Interactive math worksheet for approximating square roots to the nearest hundredth, aligned with 8.NS.2.

The task in the image is to approximate the square root of 135 to the nearest hundredth. The method used here is an estimation technique based on identifying perfect squares close to the given number and then refining the approximation. Let's break it down step by step:

---

Step 1: Identify Perfect Squares Close to 135

- The perfect square less than 135 is \( 121 \), which is \( 11^2 \).
- The perfect square greater than 135 is \( 144 \), which is \( 12^2 \).

Thus:
\[
\sqrt{121} = 11 \quad \text{and} \quad \sqrt{144} = 12
\]
This tells us that:
\[
11 < \sqrt{135} < 12
\]

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Step 2: Refine the Approximation

To refine the approximation, we use a linear interpolation method between 11 and 12. The idea is to determine how far 135 lies between 121 and 144.

#### Calculate the Differences:
- The difference between 135 and 121 is:
\[
135 - 121 = 14
\]
- The difference between 144 and 121 is:
\[
144 - 121 = 23
\]

#### Proportion Calculation:
The proportion of 135 relative to the range between 121 and 144 is:
\[
\frac{\text{Difference from lower perfect square}}{\text{Total range}} = \frac{14}{23}
\]

#### Refine the Square Root:
Since \( \sqrt{135} \) is closer to 11 than to 12, we add this proportion to 11:
\[
\sqrt{135} \approx 11 + \frac{14}{23}
\]

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Step 3: Perform the Division

Now, calculate \( \frac{14}{23} \):
\[
\frac{14}{23} \approx 0.6087
\]

Add this to 11:
\[
\sqrt{135} \approx 11 + 0.6087 = 11.6087
\]

---

Step 4: Round to the Nearest Hundredth

Finally, round 11.6087 to the nearest hundredth:
\[
11.6087 \approx 11.61
\]

---

Final Answer:

\[
\boxed{11.61}
\]

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Explanation of the Method:

This method works by leveraging the fact that the square root function is continuous and increasing. By identifying the perfect squares closest to the target number (135), we can estimate the square root as a weighted average between the square roots of those perfect squares. The proportion is determined by how far the target number lies between the two perfect squares.

This approach is particularly useful for approximating square roots without a calculator, providing a reasonable estimate with minimal computation.