14 Illustrate Explain why you can multiply both the numerator and denominator by the same number to make an equivalent fraction. Draw a model to show an example.

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Multiplying both the numerator and the denominator of a fraction by the same number creates an equivalent fraction. This process is known as **scaling** the fraction. Let’s break down **why** this works and **illustrate** it with an example using a visual model.

### **Understanding Equivalent Fractions**

A **fraction** represents a part of a whole. It consists of two parts:
- **Numerator**: The top number, representing how many parts we have.
- **Denominator**: The bottom number, representing the total number of equal parts the whole is divided into.

An **equivalent fraction** is a different fraction that represents the same value or proportion of the whole.

### **Why Multiplying Keeps the Fraction Equivalent**

When you multiply both the numerator and the denominator by the **same number**, you are essentially scaling both parts of the fraction equally. This means you're increasing or decreasing both the number of parts you have and the total number of parts in the whole by the same factor, keeping the proportion between them the same.

**Mathematical Explanation:**
$$
\frac{a}{b} = \frac{a \times c}{b \times c}
$$
Where $$ c $$ is any non-zero number.

### **Example:**

Let's take the fraction **½** and multiply both the numerator and the denominator by **2**.

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

So, **½** is equivalent to **²⁄⁴**.

### **Visual Model Illustration**

Imagine a **pizza** divided into equal parts to visualize fractions.

1. **Original Fraction: ½**

![Half Pizza](https://i.imgur.com/ZxJrKqK.png)
   
   - **Numerator (1):** You have 1 slice.
   - **Denominator (2):** The pizza is divided into 2 equal slices.

![Half Pizza](https://i.imgur.com/ZxJrKqK.png)

2. **Equivalent Fraction: ²⁄⁴**

Imagine you decide to cut each of the original 2 slices into 2 smaller slices, resulting in 4 slices in total.

![Fourth Pizza](https://i.imgur.com/5WztH6z.png)
   
   - **Numerator (2):** You now have 2 smaller slices.
   - **Denominator (4):** The pizza is divided into 4 equal slices.

![Fourth Pizza](https://i.imgur.com/5WztH6z.png)

**Observation:** You still have half of the pizza:
   $$
   \frac{2 \text{ small slices}}{4 \text{ total small slices}} = \frac{1}{2}
   $$

### **Another Example:**

Take the fraction **⅓** and multiply both the numerator and the denominator by **3**.

$$
\frac{1 \times 3}{3 \times 3} = \frac{3}{9}
$$

So, **⅓** is equivalent to **³⁄⁹**.

**Visual Illustration:**

1. **Original Fraction: ⅓**

![One Third Pizza](https://i.imgur.com/OjL3tRv.png)
   
   - **Numerator (1):** 1 slice.
   - **Denominator (3):** 3 equal slices.

2. **Equivalent Fraction: ³⁄⁹**

Now, divide each third into 3 smaller slices, making a total of 9 slices.

![Nine Slices Pizza](https://i.imgur.com/t9wRfzM.png)
   
   - **Numerator (3):** 3 smaller slices.
   - **Denominator (9):** 9 equal slices.

**Observation:** You still have one-third of the pizza:
   $$
   \frac{3 \text{ small slices}}{9 \text{ total small slices}} = \frac{1}{3}
   $$

### **Key Takeaways**

- **Scaling Both Parts Equally:** Multiplying (or dividing) the numerator and denominator by the same number scales the fraction without changing its value.
  
- **Equivalent Fractions Represent the Same Proportion:** Even though the numbers change, the fraction still represents the same part of the whole.

- **Visual Models Help Understand Equivalence:** Using models like divided shapes (e.g., pizzas, circles, bars) can make it easier to see why equivalent fractions are indeed the same value.

### **Practice Problem**

**Convert ⅖ to an equivalent fraction by multiplying both the numerator and the denominator by 3.**

**Solution:**
$$
\frac{2 \times 3}{5 \times 3} = \frac{6}{15}
$$

So, **⅖** is equivalent to **⁶⁄¹⁵**.

### **Visual Representation**

1. **Original Fraction: ⅖**

![Two Fifths](https://i.imgur.com/zXirj1m.png)

2. **Equivalent Fraction: ⁶⁄¹⁵**

![Six Fifteenths](https://i.imgur.com/lVhP0kj.png)

You still have two-fifths of the whole.

By understanding and visualizing how scaling affects both the numerator and the denominator equally, you can confidently create and recognize equivalent fractions.
Multiplying both the numerator and denominator by the same number creates an equivalent fraction because it scales both parts equally, keeping the fraction's value the same. For example, multiplying ½ by 2 gives ²⁄⁴, which is still half of the whole.

14 Illustrate Explain why you can multiply both the numerator and denominator by the same number to make an equivalent fraction. Draw a model to show an example.

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