Reasoning · Grade 6-2 Loci and Traced Paths
Problem
The triangle tips 4 times.
Counting the tips is the easiest place to be off by one, so it pays to mark 0, 2, 4, 6, 8, 10 on the road and watch where the bottom side lands rather than dividing 10 by 2 without thinking.
4.OA.A.3Draw A DiagramOnly the non-pivot corners move in a tip.
Turning about a fixed point keeps every distance to that point the same, which is exactly what makes the path a circular arc rather than something complicated.
4.G.A.1Create A Physical RepresentationEach tip turns 120 degrees.
The angles of a triangle add to 180 degrees, so each corner of an equilateral triangle is 180 / 3 = 60 degrees; the tip then swings the triangle through whatever is left of the straight angle, and that leftover is the exterior angle.
8.G.A.5Draw A DiagramOne tip turns the triangle through 120 degrees, whatever corner it pivots on.
At the pivot the corner's own 60 degrees and the turn together fill a straight line, so the turn is what is left of 180.
Three such tips would turn the triangle through one complete turn, which is the check that 120 is the right amount.
In one of the four it is the pivot.
Writing one row per tip and naming the pivot corner is what catches the tip where A stands still. Trying to hold four turns in your head is where this problem goes wrong.
4.G.A.1Make A Systematic ListThe pivot comes round in a regular pattern.
Three corners rolling in rotation means each corner takes a turn as pivot once every three tips, so you can predict the resting tips instead of tracking them one by one.
4.OA.A.3Look For A PatternOne arc is radius 2 cm through 120 degrees.
An arc is just a fraction of a circumference: work out the whole circle, then take the same fraction of it that the angle is of 360 degrees.
7.G.B.4Draw A DiagramThree such arcs make 12.56 cm.
Three one-third arcs of the same radius join up to one whole circle, so the arithmetic collapses to a single multiplication by 3.14.
5.NBT.B.7Look For A PatternWhen a shape rolls, the marked corner rests on the tips where it is the one doing the pivoting -- count those out before you add up any arcs.