AMC 8 · 2015 · #12
Grade 4 geometry-3dcounting
Pick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Trying to scan all 12 edges and check every pair would mean checking 66 pairs — too many to do safely by eye. Tool #10 (Physical Representation) helps first: pick up any box or cube and look at it. The 12 edges fall into exactly 3 directions (length, width, height), with 4 parallel edges in each direction. Tool #7 (Identify Subproblems) turns the big count into three identical mini-counts: "how many pairs from 4 edges?" Tool #2 (Systematic List) handles each mini-count by listing the pairs in order, so we never double-count or miss one. Add the three mini-counts and we're done.
Grab any box: a cube's 12 edges point in just 3 directions, 4 per direction, so they split into 3 groups of 4.
Touching the cube makes the three directions obvious — and "same direction" is exactly what "parallel" means.
4.G.A.1Create A Physical RepresentationTreat one direction at a time: the whole count becomes three copies of the same small question, how many pairs from 4 edges?
Splitting one hard count into three identical easy counts is the Tool #7 subproblems move.
4.OA.A.3Identify SubproblemsList one group's pairs in order: {1,2},{1,3},{1,4},{2,3},{2,4},{3,4} — that's 6 pairs, with none missed or repeated.
A systematic list with a clear ordering rule is the safest way to count pairs without double-counting.
4.OA.A.3Make A Systematic ListThe 3 directions each give the same 6 non-overlapping pairs, so the total is 3 × 6 = 18 pairs.
"3 groups of 6" is exactly what multiplication means in Grade 3.
3.OA.A.1Identify SubproblemsPick up a box and look at its edges — they only point in 3 directions, so the cube problem is really just "6 pairs, three times" — pure Grade 3 multiplication!
More like this
Same archetype — closest grade level first.
