Task restatement (simple)
Four people stand on the four corners of a rectangular room. The pairs are opposite: F and I are opposite corners; D and A are the other opposite corners. From F, the distance to D is 3 m and the distance to I is 5 m. What is the minimum possible distance from F to A (in metres)?
Step-by-step solution (clear, with an implied diagram)
- Place the rectangle on a coordinate grid for clarity. Let F = (0, 0). Since D is 3 m from F along one side, place D = (3, 0).
- F and I are opposite corners, so I is the vertex diagonally across the rectangle. The diagonal FI has length 5 m, so the diagonal from (0,0) to (3, y) must have length 5.
- Use the Pythagorean theorem on the right triangle with legs 3 and y and hypotenuse 5:
3^2 + y^2 = 5^2
9 + y^2 = 25
y^2 = 16
y = 4 (positive length) - Thus the other side length is 4 m. The corner A adjacent to F (and opposite D) is at (0, 4), so the distance from F to A is 4 m.
- Answer: 4 metres. (Units must be metres.)
Why this is the minimum possible distance
Given FD = 3 and FI = 5 with F and I opposite, the rectangle's side-lengths are fixed by the Pythagorean relation; the side adjacent to F must be 4 m. There is no smaller value consistent with those distances, so 4 m is the minimum (and only) possible distance from F to A.
Report on the student’s performance (Ally McBeal cadence — light rhythm, precise content)
Student response: wrote the numeric answer “4” (no units, no diagram, no written working). When asked to rework and explain aloud, the student verbally justified the reasoning fully: identified opposite corners, used Pythagoras to find the missing side (4 m), and concluded FA = 4 m.
ACARA v9 alignment (Year 8 measurement & geometry)
Learning focus: apply properties of rectangles and Pythagoras to solve problems and communicate results with appropriate units and diagrams. The student demonstrated correct procedural and conceptual understanding of Pythagoras and rectangle geometry when explaining verbally.
Assessment, strengths and areas to improve
- Correctness: Correct numerical answer (4). Student understands geometric relationships and used Pythagoras correctly when explaining verbally.
- Communication: Initial written response lacked units (metres) and omitted the requested diagram and working. These are important for full written communication and for awarding full written-credit evidence.
- Consistency: Verbal justification showed complete reasoning; this compensates for the missing written work but does not replace the requirement to show working in written form for assessment records.
Recommended next steps
- Require a corrected written submission that includes: a labelled diagram, step-by-step working, and units (m).
- Reinforce habit: always include units and a diagram for geometry problems — this is part of the expected demonstration of understanding in ACARA v9.
- For future tasks, award full conceptual marks when correct reasoning is demonstrated verbally only if a subsequent written record is provided; otherwise apply partial credit for correct answer with insufficient written evidence.
Legal-brief conclusion on student performance
Conclusion: On the balance of evidence, the student has demonstrated the required mathematical understanding to determine the minimum distance from F to A as 4 metres by correctly applying the Pythagorean theorem. However, the student failed to provide the required written supporting evidence (diagram, working and units) in the first instance. Remedial action: accept the student’s full conceptual demonstration contingent upon a corrected written submission that includes diagram and units. In assessment terms, award conceptual competence; reserve full written-credit pending resubmission.
Evaluator: Homeschool teacher — recommended resubmission requested.