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Let's go through each part of the activity step-by-step.

### (a) Use the model to fill in the table below.
The model given is \( A = s^2 \), where:
- \( s \) is the side length in centimeters (cm).
- \( A \) is the area in square centimeters (cm\(^2\)).

We will calculate \( A \) for each value of \( s \):

1. For \( s = 1 \): \( A = 1^2 = 1 \) cm\(^2\)
2. For \( s = 2 \): \( A = 2^2 = 4 \) cm\(^2\)
3. For \( s = 3 \): \( A = 3^2 = 9 \) cm\(^2\)
4. For \( s = 4 \): \( A = 4^2 = 16 \) cm\(^2\)
5. For \( s = 5 \): \( A = 5^2 = 25 \) cm\(^2\)

So, the table will look like this:

| Side Length, \( s \) (cm) | 1 | 2 | 3 | 4 | 5 |
|---------------------------|---|---|---|---|---|
| Area, \( A \) (cm\(^2\)) | 1 | 4 | 9 | 16 | 25 |

### (b) What would change if the side lengths of the squares were measured in feet instead of centimeters?
If the side lengths were measured in feet, the area would be in square feet instead of square centimeters. The numerical values would remain the same since the area depends on the square of the side length, but the units would change to square feet (\( \text{ft}^2 \)).

### (c) Sketch a graph showing the relationship between the area of a square and its side length.
To sketch this graph:
- Plot the side length \( s \) on the x-axis.
- Plot the area \( A \) on the y-axis.
- The points to plot based on the table are \((1, 1)\), \((2, 4)\), \((3, 9)\), \((4, 16)\), and \((5, 25)\).
  
The graph will show a parabolic curve starting from the origin, curving upward, indicating the quadratic relationship between side length and area.

### (d) Explain how we know that this isn’t a linear model.
A linear model has a constant rate of change, meaning as \( s \) increases by 1 unit, \( A \) would also increase by a constant amount. In this model, however, as \( s \) increases, \( A \) increases by progressively larger amounts (e.g., from \( s = 1 \) to \( s = 2 \), \( A \) increases by 3; from \( s = 2 \) to \( s = 3 \), it increases by 5). This indicates a quadratic relationship rather than a linear one.

### (e) Explain how we know that this isn’t an exponential model.
In an exponential model, \( A \) would increase by a constant multiplier as \( s \) increases (e.g., \( A \) might double or triple for each increase in \( s \)). Here, \( A \) increases by the square of \( s \), not by a constant multiple. The pattern \( A = s^2 \) reflects a power relationship, not an exponential one, which would look like \( A = b^s \) for some base \( b \).

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Would you like more details or have questions on any part of this solution?

Here are some related questions you might explore:
1. How would the table change if \( s \) was measured in millimeters?
2. Can we determine a formula if the side length is doubled? How does the area change?
3. What would happen if we extended the table to \( s = 10 \)? How would the area look then?
4. How does the shape of the graph compare to an exponential growth graph?
5. Can you identify another real-world situation where an area is calculated using a power model?

**Tip:** When working with functions, identify the rate of change to understand if the relationship is linear, quadratic, or exponential.

6.2 Power Models Prep Activities: Given a model for the area of a square, A = s^2, where s is the side length in centimeters, fill in a table with the area for side lengths 1 to 5 cm, answer questions about unit changes, and determine if the model is linear or exponential.

Explore power models through the area of a square formula, A = s^2. Fill in area calculations for side lengths in centimeters, examine unit changes, and analyze the quadratic relationship. This activity includes a detailed solution suitable for Grades 8-10.

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