Fractional Reasoning and Equivalence

Defining the Whole and Equal Partitions

Fractions help us describe parts of something. But before we can name a fraction, we must know the whole.

A whole is the entire object, set, length, or amount we are talking about. If the whole changes, the fraction can change too.

Fractions also only work when the whole is split into equal parts. Equal parts are parts that are the same size.

If the parts are not equal, we cannot correctly name them as fractions like halves, thirds, or fourths.

For example, if a sandwich is cut into 2 pieces, but one piece is much bigger than the other, the pieces are not halves. Halves must be 2 equal parts.

Main Idea 1: First define the whole

Sometimes the whole is one shape, like one pizza. Sometimes the whole is a group, like 12 counters. Sometimes the whole is a length, like one ribbon.

We always ask: What is the whole?

• If the whole is 1 cookie, then part of that cookie can be a fraction of 1 cookie.
• If the whole is 3 cookies together, then 1 cookie is only part of the whole group.
• If the whole is a strip of paper, then each equal section is a fraction of that strip.

This means the same piece can have a different name if the whole changes.

Main Idea 2: Fractions need equal partitions

A partition means to split something into parts. A fraction is made when a whole is partitioned into equal parts.

Here are some common equal partitions:

• 2 equal parts = halves
• 3 equal parts = thirds
• 4 equal parts = fourths
• 6 equal parts = sixths

If 1 whole is split into 4 equal parts, each part is one-fourth, written as \(\frac{1}{4}\).

The bottom number tells how many equal parts the whole has. The top number tells how many of those equal parts we are talking about.

So in \(\frac{3}{4}\):

• the 4 means the whole is split into 4 equal parts
• the 3 means we have 3 of those equal parts

Main Idea 3: Equal does not always mean same shape

Equal parts must be the same amount, but they do not always have to look exactly the same.

For example, a rectangle can be split into 2 equal parts up and down, or into 2 equal parts side to side. The parts may look different from another drawing, but if they are the same size, they are equal.

Sometimes shapes are cut in different ways and still make equal parts. What matters is that each part is the same size as the others.

Main Idea 4: Fractions can describe area, sets, and lengths

Fractions are not only for shapes.

• Area model: a shape split into equal parts
• Set model: a group of objects split into equal groups
• Length model: a strip, line, or number line split into equal lengths

In every model, the rule stays the same: first define the whole, then make sure the parts are equal.

Worked Example 1: One shape split into equal parts

A square is split into 4 equal small squares. What fraction is one small square?

Step 1: Define the whole. The whole is 1 square.

Step 2: Count the equal parts. The whole square is split into 4 equal parts.

Step 3: Name one part. One of 4 equal parts is \(\frac{1}{4}\).

Answer: One small square is \(\frac{1}{4}\) of the whole.

We can write:

$$1\text{ part out of }4\text{ equal parts }= \frac{1}{4}$$

Worked Example 2: Unequal parts are not fractions

A circle is cut into 3 pieces, but one piece is large and the other 2 pieces are smaller. Can we say each piece is a third?

Step 1: Ask if the parts are equal.

They are not equal because the pieces are different sizes.

Step 2: Decide if a fraction name works.

No. We cannot call the pieces thirds because thirds must be 3 equal parts.

Answer: No, these pieces are not thirds.

Important: Just counting 3 pieces is not enough. The 3 pieces must be equal.

Worked Example 3: The whole changes the fraction

There are 8 crayons in a box. 4 are blue. What fraction of the crayons are blue?

Step 1: Define the whole. The whole is all 8 crayons.

Step 2: Look at how many equal items are in the whole. Each crayon is 1 equal item in the set.

Step 3: Count the blue crayons. There are 4 blue crayons.

Step 4: Write the fraction. 4 out of 8 crayons are blue, so the fraction is \(\frac{4}{8}\).

Answer: \(\frac{4}{8}\) of the crayons are blue.

Now imagine only the 4 blue crayons are the whole group. Then 1 blue crayon would be \(\frac{1}{4}\) of that new whole.

This shows why defining the whole is so important.

Worked Example 4: Fractions on a length

A strip of paper is split into 5 equal parts. How much of the strip is 2 parts?

Step 1: Define the whole. The whole is the entire strip of paper.

Step 2: Count the equal parts. There are 5 equal parts.

Step 3: Count the parts we have. We have 2 of those parts.

Step 4: Write the fraction. 2 out of 5 equal parts is \(\frac{2}{5}\).

Answer: 2 parts of the strip is \(\frac{2}{5}\) of the whole strip.

How to check if a fraction makes sense

1. Ask, What is the whole?
2. Ask, Is the whole split into equal parts?
3. Count the total number of equal parts.
4. Count how many parts you are talking about.
5. Write the fraction.

Things to remember

• A fraction names equal parts of a whole.
• You must know what the whole is first.
• Parts must be equal in size or amount.
• If parts are not equal, the fraction name is not correct.
• The whole can be a shape, a set of objects, or a length.

Quick Compare

Look at these two situations:

• A pizza cut into 4 equal slices: each slice is \(\frac{1}{4}\).
• A pizza cut into 4 uneven slices: the slices are not fourths.

The number of pieces alone does not make a fraction. The pieces must be equal.

Summary

Fractions tell about parts of a whole. To name a fraction correctly, you must first know the whole. Then check that the whole is split into equal parts. Equal partitions let us use fraction names like halves, thirds, fourths, and fifths correctly.

Unit Fractions as Building Blocks

Fractions help us talk about parts of a whole. A unit fraction is a fraction with a numerator of 1, like \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{4}\).

A unit fraction means one equal part of a whole. The denominator tells how many equal parts the whole is cut into.

For example:

• \(\frac{1}{2}\) means 1 out of 2 equal parts.
• \(\frac{1}{3}\) means 1 out of 3 equal parts.
• \(\frac{1}{5}\) means 1 out of 5 equal parts.

Unit fractions are called building blocks because we can put them together to make other fractions.

If we have several copies of the same unit fraction, we can name the fraction by counting how many copies we have.

For example, if a whole is split into 4 equal parts, then one part is \(\frac{1}{4}\). Two of those parts make \(\frac{2}{4}\). Three of those parts make \(\frac{3}{4}\).

We can think of it like this:

$$ \frac{1}{4}+\frac{1}{4}=\frac{2}{4} $$ $$ \frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4} $$

The denominator stays the same because the whole is still split into the same number of equal parts. The numerator tells how many parts we have.

Important idea: Fractions must come from equal parts. If the parts are not equal, they do not make a correct fraction of the whole.

Unit fractions also help us on a number line. If the distance from 0 to 1 is split into equal parts, each part is a unit fraction.

For example, if 0 to 1 is split into 3 equal jumps, each jump is \(\frac{1}{3}\). Starting at 0:

• 1 jump is \(\frac{1}{3}\)
• 2 jumps is \(\frac{2}{3}\)
• 3 jumps is \(\frac{3}{3}=1\)

This shows that larger fractions can be made by joining unit fractions again and again.

Main Teaching Points

• A unit fraction has a numerator of 1.
• The denominator tells how many equal parts make the whole.
• One equal part is the unit fraction.
• Other fractions are made by putting unit fractions together.
• The numerator tells how many unit fractions we have.

Here is a helpful pattern:

$$ \frac{1}{n} $$

This means 1 part when the whole is split into \(n\) equal parts.

Then we can build more fractions:

$$ \frac{1}{n}+\frac{1}{n}=\frac{2}{n} $$ $$ \frac{1}{n}+\frac{1}{n}+\frac{1}{n}=\frac{3}{n} $$

You do not need to memorize that pattern. Just remember: count how many same-size parts you have.

Worked Example 1: Naming a unit fraction

A sandwich is cut into 4 equal pieces. You have 1 piece. What fraction of the sandwich do you have?

Step 1: Count the total equal parts. There are 4.

Step 2: Count how many parts you have. You have 1.

So the fraction is \(\frac{1}{4}\).

This is a unit fraction because the numerator is 1.

Worked Example 2: Building a fraction from unit fractions

A pizza is cut into 8 equal slices. You eat 3 slices. What fraction of the pizza did you eat?

One slice is \(\frac{1}{8}\).

If you eat 3 slices, you have 3 copies of \(\frac{1}{8}\):

$$ \frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8} $$

So you ate \(\frac{3}{8}\) of the pizza.

Worked Example 3: Using a number line

The space from 0 to 1 is split into 5 equal parts on a number line. What fraction is at the second mark after 0?

Each jump is \(\frac{1}{5}\).

The first mark is \(\frac{1}{5}\).

The second mark is 2 jumps of \(\frac{1}{5}\):

$$ \frac{1}{5}+\frac{1}{5}=\frac{2}{5} $$

So the second mark is \(\frac{2}{5}\).

Worked Example 4: Finding how many unit fractions make a fraction

How many \(\frac{1}{6}\) pieces are in \(\frac{4}{6}\)?

The denominator is 6, so the unit fraction is \(\frac{1}{6}\).

The numerator is 4, so there are 4 pieces of size \(\frac{1}{6}\).

$$ \frac{4}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} $$

So \(\frac{4}{6}\) is made of 4 unit fractions of size \(\frac{1}{6}\).

Things to Remember

• Look at the denominator to know the size of each equal part.
• Look at the numerator to know how many parts there are.
• \(\frac{1}{4}\) means one fourth-size piece.
• \(\frac{3}{4}\) means three fourth-size pieces.
• Fractions are built from unit fractions.

Quick Check

1. If a whole is split into 3 equal parts, what is one part called?
2. If you have 2 pieces and each piece is \(\frac{1}{7}\), what fraction do you have?
3. On a number line split into 4 equal parts from 0 to 1, what fraction is the third mark?

Answers

1. One part is \(\frac{1}{3}\).
2. You have \(\frac{2}{7}\).
3. The third mark is \(\frac{3}{4}\).

Summary

A unit fraction is a fraction with a numerator of 1. It names one equal part of a whole.

We use unit fractions as building blocks to make other fractions. For example, \(\frac{3}{5}\) means 3 pieces, and each piece is \(\frac{1}{5}\).

When you see a fraction, think: What is the unit fraction? Then count how many of those equal parts there are.

Composing Non-Unit Fractions means building a fraction by putting together equal fractional parts.

A unit fraction is a fraction with 1 on top, like \(\frac{1}{2}\), \(\frac{1}{3}\), or \(\frac{1}{4}\). It means one equal part of a whole.

A non-unit fraction has a number bigger than 1 on top, like \(\frac{2}{3}\), \(\frac{3}{4}\), or \(\frac{5}{6}\). It means more than one equal part of the same size.

When we compose a non-unit fraction, we are putting unit fractions together. For example, \(\frac{3}{4}\) means three copies of \(\frac{1}{4}\).

We can write that like this:

$$\frac{3}{4}=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}$$

This is an important idea: the bottom number tells the size of each part, and the top number tells how many of those parts we have.

So in \(\frac{3}{4}\):

• the 4 means the whole is split into 4 equal parts,
• the 3 means we have 3 of those parts.

All the parts must be equal. Fractions only work correctly when the whole is divided into equal pieces.

Think of a pizza. If a pizza is cut into 4 equal slices, then each slice is \(\frac{1}{4}\). If you have 3 slices, you have \(\frac{3}{4}\) of the pizza.

You can also think about fractions on a number line. If the space from 0 to 1 is split into 4 equal jumps, each jump is \(\frac{1}{4}\). Three jumps of \(\frac{1}{4}\) land on \(\frac{3}{4}\).

$$\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}$$

This shows that a non-unit fraction is made by repeating the same unit fraction.

How to compose a non-unit fraction

1. Look at the denominator, the bottom number.
2. That tells the size of each unit fraction.
3. Look at the numerator, the top number.
4. That tells how many times to use that unit fraction.

For example, to make \(\frac{5}{6}\):

• the denominator is 6, so each part is \(\frac{1}{6}\),
• the numerator is 5, so we need 5 parts of \(\frac{1}{6}\).

So:

$$\frac{5}{6}=\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}$$

Worked Example 1

Compose \(\frac{2}{5}\).

Step 1: The denominator is 5, so each part is \(\frac{1}{5}\).

Step 2: The numerator is 2, so we need 2 parts.

Step 3: Put them together.

$$\frac{2}{5}=\frac{1}{5}+\frac{1}{5}$$

So \(\frac{2}{5}\) is two copies of \(\frac{1}{5}\).

Worked Example 2

Compose \(\frac{4}{3}\) using unit fractions.

The denominator is 3, so each part is \(\frac{1}{3}\).

The numerator is 4, so we need 4 parts of \(\frac{1}{3}\).

$$\frac{4}{3}=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}$$

This fraction is more than 1 whole, because 3 thirds make 1 whole, and there is 1 more third left.

Worked Example 3

A ribbon is cut into 8 equal parts. Mia uses 3 parts. What fraction of the ribbon does she use?

Each part is \(\frac{1}{8}\).

Mia uses 3 parts, so she uses:

$$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{3}{8}$$

Mia uses \(\frac{3}{8}\) of the ribbon.

Worked Example 4

On a number line from 0 to 1, the whole is split into 6 equal parts. You make 5 jumps of size \(\frac{1}{6}\). Where do you land?

Each jump is \(\frac{1}{6}\).

After 5 jumps, you have:

$$\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=\frac{5}{6}$$

You land on \(\frac{5}{6}\).

Important ideas to remember

• A unit fraction has 1 on top.
• A non-unit fraction is made of more than one unit fraction.
• The denominator tells the size of each equal part.
• The numerator tells how many equal parts you have.
• \(\frac{3}{4}\) means 3 copies of \(\frac{1}{4}\).

Common mistake

Sometimes students think \(\frac{3}{4}\) means 3 and 4 are separate numbers that do not work together. But in a fraction, the numbers have jobs:

• 4 tells the whole is split into 4 equal parts,
• 3 tells how many of those parts are chosen.

Another mistake is adding unlike parts, such as saying \(\frac{1}{4}+\frac{1}{3}=\frac{2}{7}\). That is not how composing works. When we compose a non-unit fraction here, we are adding the same unit fraction again and again.

For example:

$$\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{3}{5}$$

All the parts are fifths, so the result is still in fifths.

Try thinking this way

• \(\frac{2}{7}\) is 2 copies of \(\frac{1}{7}\)
• \(\frac{6}{8}\) is 6 copies of \(\frac{1}{8}\)
• \(\frac{3}{2}\) is 3 copies of \(\frac{1}{2}\)

We can write these as:

$$\frac{2}{7}=\frac{1}{7}+\frac{1}{7}$$

$$\frac{6}{8}=\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}$$

$$\frac{3}{2}=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}$$

Summary

Composing non-unit fractions means building a fraction by joining equal unit fractions. A fraction like \(\frac{3}{4}\) is made from three \(\frac{1}{4}\) parts. If you know the size of one part and how many parts there are, you can understand and build any non-unit fraction.

Fractions of Continuous vs. Discrete Quantities

Fractions help us talk about parts of a whole. Sometimes the whole is one thing, like a pizza, a ribbon, or a rectangle. Sometimes the whole is a group of objects, like 12 marbles or 8 apples.

In this lesson, we will learn the difference between finding a fraction of a continuous quantity and finding a fraction of a discrete quantity.

Continuous quantity means one whole thing that can be divided into equal parts. Examples are a cake, a bar of soap, a strip of paper, or a shape.

Discrete quantity means a set of separate objects that can be counted. Examples are crayons, blocks, cookies, or toy cars.

No matter what kind of whole we have, fractions only work correctly when the whole is split into equal parts.

Remember: In a fraction, the denominator tells how many equal parts the whole is split into, and the numerator tells how many of those parts we are talking about.

For example, in \(\frac{3}{4}\):

• The \(4\) means the whole is split into 4 equal parts.
• The \(3\) means we are using 3 of those parts.

1. Fractions of a Continuous Quantity

When we find a fraction of a continuous quantity, we start with one whole object or shape.

If we want \(\frac{1}{2}\) of a rectangle, we divide the rectangle into 2 equal parts and take 1 part.

If we want \(\frac{3}{4}\) of a strip of paper, we divide the strip into 4 equal parts and take 3 parts.

The parts must be equal in size. If one part is bigger than another part, the fraction model is not correct.

2. Fractions of a Discrete Quantity

When we find a fraction of a discrete quantity, we start with a set of countable objects.

If we want \(\frac{1}{3}\) of 12 marbles, we split the 12 marbles into 3 equal groups. Then we take 1 of those groups.

Because \(12 \div 3 = 4\), each group has 4 marbles. So \(\frac{1}{3}\) of 12 is 4.

For sets of objects, we are not cutting the objects into pieces. We are making equal groups of whole objects.

How are they different?

• Continuous quantity: one whole thing is divided into equal parts.
• Discrete quantity: a group of separate objects is split into equal groups.

How are they the same?

• Both need a clear whole.
• Both must use equal parts or equal groups.
• Both use the numerator and denominator in the same way.

Worked Example 1: Continuous Shape

A rectangle is divided into 4 equal parts. 3 parts are shaded. What fraction of the rectangle is shaded?

Step 1: Count the total equal parts. There are 4.

Step 2: Count the shaded parts. There are 3.

Answer: The shaded fraction is \(\frac{3}{4}\).

We can write:

$$\frac{\text{shaded parts}}{\text{total equal parts}} = \frac{3}{4}$$

Worked Example 2: Discrete Set

There are 10 stars. What is \(\frac{1}{5}\) of 10 stars?

Step 1: The denominator is 5, so split the 10 stars into 5 equal groups.

Step 2: Find how many are in each group.

$$10 \div 5 = 2$$

Step 3: The numerator is 1, so take 1 group.

Answer: \(\frac{1}{5}\) of 10 stars is 2 stars.

Worked Example 3: Continuous Quantity

A ribbon is split into 8 equal parts. Sara uses \(\frac{5}{8}\) of the ribbon. How many parts does she use?

Step 1: The denominator 8 tells us the ribbon is split into 8 equal parts.

Step 2: The numerator 5 tells us to count 5 of those parts.

Answer: Sara uses 5 of the 8 equal parts, so she uses \(\frac{5}{8}\) of the ribbon.

This is a continuous model because the ribbon is one whole object.

Worked Example 4: Discrete Quantity

A bag has 16 marbles. What is \(\frac{3}{4}\) of 16 marbles?

Step 1: The denominator is 4, so split 16 marbles into 4 equal groups.

$$16 \div 4 = 4$$

Each group has 4 marbles.

Step 2: The numerator is 3, so take 3 groups.

$$3 \times 4 = 12$$

Answer: \(\frac{3}{4}\) of 16 marbles is 12 marbles.

A helpful way to think

1. Ask: Is the whole one object or a group of objects?
2. If it is one object, it is continuous.
3. If it is a group of countable objects, it is discrete.
4. Then make equal parts or equal groups.
5. Use the denominator first, then the numerator.

Watch out for these mistakes

• Unequal parts: Fractions must show equal parts.
• Forgetting the whole: You must know what the whole is before naming the fraction.
• Mixing up numerator and denominator: The denominator tells how many equal parts or groups. The numerator tells how many to take.
• Cutting objects in a set: For discrete quantities, we usually make equal groups of whole objects, not pieces.

Let's compare

If you see \(\frac{1}{2}\) of a pizza, you split one pizza into 2 equal parts and take 1 part.

If you see \(\frac{1}{2}\) of 6 cookies, you split 6 cookies into 2 equal groups and take 1 group. Since \(6 \div 2 = 3\), \(\frac{1}{2}\) of 6 cookies is 3 cookies.

Both show one-half, but one uses a single whole object and the other uses a set of objects.

Summary

A fraction shows equal parts of a whole. A continuous quantity is one whole thing, like a shape or ribbon, divided into equal parts. A discrete quantity is a set of separate objects, like marbles or apples, divided into equal groups.

To find a fraction, look at the denominator first to make equal parts or groups. Then use the numerator to choose how many parts or groups to take. If the parts are equal, the fraction is fair and correct.

Fractions as Numbers on a Number Line

Sometimes we see fractions as parts of a pizza or parts of a shape. That is helpful, but fractions are also numbers. A fraction can show an exact place on a number line.

In this lesson, we will learn how to place fractions between 0 and 1 on a number line. We will see that each fraction names one exact point.

What is a number line?

A number line is a straight line that shows numbers in order. Numbers get bigger as we move to the right.

When we work with fractions from 0 to 1, we look at the space between 0 and 1 and split it into equal parts.

Fractions must be made from equal parts

If the space from 0 to 1 is not split into equal parts, then we cannot correctly name the fractions.

For example, to show fourths, the space from 0 to 1 must be divided into 4 equal parts. Then the marks are:

\(\frac{1}{4}\), \(\frac{2}{4}\), \(\frac{3}{4}\), and then \(1\), which is the same as \(\frac{4}{4}\).

What do the two numbers in a fraction mean?

• Denominator: the bottom number tells how many equal parts the whole is split into.
• Numerator: the top number tells how many of those equal parts we count from 0.

In \(\frac{3}{5}\):

• the 5 means the space from 0 to 1 is split into 5 equal parts,
• the 3 means we count 3 of those parts from 0.

How to plot a fraction on a number line

1. Find 0 and 1.
2. Look at the denominator. Split the space from 0 to 1 into that many equal parts.
3. Start at 0 and count the number of parts shown by the numerator.
4. Put a point on that mark. That point is the fraction.

Important idea: A fraction is not just a piece of something. It is also a number with a place on the number line.

Example 1: Plot \(\frac{1}{2}\)

We want to place \(\frac{1}{2}\) on the number line from 0 to 1.

• The denominator is 2, so split the space into 2 equal parts.
• The numerator is 1, so count 1 part from 0.

That point is \(\frac{1}{2}\).

It is right in the middle of 0 and 1.

It can look like this:

$$0 \quad | \quad 1$$

$$\phantom{0} \quad \frac{1}{2}$$

Example 2: Plot \(\frac{3}{4}\)

Now let’s place \(\frac{3}{4}\).

• The denominator is 4, so split the space from 0 to 1 into 4 equal parts.
• The numerator is 3, so count 3 equal parts from 0.

The third mark is \(\frac{3}{4}\).

The marks are:

$$0 \quad \frac{1}{4} \quad \frac{2}{4} \quad \frac{3}{4} \quad 1$$

So \(\frac{3}{4}\) is close to 1, but it is not at 1 yet.

Example 3: Plot \(\frac{2}{3}\)

Let’s try thirds.

• The denominator is 3, so split the space from 0 to 1 into 3 equal parts.
• The numerator is 2, so count 2 parts from 0.

The second mark is \(\frac{2}{3}\).

It looks like this:

$$0 \quad \frac{1}{3} \quad \frac{2}{3} \quad 1$$

This shows that \(\frac{2}{3}\) is more than \(\frac{1}{3}\) and less than 1.

Example 4: Which fraction is at this point?

Suppose the space from 0 to 1 is split into 5 equal parts. A point is on the 4th mark after 0. What fraction is it?

• There are 5 equal parts, so the denominator is 5.
• The point is at the 4th mark from 0, so the numerator is 4.

The fraction is \(\frac{4}{5}\).

The number line marks are:

$$0 \quad \frac{1}{5} \quad \frac{2}{5} \quad \frac{3}{5} \quad \frac{4}{5} \quad 1$$

How to think about fractions between 0 and 1

• Fractions closer to 0 are smaller.
• Fractions closer to 1 are bigger.
• Each fraction has one exact spot.
• The whole space from 0 to 1 must be split into equal parts.

Helpful reminders

• Always check the denominator first.
• Make equal jumps or equal spaces on the number line.
• Then count the number of parts named by the numerator.
• Remember that \(1\) can also be written as a fraction, like \(\frac{2}{2}\), \(\frac{3}{3}\), or \(\frac{4}{4}\).

Common mistake to avoid

Do not make uneven spaces between 0 and 1. Fractions only work correctly on a number line when the parts are equal.

Another common mistake is mixing up the numerator and denominator. The denominator tells how many equal parts there are in all. The numerator tells how many parts to count from 0.

Let’s review

A fraction is a number that names a point on a number line. To place a fraction between 0 and 1, divide the space into equal parts using the denominator, then count parts from 0 using the numerator.

When you understand fractions on a number line, you can see that fractions are real numbers, just like whole numbers. They show exact places, not just pieces of objects.

Fractions Greater Than One

We often learn fractions by cutting 1 whole into equal parts. For example, if a pizza is cut into 4 equal slices, then 1 slice is \(\frac{1}{4}\).

But fractions do not stop at 1 whole. Sometimes we have more than one whole. A fraction can show that too. These are called fractions greater than one.

For example, \(\frac{5}{4}\) means 5 pieces, and each piece is \(\frac{1}{4}\) of a whole. Since 4 fourths make 1 whole, 5 fourths is 1 whole and 1 fourth.

This lesson will help you understand fractions greater than 1 using pictures, equal parts, and the number line.

1. Remember what a fraction means

A fraction has two numbers:

• The top number tells how many parts we have.
• The bottom number tells how many equal parts make 1 whole.

In \(\frac{3}{4}\):

• 3 means 3 parts
• 4 means 4 equal parts make 1 whole

If the top number is bigger than the bottom number, the fraction is greater than 1.

For example:

• \(\frac{4}{4} = 1\)
• \(\frac{5}{4} > 1\)
• \(\frac{7}{3} > 1\)

2. How to know when a fraction is greater than 1

Think about how many parts make 1 whole.

If the denominator is 6, then 6 sixths make 1 whole:

$$\frac{6}{6} = 1$$

So:

• \(\frac{1}{6}, \frac{2}{6}, \frac{3}{6}\) are less than 1
• \(\frac{6}{6}\) is exactly 1
• \(\frac{7}{6}, \frac{8}{6}\) are greater than 1

A good way to check is this:

• If the numerator is smaller than the denominator, the fraction is less than 1.
• If the numerator is the same as the denominator, the fraction is 1.
• If the numerator is greater than the denominator, the fraction is greater than 1.

3. Fractions greater than 1 with visual models

Let us use circles, bars, or rectangles split into equal parts.

Suppose each whole is divided into 3 equal parts. Each part is \(\frac{1}{3}\).

If we shade 4 thirds, we have:

• 3 thirds = 1 whole
• 1 more third = \(\frac{1}{3}\)

So \(\frac{4}{3}\) means:

$$\frac{4}{3} = 1\frac{1}{3}$$

This is called a mixed number. A mixed number has a whole number and a fraction.

Another example: if each whole is split into 5 equal parts and we have 8 parts, then:

• 5 fifths = 1 whole
• 3 fifths are left

So:

$$\frac{8}{5} = 1\frac{3}{5}$$

4. Fractions greater than 1 on a number line

A number line helps us see where fractions belong.

Let us count in fourths:

$$0, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{4}{4}, \frac{5}{4}, \frac{6}{4}, \frac{7}{4}, \frac{8}{4}$$

Since \(\frac{4}{4} = 1\), the next fraction, \(\frac{5}{4}\), is one step past 1.

On the number line:

• \(\frac{4}{4}\) is at 1
• \(\frac{5}{4}\) is just after 1
• \(\frac{6}{4}\) is two fourths after 1
• \(\frac{8}{4}\) is at 2

This shows that fractions can keep going after 1, just like whole numbers do.

5. Connecting improper fractions and mixed numbers

A fraction greater than 1 can often be written in two ways:

• as a fraction like \(\frac{5}{4}\)
• as a mixed number like \(1\frac{1}{4}\)

These names show the same amount.

To change a fraction greater than 1 into a mixed number:

1. Ask how many parts make 1 whole.
2. Make as many wholes as you can.
3. See how many parts are left.

Example: \(\frac{9}{4}\)

• 4 fourths = 1 whole
• Another 4 fourths = 1 more whole
• That uses 8 fourths
• 1 fourth is left

So:

$$\frac{9}{4} = 2\frac{1}{4}$$

Worked Example 1

Is \(\frac{6}{5}\) greater than 1?

Step 1: 5 fifths make 1 whole.

Step 2: We have 6 fifths, which is 1 more fifth than 1 whole.

So yes, \(\frac{6}{5}\) is greater than 1.

It can also be written as:

$$\frac{6}{5} = 1\frac{1}{5}$$

Worked Example 2

Write \(\frac{7}{3}\) as a mixed number.

Step 1: 3 thirds make 1 whole.

Step 2: Use 6 thirds to make 2 wholes.

$$\frac{6}{3} = 2$$

Step 3: 1 third is left.

So:

$$\frac{7}{3} = 2\frac{1}{3}$$

Worked Example 3

Where is \(\frac{5}{2}\) on the number line?

Step 1: Count in halves:

$$0, \frac{1}{2}, \frac{2}{2}, \frac{3}{2}, \frac{4}{2}, \frac{5}{2}$$

Step 2: \(\frac{2}{2} = 1\) and \(\frac{4}{2} = 2\).

Step 3: \(\frac{5}{2}\) is one half after 2.

So:

$$\frac{5}{2} = 2\frac{1}{2}$$

Worked Example 4

Compare \(\frac{3}{4}\) and \(\frac{5}{4}\).

Both fractions use fourths, so we can compare the numerators.

• \(\frac{3}{4}\) is less than 1
• \(\frac{5}{4}\) is greater than 1

So:

$$\frac{5}{4} > \frac{3}{4}$$

6. Tips for thinking about fractions greater than 1

• First find out how many equal parts make 1 whole.
• If you have more parts than that, the fraction is greater than 1.
• Use pictures to group parts into wholes.
• Use a number line to count equal jumps past 1.
• Remember: \(\frac{4}{4} = 1\), \(\frac{5}{4} = 1\frac{1}{4}\), and \(\frac{8}{4} = 2\).

7. Common mistakes to avoid

• Mistake: Thinking all fractions are less than 1.
  Fractions can be less than 1, equal to 1, or greater than 1.
• Mistake: Forgetting that the denominator tells the size of the parts.
  In \(\frac{5}{4}\), the pieces are fourths, not fifths.
• Mistake: Not making a full whole first.
  Always look for a full set, like 4 fourths or 3 thirds.

Summary

A fraction greater than 1 means you have more than one whole. This happens when the numerator is greater than the denominator.

You can show fractions greater than 1 with pictures, by grouping equal parts into wholes, or on a number line by counting past 1.

You can also write a fraction greater than 1 as a mixed number. For example, \(\frac{5}{4} = 1\frac{1}{4}\) and \(\frac{7}{3} = 2\frac{1}{3}\).

When you see a fraction, ask: How many parts make one whole? Then see whether the fraction is less than 1, equal to 1, or greater than 1.

Whole Numbers as Fractions

Sometimes we see fractions and whole numbers as different kinds of numbers. But they are connected! A whole number can be written as a fraction.

This is helpful because it shows that whole numbers and fractions belong to the same number family. It also helps us compare numbers and understand equivalent fractions.

Important idea: A fraction tells how many equal parts we have.

In a fraction, the top number tells how many parts we have, and the bottom number tells what kind of parts they are.

For example, in \(\frac{3}{4}\):

• the 3 means 3 parts,
• the 4 means the whole is split into 4 equal parts.

Now let’s think about whole numbers.

If we have 1 whole, we can write it as a fraction. One whole means we have 1 out of 1 equal part.

So,

$$1 = \frac{1}{1}$$

We can also make 1 whole in other ways. If a whole is split into 2 equal parts and we have both parts, that is still 1 whole.

$$1 = \frac{2}{2}$$

If a whole is split into 3 equal parts and we have all 3 parts, that is also 1 whole.

$$1 = \frac{3}{3}$$

So any fraction with the same top and bottom number equals 1.

Examples:

• \(\frac{4}{4} = 1\)
• \(\frac{5}{5} = 1\)
• \(\frac{10}{10} = 1\)

What about other whole numbers?

Every whole number can be written as a fraction with a denominator of 1.

That is because if the whole is split into 1 equal part, then each whole is just one big part.

So:

• \(2 = \frac{2}{1}\)
• \(3 = \frac{3}{1}\)
• \(7 = \frac{7}{1}\)

This means:

$$\text{whole number} = \frac{\text{that number}}{1}$$

For example, \(5\) means 5 wholes. As a fraction, that is \(\frac{5}{1}\).

Why does this work?

The denominator of 1 means each whole is divided into 1 equal part. So each part is a whole. If we have 5 of those parts, we have 5 wholes.

$$\frac{5}{1} = 5$$

Whole numbers can also be made from fractions greater than 1.

If we have enough equal parts to make full wholes, the fraction can name a whole number.

For example, if each whole is split into 2 equal parts, then 6 halves make 3 wholes.

$$\frac{6}{2} = 3$$

Why? Because:

• 2 halves make 1 whole,
• 4 halves make 2 wholes,
• 6 halves make 3 wholes.

Another example:

$$\frac{8}{4} = 2$$

That is because 4 fourths make 1 whole, so 8 fourths make 2 wholes.

A helpful pattern:

• If the numerator and denominator are the same, the fraction equals 1.
• If the numerator is a multiple of the denominator, the fraction can equal a whole number.
• Any whole number can be written with denominator 1.

Let’s look at some worked examples.

Worked Example 1

Write 3 as a fraction.

We put 3 over 1:

$$3 = \frac{3}{1}$$

Answer: \(\frac{3}{1}\)

Worked Example 2

What whole number does \(\frac{4}{4}\) equal?

The top and bottom numbers are the same. That means the fraction equals 1 whole.

$$\frac{4}{4} = 1$$

Answer: 1

Worked Example 3

What whole number does \(\frac{10}{5}\) equal?

Since 5 fifths make 1 whole, 10 fifths make 2 wholes.

$$\frac{10}{5} = 2$$

Answer: 2

Worked Example 4

Fill in the blank: \(2 = \frac{\square}{1}\)

Any whole number can be written over 1.

So:

$$2 = \frac{2}{1}$$

Answer: 2

Let’s compare a few more examples.

• \(1 = \frac{1}{1} = \frac{2}{2} = \frac{3}{3}\)
• \(4 = \frac{4}{1}\)
• \(\frac{12}{3} = 4\)
• \(\frac{15}{5} = 3\)

Things to remember

1. A whole number can be written as a fraction.
2. To write a whole number as a fraction, put it over 1.
3. A fraction with the same numerator and denominator equals 1.
4. Some fractions greater than 1 equal whole numbers.

Quick check

• \(6 = \frac{\square}{1}\)
• \(\frac{7}{7} = \square\)
• \(\frac{9}{3} = \square\)

Answers:

• \(6 = \frac{6}{1}\)
• \(\frac{7}{7} = 1\)
• \(\frac{9}{3} = 3\)

Summary

Whole numbers and fractions are connected. Every whole number can be written as a fraction with 1 on the bottom, like \(4 = \frac{4}{1}\). Fractions like \(\frac{4}{4}\) equal 1, and fractions like \(\frac{8}{4}\) can equal other whole numbers too.

Generating Equivalent Fractions Visually

Fractions can have different names and still mean the same amount. These are called equivalent fractions.

For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent fractions. They look different, but they cover the same part of a whole and land on the same spot on a number line.

In this lesson, we will learn how to see equivalent fractions by using:

• Area models, like shaded rectangles or circles
• Number lines, where fractions are shown as points between 0 and 1

1. What is a fraction?

A fraction shows equal parts of a whole.

In the fraction \(\frac{1}{2}\):

• The bottom number tells how many equal parts the whole is split into.
• The top number tells how many of those parts we have.

So \(\frac{1}{2}\) means 1 out of 2 equal parts.

2. What does equivalent mean?

Equivalent means equal in value. Equivalent fractions may have different numbers, but they name the same amount.

Here are some equivalent fractions:

• \(\frac{1}{2} = \frac{2}{4}\)
• \(\frac{1}{3} = \frac{2}{6}\)
• \(\frac{2}{3} = \frac{4}{6}\)

3. Using area models to see equivalent fractions

An area model is a shape split into equal parts. We can shade parts to show a fraction.

Let’s look at \(\frac{1}{2}\) and \(\frac{2}{4}\).

Imagine one rectangle split into 2 equal parts. If 1 part is shaded, that shows \(\frac{1}{2}\).

Now imagine the same size rectangle split into 4 equal parts. If 2 parts are shaded, that shows \(\frac{2}{4}\).

Even though the second rectangle has more pieces, the shaded amount is still the same. Both show half of the whole.

That is why:

$$\frac{1}{2} = \frac{2}{4}$$

A good way to think about this is: we did not change the whole. We only split the same whole into smaller equal parts.

4. Using number lines to see equivalent fractions

A number line shows fractions as points. Fractions that are equivalent land on the same point.

Think about the distance from 0 to 1.

• If we split that distance into 2 equal parts, the middle point is \(\frac{1}{2}\).
• If we split that same distance into 4 equal parts, the second point is \(\frac{2}{4}\).

Both points are in the same place: halfway between 0 and 1.

So on a number line:

$$\frac{1}{2} = \frac{2}{4}$$

This helps us prove that equivalent fractions are the same size.

5. A helpful pattern

Sometimes we can make an equivalent fraction by making more equal parts.

For example, start with \(\frac{1}{3}\). If each third is split into 2 smaller equal parts, the whole now has 6 equal parts. The 1 shaded third becomes 2 shaded sixths.

So:

$$\frac{1}{3} = \frac{2}{6}$$

We can see this visually because both fractions shade the same amount of the same whole.

Worked Example 1

Are \(\frac{1}{2}\) and \(\frac{2}{4}\) equivalent?

Step 1: Draw or imagine the same whole two times.

• Split the first whole into 2 equal parts and shade 1 part.
• Split the second whole into 4 equal parts and shade 2 parts.

Step 2: Compare the shaded amounts.

Both show the same amount of the whole.

Answer: Yes, they are equivalent.

$$\frac{1}{2} = \frac{2}{4}$$

Worked Example 2

Are \(\frac{1}{3}\) and \(\frac{2}{6}\) equivalent?

Step 1: Use area models.

• Split one shape into 3 equal parts and shade 1 part.
• Split another same-size shape into 6 equal parts and shade 2 parts.

Step 2: Look at how much is shaded.

The shaded parts cover the same amount.

Answer: Yes, they are equivalent.

$$\frac{1}{3} = \frac{2}{6}$$

Worked Example 3

Are \(\frac{2}{3}\) and \(\frac{4}{6}\) equivalent?

Step 1: Think about a rectangle.

• Split one rectangle into 3 equal parts and shade 2 parts.
• Split the same-size rectangle into 6 equal parts and shade 4 parts.

Step 2: Compare the shaded amount.

Both show the same amount of the whole.

Step 3: Check on a number line.

Both fractions land at the same point between 0 and 1.

Answer: Yes, they are equivalent.

$$\frac{2}{3} = \frac{4}{6}$$

Worked Example 4

Are \(\frac{1}{2}\) and \(\frac{3}{4}\) equivalent?

Step 1: Draw or imagine the same whole.

• For \(\frac{1}{2}\), shade 1 out of 2 equal parts.
• For \(\frac{3}{4}\), shade 3 out of 4 equal parts.

Step 2: Compare the shaded amount.

\(\frac{3}{4}\) covers more than \(\frac{1}{2}\).

Step 3: Check on a number line.

\(\frac{1}{2}\) is at the middle. \(\frac{3}{4}\) is farther to the right.

Answer: No, they are not equivalent.

6. How to tell if fractions are equivalent visually

Ask yourself these questions:

1. Are the wholes the same size?
2. Are the parts split into equal pieces?
3. Do the shaded parts cover the same amount?
4. Do the fractions land on the same point on a number line?

If the answer is yes, the fractions are equivalent.

7. Important things to remember

• Equivalent fractions have different names but the same value.
• Area models help us compare shaded parts of the same whole.
• Number lines help us compare where fractions are located.
• The whole must be the same size when we compare fractions.
• The parts must be equal parts.

8. Quick practice to think about

Try these with a drawing or a number line:

• Is \(\frac{2}{4}\) the same as \(\frac{1}{2}\)?
• Is \(\frac{3}{6}\) the same as \(\frac{1}{2}\)?
• Is \(\frac{2}{6}\) the same as \(\frac{1}{3}\)?
• Is \(\frac{2}{4}\) the same as \(\frac{3}{4}\)?

Summary

Equivalent fractions are fractions that are the same size, even if they have different numbers. We can prove this by using area models and number lines.

If two fractions shade the same amount of the same whole, or land on the same point on a number line, they are equivalent fractions.

Comparing Fractions with Shared Denominators

Fractions tell us about equal parts of a whole. When we compare fractions, we are deciding which fraction is greater, which is less, or if they are equal.

In this lesson, we will learn how to compare fractions that have the same denominator. For example, we might compare \(\frac{3}{8}\) and \(\frac{5}{8}\).

First, let’s remember what a fraction means.

• The denominator is the bottom number. It tells how many equal parts the whole is cut into.
• The numerator is the top number. It tells how many of those equal parts we have.

In the fraction \(\frac{3}{8}\):

• the denominator is 8, so the whole is split into 8 equal parts
• the numerator is 3, so we have 3 of those parts

What does “shared denominators” mean?

Shared denominators means the fractions have the same bottom number. That means the parts are the same size.

For example, in \(\frac{2}{6}\) and \(\frac{5}{6}\), both fractions have a denominator of 6. Both wholes are cut into 6 equal parts. Since the parts are the same size, we only need to compare how many parts each fraction has.

The big idea:

When two fractions have the same denominator, the fraction with the greater numerator is the greater fraction.

Why? Because if the pieces are the same size, having more pieces means having more of the whole.

Here is the rule:

For fractions with the same denominator:

$$ \text{Compare the numerators.} $$

• If the numerator is bigger, the fraction is bigger.
• If the numerator is smaller, the fraction is smaller.
• If the numerators are the same, the fractions are equal.

You can also think about a picture.

Imagine two same-sized pizzas, and each pizza is cut into 8 equal slices.

• \(\frac{3}{8}\) means 3 slices
• \(\frac{5}{8}\) means 5 slices

Since each slice is the same size, 5 slices is more than 3 slices. So:

$$ \frac{3}{8} < \frac{5}{8} $$

You can also compare fractions on a number line.

Fractions with the same denominator can be placed on a number line between 0 and 1. The fraction farther to the right is greater.

If we mark eighths on a number line, \(\frac{5}{8}\) is to the right of \(\frac{3}{8}\), so \(\frac{5}{8}\) is greater.

Worked Example 1

Compare \(\frac{1}{4}\) and \(\frac{3}{4}\).

1. Look at the denominators: both are 4.
2. That means the parts are the same size.
3. Compare the numerators: 1 and 3.
4. Since 3 is greater than 1, \(\frac{3}{4}\) is greater.

So:

$$ \frac{1}{4} < \frac{3}{4} $$

Worked Example 2

Compare \(\frac{5}{6}\) and \(\frac{2}{6}\).

1. The denominators are both 6.
2. So the parts are the same size.
3. Compare the numerators: 5 and 2.
4. Since 5 is greater than 2, \(\frac{5}{6}\) is greater.

So:

$$ \frac{5}{6} > \frac{2}{6} $$

Worked Example 3

Compare \(\frac{4}{7}\) and \(\frac{4}{7}\).

1. The denominators are both 7.
2. The numerators are both 4.
3. Since both fractions are exactly the same, they are equal.

So:

$$ \frac{4}{7} = \frac{4}{7} $$

Worked Example 4

Compare \(\frac{6}{10}\) and \(\frac{8}{10}\).

1. The denominators are both 10, so the pieces are the same size.
2. Compare the numerators: 6 and 8.
3. Since 8 is greater than 6, \(\frac{8}{10}\) is greater.

So:

$$ \frac{6}{10} < \frac{8}{10} $$

Helpful steps to remember

• Step 1: Check the denominators.
• Step 2: If the denominators are the same, compare the numerators.
• Step 3: Use \(>\), \(<\), or \(=\).

Math symbols

• \(>\) means greater than
• \(<\) means less than
• \(=\) means equal to

Common mistake to avoid

Do not compare the denominators when the denominators are already the same. If the denominators match, the pieces are already the same size. Then you only need to look at the numerators.

For example, with \(\frac{2}{9}\) and \(\frac{7}{9}\), both fractions are made of ninths. Since 7 parts is more than 2 parts, \(\frac{7}{9}\) is greater.

Try thinking with words

\(\frac{3}{5}\) means “3 fifths.”

\(\frac{4}{5}\) means “4 fifths.”

If both are fifths, then 4 fifths is more than 3 fifths.

Summary

When fractions have the same denominator, they are split into equal parts of the same size. To compare them, look at the numerators. The fraction with more equal parts is greater. If the numerators are the same, the fractions are equal.

Comparing Fractions with Shared Numerators

Fractions tell us about parts of a whole. When we compare fractions, we ask: Which fraction is bigger? or Which fraction is smaller?

In this lesson, we will learn how to compare fractions that have the same numerator. The numerator is the top number in a fraction. For example, in \(\frac{2}{3}\), the numerator is 2.

When fractions have the same numerator, they are talking about the same number of pieces. The important question is: How big is each piece?

Main Idea: If two fractions have the same numerator, the fraction with the smaller denominator is greater, because the whole is cut into fewer parts, so each part is bigger.

For example, compare \(\frac{2}{3}\) and \(\frac{2}{5}\).

Both fractions have a numerator of 2, so both mean 2 pieces. But thirds are bigger pieces than fifths, because when you split a whole into 3 equal parts, each part is bigger than when you split it into 5 equal parts.

So:

$$\frac{2}{3} > \frac{2}{5}$$

Why does this happen?

Think about sharing one sandwich.

• If the sandwich is cut into 3 equal pieces, each piece is pretty big.
• If the sandwich is cut into 5 equal pieces, each piece is smaller.

If you get 2 pieces, you would rather have 2 thirds than 2 fifths, because the thirds are bigger pieces.

Important Rule:

• Same numerator
• Compare the denominators
• Bigger denominator = smaller pieces
• Smaller denominator = bigger pieces

So when the numerators are the same:

$$\text{smaller denominator} \rightarrow \text{larger fraction}$$

and

$$\text{larger denominator} \rightarrow \text{smaller fraction}$$

How to Compare Fractions with the Same Numerator

1. Look at the numerators. Make sure they are the same.
2. Look at the denominators.
3. Remember: more parts means smaller pieces.
4. The fraction with the smaller denominator is greater.

Worked Example 1

Compare \(\frac{1}{2}\) and \(\frac{1}{4}\).

The numerators are both 1, so we compare the denominators: 2 and 4.

A whole cut into 2 equal parts gives bigger pieces than a whole cut into 4 equal parts.

So one half is bigger than one fourth.

$$\frac{1}{2} > \frac{1}{4}$$

Worked Example 2

Compare \(\frac{3}{4}\) and \(\frac{3}{8}\).

The numerators are both 3, so both fractions mean 3 pieces.

Now compare the denominators: 4 and 8.

Fourths are bigger pieces than eighths. So 3 fourths is bigger than 3 eighths.

$$\frac{3}{4} > \frac{3}{8}$$

Worked Example 3

Compare \(\frac{2}{6}\) and \(\frac{2}{3}\).

The numerators are both 2.

Now compare the denominators: 6 and 3.

Sixths are smaller pieces than thirds, because 6 parts make smaller pieces than 3 parts.

So 2 sixths is smaller than 2 thirds.

$$\frac{2}{6} < \frac{2}{3}$$

Worked Example 4

Compare \(\frac{5}{6}\) and \(\frac{5}{7}\).

The numerators are both 5, so we look at the denominators: 6 and 7.

Sixths are bigger pieces than sevenths.

So 5 sixths is bigger than 5 sevenths.

$$\frac{5}{6} > \frac{5}{7}$$

Number Line Thinking

Fractions can also be shown on a number line. Fractions farther to the right are greater.

If you mark \(\frac{2}{3}\) and \(\frac{2}{5}\) on a number line from 0 to 1, \(\frac{2}{3}\) will be farther to the right because 2 thirds is more than 2 fifths.

This matches our rule: when the numerators are the same, the fraction with the smaller denominator is greater.

Try to Think About the Size of the Pieces

• \(\frac{4}{5}\) means 4 big pieces when the whole is cut into 5 parts.
• \(\frac{4}{9}\) means 4 smaller pieces when the whole is cut into 9 parts.

Because fifths are bigger than ninths:

$$\frac{4}{5} > \frac{4}{9}$$

A Quick Comparison Trick

If the top numbers match, look at the bottom numbers.

• Smaller bottom number → bigger fraction
• Bigger bottom number → smaller fraction

Be Careful!

Sometimes students think a bigger denominator means a bigger fraction. But that is not true when the numerators are the same.

For example, \(\frac{2}{8}\) is not bigger than \(\frac{2}{4}\). Eighths are smaller pieces than fourths.

So:

$$\frac{2}{8} < \frac{2}{4}$$

Summary

When two fractions have the same numerator, they have the same number of pieces. To compare them, look at the denominator. The denominator tells how many equal parts the whole is split into.

If the whole is split into more parts, each part is smaller. So with the same numerator, the fraction with the smaller denominator is the larger fraction.

Remember:

$$\text{Same numerator} \Rightarrow \text{smaller denominator means bigger fraction}$$