Chapter Notes: Fractions

Table of Contents
1. Introduction to Fractions
2. Comparison of Fractions with Different Denominators
3. Playing with a Grid
4. Fun with Fraction Kit
5. Making Equivalent Fractions
6. Comparing Fractions-Same Denominator
7. Comparing Fractions - Same Numerator
8. Fractions Greater Than 1
9. Comparing Fractions With Reference to 1
10. Comparing Fractions with Reference to 1/2
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Introduction to Fractions

A fraction is a way of representing a part of a whole.

When a complete item is split into equal sections, each section is known as a fraction of the whole.

It has two parts:

• Numerator (the top number): shows how many equal parts are taken.

• Denominator (the bottom number): shows into how many equal parts the whole is divided.

Comparison of Fractions with Different Denominators

• Fractions only make sense when we know what the whole is.

• Two fractions like $\frac{1}{2}\backslash tfrac\{1\}\{2\}$ and $\frac{1}{3}\backslash tfrac\{1\}\{3\}$ can be directly compared only if they come from the same whole.

• This means that when fractions have different denominators, you cannot compare them directly. To make them comparable, we convert them into equivalent fractions with the same denominator.

Let's understand this using an example 

Tamanna is a fifth-grade student. She has two chocolates of different sizes

She says that:

We can see that chocolate 2 is a lot bigger than chocolate 1; they are of different sizes, so we can't really compare them.

Let's learn about the rule for comparing fractions!

To compare fractions correctly:

• The whole must be the same.

• Example: Compare $\frac{1}{2}\backslash tfrac\{1\}\{2\}$ and $\frac{1}{3}\backslash tfrac\{1\}\{3\}$ of the same chocolate. Here, $\frac{1}{2}\backslash tfrac\{1\}\{2\}$ is always bigger.

• But if the wholes are different, the result depends on the size of the wholes.

Therefore, before comparing fractions, make sure they are taken from the same whole object.

Playing with a Grid

Fractions can sometimes feel tricky, but grids make them easy to see and understand.
By shading parts of a grid, we can visualise how much of a whole a fraction represents.

What is a grid?

• A grid is divided into equal parts (like boxes in a chocolate bar).Grid A

• Each small box represents one equal part of the whole.

• When we shade some parts, we show a fraction of the total.

Example 1: Shade 1/3 on the grid A red 

• Divide a grid into 3 equal parts.

• Shade 1 out of 3 parts.

• This shows 1/3, or "one-third of the whole."

Example 2:  Shade 2/3 on the grid A black

• Divide a grid into 3 equal parts.

• Shade 2 out of 3 parts.

This shows 2/3, or "two-thirds of the whole."

Understanding Equivalent Fractions 

These are fractions that represent the same value or part of a whole, even though they have different numerators and denominators.

For example:

Let's Check!

Look at the grid below. The grid is divided into 6 equal parts.

Is  equal to ? Explain using the shaded parts.

Solution:

Step 1: Shade 2 out of 6 parts of the grid

Step 2: Now, let's divide the same grid into 3 equal parts.

Step 3: Shade 1 out of 3 parts of the grid

Now, compare the shaded regions. In both cases, the same amount of the bar is shaded.

Fractions that represent the same portion of a whole are called Equivalent Fractions.

Fun with Fraction Kit

Using a fraction kit helps us understand how fractions work. A fraction kit contains strips divided into equal parts.

• A whole can be divided into many equal parts: halves, thirds, fourths, fifths, sixths, etc.

• The denominator tells us how many equal parts the whole is divided into.

• Adding all the equal parts together gives 1 whole

Making a Whole with Fractions

• A whole can be made using smaller fraction pieces.

• Example: If we use $\frac{1}{5}\backslash tfrac\{1\}\{5\}$ pieces, we need 5 pieces to make 1 whole.

Combining Different Fraction Pieces

Sometimes, we can use different fractions together to form a whole.

• Example: One piece of 1/2 and two pieces of 1/4 make a whole

• This shows us: 

• A half ($\frac{1}{2}\backslash tfrac\{1\}\{2\}$) can be divided into two quarters ($\frac{1}{4}+\frac{1}{4}\backslash tfrac\{1\}\{4\}\; +\; \backslash tfrac\{1\}\{4\}$).

• Therefore, (equivalent fractions)

Making Equivalent Fractions

To make equivalent fractions, we change the numerator (top number) and denominator (bottom number) in the same way.

This is done by multiplying or dividing both by the same number.

1. Using Multiplication

If we multiply the numerator and denominator of a fraction by the same number, the fraction's value does not change.$\backslash tfrac\{1\}\{2\}\; =\; \backslash tfrac\{2\}\{4\}\; =\; \backslash tfrac\{4\}\{8\}$

2. Using Division (Simplifying)

If the numerator and denominator have a common factor, we can divide both by the same number.$\backslash tfrac\{3\}\{4\}$

Sameer Discovers Equivalent Fractions

Sameer shaded one-third () of some shapes.

Case 1: When he split the shapes into smaller equal parts by drawing lines, he found fractions like:

• , , - all exactly cover the same shaded area as 1/3.

• This means  =  =  =

• These are equivalent fractions.

Example: Write the next 4 fractions equivalent to 2/5.

Comparing Fractions-Same Denominator

• Fractions have two parts: numerator (top number) and denominator (bottom number).

• When fractions have the same denominator, it means the whole is divided into the same number of equal parts.

• To compare such fractions, we only need to look at the numerators.

• In such cases, the fraction with the larger numerator is the greater fraction.

Example: Sevi and Shami divided a piece of chikki between themselves. Sevi ate $\frac{1}{3}\backslash tfrac\{1\}\{3\}$ and Shami ate the rest, that is, $\frac{2}{3}\backslash tfrac\{2\}\{3\}$. Who ate more?

The whole chikki is divided into 3 equal parts (so the denominator is 3).

• Sevi ate 1 part → $\frac{1}{3}\backslash tfrac\{1\}\{3\}$

• Shami ate 2 parts → $\frac{2}{3}\backslash tfrac\{2\}\{3\}$

• Since the denominators are the same (3), we only need to compare the numerators.

• Numerator for Sevi = 1

• Numerator for Shami = 2

• Clearly, 2 > 1.

Comparing Fractions - Same Numerator 

• When fractions have the same numerator (same number of parts), but different denominators (different total parts), the bigger fraction is the one with the smaller denominator.

• This is because smaller parts make each piece bigger.

Example: Between Sevi and Shami, can you tell who ate more?

Rule: When numerators are the same, the fraction with the smaller denominator is bigger.

Comparing who ate more paratha

Fractions Greater Than 1

We know, a fraction tells us how many parts of a whole we have. 
Let's take an example. For instance, if a paratha (a soft flatbread) is cut into equal parts:

• $\frac{1}{2}$ (one-half) means the paratha is cut into 2 equal parts, and you have one of those parts.

• $\frac{1}{4}$ (one-fourth or a quarter) means the paratha is cut into 4 equal parts, and you have one of those parts.

Sometimes, you can eat more than one whole paratha. This happens when you eat many pieces of paratha, so the total amount is more than one whole. 

This is called a "fraction greater than 1."

Let's see this with some examples.

Example 1: Imagine Raman's father cuts each paratha into halves, that is, 2 equal parts.

• If Maa took 5 pieces of $\frac{1}{2}$ paratha, how many whole parathas did she eat? 

• Since 2 halves make 1 whole paratha.

• Then 5 halves are:

So, Maa ate 2 and a half parathas.

Let's Use a Number Line to Understand

• If you draw a line and divide the space between 0 and 1 into 2 equal parts (because halves),

• Then each part is $\frac{1}{2}$.

• Moving 5 steps of $\frac{1}{2}$ along the line gets you to 2 and $\frac{1}{2}$.

Let's check Radhika's halves

Radhika took 6 pieces of $\frac{1}{2}$ parathas.

$\frac{6}{2}=\mathrm{3\; parathas}$

Example 2: Now imagine some day, all parathas were cut into 4 pieces each (fourths).

• Dadaji took 9 pieces of $\frac{1}{4}$ paratha. 

So, he ate 2 and one-fourth parathas.

Example 4: Sharing Pizzas

The family ordered 2 pizzas, and each pizza is cut into 3 equal slices.

• Total slices = 2 pizzas × 3 slices = 6 slices

• 6 family members need 1 slice each.

• Dadiji and Dadaji gave their slices to Raman.

• Maa and Baba gave theirs to Radhika.

That means:

Each slice is:

$\frac{1}{3}(one slice out of 3)$ (one slice out of 3)

Calculating Raman's Share

Raman gets his slice + 2 slices from Dadiji and Dadaji = 3 slices

• Raman's total pizza = 1
  

• Representation on the number line:

• Therefore,
  

Now Raman couldn't finish and was left with one slice, so he gave it to Radhika 

Calculating Radhika's Share

Radhika gets her slice + 2 slices from Maa and Baba + 1 slice from Raman = 4 slices

• Radhika's total pizza = 1$\frac{1}{3}$

  Representation on the number line:

  

Therefore,

Comparing Fractions With Reference to 1

• When comparing fractions, check whether they are less than 1 or greater than 1.

• Improper fractions (like $\frac{8}{6}\backslash tfrac\{8\}\{6\}$) are always more than 1 whole.

Let us compare some more fractions. Between Sevi and Shami, can you tell who ate less?

• Sevi says: "I ate $\frac{7}{8}\backslash tfrac\{7\}\{8\}$paratha yesterday evening."

• Sami  says: "I ate $\frac{8}{6}\backslash tfrac\{8\}\{6\}$paratha yesterday evening."

Let's take a closer look.

• $\frac{7}{8}\backslash tfrac\{7\}\{8\}$ means 7 out of 8 equal pieces → this is less than 1 whole paratha.

• $\frac{8}{6}$ means 8 out of 6 equal pieces → this is more than 1 whole paratha (since 6 parts make 1 paratha, and 2 parts are extra).

On comparing, we get 

The Sami ate more paratha than the Sevi. 

Comparing Fractions with Reference to 1/2

• Fractions can often be compared by relating them to $\frac{1}{2}$, 1, or other simple benchmarks.

• Here, we will use $\frac{1}{2}\backslash tfrac\{1\}\{2\}$ as a reference to decide which fraction is greater.

Who do you think ate more paratha?

• The Sevi says: "I ate $\frac{5}{8}\backslash tfrac\{5\}\{8\}$ paratha yesterday evening."

• The Sami says: "I ate $\frac{3}{6}\backslash tfrac\{3\}\{6\}$ paratha yesterday evening."

Let's take a closer look.

1 means half of the whole.

• $\frac{3}{6}=\frac{1}{2}\backslash tfrac\{3\}\{6\}\; =\; \backslash tfrac\{1\}\{2\}$  (since 3 parts out of 6 equal parts = half).

• $\frac{5}{8}\backslash tfrac\{5\}\{8\}$ is more than half (because 4 out of 8 parts = half, and here 5 parts are taken).

Therefore,