Fractions and Partitioning

Partitioning Shapes into Equal Shares

Today we will learn how to split shapes into equal shares.

When we partition a shape, we cut or divide it into parts.

Equal shares means each part is the same size. No part is bigger or smaller than the others.

We can partition shapes like circles and rectangles into equal shares.

This helps us understand early fraction ideas, like sharing fairly.

Why equal shares matter

If 2 children share 1 sandwich fairly, each child should get the same amount.

If one child gets a big piece and the other gets a small piece, the shares are not equal.

Equal shares are about fair sharing.

Main ideas to remember

• A shape can be split into 2 equal shares.
• A shape can be split into 4 equal shares.
• Equal shares must cover the same amount of space.
• The parts can look different ways, but they must still be the same size.
• More shares means each share is smaller.

For example, 1 rectangle split into 2 equal shares has bigger parts than the same rectangle split into 4 equal shares.

If we compare the shares:

Ways to split a rectangle

A rectangle can be split into 2 equal shares by drawing one line down the middle.

You can draw the line up and down or side to side. Both ways can make equal shares if the two parts are the same size.

A rectangle can also be split into 4 equal shares by drawing 2 lines that make 4 same-size parts.

Ways to split a circle

A circle can be split into 2 equal shares by drawing a line through the middle.

A circle can be split into 4 equal shares by drawing 2 lines through the middle to make 4 same-size parts.

Just like cutting a pizza fairly, each piece should be the same size.

How to check for equal shares

1. Look at the whole shape.
2. Count how many parts it has.
3. Ask, “Are all the parts the same size?”
4. If yes, the shape has equal shares.
5. If no, the shape does not have equal shares.

Worked Example 1: Split a rectangle into 2 equal shares

Suppose you have 1 rectangle.

You draw 1 line right through the middle from top to bottom.

Now the rectangle has 2 parts.

If both parts are the same size, then the rectangle is partitioned into 2 equal shares.

We can say:

Worked Example 2: Is this fair?

A circle is split into 2 parts, but one part is big and one part is small.

Are these equal shares?

No.

Even though there are 2 parts, they are not the same size.

So the circle is not partitioned into equal shares.

Worked Example 3: Split a circle into 4 equal shares

Take 1 circle.

Draw 1 line through the middle from top to bottom.

Then draw 1 line through the middle from side to side.

Now the circle has 4 parts.

If all 4 parts are the same size, then the circle is partitioned into 4 equal shares.

We can say:

Worked Example 4: Which has smaller shares?

Look at 2 same-size rectangles.

• Rectangle A is split into 2 equal shares.
• Rectangle B is split into 4 equal shares.

Which rectangle has smaller shares?

Rectangle B has smaller shares.

Why?

Because the same whole shape is being split into more equal parts.

When the number of equal shares gets bigger, each share gets smaller.

Let’s practice thinking

Ask yourself these questions when you see a shape:

• How many parts are there?
• Are all the parts equal in size?
• Is the sharing fair?
• Would more parts mean smaller pieces?

Important note

Equal shares do not always have to look exactly the same shape, but in 1st Grade we usually look at simple shapes cut into matching parts.

The most important idea is that each share covers the same amount of the whole shape.

Summary

Partitioning means splitting a shape into parts.

Equal shares are parts that are the same size.

Circles and rectangles can be split into 2 equal shares or 4 equal shares.

When a shape is split into more equal shares, each share is smaller.

Always check: Are all the parts the same size? If yes, they are equal shares.

Recognizing Unequal Shares

Sometimes we split a shape into parts so people can share it. When the parts are the same size, the shares are equal shares. When the parts are not the same size, the shares are unequal shares.

Equal shares are a fair share. Unequal shares are not fair, because one part is bigger or smaller than another part.

Let’s learn how to tell if shares are equal or unequal.

What does unequal mean?

Unequal means not the same. If one piece is big and one piece is small, the pieces are unequal.

You can look at a shape and ask:

• Are the parts the same size?
• Does each person get a fair share?

If the answer is no, then the shares are unequal shares.

How to recognize unequal shares

1. Look at all the parts.
2. Compare the sizes.
3. Ask: Are they the same size?
4. If they are not the same size, they are unequal shares.

You do not need to count only the number of parts. A shape can have 2 parts, 3 parts, or more parts. What matters is whether the parts are the same size.

For example, a shape cut into 2 parts is not always equal. One part could be bigger than the other part.

Important idea

A shape can be split into parts, but if the parts are different sizes, they are unequal shares.

Equal shares means:

• same size
• fair

Unequal shares means:

• different sizes
• not fair

Worked Example 1

A sandwich is cut into 2 pieces. One piece is big, and one piece is small.

Are these equal shares or unequal shares?

Step 1: Look at the 2 pieces.

Step 2: Compare their sizes.

They are not the same size.

Answer: These are unequal shares.

Worked Example 2

A rectangle is split into 2 parts right down the middle. Both parts match.

Are these equal shares or unequal shares?

Step 1: Look at both parts.

Step 2: Compare their sizes.

They are the same size.

Answer: These are equal shares, so they are not unequal shares.

Worked Example 3

A circle is split into 3 pieces. Two pieces are small, and one piece is large.

Are these equal shares or unequal shares?

Step 1: Look at all 3 pieces.

Step 2: Compare the sizes.

One piece is bigger than the others.

The pieces are different sizes.

Answer: These are unequal shares.

Worked Example 4

A brownie pan is cut into 4 pieces. All 4 pieces are the same size.

Are these equal shares or unequal shares?

Step 1: Look at the 4 pieces.

Step 2: Compare their sizes.

All 4 pieces are the same size.

Answer: These are equal shares.

Things to remember

• More pieces does not mean equal shares.
• Only looking at the number of parts is not enough.
• You must check the size of the parts.
• If the sizes do not match, the shares are unequal.

Try thinking about these

• If 2 kids share a cookie, but one kid gets a big piece and one kid gets a tiny piece, is that fair?
• If a shape has 4 parts, but one part is larger than the others, are the shares equal?
• If all parts match in size, are the shares equal or unequal?

Quick check answers

• Big piece and tiny piece: unequal shares
• One part larger than the others: unequal shares
• All parts match in size: equal shares

Summary

Unequal shares are parts that are not the same size. When shares are unequal, the split is not fair. To recognize unequal shares, look at the parts and compare their sizes. If the parts are different sizes, they are unequal shares.

Identifying Halves

Today we will learn about halves.

A whole is one full shape or one full object. When we split a whole into 2 equal parts, each part is called a half.

Equal means the parts are the same size. If the two parts are not the same size, they are not halves.

We can show one whole split into 2 equal parts like this:

$$1\ \text{whole} = 2\ \text{halves}$$

Each half is one of the two equal parts:

$$\text{half} + \text{half} = 1\ \text{whole}$$

How do we find halves?

• Start with one whole shape or object.
• Split it into 2 parts.
• Check: Are the 2 parts equal?
• If yes, each part is a half.
• If no, they are not halves.

You can find halves in many shapes.

• A circle can be split into 2 equal parts.
• A square can be split into 2 equal parts.
• A rectangle can be split into 2 equal parts.

The line does not always have to go the same way. A shape can be split up-and-down or side-to-side. What matters is that the 2 parts are the same size.

Worked Example 1

A sandwich is cut into 2 pieces. The 2 pieces are the same size.

Are the pieces halves?

Yes. The sandwich was cut into 2 equal parts, so each piece is a half.

Worked Example 2

A rectangle is split into 2 parts. One part is big, and one part is small.

Are the parts halves?

No. The parts are not equal, so they are not halves.

Worked Example 3

Look at a circle split into 2 equal parts.

How many halves are in the whole circle?

There are 2 halves in one whole circle.

We can say:

$$2\ \text{halves} = 1\ \text{whole}$$

Worked Example 4

A square is folded down the middle so both sides match exactly.

What does the fold make?

The fold makes 2 equal parts. Each part is a half.

Things to remember about halves

• Halves always mean 2 parts.
• The 2 parts must be equal.
• Two halves make 1 whole.
• If the parts are not the same size, they are not halves.

Let's think together

If you cut an apple into 2 same-size pieces, each piece is a half.

If you cut an apple into 2 different-size pieces, the pieces are not halves.

So when you look for halves, always ask:

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

If the answer to both questions is yes, then the parts are halves.

Summary

A half is one of 2 equal parts of a whole. To identify halves, check that the whole is split into exactly 2 parts and that both parts are the same size. Remember: 2 halves make 1 whole.

Identifying Fourths (Quarters)

Today we will learn about fourths, which are also called quarters.

A whole means one full shape or one full object. When a whole is split into 4 equal parts, each part is called a fourth or a quarter.

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

You can think of it like this:

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

And each one part is:

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

This is read as one fourth or one quarter.

What makes fourths?

• There must be 4 parts.
• All 4 parts must be equal.
• The 4 parts together make 1 whole.

If a shape has 4 parts but one part is bigger or smaller, the parts are not fourths.

Look for these clues:

• Count the parts. Are there 4?
• Check the size. Are they equal?
• If yes, each part is a fourth.

Example 1: A square split into 4 same small squares

Imagine a big square cut into 4 little squares that are all the same size.

• There are 4 parts.
• All parts are equal.

So each small part is one fourth.

We can say:

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

All together, the 4 fourths make 1 whole.

Example 2: A circle split into 4 equal slices

Think of a pie cut into 4 same-size slices.

• There are 4 slices.
• Each slice is the same size.

Each slice is one quarter of the whole pie.

If you have 1 slice, you have \(\frac{1}{4}\).

If you have all 4 slices, you have the whole pie again.

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

Example 3: Are these fourths?

A rectangle is split into 4 parts, but 2 parts are big and 2 parts are small.

• Yes, there are 4 parts.
• No, they are not equal.

So these parts are not fourths.

Remember: 4 parts is not enough. The parts must be equal, too.

Example 4: How many fourths make a whole?

Let’s count the equal parts:

1. 1 fourth
2. 2 fourths
3. 3 fourths
4. 4 fourths

When we get to 4 fourths, we have 1 whole.

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

How to identify fourths

1. Start with 1 whole shape.
2. See if it is split into 4 parts.
3. Check if all 4 parts are the same size.
4. If they are, each part is a fourth or quarter.

Words to remember

• Whole: one full shape or object
• Equal parts: parts that are the same size
• Fourth: one of 4 equal parts
• Quarter: another word for fourth

Let’s think together

If a sandwich is cut into 4 equal pieces, each piece is one quarter.

If a sandwich is cut into 4 pieces that are not the same size, the pieces are not quarters.

Summary

A whole can be divided into 4 equal parts. Each equal part is called a fourth or a quarter. Four fourths make 1 whole.

Share Size Relative to Total Shares

Today we will learn about sharing a whole into equal parts.

A whole is one full thing, like one pizza, one sandwich, or one cookie.

Sometimes we cut or split a whole into equal shares. Equal shares are parts that are the same size.

Here is the big idea:

If a whole is split into more equal shares, each share gets smaller.

If a whole is split into fewer equal shares, each share gets bigger.

Let’s think about one cookie.

• If we split the cookie into 2 equal shares, each piece is pretty big.
• If we split the same cookie into 4 equal shares, each piece is smaller.

The cookie did not get bigger. It is still the same whole. We just made more pieces, so each piece is smaller.

We can show this idea with numbers:

One whole split into 2 equal shares gives shares of size \(\frac{1}{2}\).

One whole split into 4 equal shares gives shares of size \(\frac{1}{4}\).

And \(\frac{1}{4}\) is smaller than \(\frac{1}{2}\).

So:

• 2 equal shares  bigger pieces
• 4 equal shares  smaller pieces

Important: This only works when the shares are equal. If the parts are not the same size, they are not equal shares.

Let’s learn how to think about it.

1. Look at the whole.
2. Count how many equal shares it is split into.
3. Ask: Are there more shares or fewer shares?
4. Remember: More equal shares means smaller parts.

Worked Example 1

One sandwich is cut into 2 equal shares.

How big is each share?

Each share is big, because there are only 2 equal shares.

Each part is one out of 2 equal parts:

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

So each share is one-half.

Worked Example 2

Now take the same size sandwich. This time it is cut into 4 equal shares.

How big is each share now?

Each share is smaller than before, because the same whole is split into more equal shares.

Each part is one out of 4 equal parts:

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

So each share is one-fourth.

Compare the shares:

• \(\frac{1}{2}\) is bigger
• \(\frac{1}{4}\) is smaller

Worked Example 3

A pizza is cut in two different ways:

• Pizza A is cut into 3 equal shares.
• Pizza B is cut into 6 equal shares.

Which pizza has the smaller share size?

Answer: Pizza B has the smaller share size.

Why?

Because 6 equal shares is more than 3 equal shares. When we make more equal shares from the same whole, each share gets smaller.

So:

• 3 equal shares  bigger pieces
• 6 equal shares  smaller pieces

Worked Example 4

Jada and Ben each have one same-size brownie.

• Jada cuts her brownie into 2 equal shares.
• Ben cuts his brownie into 5 equal shares.

Whose one share is bigger?

Answer: Jada’s one share is bigger.

Why?

Her brownie is split into fewer equal shares. Fewer equal shares means bigger pieces.

Ben has more equal shares, so each of his pieces is smaller.

Let’s compare with simple ideas

• 1 whole split into 2 equal shares  bigger shares
• 1 whole split into 3 equal shares  smaller shares
• 1 whole split into 4 equal shares  even smaller shares

As the number of equal shares goes up, the size of each share goes down.

This means they change in opposite ways:

• More shares  smaller share size
• Fewer shares  bigger share size

Watch out!

Do not just count pieces if the pieces are not equal.

If one cake is cut into 4 pieces, but one piece is giant and one piece is tiny, those are not equal shares.

To talk about fair share size, the parts must be the same size.

Try thinking with questions

• Is it one whole?
• Are the parts equal?
• How many equal shares are there?
• If there are more shares, will each share be bigger or smaller?

Practice ideas

Imagine one apple pie.

• If it is cut into 2 equal shares, each share is bigger.
• If it is cut into 8 equal shares, each share is smaller.

Imagine one paper rectangle.

• Fold it into 2 equal parts. Each part is big.
• Fold the same size rectangle into 4 equal parts. Each part is smaller.

Summary

A whole can be split into equal shares.

When the whole is split into more equal shares, each share is smaller.

When the whole is split into fewer equal shares, each share is bigger.

Always make sure the parts are equal before you compare share sizes.