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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

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

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?

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

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Discussion

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

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 $b=7.$ *expand_more*
*expand_more*

$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

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 $3.$

2

Move $b$ Units to the Right-Hand Side of $a$

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 $3.$

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 $b$ 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.
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

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.$

a What is the total cost of the box?

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

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c Magdalena had $-$1$ of her allowance left after the purchase. How much money did she receive for her allowance?

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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.

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 $$10$ as a point on a number line.
Next, move six units to the right-hand side of $10$ to add the $$6$ that Magdalena paid.

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 $$10$ that Tadeo spent on the box.
This means that Tadeo has $$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.
Magdalena's allowance before the purchase is then $$5.$

$Magdalena’s Monthly Allowance -$1+$6=$5 $

Example

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 $67_{∘}F.$

The weather app says the temperature is expected to cool down by $9_{∘}F$ later in the afternoon. What will the temperature be then?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7109375em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">F<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["58"]}}

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Cool down

suggests a subtraction. Subtract $9$ from $67$ to find the expected temperature for the afternoon.

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 downimplies 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 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.
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 $-b$ 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 $-b=-5.$
*expand_more*
*expand_more*
*expand_more*

$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 $b$

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

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 $4.$

3

Move $b$ Units to the Left-Hand Side of $a$

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 $4.$

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 $-b$ 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.
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

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.

The siblings want to figure out how deep the dolphin is below the surface. Help them find the answer.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":null,"formTextAfter":"feet","answer":{"text":["-17"]}}

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.

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 -10ft $

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 -10ft+(-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 -10ft+(-7)ft=? ⇕-10ft−7ft=? $

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.
The dolphin's depth is $-17.$ This means that the dolphin is $17$ feet below sea level.

$Dolphin’s Depth -10ft+(-7)ft=-17ft ⇕-10ft−7ft=-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

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.

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 $3?$

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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?

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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 $3.$
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.
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

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.

Closure

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.

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'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.

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 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.
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 | $-4$ |

$Tadeo’s Final Position 1+1+(-$