6th Grade · 6.NS.C.6

The number line with negative numbers in 6th grade

A number line with negative numbers runs both ways from 0, with positive numbers to the right and negative numbers to the left. In 6th grade, your child places whole numbers, fractions and decimals on that line, including negative ones, and learns what opposites are. The same idea carries over to the coordinate grid, where points can now have negative coordinates.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.NS.C.6 Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
6.NS.C.6.a Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
6.NS.C.6.b Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
6.NS.C.6.c Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

What it means, in plain words

Before 6th grade, number lines start at 0. A number line with negative numbers keeps going to the left of 0, and the numbers there are $-1$ , $-2$ , $-3$ and so on. The farther left a number is, the smaller it is. A vertical number line works the same way, like a thermometer: numbers above 0 are positive and numbers below 0 are negative.

Opposites are two numbers on different sides of 0 at the same distance from it, such as 6 and $-6$ . The minus sign in front of a number can be read as "the opposite of". So $-(-6)$ means the opposite of $-6$ , which is 6. Zero is its own opposite, because it sits at 0.

Rational numbers on a number line include fractions and decimals, not only integers (whole numbers and their negatives). A rational number is any number that can be written as a fraction, such as $-2.5$ or $-\frac{3}{4}$ . On the coordinate grid, the sign of each coordinate tells the direction: $(4, -2)$ is 4 units to the right of $(0, 0)$ and 2 units down. Two points such as $(4, -2)$ and $(4, 2)$ differ only in one sign, so each one is the mirror image of the other across the x-axis.

Your child should be able to:

Place integers, fractions and decimals, including negative ones, on a horizontal or vertical number line
Name the opposite of a number and explain why $-(-6) = 6$
Plot points with negative coordinates on the coordinate grid
Tell which points are mirror images when their coordinates differ only in sign

Example

Where do $-\frac{3}{4}$ and its opposite go on a number line?

$-\frac{3}{4}$ is negative, so it goes to the left of 0. It is less than 1 away from 0, so it sits between $-1$ and 0.
Split the space from 0 to $-1$ into 4 equal parts and count 3 parts to the left of 0.
The opposite of $-\frac{3}{4}$ is $\frac{3}{4}$ . Count 3 parts of the same size to the right of 0.
So $-\frac{3}{4}$ is 3 parts to the left of 0 and $\frac{3}{4}$ is 3 parts to the right, the same distance on opposite sides.

Example

The point $(5, 3)$ is reflected across the x-axis. Where does it land? Where would it land after a reflection across the y-axis instead?

A reflection across the x-axis moves the point from above the axis to the same distance below it. The x-coordinate stays 5 and the y-coordinate changes sign.
So $(5, 3)$ lands at $(5, -3)$ .
A reflection across the y-axis moves the point from the right side to the left side, so the x-coordinate changes sign instead.
So $(5, 3)$ would land at $(-5, 3)$ .

Where children usually get stuck

Reading from left to right, your child labels the marks before 0 as $-1$ , $-2$ , $-3$ , so $-3$ ends up right next to 0.
Numbers have always been written in counting order from left to right, so the child counts the negative side in reading order too. On the negative side, the counting runs outward from 0, toward the left.
What helps. Start at 0 together and step left one mark at a time, saying each label as you go: negative 1, negative 2, negative 3.
Your child puts $-2.5$ between $-2$ and $-1$ .
The child finds $-2$ and then moves half a step to the right, the way 2.5 sits to the right of 2. For a negative number, the extra half takes you farther from 0, so $-2.5$ is to the left of $-2$ .
What helps. Ask whether $-2.5$ is more or less than 2 steps away from 0. More than 2 steps away means it is past $-2$ , between $-2$ and $-3$ .
On a vertical number line, your child marks $-4$ above 0.
The child learned that negative numbers go to the left, and a vertical line has no left. Nothing told the child which way is negative when the line stands up.
What helps. Use a thermometer. Ask where 4 degrees below zero is. Below zero on the thermometer means below 0 on the number line too.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
A student wrote the statement "0 is its own opposite." Is this statement true or false?
A
True
B
False
Show the answer and how to solve it
True. Zero is its own opposite because 0 reflects to 0 on the number line.
1. 1Read the claim: "0 is its own opposite."
2. 2The opposite of 0 is 0 — 0 sits at the center of the number line and reflects back to itself. So 0 is indeed its own opposite. The statement is true.
Common mistake. A common mistake is thinking 0 cannot be an opposite at all since it is neither positive nor negative. The opposite operation reflects any number across 0, and 0 reflects right back to 0.
The idea underneath. Opposite numbers are mirror positions around 0, and 0 is already at the center, so its opposite stays 0.
2On level
Look at the numbers in the table below.
Number
3
-3
0
0.5
$\displaystyle \frac{2}{5}$
Enter how many numbers are equal to their own opposites.
There arenumbers
Show the answer and how to solve it
Answer. 1
Only 0 equals its own opposite, so exactly 1 listed number fits the rule.
1. 1Look down the table and ask whether each number equals its own opposite.
2. 2Any nonzero number has a different opposite, so none of these four entries count.
3. 3The entry 0 equals its own opposite because the opposite of 0 is 0. That is exactly 1 entry.
Common mistake. A common mistake is counting 3 and -3 together because they are opposites of each other. Neither one equals its own opposite — 3's opposite is -3, not 3. Only a number whose opposite is itself (that is, only 0) counts.
The idea underneath. Being opposite means the same distance from 0 on opposite sides of 0, so the only number that can land on itself is 0.
3On level
Point A and point B are shown on the number line below. Marcus's claim: "Point B is the opposite of point A."
Select the explanation that best describes the claim.
A
The claim is not correct because point B is on the same side of 0 as point A.
B
The claim is not correct because point B is 0, and 0 is not the opposite of a nonzero point.
C
The claim is correct because point B is the same distance from 0 as point A and on the other side of 0.
D
The claim is not correct because point B is on the other side of 0 from point A, but it is not the same distance from 0.
Show the answer and how to solve it
The correct explanation is the one that says point B is on the same side of 0 as point A. Point A is at -5 and point B is at -2, both on the left side of 0, so point B is not the opposite of point A and Marcus's claim is not correct.
1. 1Look at the number line. Point A is at -5 and point B is at -2.
2. 2Opposite points must be on opposite sides of 0 and the same distance from 0. For point B to be the opposite of point A, it would need to be at 5, not at -2.
3. 3Point A is at -5 and point B is at -2, so both points are on the same side of 0, the left side. Marcus's claim is not correct. The correct explanation is the one that says point B is on the same side of 0 as point A.
Common mistake. A common mistake is misreading the diagram and thinking point B is on the other side of 0. Point A is at -5 and point B is at -2, both to the left of 0, so they are on the same side.
The idea underneath. Opposite points are mirror images across 0: they are the same distance from 0, but on opposite sides. When both points are on the same side of 0, neither can be the opposite of the other.
4Stretch
Sam's work is shown below.
$\displaystyle -\left(-\frac{7}{4}\right)$
$\displaystyle = -\left(\frac{7}{4}\right)$
$\displaystyle = -\frac{7}{4}$
Select the best description of Sam's work.
A
The work is correct.
B
Sam kept the fraction negative. The opposite of $-\frac{7}{4}$ is $\frac{7}{4}$ .
C
Sam treated the two negative signs as if they made 0.
D
Sam changed the size of the number, but only the sign should change.
Show the answer and how to solve it
Sam kept the fraction negative, but the opposite of $-\frac{7}{4}$ is $\frac{7}{4}$ .
1. 1Look at Sam's work. Sam writes $-\left(-\frac{7}{4}\right)$ , then $-\left(\frac{7}{4}\right)$ , and finally $-\frac{7}{4}$ .
2. 2The outside negative sign means to take the opposite of everything inside the parentheses. The opposite of $-\frac{7}{4}$ is needed.
3. 3Sam's first equality is incorrect. Removing only the inner negative changes the value: $-\left(\frac{7}{4}\right)$ is negative, but the starting expression is positive.
4. 4Taking the opposite changes only the sign, so the final result should be positive $\frac{7}{4}$ .
5. 5Sam stopped after removing the inner negative and left the outer negative in place, writing $-\frac{7}{4}$ instead of $\frac{7}{4}$ . The description saying Sam kept the fraction negative when the opposite of $-\frac{7}{4}$ is $\frac{7}{4}$ best fits the mistake.
Common mistake. A common mistake is removing the inner negative sign but leaving the outer negative in place. In $-\left(-\frac{7}{4}\right)$ , both negatives must be resolved, giving the positive result $\frac{7}{4}$ .
The idea underneath. Taking an opposite changes only the sign, so the opposite of a negative number is the matching positive number.

Number
3
-3
0
0.5
$\displaystyle \frac{2}{5}$