Concept

A lift in a very large shopping centre has floors above and below ground level. If we consider the ground floor (street level) as $0$:

• a) What floor do we reach if we are in basement $-3$ and go up $5$ floors?

• b) What floor do we reach if we are on floor $+2$ and go down $4$ floors?

• c) If we are in basement $-2$ and want to go up to floor $+4$, how many floors must we go up?

In everyday life, we use integers to express opposite situations relative to a reference point or "zero point". Match each of the following statements with the integer (positive or negative) that best represents it:

• a) A diver is at a depth of $15$ metres below sea level.

• b) A hiker has climbed to the top of a mountain $2300$ metres high.

• c) The balance on your Leap Card is $4$ euros in debt.

• d) An aeroplane is flying at an altitude of $9500$ metres.

• e) The temperature at a ski resort is $7$ degrees below zero.

Observe the following number line where several points are marked with letters. Indicate which integer corresponds to each of the letters represented:

Considering that the distance between two consecutive small marks is $1$ unit, determine the values of $A$, $B$, $C$ and $D$.

A team of underwater archaeologists is exploring the remains of an ancient galleon. They use a small research submarine to document the find. Express the following situations using integers:

1. a) The mothership is at sea level.
2. b) The galleon is resting on the seabed, at a depth of $45$ metres.
3. c) The submarine descends from the surface to a crevice located $12$ metres below the galleon.
4. d) After exploring the crevice, the submarine rises $25$ metres to photograph the ship's mast.

Represent these four positions on a scaled vertical line where the origin is sea level.

The absolute value (or modulus) of an integer is the distance separating it from zero on the number line, regardless of direction. Calculate the following absolute values:

• a) $|-8| =$

• b) $|+12| =$

• c) $|-35| =$

• d) $|0| =$

• e) $|+7| =$

Two integers are opposites if they are at the same distance from zero but on opposite sides of the number line. Write the opposite of each of the following numbers and represent them in pairs on a small number line:

• a) $+5$

• b) $-3$

• c) $-1$

• d) $+4$

State whether the following statements are true or false. If a statement is false, give an example (counterexample) to prove it:

1. a) The modulus of a number is always greater than the number itself.
2. b) If two numbers are negative, the one with the smaller modulus is greater.
3. c) The opposite of an integer is always a negative number.
4. d) The distance between a number and its opposite is always double its modulus.

During a scientific expedition, a weather balloon probe flies over a volcano at an altitude of $+125$ meters above sea level, while a robotic probe is exploring the bottom of a nearby lake at a depth of $-84$ meters.

1. a) Represent both positions on a vertical number line, placing sea level at the origin.

1. b) Calculate the total distance, in absolute value, between the balloon and the probe.

1. c) If the probe rises $30$ meters, what will its new position be? Express it as an integer.

In a town in the midlands of Ireland, the following minimum temperatures were recorded during the first week of January:

• Monday: $-4$ ºC

• Tuesday: $-7$ ºC

• Wednesday: $0$ ºC

• Thursday: $+2$ ºC

• Friday: $-1$ ºC

• Saturday: $+3$ ºC

• Sunday: $-5$ ºC

Order these days from coldest to warmest (that is, from lowest to highest temperature) using the symbol $<$ between the values.

Compare the following pairs of integers using the symbols $>$ (greater than) or $<$ (less than):

• a) $-10 \quad \square \quad -2$

• b) $+5 \quad \square \quad -8$

• c) $-1 \quad \square \quad 0$

• d) $-15 \quad \square \quad -20$

• e) $|-12| \quad \square \quad +10$

Order the following values from smallest to largest using the symbols $<$ or $>$. We recommend calculating the numerical value of each expression first:

$A = |−12|$

$B = op(+7)$

$C = −|−5|$

$D = op(−2)$

$E = −10$

$F = |op(−4)|$

Write all integers $x$ that satisfy the following conditions at the same time:

1. Their modulus is less than $6$ ($|x| < 6$).

2. They are numbers less than $2$ ($x < 2$).

3. They are numbers greater than or equal to $-3$ ($x \geq -3$).

Represent them on a number line.

Consider two integers $a$ and $b$ for which we know the following:

• The modulus of $a$ is $12$.

• The opposite of $b$ is $-15$.

• The distance between $a$ and $b$ on the number line is as small as possible.

Based on these data:

1. a) Identify the values of $a$ and $b$.
2. b) What is the largest integer $k$ such that $a < k < b$?
3. c) If we multiply $a$ by a negative number, will the result be greater or less than $b$? Give reasons for your answer.

Observe the following number line where three points, $X$, $Y$ and $Z$, have been marked.

If each division on the number line represents $2$ units:

1. a) Write the coordinates of the points $X$, $Y$ and $Z$.
2. b) Calculate the distance between point $X$ and point $Z$.
3. c) Write the opposite of $X$ and represent it on the line.

On a set of Cartesian coordinate axes, we have drawn three vertices of a rectangle: $A(−3, 2)$, $B(4, 2)$ and $C(4, −1)$.

1. a) Plot the points on the plane.
2. b) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
3. c) Which quadrant is each of the vertices in?
4. d) Calculate the perimetre of the rectangle by counting the grid units.

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

In a strategy video game, the coordinates of your base are $A(-3, 4)$. You must place an outpost at a point $B$ that meets the following conditions:

1. Its abscissa ($x$-coordinate) is the opposite of the abscissa of $A$.

2. Its ordinate ($y$-coordinate) is a negative integer whose absolute value is $2$.

Use the grid to plot both points (each small square measures one unit):

1. a) Determine the coordinates of point $B$.

1. b) Plot $A$ and $B$ on the Cartesian plane and state which quadrant each one is in.

1. c) What are the horizontal and vertical displacements needed to go from $A$ to $B$?