like the instruction manual that tells you how to assemble those bricks so your final model looks exactly right.

2. The Core Material

You already know the basic four arithmetic operations:

Addition (+)

Photo by BOOM 💥 Photography on Pexels

This is about combining quantities. If you have 3 apples and get 2 more, you now have $3 + 2 = 5$ apples. It's about increasing a number.

Subtraction (-)

Photo by ClickerHappy on Pexels

This is about taking away or finding the difference. If you have 5 apples and eat 2, you have $5 - 2 = 3$ apples left. It's about decreasing a number.

Multiplication (× or *)

Photo by https://kaboompics.com/ on Pexels

This is like repeated addition. $3 \times 2$ means adding 3 two times ($3+3$) or adding 2 three times ($2+2+2$), both giving 6. It's a faster way to add the same number multiple times.

Division (÷ or /)

Photo by Jakub Zerdzicki on Pexels

This is about splitting a quantity into equal groups or finding how many times one number fits into another. If you have 6 apples and share them equally among 3 friends, each friend gets $6 \div 3 = 2$ apples. It's the inverse of multiplication.

The Order of Operations (PEMDAS/BODMAS)

When you have an expression with more than one operation, you can't just calculate from left to right. There's a specific order you must follow. This order is often remembered by acronyms:

• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• BODMAS: Brackets, Orders (powers/roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

They both mean the same thing! Here's a breakdown:

1. Parentheses/Brackets: Always do anything inside parentheses ( ) or brackets [ ] first. They act like a "do me first" sign.
2. Exponents/Orders: Next, calculate any exponents (like $2^3$) or roots.
3. Multiplication and Division: These are done next. If you have both in the same expression, you work from left to right. They have equal priority.
4. Addition and Subtraction: Finally, perform all additions and subtractions. Again, if you have both, work from left to right. They also have equal priority.

Let's visualize this flow:

graph TD
    Start("Start Calculation") --> P("1. Parentheses/Brackets ( )")
    P --> E("2. Exponents/Orders (e.g., ^2)")
    E --> MD("3. Multiplication (*) and Division (/)")
    MD --> AS("4. Addition (+) and Subtraction (-)")
    AS --> End("End Calculation")

    subgraph Equal Priority (Left to Right)
        MD -->|If both exist| M(Multiplication)
        MD -->|If both exist| D(Division)
        AS -->|If both exist| A(Addition)
        AS -->|If both exist| S(Subtraction)
    end

3. Worked Example

Let's solve $10 + 4 \times (6 - 2) \div 2$.

1. Parentheses first: $(6 - 2) = 4$.
   The expression becomes: $10 + 4 \times 4 \div 2$.
2. Exponents: None in this problem.
3. Multiplication and Division (from left to right):
   First, $4 \times 4 = 16$.
   The expression becomes: $10 + 16 \div 2$.
   Next, $16 \div 2 = 8$.
   The expression becomes: $10 + 8$.
4. Addition and Subtraction (from left to right):
   Finally, $10 + 8 = 18$.

So, $10 + 4 \times (6 - 2) \div 2 = 18$.

4. Key Takeaways

• Basic operations (add, subtract, multiply, divide) are the building blocks of math.
• The order of operations (PEMDAS/BODMAS) ensures you get the single correct answer for any expression.
• Parentheses or brackets always take precedence, meaning you solve what's inside them first.
• Exponents (powers) are solved after parentheses.
• Multiplication and division have equal priority; work from left to right when both are present.
• Addition and subtraction also have equal priority; work from left to right when both are present.

Common Mistakes to Avoid:

• Calculating strictly left-to-right: Don't do $2+3 \times 4$ as $(2+3) \times 4 = 20$. It should be $2 + (3 \times 4) = 14$.
• Ignoring parentheses: Don't distribute or ignore them; solve what's inside first.
• Mixing up multiplication/division priority with addition/subtraction: Multiplication/division always come before addition/subtraction.
• Confusing subtraction of a negative with addition: $5 - (-2)$ is $5+2=7$, not $5-2=3$.

5. Now Try It

Solve the following expression: $25 - 3 \times (8 + 2) \div 5$.

What to do: Carefully follow the PEMDAS/BODMAS steps, writing down each intermediate step.
What success looks like: You should arrive at a single, correct numerical answer by applying each rule in the correct sequence.