Patterns in 3D Structures

What This Investigation Is About

Patterns in 3D Structures is a Criterion B investigation task in which students explore how geometric properties — such as the number of faces, edges, vertices, or total volume — change as a 3D arrangement grows according to a rule. The task builds skills in pattern recognition, conjecture, algebraic generalisation, and justification.

The Structure of the Investigation

Students are typically given the first few stages of a growing 3D arrangement — for example, cubes stacked in an L-shape that extends at each stage, or a tetrahedral arrangement that adds a layer at each step. They record measurements, identify patterns, and work towards a general formula.

What Students Are Expected to Do

• Collect data systematically for the first four or five stages of the pattern
• Identify numerical patterns in sequences of results (e.g., differences, ratios)
• State a conjecture in words before attempting an algebraic form
• Test the conjecture against further stages not used to construct it
• Derive or justify the general formula — not just state it

How Criterion B Is Assessed

Criterion B assesses the investigation process, not just the final formula. Achievement descriptors at higher levels require students to form a general rule consistent with the pattern, verify it independently, and provide a justification of why it works — not just evidence that it works for a few cases.

Common Weaknesses in Student Responses

• Testing only the cases that were used to construct the formula — independent verification requires a new case
• Stating a conjecture in purely numerical terms rather than relating it to the geometric structure
• Producing a correct formula without any explanation of why it is correct
• Missing the recursive relationship and going straight to a trial-and-error formula

Connecting Geometry to Algebra

The strength of a Patterns in 3D Structures response lies in connecting what is happening geometrically to the algebraic rule. Students who explain why each new stage adds a particular number of cubes or faces are producing a justification, not just a description.

Practice Approach

Practise drawing the first four stages of different 3D growing patterns before working through the investigation task formally. Focus on organising data in a table with a column for differences between consecutive terms — this often reveals the structure of the general formula.