1. **State the problem:** Calculate the value of the expression $1 \div \frac{3}{2} + \frac{4}{9} \times \frac{3}{6}$. 2. **Recall the order of operations:** Division and multiplication are performed from left to right before addition. 3. **Calculate the division:** $1 \div \frac{3}{2} = 1 \times \frac{2}{3} = \frac{2}{3}$. 4. **Calculate the multiplication:** $\frac{4}{9} \times \frac{3}{6} = \frac{4 \times 3}{9 \times 6} = \frac{12}{54}$. 5. **Simplify the fraction:** $\frac{12}{54} = \frac{\cancel{12}^{2} \times 1}{\cancel{54}^{2} \times 3} = \frac{2}{9}$. 6. **Add the results:** $\frac{2}{3} + \frac{2}{9}$. 7. **Find common denominator:** The least common denominator of 3 and 9 is 9. Convert $\frac{2}{3}$ to $\frac{6}{9}$. 8. **Add fractions:** $\frac{6}{9} + \frac{2}{9} = \frac{8}{9}$. 9. **Final answer:** The value of the expression is $\frac{8}{9}$.