maths.free › Arithmetic › 3. Integers › Add Integers

Add Integers

Model addition of integers

Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of $2-1.$ Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in .

We will model four addition facts using the numbers $5,-5\ \text{and}\ 3\ ,-3.$

\[5+3\ -5+(-3)\ -5+3\ 5+(-3)\]

Example

Try it.

Model: $5+3.$

Solution

Interpret the expression.	$5+3$ means the sum of $5$ and $3$.
Model the first number. Start with 5 positives.
Model the second number. Add 3 positives.
Count the total number of counters.
The sum of 5 and 3 is 8.	$5+3=8$

Example

Try it.

Model: $-5+(-3).$

Solution

Interpret the expression.	$-5+(-3)$ means the sum of $-5$ and $-3$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 negatives.
Count the total number of counters.
The sum of −5 and −3 is −8.	$-5+-3=-8$

Example

Try it.

Model: $-5+3.$

Solution

Interpret the expression.	$-5+3$ means the sum of $-5$ and $3$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 positives.
Remove any neutral pairs.
Count the result.
The sum of −5 and 3 is −2.	$-5+3=-2$

Notice that there were more negatives than positives, so the result is negative.

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.

For example, if you want to add $37+(-53),$ you don’t have to count out $37$ blue counters and $53$ red counters.

Picture $37$ blue counters with $53$ red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because $53-37=16,$ there are $16$ more negative counters.

\[37+(-53)=-16\]

Let’s try another one. We’ll add $-74+(-27).$ Imagine $74$ red counters and $27$ more red counters, so we have $101$ red counters all together. This means the sum is $\text{-101.}$

\[-74+(-27)=-101\]

Look again at the results of - .

$5+3$	$-5+(-3)$
both positive, sum positive	both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
$-5+3$	$5+(-3)$
different signs, more negatives	different signs, more positives
Sum negative	sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

Example

Try it.

Simplify:

ⓐ $\ 19+(-47)\$
ⓑ $\ -32+40$

Solution

ⓐ Since the signs are different, we subtract $19$ from $47.$ The answer will be negative because there are more negatives than positives.

\[\begin{array}{l}19+(-47) \\ -28\end{array}\]

ⓑ The signs are different so we subtract $32$ from $40.$ The answer will be positive because there are more positives than negatives

\[\begin{array}{l}-32+40 \\ 8\end{array}\]

Example

Try it.

Simplify: $-14+(-36).$

Solution

Since the signs are the same, we add. The answer will be negative because there are only negatives.

\[\begin{array}{l}-14+(-36) \\ -50\end{array}\]

The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

Example

Try it.

Simplify: $-5+3(-2+7).$

Solution

	$-5+3(-2+7)$
Simplify inside the parentheses.	$-5+3(5)$
Multiply.	$-5+15$
Add left to right.	$10$

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.

Example

Try it.

Evaluate $x+7\ \text{when}\$

ⓐ $\ x=-2\$
ⓑ $\ x=-11.$

Solution

ⓐ Evaluate $x+7$ when $x=-2$


Simplify.

ⓑ Evaluate $x+7$ when $x=-11$


Simplify.

Example

Try it.

When $n=-5,$ evaluate 

ⓐ $\ n+1\$
ⓑ $\ -n+1.$

Solution

ⓐ Evaluate $n+1$ when $n=-5$


Simplify.

ⓑ Evaluate $-n+1$ when $n=-5$


Simplify.
Add.

Next we'll evaluate an expression with two variables.

Example

Try it.

Evaluate $3a+b$ when $a=12$ and $b=-30.$

Solution



Multiply.
Add.

Example

Try it.

Evaluate ${(x+y)}^{2}$ when $x=-18$ and $y=24.$

Solution

This expression has two variables. Substitute $-18$ for $x$ and $24$ for $y.$

	${(x+y)}^{2}$
	${(-18+24)}^{2}$
Add inside the parentheses.	${(6)}^{2}$
Simplify	$36$

Translate Word Phrases to Algebraic Expressions

All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.

Example

Try it.

Translate and simplify: the sum of $-9$ and $5.$

Solution

The sum of −9 and 5 indicates addition.	the sum of $-9$ and $5$
Translate.	$-9+5$
Simplify.	$-4$

Example

Try it.

Translate and simplify: the sum of $8$ and $-12,$ increased by $3.$

Solution

The phrase increased by indicates addition.

	The sum of $8$ and $-12$, increased by $3$
Translate.	$[8+(-12)]+3$
Simplify.	$-4+3$
Add.	$-1$

Add Integers in Applications

Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of $\text{\$5}$ could be represented as $\text{-\$5.}$ Let’s practice translating and solving a few applications.

Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.

Example

Try it.

The temperature in Buffalo, NY, one morning started at $7\ \text{degrees}$ below zero Fahrenheit. By noon, it had warmed up $12\ \text{degrees}.$ What was the temperature at noon?

Solution

We are asked to find the temperature at noon.

Write a phrase for the temperature.	The temperature warmed up 12 degrees from 7 degrees below zero.
Translate to math notation.	−7 + 12
Simplify.	5
Write a sentence to answer the question.	The temperature at noon was 5 degrees Fahrenheit.

Example

Try it.

A football team took possession of the football on their $\text{42-yard line.}$ In the next three plays, they lost $\text{6 yards,}$ gained $\text{4 yards,}$ and then lost $\text{8 yards.}$ On what yard line was the ball at the end of those three plays?

Solution

We are asked to find the yard line the ball was on at the end of three plays.

Write a word phrase for the position of the ball.	Start at 42, then lose 6, gain 4, lose 8.
Translate to math notation.	42 − 6 + 4 − 8
Simplify.	32
Write a sentence to answer the question.	At the end of the three plays, the ball is on the 32-yard line.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Addition of Positive and Negative Integers

$5+3$	$-5+(-3)$
both positive, sum positive	both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
$-5+3$	$5+(-3)$
different signs, more negatives	different signs, more positives
Sum negative	sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

Add Integers

Model Addition of Integers

In the following exercises, model the expression to simplify.

Try it.

$7+4$

Solution

Try it.

$8+5$

Try it.

$-6+(-3)$

Solution

−9

Try it.

$-5+(-5)$

Try it.

$-7+5$

Solution

−2

Try it.

$-9+6$

Try it.

$8+(-7)$

Solution

Try it.

$9+(-4)$

Simplify Expressions with Integers

In the following exercises, simplify each expression.

Try it.

$-21+(-59)$

Solution

−80

Try it.

$-35+(-47)$

Try it.

$48+(-16)$

Solution

Try it.

$34+(-19)$

Try it.

$-200+65$

Solution

−135

Try it.

$-150+45$

Try it.

$2+(-8)+6$

Solution

Try it.

$4+(-9)+7$

Try it.

$-14+(-12)+4$

Solution

−22

Try it.

$-17+(-18)+6$

Try it.

$135+(-110)+83$

Solution

108

Try it.

$140+(-75)+67$

Try it.

$-32+24+(-6)+10$

Solution

−4

Try it.

$-38+27+(-8)+12$

Try it.

$19+2(-3+8)$

Solution

Try it.

$24+3(-5+9)$

Evaluate Variable Expressions with Integers

In the following exercises, evaluate each expression.

Try it.

$x+8$ when

ⓐ $\ x=-26$
ⓑ $\ x=-95$

Solution

ⓐ −18
ⓑ −87

Try it.

$y+9$ when

ⓐ $\ y=-29$
ⓑ $\ y=-84$

Try it.

$y+(-14)$ when

ⓐ $\ y=-33$
ⓑ $\ y=30$

Solution

ⓐ −47
ⓑ 16

Try it.

$x+(-21)$ when

ⓐ $\ x=-27$
ⓑ $\ x=44$

Try it.

When $a=-7,$ evaluate:

ⓐ $\ a+3$
ⓑ $\ -a+3$

Solution

ⓐ −4
ⓑ 10

Try it.

When $b=-11,$ evaluate:

ⓐ $\ b+6$
ⓑ $\ -b+6$

Try it.

When $c=-9,$ evaluate:

ⓐ $\ c+(-4)$
ⓑ $\ -c+(-4)$

Solution

ⓐ −13
ⓑ 5

Try it.

When $d=-8,$ evaluate:

ⓐ $\ d+(-9)$
ⓑ $\ -d+(-9)$

Try it.

$m+n$ when, $m=-15$, $n=7$

Solution

−8

Try it.

$p+q$ when, $p=-9$, $q=17$

Try it.

$r-3s$ when, $r=16$, $s=2$

Solution

Try it.

$2t+u$ when, $t=-6$, $u=-5$

Try it.

${(a+b)}^{2}$ when, $a=-7$, $b=15$

Solution

Try it.

${(c+d)}^{2}$ when, $c=-5$, $d=14$

Try it.

${(x+y)}^{2}$ when, $x=-3$, $y=14$

Solution

121

Try it.

${(y+z)}^{2}$ when, $y=-3$, $z=15$

Translate Word Phrases to Algebraic Expressions

In the following exercises, translate each phrase into an algebraic expression and then simplify.

Try it.

The sum of $-14$ and $5$

Solution

−14 + 5 = −9

Try it.

The sum of $-22$ and $9$

Try it.

$8$ more than $-2$

Solution

−2 + 8 = 6

Try it.

$5$ more than $-1$

Try it.

$-10$ added to $-15$

Solution

−15 + (−10) = −25

Try it.

$-6$ added to $-20$

Try it.

$6$ more than the sum of $-1$ and $-12$

Solution

[−1 + (−12)] + 6 = −7

Try it.

$3$ more than the sum of $-2$ and $-8$

Try it.

the sum of $10$ and $-19,$ increased by $4$

Solution

[10 + (−19)] + 4 = −5

Try it.

the sum of $12$ and $-15,$ increased by $1$

Add Integers in Applications

In the following exercises, solve.

Try it.

Temperature The temperature in St. Paul, Minnesota was $-19\text{^{\circ}F}$ at sunrise. By noon the temperature had risen $\text{26^{\circ}F.}$ What was the temperature at noon?

Solution

7°F

Try it.

Temperature The temperature in Chicago was $-15\text{^{\circ}F}$ at 6 am. By afternoon the temperature had risen $\text{28^{\circ}F.}$ What was the afternoon temperature?

Try it.

Credit Cards Lupe owes $\text{\$73}$ on her credit card. Then she charges $\text{\$45}$ more. What is the new balance?

Solution

−$118

Try it.

Credit Cards Frank owes $\text{\$212}$ on his credit card. Then he charges $\text{\$105}$ more. What is the new balance?

Try it.

Football A team lost $\text{3 yards}$ the first play. Then they lost $\text{2 yards,}$ gained $\text{1 yard,}$ and then lost $\text{4 yards.}$ What was the change in overall yardage over the four plays?

Solution

−8 yards

Try it.

Card Games April lost $\text{5 cards}$ the first turn. Over the next three turns, she lost $\text{3 cards,}$ gained $\text{2 cards,}$ and then lost $\text{1 card.}$ What was the change in cards over the four turns?

Try it.

Football The Rams took possession of the football on their own $\text{35-yard line.}$ In the next three plays, they lost $\text{12 yards,}$ gained $\text{8 yards,}$ then lost $\text{6 yards.}$ On what yard line was the ball at the end of those three plays?

Solution

25-yard line

Try it.

Football The Cowboys began with the ball on their own $\text{20-yard line.}$ They gained $\text{15 yards,}$ lost $\text{3 yards}$ and then gained $\text{6 yards}$ on the next three plays. Where was the ball at the end of these plays?

Try it.

Scuba Diving A scuba diver swimming $\text{8 feet}$ below the surface dove $\text{17 feet}$ deeper; the pressure got to them and they rose five feet. What is their new depth?

Solution

20 feet

Try it.

Gas Consumption: Ozzie rode their motorcycle for $\text{30 minutes,}$ using $\text{168 fluid ounces of gas.}$ Then they stopped and got $\text{140-fluid ounces of gas.}$ Represent the change in gas amount as an integer.

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Evaluate $x+8$ when $x=6.$
If you missed this problem, review .

Жавобни кўрсатиш

$14$
Simplify: $8+2(5+1).$
If you missed this problem, review .

Жавобни кўрсатиш

$20$
Translate the sum of $3$ and negative $7$ into an algebraic expression.
If you missed this problem, review

Жавобни кўрсатиш

$3+(-7)$

Model: $5+3.$

Жавобни кўрсатиш

Interpret the expression.	$5+3$ means the sum of $5$ and $3$.
Model the first number. Start with 5 positives.
Model the second number. Add 3 positives.
Count the total number of counters.
The sum of 5 and 3 is 8.	$5+3=8$

Model the expression.
$2+4$

Жавобни кўрсатиш

6
Model the expression.
$2+5$

Жавобни кўрсатиш

7

Model: $-5+(-3).$

Жавобни кўрсатиш

Interpret the expression.	$-5+(-3)$ means the sum of $-5$ and $-3$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 negatives.
Count the total number of counters.
The sum of −5 and −3 is −8.	$-5+-3=-8$

Model the expression.
$-2+(-4)$

Жавобни кўрсатиш

−6
Model the expression.
$-2+(-5)$

Жавобни кўрсатиш

−7

Model: $-5+3.$

Жавобни кўрсатиш

Interpret the expression.	$-5+3$ means the sum of $-5$ and $3$.
Model the first number. Start with 5 negatives.
Model the second number. Add 3 positives.
Remove any neutral pairs.
Count the result.
The sum of −5 and 3 is −2.	$-5+3=-2$

Notice that there were more negatives than positives, so the result is negative.

Model the expression, and then simplify:
$2+(-4)$

Жавобни кўрсатиш

−2
Model the expression, and then simplify:
$2+(-5)$

Жавобни кўрсатиш

−3

Model: $5+(-3).$

Жавобни кўрсатиш

Interpret the expression.	$5+(-3)$ means the sum of $5$ and $-3$.
Model the first number. Start with 5 positives.
Model the second number. Add 3 negatives.
Remove any neutral pairs.
Count the result.
The sum of 5 and −3 is 2.	$5+(-3)=2$

Model the expression, and then simplify:
$(-2)+4$

Жавобни кўрсатиш

2
Model the expression:
$(-2)+5$

Жавобни кўрсатиш

3
1. ⓐ 4 + 2
2. ⓑ −3 + 6
3. ⓒ 4 + (−5)
4. ⓓ -2 + (−3)
Жавобни кўрсатиш

ⓐ
$4+2$
Start with 4 positives.
Add two positives.
How many do you have? $4+2=6$
ⓑ
$-3+6$
Start with 3 negatives.
Add 6 positives.
Remove neutral pairs.
How many are left?
$3$. $-3+6=3$
ⓒ
$4+(-5)$
Start with 4 positives.
Add 5 negatives.
Remove neutral pairs.
How many are left?
$-1$. $4+(-5)=-1$
ⓓ
$-2+(-3)$
Start with 2 negatives.
Add 3 negatives.
How many do you have? $-5$. $-2+(-3)=-5$
Simplify:
1. ⓐ $\ 19+(-47)\$
2. ⓑ $\ -32+40$
Жавобни кўрсатиш

ⓐ Since the signs are different, we subtract $19$ from $47.$ The answer will be negative because there are more negatives than positives.
\[\begin{array}{l}19+(-47) \\ -28\end{array}\]
ⓑ The signs are different so we subtract $32$ from $40.$ The answer will be positive because there are more positives than negatives
\[\begin{array}{l}-32+40 \\ 8\end{array}\]
Simplify each expression:
1. ⓐ $\ 15+(-32)\$
2. ⓑ $\ -19+76$
Жавобни кўрсатиш
1. ⓐ −17
2. ⓑ 57
Simplify each expression:
1. ⓐ $\ -55+9\$
2. ⓑ $\ 43+(-17)$
Жавобни кўрсатиш
1. ⓐ −46
2. ⓑ 26
Simplify: $-14+(-36).$

Жавобни кўрсатиш

Since the signs are the same, we add. The answer will be negative because there are only negatives.
\[\begin{array}{l}-14+(-36) \\ -50\end{array}\]
Simplify the expression:
$-31+(-19)$

Жавобни кўрсатиш

−50
Simplify the expression:
$-42+(-28)$

Жавобни кўрсатиш

−70
Simplify: $-5+3(-2+7).$

Жавобни кўрсатиш

$-5+3(-2+7)$
Simplify inside the parentheses. $-5+3(5)$
Multiply. $-5+15$
Add left to right. $10$
Simplify the expression:
$-2+5(-4+7)$

Жавобни кўрсатиш

13
Simplify the expression:
$-4+2(-3+5)$

Жавобни кўрсатиш

0
Evaluate $x+7\ \text{when}\$
1. ⓐ $\ x=-2\$
2. ⓑ $\ x=-11.$
Жавобни кўрсатиш

ⓐ Evaluate $x+7$ when $x=-2$
Simplify.
ⓑ Evaluate $x+7$ when $x=-11$
Simplify.
Evaluate each expression for the given values:
$x+5\ \text{when}\$
1. ⓐ $\ x=-3\ \text{and}\$
2. ⓑ $\ x=-17\$
Жавобни кўрсатиш
1. ⓐ 2
2. ⓑ −12
Evaluate each expression for the given values: $y+7\$ when
1. ⓐ $\ y=-5\$
2. ⓑ $\ y=-8\$
Жавобни кўрсатиш
1. ⓐ 2
2. ⓑ −1
When $n=-5,$ evaluate 
1. ⓐ $\ n+1\$
2. ⓑ $\ -n+1.$
Жавобни кўрсатиш

ⓐ Evaluate $n+1$ when $n=-5$
Simplify.
ⓑ Evaluate $-n+1$ when $n=-5$
Simplify.
Add.
When $n=-8,$ evaluate
1. ⓐ $\ n+2\ \$
2. ⓑ $\ -n+2\$
Жавобни кўрсатиш
1. ⓐ −6
2. ⓑ 10
$\text{When}\ y=-9,\ \text{evaluate}\$
1. ⓐ $\ y+8\$
2. ⓑ $\ -y+8.$
Жавобни кўрсатиш
1. ⓐ −1
2. ⓑ 17
Evaluate $3a+b$ when $a=12$ and $b=-30.$

Жавобни кўрсатиш

Multiply.
Add.
Evaluate the expression:
$a+2b\ \text{when}\ a=-19\ \text{and}\ b=14.$

Жавобни кўрсатиш

9
Evaluate the expression:
$5p+q\ \text{when}\ p=4\ \text{and}\ q=-7.$

Жавобни кўрсатиш

13
Evaluate ${(x+y)}^{2}$ when $x=-18$ and $y=24.$

Жавобни кўрсатиш

This expression has two variables. Substitute $-18$ for $x$ and $24$ for $y.$

${(x+y)}^{2}$
${(-18+24)}^{2}$
Add inside the parentheses. ${(6)}^{2}$
Simplify $36$
Evaluate:
${(x+y)}^{2}$ when $x=-15$ and $y=29.$

Жавобни кўрсатиш

196
Evaluate:
${(x+y)}^{3}$ when $x=-8$ and $y=10.$

Жавобни кўрсатиш

8
Translate and simplify: the sum of $-9$ and $5.$

Жавобни кўрсатиш

The sum of −9 and 5 indicates addition. the sum of $-9$ and $5$
Translate. $-9+5$
Simplify. $-4$

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Add Integers

Model addition of integers
Simplify expressions with integers
Evaluate variable expressions with integers
Translate word phrases to algebraic expressions
Add integers in applications
ⓐ 4 + 2
ⓑ −3 + 6
ⓒ 4 + (−5)

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Ўзингизни синаб кўринг

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Кўпроқ Arithmetic

Order of operations Fractions Percentages Powers and roots

	\(-5+3(-2+7)\)
Simplify inside the parentheses.	\(-5+3(5)\)
Multiply.	\(-5+15\)
Add left to right.	\(10\)

	\({(x+y)}^{2}\)
	\({(-18+24)}^{2}\)
Add inside the parentheses.	\({(6)}^{2}\)
Simplify	\(36\)

	The sum of \(8\) and \(-12\), increased by \(3\)
Translate.	\([8+(-12)]+3\)
Simplify.	\(-4+3\)
Add.	\(-1\)

	\(-5+3(-2+7)\)
Simplify inside the parentheses.	\(-5+3(5)\)
Multiply.	\(-5+15\)
Add left to right.	\(10\)

	\({(x+y)}^{2}\)
	\({(-18+24)}^{2}\)
Add inside the parentheses.	\({(6)}^{2}\)
Simplify	\(36\)

Interpret the expression.	\(5+3\) means the sum of \(5\) and \(3\).
Model the first number. Start with 5 positives.
Model the second number. Add 3 positives.
Count the total number of counters.
The sum of 5 and 3 is 8.	\(5+3=8\)

Interpret the expression.	\(-5+(-3)\) means the sum of \(-5\) and \(-3\).
Model the first number. Start with 5 negatives.
Model the second number. Add 3 negatives.
Count the total number of counters.
The sum of −5 and −3 is −8.	\(-5+-3=-8\)

Interpret the expression.	\(-5+3\) means the sum of \(-5\) and \(3\).
Model the first number. Start with 5 negatives.
Model the second number. Add 3 positives.
Remove any neutral pairs.
Count the result.
The sum of −5 and 3 is −2.	\(-5+3=-2\)

\(5+3\)	\(-5+(-3)\)
both positive, sum positive	both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
\(-5+3\)	\(5+(-3)\)
different signs, more negatives	different signs, more positives
Sum negative	sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

The sum of −9 and 5 indicates addition.	the sum of \(-9\) and \(5\)
Translate.	\(-9+5\)
Simplify.	\(-4\)

Interpret the expression.	\(5+(-3)\) means the sum of \(5\) and \(-3\).
Model the first number. Start with 5 positives.
Model the second number. Add 3 negatives.
Remove any neutral pairs.
Count the result.
The sum of 5 and −3 is 2.	\(5+(-3)=2\)

ⓐ
	\(4+2\)
Start with 4 positives.
Add two positives.
How many do you have?	\(4+2=6\)

ⓑ
	\(-3+6\)
Start with 3 negatives.
Add 6 positives.
Remove neutral pairs.
How many are left?
	\(3\). \(-3+6=3\)

ⓒ
	\(4+(-5)\)
Start with 4 positives.
Add 5 negatives.
Remove neutral pairs.
How many are left?
	\(-1\). \(4+(-5)=-1\)

ⓓ
	\(-2+(-3)\)
Start with 2 negatives.
Add 3 negatives.
How many do you have?	\(-5\). \(-2+(-3)=-5\)