Adding Integers
The addition of integers can be done in a variety of ways, such as using number lines,
manipulatives and a T-chart, calculators or shortcuts.
Parentheses (or brackets) are often used around integers and their positive/negative
signs to make things less confusing when doing mathematical operations.
Adding Integers Using a Number Line
Step 1: Draw a number line.
Step 2: Locate the first number on the number line.
Step 3: Move the number of spaces as shown by the second number.
• Move to the right if the second integer is positive.
• Move to the left if the second integer is negative.
The number that you land on after moving the correct number of spaces to the left or
right is the answer.
Examples
A) (–3) + (+5)
Step 1: Draw a number line.
–5 –4
–3
–2
–1
0
+1
+2
+3
+4
+5
+4
+5
Step 2: Locate the first number on the number line.
–5 –4
–3
–2
–1
0
+1
+2
+3
Step 3: From the first number, move five spaces to the right, because the second
number is positive.
–5 –4
–3
–2
–1
0
+1
+2
+3
+4
+5
Conclusion: (–3) + (+5) = +2
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B) (–2) + (–3)
–5
–4
–3
–2
–1
0
+1
+2
+3
+4
+5
(–2) + (–3) = –5
Move three places to the left of the first number, because the second number is
negative.
C) (+4) + (–6)
(+4) + (–6) = –2
–5
–4
–3
–2
–1
0
+1
+2
+3
+4
+5
Move six places to the left of the first number, because the second number is
negative.
D) (–5) + (+3)
(–5) + (+3) = –2
–5
–4
–3
–2
–1
0
+1
+2
+3
+4
+5
Move three places to the right of the first number, because the second number is
positive.
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You can also use a vertical number line to help you think about adding integers.
Think of a hot air balloon floating through the points on a vertical number line.
Positive numbers are like puffs of air to lift the hot air balloon up.
Negative numbers are like sandbags to weigh the hot air balloon down.
To add integers, place the first number on the number line. Look at what the second
number is (positive = a puff of air; negative = a sandbag). Move along the number line
the number of spaces as shown by the second number.
Examples
+1
A) (–6) + (+4) = ___
0
–1
You land here on (–2).
–2
–3
–4
–5
Start the balloon
here, at the first
number.
Conclusion:
The operation says to
add a positive number
(+4). This means you
have 4 puffs of air lifting
the balloon up.
–6
(–6) + (+4) = (–2)
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B)
(+5) + (–7) = ___
What does this mean?
Think of the second number as sandbags.
Start with (+5) and ADD (–7) or sandbags, which will bring down your hot air balloon.
Look on the vertical number line.
Start here with
the balloon.
+5
+4
+3
+2
+1
0
The operation says to add
a negative number (–7).
This means you have 7
sandbags weighing the
balloon down.
–1
You land here on (–2).
–2
–3
–4
Conclusion: (+5) + (–7) = (–2)
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Adding Integers Using Manipulatives and a T-Chart
Manipulatives are objects or shapes, such as coloured counters or chips, that are used
to represent positive and negative numbers.
Manipulatives can be created using a computer. For example, copy a simple
shape onto a word processing document.
Colour or shade the shape and make 10 copies. Each of these shapes will
represent positive one (+1).
Create another shape and shade it a different colour. Make 10 copies. Each
of these shapes will represent negative one (–1).
E.g.,
+1
–1
One positive object and one negative object equal ZERO.
E.g.,
= 0
+
(+1) + (–1) = 0 can be shown on the number line below.
–3
–2
–1
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0
+1
+2
+3
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Placing manipulatives on a chart like the one below, or on opposite sides of a folded
sheet of paper, helps in the addition of integers.
–
E.g.,
+
Negative objects are placed on
this side of the chart.
Examples
–
+
–
+
Positive objects are placed on
this side of the chart.
A) (–4) + (+7)
Step 1: Place objects on the chart to represent
the value of each integer.
Step 2: Cross off or remove pairs of positive and
negative objects. Remember:
=0
–
+
The solution to the problem is the number of objects
remaining. The sign is the same as the colour of the objects
remaining and/or the sign of the column of the T-chart.
(–4) + (+7) = +3
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B) The temperature was –4°C, then dropped another 3°C. What is the final temperature
now? (–4) + (–3)
–
Add objects to the chart to represent the value of each integer.
+
There are no pairs of positive and negative integers, therefore the
solution is the number of objects on the chart.
(–4) + (–3) = –7
The temperature is now –7°C.
C) A golfer scored +3 on one hole and then –1 on the next hole. What is her score so
far? (–1) + (+3)
Add objects to the chart to represent the value of each integer.
–
–
+
+
Cross off or remove pairs of positive and negative objects.
The solution to the addition problem is the number of objects
remaining on the chart, with the sign of the integer matching the
colour of the counters/chips left.
(+3) + (–1) = +2
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The golfer’s score so far is +2.
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Adding Integers Using a Calculator
Calculators can be used to add integers if the calculator has
an integer button. On many models, the integer button
looks like this:
+
Addition of integers using a calculator can be done using one of two methods,
depending on the model of calculator.
Calculator Method #1
Example
(–13) + (+32)
1. Enter the value of the first integer, followed by the
negative.
1
2. Press the
+
3
+ / – key only if the integer is
+/–
–13
key.
3. Enter the value of the second integer, followed by the
+/–
key only if the integer
is negative. In this example, the second number is not negative, so the
+/–
key
is not pushed.
4. Press the
32
3
2
=
key to display the answer.
19
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(–13) + (+32) = +19
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Calculator Method #2
Example
(–3) + (+12)
1. If the first integer is negative, press the + / –
integer.
key, then enter the value of the first
3
2. Press the
–3
key.
+
3. If the second integer is negative, press the + / –
key then enter the value of the
second integer. In this example, the second integer is positive, so the
key
+/–
is not pushed.
1
=
4. Press the
12
2
key to display the answer.
(–3) + (+12) = +9
9
Example
Check out this example of adding integers using a calculator.
(–28) + (+12)
Method #1
Method #2
2
+/–
8
+/–
+
1
2
=
–16
8
+
1
2
=
–16
2
(–28) + (+12) = –16
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Adding Integers Using Shortcuts
Two shortcuts can be applied to help with the addition of integers.
Adding integers with different signs
Example
(+5) + (–2)
Ignore the positive and negative signs for the moment and subtract the numbers.
5–2=3
The sign of the answer is the sign of the largest number in the question.
5>2
The sign of the answer will be the sign in front
of the 5 in the question.
(+5) + (–2) = +3
Proof:
–5 –4
–3
–2
–1
0
+1
+2
+3
+4
+5
Adding integers with the same signs
Example
(–1) + (–3)
If both numbers are positive or negative, add the numbers.
1+3=4
The sign of the numbers in the question becomes the sign of the answer.
The signs of both numbers in the question were negative, so the sign of the
answer will also be negative.
(–1) + (–3) = –4
Proof:
–5 –4
–3
–2
–1
0
+1
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+2
+3
+4
+5
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Practice: Adding Integers
1. What is the opposite integer? The first one is done for you.
a)
c)
Integer
Opposite
+13
–13
Integer
Opposite
b)
Integer
Opposite
–23
d)
+40
Integer
Opposite
–55
2. Find the missing integer.
a) (+4) + (
) = +10
b) (+6) + (
) = +4
c) (–5) + (
) = –9
d) (–7) + (
) = –5
e) (+9) + (
) = +3
f) (–4) + (
) = –11
3. Use a number line or manipulatives to solve the following.
a) (+2) + (+6) + (–2) = ______________
b) (–3) + (+8) + (+2) = ____________
c) (–6) + (–1) + (–7) = ______________
d) (+9) + (–6) + (+4) = ____________
e) (+1) + (–7) + (+5) = ______________
f) (–7) + (+4) + (–12) = ____________
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4. Write each problem in mathematical terms using integers. Choose a method to
solve for the unknown, such as a number line, manipulatives or a calculator. The
first one is done for you.
Imran has $500.00 in his bank account. He purchased a personal CD player for
$295.00. How much money does Imran have left in his account?
(+$500.00) + (–$295.00) = $205.00
Imran has $205.00 left in his account.
a) Chong woke up in the morning and checked the thermometer outside her window. It
was –20°C. Four hours later, the temperature dropped 7°C. What is the outside
temperature now?
b) David cuts and rakes grass in the neighbourhood. He earned $40.00 on Monday,
spent $10.00 on Tuesday, spent $5.00 on Wednesday, earned $20.00 on Thursday,
and earned $10.00 on Friday. How much money did David have at the end of the
week?
c) A submarine starts at sea level and dives 50 m below the surface, then rises 20 m,
and then finally dives 84 m. What is the submarine’s final depth?
d) Nina and her friends took up the sport of golf. They played their best on hole 8.
Their scores for hole 8 were: +5, –1, 0, +4, –2, –1, +3. What was their combined
score on the eighth hole?
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