Moving Mean

I've decided to find a math editor to submit my assignments. My handwriting isn't just atrocious—it's also SLOW! :-)

My process is embedded in my notes, but a short explanation is as follows:

I started by defining "What I know," "What I want," and "What can I change?" This borrowed from Mason’s approach and made the problem approachable. For example:

• I knew the mean had to be 5, and that 16 was in the list.

• I wanted to find the largest number possible in the list.

• I explored what could be varied, like the list size or the values of the remaining numbers, while maintaining the required mean.

This breakdown allowed me to identify the relationship between the mean, sum, and the removal of 16, leading to a clear path toward the solution.

I started by “specializing” by working with a small list size, which helped me arrive at the smallest valid list size for the problem. I increased the list size from here, exploring how the sum and the mean changed as I added more numbers.

Specializing was key in helping me establish a base understanding of the relationships between the list size, the sum, and the mean. It allowed me to observe patterns that could be generalized as I moved on to larger lists.

I worked backward when I started from the target mean of 4 (after removing 16) and considered what combinations of numbers could lead to this mean. This helped me check whether certain list sizes would work or not, allowing me to test and refine conjectures.

Once I worked out the conditions for the smallest possible list size, I applied the general pattern discovered to larger list sizes. This allowed me to determine that the mean increased as the list size grew, and I found that the sum and the remaining mean after removing 16 had consistent patterns across different values of n.

Finally, I used the equations to understand the relationship between the total sum, the number of elements in the list, and how removing 16 affected the remaining mean. These equations helped me calculate how the sum and mean changed as elements were added or removed, allowing me to solve for the total number of elements n=12 and determine the largest possible number in the list. By combining these equations, I ensured that both the original and remaining means met the conditions of the problem.

This structured approach, rooted in Mason’s problem-solving framework, allowed me to solve the problem mathematically and conceptually while reflecting on how patterns emerged as I changed the elements of the list.

The total number of numbers in the original list was 12.

The largest possible number in the original list was 34.