Fractions on a number line

You meet frac­tions on lines all the time: the mark­ings on a mea­sur­ing jug, the half-way line on a cricket pitch, the scale on a ther­mome­ter. Putting a frac­tion on a num­ber line shows how big it is com­pared with the whole num­bers around it. It also shows at a glance that $\displaystyle \dfrac{7}{4}$ is more than $1$ but less than $2$. This les­son shows you how to do it for halves and quar­ters, then past $1$ and below $0$.

A num­ber line is a straight line with num­bers marked along it. It works like a ruler. The whole num­bers sit in order: 0, then 1, then 2.

Frac­tions live on the line too. A frac­tion is not a strange new thing. It just sits between the whole num­bers.

The bot­tom num­ber cuts, the top num­ber counts

Pic­ture the space from 0 to 1 as one long bar of choco­late. The bot­tom num­ber of a frac­tion tells you how many equal pieces to cut. The top num­ber tells you how many pieces to count.

Take $\displaystyle \frac{3}{4}$. The bot­tom num­ber is 4. So you cut the bar into 4 equal pieces. The top num­ber is 3. So you count 3 pieces from the start.

Remem­ber. The bot­tom num­ber cuts the whole up. The top num­ber counts the parts you take.

One whole cut into 2 equal parts

Draw a line. Put 0 near the left end. Put 1 fur­ther along to the right. Leave a wide gap between them.

Now cut that gap into 2 equal parts. One mark does it. Put the mark exactly in the mid­dle.

The mid­dle mark is $\displaystyle \frac{1}{2}$. We say that as one half. It has to be the mid­dle. Both parts must be the same size, and only the mid­dle gives that.

Here are the marks in order, read­ing from left to right.

One whole in halves

$$\displaystyle \begin{aligned}1 &= \frac{2}{2} \\ &= \frac{1}{2} + \frac{1}{2}\end{aligned}$$

Two halves fill the gap. So two halves make one whole. The mark for $\displaystyle \frac{2}{2}$ is the same spot as 1.

One whole cut into 4 equal parts

Use the same gap from 0 to 1. This time cut it into 4 equal parts. That takes 3 marks inside the gap.

Why 3 marks and not 4? One cut makes 2 parts. Two cuts make 3 parts. So three cuts make 4 parts.

When you cut a whole into 4 equal parts, we call one of those parts a quar­ter. So each part here is one quar­ter wide. Now count from 0. Each step to the right adds one more quar­ter.

$\displaystyle \frac{0}{4}$ means you have cut the whole into 4 parts but taken none of them yet. You are still stand­ing on 0.

Here is that same gap, mark by mark.

One whole in quar­ters

$$\displaystyle \begin{aligned}1 &= \frac{4}{4} \\ &= \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\end{aligned}$$

Remem­ber. $\displaystyle \frac{4}{4}$ and 1 are the same point. Four quar­ters fill one whole, so they end in the same place.

Mark­ing three quar­ters

Start at 0. Take 3 steps, each one quar­ter long. You land on the third mark.

That point is $\displaystyle \frac{3}{4}$. It sits just before 1. You have taken 3 parts of the 4, so one part is still to come.

Going past 1

Some frac­tions are big­ger than one whole. $\displaystyle \frac{7}{4}$ is one of them. Seven quar­ters will not fit between 0 and 1.

So carry the line on past 1, towards 2. Cut that whole into 4 equal parts as well. Every whole on the line gets the same num­ber of parts.

They have to match. A quar­ter is a quar­ter wher­ever you are on the line. If the sec­ond whole had a dif­fer­ent num­ber of parts, your steps would change size half way along. Then your count­ing would go wrong.

Now the quar­ters carry on: $\displaystyle \frac{5}{4}$, then $\displaystyle \frac{6}{4}$, then $\displaystyle \frac{7}{4}$. Then comes 2.

Here is the sec­ond whole, mark by mark.

Where seven quar­ters sits

$$\displaystyle \begin{aligned}\frac{7}{4} &= \frac{4}{4} + \frac{3}{4} \\ &= 1 + \frac{3}{4}\end{aligned}$$

So $\displaystyle \frac{7}{4}$ is one whole, then 3 more quar­ters. On the line it is the third mark after 1. It is near 2, but not at 2.

You can also count the whole way. Take 7 steps of one quar­ter from 0. Step 4 lands you on 1. Three more steps bring you to $\displaystyle \frac{7}{4}$.

Two wholes hold 8 quar­ters. So $\displaystyle \frac{8}{4}$ and 2 are the same point on the line.

Going left of 0

The line does not stop at 0. It car­ries on to the left. Num­bers to the left of 0 are less than 0. We write them with a minus sign in front.

Cut the gap from $-1$ to 0 into 4 equal parts too. Then count back from 0, one quar­ter at a time.

One step left is $\displaystyle -\frac{1}{4}$. Two steps left is $\displaystyle -\frac{2}{4}$. Three steps left is $\displaystyle -\frac{3}{4}$. Four steps left is $-1$.

Here is the gap to the left of 0, mark by mark.

So $\displaystyle -\frac{3}{4}$ is the mark just to the right of $-1$. You have gone 3 parts of the 4, with one part still to go.

Remem­ber. The gap from $-1$ to 0 is the same width as the gap from 0 to 1. You cut it into the same num­ber of parts, so each step is the same size. Only the way you walk changes: right for plus, left for minus.

More worked exam­ples

Exam­ple 1: halves past 2

Mark $\displaystyle \dfrac{5}{2}$ on a num­ber line.

The bot­tom num­ber is $2$, so cut every whole into $2$ equal parts. Two halves make one whole, so four halves make $2$.

$$\displaystyle \begin{aligned}\frac{5}{2} &= \frac{4}{2} + \frac{1}{2} \\ &= 2 + \frac{1}{2}\end{aligned}$$

So carry the line on to $3$. The point $\displaystyle \dfrac{5}{2}$ is the mid­dle mark between $2$ and $3$.

Exam­ple 2: a longer walk

Mark $\displaystyle \dfrac{11}{4}$ on a num­ber line.

Four quar­ters make one whole, so eight quar­ters make $2$. That leaves $11 - 8 = 3$ quar­ters.

$$\displaystyle \begin{aligned}\frac{11}{4} &= \frac{8}{4} + \frac{3}{4} \\ &= 2 + \frac{3}{4}\end{aligned}$$

Go to $2$, then take three more quar­ter steps. You land on the third mark after $2$, just before $3$.

Exam­ple 3: past minus one

Mark $\displaystyle -\dfrac{5}{4}$ on a num­ber line.

Walk left from $0$. Four quar­ter steps take you to $-1$. One more quar­ter step takes you fur­ther left.

$$\displaystyle \begin{aligned}-\frac{5}{4} &= -\frac{4}{4} - \frac{1}{4} \\ &= -1 - \frac{1}{4}\end{aligned}$$

So $\displaystyle -\dfrac{5}{4}$ is the first mark to the left of $-1$. You need to carry the line on past $-1$ towards $-2$ to show it.

How to draw one your­self

1. Draw a straight line with a ruler. Leave spare room at both ends.
2. Mark 0 some­where near the mid­dle.
3. Mark 1 to the right of 0. Then mark 2 the same dis­tance again.
4. Mark $-1$ to the left of 0, using that same dis­tance.
5. Look at the bot­tom num­ber of your frac­tion. Cut every whole gap into that many equal parts.
6. Check the parts by eye. They should all look the same width.
7. Start at 0. Count the steps given by the top num­ber. Go right for a plus, left for a minus.

The equal parts mat­ter most. Dis­tance along the line is what tells you how big a num­ber is. Marks on a ruler work the same way. If one part is fat­ter than the rest, the marks tell you the wrong thing.

Prac­tice

Draw a line from $-1$ to 2. Cut every whole into 4 equal parts. Mark each of these num­bers on it.

Extra prac­tice with thirds

The line above uses quar­ters, as your chap­ter does. Thirds work in just the same way. Try this set once the quar­ters feel easy.

Draw a fresh line from $-1$ to 2. Cut every whole into 3 equal parts this time. Mark each of these num­bers on it.

Every num­ber here sits on the line you drew. Check your marks below. Each answer names the whole num­ber you reach first. It then gives the extra parts.

Remem­ber. Cut each whole into as many equal parts as the bot­tom num­ber. Then count along. That is all a frac­tion on a line means.

Com­mon mis­takes

• Cut­ting the whole into the top num­ber of parts instead of the bot­tom num­ber. For $\displaystyle \dfrac{3}{4}$, cut into $4$, then count $3$.
• Draw­ing 4 marks inside the gap to make quar­ters. Three marks inside make four parts.
• Mak­ing the parts dif­fer­ent sizes. Every part must be the same width, or the marks tell you the wrong size.
• Count­ing the $0$ mark as the first step. The first step ends at the first mark after $0$.
• Walk­ing right for a neg­a­tive frac­tion. Minus means left of $0$.
• Cut­ting the sec­ond whole into a dif­fer­ent num­ber of parts from the first.

Key terms

Num­ber line

A straight line with num­bers marked in order at equal dis­tances.

Frac­tion

A num­ber such as $\displaystyle \dfrac{3}{4}$ that names equal parts of a whole.

Bot­tom num­ber (denom­i­na­tor)

How many equal parts each whole is cut into.

Top num­ber (numer­a­tor)

How many of those parts you count.

Half

One of two equal parts of a whole.

Quar­ter

One of four equal parts of a whole.

Neg­a­tive frac­tion

A frac­tion less than $0$, marked to the left of $0$.

Answers

Quar­ters, on a line from $-1$ to $2$

1. $\displaystyle \dfrac{1}{4} = 0 + \dfrac{1}{4}$: the first mark after $0$.
2. $\displaystyle \dfrac{3}{4} = 0 + \dfrac{3}{4}$: the third mark after $0$, just before $1$.
3. $\displaystyle \dfrac{5}{4} = 1 + \dfrac{1}{4}$: the first mark after $1$.
4. $\displaystyle \dfrac{7}{4} = 1 + \dfrac{3}{4}$: the third mark after $1$, just before $2$.
5. $\displaystyle -\dfrac{2}{4} = 0 - \dfrac{2}{4}$: two marks left of $0$, half-way between $-1$ and $0$.
6. $\displaystyle -\dfrac{3}{4} = 0 - \dfrac{3}{4}$: three marks left of $0$, the mark just to the right of $-1$.

Thirds, on a line from $-1$ to $2$

1. $\displaystyle \dfrac{2}{3} = 0 + \dfrac{2}{3}$: the sec­ond mark after $0$, just before $1$.
2. $\displaystyle \dfrac{4}{3} = 1 + \dfrac{1}{3}$: the first mark after $1$.
3. $\displaystyle \dfrac{5}{3} = 1 + \dfrac{2}{3}$: the sec­ond mark after $1$, just before $2$.
4. $\displaystyle \dfrac{6}{3} = 2$: the same point as $2$.
5. $\displaystyle -\dfrac{1}{3} = 0 - \dfrac{1}{3}$: the first mark left of $0$.
6. $\displaystyle -\dfrac{2}{3} = 0 - \dfrac{2}{3}$: the sec­ond mark left of $0$, just to the right of $-1$.