Open-Top Box Optimiser

Length L30 cmWidth W20 cmCut x4 cm

physical geometry

volume graph · drag the marker

ℓ = L−2x = 22 cmw = W−2x = 12 cmh = x = 4 cm

model

V(x) = x(L − 2x)(W − 2x)

= 4(30 − 2·4)(20 − 2·4)

stage 1 calculus investigation

Build evidence, then prove the pattern.

Task sheet · DOCX Rubric · DOCX

01 · Hypothesise

Will square and rectangular sheets share the same cut-size rule? Choose side lengths and ratios for a reason.

02 · Gather

Test the 5×5 square, at least three more squares, then 2:1 and two other rectangle ratios. Record technology evidence.

03 · Prove

Differentiate the cubic, solve V′(x)=0 with the quadratic formula, and show the valid stationary point is a maximum.

04 · Evaluate

Interpret the result in context; discuss domain, material thickness, folds, measurement error and what ratio should be tested next.

technology evidence log

Capture the current model. Aim for comparable lengths and ratios, not random clicks.

No trials yet · current sheet is 3:2.

Assessment lens · what strong evidence looks like

Concepts & techniques · accurate model, efficient calculus and quadratic-formula work, purposeful technology.

Reasoning & communication · interpret results, test a valid conjecture, justify conclusions, and state limitations using precise notation.

Report conditions · three weeks; typed report up to 8 A4 pages, minimum 10 pt; include neat working, screenshots/figures, appendices and bibliography.

For the teacher

This studio mirrors the Stage 1 investigation rather than replacing it. Begin with the 5 cm square from Part A, then use the presets and dimension controls to choose at least three more squares. The square pattern x = side/6 becomes a conjecture students can see before proving. Part B is intentionally different: hold a ratio such as 2:1, scale it, and compare it with two other ratios before generalising. Every displayed number comes from the same model V(x)=x(L−2x)(W−2x), and the graph marker can be dragged directly. Reveal calculus adds the tangent and the derivative; Find maximum animates to the physically valid stationary point, but students still need to show the quadratic-formula work. Record trial builds a compact technology-evidence table. The final panel converts the assessment criteria into four actions: hypothesise, gather, prove and evaluate, with direct downloads of the original task and rubric.