Exercises: Representation on the number line

Look at the following number line and identify which integer corresponds to each of the letters marked with a dot:

Draw a number line in your notebook and represent the following integers on it: $+4$, $-3$, $0$, $+1$ and $-5$. Use a distance of $1$ cm between each number.

On a thermometer, temperatures above zero are considered positive and below zero, negative. Represent the following temperatures from a winter week on the vertical scale of the thermometer: $-2 ^\circ C$, $+5 ^\circ C$, and $0 ^\circ C$.

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

State which integer is the opposite of the one represented by the orange dot on the line. Also represent that opposite on the same line.

A diver is at $-10$ meters depth (below sea level). If they rise $4$ meters, what position are they in now? Represent it on a vertical number line where $0$ is sea level.

Look at the number line and write the missing integers in the empty boxes, knowing that the distance between each mark is $1$ unit.

Remember that on the number line, the further to the right a number is, the greater it is. Based on this, use the symbols $>$ (greater than) or $<$ (less than) to compare these pairs of numbers:

1. a) $-4$ _____ $+2$
2. b) $-1$ _____ $-5$
3. c) $+3$ _____ $0$
4. d) $-2$ _____ $+1$

Write the integers whose absolute value is equal to $3$. Then, represent them on a number line like the one in the figure and explain what you notice about their position relative to zero.

In an exam, Marta had to calculate the distance on the number line between the number $-6$ and the number $+3$. On her answer sheet, she wrote:

"Since distance is the difference between the numbers, I do $+3 - 6 = -3$. Therefore, the distance between both is $-3$ units".

1. a) Identify the two mistakes Marta made (one conceptual about the result of a distance and another about how to set up the subtraction with negative numbers).
2. b) Explain in your own words why a distance can never be negative on the number line.
3. c) Write the correct operation and calculate the actual distance between $-6$ and $+3$.

Two integers, $A$ and $B$, are marked on a number line. We know that the exact midpoint between both is the number $-2$. In addition, you are given that point $A$ is the number $+4$.

1. a) Calculate the distance from point $A$ to the midpoint $-2$.
2. b) Knowing that the midpoint is equidistant from both endpoints, deduce with reasoning at which integer point $B$ must be located.
3. c) Apply the midpoint formula, $\dfrac{A+B}{2}$, substituting the values of $A$ and $B$ to check that your result gives exactly $-2$.

Look at the following number line on which three points, $A$, $B$, and $C$, have been marked. Taking into account the positions of the numbers $0$ and $+2$, identify which integer each letter represents.

Draw a number line and represent on it all the integers $x$ that simultaneously satisfy these two conditions:

1. Their absolute value is less than $5$, that is: $|x| < 5$.

2. They are numbers strictly less than $-1$.

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

A thermal inspection drone makes the following movements along the vertical axis of an industrial chimney. If we consider the top of the chimney to be at $0$ metres:

• It starts at a height of $+12$ m.

• It descends $15$ metres to inspect a crack.

• It climbs $6$ metres again to take a reading.

Represent the path on a vertical line and mark the final position of the drone with a highlighted dot. What height is it at relative to the top of the chimney?

On the number line, the midpoint between two integers is the position that is exactly the same distance from both.

• Represent the numbers $-6$ and $+2$.

• Visually find the midpoint between them and state what number it represents.

• What is the distance (in units) from that midpoint to either of the two endpoints?

Represent the following integers on a number line. Choose a suitable scale (where each mark on the line can represent more than 1 unit) so that they all fit in your drawing:

$S = \{ -15, +5, -20, +25, 0, -5 \}$

Identify which integers are represented by points $M$ and $N$ on this number line, knowing that the distance between consecutive marks is always the same.

Three scientists are studying an underwater volcano. Their positions relative to sea level ($0$ m) are:

• Researcher A: Is in a submarine at $-450$ metres.

• Researcher B: Is at the summit of the volcano, $+150$ metres above sea level.

• Researcher C: Is on a boat at $0$ metres.

Represent these three positions on a vertical line. If Researcher A wants to ascend to the position of Researcher B, how many metres must they climb in total? Explain how you see this on the line.

Draw a number line and plot the number $-3$ on it. Next, represent:

• Point $A$, which is the opposite of $-3$.

• Point $B$, which is $5$ units to the left of $-3$.

• Point $C$, which is $2$ units to the right of $A$.

Which number does point $C$ represent? Which is the smallest of the four numbers marked on the line?

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

The State Meteorological Agency (AEMET) records at a mountain station in Teruel a minimum temperature of $-7\text{ }^\circ\text{C}$ at 7:00 h and a maximum of $+3\text{ }^\circ\text{C}$ at 15:00 h.

1. a) Before doing any calculations, mentally place both numbers on the number line and estimate whether the temperature of the midpoint between the minimum and the maximum will be positive, negative, or exactly zero. Justify your prediction by explaining which of the two temperatures is farther from $0$.
2. b) Calculate the total thermal distance (temperature range) between the minimum and the maximum using absolute value.
3. c) Calculate the exact thermal midpoint by applying the formula $\dfrac{a+b}{2}$ and check whether your estimate from section a) was correct.

In an intermodal transport station in Madrid, we take street level as elevation $0\text{ m}$. The underground Cercanías platform is located at an elevation of $-12\text{ m}$, while the upper access walkway to the buses is located at an elevation of $+8\text{ m}$.

1. a) Model this situation on a number line and calculate how many metres of distance a panoramic lift travels going directly from the Cercanías platform to the bus walkway.
2. b) The station management wants to place an information board at the exact midpoint of the lift's vertical travel. At what elevation (in metres) should it be installed? Will it be above or below street level?
3. c) A maintenance technician is working at a point that is exactly $5\text{ m}$ away from the Cercanías platform (elevation $-12\text{ m}$). At which two different elevations could this technician be located? Justify both options.

An explorer is looking for a treasure buried on a straight path that we represent as a number line. The map gives him the following clues:

1. The treasure is at a position represented by an integer $x$.

2. The distance between $x$ and the number $-2$ is exactly $5$ units.

3. The absolute value of $x$ is greater than $4$.

Represent on the number line the possible locations of the treasure and determine at which exact number or numbers it could be. You can copy this number line to work on it:

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

A surveillance drone moves along a horizontal axis. Its charging base is at point $0$. It makes the following consecutive movements:

• It starts at the base and moves to point $+7$.

• From there, it moves back $12$ units.

• Finally, it moves to a position $P$ such that the total distance travelled from the start is $25$ units.

If the drone always moves in a straight line, at which integer on the line is point $P$ located? Is there more than one possible solution? Represent it graphically.

We are designing a special thermometer for a physics experiment. We must mark four critical points: $A$, $B$, $C$ and $D$, which meet these conditions:

• $A$ is the opposite of $+3$.

• $B$ has an absolute value of $5$ and is to the left of $A$.

• $C$ is an integer such that $-1 < C < 4$ and its distance to $A$ is the same as its distance to $D$.

• $D$ is the largest possible integer that is less than $+6$.

Represent the four points on the same number line (you can use the line in the figure as a model) and order them from least to greatest absolute value. If two points have the same absolute value, point it out.

Two towns, Villafría and Villacalor, are connected by a straight road that we imagine as a number line. Villafría is at point $-15$ and Villacalor is at point $+21$.

A mobile phone mast is to be installed at a point $M$ that is exactly equidistant from both towns.

1. What integer represents the position of the mast $M$?

2. If a car leaves Villafría heading towards the mast and breaks down when it has covered $\frac{1}{3}$ of the total distance between the town and the mast, at which number on the line has it stopped?

In a video game, a character moves along a line of squares numbered with integers. The player must place "traps" on all integers $n$ that simultaneously satisfy these two conditions:

• The absolute value of the number is less than or equal to $6$: $|n| \le 6$.

• The distance between the number $n$ and the number $-3$ is greater than $4$ units.

Draw a number line and mark with an orange "X" all the positions where the player must place a trap. How many traps are there in total?

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

A historian is creating a digital timeline. They define year $0$ as the time of a great discovery.

• The Alpha civilisation emerged in year $-450$ and lasted for $300$ years.

• The Beta civilisation began just as Alpha ended and lasted twice as long.

• The Gamma civilisation is placed on the line such that its starting year is the midpoint between the start of Alpha and the end of Beta.

Represent the start and end points of each civilisation on a number line (using a suitable scale, for example, 1 unit = 100 years). In which year did the Gamma civilisation begin?

Consider three integers $a, b$ and $c$ located on the number line about which we know the following:

• $a$ is the opposite of the absolute value of $-8$.

• $c$ is a number such that its distance to $0$ is half the distance from $a$ to $0$, and $c$ is positive.

• $b$ is a number that is the same distance from $a$ as from $c$.

Draw the line, plot the three points, and calculate the value of the following expression using their positions:

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

Imagine that the number line is a taut rope. We tie a weight at the number $0$ and another at the number $-12$.

1. Find the position of a point $P$ (an integer) that is at a distance of $4$ units from point $-6$. How many solutions are there?

2. If we now search for a point $Q$ such that its distance to $+2$ is double its distance to $-1$, which integers could $Q$ be? Identify them and mark them on the number line.

We want to find all integers that are at a distance of exactly $6$ units from the number $-2$ on the number line.

Two 1st ESO students propose two different paths:

• Alba's method: Draw a line graduated by $1$s, locate the position of $-2$, and visually count $6$ divisions to the right and then $6$ divisions to the left to see which marks it lands on.

• Sergio's method: Translate the problem into direct arithmetic operations: since moving to the right is adding and to the left is subtracting, calculate $(-2) + 6$ and $(-2) - 6$.

1. a) Apply both methods step by step and verify that both yield the same numbers.
2. b) Now imagine the statement changes to: "Find the integers located at a distance of $235$ units from the number $-148$". Compare both methods and justify which of the two you would choose for being more efficient and avoiding counting errors.
3. c) Solve the case of $235$ units of distance from $-148$ using the method you selected in section b).

We are looking for two unknown integers, $x$ and $y$ (where $x < y$), represented on the number line. To find them, you only have four clues:

1. The distance between $x$ and $y$ is $14$ units.

2. The product of both numbers is strictly negative (that is, $x \cdot y < 0$, which implies that they are on opposite sides of $0$).

3. The midpoint between $x$ and $y$ is a negative integer.

4. The absolute value of one of the two numbers is exactly $4$.

1. a) Deduce with reasoning which of the two numbers, the positive or the negative one, must be the one with an absolute value of $4$. (Hint: think about which number must be farther from $0$ on the line so that the midpoint turns out negative).
2. b) Based on your deduction, determine the exact values of $x$ and $y$.
3. c) Calculate the midpoint between the values you found and numerically verify that all four conditions of the statement are satisfied.