AMC 8 · 2011 · #8
Grade 3 countingPick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only 3 × 3 = 9 ways to pick one chip from each bag, so Tool #11 (Make a Table) is the cleanest way to lay every sum out in a 3 × 3 addition grid — nothing can be missed. Tool #1 (Find a Pattern) then explains the structure of the answer: odd + even is always odd, and the sums are evenly spaced from the smallest (1+2=3) to the largest (5+6=11), so the distinct sums must be 3, 5, 7, 9, 11.
Build a 3 × 3 addition table: Bag A values down the side, Bag B across the top, each cell the sum a + b.
An addition table is the Grade 3 way to organize "each from this set combined with each from that set" — every possible pair gets exactly one cell.
3.OA.A.1Work BackwardsThe table's nine sums are 3, 5, 7, 5, 7, 9, 7, 9, 11; collapsing the repeats leaves 3, 5, 7, 9, 11.
Spotting that sums repeat in a regular pattern — and that the set of different sums forms a Grade 3 arithmetic pattern — is the Tool #1 move.
3.OA.D.9Draw A DiagramThe set 3, 5, 7, 9, 11 has 5 members, so there are 5 different possible sums.
Counting the distinct entries that appear in the addition table is the same equal-groups counting move from Grade 3 multiplication.
3.OA.A.1Work BackwardsThis AMC 8 problem only needs Grade 3 skills — make an addition table, spot the pattern, and count — that you already know!