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5. Add and Subtract Integers
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Pre-Algebra /
Chapter 1
5. 

Add and Subtract Integers

This lesson explores the fundamentals of adding and subtracting integers, particularly focusing on positive and negative integers. It uses relatable scenarios, like Tadeo and Magdalena's adventures in spending their allowances and playing board games, to illustrate the concepts. For example, the material explains how to add or subtract money amounts and temperatures, making it relevant for everyday calculations. It also discusses the importance of understanding the number line for these operations. The lesson aims to make you adept at integer operations through engaging examples and clear, step-by-step explanations.

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Student Learning Objectives:
  • Add and subtract integers on a number line
  • Add and subtract negative integers
12 Theory slides
9 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
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Add and Subtract Integers
Slide of 12
Using a positive integer is a good way to represent the savings in a piggy bank. Adding to this number will give the final amount after adding money to the bank. On the other hand, subtracting the number will give the final amount after withdrawing money from the savings. Throughout this lesson, a variety of examples like this will be used to show the process of adding and subtracting integers.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

Discovering What Is Inside the Mystery Box

Siblings Tadeo and Magdalena bought a mystery box online. A curious board game inside caught their attention.

Mystery box
In this game, each player starts at 0. Players take turns rolling a die and moving backward or forward based on the outcome of the roll. The rules for each roll outcome are shown in the table.

Outcome Action
1 Move 1 step forward
2 Move 2 steps backward
3 Move 3 steps forward
4 Move 4 steps backward
5 Move 5 steps backward
6 Move 6 steps forward

The first player to reach 36 is the winner.

a

Magdalena rolls the die three times and gets 3, 5, and 1, respectively. What is the final position of her game piece on the board?

b

Tadeo got 1, 1, and 4 in his three rolls. What is the final position of his game piece on the board?

Discussion

Adding and Subtracting Integers

Integers can be added or subtracted to find greater or smaller values.

Adding a Positive Integer

To add a positive integer b to an integer a, move b units to the right of a on a number line. Consider a= 3 and b= 7. a= 3, b= 7 ⇓ 3+ 7=? The process of adding a and b will be illustrated using these values.

1
Plot a on a Number Line
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Begin by graphing a on a number line. In this case, the value of a= 3. Move 3 units to the right side of 0 to plot 3.

Graphing 3 on a number line.

2
Move b Units to the Right-Hand Side of a
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Starting from a, move b units to the right to add b units to a. For this example, move seven units to the right of 3 to add 7 to 3.

Moving 7 units to the right-hand side of 3.

The point is now at 10. This means that the sum of 3 and 7 is 10. 3+ 7=10

Discussion

Subtracting a Positive Integer

Subtracting a positive integer b from a is similar. In this case, move b units to the left of a on a number line. Consider the subtraction of b from a using the same example values. a= 3, b= 7 ⇓ 3- 7=? This process can be performed on the number line.

Moving seven units to the left-hand side of 3 to subtract 7 from 3.
Now the point is at -4. The result of subtracting 7 from 3 is -4. 3- 7=-4

This process applies to adding or subtracting any positive integer b from a positive or negative integer a. The result can be a negative integer, a positive integer, or 0.
Example

Money Spent on the Mystery Box

Tadeo and Magdalena bought the mystery box by combining their monthly allowances. Tadeo paid $10 out of the total cost of the box and Magdalena paid only $6.

Dollar
a

What is the total cost of the box?

b

Tadeo had $12 before they purchased the box. How much does he have left after the purchase?

c

Magdalena paid $ 6 toward the box, but she had to use $ 1 from her savings in addition to her entire allowance. How much did she get for her allowance?

Hint
a

Plot the amount Tadeo paid out of the total cost of the box as a point on a number line. Move six units to the right of the point to add the $6 that Magdalena paid. The end point represents the total cost of the box.

b

Graph the starting amount of money that Tadeo had as a point on a number line. Move ten units to the left side of the point to subtract the amount he paid for the box. The end point represents the amount of allowance he has left after the purchase.

c

Start with the amount Magdalena wanted to pay. Then subtract the money she was missing.

Solution
a

We can find the total cost of the box by adding the amounts of money Tadeo and Magdalena paid. Remember that Tadeo paid $10 and Magdalena $6.

$10+ $6=? This expression is the sum of two positive integers. First, we graph $ 10 on the number line. Then we move 6 units to the right starting from 10.

Illustrating the sum of 10 and 6 on a number line.

Tadeo and Magdalena paid $16 for the box.

b

We can find how much money Tadeo has left after buying the box by subtracting the $10 he spent on the mystery box from the $12 of allowance that he had.

$12- $10=? This is a subtraction of a positive integer from another integer. To find the difference, we move 10 units to the left from 12.

Illustrating $12-$10 on a number line.

Tadeo has $2 after buying the box.

c

When Magdalena wanted to use her allowance to pay $6 for the box, she found out that she was missing $1. She used $1 from her savings to pay the missing amount.

Paid:& $ 6 Missing:& $ 1 This means that her allowance is the difference between the amount she wanted to pay and the amount she took from her savings. $ 6 - $ 1 = ? To find it, we move 1 unit to the left from 6.

Illustrating $6-$1 on a number line.

Magdalena got $5 for her allowance.

Example

Watch a TV Show or Play Soccer?

Tadeo and Magdalena are deciding whether to play soccer or stay in the house to watch a movie. The kids agree that they will stay inside if the weather is colder than 62^(∘) F in the afternoon. The current temperature is 67^(∘) F.

Weather app

The weather app says the temperature is expected to cool down by 9^(∘)F later in the afternoon. What will the temperature be then?

What will the kids most likely do during the afternoon?

Hint
Cool down suggests a subtraction. Subtract 9 from 67 to find the expected temperature for the afternoon.

Solution
“Cool down” means the temperature will get lower. To find the expected temperature later, we subtract 9^(∘)F from the current temperature. 67^(∘) F- 9^(∘) F = ? This is a subtraction of a positive integer from another integer. First, we graph 67^(∘)F on the number line. Then we move 9 units to the left starting from 67.

Subtracting 9 from 67 on a number line.

The predicted temperature for later in the afternoon is 58^(∘) F. 67^(∘) F- 9^(∘) F=58^(∘) F The temperature in the afternoon is expected to be below 62^(∘) F. This means that the kids will probably stay inside and watch a movie.

Discussion

Adding a Negative Integer

Adding a negative integer - b to an integer a requires changing the addition sign to a subtraction sign and changing - b to its additive inverse, b. To illustrate this, consider a= 4 and - b= - 5. a= 4, - b= - 5 ⇓ 4+( -5)=? This process is illustrated with this pair of integers.

1
Change the Addition Sign to a Subtraction Sign and Change - b to b
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Change the addition sign to a subtraction sign and then change - b to its opposite b. In this example, - b = -5 and its opposite is 5. 4 + ( -5) ⇔ 4 - 5 The addition of an integer and a negative integer is now turned into a subtraction of an integer and a positive integer.
2
Plot a on a Number Line
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The process now is similar to subtracting one positive integer from another. First, plot a on a number line. The value of a, in this case, is 4.

Graphing 4 on a number line.

3
Move b Units to the Left of a
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Now move b units to the left of a to subtract b from a. In this example, move 5 units to the left of 4 to subtract 5 from 4.

Moving five units to the left of 4 to subtract 5 from 4.

The point is now at -1. This means that the subtraction of 5 from 4 is -1. This is also the result of adding -5 to 4. 4+( -5)=-1

Discussion

Subtracting a Negative Integer

Subtracting a negative integer - b from an integer a requires changing the subtraction sign to an addition sign and changing - b to its additive inverse. To illustrate this, consider a= -2 and - b= - 6 a - ( - b) ⇔ a + b -2 - ( -6) ⇔ -2 + 6 The result is the sum of an integer and a positive integer.

Sum of 4 and 5.
The point is now at 4. The result of subtracting - 6 from -2 is 4. -2 - ( -6)=4 ⇔ -2 + 6=4

This process applies to adding or subtracting any negative integer - b from a positive or negative integer a. The result can be a negative integer, a positive integer, or 0.
Example

Swimming with the Dolphins

A diver notices a dolphin swimming in the water. The diver swims at 10 feet below sea level, but the dolphin is 7 feet deeper. How far below the surface is the dolphin?

Hint
Start by representing the diver's depth as a negative value. The seven feet further down to reach the dolphin is also negative. When adding a negative integer to another integer, change the addition sign to a subtraction sign and the negative integer added to its opposite.

Solution
If we think of the sea level as level 0, we can represent the depth with negative numbers. Since the diver is 10ft below sea level, his depth is - 10ft. Diver: -10 ft The dolphin is 7 feet deeper than the diver. This means the dolphin is - 7 feet relative to the diver. To find the dolphin’s depth, we add - 7 to the diver’s depth. -10 ft +( - 7 ft) We are adding a negative integer to another negative integer. First, we change the addition sign to a subtraction sign. At the same time, we change -7 to its opposite, 7. -10 ft + ( -7) ft = -10 ft - 7 ft The expression is now a subtraction of a positive integer from another integer. To solve it, we plot -10 on a vertical number line and move 7 units down.

Subtracting 7 from -10 on a vertical number line.

The position of the dolphin is -17ft, so it is 17ft below sea level.

Example

Hide and Seek

Tadeo and Magdalena are playing hide-and-seek in the public areas of their apartment building. The lobby is represented by Floor 0. Floors above the lobby are represented by positive integers, and parking levels below the lobby are represented by negative integers. Magdalena starts counting in the community room on Floor 3.

Magdalena and Tadeo searching Luna.
a

Tadeo hides among the cars in Parking Level 3. How many floors does he go down to reach Parking Level 3?

b

Magdalena quickly finds her brother, so now it's her turn to hide. She makes her way to Parking Level 1. How many floors up did she go?

Hint
a

The parking levels are underground, so the floor numbers are negative. Subtract the Parking Level 3 floor from the community room. When subtracting a negative integer from another integer, change the subtraction sign to an addition sign and the subtrahend to its opposite.

b

Write Magdalena's hiding spot as an integer. Subtract this integer from the integer that represents Tadeo's hiding spot on the Parking Level 3 floor. Apply the absolute value to the result to get the distance.

Solution
a

Since the community room is on Floor 3 and the lobby is on Floor 0, Tadeo first goes down 3 floors to reach the lobby. Then, he continues down to the parking levels.

Community Room to the Lobby 3 Next, he continues down to the third subbasement, Parking Level 3. Negative floor numbers commonly represent floors below ground level. Since Tadeo is going 3 levels below the lobby, this floor number would be -3. Parking Level3 -3 The difference between these numbers represents the number of floors that Tadeo went down to reach his hiding place on Parking Level 3. Number of Floors Descended 3 - ( -3)=? This situation represents the subtraction of a negative integer from another integer. The first step to finding this difference is to change the subtraction sign to an addition sign and change -3 to its opposite, 3. Number of Floors Descended 3 - ( -3)=? ⇕ 3 + 3=? The difference is now turned into the addition of two positive integers. On a vertical number line, move 3 units up starting from 3 to find the number of floors that Tadeo went down to reach the third underground floor, Parking Level 3.

Adding 3 to 3 on a vertical number line.

Tadeo went down six floors to reach Parking Level 3. Number of Floors Descended 3 - ( -3)=6 ⇕ 3 + 3=6

b

Recall that Parking Level 3 is represented by the number -3. Magdalena hid on the first floor of the underground parking structure, one floor below the lobby. This position is -1. Subtracting her position from Parking Level 3 will give how many floors the she went up.

-3 - ( -1)=? This expression is also a subtraction of a negative integer from another integer. Change the subtraction sign to an addition sign and change -1 to its opposite 1. -3 - ( -1)=? ⇕ -3 + 1=? The situation now turns into a sum of a positive integer to another integer. Use the vertical number line again to find this sum.

Adding 1 to -3 on a vertical number line.

This sum is another negative value, -2. -3 - ( -1)=- 2 ⇕ -3 + 1=- 2 Now, because the number of floors to go up is a distance and distances cannot be negative, find the absolute value to this result to find the number of floors they must go up. |-2|=2 This means that Magdalena went up two floors up to get to her hiding spot.

Pop Quiz

Finding the Sum or Subtraction of Random Integers

Find the given sum or difference of integer numbers. Remember that adding a negative number is the same as subtracting its opposite, while subtracting a negative integer is the same as adding its opposite.

An applet asking for the additive inverse of random integer numbers.
Closure

Who Wins the Board Game That Comes in the Mystery Box?

In this lesson, we learned how to add and subtract integer numbers. Now, we will use what we learned to find where Tadeo’s and Magdalena’s pieces are on the curious board game from their mystery box. Let’s start by looking at the board game.

The curious board game

In this game, players start at 0 and take turns rolling a die to move forward or backward based on the outcome of the roll. The rules for each of these outcomes are shown in the table.

Outcome Action
1 Move 1 step forward
2 Move 2 steps backward
3 Move 3 steps forward
4 Move 4 steps backward
5 Move 5 steps backward
6 Move 6 steps forward

The first player to reach 36 is the winner.

a

Magdalena rolls the die three times and gets 3, 5, and 1, respectively. What is the final position of her game piece on the board?

b

Tadeo got 1, 1, and 4 in his three rolls. What is the final position of his game piece on the board?

Hint
a

Magdalena's game piece's final position on the board is the sum of the three integers representing the actions for her rolls. Use the opposite numbers to simplify calculations. Then, perform the calculations from left to right on a number line.

b

Follow the same steps as in Part A.

Solution
a

We start by representing each number Magdalena rolled as an integer. If Magdalena moves forward, we use a positive integer. If she moves backward, we use a negative integer.

Outcome Action Integer
3 Move 3 steps forward 3
5 Move 5 steps backward -5
1 Move 1 step forward 1

Her final position is the sum of the integers that represent the moves for each number she rolled. 3 + ( -5)+ 1 First, we add a negative integer to another integer. To make it easier, we change the addition sign to a subtraction sign and change -5 to its opposite, 5. 3 + ( -5)+ 1 = 3 - 5+ 1 Next, we calculate from left to right. We find the difference between 3 and 5, then add 1 to the result. We can also use a number line to help us see each step.

Calculating 3-5+1 using a number line.

Her moves add up to -1, so she ends on the square labeled -1.

b

To find the position of Tadeo's game piece, we follow the same steps as in Part A. First, we write each roll as an integer. We use a positive integer for moving forward and a negative integer for moving backward.

Outcome Action Integer
1 Move 1 step forward 1
1 Move 1 step forward 1
4 Move 4 steps backward -4

His final position is the sum of these three integers. 1+ 1 + ( -4) The sum of the last two numbers adds a negative integer to another integer. We change the addition sign to a subtraction sign and -4 to its opposite, 4. 1+ 1 + ( -4) = 1+ 1 - 4 Again, we can use a number line. We plot 1, then move one unit to the right to add another 1. Finally, we move 4 units to the left to subtract 4.

Tadeo is a little behind because he is on the square labeled -2. The siblings keep playing, and Tadeo makes a great comeback to beat Magdalena and win the game. Good job!



Level 1
Level 2
Level 3
Add and Subtract Integers
Exercise 1.1
Find the given sum of integers. 27+2

33+(-5)

-11+(-6)

Consider the given sum. 27+ 2 This sum is the addition of a positive integer to another integer. Let's move two units to the right starting from 27 to find the value these two numbers sum up.

The sum of 27 and 2 is 29. 27+ 2=29


Look at the given sum. 33 + ( -5) In this case, this is a sum of a negative integer to another integer. We must change the positive sign to a subtraction sign and -5 to its opposite 5. 33 + ( -5) ⇕ 33 - 5 The result is the subtraction of a positive integer from another integer. Let's move 5 units on a number line to the left starting from 33 to find the result of this subtraction.

The difference is 28, which is also the result of the initial sum. 33 + ( -5)=28 ⇕ 33 - 5=28


Let's look at the last sum. -11 + ( -6) This is also a sum of a negative integer to another integer. Let's change the addition sign to a subtraction sign and -6 to its opposite 6. -11 + ( -6) ⇕ -11 - 6 We have now the subtraction of a positive integer from another integer. Let's find the result of this subtraction on a number line.

The result of this subtraction is -17. This is also the result of the initial sum. -11 + ( -6)=-17 ⇕ -11 - 6=-17

E
Hint & Answer
S
Solution
Find the result of the given subtraction of integers. 4-5

12-(-8)

-6-(-7)

Consider the given subtraction. 4- 5 We have a subtraction of a positive integer from another integer. We can find the result of this subtraction by moving 5 units to the left on a number line starting from 4.

The result of this subtraction is -1. 4- 5=-1


Look at the given subtraction. 12 - ( -8) In this case, we are subtracting a negative integer from another integer. We need to change the subtraction sign to an addition sign and -8 to its opposite 8 to perform this subtraction. 12 - ( -8) ⇕ 12 + 8 We came to a sum of two positive integers. We can then move 8 units to the right starting from 12 to find the result of this sum.

These values add up to 20, which is also the result of the initial difference. 12 - ( -8)=20 ⇕ 12 + 8=20


Let's look at the last subtraction. -6 - ( -7) Again, we have a subtraction of a negative integer from another integer. Let's change the subtraction sign to an addition sign and -7 to its opposite 7. -6 - ( -7) ⇕ -6 + ( 7) The subtraction simplifies to an addition of two integers. Let's find this sum.

This sum equals 1, which is the result of the initial subtraction. -6 - ( -7)=1 ⇕ -6 + ( 7)=1

E
Hint & Answer
S
Solution

Dominika is practicing sum and subtraction with integers. She did the following procedure for the given sum.

Dominika's procedure to calculate the sum and subtraction of three integers.

There is a mistake in Dominika's procedure. Which step has the mistake?

We need to find the mistake in Dominika's procedure. In step A, she added 12 and 2. These values sum up to 14, which Dominika calculated correctly. 12+2-(-5) ⇓ 14-(-5) After performing this sum, the expression simplifies to a subtraction of a negative integer from another integer. In this case, we must continue simplifying the resulting expression by changing the subtraction sign to an addition sign and -5 to its opposite 5. 14-(-5) ⇕ 14+5 We can see from Dominika's procedure that she correctly changed -5 to its opposite. However, she did not change the subtraction sign to an addition sign. This means that step step B is the one that has the mistake. Let's fix this in Dominika's procedure and find the result of this sum.

E
Hint & Answer
S
Solution
Find if the given statement is always, sometimes, or never true.

The sum of two positive integers is positive.

The sum of an integer and its absolute value is 0.

Consider the given statement.

The sum of two positive integers is positive.

Let's test some pairs of numbers and find their sum using a number line. That can help us to discover if this statement is always, sometimes, or never true. 3+7 In this case, we move seven units forward starting from 3 to find the sum.

The result of adding these two positive integers is also positive. Let's try with another pair of numbers. 77+5 Again, let's look at this sum on a number line.

The result is also a positive integer. Note that when adding a positive number to another positive, we always move forward starting from the first positive integer. This means that the sum will always be positive when adding two positive integers.


Let's look at the given statement.

The sum of an integer and its absolute value is 0.

We can follow a similar reasoning to find out if this statement is always, sometimes, or never true. Let's begin by considering the sum of a positive integer and its absolute value. 7 + |7| Recall that the absolute value of a positive integer is itself. This means that the absolute value of |7|=7. The previous sum simplifies to a sum of two positive integers. 7 + |7| ⇕ 7 + 7 = 14 In this case, the statement is false. Now, let's consider a negative integer and its absolute value. -3+|-3| We can simplify this sum by recalling that the absolute value of a negative integer is its opposite. The opposite of -3 is 3. -3+|-3| ⇕ -3 + 3 The sum simplifies to the sum of -3 and its opposite. Consider that the sum of an integer and its opposite is 0. -3 + 3 = 0 ⇕ -3+|-3| = 0 We have an example where the statement is false and one when the statement is true. We can then say that the statement is sometimes true.

E
Hint & Answer
S
Solution

Add and Subtract Integers

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