Concept

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 public transport 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.

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.

In a city in northern Spain, 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:

$A = |−12|$

$B = op(+7)$

$C = −|−5|$

$D = op(−2)$

$E = −10$

$F = |op(−4)|$

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.