49. A third-grade teacher displays a multiplication chart. The teacher highlights the multiples of 6 and asks the students to identify patterns they see in the numbers that are highlighted. Which THREE of the following are correct student responses?
Answer: A,B,C
The sums of the digits are all multiples of 3, all the highlighted numbers are even, and no matter which order you multiply the numbers in, you get the same answer.
Students can identify that the highlighted multiples of 6 exhibit specific patterns: the sums of the digits of these multiples are indeed multiples of 3, all highlighted numbers are even, and multiplication is commutative, meaning the order of factors does not affect the product.
A) The sums of the digits are all multiples of 3.
This statement is correct because multiples of 6, such as 6, 12, 18, etc., yield digit sums that are also multiples of 3. For example, 12 has a digit sum of 1 + 2 = 3, which is a multiple of 3, confirming the observed pattern.
B) All the highlighted numbers are even.
This statement is also correct because every multiple of 6 is even. As all highlighted numbers fall within the multiples of 6 (6, 12, 18, etc.), they are guaranteed to be even, reinforcing this pattern.
C) No matter which order you multiply the numbers in, you get the same answer.
This statement is correct as well, reflecting the commutative property of multiplication. Regardless of the order in which the multiples of 6 are multiplied, the product remains the same, an essential concept in multiplication.
D) Each digit in the tens column stays the same for 2 multiples and then increases by one.
This statement is incorrect. In the case of multiples of 6, the pattern of digits does not follow a consistent rule where the tens digit remains the same for two multiples before increasing. For instance, moving from 6 to 12 to 18, the tens digit does not stay consistent in that manner.
E) The highlighted numbers are each divisible by 12.
This statement is incorrect as well. While some multiples of 6 are divisible by 12 (e.g., 12, 24), not all multiples of 6 are divisible by 12 (e.g., 6 and 18). Therefore, this cannot be generalized for all highlighted numbers.
Conclusion
The correct responses accurately highlight the mathematical properties of multiples of 6: they are even, their digit sums are multiples of 3, and multiplication of these multiples is commutative. Other options fail to represent consistent patterns or properties associated with the multiples of 6, making them incorrect in this context.