Reasoning · Grade 6-2 Loci and Traced Paths
Problem
The centre stays 1 cm out from the track.
This is just what a radius is: the fixed distance from the centre to the rim. Once you see the centre as a dot sliding 1 cm outside the track, the rest of the problem is measuring that dot's route.
4.G.A.1Draw A DiagramAs the coin rolls, its centre stays exactly one radius out from the track the whole way round.
The centre is always one radius from the point of the rim that touches, because every point of a circle sits that far from its centre.
Along a straight stretch the centre's path keeps that same distance from the track, so it runs parallel to it.
In (1) the centre traces a circle of radius 4 cm.
Two circles touching on the outside always have their centres a sum-of-radii apart, and 'always the same distance from one point' is the definition of a circle -- no new machinery needed.
7.G.B.4Solve An Easier Related ProblemThat circle measures 25.12 cm.
The circumference of a circle is 2 x radius x pi, and multiplying 8 by 3.14 is ordinary decimal multiplication.
7.G.B.4Identify SubproblemsIn (2) the straight pieces total 20 cm.
Sliding a segment 1 cm sideways does not make it longer or shorter, so the four straight pieces are just the four sides of the square moved outward, and adding four 5 cm sides is the ordinary perimeter of a square.
3.MD.D.8Draw A DiagramAt a corner the centre traces an arc.
Picture a coin pivoting round the corner of a table: the coin's middle swings round the corner on a small circle whose radius is the coin's own radius.
4.G.A.1Visualize Spatial RelationshipsEach arc turns 90 degrees.
The turn at a corner is the supplement of that corner's angle, because the centre's radius starts perpendicular to one side and ends perpendicular to the next. A square's right angle turns you a right angle.
7.G.B.5Identify SubproblemsThe four arcs make one whole circle.
Angles at a point add up, and a shape you walk right round turns you through one full 360 degree turn. That makes the four quarter circles reassemble into a single circle you can measure in one go.
4.MD.C.7Identify SubproblemsAdding gives (2) as 26.28 cm.
Lining up the decimal points and adding hundredths is the last step; the geometry is already done.
5.NBT.B.7Identify SubproblemsRoll a circle once all the way around any track and its centre travels the length of the track plus exactly one extra circle of the roller's own size.