AMC 8 · 2004 · #23
Grade 8 rate-ratio
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Five concrete graphs are offered, so Tool #3 (Eliminate Possibilities) is the natural AMC multiple-choice move: list three qualitative features the right graph must have, then strike any graph that fails one. Tool #1 (Draw a Diagram) makes those features visible — sketching the rectangle and marking distances at the four corners reveals the rise-peak-fall shape. Tool #9 (Solve an Easier Related Problem) replaces the curved Pythagorean pieces with their easier endpoint values: we only need the corner distances (0, side JK, diagonal JL, side JM, 0), not a calculus-style formula.
Sketch JKLM and mark the distance from J at each corner in order: 0, side JK, diagonal JL (the farthest), side JM, then 0 again.
Plotting distance-from-home at the four corners on coordinate axes is Grade 5 "graph real-world points" work.
5.G.A.2Draw A DiagramTurn those into three rules for the graph: start and end at 0, one single peak at the midpoint, and never flat anywhere.
Describing a function as "increases, then decreases, with one maximum" is the Grade 8 qualitative-graph reading move.
8.F.B.5Solve An Easier Related ProblemReject (A): it only climbs and never returns to 0, but Tess runs back home.
Reading the right-hand endpoint off a coordinate graph is Grade 8 function-interpretation work.
8.F.B.5Eliminate PossibilitiesReject (B): its flat stretches mean constant distance, which needs a circle around J — but she runs straight sides.
A horizontal segment means the function is constant on that interval — Grade 8 qualitative reading.
8.F.B.5Eliminate PossibilitiesReject (C): two peaks would mean the farthest point is hit twice, but only L is farthest.
Counting maxima on a graph is Grade 8 function-feature reading.
8.F.B.5Eliminate PossibilitiesReject (E): it has flat stretches and never returns to 0 — it breaks two rules at once.
Same Grade 8 check — flat means constant, and the right end must touch 0.
8.F.B.5Eliminate PossibilitiesConfirm (D): it rises to one midpoint peak, falls back to 0, splits into four segments, and never flattens — every rule holds.
After three failed tests, the last survivor is the answer — Grade 8 graph interpretation seals it.
8.F.B.5Eliminate PossibilitiesWhen the answer choices are graphs, list two or three must-have features (starts at 0, one peak, ends at 0) and cross off any graph that breaks even one — Grade 8 qualitative graph reading is enough!