Reasoning · Grade 6-2 Loci and Traced Paths
Problem
Name the corners and follow the string.
Actually doing it with card and thread makes the rule obvious before any calculation: the string turns about one point until it meets an obstacle, then the obstacle becomes the new turning point.
4.G.A.1Create A Physical RepresentationRound the tied corner it sweeps 300 degrees.
The three angles of any triangle add to 180 degrees, so each angle of an equilateral triangle is 180 / 3 = 60 degrees. Subtracting that one blocked wedge from the full 360 degrees around a point is the only angle work the big arc needs.
8.G.A.5Draw A DiagramOnce caught, 2 cm of string is left.
Every time the string catches on a corner it loses exactly one side length, because that much of it is now lying along the side. Subtracting one side length is the whole rule.
4.MD.C.7Identify SubproblemsAt that corner it sweeps 120 degrees.
The turn at a corner is always the supplement of the corner's angle, because the string arrives pointing straight on past the corner and leaves lying along the next side. Sharp corner, big turn.
7.G.B.5Draw A DiagramThe other corner does exactly the same.
The triangle looks the same from both sides of A, so whatever happens at B has to happen at C too. Spotting the mirror saves repeating the reasoning.
7.G.B.5Visualize Spatial RelationshipsA middle 1 cm stays out of reach.
Comparing the 2 cm of free string against the 5 cm side is a single comparison of two lengths, and it is the check that stops you from drawing extra arcs that cannot exist.
4.MD.C.7Identify SubproblemsThe figure is three sectors joined.
Each piece of the drawing is one compass setting held for one measured turn, so a compass and a protractor are the only tools you need.
4.G.A.1Draw A DiagramThe turning totals 540 degrees.
Angles at a point simply add, so totalling the turns is an honest check that no piece of the sweep has been left out or counted twice.
4.MD.C.7Identify SubproblemsThe turning angles of the three arcs must add up correctly, which checks the whole sweep at once.
The string turns around the corners of the shed, and the turns it makes are fixed by how much of a full turn the corners leave free.
The swept region is the separate arcs' sectors put together with no gap and no overlap, so their angles account for the whole sweep.
Every time a taut string catches on a corner it loses one side length and swings again through the outside angle -- so just keep asking 'how long is left, and how far can it turn?'