Reasoning · Grade 6-2 Area of the Shaded Region
Problem
Slide it and watch where the ink sweeps.
Sliding paper never changes its size or shape, so the traced-off arc is an exact copy of the first arc, just moved down. That single fact is what the whole solution will rest on.
8.G.A.1Create A Physical RepresentationThe band is the region under the starting arc minus the moved semicircle.
When a shape is hard to measure but the shape around it is easy, measure the easy one and subtract. Working out what is left over is often far less trouble than working out the leftover directly.
6.G.A.1Change Focus Count The ComplementThat region splits into a semicircle and a rectangle.
One extra line turns a lumpy shape into two shapes you have known how to handle for years. The rectangle is 12 by 3 because a straight slide keeps the diameter horizontal and the same length.
6.G.A.1Identify SubproblemsThe semicircles cancel, leaving just the rectangle.
This is the whole idea of the problem: a slide moves area around without creating or destroying any, so the moved copy cancels the original and the only new area is the strip the shape swept across.
8.G.A.1Change Focus Count The ComplementThe two semicircles cancel, so the swept band is exactly the rectangle the shape slid across.
Sliding the shape does not change its size, so the arc at the finish is an exact copy of the arc at the start.
The same semicircle is added at one end and taken away at the other, so it leaves the difference between the two regions untouched.
That rectangle is 36 cm².
Once the curved parts have cancelled, the only calculation left is a Grade 4 rectangle area.
4.MD.A.3Identify SubproblemsThe long way also gives 36 cm².
Running the arithmetic once confirms the shortcut is real and not wishful thinking, and it makes clear that the 3.1 in the question is there to test whether you notice it is not needed.
7.G.B.4Draw A DiagramSliding a shape never makes or loses area, so the two semicircles cancel and the inked band is just the 12 cm by 3 cm rectangle it swept.