Integer Word Problems and Finding Sign Errors | Grade 7 Math | LevelKite

Temperatures drop below zero. Divers swim below sea level. People owe money. Stories like these use integers: the counting numbers $1, 2, 3, \ldots$ , their opposites $-1, -2, -3, \ldots$ , and $0$ . In a story, a number's sign tells you a direction.

Before you compute anything in a story problem, name three things:

The zero point. What does $0$ stand for? It could be sea level, $0$ °C, or owing nothing.
The positive direction. Which way counts as positive: up, warmer, money gained?
The unit. Meters, degrees, dollars?

With those named, you can write every value in the story as a positive number, a negative number or zero, and every change as an addition or a subtraction.

This lesson has two parts. First you will turn a story into an expression (numbers joined by operation signs) and find its value. Then you will check finished solutions line by line and find exactly where a sign went wrong.

Story words as integers, with up, warmer and having money chosen as positive

Story words	What they describe	Write it as
47 m below sea level	A value. Sea level is $0$ .	$-47$ m
4 degrees below zero	A value. $0$ °C is the zero point.	$-4$ °C
You owe 12 dollars	A value. Owing nothing is $0$ .	$-12$ dollars
It rises 11 degrees	A change in the positive direction	Add $11$
It falls 15 degrees	A change in the negative direction	Add $-15$ , which equals subtracting $15$
How much higher is one value, $a$ , than another, $b$ ?	A comparison of two values in the same unit	$a - b$

Key concept

Start, distance, direction

Name the zero point, the positive direction and the unit. Then every step of a story, and every line of a solution, is the same kind of move: start, distance, direction. Two facts you already know make this work for any integers $p$ and $q$ :

$p + q$ is the number $\lvert q \rvert$ units from $p$ . Here $\lvert q \rvert$ , the absolute value of $q$ , is the distance of $q$ from $0$ . Go toward greater numbers when $q$ is positive and toward lesser numbers when $q$ is negative. When $q = 0$ , stay at $p$ .
Subtracting $q$ gives the same result as adding its additive inverse (its opposite), $-q$ . Opposites sum to zero: $q + (-q) = 0$ .

p - q = p + (-q)

Worked example

A temperature that rises and falls

At 6 a.m. the temperature is 4 degrees below zero on the Celsius scale. By noon it has risen 11 degrees. By midnight it has fallen 15 degrees from the noon reading. What is the temperature at midnight?

Name the reference. Let 0 stand for 0 °C, let warmer be the positive direction, and measure in degrees Celsius. Then 4 degrees below zero is negative 4. Let T be the midnight temperature in degrees Celsius.
$\text{6 a.m. temperature} = -4$
Write the story in order. Start at negative 4. A rise is a change in the positive direction, so add 11. A fall is a change in the negative direction, so subtract 15.
$T = -4 + 11 - 15$
Rewrite the subtraction as adding the additive inverse. Subtracting 15 gives the same result as adding negative 15, and this is true for any integers, so the value of the expression stays the same.
$T = -4 + 11 + (-15)$
Add from left to right. Adding 11 moves 11 units toward greater numbers from negative 4. The first 4 units reach 0, and the other 11 minus 4, or 7, units go above it.
$-4 + 11 = 7$
Adding negative 15 moves 15 units toward lesser numbers from 7. The first 7 units reach 0, and the other 15 minus 7, or 8, units go below it.
$T = 7 + (-15) = -8$
Check by a second route. Every operation is now an addition, and addition can be regrouped (the associative property), so combine the two changes into one net change first. Moving 15 units toward lesser numbers from 11 reaches 0 after 11 units and ends 4 below it. A net change of negative 4 takes the start, negative 4, down 4 more units to negative 8. Both routes agree.
$11 + (-15) = -4, \qquad -4 + (-4) = -8$

Answer

The temperature at midnight is $-8$ °C, which is 8 degrees below zero.

Some stories do not ask where you end up. They ask you to compare two values: how much warmer, how much higher, how much more money.

You answer a how much more question with a subtraction. To find how much greater a value $a$ is than a value $b$ , with both in the same unit and measured from the same zero point, compute the difference $a - b$ . A difference is the result of a subtraction. It is positive when $a$ is greater than $b$ , negative when $a$ is less than $b$ , and $0$ when they are equal.

Look closely at the case where $a$ is positive and $b$ is negative. The gap from $b$ up to $a$ crosses zero, so it is the distance from $b$ to $0$ plus the distance from $0$ to $a$ . The subtraction agrees, because subtracting a negative number adds its positive opposite.

Worked example

Comparing two elevations

Elevation is height above or below sea level. A hiker stands on a ridge 132 m above sea level. A diver is 47 m below sea level. How many meters higher is the hiker than the diver?

Name the reference. Let 0 stand for sea level, let up be the positive direction, and measure in meters. The hiker is at 132 and the diver is at negative 47. Let d be the number of meters by which the hiker is higher than the diver.
$\text{hiker} = 132, \qquad \text{diver} = -47$
Write the comparison. To find how much greater the first value is than the second, subtract the second from the first. Here that is the hiker's elevation minus the diver's elevation.
$d = 132 - (-47)$
Rewrite the subtraction as adding the additive inverse, which keeps the value the same for any integers. The additive inverse of negative 47 is 47, because negative 47 plus 47 equals 0. This matches the story: the gap is 132 m down to sea level plus 47 m more down to the diver.
$d = 132 + 47$
Add. Adding 47 moves 47 units toward greater numbers from 132. Adding 40 gives 172, and adding 7 more gives 179.
$d = 132 + 40 + 7 = 179$
Check by adding back. A difference is correct exactly when adding it to the number you subtracted returns the number you started with. Moving 179 units toward greater numbers from negative 47 uses 47 units to reach 0, and the other 179 minus 47, or 132, units go above it.
$-47 + 179 = 132$

Answer

The hiker is $179$ m higher than the diver.

Error analysis means reading a finished solution line by line, finding the first line that is wrong, naming the slip, and repairing the work from there. You can use it on a classmate's work and, above all, on your own.

Follow the same routine every time:

Check the setup against the story. Does each value have the right sign for the zero point and the positive direction? Is each change added or subtracted correctly?
Recompute each line from the line above it. When a solution simplifies one expression step by step, every line must have the same value as the line before. The first line whose value changes holds the first computing error.
Name the slip. Was a sign dropped? Did a move go the wrong way? Was the order switched?
Repair and finish. Redo the work from the wrong line on, then check the final answer.

Four common slips and a check that catches each one

Wrong line	The slip	Check that catches it	Correct line
$5 - (-3) = 2$	The negative sign on $-3$ was dropped, so $5 - 3$ was computed.	Subtracting a negative number must give more than $5$ , but $2$ is less.	$5 - (-3) = 5 + 3 = 8$
$-7 + 4 = -11$	The move went the wrong way. $-11$ is the value of $-7 - 4$ .	Adding a positive number must give more than $-7$ , but $-11$ is less.	$-7 + 4 = -3$
$-6 - 2 = -4$	The move went the wrong way. $-4$ is the value of $-6 + 2$ .	Subtracting a positive number must give less than $-6$ , but $-4$ is greater.	$-6 - 2 = -6 + (-2) = -8$
$3 - 8 = 5$	The order was switched to $8 - 3$ , which gives the opposite of $3 - 8$ .	Add back: $5 + 8 = 13$ , not $3$ .	$3 - 8 = 3 + (-8) = -5$

Two minus signs do not always make a plus. Look at $-5 - 3$ . It is tempting to say two negatives make a positive and write $-5 - 3 = 8$ .

This fails because the two signs do different jobs. The first belongs to the number $-5$ . The second is the operation subtract, and the number being subtracted is positive $3$ . Subtracting $3$ gives the same result as adding its opposite, $-3$ : $-5 + (-3)$ . From $-5$ , move $3$ units toward lesser numbers to reach $-8$ . In a money story, you owe 5 dollars and spend 3 more, so you owe 8 dollars.

When you add and subtract, two minus signs turn into a plus only when you subtract a negative number, because its opposite is positive: $-5 - (-3) = -5 + 3 = -2$ .

Worked example

Find and fix the first wrong line

A submarine is 120 m below sea level. It rises 45 m and then descends 70 m. Maya finds its final elevation in three lines. Line 1 is $-120 + 45 - 70$ . Line 2 is $-75 - 70$ . Line 3 is $-5$ . She says the submarine ends 5 m below sea level. Find the first wrong line, explain the slip, and give the correct final elevation.

Check the setup against the story. Let 0 stand for sea level, let up be the positive direction, and measure in meters. The start is negative 120, the rise adds 45, and the descent subtracts 70. Line 1 matches the story.
$-120 + 45 - 70$
Recompute Line 2 from Line 1. Adding 45 moves 45 units toward greater numbers from negative 120. Because 45 is less than 120, the result stays below 0. It is 120 minus 45, or 75, units below 0. Line 2 is correct.
$-120 + 45 = -75$
Recompute Line 3 from Line 2. Subtracting 70 gives the same result as adding its additive inverse, negative 70, so move 70 units toward lesser numbers from negative 75. That lands 75 plus 70, or 145, units below 0. Maya wrote negative 5, so Line 3 is the first wrong line.
$-75 - 70 = -75 + (-70) = -145$
Name the slip. Negative 5 is what you get by moving 70 units toward greater numbers from negative 75, so Maya moved the wrong way. The direction check shows it: subtracting a positive number must give less than negative 75, but negative 5 is greater.
$-75 + 70 = -5, \qquad -5 > -75$
Confirm the repair by a second route. Every operation in Line 1 can be written as an addition, so you may combine the two changes first. Moving 70 units toward lesser numbers from 45 reaches 0 after 45 units, and the other 25 go below it. This net change of negative 25 moves the start, negative 120, down to 120 plus 25, or 145, units below 0. Both routes agree.
$45 + (-70) = -25, \qquad -120 + (-25) = -145$

Answer

Line 3 is the first wrong line, because Maya moved 70 units the wrong way. The submarine's final elevation is $-145$ m, which is 145 m below sea level.

Here is the idea to keep. A signed-number story has a zero point, a positive direction and a unit. Once you name them, each value gets its place above, below or at zero, each change becomes an addition, and each subtraction becomes adding the opposite: $p - q = p + (-q)$ for any integers $p$ and $q$ .

The same idea lets you check any solution, including your own. Read each line as start, distance, direction. Ask whether the result moved the right way, and add back to test a difference. The first line that fails is where the repair begins.