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Players typically lose and win points based on their actions in video games. The wins can be represented by natural and whole numbers but not losses. *Integers,* however, can represent losses. This lesson presents attributes and uses of *integer numbers* used in such situations. ### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Look at the thermometer and play with it to measure different temperatures.

How can temperatures below zero be expressed?

Discussion

Integers are whole numbers and negative numbers without fractions or decimal parts. The following are some examples of integers and non-integers.

Integers | Non-integers |
---|---|

$17,$ $0,$ $-1,$ $121,$ $-67$ | $π,$ $-43 ,$ $1.5,$ $-12.9,$ $21 $ |

An integer can be positive, negative or zero. The set of all integers is denoted by $Z.$

$Z={…-2,-1,0,1,2…}$

Discussion

Similar to natural and whole numbers, integer numbers can be displayed on a number line. This helps to compare numbers and visualize which are greater and smaller.

Concept

A number line can display integer numbers by extending it on the left- and right-hand side of zero. The positive integers or natural numbers will be on the right-hand side, and the negative integers will be on the left-hand side of zero.

Pop Quiz

Determine the integer represented by the point on the number line.

Example

Maya is attending a teen-based summer camp. She is now taking part in a nature scavenger hunt. In this game, players are divided into teams; each has a map, a list of items to find, and tasks to complete. Each action, not only completed task, is worth points.

Action or Task | Points |
---|---|

Find a hidden item (I) | $2$ |

Cheating (C) | $-7$ |

Complete a physical challenge (P) | $4$ |

Solve a puzzle (S) | $7$ |

Take a wrong turn (W) | $-3$ |

Run out of time before finishing a task (T) | $-5$ |

The team that gets more points will be the winner. Maya wants a better idea of the meaning of these points to help her team win. She thinks that plotting them on a number line should help. Which of the following graphs matches Maya's?

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Identify if the points given by each action or task are negative or positive. When the number is positive, count that number of units going to the right of $0$ on the number line. On the other hand, if the number of points is negative, count that number of units to the left of $0.$

The points for each task or action are integers. Display each on a number line and label it. Consider that finding a hidden item (I) gives $2$ points. That requires moving two units to the right of $0$ to display the points earned.
**B**. Now that Maya has a better understanding of the game, she can help her team to win this wonderful adventure.

Next, cheating (C) is represented by $-7.$ This means that the team losses $7$ points if they cheat. Move seven units to the left of zero to represent the points given by this action.

Follow a similar method to graph the remaining points.

This graph matches option

Example

Maya is loving the summer camp. The staff even gave her what they call an explorer's box. One of the items in it is a thermometer. Her grandma used one just like it! Maya is excited to record temperatures all week wherever they go.

She created a table with the temperatures she recorded.

Temperatures Measured |
---|

$-1_{∘}C$ |

$9_{∘}C$ |

$-4_{∘}C$ |

$3_{∘}C$ |

$12_{∘}C$ |

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A number line helps to compare numbers. The smaller numbers are further left and the greater ones are further right. Now, because the list contains positive and negative integers, the number line must contain $0.$

The first recorded temperature is $-1_{∘}C.$ Move one unit to the left of zero and draw a point over the number line to draw this temperature.

The second recorded temperature is $9_{∘}C.$ Plot this temperature by following a similar method. In this case, move $9$ units to the right-hand side of $0.$

The graph shows that the first temperature Maya recorded is less than the second. Now, plot the remaining temperatures following a similar reasoning.

The number line shows that the coldest recorded temperature was $-4_{∘}C$ and the warmest was $12_{∘}C.$ The temperatures can now be written from the coldest to the warmest.$-4_{∘}C,-1_{∘}C,3_{∘}C,9_{∘}C,12_{∘}C $

Discussion

Consider the lucky number $-7.$ Well, perhaps the sign needs to change to get $7$ for it to be lucky. Moving on, what happens when $7$ is added to $-7.$
*additive inverse* of $-7.$ This property of integer numbers is worth exploring deeper! ## Additive Inverse

$-7+7=0 $

Notice that $7$ cancels $-7.$ The reason for that is $7$ is the Concept

The additive inverse of a number is another number such that their sum equals $0.$ If $y$ is the additive inverse of $x,$ then the following equation holds true.

$x+y=0$

Given a number, its additive inverse — also called opposite number — can be found by changing the sign. Some examples of additive inverses are listed in the table.

Number | Additive Inverse | Sum |
---|---|---|

$5$ | $-5$ | $5+(-5)=0$ |

$-21$ | $21$ | $-21+21=0$ |

$0$ | $-0$ | $0+(-0)=0$ |

$a$ | $-a$ | $a+(-a)=0$ |

$-b$ | $b$ | $-b+b=0$ |

Pop Quiz

The additive inverse of a number is another number such that the sum of these two numbers equals $0.$ Use this information to find the additive inverse of the indicated number.

Discussion

It is common to compare quantities that express distances. They help identify which quantity is further than another. Knowing which number represents a further distance is simple if the numbers are positive. What if they are negative? No fears — the *absolute value* comes to the rescue.

Concept

The absolute value of a number $a$ is the distance between $a$ and $0$ on the number line. It is denoted as $∣a∣$ and it is always a non-negative value.

The absolute value of a negative number is its opposite value, while the absolute value of a positive number is equal to itself.

$∣a∣={a-a ifa≥0ifa<0 $

For any integer number $a,$ these two properties hold true.

Property | Algebraic Representation | Example |
---|---|---|

Non-negativity | $∣a∣≥0$ | $∣7∣=7$ and $7≥0$ |

Symmetry | $∣-a∣=∣a∣$ | $∣-7∣=7$ and $∣7∣=7∣-7∣=∣7∣$ |

Pop Quiz

Time to identify and compare numbers. See the given statement and evaluate if it is true or false. Some statements involve absolute values.

Example

Near the camp is an animal rehabilitation center. There, the animals have space to roam. Maya gets the chance to visit! She takes photos of the animals and notes their elevation compared to her standing position.

Animal | Elevation (In Feet) |
---|---|

Kingfisher | $10$ feet |

River otter | $-8$ feet |

Porcupine | $7$ feet |

Sturgeon | $-12$ feet |

Painted turtle | $-3$ feet |

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Use the absolute values to order the elevations of the animals from Maya's position.

The elevations contain negative and positive integers. Apply the absolute value of the elevations to get the distances the animals are from Maya's position.

Animal | Elevation (In Feet) |
---|---|

Kingfisher | $10$ feet |

River otter | $-8$ feet |

Porcupine | $7$ feet |

Sturgeon | $-12$ feet |

Painted turtle | $-3$ feet |

The absolute value of a negative number is its additive inverse. In contrast, the absolute value of a positive number is equal to itself. Now, move $10$ units up from $0$ in a vertical number line to display the distance of the Kingfisher.

Next, the river otter's elevation is $-8$ feet. Its absolute value is $∣-8∣=8.$ That means the river otter's distance is $8$ units from $0$ in the positive direction.

Use a similar reasoning to identify the remaining distances.

The graph shows that the distance of the sturgeon is the furthest. That is even with a negative elevation! On the other side of the spectrum, the painted turtle is the closest.

Closure

Many campers trust Maya's math skills and entrust their money to her. Vincenzo, has already given her $$20.$ He then goes to the gift shop and orders a custom bear sculpture. Before paying, he runs to Maya to retrieve his $$20.$ Maya sees the ticket and notices that the bear sculpture costs $$30.$ She shows his total balance to Vincenzo.

What does this amount represent? The negative amount in Vincenzo's balance means he has a debt or money owed. The amount he is in debt is given by the absolute value of the total.$Total Vincenzo’s Debt:∣-10∣=$10 $

What if Vincenzo's balance has a positive amount? That means he would still have cash and credit to buy more items. What if the balance is at $0?$ That means Vincenzo has no debt and no money. Balance | Meaning |
---|---|

Negative | Debt or money owed |

$0$ | No debt and no credit |

Positive | Cash and credit |

Vincenzo needs to pay that $$10$ because the bear sculpture costs more than he actually has. Scouring for loose change, he finds $$10$ in a secret pocket in his backpack. That bear sculpture is all his!

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