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5. Add and Subtract Integers
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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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9 Exercises - Grade E - A
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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 it 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

Addition and subtraction are two primary operations performed between two or more integers to increase or decrease their values.

Adding a Positive Integer

To add a positive integer b to an integer a, move b units to the right-hand side 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
expand_more
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
expand_more
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-hand side 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 had -$1 of her allowance left after the purchase. How much money did she receive 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-hand side 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-hand 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 remaining amount of money that Magdalena has. Add the amount of money she paid for the box to this amount.

Solution
a The sum of the amount of money paid by Tadeo and by Magdalena gives the total cost of the box. Remember that that Tadeo paid $10 and Magdalena $6.

Total Cost of the Box $10+ $6=? This expression is the sum of two positive integers. Move 10 units to the right-hand side of 0 to first graph $10 as a point on a number line.

Graphing $10 on a number line.

Next, move six units to the right-hand side of 10 to add the $6 that Magdalena paid.

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

The end point is 16. This means that, in total, the siblings paid $16 for the box. Total Cost of the Box $10+ $6=$16

b Tadeo's started with $12 of allowance.

Tadeo's Monthly Allowance $12 He then spent $10 of that money on the mystery box. The amount of money left after buying the box can be found by subtracting $10 from $12. Money Left After Buying the Box $12- $10=? This is a subtraction of a positive integer from another integer. Move 10 units to the left-hand side of 12 to subtract the $10 that Tadeo spent on the box.

Illustrating $12-$10 on a number line.

This means that Tadeo has $2 money left after buying the box. Money Left After Buying the Box $12- $10=$2

c Begin by writing how much of her allowance Magdalena has left.

Magdalena's Remaining Allowance -$ 1 This amount is negative because Magdalena spent more than her monthly allowance. She might have had some money leftover from last month, or maybe she borrowed from her parents. Either way, add the amount Magdalena paid out of the total cost of the box to this number to find how much money Magdalena gets for her monthly allowance. Magdalena's Monthly Allowance -$1+ $6=? Move six units to the right-hand side of -1 to find this amount.

Illustrating -$1+$6 on a number line.

Magdalena's allowance before the purchase is then $5. Magdalena's Monthly Allowance -$1+ $6=$5

Example

Watch a TV Show or Play Soccer?

Tadeo and Magdalena take a break from unpacking the box. They are now deciding whether to play soccer or stay in the house to watch a movie in the afternoon. The kids agree that they will stay inside if the weather is colder than 62^(∘) F (degrees Fahrenheit) 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
Tadeo and Magdalena know that the current temperature is 67^(∘) F. Current Temperature 67^(∘) F They want to know the temperature later in the afternoon, given that it is expected to cool down by 9^(∘)F. Cool down implies subtraction — in other words, subtracting 9 degrees from the current temperature will give the temperature for later in the afternoon. Afternoon Temperature 67^(∘) F- 9^(∘) F=? This expression is a subtraction of a positive integer from another integer. This can be represented on a number line. Start by locating the current temperature on the line. Next, move 9 units to the left-hand side of 67 to subtract the 9 degrees from the current temperature.

Adding 6 to 17 on a number line.

This means that the predicted temperature for later in the afternoon is 58^(∘) F. Afternoon Temperature 67^(∘) F- 9^(∘) F=58^(∘) F Since the temperature for the afternoon is expected to be less than 62^(∘) F, the kids will most likely stay inside to watch a movie. Pass the popcorn!

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)=? Next, this process is shown with these 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
expand_more
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-Hand Side of a
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Now move b units to the left-hand side 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-hand side 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 netative integer - b from a positive or negative integer a. The result can be a negative integer, a positive integer, or 0.
Example

Rescuing the Dolphin

Magdalena and Tadeo spend the rest of the day watching a fantastic movie together. The movie is about a dolphin stuck in a net in the ocean. The siblings are intrigued because the main character, a kid like them, wants to save the dolphin. The kid swims 10 feet below sea level, but the dolphin is 7 feet deeper.

A kid swimming toward a stuck dolphin in a net
External credits: @freepik, @upklyak

The siblings want to figure out how deep the dolphin is below the surface. Help them find the answer.

Hint
Start by representing the main character'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
Negative values usually represent elevations below sea level, or depths below sea level. It is given that the kid is 10 feet below sea level. His depth would then be -10. Kid's Depth -10 ft Now, the dolphin is deeper down than the kid is, which means that this extra depth must be added to the kid's depth to find how far down the dolphin is. The dolphin is seven feet deeper than the main character, so this depth can be written as written as -7. Dolphin's Depth -10 ft + ( -7) ft = ? The resulting expression is an addition of a negative integer to another integer. The first step to find this sum is to change the addition sign to a subtraction sign. At the same time, -7 must be changed to its opposite, 7. Dolphin's Depth -10 ft + ( -7) ft = ? ⇕ -10 ft - 7 ft = ? The expression is now a subtraction of a positive integer from another integer. This can be solved on a number line by plotting -10 on a vertical number line, then moving 7 units down from this position. The ending value will be the depth of the dolphin.

Subtracting 7 from -10 on a vertical number line.

The dolphin's depth is -17. This means that the dolphin is 17 feet below sea level. Dolphin's Depth -10 ft + ( -7) ft = -17ft ⇕ -10 ft - 7 ft = -17ft The siblings are happy because the main character saved the dolphin. The movie had a great lesson for everyone — avoid throwing trash away irresponsibly, especially in the ocean, because it is hazardous for all living beings.

Example

Where Is Luna, the Family Pet?

Oh no! Magdalena and Tadeo were distracted by the movie and did not notice that Luna, the family pet, escaped. They must find Luna before their parents get home.

Magdalena and Tadeo searching Luna.

Tadeo and Magdalena live in a third-floor apartment. They decide start their search in the underground parking structure, which is a common place that Luna goes to. She particularly likes the third floor down of the parking structure, Subbasement 3.

a How many floors do they go down to reach Subbasement 3?

b Luna was not in Subbasement 3, so the siblings started working their way back up. They found her on the first floor below ground level, Subbasement 1. How many floors did they go up before coming across Luna?

Hint
a The parking structure is underground, so the floor numbers are negative. Subtract the Subbasement 3 floor from the family's apartment floor. When subtracting a negative integer from another integer, change the subtraction sign to an addition sign and the subtrahend to its opposite.
b Write Luna's location as an integer. Subtract this integer from the integer that represents where Tadeo and Magdalena are on the Subbasement 3 floor. Apply the absolute value to the result to get the distance.

Solution
a Since the siblings live on the third floor, they go down 2 floors to reach the ground floor. If the ground floor is represented by 0 on the number line, the family apartment floor will be represented by 2 on the line.

Family Apartment 2 Next, they descended to the third floor underground, Subbasement 3, to start the search for Luna. Negative values commonly represent floors below ground level. In this case, the number -3 would represent this location. Subbasement3 -3 The difference between these numbers represents the number of floors the siblings went down to reach Subbasement 3. Number of Floors Descended 2 - ( -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. Then, -3 must be changed to its opposite, 3. Number of Floors Descended 2 - ( -3)=? ⇕ 2 + 3=? The difference is now turned into an addition of two positive integers. On a vertical number line, move 3 units up starting from 2 to find the number of floors the siblings went down to reach the third underground floor, Subbasement 3.

Adding 3 to 2 on a vertical number line.

Magdalena and Tadeo went down five floors to reach Subbasement 3. Number of Floors Descended 2 - ( -3)=5 ⇕ 2 + 3=5

b Recall that Subbasement 3 is represented by the number -3. Luna is on the first floor of the underground parking structure, one floor below ground. This position is -1. Subtracting Luna's position from Subbasement 3 will give how many floors the siblings must go up in order to find Luna.

-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 the siblings must go two floors up to find Luna. Great news, they found Luna and can go home now. However, they learned another lesson today. They must be more careful from now on because having a pet is a great responsibility!

Pop Quiz

Finding the Sum or Subtraction of Random Integers

Find the given sum or difference of integer numbers. Consider that adding a negative number is the same as subtracting its opposite. In contrast, 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, how to add and subtract integer numbers was explained. Use this information to find the positions of Tadeo's and Magdalena's pieces on the curious board game that came in the mystery box they bought. Begin 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 Begin by representing each number Magdalena rolled as an integer value. If the action of to a given outcome is to move forward 3 steps, for example, this would be represented by the positive integer 3. On the other hand, if the action is to move backward 5 steps, it would be represented as -5.
Outcome Action Integer
3 Move 3 steps forward 3
5 Move 5 steps backward -5
1 Move 1 step forward 1

Magdalena's final position is the sum of the integers representing each number she rolled on the die. Magdalena's Final Position 3 + ( -5)+ 1=? Now, rewrite the sum of the first two numbers. Note that the sum of these two numbers is a sum of a negative integer to another integer. To add them, change the addition sign to a subtraction sign and change -5 to its opposite number, 5. Magdalena's Final Position 3 + ( -5)+ 1=? ⇕ 3 - 5+ 1=? Perform the calculations from left to right. In other words, calculate the difference between 3 and 5 first, then add 1 to this result. A number line can also be used to represent these calculations.

Calculating 3-5+1 using a number line.

Magdalena's game piece ends up on the square labeled -1 because this is the sum of her three rolls. Magdalena's Final Position 3 + ( -5)+ 1=-1 ⇕ 3 - 5+ 1=-1

b Follow a similar process to find the position of Tadeo's game piece. First, represent each of his rolls as an integer value.
Outcome Action Integer
1 Move 1 step forward 1
1 Move 1 step forward 1
4 Move 4 steps backward -4

The sum of these three integers represents his final position on the board. Tadeo's Final Position 1+ 1 + ( -4)=? Note that the sum of the last two numbers is the sum of a negative integer to another integer. Rewrite this sum by changing the addition sign to a subtraction sign and -4 to its opposite, 4. Tadeo's Final Position 1+ 1 + ( -4)=? ⇕ 1+ 1 - 4=? Again, a number line might be helpful to finding the result of this expression.

Wow! Tadeo is a little behind because he is on the square labeled -2. The siblings continued playing, and Tadeo made a remarkable 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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