1. **Problem statement:** Determine between which two consecutive integers the number $\sqrt{7}$ lies without using a calculator. 2. **Recall the concept:** The square root of a number $n$ is a value $x$ such that $x^2 = n$. 3. **Identify perfect squares near 7:** The perfect squares closest to 7 are $2^2 = 4$ and $3^2 = 9$. 4. **Compare 7 with these perfect squares:** Since $4 < 7 < 9$, it follows that $2^2 < 7 < 3^2$. 5. **Conclude the range for $\sqrt{7}$:** Taking square roots on all parts of the inequality, we get $2 < \sqrt{7} < 3$. 6. **Final answer:** $\sqrt{7}$ lies between the consecutive integers 2 and 3. This method uses the fact that the square root of a number lies between the square roots of the perfect squares immediately below and above it.