This worksheet challenges students to distinguish between rational and irrational numbers and accurately plot them on a number line.

To solve the problem, we need to determine whether each given number is rational or irrational and then place it on the correct number line. Let's analyze each number step by step.

Definitions:

- Rational Numbers: Numbers that can be expressed as a ratio of two integers \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). Rational numbers include integers, fractions, terminating decimals, and repeating decimals.
- Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. Irrational numbers include non-terminating, non-repeating decimals and certain roots (e.g., \( \sqrt{2} \)).

Given Numbers:

1. \( -\frac{4}{5} \)
2. \( 0.672 \)
3. \( \frac{\pi}{2} \)
4. \( \frac{8}{3} \)
5. \( 3.5 \)
6. \( \sqrt{7} \)
7. \( -\sqrt{7} \)
8. \( \frac{6 + \sqrt{2}}{6} \)
9. \( \frac{9 + \sqrt{1}}{9} \)
10. \( 1.777... \)
11. \( -2.701... \)

Analysis:

#### 1. \( -\frac{4}{5} \)
- This is a fraction of two integers.
- Rational.

#### 2. \( 0.672 \)
- This is a terminating decimal.
- Rational.

#### 3. \( \frac{\pi}{2} \)
- \( \pi \) is an irrational number.
- Dividing \( \pi \) by 2 does not make it rational.
- Irrational.

#### 4. \( \frac{8}{3} \)
- This is a fraction of two integers.
- Rational.

#### 5. \( 3.5 \)
- This can be written as \( \frac{7}{2} \), which is a fraction of two integers.
- Rational.

#### 6. \( \sqrt{7} \)
- \( 7 \) is not a perfect square, so \( \sqrt{7} \) is not an integer.
- \( \sqrt{7} \) is a non-terminating, non-repeating decimal.
- Irrational.

#### 7. \( -\sqrt{7} \)
- The negative sign does not change the irrationality of \( \sqrt{7} \).
- Irrational.

#### 8. \( \frac{6 + \sqrt{2}}{6} \)
- \( \sqrt{2} \) is irrational.
- Adding \( 6 \) (a rational number) to \( \sqrt{2} \) results in an irrational number.
- Dividing an irrational number by a rational number (\( 6 \)) still results in an irrational number.
- Irrational.

#### 9. \( \frac{9 + \sqrt{1}}{9} \)
- \( \sqrt{1} = 1 \), which is rational.
- \( 9 + \sqrt{1} = 9 + 1 = 10 \), which is rational.
- Dividing \( 10 \) by \( 9 \) gives \( \frac{10}{9} \), which is rational.
- Rational.

#### 10. \( 1.777... \)
- This is a repeating decimal, which can be expressed as a fraction.
- Let \( x = 1.777... \). Then \( 10x = 17.777... \). Subtracting these gives \( 9x = 16 \), so \( x = \frac{16}{9} \).
- Rational.

#### 11. \( -2.701... \)
- This is a non-terminating, non-repeating decimal.
- Irrational.

Summary of Rational and Irrational Numbers:

- Rational Numbers: \( -\frac{4}{5}, 0.672, \frac{8}{3}, 3.5, \frac{9 + \sqrt{1}}{9}, 1.777... \)
- Irrational Numbers: \( \frac{\pi}{2}, \sqrt{7}, -\sqrt{7}, \frac{6 + \sqrt{2}}{6}, -2.701... \)

Placement on Number Lines:

- Rational Numbers Line:
- \( -\frac{4}{5} \approx -0.8 \)
- \( 0.672 \)
- \( \frac{8}{3} \approx 2.67 \)
- \( 3.5 \)
- \( \frac{9 + \sqrt{1}}{9} = \frac{10}{9} \approx 1.11 \)
- \( 1.777... = \frac{16}{9} \approx 1.78 \)

- Irrational Numbers Line:
- \( \frac{\pi}{2} \approx 1.57 \)
- \( \sqrt{7} \approx 2.65 \)
- \( -\sqrt{7} \approx -2.65 \)
- \( \frac{6 + \sqrt{2}}{6} \approx 1.24 \)
- \( -2.701... \)

Final Answer:

\[
\boxed{
\begin{array}{l}
\text{Rational Numbers: } -\frac{4}{5}, 0.672, \frac{8}{3}, 3.5, \frac{9 + \sqrt{1}}{9}, 1.777... \\
\text{Irrational Numbers: } \frac{\pi}{2}, \sqrt{7}, -\sqrt{7}, \frac{6 + \sqrt{2}}{6}, -2.701...
\end{array}
}
\]