Problem
Count segments joining points on a loop
Six dots sit evenly around a circle. Two dots are joined by a straight segment, and A-B is the same segment as B-A. Find how many different segments can be drawn.
Reasoning · Grade 2-1 Connecting Points to Make Shapes
Your answer
Watch out Five per dot suggests 30, but every segment was counted once from each end — halve it.
How to solve
Strategy Make a Systematic List — We label the 6 points and list every pair without repeating, which guarantees we miss none and count none twice. Drawing helps us see the segments, and a counting pattern (each point joins to 5 others) lets us check the total.
1STEP 1
Label the points
Number the dots 1 to 6 and join only smaller to larger.
A segment is just two endpoints joined by a straight line, so naming the endpoints lets us track every line we draw.
4.G.A.1Draw A Diagram2STEP 2
List the segments from each point in order
Then each dot contributes 5, 4, 3, 2, 1 new segments.
5 + 4 + 3 + 2 + 1
Only listing pairs in increasing order automatically skips the duplicate (1-2 and 2-1 are the same line), so the list is complete with no repeats.
2.OA.B.2Make A Systematic List3STEP 3
Add up the list
Adding them: 5 + 4 + 3 + 2 + 1 = 15.
5 + 4 + 3 + 2 + 1 = 15
Each line was listed exactly once, so adding the group sizes gives the exact total.
2.OA.B.2Make A Systematic List4STEP 4
Check with a counting pattern
Halving 6 × 5 = 30 also lands on 15.
(6 × 5)/2 = 30/2 = 15
Counting from both ends doubles every segment, so halving fixes the double count and confirms 15.
3.OA.A.3Look For A PatternCounting the lines from every point in turn counts each segment twice, so halving that total confirms 15.
Why?
A segment has two endpoints, and it gets counted once at each of them, so the sweep over all points delivers exactly double the truth.
🧱Counted twice, so halveIf every item got counted from both of its two ends, the total is exactly double the truth, so halving fixes it.
Why?
Each segment is named by its pair of endpoints and each pair names one segment, so counting segments is counting pairs.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Answer
15 segments
5 + 4 + 3 + 2 + 1 = 15
Two ways of counting (the 5+4+3+2+1 list and the 6x5 then halve method) both give 15, and 15 is a sensible amount of lines among 6 points - not too few, not impossibly many.
Takeaway
List the pairs in order (5, then 4, then 3...) and you will never count a line twice - that is all it takes!
- Label the points
- List the segments from each point in order
- Add up the list
- Check with a counting pattern