Add and subtract by walking the number line — and one rule that turns subtraction into addition.
Point 3 of 5 in this lesson: 5.4.3 The laws of addition — reorder to make life easy
In Lesson 5.3 you learned to place every rational number on the line and tell which is bigger. Now you will make those numbers move. Adding two signed numbers is nothing more mysterious than a walk: start at the first number, then take a step — to the right if the second number is positive, to the left if it is negative — and wherever you stop is the answer. Subtraction turns out to be the very same walk done backward, which is exactly why every subtraction a − b can be rewritten as the addition a + (−b).
By the end you will be able to add any two signed numbers with or without a picture, use the same-sign and different-sign rules, reorder a long string to pair off opposites, and turn any subtraction into an addition — including the famous "minus a negative." We keep the steady color habit: positive numbers are teal and mean a step right, negative numbers are red and mean a step left, zero is slate, and an absolute value or distance is amber.
To add two signed numbers, treat the first one as your starting spot and the second one as a step. The sign of the step tells you which way to face: a positive step walks you right, a negative step walks you left. The size of the step is just its absolute value — how many units to walk. Where your feet stop is the sum.
Take (+3) + (−5). Start at +3. The step is −5, so face left and walk 5 units. You pass through +2, +1, the origin 0, then −1, and stop at −2. So (+3) + (−5) = −2.
The picture also explains why the answer can flip sign. The first number got you a little way to the right; the second step was bigger and pointed left, so it carried you back past zero into negative territory. The further-reaching step wins the tug-of-war.
Adding is a walk: start at the first number, then step the second number's distance — right if it is positive, left if it is negative. Where you land is the sum.
Set the two addends. The first arrow walks from 0 to the first number; the second arrow steps from there. Watch where you land — that landing spot is the sum.
The walk is the truth, but you cannot draw a line for every problem on a test. Luckily the picture boils down to two short rules, depending on whether the two numbers point the same way or different ways.
Same signs — both positive or both negative. The two steps point the same direction, so they pile up. Add the absolute values and keep the shared sign. For example (−4) + (−3): both red, so 4 + 3 = 7, and the answer stays negative → −7. (Owe \$4, then owe \$3 more, and you owe \$7.)
Different signs — one positive, one negative. The steps fight, and the longer one wins by the difference. Subtract the smaller absolute value from the larger, and take the sign of the number whose absolute value is larger. For (−7) + (+3): the absolute values are 7 and 3, so 7 − 3 = 4; since −7 has the larger absolute value, the answer is −4. And when the two absolute values are equal, the steps cancel exactly: 6 + (−6) = 0.
| The two signs | What you do with the absolute values | Sign of the answer |
|---|---|---|
| + and + | add them | positive |
| − and − | add them | negative |
| different (+ & −) | subtract smaller from larger | sign of the bigger absolute value |
Find (+9) + (−14).
Different signs, so subtract: 14 − 9 = 5. The number with the larger absolute value is −14, which is negative, so the answer takes a minus sign: −5.
Set two signed numbers. The machine tells you which case you are in, does the absolute-value arithmetic, and prints the signed answer — no number line needed.
Addition obeys two laws you already trust for ordinary numbers, and they still hold once signs appear. Addition is commutative — a + b = b + a, you may swap the order — and associative — (a + b) + c = a + (b + c), you may regroup which sum you do first. Together they mean you can shuffle a string of additions into any order you like and the total never changes.
That freedom is a gift, because some orders are far easier than others. The smart move is to hunt for opposites — pairs like 8 and −8 that cancel to 0 — and for groupings that make round numbers. Compare the two ways to do (−8) + 5 + 8:
Because addition is commutative and associative, you may reorder and regroup freely. Pair off opposites (they vanish to 0) and build round numbers before adding the rest.
Here is a string of signed terms. Tap two terms to add them in whichever order you like; cancel the opposites first and watch the total drop out fast. The total never changes — only the effort does.
Here is the rule that makes everything from here on easy: subtracting a number is the same as adding its opposite. In symbols, a − b = a + (−b). On the number line, the minus sign simply turns you around before you take the step.
Try 4 − 9. Rewrite it as 4 + (−9): start at 4, face left, walk 9 units, and land on −5. So 4 − 9 = −5. Subtraction never needed its own set of rules — it just borrows addition's.
The rule earns its keep when you subtract a negative. What is 3 − (−5)? The opposite of −5 is +5, so the problem becomes 3 + 5 = 8. Two minus signs in a row turn you around twice — and turning around twice means you face right again, so you add. (Take away a \$5 debt and you are \$5 richer.)
Do not let two minus signs scare you. 3 − (−5) is not −8 and not −2. Rewrite it the safe way every single time: subtraction means add the opposite, so 3 − (−5) = 3 + 5 = 8. On the line, the second minus simply turns you back around to face right.
Set a and b (b may be negative). The widget rewrites the subtraction as an addition of the opposite, then walks it on the line. Try a negative b and watch the step flip to the right.
Real problems mix pluses and minuses in one long string. The reliable recipe is three steps: first rewrite every subtraction as adding the opposite, so the whole thing becomes a pure sum of signed terms; next keep each sign glued to the number right after it; then reorder freely to gather the positives together and the negatives together.
The "glue the sign to its number" idea is the safety rail. Once 7 − 10 + 4 − 2 is read as the four terms +7, −10, +4, −2, you can scoop them up in any order without ever losing track of who is positive and who is negative.
Evaluate 7 − 10 + 4 − 2.
Rewrite as a sum: 7 + (−10) + 4 + (−2).
Group same signs: (7 + 4) + (−10 + −2) = 11 + (−12).
Finish with the different-sign rule: 12 − 11 = 1, and −12 has the larger absolute value, so the answer is −1.
For a long string: rewrite every − as + (the opposite), keep each sign glued to its number, then reorder to add all the positives, add all the negatives, and combine the two totals once.
Set the four terms. The widget glues each sign to its number, sums the positives and the negatives separately, then combines them with the different-sign rule.
Adding two signed numbers is a walk: start at the first, step the second's distance right if positive or left if negative, and read where you land. Without drawing, use the rules: same signs add the absolute values and keep the sign; different signs subtract the smaller absolute value from the larger and take the sign of the bigger one; opposites like 6 and −6 cancel to 0. Addition is commutative and associative, so reorder to pair opposites and build round numbers. Every subtraction is secretly an addition — a − b = a + (−b) — so 4 − 9 = −5 and 3 − (−5) = 8. For a mixed string, glue each sign to its number, then add the positives, add the negatives, and combine once.
You can move along the line by stepping. Next you will learn what happens when you multiply and divide signed numbers — why a negative times a negative comes out positive, and how that flows straight from the patterns you just built.
Work each one out first, then open the answer to check your thinking.
Six questions to lock it in. Tap the answer you think is right.
This lesson is the heart of 7.NS.A.1 — adding and subtracting rational numbers on the number line. It develops all four sub-standards: 7.NS.A.1a (a sum p + q is found by walking |q| units from p, right or left by the sign of q); 7.NS.A.1b (a number and its opposite are additive inverses that sum to 0, and a + (−a) = 0); 7.NS.A.1c (subtraction as adding the additive inverse, a − b = a + (−b), and the distance between p and q on the line is |p − q|); and 7.NS.A.1d (the commutative and associative properties let you reorder and regroup). It builds on the ordering and absolute-value work of 6.NS.C.5–7. The single most common misconception is mishandling subtracting a negative — students read 3 − (−5) as −8 or −2. The antidote, repeated throughout Section 5.4.4 and the quiz, is a two-word rule plus a picture: subtraction means add the opposite, and on the number line the second minus simply turns you around to face right.