To determine between which two numbers the square root of 37 ($\sqrt{37}$) lies on a number line, we need to find two perfect squares that are close to 37.

1. The perfect square less than 37 is $36$, and the square root of 36 is $6$.
2. The perfect square greater than 37 is $49$, and the square root of 49 is $7$.

Since 37 is between 36 and 49, the square root of 37 ($\sqrt{37}$) lies between 6 and 7.

So, $\sqrt{37}$ lies between the numbers 6 and 7 on a number line.

Would you like details or have any questions? Here are some related questions:

1. What is the approximate decimal value of $\sqrt{37}$?
2. How do you estimate square roots of non-perfect squares?
3. What other square roots lie between 6 and 7?
4. What is the square of 6.5, and how close is it to 37?
5. How can you use the method of successive approximations to find $\sqrt{37}$?

Tip: When estimating square roots of non-perfect squares, look for the nearest perfect squares to quickly determine the range in which the square root lies.