AMC 8 · 2004 · #23
Grade 8 rate-ratioProblem
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?
Pick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Tess runs counterclockwise around the rectangular block $JKLM$, starting and ending at her home corner $J$. We must pick the graph (out of A–E) whose vertical axis — Tess's straight-line distance from $J$ — rises, peaks at the corner opposite $J$, and falls back to $0$ as time goes by.
Givens: $JKLM$ is a rectangle with Tess's home at corner $J$; Tess runs at a steady speed counterclockwise: $J \to K \to L \to M \to J$; The vertical axis on each candidate graph is straight-line distance from $J$ (not distance run); Five candidate graphs A–E are sketched with different shapes
Unknowns: Which lettered graph (A, B, C, D, or E) correctly represents her distance from home over time
Understand
Restated: Tess runs counterclockwise around the rectangular block $JKLM$, starting and ending at her home corner $J$. We must pick the graph (out of A–E) whose vertical axis — Tess's straight-line distance from $J$ — rises, peaks at the corner opposite $J$, and falls back to $0$ as time goes by.
Givens: $JKLM$ is a rectangle with Tess's home at corner $J$; Tess runs at a steady speed counterclockwise: $J \to K \to L \to M \to J$; The vertical axis on each candidate graph is straight-line distance from $J$ (not distance run); Five candidate graphs A–E are sketched with different shapes
Plan
Primary tool: #3 Eliminate Possibilities
Secondary: #1 Draw a Diagram, #9 Solve an Easier Related Problem
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.
Execute — Answer: D
5.G.A.2 Step 1 - Draw the rectangle $JKLM$ and mark Tess's straight-line distance from $J$ at each corner she reaches in order.
- At $J$ the distance is $0$; at $K$ it is the side length $JK$; at $L$ it is the diagonal $JL$ (the farthest she ever gets); at $M$ it is the side length $JM$; back at $J$ it is $0$ again.
💡 Plotting distance-from-home at the four corners on coordinate axes is Grade 5 "graph real-world points" work.
8.F.B.5 Step 2 - Use the corner distances to write down three must-have features for the correct graph.
- (1) It starts at $0$ and ends at $0$.
- (2) It has a single maximum value, the diagonal, reached in the middle of the trip — once Tess has gone halfway around (to $L$).
- (3) It strictly rises from $J$ to $L$ and strictly falls from $L$ back to $J$ — no flat (constant-distance) stretch, because $J$ is a corner, not the center of a circle.
💡 Describing a function as "increases, then decreases, with one maximum" is the Grade 8 qualitative-graph reading move.
8.F.B.5 Step 3 - Eliminate (A).
- Graph (A) is a single straight line that keeps rising all the way to the end.
- That violates the must-end-at-$0$ rule — Tess goes back home, so her distance has to come back down.
💡 Reading the right-hand endpoint off a coordinate graph is Grade 8 function-interpretation work.
8.F.B.5 Step 4 - Eliminate (B).
- Graph (B) has flat horizontal pieces — long stretches of "distance does not change." Tess's distance from $J$ would only stay constant if she were running on a circle centered at $J$, but she runs on straight sides of a rectangle whose corner is $J$.
- So her distance changes every moment.
💡 A horizontal segment means the function is constant on that interval — Grade 8 qualitative reading.
8.F.B.5 Step 5 - Eliminate (C).
- Graph (C) shows two separate peaks with a flat region in between.
- The path has only one farthest point ($L$), so there can be only one maximum.
- Two peaks would mean Tess hits her greatest distance twice, which is impossible.
💡 Counting maxima on a graph is Grade 8 function-feature reading.
8.F.B.5 Step 6 - Eliminate (E).
- Graph (E) also has flat horizontal pieces (constant distance), and it never comes back down to $0$.
- Both features clash with the rules from Step 2.
💡 Same Grade 8 check — flat means constant, and the right end must touch $0$.
8.F.B.5 Step 7 - Confirm (D).
- Graph (D) starts at $0$, rises to a single peak at the middle of the time interval, then falls back to $0$.
- It also shows four piecewise segments — one per side of the rectangle — and never flattens.
- Every must-have feature checks out, so (D) is the only graph that could represent Tess's distance from home.
💡 After three failed tests, the last survivor is the answer — Grade 8 graph interpretation seals it.
5.G.A.2 Draw the rectangle $JKLM$ and mark Tess's straight-line distance from $J$ at eac 8.F.B.5 Use the corner distances to write down three must-have features for the correct 8.F.B.5 Eliminate (A). Graph (A) is a single straight line that keeps rising all the way 8.F.B.5 Eliminate (B). Graph (B) has flat horizontal pieces — long stretches of "distanc 8.F.B.5 Eliminate (C). Graph (C) shows two separate peaks with a flat region in between. 8.F.B.5 Eliminate (E). Graph (E) also has flat horizontal pieces (constant distance), an 8.F.B.5 Confirm (D). Graph (D) starts at $0$, rises to a single peak at the middle of th Review
Reasonableness: Sanity-check with a $6 \times 4$ block: side $JK = 4$, diagonal $JL = \sqrt{4^2+6^2} \approx 7.21$, side $JM = 6$. The corner distances $0, 4, 7.21, 6, 0$ rise to a single peak in the middle and fall back to $0$ — exactly the shape of graph (D). The two side-runs ($JK$ and $MJ$) give straight-line pieces on the graph, and the two cross-runs ($KL$ and $LM$) give pieces that curve gently toward the peak. None of them is ever flat, which is why graphs (B), (C), (E) had to go.
Alternative: Tool #10 (Make It Physical) is a fast classroom check: have someone walk counterclockwise around a rectangular table while a partner holds a measuring tape from the starting corner. The tape length grows to its longest at the opposite corner, then shrinks back to $0$ — a single rise-and-fall with no flat stretches. Among A–E only (D) shows that shape.
CCSS standards used (min grade 8)
5.G.A.2Represent real-world and mathematical problems by graphing points in the first quadrant (Plotting Tess's distance from $J$ at each corner on a time–distance coordinate grid to set up the qualitative shape of the graph.)8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (Reading each candidate graph for increasing/decreasing intervals, constant (flat) segments, number of maxima, and start/end values, then eliminating any graph whose qualitative features clash with the rise-peak-fall story.)
⭐ When 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!
⭐ When 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!
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