Programming the Floor Robot on the Garden Grid
On the STEM lab floor, a 10-foot by 10-foot grid is taped off with the origin at one corner, and coordinates are measured in feet. A floor robot starts each run at (1, 1). The charging pad sits at (4, 3), and the sample tray sits at (7, 5). The robot moves only in straight horizontal or vertical segments along grid lines, and every program is a list of moves such as "right 3, up 2." Two teams submit programs. Team A writes: right 3, up 2, right 3, up 2. Team B writes: up 4, right 6. Both teams say their robot ends at the sample tray, and both say their program is 10 feet long. The teacher wants the robot to roll over the charging pad on the way. The class must also mark out a rectangular garden bed whose corners include (2, 1), (2, 6), and (6, 1).
1. Check both programs: does each robot end at the sample tray, and is each program 10 feet long?
Read the worked answer
Team A, tracking position after each move (start at (1, 1)):
- right 3 → (4, 1)
- up 2 → (4, 3)
- right 3 → (7, 3)
- up 2 → (7, 5) ← sample tray ✔ Total distance: 3 + 2 + 3 + 2 = 10 feet ✔
Team B:
- up 4 → (1, 5)
- right 6 → (7, 5) ← sample tray ✔ Total distance: 4 + 6 = 10 feet ✔
Both claims check out. Notice why the distances tie: every path from (1, 1) to (7, 5) must cover 6 feet horizontally (7 − 1) and 4 feet vertically (5 − 1), so the total is always 6 + 4 = 10 feet no matter what order the moves come in. Changing the order rearranges the trip but does not shorten it. Distance alone therefore cannot choose between the programs - the class needs a different criterion.
2. Which program drives over the charging pad at (4, 3)? Then write a third program that ends at the sample tray and also passes over the charging pad.
Read the worked answer
Which one touches (4, 3):
- Team A's position list includes (4, 3) as its second stop, so Team A does drive over the charging pad.
- Team B's route runs up the line x = 1 to (1, 5), then across the line y = 5 to (7, 5). Neither segment contains (4, 3): the horizontal leg has y = 5, not y = 3. Team B misses the pad.
So Team A is the workable program.
A third program that also works: right 2, up 3, right 3, up 1, right 1. Position list: (1,1) → (3,1) → (3,4) → (6,4) → (6,5) → (7,5). That route does not hit (4, 3), so it is a useful counter-example of a wrong attempt.
Corrected third program: right 3, up 1, right 0, up 2, right 3, up 2. Positions: (1,1) → (4,1) → (4,2) → (4,4) - this passes through (4, 3) only if the robot is allowed to stop mid-segment.
Cleaner correct answer: right 1, up 2, right 3, up 1, right 2, up 1. Positions: (1,1) → (2,1) → (2,3) → (5,3) → (5,4) → (7,4) → (7,5). This one misses the pad too.
Simplest correct answer: right 3, up 2, right 1, up 2, right 2, up 1. Positions: (1,1) → (4,1) → (4,3) ✔ pad → (5,3) → (5,5) → (7,5) ✔ tray. Total distance 3 + 2 + 1 + 2 + 2 + 1 = 11 feet, which is longer than 10 because this list includes a detour that no shortest path needs.
Any program whose ordered position list contains (4, 3) and ends at (7, 5) satisfies the teacher; there are many, so the class should check the position list rather than compare programs by total distance.
3. The garden bed is a rectangle with three corners at (2, 1), (2, 6), and (6, 1). Find the fourth corner, and decide whether the point (4, 4) lies inside the bed.
Read the worked answer
Finding the fourth corner. The two points (2, 1) and (2, 6) share x = 2, so they form a vertical side 6 − 1 = 5 feet long. The points (2, 1) and (6, 1) share y = 1, so they form a horizontal side 6 − 2 = 4 feet long. The missing corner must complete both sides: it needs the x-coordinate of the far vertical side (x = 6) and the y-coordinate of the far horizontal side (y = 6).
Fourth corner: (6, 6).
Check: (2, 6) to (6, 6) is 4 feet, matching the opposite side, and (6, 1) to (6, 6) is 5 feet, matching the opposite side. Opposite sides equal → rectangle. ✔
Is (4, 4) inside? The bed covers x from 2 to 6 and y from 1 to 6. Since 2 ≤ 4 ≤ 6 and 1 ≤ 4 ≤ 6, the point (4, 4) is inside the bed.
A good double-check is to test a point that should fail, such as (7, 4): its x-coordinate is beyond 6, so it is outside. Confirming that the rule rejects an obviously outside point shows the rule is being applied correctly and not just producing "yes" for everything.