Question

which model represents 2/3 x 1/2

Answer 1

how would you model a multiplication between fractions

Answer 2

simpler and shorter please

Answer 3

To understand the product of two fractions, such as \( \frac{2}{3} \) and \( \frac{1}{2} \), we can use a visual model. Here’s a step-by-step way to model it:

1. **Fraction Representation**:
- Start with a rectangle to represent a whole.

2. **Dividing into Equal Parts**:
- Divide the rectangle into 3 equal vertical sections to represent the denominator of \( \frac{2}{3} \). Shade 2 of those sections to represent \( \frac{2}{3} \).

3. **Further Division**:
- Now take the shaded \( \frac{2}{3} \) and divide each of those sections into 2 equal horizontal parts (since you are also multiplying by \( \frac{1}{2} \)). Each vertical section that was previously shaded is now divided into 2 equal parts.

4. **Counting the Overlapping Area**:
- The total area of the rectangle can be viewed as a whole (1).
- The area that remains shaded after these divisions represents the product \( \frac{2}{3} \times \frac{1}{2} \).
- You will end up with 2 shaded parts out of 6 equal parts in total.

5. **Result**:
- So, \( \frac{2}{3} \times \frac{1}{2} = \frac{2}{6} \), which simplifies to \( \frac{1}{3} \).

In summary, visually representing \( \frac{2}{3} \times \frac{1}{2} \) shows you how the area representation confirms that the product indeed equals \( \frac{1}{3} \).

Answer 4

To model the multiplication of fractions, we can use a visual approach, often involving area models or number lines. Below is a step-by-step guide on how to model the multiplication of two fractions using an area model.

### Steps to Model Multiplication of Fractions

**Example: Multiply \( \frac{a}{b} \times \frac{c}{d} \)**

1. **Draw a Rectangle**:
- Start by drawing a rectangle to represent 1 whole.

2. **Divide the Rectangle Vertically**:
- Divide the rectangle into \( b \) equal vertical sections. Each section represents \( \frac{1}{b} \).
- Shade \( a \) of those sections to represent \( \frac{a}{b} \).

3. **Divide the Rectangle Horizontally**:
- Now, take the whole rectangle and divide it into \( d \) equal horizontal sections. Each section represents \( \frac{1}{d} \).
- From the first shaded sections representing \( \frac{a}{b} \), shade \( c \) out of the \( d \) horizontal sections.

4. **Find the Overlapping Area**:
- The area that is both vertically shaded (representing \( \frac{a}{b} \)) and horizontally shaded (representing \( \frac{c}{d} \)) illustrates the product of the two fractions.

5. **Count the Shaded Parts**:
- The area of the total rectangle represents the whole (1), and each intersection of the vertical and horizontal divisions represents a smaller section of \( \frac{1}{(b \cdot d)} \).
- The resulting shaded area will represent the product \( \frac{a \cdot c}{b \cdot d} \).

### Example with Actual Numbers

Let's model \( \frac{2}{3} \times \frac{1}{2} \):

1. **Draw a rectangle**.
2. **Divide vertically into 3 parts**:
- Shade 2 out of 3 sections (representing \( \frac{2}{3} \)).
3. **Divide horizontally into 2 parts**:
- The two vertical shaded strips should be further divided into 2.
4. **Shade half of the already shaded area**:
- This represents \( \frac{1}{2} \) of the \( \frac{2}{3} \).

#### Completed Area Model

- You will see that the entire rectangle is now divided into \( 6 \) equal smaller rectangles (3 vertical and 2 horizontal).
- The newly shaded area will cover \( 2 \) of those \( 6 \) smaller rectangles, representing \( \frac{2}{6} \), which simplifies to \( \frac{1}{3} \).

### Conclusion

Using this visual area model helps clarify how multiplying fractions involves scaling parts of wholes, and illustrates that \( \frac{2}{3} \times \frac{1}{2} \) indeed equals \( \frac{1}{3} \). This method can be adapted for any other fractions!

Answer 5

Certainly! Here’s a simpler and shorter way to model multiplication of fractions:

### Steps to Model \( \frac{a}{b} \times \frac{c}{d} \)

1. **Draw a Rectangle**: This represents 1 whole.

2. **Divide Vertically for the First Fraction**:
- Divide the rectangle into \( b \) equal parts and shade \( a \) parts to represent \( \frac{a}{b} \).

3. **Divide Horizontally for the Second Fraction**:
- Divide the whole rectangle into \( d \) equal parts. Now shade \( c \) parts across the already shaded vertical sections to represent \( \frac{c}{d} \).

4. **Find the Overlapping Area**:
- The area that is shaded from both divisions represents the product.

### Example: \( \frac{2}{3} \times \frac{1}{2} \)

1. Draw a rectangle.
2. Divide vertically into 3 parts; shade 2.
3. Divide horizontally into 2 parts; shade 1 part across the 2 shaded sections.
4. You will have 2 shaded sections out of 6 total sections.

### Result:
The final shaded area represents \( \frac{2}{6} \) which simplifies to \( \frac{1}{3} \).

This effectively models how multiplying fractions yields a smaller fraction!