Robot on the Grid: Same Endpoint, Different Path
A fifth-grade robotics club programs a floor robot on a coordinate grid marked in decimeters, starting at (0, 0). The program repeats one instruction block four times: "move right 3, move up 2." A student predicts the robot will finish at (12, 8) and writes the whole trip as "right 12, up 8." The club president says those are two different descriptions and wants the endpoint confirmed, the total distance the robot travels along its path, and whether the robot ever stands on a point where the x-coordinate equals the y-coordinate. The club then writes a second program that starts at (1, 0) and repeats the rule "add 2 to x, add 3 to y." They want to know whether that robot reaches (7, 9), where it stops after five blocks, and how to tell without tracing the whole grid by hand.
1. Find the first robot's endpoint and its total distance traveled, then explain precisely how "right 3, up 2, four times" differs from "right 12, up 8."
Read the worked answer
Endpoint: four blocks of (right 3, up 2) give right 4 × 3 = 12 and up 4 × 2 = 8, so the robot stops at (12, 8). The student's prediction is correct.
Distance traveled: each block covers 3 + 2 = 5 decimeters, so 4 × 5 = 20 decimeters. The shortcut description "right 12, up 8" also totals 20, so distance alone cannot distinguish the two.
The difference is the route, not the endpoint. The program makes a staircase, passing through (0,0), (3,2), (6,4), (9,6), (12,8). "Right 12, up 8" is one long L that reaches only the corners. The two descriptions agree on start, finish, and total distance but disagree on which grid points the robot occupies - information the club needs if the robot is carrying a sensor or must avoid an obstacle.
2. Does the first robot ever occupy a point where the x-coordinate equals the y-coordinate? Justify your answer by listing the points.
Read the worked answer
List every stopping point: (0,0), (3,2), (6,4), (9,6), (12,8). Compare coordinates at each: only (0,0) has x = y. The others have x larger than y by 1, 2, 3, and 4 respectively.
Check the in-between points too, since the robot passes through them. Moving right from (0,0) visits (1,0) and (2,0), whose y is 0 - not equal to x. Moving up from (3,0) visits (3,1), not equal. Repeating the same check at each block gives the same result: during a rightward move the y-coordinate is fixed at the previous value, and during an upward move x is fixed, so no new equality appears.
Conclusion: the robot stands on x = y only at the start, (0,0).
3. For the second program starting at (1, 0) with the rule "add 2 to x, add 3 to y," determine whether it reaches (7, 9), state its position after five blocks, and describe a method that avoids guess-and-check.
Read the worked answer
Generate the first few points: (1,0) → (3,3) → (5,6) → (7,9) → (9,12) → (11,15).
It reaches (7, 9) after three blocks.
After five blocks it is at (11, 15).
A rule-based method is faster than plotting: after n blocks, x = 1 + 2n and y = 0 + 3n. To test (7, 9), solve 1 + 2n = 7, giving n = 3; then check the y-rule at that same n: 3 × 3 = 9, which matches. Because a single n satisfies both coordinates, the point is on the path. Testing (10, 12) the same way fails, since n = 4.5 from x but n = 4 from y - mismatched n values mean the point is never visited.