Evaluate, Apply and Check Rational Number Operations

Look at $2 - 6 \times \frac{1}{2}$. If you subtract first, you get $2 - 6 = -4$ and then $-4 \times \frac{1}{2} = -2$. If you multiply first, you get $6 \times \frac{1}{2} = 3$ and then $2 - 3 = -1$. One expression should name one number, so everyone follows a single agreed sequence, called the order of operations. Grouping symbols, such as parentheses, enclose a part of an expression that is worked out as one number. For an expression built from the four operations and grouping symbols:

1. Work inside grouping symbols first, following steps 2 and 3 there.
2. Then multiply and divide, moving from left to right.
3. Then add and subtract, moving from left to right.

So $2 - 6 \times \frac{1}{2} = 2 - 3 = -1$. Parentheses that hold only a negative number, as in $\left(-\frac{2}{3}\right)$, have no operation inside. They just keep the sign attached to its number.

Each single operation rests on a fact from your earlier lessons, so pin down a few words first. The additive inverse of a number is its opposite: the number that adds with it to make $0$. The additive inverse of $\frac{3}{4}$ is $-\frac{3}{4}$, and the additive inverse of $-2.5$ is $2.5$. A factor is one of the numbers being multiplied, and the product is the result. The reciprocal of a nonzero number is the number that multiplies with it to make $1$. The reciprocal of $-\frac{2}{3}$ is $-\frac{3}{2}$, because $-\frac{2}{3} \times \left(-\frac{3}{2}\right) = 1$. Zero has no reciprocal, because zero times any number is $0$, never $1$. That is why a divisor, the number you divide by, can never be $0$: division by zero is undefined.

In the table, $a$, $b$, $c$ and $d$ stand for rational numbers. Each fact is true only under the condition written beside it.

A word problem hides an expression inside a story, and your first job is to pull it out. Signs do much of that work, because a sign records direction. Once you choose which direction counts as positive, the opposite direction is negative. The usual choice makes a rise positive and a drop negative, and makes a height above a zero level positive and a depth below it negative.

A rate tells how much one quantity changes for each single unit of another, such as meters per minute. You find a rate by dividing, so the amount you divide by must not be zero.

To solve a word problem:

• Say what each number means, with its unit and its sign.
• Choose the operations. Combining changes is addition. The change from a start to an end is the end minus the start. Repeating an equal change is multiplication. Splitting a change equally, or finding a rate, is division.
• Write one expression, evaluate it in the agreed order, and answer in a sentence with the unit. Then check that the sign and size of your answer fit the story.

Now use the same ideas to check someone else's work. In a correct evaluation, every line is another name for the same number, so each line must equal the line above it. An error is a step that breaks this chain. To find and correct one:

1. Read down the work and test each line against the line above it. Was the right operation chosen next, and was it carried out correctly?
2. Stop at the first line that changes the value. Name what went wrong, such as a wrong order, a lost sign or a wrong reciprocal.
3. Redo the work from the last correct line, then compare your result with the original answer.

Lines after an error can follow correctly from it and still give the wrong value, because they carry the error forward. That is why you look for the first line that fails.

Here is the idea to keep. A numerical expression that combines rational numbers with the four operations and grouping symbols, and never divides by zero, has exactly one value. You reach it one operation at a time: grouping symbols first, then multiplication and division from left to right, then addition and subtraction from left to right. Every line you write equals the line before it. That one fact lets you evaluate an expression, answer a word problem once you have written its expression with signs and units, and find an error by spotting the first line that changes the value.