Question 1: Construct a factor tree for the number 48. Identify any errors in your tree and correct them.

Answer: A correct factor tree for 48 involves dividing into 6 and 8, then further dividing to primes, resulting in 2, 2, 2, 2, 3.

Question 2: Below is a flawed factor tree for the number 36: 36 / \ 4 9 / \ / \ 2 2 3 3 Identify the mistake and correct the factor tree.

Answer: The mistake is dividing 36 into 4 and 9 directly, which is correct, but if students assumed 4 and 9 are primes, they'd be wrong. The correction involves breaking down 4 into 2 and 2, 9 into 3 and 3, which are primes.

Question 3: Using the factor tree method, find the prime factorization of 84. Show all steps.

Answer: 84 = 2 × 42; 42 = 2 × 21; 21 = 3 × 7; so prime factors are 2, 2, 3, 7.

Question 4: Explain why dividing by a composite number (e.g., 6) in a factor tree might lead to errors in prime factorization. Provide an example.

Answer: Dividing by a composite number skips the step of breaking it down into prime factors, leading to an incomplete or incorrect prime factorization. For example, dividing 24 directly by 6 gives 4 and 6, but 6 is not prime; it needs to be broken into 2 and 3 first.

Question 5: Construct the correct factor tree for 100 and explain the common misconception that might occur.

Answer: Correct factor tree: 100 = 10 × 10; then 10 = 2 × 5, so prime factors are 2, 2, 5, 5. A common misconception is dividing 100 directly into 4 and 25 without breaking 25 into primes, leading to incomplete prime factorization.

Question 6: Construct a factor tree for the number 81. What mistake might students make, and how do you correct it?

Answer: 81 = 3 × 27; then 27 = 3 × 9; then 9 = 3 × 3; students might mistakenly divide 81 into 9 and 9 directly. The correction is to break down 81 step-by-step into prime factors: 3, 3, 3, 3, 3.

Question 7: A student constructs a factor tree for 50 but mistakenly stops at 5 and 10. Identify and explain the error.

Answer: The error is stopping at 10, which is not prime. The correct approach is to further factor 10 into 2 and 5, both primes, to complete the prime factorization.

Question 8: Using a factor tree, find the prime factorization of 120. Then, identify the common mistake that can occur if students divide by non-prime factors initially.

Answer: 120 = 10 × 12; then 10 = 2 × 5; 12 = 2 × 6; mistake: dividing by 6 directly, which is not prime. Correct prime factors are 2, 2, 2, 3, 5.

Question 9: Construct a factor tree for 45. Some students incorrectly stop after dividing into 9 and 5. Explain why this is incomplete and how to fix it.

Answer: Stopping at 9 and 5 is incomplete because 9 is not prime. The correct process is to break down 9 into 3 and 3, resulting in prime factors 3, 3, 5.

Question 10: Identify the common error in constructing the factor tree for 60 when a student divides directly into 6 and 10, and explain the correction.

Answer: The error is dividing into 6 and 10 without breaking down 6 and 10 into their prime factors. Correct approach: 60 = 6 × 10, then 6 = 2 × 3, 10 = 2 × 5.