Reasoning · Grade 3-2 Circles and Counting Shapes
Problem
A new segment adds one piece, plus one more per crossing.
Every crossing point is a new place where a region gets split, so crossings are exactly what create extra pieces.
2.OA.B.2Solve An Easier Related ProblemEach new crossing point splits one region into two, so crossings are exactly what create the extra pieces.
A segment entering a region and leaving it cuts that region into two, so one crossing adds exactly one piece.
The pieces fill the circle with no gap and no overlap, so the total count is the first piece plus one for every cut made.
The 4 and 7 for three segments match this rule.
Because the rule reproduces both numbers the problem already gave us, we can trust it for 4 segments.
3.OA.D.9Look For A PatternFour non-crossing segments give the fewest, 5, and all six crossings the most, 11 pieces.
Going from 3 to 4 segments raises the most-pieces count from 7 to 11, exactly 4 more, because the new 4th segment can cross all 3 old ones plus add its own piece.
3.NBT.A.2Look For A PatternSo each target needs target minus 5 crossings.
Since one crossing equals one extra piece, choosing the target just means choosing how many of the 4 segments are allowed to cross.
3.OA.A.3Draw A DiagramDrawing them out hits every count from 5 to 11.
Drawing one extra crossing at a time lets us climb the targets one piece at a time, so every count from 5 up to 11 is reachable.
2.G.A.1Draw A DiagramEvery time two cuts cross, you get one more piece — so to hit your target, just count out exactly how many crossings you need!