AMC 8 · 2012 · #22
Grade 6 arithmeticPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Instead of analyzing all ways to pick the three unknown integers, solve the easier extreme cases first (Tool #9): What is the smallest median we can force? What is the largest? Once we have the range, use Tool #2 to systematically check each integer in that range and confirm it is reachable. The median of 9 numbers is always the 5th element after sorting, so the question reduces to: which positions in the sorted known list can become the 5th slot when we slide three free integers into the line?
With 9 distinct integers in ascending order the median is the 5th number, so we hunt for every integer that can occupy that slot.
Reducing "median of a 9-element set" to "the 5th smallest" is the Grade 6 definition of median for an odd-sized data set.
6.SP.B.5Solve An Easier Related ProblemSlide all three unknowns below 2 so the known six shift up; the 5th slot then lands on 3 — the smallest possible median.
Solving the extreme "how small can it be?" first is the Tool #9 move — replace the general question with an easier boundary version.
6.SP.B.5Solve An Easier Related ProblemSlide all three unknowns above 14 so the known six drop down; the 5th slot then lands on 9 — the largest possible median.
The companion extreme — "how large can it be?" — caps the range. Together the two extremes bound every possible median between 3 and 9.
6.SP.B.5Solve An Easier Related ProblemBetween the extremes every integer works: give each value in 3 to 9 a witness arrangement by dropping the unknowns into the right gaps.
A systematic list of one witness per value (Tool #2) shows nothing in {3, …, 9} gets skipped.
6.SP.B.5Make A Systematic ListCount the integers from 3 to 9 inclusive: 9 - 3 + 1 = 7 possible medians.
Counting consecutive integers from a to b as b - a + 1 is a standard Grade 4 word-problem move.
4.OA.A.3Make A Systematic ListThis AMC 8 problem only needs the Grade 6 definition of median — the middle number of a sorted list — that you already know!