Hexagonal Numbers – MYP Year 5 Extended Pattern Investigation
MYP Year 5 Extended hexagonal numbers investigation: building, generalising, and justifying quadratic figurate number patterns. Criterion B focused.
What Are Hexagonal Numbers?
Hexagonal numbers are a family of figurate numbers — numbers that can be represented as a geometric arrangement of dots or tiles in a hexagonal shape. The sequence begins 1, 7, 19, 37, 61 … and the nth hexagonal number is given by Hₙ = 3n² − 3n + 1. Year 5 Extended students are not expected to memorise this formula — they are expected to derive it.
Why This Topic Appears in MYP
Hexagonal numbers are a classic Criterion B investigation context. The pattern is visual, the algebra is non-trivial (it produces a quadratic, not a linear rule), and the justification step is genuinely demanding. This makes the topic well-suited for assessing the full range of Criterion B descriptors.
What Students Are Expected to Do
Stage 1 – Building the Sequence
Students typically work from diagrams or are asked to draw the first several hexagonal arrangements. Counting carefully and tabulating (stage number vs total dots) is the essential first step.
Stage 2 – Identifying the Pattern Type
First differences in the sequence (6, 12, 18, 24 …) are not constant — they form their own arithmetic sequence. This tells students the rule is quadratic. Second differences (all equal to 6) confirm this. Recognising and using second differences is an Extended-level skill.
Stage 3 – Finding the General Rule
Students use the second difference method or systems of equations to determine a, b, and c in the quadratic uₙ = an² + bn + c. They then simplify and verify against their table.
Stage 4 – Justifying the Rule
The highest Criterion B marks require justification. A strong response connects the algebraic formula back to the geometric structure — for example, explaining that each outer ring of a hexagonal arrangement adds 6n − 6 dots, making the quadratic growth visible from the shape itself.
Common Mistakes
- Assuming the pattern is linear after seeing only two terms
- Making an arithmetic error in the second-difference calculation and then working with a wrong formula
- Verifying only for n = 1 and n = 2 (the values already in the table)
- Offering a formula without any geometric or algebraic justification