Problem
Drawing them out: 1 segment gives 2 regions, 2 give 4, and 3 give 7.
Drawing the first few cases lets a 5th grader see the shapes instead of imagining them.
4.OA.C.5Draw A DiagramThe k-th segment crosses the k-1 before it, so it splits into k pieces and adds k regions.
Tracking one new segment at a time turns a scary picture into easy counting.
4.OA.C.5Solve An Easier Related ProblemThe k-th segment drawn adds exactly k new regions.
To make as many regions as possible the new segment crosses every one of the k-1 segments already drawn, and those crossings break it into k parts.
Each crossing splits one part of the new segment into two, so k-1 crossings turn one part into k parts.
Each of those k parts runs through one existing region and cuts it in two, so each part adds one region.
The region it passes through is divided into two pieces that together fill exactly the same space as before.
So the total is 1 for the whole circle plus 1 + 2 + 3 and on up to the number of segments.
The segment number and the regions added march together: 1 adds 1, 2 adds 2, 3 adds 3 — a clear correspondence.
5.OA.B.3Look For A PatternThe circle counts 1, and 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45, so the total is 46.
Once the rule is clear, the answer is just an addition a 5th grader can do.
4.OA.C.5Look For A PatternEach new line that crosses all the others adds as many regions as its own number, so just add 1+2+3+...+9 and one for the circle.