The set of integers

In a survival video game session, your character's actions affect their experience points. Express the following situations using an integer:

1. a) You find a chest and gain $450$ points.
2. b) You are defeated by an enemy and lose $125$ points.
3. c) You fall into a trap and your score drops by $50$ points.
4. d) You complete a side quest and receive $80$ points.

State whether the following statements are true or false:

1. a) The number $-7$ is a natural number ($\mathbb{N}$).
2. b) All natural numbers are also integers ($\mathbb{Z}$).
3. c) The absolute value of a number is always positive or zero.
4. d) The opposite of a negative number is always negative.

Associate an integer with each of the following situations taken from a teenager's daily life:

1. a) The balance on your travel card after spending $5$ € on trips and having a previous top-up of $2$ €.
2. b) The position of an exploration submarine in a video game that is $150$ metres below sea level.
3. c) The temperature at a ski resort reading $4$ degrees below zero.
4. d) The year in which the abacus is estimated to have been invented, approximately $500$ years BC.
5. e) Going down in a shopping centre lift from the cinema floor (floor $3$) to the car park located two floors below.

A group of friends is measuring the temperature in different European cities on a winter day:

• Berlin: $-4 \text{ °C}$

• Dublin: $+9 \text{ °C}$

• Oslo: $-12 \text{ °C}$

• Rome: $+15 \text{ °C}$

• Zurich: $-1 \text{ °C}$

Order the cities from coldest to warmest.

In a high-mountain town, the weather service records the following temperatures throughout a winter's day:

• At 06:00: $-8$ ºC.

• At 12:00: The temperature has risen by $10$ ºC compared to 06:00.

• At 20:00: The temperature has fallen by $5$ ºC compared to 12:00.

• At 00:00: The temperature is the opposite of the one recorded at 12:00.

Represent these four temperatures on a number line and order them from lowest to highest using the $<$ symbol.

Given the integers $a = -12$ and $b = +5$:

1. a) Calculate the distance between them on the number line.
2. b) Find the number $c$ that is at exactly the same distance from $a$ as from $b$. Is $c$ an integer? Justify your answer.
3. c) If $d$ is a number such that $|d| = |a| + |b|$, what values can $d$ take?

Calculate the absolute value and the opposite of the following integers:

1. a) $-14$
2. b) $+21$
3. c) $-5$
4. d) $+9$

Consider two integers $a$ and $b$. It is known that $a$ is a negative number and $b$ is a positive number. State whether the following statements are always true, sometimes true, or always false, justifying your answer with an example:

1. a) $|a|$ is always greater than $b$.
2. b) The opposite of $b$ is always less than $a$.
3. c) The absolute value of $a$ is always a positive number.
4. d) $b$ is always greater than $a$.

Reason whether the following statements are true or false, providing a counterexample if they are false:

1. a) If an integer $a$ is less than $b$, then the modulus of $a$ is always less than the modulus of $b$ ($|a| < |b|$).
2. b) The opposite of the modulus of any integer (other than zero) is always a negative number.
3. c) There is no integer whose modulus is equal to its opposite.

Write down all the integers $x$ that satisfy both of the following conditions simultaneously:

1. Their absolute value is less than or equal to $4$: $|x| \le 4$.

2. They are greater than the opposite of $2$: $x > Op(+2)$.

We want to define a set of unknown integers using several simultaneous conditions, such that the only numbers in all of $\mathbb{Z}$ that satisfy all of them at once are exactly $\{-6, -4, -2\}$.

Design three conditions that an integer $x$ must satisfy to obtain this result:

• The first condition must define a numerical range using the absolute value $|x|$.

• The second condition must refer to belonging or not belonging to the sets $\mathbb{N}$ and $\mathbb{Z}$.

• The third condition must require a property regarding the divisibility or parity of the number.

Once you have written your three conditions:

1. a) Check step by step that the numbers $-6$, $-4$, and $-2$ satisfy all three simultaneously.
2. b) Write an integer that satisfies the first two conditions but not the third, showing that your third condition is essential so that other solutions do not slip in.

Determine all the integers $x$ that simultaneously satisfy the following three conditions:

• Its absolute value is greater than $5$ and less than or equal to $9$.

• The number $x$ does not belong to the set of natural numbers $\mathbb{N}$.

• $x$ is an even number.

Graphically represent on the number line the set of solutions you obtained. You can use this unmarked number line as a guide:

You can do it on paper and send a photo with your answer.

Lucas has the point $A(-4, 6)$ located in a Cartesian coordinate system and wants to find its reflection across the $x$-axis. In his notebook, he writes the following explanation:

"Since the $x$-axis is the horizontal line, to reflect across it I have to change the sign of the first coordinate, which is the horizontal one. Therefore, the reflected point is $A'(4, 6)$".

1. a) Explain with reasons what conceptual mistake Lucas is making, relying on the distance from the point to the horizontal axis of symmetry.
2. b) Write the correct coordinates of the reflection of $A$ across the $x$-axis and also those of its reflection across the $y$-axis.
3. c) If a point $B(0, -5)$ lies on the $y$-axis, what is its reflection across that same axis? Justify your answer using the concept of opposite number.

On a coordinate grid, three of the vertices of a rectangle are the points $A(-4, 3)$, $B(2, 3)$ and $C(2, -1)$.

1. a) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
2. b) Calculate the perimetre of the rectangle knowing that each grid unit is equivalent to $1$ metre.
3. c) What is the symmetric point of vertex $A$ with respect to the x-axis (horizontal axis)? State its coordinates using integers.

In a 2nd Year class, a question arises about how to find the coordinates of the resulting point after applying two consecutive reflections to a generic point $P(x, y)$ with non-zero integer coordinates: first across the $x$-axis and then, to the resulting point, across the $y$-axis. Two students propose different approaches:

• Seán's method: He chooses a specific point, for example $P(-3, 5)$. He applies the reflection across the $x$-axis, obtaining $P_1(-3, -5)$, and then the reflection across the $y$-axis to $P_1$, obtaining $P_2(3, -5)$. Since he observes that both coordinates have changed sign compared to $P$, he concludes that this rule works for any point in the plane.

• Aoife's method: She applies algebraic rules with letters: across the $x$-axis, the point $(x, y)$ transforms into $(x, op(y)) = (x, -y)$; then, across the $y$-axis, a point $(a, b)$ transforms into $(op(a), b) = (-a, b)$, so applying it to $(x, -y)$ results in $(-x, -y)$.

1. a) Evaluate both methods: which of the two demonstrates with mathematical rigor that the property is true for all integers in the plane, and why does the other method only provide a particular conjecture?
2. b) Apply the general conclusion to directly find the final coordinates of $Q(15, -42)$ after these two successive reflections, without calculating the intermediate point.
3. c) If a third reflection were then applied, once again across the $x$-axis, what coordinates would the final point have in terms of the initial $(x, y)$? What single reflection are these three reflections together equivalent to?