Determine what two whole numbers the square root is between without using a calculator or table. Then use a calculator or table to check.
\[
\sqrt{117}
\]
$\sqrt{117}$ is between $\square$ and $\square$. (Type whole numbers. Use ascending order.)

Step 1 ：The square root of a number is the value that, when multiplied by itself, gives the original number. To find out which two whole numbers the square root of 117 is between, we need to find two perfect squares that 117 is between. Perfect squares are the squares of whole numbers. So, we need to find two whole numbers such that the square of the first one is less than 117 and the square of the second one is more than 117.

Step 2 ：We can start by squaring whole numbers starting from 1 and continue until we find the two whole numbers. Let's start with 1. The square of 1 is 1. This is less than 117. So, we continue with the next whole number.

Step 3 ：The square of 2 is 4. This is also less than 117. We continue with the next whole number.

Step 4 ：We continue this process until we find the two whole numbers.

Step 5 ：The process returned (10, 11). This means that the square root of 117 is between 10 and 11.

Step 6 ：To confirm this, we can calculate the squares of 10 and 11. The square of 10 is 100 and the square of 11 is 121. Since 117 is between 100 and 121, the square root of 117 is indeed between 10 and 11.

Step 7 ：Final Answer: \( \sqrt{117} \) is between \( \boxed{10} \) and \( \boxed{11} \).