Reasoning · Grade 4-2 Tessellation
Problem
Angles at a point must total 360 degrees.
Angles that sit side by side around a point simply add, and one whole turn measures 360 degrees - the same additive idea used to find a missing angle in a diagram.
4.MD.C.7Draw A DiagramA shape tiles the plane only if copies of its corners can be gathered at a point to make exactly a full turn.
Angles gathered around one point must fill 360 degrees, or a gap or an overlap appears at that point.
A single shape whose corner angle divides into 360 with something left over is refuted on the spot, which is how the pentagon falls.
Triangles 1 and 2 both work.
Knowing that a triangle's angles total 180 degrees turns an infinite tiling question into one small addition, and the two-copies-make-a-parallelogram picture can be checked with two paper triangles.
8.G.A.5Solve An Easier Related ProblemQuadrilaterals 3, 4 and 6 work too.
Cutting a four-sided shape along a diagonal into two triangles is a Grade-level move that immediately gives the 360 degree total, and pairing an upright house with an upside-down one is easy to check with two paper cut-outs.
8.G.A.5Solve An Easier Related ProblemHexagon 7 fits three 120s, so it works.
The formula is just the triangle fact used repeatedly: a hexagon splits into 4 triangles, so its angles total 720 degrees, and the six equal angles are 720 divided by 6.
8.G.A.5Solve An Easier Related ProblemPentagon 5's 108 degrees never fills 360.
Checking whether 108 divides 360 evenly is an ordinary division question, and once it fails there is nothing left to try - the shape is eliminated for good.
8.G.A.5Eliminate PossibilitiesCircle 8 has no angles, so gaps remain.
Pushing three coins together and looking at the curvy hole in the middle settles this without any calculation at all.
4.MD.C.7Draw A DiagramThe ones that cannot tile are 5 and 8.
Walking through all eight shapes in order guarantees the answer is complete - no shape was skipped, and every one has a recorded reason.
4.MD.C.7Make A Systematic ListTiles only fit if the corners meeting at a point add to exactly 360 degrees - the pentagon's 108 degree corners never do, and a circle has no corners at all.