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What is the relationship between the patterns in each design. Create an equation for each one.

The patterns in each design option represent a series of grids, where each subsequent pattern seems to expand in a consistent manner. Let's examine each design option and identify the relationship between the patterns.

### Design Option #1
- **Pattern #1**: 4x4 grid, center 2x2 square shaded.
- **Pattern #2**: 6x6 grid, center 4x4 square shaded.
- **Pattern #3**: 8x8 grid, center 6x6 square shaded.

#### Equation for Design Option #1
Each pattern increases the grid size by 2 units on each side, and the shaded area is always the grid size minus 2 units on each side:
\[ \text{Grid Size} = 2n + 4 \]
\[ \text{Shaded Area Size} = 2n \]
where \( n \) is the pattern number (starting from 1).

### Design Option #2
- **Pattern #1**: 5x5 grid, center 3x3 square shaded.
- **Pattern #2**: 7x7 grid, center 5x5 square shaded.
- **Pattern #3**: 9x9 grid, center 7x7 square shaded.

#### Equation for Design Option #2
Each pattern increases the grid size by 2 units on each side, and the shaded area is always the grid size minus 2 units on each side:
\[ \text{Grid Size} = 2n + 3 \]
\[ \text{Shaded Area Size} = 2n + 1 \]
where \( n \) is the pattern number (starting from 1).

### Design Option #3
- **Pattern #1**: 6x6 grid, center 2x2 and four 2x2 corners shaded.
- **Pattern #2**: 8x8 grid, center 4x4 and four 2x2 corners shaded.
- **Pattern #3**: 10x10 grid, center 6x6 and four 2x2 corners shaded.

#### Equation for Design Option #3
Each pattern increases the grid size by 2 units on each side, and the shaded area in the center increases similarly while the corner areas remain 2x2:
\[ \text{Grid Size} = 2n + 4 \]
\[ \text{Center Shaded Area Size} = 2n \]
where \( n \) is the pattern number (starting from 1).

### Summary of Equations
1. **Design Option #1**:
   - Grid Size: \( 2n + 4 \)
   - Shaded Area Size: \( 2n \)

2. **Design Option #2**:
   - Grid Size: \( 2n + 3 \)
   - Shaded Area Size: \( 2n + 1 \)

3. **Design Option #3**:
   - Grid Size: \( 2n + 4 \)
   - Center Shaded Area Size: \( 2n \)

Would you like more details or have any questions?

### Additional Questions
1. How would the shaded area change if the pattern number increases indefinitely?
2. Can you calculate the number of shaded squares for the 4th pattern in each design option?
3. How does the ratio of shaded to unshaded squares change with each pattern?
4. What is the total number of squares in the grid for each pattern?
5. How can these patterns be represented graphically?
6. Can we derive a formula for the unshaded area in each design option?
7. How do the patterns compare in terms of growth rate?
8. What would be the effect of starting with different initial grid sizes?

### Tip
When analyzing patterns, it’s often useful to express relationships in terms of equations. This helps in predicting future patterns and understanding the underlying structure.

Explore the relationship between patterns in three design options with detailed equations. Each design option increases grid size and shaded area in a structured manner. Suitable for Grades 6-8.

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