Adding and Subtracting Integers in Context | Grade 8 Math | LevelKite

You already know how to compare and order integers on a number line whose numbers increase to the right. Now you will add and subtract them. The integers are the counting numbers $1, 2, 3, \dots$ , their negatives $-1, -2, -3, \dots$ , and $0$ . Every integer except $0$ has a sign, positive or negative; $0$ has neither.

Start with what adding means. A sum is the result of an addition. To find the sum of two integers, stand at the first number on the number line and let the second number tell you how to move:

Adding a positive integer such as $5$ moves you $5$ units to the right.
Adding a negative integer such as $-5$ moves you $5$ units to the left.
Adding $0$ does not move you.

For example, $-3 + 5$ says: start at $-3$ and move $5$ units right. You pass $-2$ , $-1$ , $0$ and $1$ , and you land on $2$ . So $-3 + 5 = 2$ . The next table works six more sums the same way.

Adding integers as moves on the number line

Sum	Start at	Move	Land on
$4 + 3$	$4$	$3$ units right	$7$
$-4 + (-3)$	$-4$	$3$ units left	$-7$
$4 + (-7)$	$4$	$7$ units left	$-3$
$-4 + 7$	$-4$	$7$ units right	$3$
$-4 + 4$	$-4$	$4$ units right	$0$
$-4 + 0$	$-4$	no move	$-4$

You can find these sums without counting units. Reaching the first number is itself a move from $0$ : right for a positive number, left for a negative one. So a sum is two moves in a row. The absolute value of a number is its distance from $0$ , so it tells you how long each move is. You write it with bars: $|-4| = 4$ and $|7| = 7$ . A distance is never negative.

Same sign (both positive or both negative): both moves go the same way, so the distances add. Add the absolute values and keep the shared sign. For $-4 + (-3)$ : $4 + 3 = 7$ , so the sum is $-7$ .
Different signs, unequal absolute values: the moves go opposite ways, and the shorter move undoes part of the longer one. Subtract the smaller absolute value from the larger and use the sign of the number with the larger absolute value. For $4 + (-7)$ : $7 - 4 = 3$ , and $-7$ is farther from $0$ , so the sum is $-3$ .
Different signs, equal absolute values: the numbers are opposites, such as $-4$ and $4$ . The moves cancel exactly, so the sum is $0$ .
Zero: if either number is $0$ , the sum is the other number.

Worked example

Adding integers with different signs

What is $-14 + 9$ ?

Compare the signs, because they decide which addition case applies. The first number is negative and the second is positive, so the signs are different and the two moves go in opposite directions.
Find each absolute value, the distance of each number from 0, so you know how long each move is.
$|-14| = 14 \quad \text{and} \quad |9| = 9$
The absolute values are not equal, so subtract the smaller from the larger. The 9 units to the right undo 9 of the 14 units to the left, which leaves the sum 5 units from 0.
$14 - 9 = 5$
Choose the sign. The negative number has the larger absolute value, so the move to the left is the longer one and the sum is negative.
$-14 + 9 = -5$
Check by moving on the number line. Start at negative 14 and go 9 units right. You stop 5 units before 0, at negative 5, which agrees with the rule.

Answer

The sum $-14 + 9$ is $-5$ .

Subtraction asks a question about addition. A difference is the result of a subtraction, and $7 - 3$ is the number you must add to $3$ to get $7$ . That number is $4$ , because $3 + 4 = 7$ .

Ask the same question when the number you subtract is negative. The difference $3 - (-5)$ is the number you must add to $-5$ to get $3$ . From $-5$ you move $5$ units right to reach $0$ , then $3$ more units right to reach $3$ . That is $8$ units right in all, so $3 - (-5) = 8$ .

Now notice that $3 + 5 = 8$ as well. Subtracting $-5$ gave the same result as adding its opposite, $5$ . That is not a coincidence, and you are about to see why.

Six subtractions, each beside adding the additive inverse; every check returns the first number

Subtraction	Add the additive inverse	Result	Check: result plus the number you subtracted
$7 - 3$	$7 + (-3)$	$4$	$4 + 3 = 7$
$3 - 7$	$3 + (-7)$	$-4$	$-4 + 7 = 3$
$3 - (-5)$	$3 + 5$	$8$	$8 + (-5) = 3$
$-3 - 5$	$-3 + (-5)$	$-8$	$-8 + 5 = -3$
$-3 - (-5)$	$-3 + 5$	$2$	$2 + (-5) = -3$
$0 - (-4)$	$0 + 4$	$4$	$4 + (-4) = 0$

Key concept

Subtraction adds the additive inverse

For any integers $a$ and $b$ , subtracting $b$ gives the same result as adding the additive inverse of $b$ . The table shows six cases; this reasoning covers every case.

Why: $a - b$ means the number you add to $b$ to get $a$ . Test $a + (-b)$ : adding $b$ to it gives $a + (-b) + b$ . Because $-b + b = 0$ , that sum is $a + 0 = a$ . So $a + (-b)$ is the difference $a - b$ .

How: keep the first number, change the subtraction to addition, and replace the number you subtract with its additive inverse. Then add with the addition cases you already have. To check, add the number you subtracted to your result; you must get the first number back.

a - b = a + (-b)

Worked example

Subtracting a negative integer

What is $-6 - (-11)$ ?

Rewrite the subtraction as an addition, because subtraction adds the additive inverse. The additive inverse of negative 11 is 11, since their sum is 0. The first number, negative 6, stays the same.
$-6 - (-11) = -6 + 11$
Now add. The signs are different and the absolute values, 6 and 11, are not equal, so subtract the smaller absolute value from the larger.
$11 - 6 = 5$
Choose the sign. The positive number, 11, has the larger absolute value, so the sum is positive.
$-6 + 11 = 5$
Check by adding the number you subtracted, negative 11, to the result. You get the first number back, so the difference is correct.
$5 + (-11) = -6$

Answer

The difference $-6 - (-11)$ is $5$ .

Integers measure quantities that can lie on either side of a reference point, the level you call $0$ . Elevation, which is height above or below sea level, is one example. A word problem needs a short setup before any arithmetic:

Name the reference point and the unit. For example, sea level is $0$ and elevations are in meters (m).
Say what the signs mean. For example, above sea level is positive and below sea level is negative.
Add for a change. An increase adds a positive number and a decrease adds a negative number. A decrease of $9$ adds $-9$ , which equals subtracting $9$ , because subtraction adds the additive inverse.
Subtract to compare. The change from a starting value to a final value is the final value minus the starting value. To find how far one value is above another, subtract the second value from the first.

Finish with a sentence that includes the unit, and ask whether the sign of your answer fits the situation.

From a situation to a calculation

Situation	Reference point, unit and signs	Calculation	Answer in context
Your quiz-game score is $-30$ points. You earn $50$ points. What is your new score?	A score of $0$ points is the reference; points earned are positive, points lost are negative	$-30 + 50 = 20$	Your new score is $20$ points.
A diver is at $-12$ m and swims $9$ m deeper. Where is the diver now?	Sea level is $0$ m; below sea level is negative, so going deeper is a decrease	$-12 + (-9) = -21$	The diver is at $-21$ m, which is $21$ m below sea level.
A hilltop is at $40$ m and a cave floor is at $-25$ m. How far is the hilltop above the cave floor?	Sea level is $0$ m; above sea level is positive, below sea level is negative	$40 - (-25) = 40 + 25 = 65$	The hilltop is $65$ m above the cave floor.
A lake's level goes from $3$ cm to $-4$ cm. What is the change in level?	The usual level is $0$ cm; above it is positive, below it is negative	$-4 - 3 = -4 + (-3) = -7$	The change is $-7$ cm, so the level fell $7$ cm.

Worked example

Temperature through a day

At 6 a.m. the temperature in a mountain village is $-9$ °C. By 2 p.m. it has risen $14$ °C. From 2 p.m. to midnight it falls $11$ °C. What is the temperature at midnight, and what is the change in temperature from 6 a.m. to midnight?

Set up the meaning of the numbers. Every temperature is in degrees Celsius, and 0 degrees Celsius is the reference point. Temperatures above it are positive and temperatures below it are negative. A rise is an increase, so it adds a positive number. A fall is a decrease, so it adds a negative number.
Find the 2 p.m. temperature by adding the rise to the 6 a.m. temperature. The signs are different, 14 minus 9 is 5, and the positive number has the larger absolute value, so the sum is positive.
$-9 + 14 = 5$
Find the midnight temperature. A fall of 11 means subtracting 11, and subtraction adds the additive inverse, negative 11. The signs are different, 11 minus 5 is 6, and the negative number has the larger absolute value, so the result is negative.
$5 - 11 = 5 + (-11) = -6$
Find the change from 6 a.m. to midnight, which is the final value minus the starting value. Subtracting negative 9 adds its additive inverse, 9. The signs are different, 9 minus 6 is 3, and the positive number has the larger absolute value, so the change is positive.
$-6 - (-9) = -6 + 9 = 3$
Check the change a second way by combining the rise and the fall. Both methods give 3. A positive change also fits the situation: negative 6 is greater than negative 9, so midnight is warmer than 6 a.m.
$14 + (-11) = 3$

Answer

At midnight the temperature is $-6$ °C. The change from 6 a.m. to midnight is $3$ °C, so midnight is $3$ °C warmer than 6 a.m.

Takeaway. Adding integers combines moves on the number line: numbers with the same sign add their distances from $0$ and keep that sign, while numbers with different signs cancel partly, or fully when they are opposites. Subtraction adds the additive inverse: for any integers $a$ and $b$ , $a - b = a + (-b)$ , so the addition cases handle both operations. In a word problem, set the reference point, the unit and the meaning of the signs first; then add for a change and subtract to compare. Next, you will multiply and divide integers.