Integers and Opposites on a Number Line | Grade 6 Math | LevelKite

You already know positive and negative numbers. In this lesson you will place integers on a number line. Then you will use the line to learn what the opposite of a number is.

A number line is a straight line on which each point stands for one number. On every number line in this lesson, short marks called tick marks are evenly spaced. Moving from one tick mark to the next is one step. Because the tick marks are evenly spaced, every step on the same line changes the number by the same amount. That amount is what one step is worth.

The integers are the numbers $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ . They are the counting numbers, zero, and the negative numbers $-1, -2, -3$ and so on. To graph a number means to mark the point on the number line that stands for that number.

Every number line in this lesson is horizontal, and its numbers get greater as you move to the right. On a line like that, positive numbers are to the right of $0$ and negative numbers are to the left of $0$ .

So the sign of a positive or negative number gives its direction from $0$ . (A number other than $0$ that is written with no sign, such as $4$ , is positive.) The number's distance from $0$ is how far away it is. You measure that distance by counting equal steps from $0$ .

To graph an integer that lands exactly on a tick mark, as every integer in this lesson does:

Find what one step is worth. Two printed numbers that are next to each other tell you.
Read the sign. Go right from $0$ for a positive integer and left from $0$ for a negative integer.
Count equal steps from $0$ , keeping track of the number at each tick mark, until you reach the integer.
Mark the tick mark where you stop.

The integer $0$ has no sign and needs no counting. Its point is the tick mark at $0$ .

Worked example

Graph two integers when every tick mark has a number

A number line runs from $-6$ to $6$ . A number is printed at every tick mark. Graph the integers $4$ and $-5$ .

Find what one step is worth. The printed numbers 0 and 1 are next to each other, with 1 step between them. They are 1 apart, so that one step is worth 1.
$1 - 0 = 1$
Graph 4. It is positive, so go right from 0. Count steps of 1, reading the number at each tick mark, until you reach 4. That takes 4 steps. Mark the tick mark where you stop. In the line of symbols, each arrow stands for one step.
$0 \to 1 \to 2 \to 3 \to 4$
Graph −5. It is negative, so go left from 0. Count steps of 1 again until you reach −5. That takes 5 steps. Mark the tick mark where you stop.
$0 \to -1 \to -2 \to -3 \to -4 \to -5$
Check both points against the printed numbers. One point is on the tick mark printed 4, to the right of 0. The other is on the tick mark printed −5, to the left of 0. The number line after this example shows both points.

Answer

The point for $4$ is 4 steps to the right of $0$ , and the point for $-5$ is 5 steps to the left of $0$ . Each step on this line is worth $1$ .

Each step is worth

1

. The marker at

4

is 4 steps to the right of

0

, and the marker at

-5

is 5 steps to the left of

0

.

Many number lines print a number at only some of the tick marks. The tick marks in between have no printed number, but you still count every step between them.

On a line like that, find what one step is worth before you count. Choose two printed numbers that are next to each other. Find how far apart they are. Then divide by the number of steps between them. This works because the tick marks are evenly spaced, so those steps share the amount equally.

When $0$ is printed, an easy pair to use is $0$ and the first printed number to its right.

Worked example

Graph integers when only some tick marks have numbers

A number line runs from $-20$ to $20$ . Numbers are printed only at $-20$ , $-10$ , $0$ , $10$ and $20$ , and there are 5 equal steps between each printed number and the next one. Graph the integers $14$ and $-14$ .

Find what one step is worth. The printed numbers 0 and 10 are next to each other, with 5 equal steps between them. They are 10 apart, and the 5 steps share that 10 equally. So divide 10 by 5.
$(10 - 0) \div 5 = 10 \div 5 = 2$
Graph 14. It is positive, so go right from 0. Each step is worth 2, so the tick marks to the right of 0 stand for 2, 4, 6 and so on. Count until you reach 14. That takes 7 steps. Mark the tick mark where you stop.
$0 \to 2 \to 4 \to 6 \to 8 \to 10 \to 12 \to 14$
Graph −14. It is negative, so go left from 0. Each step is still worth 2, so the tick marks to the left of 0 stand for −2, −4, −6 and so on. Count until you reach −14. That also takes 7 steps. Mark the tick mark where you stop.
$0 \to -2 \to -4 \to -6 \to -8 \to -10 \to -12 \to -14$
Check the count. 7 steps worth 2 each cover 14, which is how far both numbers are from 0. On the line, 14 is 2 tick marks to the right of the printed 10, and −14 is 2 tick marks to the left of the printed −10. The number line after this example shows both points.
$7 \times 2 = 14$

Answer

The point for $14$ is on the 7th tick mark to the right of $0$ , and the point for $-14$ is on the 7th tick mark to the left of $0$ . Each step on this line is worth $2$ .

Neighboring printed numbers are 10 apart with 5 steps between them, so each step is worth

10 \div 5 = 2

. The markers at

14

and

-14

are each 7 steps from

0

, on opposite sides of

0

.

Key concept

Opposites: same steps from 0, opposite sides

Two different numbers are opposites when they are on opposite sides of $0$ and the same number of equal steps from $0$ , counted on the same number line.

For example, when each step is worth $2$ , the numbers $14$ and $-14$ are each 7 steps from $0$ . One is to the right of $0$ and the other is to the left, so $14$ and $-14$ are opposites.

To find the opposite of any number other than $0$ , keep its number of steps from $0$ and switch sides. In symbols, only the sign changes: a positive number becomes negative, and a negative number becomes positive.

The number $0$ is 0 steps from $0$ and is on neither side, so $0$ is its own opposite.

Opposites are not only for integers. A rational number is a number that can be written as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers and $b$ is not $0$ . Every integer is a rational number, because you can write it over $1$ : for example, $14 = \frac{14}{1}$ and $-5 = \frac{-5}{1}$ . Fractions such as $\frac{3}{4}$ are rational numbers. So are decimals such as $-1.5$ , which equals $\frac{-3}{2}$ .

The meaning of opposite is the same for every rational number. Count the equal steps from $0$ to the number. Then go the same number of steps from $0$ on the other side. You need a number line whose tick marks land exactly on the number. For $-1.5$ , a line with a tick mark every $0.5$ works.

Worked example

Find the opposite of a negative decimal

Find the opposite of $-1.5$ . Use a number line that runs from $-2$ to $2$ , with a tick mark every $0.5$ and a number printed at every tick mark.

Find what one step is worth. The printed numbers 0 and 0.5 are next to each other, with 1 step between them. They are 0.5 apart, so that one step is worth 0.5.
$0.5 - 0 = 0.5$
Count the steps from 0 to −1.5. The number is negative, so go left from 0. It takes 3 steps of 0.5 to reach −1.5.
$0 \to -0.5 \to -1 \to -1.5$
Keep the 3 steps and switch sides, because an opposite is the same number of equal steps from 0 on the other side of 0. Count 3 steps of 0.5 to the right of 0.
$0 \to 0.5 \to 1 \to 1.5$
Check the result. 3 steps worth 0.5 each cover 1.5, so both numbers are the same distance from 0. They are on opposite sides of 0, and only the sign is different. The number line after this example shows both numbers.
$3 \times 0.5 = 1.5$

Answer

The opposite of $-1.5$ is $1.5$ .

The marker at

-1.5

is 3 steps of

0.5

to the left of

0

. The marker at its opposite,

1.5

, is 3 steps of

0.5

to the right of

0

.

More opposites. Apart from 0, each number and its opposite share a step count and sit on opposite sides of 0.

Number	Where it is	Opposite	Where the opposite is
$9$	9 steps of $1$ to the right of $0$	$-9$	9 steps of $1$ to the left of $0$
$-12$	12 steps of $1$ to the left of $0$	$12$	12 steps of $1$ to the right of $0$
$\frac{3}{4}$	3 steps of $\frac{1}{4}$ to the right of $0$	$-\frac{3}{4}$	3 steps of $\frac{1}{4}$ to the left of $0$
$-0.6$	6 steps of $0.1$ to the left of $0$	$0.6$	6 steps of $0.1$ to the right of $0$
$0$	at $0$ , which is 0 steps away	$0$	at $0$ , which is 0 steps away

Takeaway. On a number line with evenly spaced tick marks, the sign of a positive or negative number tells you which side of $0$ it is on, and counting equal steps from $0$ tells you how far away it is. To graph an integer that lands on a tick mark, find what one step is worth from two printed numbers that are next to each other, then count steps from $0$ to that tick mark. Two different numbers are opposites when they are the same number of equal steps from $0$ on opposite sides of $0$ , so they differ only in sign. The number $0$ is its own opposite.

Next, you will graph fractions and decimals on a number line and find the opposite of the opposite of a number.