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how and explain how to use the part-whole deﬁnition of a fraction to determine fractional values of a portion of an area model. The image below shows some rectangles with some shading.

Question : If the area of the whole square below counts as “1”, what is the value of the shaded region? Write your answer as a fraction. Modify the picture below (or your own sketch of the picture) to show why your answer is correct. Explain your reasoning, using the part-whole definition of a fraction to justify your answer.

Let's work through this step by step using the image provided.

### Part-whole Definition of a Fraction:
- The part-whole definition of a fraction says that a fraction represents the part (numerator) of a whole (denominator). In other words, it's the ratio of the shaded area to the total area.

### Step 1: Divide the Square
First, notice that the square is divided into smaller sections:
1. The square is split horizontally into two equal rectangles.
2. Each rectangle is further subdivided into triangular and quadrilateral sections.

### Step 2: Focus on the Shaded Region
The shaded region is a right triangle located in the top left section of the square. To determine its fractional value, we need to calculate the area of the shaded triangle relative to the whole square.

#### Step 3: Total Area of the Square
The total area of the entire square is given as \( 1 \) unit. 

#### Step 4: Analyzing the Division
- The square is first divided horizontally into two equal parts, so each rectangle has an area of \( \frac{1}{2} \) of the total square.
- The shaded triangle is part of the top rectangle, which is further divided into smaller regions, including two large right triangles.
  
#### Step 5: Area of the Shaded Triangle
- The shaded triangle occupies exactly half of the area of the top left rectangle. Since the entire rectangle has an area of \( \frac{1}{2} \), the shaded triangle covers half of this area.

So, the area of the shaded triangle is:
\[
\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}
\]

### Conclusion:
The value of the shaded region is \( \frac{1}{4} \) of the entire square.

### Explanation Using the Part-Whole Definition:
- The entire square is "1" whole unit.
- The shaded triangle is part of the top left rectangle, which is \( \frac{1}{2} \) of the square.
- Since the triangle is half of that rectangle, it represents \( \frac{1}{4} \) of the total area of the square.
  
This reasoning follows the part-whole definition of fractions: the shaded area (part) is \( \frac{1}{4} \), and the whole is the entire square which has an area of 1.

Would you like further details or visual modifications of the image?

### 5 Questions to Expand Your Understanding:
1. How would the fraction change if a different part of the square were shaded?
2. Can you explain how dividing shapes into smaller equal parts helps in determining fractions?
3. How would you calculate the area of the shaded region if the square were split into more sections?
4. What if the whole square were divided into three parts instead of two? How would that affect the fraction?
5. Could you apply the same reasoning if the figure were a circle instead of a square?

### Tip:
When working with area models, always check how many equal parts the whole is divided into before calculating the fraction. This helps you quickly identify the fraction for any shaded part.

If the area of the whole square below counts as '1', what is the value of the shaded region? Write your answer as a fraction. Modify the picture below (or your own sketch of the picture) to show why your answer is correct. Explain your reasoning, using the part-whole definition of a fraction to justify your answer.

Explore how to use the part-whole definition of a fraction to determine the area of a shaded region in a square. This problem covers key concepts in fractions and geometry, suitable for students in Grades 4-6.

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