AMC 10 · 2024 · #13
Grade 8 geometry-2dcountingPick an answer.
AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question itself names a finite collection — C(4, 2) = 6 pairs — and asks how many satisfy a yes/no condition. That is the signature of Tool #2 (Make a Systematic List): enumerate every pair, decide each one, then count the yeses. The deciding mechanism is Tool #13 (Convert to Algebra): once each geometric move is written as a coordinate rule, "do they commute" reduces to comparing two ordered pairs symbolically. Algebra is the right tool here because tracing pictures by hand for six pairs would be slow and error-prone, while the coordinate rules let each check fit on one line.
Read each geometric move off as a coordinate rule, so composing two moves becomes plain substitution.
Grade 8 transformation rules: each move becomes a formula on coordinates, so composing two moves is just substitution.
8.G.A.3Convert To AlgebraList all 6 pairs in index order so none is missed, then compose each pair both orders on (x,y) and compare.
Listing pairs in index order is the simple-event-tree habit from Grade 7 probability: every pair appears exactly once.
7.SP.C.8Make A Systematic ListPair {1,2}: translating then rotating carries the shift around, so the two orders differ — they do not commute.
Grade 6 "are these two expressions equivalent for all values" check — a single counterexample value of (x,y) would disprove it; here the expressions themselves differ.
6.EE.A.4Convert To AlgebraPair {1,3}: translation touches only x and reflection only y, so they act on disjoint coordinates and commute.
Operations on independent coordinates always commute — like first putting on a left sock and then a right shoe vs. the reverse.
6.EE.A.4Convert To AlgebraPair {1,4}: dilation from the origin scales the 2-unit shift too but pure translation does not, so they do not commute.
Dilation from the origin treats the 2-unit shift like every other length and doubles it; translation does not get a second chance to double itself.
8.G.A.3Convert To AlgebraPair {2,3}: both move both coordinates, and the two orders land on different points, so they do not commute.
Geometrically, rotate-then-reflect and reflect-then-rotate give two different reflections (across perpendicular lines), so the outputs disagree.
8.G.A.3Convert To AlgebraPair {2,4}: both fix the origin, and scaling by 2 slides through the rotation's coordinate swap, so they commute.
Scaling by a constant and applying a linear coordinate swap can be done in either order — multiplication by 2 slides through the rotation.
8.G.A.3Convert To AlgebraPair {3,4}: reflection scales y by -1 and dilation scales both by 2, and constant factors reorder freely, so they commute.
Coordinate-wise the operations are just ×(-1) on y and × 2 on both — Grade 6 properties of multiplication say constant factors can be reordered.
6.EE.A.3Convert To AlgebraOf the six pairs, the ones that commute are {1,3}, {2,4}, {3,4} — three in all, giving (C).
After listing each case as commute / not commute, the final count is a Grade 4 "how many in the list satisfy the rule" tally.
4.OA.C.5Make A Systematic ListWhen a problem asks how many out of a small collection satisfy a rule, list every case and decide each one — and once each geometric move is written as a coordinate formula, deciding "do they commute" is just checking whether two short expressions match.