Welcome to the World of Fractions!
Hello Super Mathematicians!
Have you ever had to share a pizza with your friends or a chocolate bar with your family? If you have, you've already used fractions! Fractions are just a way of talking about parts of a whole thing. They help us make sure everything is shared fairly.
In these notes, we're going to learn what fractions are and how to do something really cool with them: adding and subtracting them. Don't worry if this sounds tricky at first, we'll go step-by-step and use fun examples. You'll be a fraction expert in no time!
What is a Fraction Anyway?
Imagine you have one whole pizza. A fraction is just one or more slices of that pizza. Every fraction has two parts: a top number and a bottom number.
The Bottom Number: Denominator
The number at the bottom of a fraction is called the denominator. It tells us how many equal slices the whole pizza has been cut into.
Example: If our pizza is cut into 8 equal slices, the denominator is 8.
Memory Trick: Think 'd' for denominator and 'd' for 'down' (it's the number down below!).
The Top Number: Numerator
The number at the top of a fraction is called the numerator. It tells us how many of those slices we are talking about.
Example: If you eat 3 of the 8 slices, the numerator is 3. So, you ate $$ \frac{3}{8} $$ of the pizza!
Memory Trick: The numerator 'numerates' or 'counts' the pieces you have.
Let's see it!
This fraction is three-eighths:
$$ \frac{3}{8} $$Here, 3 is the numerator and 8 is the denominator.
Which Fraction is Bigger? (Comparing Fractions)
Before we add or subtract, let's see how to tell which fraction is bigger when they are in the same 'family'. Fractions are in the same family if they have the same denominator.
The Simple Rule
When the denominators (bottom numbers) are the same, you only need to look at the numerators (top numbers). The fraction with the bigger numerator is the bigger fraction!
Example: Which is bigger, $$ \frac{5}{8} $$ or $$ \frac{3}{8} $$?
Think about our pizza cut into 8 slices. Is having 5 slices more than having 3 slices? Yes! So...
$$ \frac{5}{8} $$ is bigger than $$ \frac{3}{8} $$
Key Takeaway
For fractions with the same denominator, bigger top number = bigger fraction. Easy peasy!
Putting the Pieces Together: Adding Fractions
Adding fractions is like getting more slices of pizza. But there is one very important rule!
The Golden Rule of Adding Fractions
You can only add fractions that have the same denominator. Think of it like this: you can add 2 apples and 3 apples to get 5 apples. But you can't add 2 apples and 3 bananas to get '5 apple-bananas'. It's the same with fractions! The denominator tells you what 'family' of pieces you have (like eighths, or fourths, or sixths).
How to Add: Step-by-Step
It’s as easy as 1-2-3!
1. Check the denominators. Make sure they are the same.
2. Add the numerators (the top numbers) together.
3. Keep the denominator the same! The size of the slices doesn't change.
Let's Try an Example!
You ate $$ \frac{2}{6} $$ of a chocolate bar. Your friend gave you $$ \frac{3}{6} $$ of the same chocolate bar. How much do you have now?
Step 1: Check the denominators. Are they the same? Yes, they are both 6!
Step 2: Add the numerators. $$ 2 + 3 = 5 $$
Step 3: Keep the denominator the same. It stays as 6.
So, the answer is:
$$ \frac{2}{6} + \frac{3}{6} = \frac{5}{6} $$You have $$ \frac{5}{6} $$ of the chocolate bar!
Adding Three Fractions
The rule works for adding more than two fractions too!
Let's add $$ \frac{1}{8} + \frac{2}{8} + \frac{4}{8} $$
Just add all the numerators: $$ 1 + 2 + 4 = 7 $$
And keep the denominator: 8
$$ \frac{1}{8} + \frac{2}{8} + \frac{4}{8} = \frac{7}{8} $$
Watch out for this common mistake!
Many people are tempted to add the denominators together. NEVER add the denominators! The number of slices the whole is cut into doesn't change.
Wrong way: $$ \frac{2}{6} + \frac{3}{6} \neq \frac{5}{12} $$ (This is incorrect!)
Key Takeaway for Addition
Add the tops, but leave the bottom alone!
Taking Pieces Away: Subtracting Fractions
You are going to love this... subtraction follows the exact same rules as addition!
If you can add fractions, you can definitely subtract them. You just need to make sure they are from the same family (they have the same denominator).
How to Subtract: Step-by-Step
1. Check the denominators. Are they the same?
2. Subtract the numerators (the top numbers).
3. Keep the denominator the same!
Let's Try an Example!
A pizza has $$ \frac{7}{8} $$ left. You decide to eat $$ \frac{4}{8} $$ of the pizza. How much is left now?
Step 1: Check the denominators. They are both 8. Perfect!
Step 2: Subtract the numerators. $$ 7 - 4 = 3 $$
Step 3: Keep the denominator the same. It stays as 8.
So, the answer is:
$$ \frac{7}{8} - \frac{4}{8} = \frac{3}{8} $$There is $$ \frac{3}{8} $$ of the pizza left.
Key Takeaway for Subtraction
Subtract the tops, but leave the bottom alone!
Chapter Summary & Fun Facts
Quick Review Box
- A fraction is a part of a whole.
- The denominator (bottom number) is how many equal parts the whole is divided into.
- The numerator (top number) is how many of those parts we have.
- To add or subtract fractions, the denominators MUST be the same.
- When you add or subtract, you only change the numerator. The denominator STAYS THE SAME!
Did you know?
The line that separates the numerator and the denominator in a fraction has a special name. It's called a vinculum! It's a Latin word that means 'bond' or 'chain', because it links the two numbers together.
Great job working through fractions today! Keep practicing, and you'll see them everywhere you look.