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Let's analyze the three proportional relationships given and find which has the greatest rate of change (slope).

1. **Option A: \( y = 7x \)**
   - The equation \( y = 7x \) represents a linear function where the slope (rate of change) is 7. This means for every increase of 1 unit in \( x \), \( y \) increases by 7 units.

2. **Option B:**
   - "The value of \( y \) increases by 12 for every increase of 4 in \( x \)." 
   - The rate of change can be found as the ratio of the change in \( y \) to the change in \( x \): 
     \[
     \text{Rate of change} = \frac{12}{4} = 3
     \]
   - So, the slope is 3 for this proportional relationship.

3. **Option C:**
   - The table provided shows the following points: 
     \[
     (0, 0), (2, 8), (4, 16), (6, 24)
     \]
   - To calculate the rate of change (slope), we can use the formula:
     \[
     \text{Slope} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
     \]
   - Let's take two points, say \( (0, 0) \) and \( (2, 8) \):
     \[
     \text{Slope} = \frac{8 - 0}{2 - 0} = \frac{8}{2} = 4
     \]
   - This gives us a rate of change of 4 for this relationship.

### Conclusion:
- **Option A** has the greatest rate of change, with a slope of **7**, compared to Option B's slope of **3** and Option C's slope of **4**.

Would you like more details on how slopes are calculated or have any further questions?

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Here are five related questions for further exploration:
1. How do you calculate the slope from two points on a line?
2. What is the relationship between the slope and the steepness of a line?
3. How would the rate of change differ if the slope was negative?
4. Can you have a proportional relationship with a slope of 0?
5. How do you find the slope from a graph without a formula?

**Tip:** The slope of a line represents the rate of change in \( y \) with respect to \( x \). The steeper the line, the greater the rate of change.

Which proportional relationship has the greatest rate of change? (Three options provided: A) y = 7x, B) y increases by 12 for every increase of 4 in x, C) table of x and y values: (0,0), (2,8), (4,16), (6,24))

Determine which proportional relationship has the greatest rate of change by analyzing three options. Learn how to calculate the slope from a linear equation, a word description, and a table of values. Ideal for Grades 6-8.

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