Reasoning · Grade 4-2 Tessellation
Problem
In the Example the angles at a point make 360 degrees.
Gaps and overlaps are decided one point at a time: if the angles around a point add to exactly a full turn of 360° the shapes close up perfectly, and any other total shows as a wedge of white paper or a pile-up.
4.MD.C.7Look For A PatternThe shapes close up with no gap and no overlap exactly when the angles meeting at a point add to a full turn.
Going all the way round one point is 360 degrees, so anything less leaves a wedge of white paper and anything more piles up.
The corner angles sit side by side with no gap between them, so their measures add up like lengths laid end to end.
At a square's corner sit 90 degrees plus two angles.
Copying the structure of the worked Example — one middle shape and two neighbours meeting at each corner — turns the new question into the same sentence with one number changed.
4.MD.C.7Draw A DiagramSo the attached shape needs a corner of 135 degrees.
One subtraction removes the part of the turn that is already used up, and one halving splits the rest between the two equal neighbours — no algebra needed.
4.MD.C.7Look For A PatternThe regular polygon with a 135-degree corner is the octagon.
Listing the shapes in order of their number of sides puts their corner angles in increasing order too, so once the list passes 135° you know you have found the only shape that fits.
8.G.A.5Make A Systematic List90 + 135 + 135 = 360 degrees, closing exactly.
Equal side lengths mean the shapes line up edge to edge, and equal angles summing to a full turn mean they close at the corners — those two checks together are exactly what "no gaps" means.
4.G.A.2Draw A DiagramNo gaps means the angles meeting at one point add to exactly 360° — take away the square's 90°, split what is left between two neighbours, and 135° names the octagon.