The problem in the image focuses on determining the approximate value of the square root of an integer based on its location on a number line.

Breakdown:

• The red dot is located between $\sqrt{144}$ (which is 12) and $\sqrt{169}$ (which is 13).
• The dot is described as slightly below halfway between 12 and 13.

Steps to Solve:

1. Square Roots of Known Integers:

   • The square root of 144 is 12.
   • The square root of 169 is 13.

2. Dot Location:

   • Since the red dot is slightly below halfway between 12 and 13, it suggests that the square root of the unknown integer is slightly below halfway between 144 and 169.

3. Find the Integer:

   • Halfway between 144 and 169 is the average: $\frac{144 + 169}{2} = \frac{313}{2} = 156.5.$
   • Since the dot is slightly below halfway, the square root we're looking for corresponds to an integer less than 157 but close to it. Testing the squares, $\sqrt{156} \approx 12.49$, fits this description.

4. Final Answer:

   • The red dot represents the square root of 156.

Filling in the Boxes:

• First Box: $144$ (The square root of 12).
• Second Box: $169$ (The square root of 13).
• Third Box: $156$ (The nearest integer, whose square root is approximated by the red dot).

Would you like more details or have any questions about this explanation?

Here are 5 related questions to expand on this topic:

1. How can you find the square root of a number that isn't a perfect square without a calculator?
2. What are the steps to approximate a square root using a number line?
3. How would the red dot move if the number were $\sqrt{150}$ instead of $\sqrt{156}$?
4. How do you calculate the average of two numbers and how is it useful here?
5. What are other methods to visually represent square roots on a number line?

Tip: To estimate the square root of non-perfect squares, knowing the closest perfect squares helps place the value more accurately.