Fractions on a Number Line: Plotting and Reading Them

Q: How do I put a fraction on a number line?

Split the space from 0 to 1 into as many equal parts as the bottom number. Then count that many parts from zero. For three fifths, make five parts and count three.

Q: What if the fraction is bigger than one?

Keep counting into the next whole. Five quarters means four parts in every whole, counted five times, so it lands one quarter past 1.

Q: Why use a number line instead of a pizza?

A number line shows that a fraction is a number with its own place, not only a piece of something. It also makes ordering visible, since further right always means bigger.

Q: How do I read a fraction off a number line?

Count the equal parts between two whole numbers to get the bottom number, then count the parts from the last whole number to the dot to get the top number.

Quick answer

To put a fraction on a number line, split the space from one whole number to the next into as many equal parts as the bottom number says, then count that many parts across from zero. Three quarters means four equal parts between 0 and 1, counting three of them. Fractions past 1 keep going into the next whole. Whichever fraction sits further right is the bigger one.

What you'll learn

Split a number line into equal parts to match a bottom number
Plot a fraction on a number line and read one off
Use position on the line to order fractions

A fraction is a number, not only a piece

Pizzas and bars are useful, but they hide something. They make a fraction feel like a piece of a thing.

A number line puts it right: a fraction is a number, and it has its own spot, the same way $3$ has its own spot.

Three quarters on a number line

How to plot one

Two steps, in this order.

The bottom number tells you how to cut. Split the space from one whole number to the next into that many equal parts.
The top number tells you how far to count. Start at $0$ and count that many parts.

For $\tfrac{3}{4}$ : cut the space from $0$ to $1$ into four equal parts, then count three of them. The dot lands on the third tick.

Why the cuts must be equal

The whole idea rests on the parts being the same size.

If you cut a bar into four pieces where one is huge, “one quarter” no longer means anything. The number line is strict about this: the ticks are evenly spaced, so every part is the same width.

That is also why the space from $0$ to $1$ matters and not the space from $0$ to the edge of your paper. The whole is one unit, and the parts split that unit.

Fractions past one

Nothing changes. Every whole gets the same number of cuts, and you keep counting.

Five quarters sits past one

For $\tfrac{5}{4}$ : quarters again, so cut each whole into four. Count five quarters from $0$ . Four of them get you to $1$ , and the fifth carries you one tick past it.

A fraction bigger than $1$ is called an improper fraction. There is nothing wrong with it. It sits on the line like any other number.

Reading one off

Work backwards.

Count the equal parts between two whole numbers. That is your bottom number.
Count the parts from the last whole number to the dot. Add that to the whole number.

Reading two fifths and seven fifths

Between $0$ and $1$ there are five gaps, so these are fifths. The first dot is two gaps along, so it is $\tfrac{2}{5}$ . The second dot is two gaps past $1$ , and $1$ is $\tfrac{5}{5}$ , so it is $\tfrac{7}{5}$ .

Order comes free

On a number line, further right always means bigger. So the line settles any comparison at a glance.

One third, one half and two thirds in order

To place all three on one line, cut into sixths, because thirds and halves both fit neatly into sixths. That is the same idea as a common denominator, drawn instead of written.

Worked examples

Common mistakes

Practice problems

A number line from $0$ to $1$ is cut into four equal parts. Which fraction is the first tick after $0$ ?

Answer

$\tfrac{1}{4}$

Full solution

Four equal parts means quarters. One gap from $0$ is one quarter.
Where does $\tfrac{3}{5}$ sit between $0$ and $1$ ?

Hint

Cut into five, count three.

Answer

On the third tick of five equal parts.

Full solution

The bottom number $5$ says cut the space into five equal parts. The top number $3$ says count three of them from $0$ .
A line from $0$ to $1$ has ten equal gaps and a dot on the seventh tick. What fraction is it?

Answer

$\tfrac{7}{10}$

Full solution

Ten gaps means tenths, and the dot is seven gaps along, so the fraction is $\tfrac{7}{10}$ .
Between which two whole numbers does $\tfrac{9}{4}$ sit?

Hint

How many quarters make one whole?

Answer

Between $2$ and $3$ .

Full solution

Four quarters make $1$ and eight quarters make $2$ . The ninth quarter takes you one tick past $2$ , so $\tfrac{9}{4}$ sits between $2$ and $3$ .
Which is further right on a number line, $\tfrac{2}{3}$ or $\tfrac{5}{6}$ ?

Answer

$\tfrac{5}{6}$

Full solution

Cut into sixths. Then $\tfrac{2}{3}$ is $\tfrac{4}{6}$ , which sits on the fourth tick, and $\tfrac{5}{6}$ sits on the fifth. The fifth tick is further right.
A dot sits exactly halfway between $1$ and $2$ on a line cut into quarters. Name the fraction two ways.

Answer

$\tfrac{6}{4}$ , which is also $\tfrac{3}{2}$ .

Full solution

Four quarters reach $1$ , and halfway to $2$ is two more quarters, giving $\tfrac{6}{4}$ .

Simplifying by dividing both parts by $2$ gives $\tfrac{3}{2}$ .
Where does $\tfrac{6}{6}$ sit?

Answer

Exactly on $1$ .

Full solution

Six sixths is all six parts of one whole, so the dot lands on $1$ .
Put $\tfrac{1}{4}$ , $\tfrac{2}{3}$ and $\tfrac{1}{2}$ in order using a number line cut into twelfths.

Hint

Rewrite all three in twelfths first.

Answer

$\tfrac{1}{4} < \tfrac{1}{2} < \tfrac{2}{3}$

Full solution

In twelfths: $\tfrac{1}{4} = \tfrac{3}{12}$ , $\tfrac{1}{2} = \tfrac{6}{12}$ and $\tfrac{2}{3} = \tfrac{8}{12}$ .

The dots sit on the third, sixth and eighth ticks, so that is the order from left to right.
Leo cuts a line from $0$ to $2$ into three equal parts and calls the first tick $\tfrac{1}{3}$ . What went wrong?

Hint

What is the whole here?

Answer

He cut the whole stretch, not each unit. The first tick is $\tfrac{2}{3}$ .

Full solution

The whole is the space from $0$ to $1$ , so thirds means cutting each unit into three, giving six parts across the line.

Leo cut a length of $2$ into three, so each of his parts is $\tfrac{2}{3}$ long, and his first tick sits at $\tfrac{2}{3}$ .
A number line is cut into eighths. Name a fraction that sits between $\tfrac{1}{2}$ and $\tfrac{3}{4}$ .

Hint

Write both endpoints in eighths and look for a tick in between.

Answer

$\tfrac{5}{8}$

Full solution

In eighths, $\tfrac{1}{2} = \tfrac{4}{8}$ and $\tfrac{3}{4} = \tfrac{6}{8}$ .

The only tick strictly between them is the fifth, so $\tfrac{5}{8}$ works.

Frequently asked questions

How do I put a fraction on a number line?

Split the space from 0 to 1 into as many equal parts as the bottom number. Then count that many parts from zero. For three fifths, make five parts and count three.

What if the fraction is bigger than one?

Keep counting into the next whole. Five quarters means four parts in every whole, counted five times, so it lands one quarter past 1.

Why use a number line instead of a pizza?

A number line shows that a fraction is a number with its own place, not only a piece of something. It also makes ordering visible, since further right always means bigger.

How do I read a fraction off a number line?

Count the equal parts between two whole numbers to get the bottom number, then count the parts from the last whole number to the dot to get the top number.

What to learn next

Mixed Numbers and Improper Fractions: Converting Both WaysHow to change an improper fraction into a mixed number by dividing, how to change it back by multiplying, and when each form is the useful one.

Key terms in this lesson

Fraction: A fraction names a number of equal parts of a whole. It is written as one number over another, and the bar between them means divide.

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.3.NF.A.2Number and Operations—FractionsUnderstand a fraction as a number on the number line; represent fractions on a number line diagram.
CCSS.MATH.CONTENT.3.NF.A.2aNumber and Operations—FractionsRepresent a fraction 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 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
CCSS.MATH.CONTENT.3.NF.A.2bNumber and Operations—FractionsRepresent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

How do I put a fraction on a number line?

What if the fraction is bigger than one?

Why use a number line instead of a pizza?

How do I read a fraction off a number line?

What to learn next

Key terms in this lesson

Standards alignment

More Fractions lessons