Problem
Segments joining points on a circle
Eight points sit evenly around a circle. All eight are used by four segments, and no two segments may cross. Same shape with different endpoints counts separately. Count the ways.
Reasoning · Grade 3-2 Circles and Counting Shapes
Your answer
Watch out Point 1 seems free to join anywhere, but leaving an odd count on one side forces a crossing.
How to solve
Strategy Make a Systematic List — Listing every non-crossing pairing by hand for 8 points is error-prone, so we first solve easier related problems (2 points, 4 points, 6 points), record the counts, look for the pattern in those counts, and then build the list for 8 points in an organized way that guarantees we miss none and repeat none.
1STEP 1
Understand the non-crossing rule with the smallest cases
Four points give 2 ways and six give 5.
2 points → 1, 4 points → 2
Starting tiny lets a third grader actually see what 'non-crossing' means before tackling 8 points.
2.G.A.1Solve An Easier Related Problem2STEP 2
Solve the 6-point case to grow the pattern
Point 1 must leave even sides, so it joins only 2, 4, 6 or 8.
6 points → 5
Each new pair of points roughly multiplies the possibilities, so we expect 8 points to give clearly more than 5.
2.G.A.1Solve An Easier Related Problem3STEP 3
See the counting idea: where can one chosen point connect?
Each choice leaves arcs that are the smaller cases already solved.
1 - 2, 1 - 4, 1 - 6, 1 - 8
A chord that leaves an odd lonely group on one side traps a point that must reach across, forcing a crossing — so only even splits survive.
3.OA.D.9Make A Systematic List4STEP 4
Count each branch and add them up
The four cases add to 5 + 2 + 2 + 5 = 14.
5 + 2 + 2 + 5 = 14
Breaking by point 1's partner sorts every arrangement into one and only one box, so adding the boxes counts each way exactly once.
3.OA.C.7Make A Systematic ListSorting the arrangements by which point the first one joins puts every arrangement in exactly one box.
Why?
Point 1 has exactly one partner in any arrangement, so an arrangement can never fall into two boxes at once.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
The boxes cover every arrangement and never overlap, so adding the box counts counts each arrangement exactly once.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
5STEP 5
Confirm the pattern of counts
The run 1, 2, 5, 14 confirms it.
1, 2, 5, 14
Seeing 14 land neatly at the end of a clean growing sequence is strong evidence we neither missed nor double-counted any arrangement.
4.OA.C.5Look For A PatternAnswer
14 ways
5 + 2 + 2 + 5 = 14
14 is larger than the 5 ways for 6 points and the 2 ways for 4 points, which makes sense because adding two more points should give more room for arrangements. It is also far smaller than the 105 total ways to pair 8 points if crossings were allowed, which is reasonable since the non-crossing rule throws most pairings out.
Takeaway
Break a big counting puzzle into the small puzzles you already solved, then add them up — that turns 8 scary points into a friendly 5 + 2 + 2 + 5 = 14!
- Understand the non-crossing rule with the smallest cases
- Solve the 6-point case to grow the pattern
- See the counting idea: where can one chosen point connect?
- Count each branch and add them up
- Confirm the pattern of counts