Fractions on the Number Line and Equivalent Fractions

Think of one pizza before anyone cuts it. That pizza is the whole. A whole can also be a ribbon, a chocolate bar, or one sandwich.

Make equal parts

Share one sandwich fairly among four children. Cut it into four parts of the same size. Each child gets one fourth, written $\frac{1}{4}$. You can also call it one quarter.

Equal parts need the same amount, but their shapes can differ. A square cut into four equal strips shows fourths. The same square cut along both diagonals makes four equal triangles. Each triangle is also one fourth of the square.

A lopsided cut needs more care. If one piece is much bigger, counting the pieces alone cannot tell you its fraction. Check their sizes before naming a share.

Read the two numbers

A unit fraction names one equal part of a whole. Its top number is 1. The bottom number names the parts:

The bottom number is the denominator. It tells how many equal parts fill the whole. The top number is the numerator. It tells how many parts you count. The line between them is the fraction bar.

In $\frac{1}{6}$, six equal parts fill the whole. You count one of them.

More parts, smaller pieces

Picture a ribbon 12 inches long. Cut it into two equal pieces. Each half is 6 inches long. Cut a matching ribbon into six equal pieces. Each sixth is only 2 inches long. More equal pieces must fit into the same length, so each piece is smaller.

This explains why $\frac{1}{8}$ is smaller than $\frac{1}{4}$ of the same pizza. The 8 counts more pieces, not bigger pieces.

The whole matters, too. Half a small pizza gives less food than half a giant pizza. Both shares are called one half, but the pizzas have different sizes.

You can count several unit-fraction pieces together. Three pieces that are each one fourth make three fourths, $\frac{3}{4}$. This is one number, with one place between 0 and 1.

Name what is shaded

Imagine a paper rectangle split into eight equal parts. Five parts are shaded blue. To write the shaded fraction:

1. Count every equal part in the rectangle. There are eight, so the denominator is 8.
2. Count the blue parts. There are five, so the numerator is 5.
3. Write $\frac{5}{8}$ and say “five eighths.”

Three parts are not blue, so the unshaded fraction is $\frac{3}{8}$. Both fractions use 8 on the bottom because both refer to the same whole rectangle.

Writing $\frac{5}{3}$ for the blue part would mix up the jobs. The denominator counts all eight parts, not just the three unshaded parts.

Say the fraction name

Read the top number first. Then name the kind of parts:

• $\frac{2}{3}$ is “two thirds.”
• $\frac{5}{6}$ is “five sixths.”
• $\frac{7}{10}$ is “seven tenths.”

For parts called halves, say “one half” or “three halves.” A non-unit fraction has a numerator other than 1, such as $\frac{2}{3}$.

Share more than one sandwich

Four children share three matching sandwiches fairly. Cut each sandwich into four equal parts. Give every child one piece from each sandwich.

Each child now has three fourth-size pieces. Together, those pieces make $\frac{3}{4}$ of one sandwich. The denominator stays 4 because four pieces fill one sandwich. The numerator is 3 because each child gets three pieces.

Count none or all

If none of four equal parts is shaded, the shaded amount is $\frac{0}{4}$. That equals 0. If all four are shaded, the amount is $\frac{4}{4}$. That equals one whole.

For any picture, check that the pieces are equal before counting them. Three unequal pieces do not let you name a shaded share just by counting one piece out of three.

A fraction is a number, so you can mark it on a number line. Its distance from 0 shows its size. To begin, find the marks labeled 0 and 1. The length from 0 to 1 is one whole.

Place one fourth

Draw a straight line and label 0 and 1. Leave enough room to split that length into equal spaces.

1. Split the length from 0 to 1 into four equal spaces.
2. Start at 0 and move across one space.
3. Label your landing point $\frac{1}{4}$.

One jump covers one of the four equal parts. The mark at 0 is the start. It is not $\frac{1}{4}$, because you have not made a jump yet.

Count spaces, not marks

Four spaces need three inside marks, plus the marks at 0 and 1. Trace the spaces with your finger:

• From 0 to the first inside mark: one space.
• From there to the next mark: two spaces so far.
• To the third inside mark: three spaces so far.
• To 1: four spaces altogether.

Drawing four marks between 0 and 1 would make five spaces. Those would be fifths, not fourths.

The spaces must match in length. Folding a paper strip in half, then in half again, makes four equal lengths. Unfold it and use the creases to guide your marks. A ruler also helps you keep the gaps equal.

Read another line

Suppose 0 to 1 is split into six equal spaces. Each space is one sixth of the whole length. The first landing after 0 is $\frac{1}{6}$. Six jumps reach the mark at 1, also called $\frac{6}{6}$.

A longer drawing does not change the rule. Even if your line goes to 2, use only the length from 0 to 1 to find the part size.

Look at a ruler

A ruler is a number line used for measuring. On a ruler with eight equal spaces per inch, one small space is $\frac{1}{8}$ inch. A centimeter ruler has ten millimeter spaces in each centimeter. One millimeter is $\frac{1}{10}$ of a centimeter.

On lines with matching 0 and 1, smaller unit fractions sit nearer 0. One eighth is nearer 0 than one fourth. One half sits halfway to 1.

Once you know the size of one space, you can use several spaces to find a fraction. The denominator names the jump size. The numerator counts jumps from 0.

Find five sixths

To place $\frac{5}{6}$, split 0 to 1 into six equal spaces. Start at 0 and make five jumps to the right. Your landings are:

$\frac{1}{6}$, $\frac{2}{6}$, $\frac{3}{6}$, $\frac{4}{6}$, $\frac{5}{6}$.

Stop at the fifth landing and label it $\frac{5}{6}$. One more sixth-size jump would reach 1, so your point should be close to 1.

Read an unlabeled point

A line has four equal spaces from 0 to 1. A dot sits at the third landing after 0.

First count spaces in one whole: four. That gives the denominator. Next count jumps from 0 to the dot: three. That gives the numerator. The dot marks $\frac{3}{4}$.

Start counting with the first jump, not with the starting mark. Counting the mark at 0 as the first part would put every label one step too far left.

Count backward

On a line in eighths, put your finger at $\frac{7}{8}$. Move one space left at a time. Say “six eighths, five eighths, four eighths.” The part name stays eighths. Only the number of parts changes.

You can count in tenths the same way. Forward from $\frac{7}{10}$ comes $\frac{8}{10}$, then $\frac{9}{10}$, then $\frac{10}{10}$, which is 1. Backward from $\frac{3}{10}$ comes $\frac{2}{10}$, then $\frac{1}{10}$, then 0.

Check the whole on a longer line

A line from 0 to 2 has eight equal spaces altogether. Four of those spaces lie between 0 and 1. Each space is therefore one fourth, not one eighth.

The denominator counts spaces per whole, even when you can see two wholes. To read the third landing on this line, use $\frac{3}{4}$.

Use familiar places

The middle of 0 to 1 is $\frac{1}{2}$. Halfway from 0 to that middle is $\frac{1}{4}$. Halfway from the middle to 1 is $\frac{3}{4}$. These places help you check a drawing before counting every space. For example, $\frac{5}{6}$ should sit to the right of halfway.

A fraction can be equal to 1, greater than 1, or equal to another whole number. The number line keeps going, and so can your fraction count.

Reach 1 and keep going

On a line in fourths, four jumps reach $\frac{4}{4}$. You have counted every part of one whole, so $\frac{4}{4}$ equals 1.

Keep the same jump size. The next landing is $\frac{5}{4}$, then $\frac{6}{4}$, $\frac{7}{4}$, and $\frac{8}{4}$. Do not start the numerator over after 1. It keeps counting every jump you made from 0.

The point $\frac{5}{4}$ is one fourth-size jump past 1. The point $\frac{8}{4}$ is at 2 because two groups of four fourths make two wholes.

Make groups of one whole

Suppose you fill a half-cup scoop six times. You have six halves of a cup, written $\frac{6}{2}$ cups.

Pair the scoops to make whole cups:

1. The first two half-cups fill one cup.
2. The next two fill a second cup.
3. The last two fill a third cup.

So $\frac{6}{2}$ equals 3. You can check a whole-number fraction by asking how many full groups fit. With thirds, three parts make one whole, and six parts make two wholes: $\frac{6}{3}$ equals 2.

Compare a fraction with 1

The 4 in $\frac{4}{4}$ counts four fourth-size parts. It does not mean four wholes.

Write a whole number as a fraction

A denominator of 1 makes each part one whole. In $\frac{3}{1}$, you count three whole ones, so $\frac{3}{1}$ equals 3. In $\frac{6}{1}$, you count six whole ones, so it equals 6.

Watch the order: $\frac{1}{3}$ is one small third, but $\frac{3}{1}$ is three wholes.

Whole numbers have other fraction names, too. At the point 1, you can write $\frac{2}{2}$, $\frac{3}{3}$, or $\frac{6}{6}$. At the point 2, you can write $\frac{4}{2}$, $\frac{6}{3}$, or $\frac{8}{4}$. Each name counts enough equal parts to reach that whole-number point.

Equivalent fractions are different fraction names for the same amount of the same whole. One half and two fourths are equivalent. The equal sign records this: $\frac{1}{2}$ = $\frac{2}{4}$.

Split the shading into smaller parts

Draw a rectangle and split it into two equal halves. Shade one half.

Now draw a line that splits each half into two equal pieces. There are four equal parts in the whole rectangle. Two of those parts are shaded. You can now name the shading $\frac{2}{4}$.

You did not shade more or erase any shading. You only changed the size of the pieces you counted. That is why $\frac{1}{2}$ and $\frac{2}{4}$ are equal.

Build a fraction wall

Make several paper strips of exactly the same length. Keep one whole. Split the others into halves, thirds, fourths, sixths, and eighths. Stack them with their left ends lined up. This stack is a fraction wall.

Look for pieces that reach the same stopping point. One half reaches as far as two fourths. It also reaches as far as three sixths and four eighths. The pieces differ in size, but each group covers the same length.

A line on the wall can help your eye follow the match. From the end of one third, trace straight down to the sixths row. You reach the end of two sixths, so $\frac{1}{3}$ = $\frac{2}{6}$.

Show two thirds another way

Take a strip cut into thirds and shade two parts. Split every third into two equal pieces.

• The whole now has six equal parts.
• The shading now covers four of those parts.
• The shaded length has stayed the same.

So $\frac{2}{3}$ = $\frac{4}{6}$. The same splitting also shows that three fourths cover six eighths.

Start with matching wholes

Two strips must have the same whole length to show equivalence by lining up their ends. If one whole strip is longer, the shaded lengths can mislead you. Use matching rectangles when comparing shaded areas, too.

Not every pair will line up. On a fraction wall, one third reaches farther than one fourth. Those fractions name different amounts.

A fraction tells you a distance from 0. Equivalent fractions give the same distance, so they land on the same point of a number line.

Line up two number lines

A double number line has one line above another. Draw them with 0 directly above 0 and 1 directly above 1. The whole length must match.

Split the top line into halves. Split the bottom line into fourths. The half mark on top lines up with the two-fourths mark below. Both points are halfway from 0 to 1, so $\frac{1}{2}$ = $\frac{2}{4}$.

Use a ruler to trace straight down from one mark to the other. You are matching a place, not just looking for fraction names with similar numbers.

Match thirds and sixths

Draw another pair of lines with matching whole lengths. Make thirds on top and sixths below. One third-size space covers two sixth-size spaces.

The matching marks are:

• $\frac{1}{3}$ and $\frac{2}{6}$.
• $\frac{2}{3}$ and $\frac{4}{6}$.
• $\frac{3}{3}$ and $\frac{6}{6}$, both at 1.

Each sixth-size jump is shorter. You need twice as many of those jumps to reach the same place.

Name the halfway point

Halfway has many fraction names: $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, $\frac{4}{8}$, $\frac{5}{10}$, and $\frac{6}{12}$. In every name, the top number is half the bottom number.

For example, a whole split into ten equal spaces has five spaces on each side of halfway. That puts $\frac{5}{10}$ at the same point as $\frac{1}{2}$. Bigger numbers in a fraction name do not always mean a bigger fraction.

Match points past 1

Extend a pair of lines to 2. Keep halves on top and fourths below. Three half-size jumps reach the same place as six fourth-size jumps: $\frac{3}{2}$ = $\frac{6}{4}$. Four halves and eight fourths both reach 2.

Before using a double number line, check both pairs of whole-number marks. Aligning only the zeros is not enough. If the ones do not line up, the two whole lengths differ and the matching marks will be wrong.

A group of equivalent fractions is a fraction family. All its names have the same value. You can build a family by splitting every piece the same way.

Follow the splitting

Start with one third of a strip shaded. Split every third into two equal pieces. The whole has six parts, and two are shaded: $\frac{1}{3}$ = $\frac{2}{6}$.

Split every third into three equal pieces instead. The whole has nine parts, and three are shaded: $\frac{1}{3}$ = $\frac{3}{9}$. The shaded amount stays one third each time.

This gives a number rule. Multiply the numerator and denominator by the same whole number greater than 0. Both counts change together because you split all the parts the same way.

For $\frac{3}{4}$, multiplying both numbers by 2 gives $\frac{6}{8}$. Multiplying both by 3 gives $\frac{9}{12}$. Those names form part of one family.

Look for familiar families

The one-half family includes $\frac{1}{2}$, $\frac{2}{4}$, $\frac{3}{6}$, $\frac{4}{8}$, $\frac{5}{10}$, and $\frac{6}{12}$. Each numerator is half its denominator.

The one-fourth family begins $\frac{1}{4}$, $\frac{2}{8}$, and $\frac{3}{12}$. In that family, four copies of the top number make the bottom number.

Join pieces to go back

You can also join small pieces into larger equal groups. Imagine six eighths shaded on a strip. Put all eight eighths into pairs. Four pairs fill the whole, so each pair is one fourth. The six shaded eighths make three pairs. The shading is now three fourths: $\frac{6}{8}$ = $\frac{3}{4}$.

Keep both counts working together

Adding 1 to the top and bottom does not keep a fraction equal. Starting with $\frac{1}{2}$ gives $\frac{2}{3}$, which reaches farther along a fraction wall.

Changing only one number also changes the amount. Doubling only the top of $\frac{1}{2}$ gives $\frac{2}{2}$, a whole. Doubling only the bottom gives $\frac{1}{4}$, a smaller piece.

Use a new name to compare

Is $\frac{3}{8}$ before or after halfway? Name halfway as $\frac{4}{8}$. Three eighths is less than four eighths, so $\frac{3}{8}$ is before halfway.

You use this idea when reading an inch ruler, too. The same mark may be called $\frac{1}{2}$ inch, $\frac{2}{4}$ inch, or $\frac{4}{8}$ inch.

Imagine two equal pizzas. One is cut into thirds. The other is cut into fifths. Which single slice is larger?

Each third is larger because only three equal pieces need to fill its pizza. Five smaller pieces fill the other pizza. So $\frac{1}{3}$ is greater than $\frac{1}{5}$.

Compare unit fractions

Unit fractions all have numerator 1. You are comparing one part each time. With the same whole, a bigger denominator means a smaller part.

Here are some unit fractions from greatest to smallest:

$\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{6}$, $\frac{1}{8}$, $\frac{1}{12}$.

On matching number lines, one half sits farthest right among these points. One twelfth sits closest to 0.

Thinking only “8 is bigger than 4” would give the wrong answer for $\frac{1}{8}$ and $\frac{1}{4}$. The denominator tells you how many pieces must fit into the whole. Eight pieces need to be smaller than four pieces.

Count several parts of each size

The same idea works when the matching numerators are bigger than 1. Compare $\frac{3}{4}$ and $\frac{3}{8}$ of two equal paper strips.

1. Both fractions count three parts.
2. Each fourth is larger than each eighth.
3. Three larger parts give more length than three smaller parts.

So $\frac{3}{4}$ is greater than $\frac{3}{8}$. You can show this by shading three parts on each strip, with their left ends lined up.

For fractions with the same numerator of 1 or more, the smaller denominator gives the greater fraction. For example, $\frac{2}{3}$ is greater than $\frac{2}{6}$. You count two parts both times, but the thirds are bigger.

Check what matches

This rule compares equal counts of parts. It does not decide every pair with different denominators. For example, $\frac{2}{8}$ and $\frac{1}{4}$ are equal. Two small eighths cover one larger fourth.

Also check the whole when comparing food or lengths. Three fourths of a tiny sandwich can be less food than two fourths of a large sandwich. Use equal-size wholes to compare the amounts by their fraction names.

When fractions have the same denominator, their parts have the same name and size. For the same whole, you can decide which amount is greater by counting those parts.

Compare two chocolate shares

Two matching chocolate bars each have ten equal pieces. One child eats seven pieces. Another eats four pieces.

Their shares are $\frac{7}{10}$ and $\frac{4}{10}$. Every piece is one tenth of a matching bar. Seven pieces give more chocolate than four pieces, so $\frac{7}{10}$ is greater than $\frac{4}{10}$.

The denominator alone does not tell you the full amount. Both shares are tenths, but the numerators tell you how many tenths each child has.

Check on a number line

Draw a line with eight equal spaces from 0 to 1. Place $\frac{3}{8}$ at the third landing and $\frac{5}{8}$ at the fifth landing.

Both jumps are eighth-size. Five jumps take you farther from 0 than three jumps. The point $\frac{5}{8}$ is farther right, which shows that it is greater.

Put several fractions in order

Order $\frac{5}{6}$, $\frac{1}{6}$, $\frac{6}{6}$, and $\frac{2}{6}$ from smallest to greatest.

1. Check the denominators. They are all 6, so each part is a sixth.
2. Order the numerators: 1, 2, 5, 6.
3. Keep sixths as the part name.

The order is $\frac{1}{6}$, $\frac{2}{6}$, $\frac{5}{6}$, $\frac{6}{6}$. The last fraction fills the whole because all six sixths are counted.

Compare past 1

The rule still works beyond one whole. On a line in fourths, $\frac{5}{4}$ is at the fifth landing and $\frac{7}{4}$ is at the seventh. Seven fourths is greater than five fourths. Both lie to the right of $\frac{4}{4}$, which equals 1.

For shares of objects, keep the whole size the same. A fourth of a large bar can contain more chocolate than a fourth of a small bar. Matching denominators give matching piece sizes when the wholes match.

A comparison tells which fraction is greater, smaller, or equal. Once you have decided, write a symbol to record your answer.

Read the signs

The wide-open side of the sign faces the greater number. Read the whole comparison from left to right. For example, say “one sixth is less than one third.”

Order points from left to right

On one number line, the smaller number is farther left. To order unit fractions from smallest to greatest, think about their part sizes. For $\frac{1}{2}$, $\frac{1}{8}$, $\frac{1}{4}$, and $\frac{1}{3}$, the order is:

$\frac{1}{8}$, $\frac{1}{4}$, $\frac{1}{3}$, $\frac{1}{2}$.

For matching numerators of 1 or more, use the same idea. Order $\frac{2}{3}$, $\frac{2}{8}$, and $\frac{2}{4}$ from greatest to smallest. Each fraction counts two parts. Thirds are the biggest parts and eighths are the smallest. The order is $\frac{2}{3}$, $\frac{2}{4}$, $\frac{2}{8}$.

Equivalent fractions share a place in an ordering. Put $\frac{1}{2}$ and $\frac{2}{4}$ at the same point, not at two different stops.

Use halfway as a helper

Some fractions have different top and bottom numbers. You can still compare them when one is below halfway and the other is above it.

Compare $\frac{3}{8}$ with $\frac{4}{6}$. Half of eight is four. Three eighths is therefore less than halfway. Half of six is three. Four sixths is more than halfway. So $\frac{3}{8}$ < $\frac{4}{6}$.

For any fraction here, compare its numerator with half its denominator. A smaller numerator puts it below $\frac{1}{2}$. A larger numerator puts it above $\frac{1}{2}$. If the numerator is larger than the whole denominator, the fraction is even past 1.

Give your reason

Say what you used, not just the answer. For $\frac{5}{8}$ > $\frac{3}{8}$, you could say, “Both fractions use eighths of the same whole. Five of those parts give more than three.” You could also draw their points on one number line and show which is farther right.