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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### Understanding Equivalent Fractions by Multiplying Numerator and Denominator

**Equivalent fractions** are different fractions that represent the same value or proportion. One common method to create equivalent fractions is by multiplying (or dividing) both the numerator (the top number) and the denominator (the bottom number) by the same non-zero number.

#### **Why Does This Work?**

When you multiply both the numerator and the denominator by the same number, you're essentially scaling the fraction by a factor of 1. Since multiplying by 1 doesn't change the value, the fraction remains equivalent.

**Mathematically:**

$$
\frac{a}{b} \times \frac{c}{c} = \frac{a \times c}{b \times c} = \frac{a \cdot c}{b \cdot c}
$$

Here, $$\frac{c}{c} = 1$$, so you're multiplying the original fraction by 1, keeping its value the same.

**Example:**

Let's take the fraction $$\frac{1}{2}$$.

If we multiply both numerator and denominator by 3:

$$
\frac{1 \times 3}{2 \times 3} = \frac{3}{6}
$$

So, $$\frac{1}{2} = \frac{3}{6}$$, which are equivalent fractions.

#### **Visual Model: Fraction Circles**

To illustrate this concept, let's use a visual model with fraction circles.

1. **Original Fraction ($$\frac{1}{2}$$):**

![Half Circle](https://i.imgur.com/6N6cXcR.png)
   
   - The circle is divided into 2 equal parts.
   - One part is shaded, representing $$\frac{1}{2}$$.

2. **Equivalent Fraction ($$\frac{2}{4}$$):**

![Two Fourths Circle](https://i.imgur.com/6N6cXcR.png)
   
   - The same circle is now divided into 4 equal parts.
   - Two parts are shaded, representing $$\frac{2}{4}$$.

**Explanation:**

- When we multiply the numerator and denominator of $$\frac{1}{2}$$ by 2, we get $$\frac{2}{4}$$.
- Both fractions shade the same proportion of the circle:
  - $$\frac{1}{2}$$ shades 1 out of 2 parts.
  - $$\frac{2}{4}$$ shades 2 out of 4 parts.
- Visually, both represent the same amount of the whole circle, confirming they are equivalent.

#### **Another Example:**

Consider the fraction $$\frac{3}{5}$$.

Multiply both numerator and denominator by 2:

$$
\frac{3 \times 2}{5 \times 2} = \frac{6}{10}
$$

So, $$\frac{3}{5} = \frac{6}{10}$$.

**Visual Representation:**

1. **Original Fraction ($$\frac{3}{5}$$):**

![Three Fifths Circle](https://i.imgur.com/NzKwCkL.png)
   
   - The circle is divided into 5 equal parts.
   - Three parts are shaded.

2. **Equivalent Fraction ($$\frac{6}{10}$$):**

![Six Tenths Circle](https://i.imgur.com/NzKwCkL.png)
   
   - The same circle is divided into 10 equal parts.
   - Six parts are shaded.

Again, both fractions shade the same proportion of the circle, demonstrating their equivalence.

### **Key Takeaways**

- **Multiplying both the numerator and denominator by the same number doesn't change the fraction's value.**
- **Equivalent fractions represent the same part of a whole, just expressed differently.**
- **Visual models like fraction circles help in understanding and confirming the equivalence of fractions.**

Understanding this concept is fundamental in various math operations, including simplifying fractions, comparing fractions, and performing arithmetic with fractions.
To create equivalent fractions, multiply both the numerator and denominator by the same number. For example, $$\frac{1}{2}$$ becomes $$\frac{3}{6}$$ when both are multiplied by 3. This works because multiplying by 1 (in the form of $$\frac{3}{3}$$) doesn't change the fraction's value.

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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