Grouping and the Order of Operations

Think about what $4 + 6 \times 5$ can stand for. You have $4$ loose markers and $6$ packs with $5$ markers in each pack. The part $6 \times 5$ means $6$ groups of $5$. It is one amount: $30$ markers. Add the $4$ loose markers and you have $34$ markers. Adding $4 + 6$ first would mix $4$ markers with $6$ packs, and that sum is not a number of markers. Division makes one amount too: $12 \div 4$ is the size of one share when $12$ is split into $4$ equal shares.

The agreed order fits this meaning. For an expression made only of numbers and the signs $+$, $-$, $\times$ and $\div$:

1. Multiply and divide first, working from left to right.
2. Then add and subtract, working from left to right.

Multiplication and division have the same rank: they sit at the same level in the order. Neither one always goes first, so you take them as they appear from left to right. Addition and subtraction share a rank in the same way.

To evaluate one step at a time, do exactly one operation on each line:

• Choose the next operation, using the order above.
• Replace that part with the number it equals.
• Copy everything else exactly as it is.

When you choose the operation this way, each new line equals the line before it. You have only swapped a part, such as $6 \times 5$, for the number it equals, $30$. So the value of the whole expression stays the same. It just gets easier to see.

Each step uses up one operation sign, so an expression with four operations takes four steps. Keep going until a single number is left. That number is the value.

Here is $4 + 6 \times 5$ written this way. It has two operations, so it takes two steps.

Grouping symbols are marks that wrap part of an expression. They come in pairs: one opens the group and one closes it. This lesson uses two kinds: parentheses, as in $(3 + 4)$, and brackets, as in $[3 + 4]$. Both mean the same thing: treat everything inside as one number, and find that number first. To find it, evaluate the inside like any other expression.

Go back to the markers. The expression $8 \times (3 + 4)$ can stand for $8$ packs that each hold $3$ red markers and $4$ blue markers. The parentheses keep $3 + 4$ together as one amount: the markers in one pack. So you add first, even though multiplication usually comes before addition. You get $3 + 4 = 7$ markers in a pack, then $8 \times 7 = 56$ markers in all.

Grouping symbols can change the value of an expression, but they do not always. The table shows both cases with the same numbers.

Sometimes one pair of grouping symbols sits inside another. The outer pair is often written as brackets so that you can tell the pairs apart, as in $[20 - (4 + 3)] \times 2$.

You cannot finish the brackets until the parentheses inside them have become a single number. So start with the innermost pair, the one with no grouping symbols inside it, and work outward. Here that gives $4 + 3 = 7$, then $20 - 7 = 13$, then $13 \times 2 = 26$.

Takeaway. The order of operations is the agreed order that lets everyone get the same value for an expression. For expressions with $+$, $-$, $\times$, $\div$ and grouping symbols, work inside the grouping symbols first, starting with the innermost pair. Then multiply and divide from left to right. Then add and subtract from left to right. Do one operation on each line, and every line equals the one before it.

Next, you will write numerical expressions from words, insert parentheses to make an equation true, and find mistakes in the order of operations.