Lesson 5.1
The Birthday Cake Problem and the Quadrilateral Family
Focus: constraint task — bucket sort — hierarchy — conditional logic thread from Unit 2
This lesson has two tasks and they work in sequence. Students encounter a square as ‘the configuration where all four sides are equal and the two diagonals are the same length’ before they encounter it as ‘a shape with four equal sides and four right angles.’ That reframing — from object to solution — is what the rest of the unit is built on.
The Birthday Cake Problem (from Peter Winkler, Mathematical Mind-Benders, 2007)) has exactly six solutions. Nearly every group misses at least one, and the most commonly missed are the specific rhombus (60°/120° angles, equivalent to two equilateral triangles sharing an edge) and the four-points-of-a-regular-pentagon configuration (trapezium). The completeness question — how do you know you haven’t missed one? — is more important than any individual solution. Push groups to justify their claim of completeness rather than simply list configurations. Get a visual of all solutions for the Guardian here.
The bucket sort follows. Groups receive physical shape manipulatives and sort them at the boards into self-defined categories. The shapes should include convex and concave quadrilaterals alongside the named types. The concave quadrilateral matters: it belongs to no special bucket except the most general one, and its presence forces a confrontation with what the word ‘quadrilateral’ actually means at its broadest.
The key consolidation moment is the buckets-within-buckets question. When a group notices that every shape in their ‘4 right angles’ bucket is also in their ‘2 pairs of parallel sides’ bucket, they have discovered that rectangles are a subset of parallelograms. Ask them to draw it that way. Then ask: can you write what you found as a conditional statement? That move — from a picture of circles inside circles to a logical claim — is the connection back to Unit 2 that carries through the whole unit.
The square will appear in more buckets than any other shape. That is exactly right. A square satisfies the most constraints, so it belongs to the most categories. Students who find this satisfying rather than confusing have understood something important about how the hierarchy works.