Progressive Tile Arrangements

What This Investigation Is About

Progressive Tile Arrangements is a Criterion B and C investigation task set in a geometric context. Students examine a growing arrangement of tiles — typically on a 2D grid — where each stage follows a consistent rule. They analyse how the arrangement grows, find a general formula, and communicate their findings with precision and mathematical justification.

The Geometric Context

Unlike a purely numerical sequence, tile arrangement investigations have a visual structure that students can exploit. The shape of the arrangement at each stage — whether it forms an L, a cross, a bordered square, or a staircase — connects directly to the algebraic formula. Students who use the geometry to explain the formula score more highly on both Criterion B and Criterion C.

What the Investigation Involves

• Observing and drawing the first several stages of the tile arrangement
• Counting tiles by category (border tiles, interior tiles, total tiles) systematically
• Identifying linear, quadratic, or other relationships between stage number and tile count
• Writing a general formula and verifying it for stages not drawn
• Justifying the formula by referring to the geometric structure — for example, explaining why a quadratic term arises from the area of an expanding square region

Criterion B: Investigating Patterns

A strong Criterion B response identifies patterns beyond the obvious, forms a precise conjecture with correct algebraic notation, and provides a justification that goes beyond testing cases. Students should explain why the pattern holds, connecting it to the geometry of the arrangement.

Criterion C: Communicating

Criterion C assesses the clarity and precision of the mathematical communication. Students should use correct notation, organise their response logically, label diagrams clearly, and write explanations that a reader unfamiliar with the task could follow.

Common Weaknesses in Student Responses

• Listing values without organising them in a table — making it difficult to spot and justify the pattern
• Finding a formula by trial and error without justifying its structure geometrically
• Using informal language ("it keeps going up by 4 each time") without converting this to algebraic form
• Not verifying the formula with an independent stage before claiming it is general
• Poor diagram labelling that makes it unclear which tiles are being counted

Practice Approach

Practise drawing and counting tile arrangements neatly and systematically. Organise data in a table with columns for stage number, tile count, first difference, and second difference — this reveals whether the relationship is linear or quadratic. Then work on writing the geometric explanation alongside the algebraic formula, as this is what distinguishes a high-scoring response from a correct-but-unexplained one.