3rd Grade · 3.NF.A.2

Fractions on a number line in 3rd grade

Fractions on a number line show that a fraction is a number, with its own place between the whole numbers. In 3rd grade, your child cuts the space from 0 to 1 into equal parts and counts those parts to find a fraction such as $\frac{2}{3}$ .

What the standard says

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

3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.A.2.a Represent a fraction $\frac{1}{b}$ on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size $\frac{1}{b}$ and that the endpoint of the part based at 0 locates the number $\frac{1}{b}$ on the number line.
3.NF.A.2.b Represent a fraction $\frac{a}{b}$ on a number line diagram by marking off a lengths $\frac{1}{b}$ from 0. Recognize that the resulting interval has size $\frac{a}{b}$ and that its endpoint locates the number $\frac{a}{b}$ on the number line.

What it means, in plain words

A number line is a straight line with numbers marked in order, like a ruler. Until now, your child has marked only whole numbers on it: 0, 1, 2, 3. Number line fractions fill in the spaces between them. A fraction number line is the same line with the space from 0 to 1 cut into equal parts.

The space from 0 to 1 is the whole. To show thirds, your child cuts it into 3 equal spaces, so each space is $\frac{1}{3}$ long. Counting from 0, the mark at the end of the first space is the number $\frac{1}{3}$ .

To find $\frac{2}{3}$ , your child counts 2 of those spaces from 0, and the mark where the second space ends is $\frac{2}{3}$ . The bottom number of a fraction says how many equal spaces fill the space from 0 to 1. The top number says how many spaces to count from 0. Placing fractions on a number line this way works for halves, thirds, fourths, sixths and eighths.

Your child should be able to:

Cut the space from 0 to 1 into 2, 3, 4, 6 or 8 equal parts
Find and label a unit fraction such as $\frac{1}{6}$ on a number line
Count equal spaces from 0 to place a fraction such as $\frac{3}{4}$
Explain what the top and bottom numbers of a fraction mean on the line

Example

Where does $\frac{2}{3}$ go on a number line from 0 to 1?

The bottom number is 3, so cut the space from 0 to 1 into 3 equal spaces.
Each space is $\frac{1}{3}$ .
Count 2 spaces from 0. The mark at the end of the second space is $\frac{2}{3}$ .
So $\frac{2}{3}$ sits on the second mark after 0, one space before 1.

Example

The space from 0 to 1 is cut into 8 equal parts. A dot sits on the seventh mark after 0. What fraction does the dot show?

8 equal parts means each space is $\frac{1}{8}$ .
The dot is 7 spaces from 0.
7 spaces of $\frac{1}{8}$ make $\frac{7}{8}$ .
The dot shows $\frac{7}{8}$ , one space before 1.

Where children usually get stuck

On a line from 0 to 1 with 5 marks, your child says the spaces are fifths.
Your child counts the marks, not the spaces between them. Four equal spaces need 5 marks, because 0 and 1 are marks too.
What helps. Put a finger on 0 and hop along the line together. Count the hops, not the marks.
Your child labels the mark at 0 as $\frac{1}{4}$ , so every label after it is one space off.
When counting objects, the first thing you touch is 1. On a number line, 0 is where you start, and you have to cross a whole space before you reach $\frac{1}{4}$ .
What helps. Write 0 under the first mark together and say "no spaces yet". Then count each space as you cross it.
Asked to show $\frac{1}{3}$ on a line from 0 to 5, your child puts a dot at 3.
Your child reads the fraction as the whole numbers 1 and 3, not as one number that is less than 1.
What helps. Ask first whether $\frac{1}{3}$ is more or less than 1 whole. Then look for it between 0 and 1.

Try four practice questions

Real 3rd 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
Look at the fraction $\frac{1}{2}$ marked on the number line below.
Use the dropdowns to complete the sentence.
The 2 in $\frac{1}{2}$ tells
, and the 1 in $\frac{1}{2}$ tells
.
Show the answer and how to solve it
The denominator tells the number of equal parts from $0$ to $1$ , and the numerator tells how many of those parts from $0$ to the marked point. For $\frac{1}{2}$ , the denominator $2$ counts the $2$ equal parts on the line, and the numerator $1$ counts the $1$ part from $0$ to the point.
1. 1Look at the number line. From $0$ to $1$ there are $2$ equal parts. The denominator of the fraction $\frac{1}{2}$ is $2$ — the bottom number names the count of equal parts.
2. 2The marked point is at the end of $1$ of those $2$ parts, counting from $0$ . The numerator of $\frac{1}{2}$ is $1$ — the top number counts how many of those parts to reach the point.
3. 3So the denominator $2$ tells the number of equal parts from $0$ to $1$ , and the numerator $1$ tells how many of those parts from $0$ to the marked point. Pick the option that names the count of equal parts in the first dropdown, and the option that names how many parts from $0$ to the point in the second dropdown.
Common mistake. Do not choose "the total number of equal parts (2 in all)" for the numerator. The numerator $1$ counts only the parts from $0$ to the point. The denominator $2$ counts the equal parts from $0$ to $1$ ; it is not the value at the point.
The idea underneath. For the fraction $\frac{1}{2}$ on this number line: the denominator $2$ partitions the whole from $0$ to $1$ into $2$ equal parts, and the numerator $1$ counts how many of those parts you go from $0$ to reach the marked point.
2On level
Which statement correctly explains how the denominator and numerator of the labeled fraction describe the marked point on the number line?
A
The numerator (1) tells the number of equal parts from 0 to 1, and the denominator (4) tells how many of those parts from 0 to the marked point.
B
The denominator (4) tells the number of equal parts from 0 to 1, and the numerator (1) tells how many of those parts from 0 to the marked point.
C
The denominator (4) tells the number of equal parts from 0 to 1, and the numerator (1) is the value of the marked point.
D
The denominator (4) tells the number of tick marks on the line, and the numerator (1) tells how many parts from 0 to the marked point.
Show the answer and how to solve it
The denominator $4$ is the number of equal parts from $0$ to $1$ , and the numerator $1$ is the number of those parts from $0$ to the marked point.
1. 1Look at the number line. It is divided from $0$ to $1$ into $4$ equal parts. The marked point is labeled $\frac{1}{4}$ .
2. 2Count the equal parts from $0$ to $1$ . There are $4$ equal parts. The denominator of $\frac{1}{4}$ is $4$ , so the denominator tells the total number of equal parts.
3. 3Now count the equal parts from $0$ to the marked point. There is $1$ equal part. The numerator of $\frac{1}{4}$ is $1$ , so the numerator tells how many of those parts you count from $0$ to the marked point.
4. 4Match this to the answer choices. The correct statement says: the denominator gives the total equal parts, and the numerator counts how many of those parts from $0$ to the marked point.
Common mistake. A common mistake is swapping the roles of the numerator and denominator — thinking the top number tells the total parts and the bottom number counts to the point. The denominator (the bottom number) is the total equal parts; the numerator (the top number) counts how many of those parts.
The idea underneath. On a number line from $0$ to $1$ , a fraction like $\frac{1}{4}$ uses the denominator $4$ to split the line into equal parts and the numerator $1$ to count how many of those parts to step from $0$ .
3On level
Which statement correctly explains how the denominator and numerator of the labeled fraction describe the marked point on the number line?
A
The numerator (2) tells the number of equal parts from 0 to 1, and the denominator (4) tells how many of those parts from 0 to the marked point.
B
The denominator (4) tells the number of equal parts from 0 to 1, and the numerator (2) is the value of the marked point.
C
The denominator (4) tells the number of equal parts from 0 to 1, and the numerator (2) tells how many of those parts from 0 to the marked point.
D
The denominator (4) tells the number of tick marks on the line, and the numerator (2) tells how many parts from 0 to the marked point.
Show the answer and how to solve it
The denominator $4$ is the number of equal parts from $0$ to $1$ , and the numerator $2$ is the number of those parts from $0$ to the marked point.
1. 1Look at the number line. It is divided from $0$ to $1$ into $4$ equal parts. The marked point is labeled $\frac{2}{4}$ .
2. 2Count the equal parts from $0$ to $1$ . There are $4$ equal parts. The denominator of $\frac{2}{4}$ is $4$ , so the denominator tells the total number of equal parts.
3. 3Now count the equal parts from $0$ to the marked point. There are $2$ equal parts. The numerator of $\frac{2}{4}$ is $2$ , so the numerator tells how many of those parts you count from $0$ to the marked point.
4. 4Match this to the answer choices. The correct statement says: the denominator gives the total equal parts, and the numerator counts how many of those parts from $0$ to the marked point.
Common mistake. A common mistake is swapping the roles of the numerator and denominator — thinking the top number tells the total parts and the bottom number counts to the point. The denominator (the bottom number) is the total equal parts; the numerator (the top number) counts how many of those parts.
The idea underneath. On a number line from $0$ to $1$ , a fraction like $\frac{2}{4}$ uses the denominator $4$ to split the line into equal parts and the numerator $2$ to count how many of those parts to step from $0$ .
4Stretch
Which statement correctly explains how the denominator and numerator of the labeled fraction describe the marked point on the number line?
A
The numerator (5) tells the number of equal parts from 0 to 1, and the denominator (6) tells how many of those parts from 0 to the marked point.
B
The denominator (6) tells the number of equal parts from 0 to 1, and the numerator (5) is the value of the marked point.
C
The denominator (6) tells the number of equal parts from 0 to 1, and the numerator (5) tells how many of those parts from 0 to the marked point.
D
The denominator (6) tells the number of tick marks on the line, and the numerator (5) tells how many parts from 0 to the marked point.
Show the answer and how to solve it
The denominator $6$ is the number of equal parts from $0$ to $1$ , and the numerator $5$ is the number of those parts from $0$ to the marked point.
1. 1Look at the number line. It is divided from $0$ to $1$ into $6$ equal parts. The marked point is labeled $\frac{5}{6}$ .
2. 2Count the equal parts from $0$ to $1$ . There are $6$ equal parts. The denominator of $\frac{5}{6}$ is $6$ , so the denominator tells the total number of equal parts.
3. 3Now count the equal parts from $0$ to the marked point. There are $5$ equal parts. The numerator of $\frac{5}{6}$ is $5$ , so the numerator tells how many of those parts you count from $0$ to the marked point.
4. 4Match this to the answer choices. The correct statement says: the denominator gives the total equal parts, and the numerator counts how many of those parts from $0$ to the marked point.
Common mistake. A common mistake is swapping the roles of the numerator and denominator — thinking the top number tells the total parts and the bottom number counts to the point. The denominator (the bottom number) is the total equal parts; the numerator (the top number) counts how many of those parts.
The idea underneath. On a number line from $0$ to $1$ , a fraction like $\frac{5}{6}$ uses the denominator $6$ to split the line into equal parts and the numerator $5$ to count how many of those parts to step from $0$ .