Reasoning · Grade 3-1 Counting Figures Formed by Joining Dots
Problem
Upright sides of 1, 2, 3, 4 give 4 kinds.
Upright squares only differ by how big they are, and a 5-dot row allows sides of 1, 2, 3, or 4 gaps.
2.G.A.1Identify SubproblemsTilted ones use a step (a, b) and need a + b of 4 or less.
Tilting a square means its side leans by 'a over, b up'; the more it leans the bigger the box it needs, so the lean is limited by the grid.
3.G.A.1Draw A DiagramThe steps (1,1), (2,1), (2,2), (3,1) give 4 kinds, all different sizes.
Each allowed lean gives a square of its own size, and the four leans that fit all give different sizes, so they are 4 separate kinds.
3.OA.A.1Make A Systematic ListNo size repeats across the two groups: 4 + 4 = 8.
Grouping every square by its size and counting one per size makes sure turned or flipped copies are never counted twice.
3.OA.C.7Make A Systematic ListCounting one square per distinct size makes sure that turned or flipped copies are never counted as new kinds.
A turn or a flip carries a square onto a square of the very same side length, so the copy is the same kind in a new position.
Each square has one definite size, so the sizes sort every square into exactly one kind and the kinds can be added up.
Since turned or flipped copies count as one kind, I just list the sizes: 4 upright squares (1, 2, 3, 4) and 4 tilted ones (the leans that still fit) — 8 different squares in all!