This answer key shows the full solution for each test question and explains how the 20 points are awarded. Partial credit is given for a valid method with one arithmetic slip, such as a missing factor in a list.
Question 1: List factors (2 points)
a) \(1 \times 28\), \(2 \times 14\), \(4 \times 7\): 1, 2, 4, 7, 14, 28.
b) \(1 \times 45\), \(3 \times 15\), \(5 \times 9\): 1, 3, 5, 9, 15, 45.
Scoring: 1 point for each complete and correct list. Deduct the point if a factor is missing or an extra number is listed.
Question 2: Prime factorizations (2 points)
a) \(84 = 4 \times 21 = 2 \times 2 \times 3 \times 7 = \)\(2^2 \times 3 \times 7\).
b) \(90 = 9 \times 10 = 3 \times 3 \times 2 \times 5 = \)\(2 \times 3^2 \times 5\).
Scoring: 1 point for each correct factorization with prime factors only.
Question 3: Greatest common factors (2 points)
a) \(30 = 2 \times 3 \times 5\), \(45 = 3^2 \times 5\). Shared: \(3 \times 5 = \)15.
b) \(64 = 2^6\), \(96 = 2^5 \times 3\). Shared: \(2^5 = \)32.
Scoring: 1 point for each correct GCF.
Question 4: Least common multiples (2 points)
a) Multiples of 10: 10, 20, 30, 40. \(8 \times 5 = 40\). LCM 40.
b) Multiples of 9: 9, 18. \(6 \times 3 = 18\). LCM 18.
Scoring: 1 point for each correct LCM.
Question 5: Rewrite sums as products (2 points)
a) GCF of 40 and 24 is 8. \(40 : 8 = 5\), \(24 : 8 = 3\). \(8(5 + 3)\). Check: \(64 = 8 \times 8\).
b) GCF of 63 and 99 is 9. \(63 : 9 = 7\), \(99 : 9 = 11\). \(9(7 + 11)\). Check: \(162 = 9 \times 18\).
Scoring: 1 point for each product that uses the GCF. Using a smaller common factor earns no point.
Question 6: Snack bags (3 points)
a) The GCF of 36 and 54 is \(2 \times 3^2 = 18\). 18 bags.
b) \(36 : 18 = 2\) granola bars and \(54 : 18 = 3\) juice boxes. 2 and 3.
c) \(36 + 54 = 18(2 + 3)\). Check: \(90 = 18 \times 5\).
Scoring: 1 point for each part.
Question 7: Flashing lights (3 points)
a) Multiples of 12: 12, 24, 36. \(9 \times 4 = 36\). The LCM is 36 seconds.
b) \(36 : 12 = \)3 flashes.
Scoring: part a, 1 point for a valid method (listing multiples or prime factors) and 1 point for 36; 1 point for part b.
Question 8: Find the error (2 points)
54 is larger than both 18 and 27, but a factor of a number can never be larger than the number. Noah found the least common multiple (\(18 \times 3 = 54\) and \(27 \times 2 = 54\)) instead of the GCF.
\(18 = 2 \times 3^2\) and \(27 = 3^3\). Shared: \(3^2 = 9\).
The GCF is 9.
Scoring: 1 point for explaining that a factor cannot exceed the number or that 54 is the LCM; 1 point for 9.
Question 9: Prime or not (2 points)
\(51 = 3 \times 17\), so 51 has factors 1, 3, 17, and 51. A prime number has exactly two factors, so 51 is not prime. A quick test: the digits add to \(5 + 1 = 6\), which is divisible by 3.
Mei is not correct.
Scoring: 1 point for a factor pair (3 and 17) or a divisibility test; 1 point for the correct conclusion.
20 points
See every chapter on the Grade 6 math page.