Foundations of Fractions and Partitioning

Understanding Equal Shares

Sometimes we split a shape into parts. When we do this fairly, each part must be the same size. These parts are called equal shares.

Equal shares are important because they help us understand halves, thirds, and fourths. If the parts are not the same size, they are not equal shares.

Think about sharing a sandwich with a friend. If one person gets a big piece and the other gets a tiny piece, that is not fair. Fair sharing means both pieces are the same size.

What does equal mean?

Equal means the same amount. When a shape is divided into equal shares, every part has the same amount of space inside it.

We are not looking only at the shape of the pieces. We are looking at whether the pieces cover the same amount of area. Sometimes pieces can look different but still be equal if they take up the same amount of space.

How can we tell if shares are equal?

• Look to see if each part is the same size.
• Check whether one part is bigger or smaller than another.
• Ask: “Would this be a fair share?”
• Remember: equal shares must cover the same amount of the whole shape.

Important idea: A shape can be divided in more than one way and still have equal shares.

For example, a rectangle can be split into 2 equal shares by a line down the middle. It can also be split into 4 equal shares by using lines that make 4 same-size parts.

Equal shares and the whole

The whole is the entire shape before it is divided. Equal shares are parts of that whole.

If a whole shape is split into 2 equal shares, each share is one of 2 equal parts. If a whole shape is split into 4 equal shares, each share is one of 4 equal parts.

We can write this idea with numbers:

$$1\text{ whole} = 2\text{ equal shares}$$

or

$$1\text{ whole} = 4\text{ equal shares}$$

What is not an equal share?

If one part is larger than another part, the shares are not equal.

For example, if a circle is cut so one piece is large and one piece is small, those 2 parts are not equal shares.

Even if there are 2 parts, they must be the same size to be equal shares. Just counting the number of parts is not enough.

Worked Example 1

A square is divided by one line straight down the middle. Now there are 2 parts.

Are the 2 parts equal shares?

Yes. The line makes 2 same-size parts. Each part takes up the same amount of space.

So, the square is divided into 2 equal shares.

Worked Example 2

A rectangle is divided by one line, but the line is not in the middle. One part is wide, and one part is narrow.

Are the 2 parts equal shares?

No. One part is bigger than the other. The parts do not have the same amount of area.

So, these are not equal shares.

Worked Example 3

A circle is divided into 4 parts. All 4 parts are the same size.

Are the 4 parts equal shares?

Yes. Since all 4 parts are the same size, they are equal shares.

Each part is one of 4 equal shares of the whole circle.

We can say:

$$4\text{ equal shares make }1\text{ whole}$$

Worked Example 4

A shape is divided into 3 parts. Two parts are small, and one part is large.

Are these equal shares?

No. All shares must be the same size. Since one part is larger, the shape is not divided into equal shares.

Things to remember

1. Equal shares means all parts are the same size.
2. The parts must come from one whole shape.
3. If one part is bigger or smaller, they are not equal shares.
4. Shapes can look different, but the shares can still be equal if they cover the same amount of space.

Try thinking about these

• If a shape has 2 parts, are they always equal shares? No. They must be the same size.
• If a shape has 4 parts and one part is bigger, are they equal shares? No.
• If all parts in a shape are the same size, are they equal shares? Yes.

Summary

Equal shares are fair parts of a whole shape. Every share must have the same amount of space inside it. When you look at a divided shape, always check whether all the parts are the same size. If they are, the shape has equal shares. If they are not, then the shares are not equal.

Partitioning Rectangles into Arrays

Today we will learn how to partition rectangles into arrays. That means we will take a rectangle and divide it into equal-sized squares using straight lines.

When we divide a rectangle this way, we make rows and columns. This helps us count squares carefully and understand how shapes can be split into equal parts.

An array is a set of objects or squares arranged in neat lines. Arrays have:

• Rows that go across
• Columns that go up and down

If a rectangle is partitioned into equal squares, each small square is the same size. That is important because partitioning means splitting a shape into equal shares.

Why do we partition rectangles?

• It helps us organize a shape into equal parts.
• It helps us count squares in a neat way.
• It helps us see rows and columns.
• It is an early step in learning about area.

Main idea: To partition a rectangle into an array, we draw straight lines across and up-and-down so the rectangle is split into equal-sized squares.

Let’s learn the important words.

• Rectangle: a shape with 4 sides and 4 corners
• Partition: to split into parts
• Equal shares: parts that are the same size
• Row: squares going across
• Column: squares going up and down
• Array: rows and columns lined up neatly

When making an array, we want all the little squares to match. If one part is bigger or smaller, the rectangle is not partitioned into equal squares.

Here is a simple way to make an array in a rectangle:

1. Start with a rectangle.
2. Draw straight vertical lines to make columns.
3. Draw straight horizontal lines to make rows.
4. Check that all small parts are the same size.
5. Count the rows and columns.

Worked Example 1

Suppose a rectangle is split into 2 rows and 3 columns. What does the array look like?

We draw 2 strips going across and 3 strips going up and down. That makes equal-sized squares.

The array has:

• 2 rows
• 3 columns

We can count the squares:

Row 1 has 3 squares.

Row 2 has 3 squares.

So there are:

$$3 + 3 = 6$$

The rectangle is partitioned into 6 equal squares.

Worked Example 2

A rectangle has 4 columns and 2 rows. How many equal squares are inside?

Each row has 4 squares. Since there are 2 rows, we count:

$$4 + 4 = 8$$

So the rectangle has 8 equal squares.

We can also say:

• 2 rows of 4
• 4 columns of 2

Both descriptions match the same array.

Worked Example 3

Look at a rectangle partitioned into 3 rows and 3 columns. Is it an array of equal squares?

Yes, if all the small parts are the same size.

Let’s count the squares:

$$3 + 3 + 3 = 9$$

So the rectangle is partitioned into 9 equal squares.

This is a square array because it has the same number of rows and columns.

Worked Example 4

A student draws a rectangle and splits it into parts, but some parts are big and some parts are small. Is this a correct partition into an array?

No. An array for this lesson must be made of equal-sized squares.

If the parts are not equal, then the rectangle is not partitioned correctly into equal shares.

How to check your work

• Are the lines straight?
• Did you make rows across?
• Did you make columns up and down?
• Are all the small squares the same size?
• Can you count the squares by rows or by columns?

Helpful thinking

If you know the number of rows and columns, you can build the array carefully.

For example:

• 1 row and 5 columns makes 5 squares
• 2 rows and 2 columns makes 4 squares
• 3 rows and 4 columns makes 12 squares

You can count by adding the number in each row.

For 3 rows and 4 columns:

$$4 + 4 + 4 = 12$$

That means the rectangle was partitioned into 12 equal squares.

Common mistakes

• Making rows but forgetting columns
• Making columns but forgetting rows
• Drawing parts that are not equal in size
• Counting squares twice

Remember:

• Rows go across.
• Columns go up and down.
• An array is neat and lined up.
• The small squares must be equal in size.

Let’s practice thinking

If a rectangle has 2 rows and 5 columns, each row has 5 squares.

Count them:

$$5 + 5 = 10$$

So there are 10 equal squares.

If a rectangle has 3 rows and 2 columns, each row has 2 squares.

Count them:

$$2 + 2 + 2 = 6$$

So there are 6 equal squares.

Summary

Partitioning rectangles into arrays means dividing a rectangle into equal-sized squares using rows and columns.

Rows go across, and columns go up and down. When all the parts are equal, we can count the squares easily and describe the rectangle as an array.

This helps us understand equal shares and prepares us for learning more about measuring space inside shapes.

Partitioning into Halves

Today we will learn about halves. A half means one of 2 equal parts.

When we split a shape into 2 parts, the parts must be the same size to be called halves. If the parts are not equal, they are not halves.

This is an important idea in fractions. We can write one half like this: \(\frac{1}{2}\).

That means:

• the whole shape is split into 2 equal parts
• we are talking about 1 of those parts

So, one half is:

Main Idea 1: Halves must be equal

Look carefully at a shape when it is divided. Ask yourself:

• Are there 2 parts?
• Are the 2 parts equal?

If the answer to both questions is yes, then the shape is partitioned into halves.

If one part is bigger and one part is smaller, then the shape is not partitioned into halves.

Main Idea 2: Shapes can be split in different ways

A circle or rectangle can be split into halves in more than one way.

• A rectangle can be split up and down into 2 equal parts.
• A rectangle can also be split across into 2 equal parts.
• A circle can be split into 2 equal parts with a line through the middle.

The line does not always have to go the same way. What matters is that the 2 parts are equal.

Main Idea 3: Each half is equal to the other half

When a whole shape is split into halves, each part is the same size as the other part.

Two halves make one whole.

We can say:

This means one half and one half together make one whole shape.

How to check for halves

1. Start with one whole shape.
2. See if it is split into 2 parts.
3. Check whether the 2 parts are the same size.
4. If they are equal, each part is a half.

Worked Example 1: Rectangle split down the middle

A rectangle is divided into 2 parts with a line straight down the middle.

Both parts are the same size.

So this rectangle is partitioned into halves. Each part is \(\frac{1}{2}\) of the rectangle.

Worked Example 2: Rectangle split into unequal parts

A rectangle is divided into 2 parts, but one part is wider than the other.

There are 2 parts, but they are not equal.

So these parts are not halves.

Worked Example 3: Circle split through the center

A circle is divided into 2 equal parts by a line through the middle.

Each part is the same size.

So the circle is partitioned into halves. Each part is \(\frac{1}{2}\) of the circle.

Worked Example 4: Finding the shaded half

A rectangle is split into 2 equal parts. One part is shaded, and one part is not shaded.

Because the rectangle is split into 2 equal parts, the shaded part is one half.

We can say the shaded amount is:

Things to remember

• A half means 1 of 2 equal parts.
• You must have 2 equal parts to make halves.
• If the parts are not the same size, they are not halves.
• Circles and rectangles can both be partitioned into halves.
• Two halves make one whole.

Let’s think

If you see a shape split into 2 pieces, do not guess right away. First ask, “Are the pieces equal?”

If they are equal, each piece is a half. If they are not equal, they are not halves.

Summary

Partitioning into halves means dividing a whole shape into 2 equal parts. Each equal part is called one half, or \(\frac{1}{2}\). We can find halves in circles and rectangles, as long as the two parts are the same size.

Partitioning into Thirds means splitting a shape into 3 equal parts.

When a shape is cut into 3 equal parts, each part is called one third.

We can write one third like this: \(\frac{1}{3}\).

The top number tells us we have 1 part. The bottom number tells us the whole shape is split into 3 equal parts.

Equal parts are very important. Equal means each part is the same size.

If the parts are not the same size, they are not thirds.

We can find thirds in shapes like:

• circles
• rectangles
• other flat shapes

Let’s learn how to look for thirds.

How to make thirds

1. Start with 1 whole shape.
2. Split the shape into 3 parts.
3. Check that all 3 parts are equal.

If all 3 parts match in size, each part is one third.

Here is the idea:

$$1\text{ whole} = 3\text{ thirds}$$

That means 3 thirds make the whole shape again.

Thirds in a rectangle

A rectangle can be split into 3 equal strips.

For example, you can draw 2 straight lines to make 3 equal boxes inside the rectangle.

If the boxes are all the same size, each box is \(\frac{1}{3}\) of the rectangle.

Thirds in a circle

A circle can also be split into 3 equal parts.

The parts might look like 3 matching slices. If all the slices are the same size, each slice is one third of the circle.

Be careful!

A shape having 3 parts does not always mean it is split into thirds.

The 3 parts must be equal.

If one part is bigger or smaller, the shape is not partitioned into thirds.

Worked Example 1

A rectangle is split into 3 equal long parts.

Is each part a third?

Yes.

Why? The rectangle has:

• 3 parts
• all equal size

So each part is one third, or \(\frac{1}{3}\).

Worked Example 2

A circle is cut into 3 slices. Two slices are small, and one slice is big.

Are these thirds?

No.

Why? There are 3 parts, but they are not equal.

So they are not thirds.

Worked Example 3

A rectangle is split into 3 equal boxes. Two boxes are shaded.

How many thirds are shaded?

Each box is \(\frac{1}{3}\).

There are 2 shaded boxes, so 2 thirds are shaded.

We can write that as:

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

So the shaded part is \(\frac{2}{3}\).

Worked Example 4

A circle is partitioned into 3 equal parts. One part is colored.

What fraction of the circle is colored?

There are 3 equal parts, and 1 part is colored.

So the colored part is one third.

We write it as \(\frac{1}{3}\).

How to check for thirds

• Do I see 3 parts?
• Are the 3 parts equal?

If the answer to both questions is yes, the shape is partitioned into thirds.

Important idea

Thirds can look different in different shapes.

A rectangle may have 3 equal strips.

A circle may have 3 equal slices.

Even if the parts look different from shape to shape, they are still thirds if the 3 parts are equal.

Let’s remember

• Thirds means 3 equal parts.
• Each part is called one third.
• One third is written as \(\frac{1}{3}\).
• Parts must be equal to be thirds.
• 3 thirds make 1 whole.

When you see a shape, ask: Is it split into 3 equal parts? If yes, it is partitioned into thirds.

Partitioning into Fourths

Today we will learn how to split shapes into four equal parts. When a shape is split into 4 equal parts, each part is called a fourth. Another name for a fourth is a quarter.

That means:

$$1\text{ whole} = 4\text{ fourths}$$

If the parts are not equal, they are not fourths. Equal means the same size.

What does partitioning mean?

Partitioning means splitting something into parts. In math, we often partition shapes. We have to be careful to make the parts equal when we are making fourths.

Here are important ideas to remember:

• A shape must be split into 4 parts.
• All 4 parts must be equal in size.
• Each equal part is called one fourth or one quarter.

Fourths in a rectangle

A rectangle can be partitioned into fourths in different ways. You can split it into 4 equal long parts, 4 equal short parts, or 4 equal box-like parts. As long as there are 4 equal parts, they are fourths.

For example, a rectangle can be split like this:

• 2 equal rows and 2 equal columns, making 4 equal small rectangles
• 4 equal vertical strips
• 4 equal horizontal strips

Fourths in a circle

A circle can also be partitioned into fourths. One common way is to split the circle into 4 equal pieces, like a pizza cut into 4 same-size slices.

If one slice is bigger or smaller than the others, then the circle is not partitioned into fourths.

How can you check for fourths?

Ask yourself these questions:

1. Are there 4 parts?
2. Are all 4 parts the same size?

If the answer to both questions is yes, then the shape is partitioned into fourths.

Worked Example 1: A rectangle split into 4 equal strips

Look at a rectangle divided into 4 equal vertical parts.

There are 4 parts, and each part is the same size.

So each part is one fourth.

We can say:

$$\text{Each part} = 1\text{ fourth}$$

Worked Example 2: A circle split into 4 equal slices

Imagine a circle cut into 4 same-size pieces.

Because there are 4 equal parts, each piece is one fourth, or one quarter.

If you color 1 piece, you have colored one fourth of the circle.

If you color 2 pieces, you have colored two fourths of the circle.

Worked Example 3: Is this shape partitioned into fourths?

A rectangle is split into 4 parts, but one part is bigger than the others.

Let us check:

• Are there 4 parts? Yes.
• Are they equal in size? No.

So the rectangle is not partitioned into fourths.

Even though there are 4 parts, they must also be equal.

Worked Example 4: Finding fourths in a square

A square is divided by one line across the middle and one line down the middle.

This makes 4 small equal squares.

Because there are 4 equal parts, each small square is one fourth of the whole square.

Things to remember

• Fourths means 4 equal parts.
• Quarters means the same thing as fourths.
• Different shapes can be partitioned into fourths.
• The parts do not have to look exactly the same way, but they must be the same size.

Try thinking about these

• If a pizza is cut into 4 equal slices, each slice is one fourth.
• If a sandwich is cut into 4 equal pieces, each piece is one quarter.
• If a brownie pan is cut into 4 equal pieces, each piece is one fourth.

Summary

Partitioning into fourths means splitting a shape into 4 equal parts. Each part is called one fourth or one quarter. To know if a shape is partitioned into fourths, make sure there are exactly 4 parts and all of them are the same size.

Inverse Relationship of Shares to Size

Today we will learn a very important idea about sharing shapes.

When we cut a whole shape into more equal shares, each share gets smaller. When we cut a whole shape into fewer equal shares, each share is bigger.

This is called an inverse relationship. That is a big way of saying: the number of shares goes up, but the size of each share goes down.

Let’s think about one whole sandwich, one whole cookie, or one whole rectangle. We start with 1 whole.

If we split the whole into 2 equal shares, each share is large. If we split the same whole into 4 equal shares, each share is smaller. The whole did not get bigger. We just made more pieces, so each piece had to be smaller.

Main Idea

• A whole is one complete shape or object.
• Equal shares means all the pieces are the same size.
• If the whole stays the same, more equal shares mean smaller pieces.
• If the whole stays the same, fewer equal shares mean bigger pieces.

We can name shares with fraction words.

• 2 equal shares are called halves.
• 3 equal shares are called thirds.
• 4 equal shares are called fourths.

Look at how the size changes:

One whole cut into 2 equal shares gives bigger pieces than one whole cut into 4 equal shares.

In math, we can write:

This means one half is bigger than one fourth when they come from the same size whole.

Why does this happen?

Imagine sharing 1 pizza.

• If 2 kids share it equally, each kid gets a big piece.
• If 4 kids share the same pizza equally, each kid gets a smaller piece.

The pizza is still just 1 whole pizza. More equal shares mean each share must be smaller.

Worked Example 1: Split a rectangle into 2 and 4 equal shares

Suppose you have 1 rectangle.

1. First, divide it into 2 equal parts.
2. Each part is one half, or \(\frac{1}{2}\).
3. Now think about the same size rectangle divided into 4 equal parts.
4. Each part is one fourth, or \(\frac{1}{4}\).

Which piece is bigger: \(\frac{1}{2}\) or \(\frac{1}{4}\)?

Answer: \(\frac{1}{2}\) is bigger.

Why? Because 2 shares is fewer than 4 shares, so each share is larger.

Worked Example 2: Sharing one brownie

One brownie is cut into 3 equal shares. Another same-size brownie is cut into 6 equal shares.

Which share is bigger?

1. The first brownie has 3 equal shares.
2. Each share is one third, \(\frac{1}{3}\).
3. The second brownie has 6 equal shares.
4. Each share is one sixth, \(\frac{1}{6}\).

Answer: \(\frac{1}{3}\) is bigger than \(\frac{1}{6}\).

Why? The whole brownies are the same size. More shares, like 6, make smaller pieces than fewer shares, like 3.

Worked Example 3: Color one share

A circle is split into 4 equal shares. A same-size circle is split into 8 equal shares.

If you color 1 share in each circle, which colored part is bigger?

1. In the first circle, 1 share is \(\frac{1}{4}\).
2. In the second circle, 1 share is \(\frac{1}{8}\).
3. Compare 4 shares to 8 shares.

Answer: The colored part in the circle split into 4 equal shares is bigger.

Why? Fourths are bigger than eighths when the wholes are the same size.

Worked Example 4: True or false?

Sentence: “If I cut the same cake into more equal pieces, each piece gets bigger.”

Answer: False.

Correct idea: If you cut the same cake into more equal pieces, each piece gets smaller.

Things to remember

• The whole must be the same size when we compare shares.
• The shares must be equal shares.
• More equal shares means smaller parts.
• Fewer equal shares means bigger parts.

Try thinking about these questions:

• Which is bigger from the same whole: \(\frac{1}{2}\) or \(\frac{1}{3}\)?
• Which is smaller from the same whole: \(\frac{1}{4}\) or \(\frac{1}{8}\)?
• If a shape is cut into 5 equal shares instead of 2 equal shares, do the pieces get bigger or smaller?

Answers:

• \(\frac{1}{2}\) is bigger than \(\frac{1}{3}\).
• \(\frac{1}{8}\) is smaller than \(\frac{1}{4}\).
• The pieces get smaller.

Quick Summary

A whole can be divided into equal shares. If the whole stays the same, cutting it into more equal shares makes each share smaller. Cutting it into fewer equal shares makes each share bigger.

So remember: more shares, smaller pieces; fewer shares, bigger pieces.

Equivalent Area in Different Shapes

Sometimes two pieces do not look the same, but they can still cover the same amount of space. In math, the amount of space inside a shape is called its area.

When we talk about fractions, we often split a whole shape into equal shares. Equal shares mean each part has the same area, even if the parts have different shapes.

This is an important idea: parts can look different and still be equal. If two parts take up the same amount of the same whole, then they are equal shares.

For example, one half of a rectangle might look like a tall piece, and another half might look like a wide piece. They are still both one-half if each piece covers the same amount of the whole.

What to remember:

• Area means how much space is inside a shape.
• Equal shares must have the same area.
• Equal shares do not always have to be the same shape.
• If the whole is the same size, the same fraction means the same amount of area.

Let’s think about one whole shape.

Imagine a rectangle. If we divide it into 2 equal shares, each share is one-half.

We can cut it straight up and down, or straight across. The pieces will look different, but each piece can still be one-half of the whole.

That means:

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

Even when the pieces look different, they are equal if they cover the same area of the same whole.

How can we tell if shares are equal?

1. Look at the whole. Is it the same whole shape or same-size whole?
2. Look at the parts. Do the parts cover the same amount of space?
3. Do not only look at the shape of the part. A skinny part and a wide part can still have the same area.

Worked Example 1: Two halves that look different

A square is split into 2 equal shares.

• Picture A: The square is cut from top to bottom.
• Picture B: The square is cut from one corner to the opposite corner.

In Picture A, the halves are rectangles. In Picture B, the halves are triangles.

Do both pictures show halves?

Yes. Both pictures split the same whole square into 2 equal shares. The pieces look different, but each piece covers half of the square.

So each part is:

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

Worked Example 2: Fourths in different shapes

A rectangle is split into 4 equal shares in two ways.

• Picture A: 4 long strips
• Picture B: 4 small boxes

Do the shares have to look the same to be fourths?

No. If the whole rectangle is the same size and each part has the same area, then each part is one-fourth.

Each share is:

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

The strips and the boxes may look different, but they can still be equal shares of the same whole.

Worked Example 3: Which parts are equal?

A rectangle is divided into 2 parts.

• Part A is a large piece.
• Part B is a small piece.

Are these equal shares?

No. The two parts do not cover the same amount of space. One part has more area than the other.

So these are not halves.

This teaches us that equal shares must have equal area, not just any two parts.

Worked Example 4: Same fraction, different shape

Two same-size circles are each split into 2 equal shares.

• In the first circle, one half is the top part.
• In the second circle, one half is the left part.

Do both shaded parts show the same amount?

Yes. Each shaded part is one-half of the same-size whole circle.

They may look different because they are in different places, but the area is the same.

So:

$$\frac{1}{2} \text{ of one same-size circle } = \frac{1}{2} \text{ of another same-size circle}$$

Tips for students

• Ask, “Is the whole the same size?”
• Ask, “Do the parts have the same area?”
• Do not decide by shape alone.
• Equal fractions of the same whole have equal area, even if they look different.

Let’s practice thinking

• If two pieces are both one-half of the same whole, they have the same area.
• If two pieces are both one-fourth of the same whole, they have the same area.
• If one piece is bigger than another, they are not equal shares.

Summary

Equivalent area means two parts cover the same amount of space.

When shapes are split into equal shares, the shares can look different but still be equal. The most important thing is not how the parts look. The most important thing is that they have the same area.

So, identical fractions of the same whole can have different shapes and still be equal.