Question 1: Calculate the square of 12.

Answer: 144

Question 2: What is the square of 25?

Answer: 625

Question 3: List all perfect squares between 1 and 100.

Answer: [1, 4, 9, 16, 25, 36, 49, 64, 81, 100]

Question 1: A number is a perfect square. When doubled, it becomes 144. What is the original number? Explain your reasoning.

Answer: 48 (original number is 36, because 36 doubled is 72, which is incorrect; the correct approach is to find x where 2x = square of some number, so x = square root of 72, which is not a perfect square, so the question is a trick; actual answer: the number is 36, since 36 doubled is 72, not 144, so reconsider: the question asks for a perfect square doubled to 144, so x^2 * 2 = 144, x^2 = 72, which is not a perfect square; thus, there is no such perfect square, indicating a misconception to identify.

Question 1: A square garden has a side length that is a perfect square number. If the side length is 16 meters, what is the area? Is this area a perfect square? Explain your reasoning.

Answer: Area is 256 square meters; yes, 256 is a perfect square (16^2).

Question 1: Prove that the square of any odd number can be expressed as 8k + 1 for some integer k. Provide a proof or counterexample.

Answer: For any odd number n = 2k + 1, (2k + 1)^2 = 4k^2 + 4k + 1 = 4k(k + 1) + 1. Since k(k + 1) is always even, 4k(k + 1) is divisible by 8, so (2k + 1)^2 = 8m + 1 for some integer m. Therefore, the square of any odd number is of the form 8k + 1.

Question 2: If a number's square is 81, what are the possible original numbers? Are there misconceptions students might have? Explain.

Answer: The original numbers are 9 and -9. A common misconception is to think only of positive roots.

Question 1: A student claims that 49 is not a perfect square because 7 is not a squared number. Identify the mistake and correct it.

Answer: Mistake: Believing 7 is not a square number. Correct: 7 is not a perfect square, but 49 is, since 7 × 7 = 49.

Question 2: Identify the error: The student squares 15 and gets 225, which is not a perfect square.

Answer: Error: 225 is actually a perfect square because 15^2 = 225. The mistake is in the student's statement.