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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 Since 58^(∘) F is less than 62^(∘)F, 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
Sea level is an altitude of 0ft. We can represent the depth below sea level with negative numbers. The diver has a depth of - 10ft. The dolphin is 7 feet deeper than the diver. To find the dolphin’s depth, we add - 7 to the diver’s depth. -10 ft +( - 7 ft) We are adding two negative integers. First, we change the addition sign to a subtraction sign and we change - 7 to its opposite. -10 ft + ( -7) ft = -10 ft - 7 ft The expression is now a subtraction of a positive integer. To simplify the expression, we plot -10 on a vertical number line and move 7 units down.

Subtracting 7 from -10 on a vertical number line.

The dolphin has a depth of -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, change the subtraction sign to an addition sign and change the negative number we subtract to its opposite.

b

Write Magdalena’s hiding spot as an integer. Subtract the integer that represents Tadeo’s hiding spot on Parking Level 3 from Magdalena’s integer. This gives the number of floors between their hiding spots.

Solution
a

Tadeo goes from the community room on Floor 3 to Parking Level 3, which is floor - 3. Since he passes through Floor 0, the difference between the starting and ending floors shows how many floors he travels.

3 - ( -3) The expression is a subtraction of a negative integer from another integer. First, we change the subtraction sign to an addition sign and change - 3 to its opposite. 3 - ( -3) = 3 + 3 Now we add the numbers on a number line by moving 3 units to the right, starting at 3.

Adding 3 to 3 on a vertical number line.

Tadeo went down 6 floors.

b

Tadeo hid on Parking Level 3, so his position is -3. Magdalena then hid on Parking Level 1, so her position is -1. Subtracting -3 from -1 shows how many floors she moved.

-1 - ( -3) Again, we can rewrite the subtraction of a negative as addition of a positive. -1 - ( -3) = -1 + 3 To find it, we move 3 units to the right from - 1 on a number line.

Adding 3 to -1 on a number line.

Magdalena went up 2 floors.

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. 3 + ( -5)+ 1 We can simplify the expression by rewriting the addition of a negative as subtraction of a positive. 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 will use a number line to help us see each step.

Calculating 3-5+1 using a number line.

Magdalena ends on -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 integers. 1+ 1 + ( -4) Once again we can rewrite the addition of a negative as subtraction of a positive. 1+ 1 + ( -4) = 1+ 1 - 4 Again, we can use a number line. We plot 1 and move 1 unit to the right. Finally, we move 4 units to the left.

Tadeo is at - 2, one position behind Magdalena.



Level 1
Level 2
Level 3
Add and Subtract Integers
Exercise 2.1

Maya is playing a special edition of darts at the local fair.

The following table shows the points earned based on which color the dart hits.

Color Points
Yellow -3
Blue 2
Red -5
Purple 7

Maya is given three darts to throw. The points of each are added. If she earns more than 3 points total, she will win a toy. Maya gives her best throws. She hit red, then blue, and finally purple. How many points did she earn?

We are told that Maya hit red, blue, and finally purple. Let's write the points that each of these colors are worth. Red:& -5 Blue:& 2 Purple:& 7 We can add these points to get the total points she earned after her three attempts. Maya's Total Points -5+ 2+ 7 We can now find the sum by making the calculations from left to right. This means that we first move 5 units left starting from 0 on a number line to represent the five points she loses by hitting red.

She then hit blue and won 2 points. We now move two units to the right from her position to add these two points to her score.

Finally, she hit purple and gained 7 points. Let's add this information to our diagram to determine Maya's total score after the three attempts.

Maya's final score after the three attempts is 4. Because she got a score greater than 3, she went home with a Teddy Bear.

E
Hint & Answer
S
Solution

Davontay is measuring the temperature outside for a school experiment. The current temperature dropped 8 degrees from the previous one Davontay recorded. If the previous temperature was 15^(∘) C, what is the current temperature?

The previous temperature was 15^(∘) C. We are told that the current temperature dropped 8^(∘) C from the previous one. We must subtract this change in temperature from the current temperature because a drop means it goes down. Current Temperature 15-8=? We got a subtraction of a positive integer from another integer. Let's display the previous temperature on a number line. We will then move eight units to its left side starting from the previous temperature to find the current temperature.

This means that after dropping 8^(∘) C, the temperature changes from 15^(∘) C to 7^(∘) C. Current Temperature 15-8=7^(∘) C

E
Hint & Answer
S
Solution

Diego is enjoying his vacation with his family at a Caribbean beach. While his parent is diving in the coral reef 13 feet below sea level, Diego sees a seagull above his parent at about 10 feet. What is the distance from Diego's parent to the seagull?

We need to find the distance between Diego's parent to the seagull. We can begin by representing the position of each relative to the sea level. Because Diego's parent is 13 feet below sea level, the negative number -13 represents his position. The seagull is 10 feet above level, the integer 10 represents its position. rc &Position Diego's Parent:& -13 Seagull: & 10 We can now use the position of each of them to find their distance to sea level. We can find their distances by using the absolute value. Recall that the absolute value of a negative integer is its additive inverse while the absolute value of a positive number is itself.

Position Distance
Diego's Parent -13 |-13|=13
Seagull 10 |10|=10

The sum of the distances result in the distance between Diego's parent to the seagull. Let's add this number. 13+10=23feet

E
Hint & Answer
S
Solution

A letter carrier must deliver two parcels to two offices in a very high building. He has to go down three floors to deliver the first parcel when being at ground level. He then has to go up eight floors to deliver the second parcel.

Postman.jpg

How many floors above ground level is the office where the postman has to deliver the second parcel?

In this situation, the ground level represents the zero point. The mail carrier moves three floors down first. We can represent this movement as -3. Next, he has to move 8 floors up to deliver the second parcel. Add 8 to -3 to find the second office's floor. Second Office's Floor [0.5em] -3+ 8 We now have a sum of a positive integer to another integer. We can find this sum using a vertical number line. Draw a point at the number zero to represent the ground level.

Next, move three units down from the ground level to represent the office where the first parcel must be delivered.

The number -3 represents the letter carrier's location when he delivers the first parcel. He then has to move eight floors up to deliver the second parcel. Move eight units up from -3 to add the 8 floors he needs to go up.

The letter carrier reaches the fifth floor. This means that the office where he must deliver the second parcel is on the fifth floor.

E
Hint & Answer
S
Solution

Add and Subtract Integers

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