Reorder, Regroup, and Find Missing Integers | Grade 7 Math | LevelKite

You already know a useful move with integers, which are the counting numbers, their opposites, and $0$ . To subtract an integer, you add its additive inverse, which is its opposite: the number that sums with it to $0$ . The additive inverse of $9$ is $-9$ , and the additive inverse of $-9$ is $9$ . So $6 - 9 = 6 + (-9)$ and $6 - (-9) = 6 + 9$ .

In this lesson you put that move to work in two ways:

You find the value of an expression (numbers joined by operation signs) with three or more integers, such as $17 - 25 + 3$ , by choosing an order that makes the arithmetic easy.
You find a missing integer, such as the integer $n$ that makes $-7 + n = 3$ true.

Both rest on the same two facts: subtracting a number gives the same result as adding its opposite, and a number plus its opposite is $0$ .

Why can you choose the order? Think of adding as moving along a number line. For any integers $p$ and $q$ , the sum $p + q$ is the number that sits $|q|$ units from $p$ . Here $|q|$ is the absolute value of $q$ , which is its distance from $0$ . You move right when $q$ is positive and left when $q$ is negative. When $q$ is $0$ , you stay at $p$ .

Now make two moves in a row. For $4 + (-6) + 5$ , start at $4$ , move left $6$ to reach $-2$ , then move right $5$ to reach $3$ . Swap the two moves to get $4 + 5 + (-6)$ : start at $4$ , move right $5$ to reach $9$ , then move left $6$ to reach $3$ . You finish at the same number.

Each number kept its own sign, and so its own direction, and the order of the moves did not change the finish. This is true for every sum of integers. A number in a sum, together with its sign, is a term of the sum. Another name for it is an addend. The table names the properties of addition you will use. In the table, $a$ , $b$ and $c$ stand for any integers.

Properties of addition that hold for all integers

Property	What it says	Example
Commutative property of addition	Changing the order of the addends does not change the sum: $a + b = b + a$ .	$-7 + 4 = 4 + (-7) = -3$
Associative property of addition	Changing which addends are grouped first does not change the sum: $(a + b) + c = a + (b + c)$ .	$(-2 + 5) + (-5) = -2 + (5 + (-5)) = -2$
Additive inverse property	A number and its opposite sum to zero: $a + (-a) = 0$ .	$9 + (-9) = 0$
Additive identity property	Adding zero leaves a number unchanged: $a + 0 = a$ .	$-6 + 0 = -6$

Key concept

Make everything addition, then reorder and regroup

For any integers $a$ and $b$ , subtracting $b$ gives the same result as adding its additive inverse, $-b$ . So you can write any string of integer additions and subtractions as a sum of signed terms.

Addition is commutative and associative, so you may add those terms in any order and in any groups, and the sum stays the same. Subtraction is different: changing the order can change the result, as in $5 - 3 = 2$ but $3 - 5 = -2$ . That is why you rewrite first, and why each term keeps its own sign when it moves.

Use this freedom to pair opposites, which sum to $0$ , to pair terms that make a round number, or to add terms with the same sign first.

a - b = a + (-b)

Worked example

Reorder to make a round number

Find the value of $17 - 25 + 3$ .

Rewrite the subtraction as adding the additive inverse. Subtracting 25 gives the same result as adding negative 25, so the expression is now a sum of three signed terms.
$17 - 25 + 3 = 17 + (-25) + 3$
Reorder and regroup so that you add 17 and 3 first. They are both positive, and they make the round number 20. The commutative and associative properties of addition allow this, and negative 25 keeps its sign when it moves.
$17 + (-25) + 3 = (17 + 3) + (-25)$
Add inside the parentheses first: 17 plus 3 is 20.
$(17 + 3) + (-25) = 20 + (-25)$
Add negative 25 to 20. The sum is 25 units from 20, to the left because negative 25 is negative. Moving 20 units left reaches 0, and the 5 units that remain reach negative 5.
$20 + (-25) = -5$

Answer

The value of $17 - 25 + 3$ is $-5$ .

Worked example

Pair opposites in a longer sum

Find the value of $-9 - (-14) + 9 - 30 + 6$ .

Rewrite each subtraction as adding the additive inverse. The additive inverse of negative 14 is 14, and the additive inverse of 30 is negative 30.
$-9 - (-14) + 9 - 30 + 6 = -9 + 14 + 9 + (-30) + 6$
Reorder and regroup the five terms, which the commutative and associative properties of addition allow. Pair the opposites negative 9 and 9, and pair 14 with 6 because they make 20.
$-9 + 14 + 9 + (-30) + 6 = (-9 + 9) + (14 + 6) + (-30)$
Add inside each pair of parentheses. The opposites sum to 0, and 14 plus 6 is 20.
$(-9 + 9) + (14 + 6) + (-30) = 0 + 20 + (-30)$
Adding 0 leaves 20 unchanged. Then add negative 30: the sum is 30 units to the left of 20. Moving 20 units left reaches 0, and the 10 units that remain reach negative 10.
$0 + 20 + (-30) = 20 + (-30) = -10$
Check by adding the five terms from left to right. The result matches, because reordering and regrouping addends does not change a sum.
$-9 + 14 + 9 + (-30) + 6 = 5 + 9 + (-30) + 6 = 14 + (-30) + 6 = -16 + 6 = -10$

Answer

The value of $-9 - (-14) + 9 - 30 + 6$ is $-10$ .

Now turn the question around. An equation is a statement that two expressions have the same value. Sometimes one integer in an equation is missing, and your job is to find it. Let the letter $n$ stand for the missing integer.

Take $-7 + n = 3$ . It asks: starting at $-7$ , which move lands on $3$ ? Moving $7$ units right reaches $0$ , and $3$ more units reach $3$ . That is $10$ units to the right in all, so $n = 10$ .

You can also find $n$ with opposites, and this method works for every equation in this lesson. Add $7$ , the opposite of $-7$ , to both sides. The two sides had the same value before, so they still have the same value after you add the same number to each. Here is the equation after that step:

7 + (-7) + n = 7 + 3

On the left, the opposites $7$ and $-7$ sum to $0$ , and adding $0$ leaves $n$ unchanged. On the right, $7 + 3 = 10$ . So $n = 10$ , the same integer as before.

This undo step works wherever the missing integer sits in an addition or subtraction equation. If the equation has a subtraction, first rewrite it as adding the additive inverse. Then look at the side that has $n$ , and add the opposite of the other term on that side to both sides. The table shows four cases.

Using opposites to find a missing integer

Equation	Written as addition	Add to both sides	Result
$n + (-12) = -5$	Already addition	$12$ , the opposite of $-12$	$n = -5 + 12 = 7$
$n - 8 = -3$	$n + (-8) = -3$	$8$ , the opposite of $-8$	$n = -3 + 8 = 5$
$n - (-6) = -15$	$n + 6 = -15$	$-6$ , the opposite of $6$	$n = -15 + (-6) = -21$
$6 - n = 10$	$6 + (-n) = 10$	$-6$ , the opposite of $6$	$-n = -6 + 10 = 4$ , so $n = -4$

Worked example

A missing integer that is being subtracted

Find the integer $n$ that makes $-4 - n = 11$ true.

Rewrite the subtraction as adding the additive inverse. Subtracting n gives the same result as adding the opposite of n.
$-4 + (-n) = 11$
Add 4 to both sides, because 4 is the opposite of negative 4. Adding the same number to both sides keeps the two sides equal.
$4 + (-4) + (-n) = 4 + 11$
Simplify each side. On the left, the opposites 4 and negative 4 sum to 0, and adding 0 leaves the opposite of n unchanged. On the right, 4 plus 11 is 15.
$-n = 15$
The opposite of n is 15, so n is the opposite of 15.
$n = -15$
Check in the original equation. Subtracting negative 15 is the same as adding 15. The sum is 15 units to the right of negative 4: moving 4 units reaches 0, and the 11 units that remain reach 11. That matches the right side.
$-4 - (-15) = -4 + 15 = 11$

Answer

The missing integer is $n = -15$ , because $-4 - (-15) = 11$ .

Takeaway: Write every subtraction as adding the additive inverse. Then you have a sum of signed terms, and the commutative and associative properties of addition let you add those terms in any order and in any groups. To find a missing integer, write the equation as addition. On the side with the missing integer, find the other term, and add its opposite to both sides, because opposites sum to $0$ . Then check your integer in the original equation.

Next, you will use these skills to solve word problems with integers and to find and fix errors in integer addition and subtraction.