Write, Group and Check Numerical Expressions

Everyone gets the same value for an expression because everyone follows the order of operations, the agreed order for the steps. Every expression in this lesson uses only $+$, $-$, $\times$, $\div$ and parentheses. For these expressions you already know the order:

1. Work out what is inside parentheses first. Use steps 2 and 3 inside them.
2. Multiply and divide from left to right.
3. Add and subtract from left to right.

Words can describe a calculation too, and words also say what happens first. Read these two descriptions. They use the same three numbers.

• Add 8 and 2, then multiply by 3. First $8 + 2 = 10$. Then $10 \times 3 = 30$.
• Multiply 2 and 3, then add the result to 8. First $2 \times 3 = 6$. Then $8 + 6 = 14$.

The descriptions give different numbers, so they need different expressions. With no parentheses, the order of operations multiplies before it adds. So $8 + 2 \times 3$ matches the second description. For the first description, the addition must happen first. Parentheses are how you say that:

Descriptions often use four special words. A sum is the result of adding. A difference is the result of subtracting. A product is the result of multiplying. A quotient is the result of dividing.

Each of these words names one number. The sum of 4 and 5 is the single number you get when the adding is done. So a named sum, difference, product or quotient acts as a group inside a longer description. It must be finished before anything else uses it.

For subtraction and division, the order of the numbers matters. Most phrases in the table keep the numbers in the order you read them. One phrase flips them: subtract 4 from 9 starts with 9, so it is $9 - 4$.

An equation is true when its two sides really have the same value. For example, $3 + 4 \times 2 = 11$ is true, because the left side is $3 + 8 = 11$. The equation $3 + 4 \times 2 = 14$ is false as written, because the left side is still $11$.

Parentheses can sometimes turn a false equation into a true one. They do not change any number or operation symbol. They change which operation happens first, and that can change the value. Here $(3 + 4) \times 2 = 7 \times 2 = 14$, so one pair of parentheses makes the second equation true.

To find where the parentheses go:

• Work out the side with the operations as it is written. Now you know your starting value.
• Choose a part that would not be done first on its own, and put parentheses around it. Parentheses around the operation that is done first anyway leave the value the same.
• Work out the new value and compare it with the other side. If the two are not equal, try another place.

The same thinking lets you check worked-out math. When someone works out an expression, they rewrite it line by line, with more of it worked out each time. In correct work, every line has the same value as the line above it. That holds as long as each step does an operation that the order of operations allows next, and does the arithmetic correctly.

So to find a mistake, read the work one line at a time. For each line, ask: Which operation should come next? Is that the one that was done? Is its arithmetic right? The first line where an answer is no holds the mistake. Later lines may follow correctly from that line and still end at a wrong value.

The table shows four order mistakes to look for. Each one does an operation before its turn.

An expression is a set of instructions, and the order of operations tells every reader how to follow them. Parentheses are how you change which instruction comes first. You used that one idea three ways. You wrote expressions from words, with parentheses around any part the words finish first but the order of operations would do later. You made an equation true by testing where parentheses could go. You found a mistake by asking, at each line, which operation should come next. Whenever you write or read an expression, ask yourself: what happens first, and is that what I mean?