Problem
Count segments and triangles from marked points
Three circles have dots spaced evenly around the rim. They carry 4 dots, 5 dots and 6 dots. A segment joins two of the dots. Count the segments possible on each circle.
Reasoning · Grade 4-2 Counting Figures Formed by Joining Dots
Your answer
Watch out Three from each dot gives 4 × 3, but that counts each segment twice.
How to solve
Strategy Make a Systematic List — With only 4 dots I can honestly list every segment and count them, which gives a trustworthy answer to check everything else against. Numbering the dots around the rim keeps that list orderly so nothing is missed or repeated. Then I look at the list a second way - counting from each dot in turn - which reveals a multiply-then-halve shortcut, and I use the pattern that shortcut produces to handle 5 and 6 dots quickly and to check that the three answers fit together.
1STEP 1
Label the dots so nothing gets lost
Giving the dots numbers keeps the list complete.
dots 1,2,3,4
Giving each segment a single agreed name is what stops 1-3 and 3-1 from being counted as two different lines, which is the mistake this whole problem is built to catch.
4.G.A.1Draw A Diagram2STEP 2
List every segment for 4 dots
Four dots give 3 + 2 + 1 = 6.
3 + 2 + 1 = 6
Working through the starting dots in order and only writing down partners further along the list guarantees every segment appears once and only once.
4.OA.A.3Make A Systematic List3STEP 3
See the same count as multiply-then-halve
Halving 4 × 3 also gives 6.
4 × 3 = 12, 12 ÷ 2 = 6
A segment has two ends, so counting from every dot counts everything exactly twice - halving is the clean way to undo that, and it works no matter how many dots there are.
3.OA.A.3Look For A PatternCounting the chords from every dot counts each chord twice, so multiplying and then halving gives the true count.
Why?
A chord has two ends and is counted once at each of them, so the sweep over all dots delivers exactly double.
🧱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 dot can be joined to every other dot with nothing forbidden, so the doubled total is one count multiplied by another.
🧱Independent choices multiplyIf a first choice can go a ways and each of those leaves b ways for a second, the pair can be made a times b ways.
4STEP 4
Apply the shortcut to 5 dots
Five dots give 10.
5 × 4 ÷ 2 = 10 and 4+3+2+1 = 10
Getting the same 10 from two methods that share no steps is strong evidence the count is right, and the star picture makes the 5 sides plus 5 diagonals easy to see.
3.OA.A.3Look For A Pattern5STEP 5
Apply the shortcut to 6 dots
Six dots give 15.
6 × 5 ÷ 2 = 15 and 5+4+3+2+1 = 15
Sorting the 15 into sides, short diagonals and long diagonals is a third independent way to reach 15, and it uses only what you can see in the drawing.
3.OA.A.3Make A Systematic List6STEP 6
Check that the three answers fit together
Each extra dot adds that many more — it checks.
6 + 4 = 10, 10 + 5 = 15
A pattern that can be explained by what the new dot actually does, rather than just noticed in the numbers, is a pattern you can rely on.
4.OA.C.5Look For A PatternAnswer
6, 10, 15 segments
4 × 3 ÷ 2 = 6, 5 × 4 ÷ 2 = 10, 6 × 5 ÷ 2 = 15
Each answer must be at least the number of dots, since the rim edges alone give 4, 5 and 6 segments, and 6, 10 and 15 all clear that. Each must also grow faster than the number of dots, because every extra dot adds several segments at once, and the jumps 6 to 10 to 15 do. For 4 dots the picture can simply be counted: 4 sides plus 2 diagonals equals 6, which settles the smallest case by eye. The answers are counts of segments, so they are whole numbers with no units, as they should be.
Takeaway
Count the lines leaving every dot, then halve - because every line gets counted once at each of its two ends.
- Label the dots so nothing gets lost
- List every segment for 4 dots
- See the same count as multiply-then-halve
- Apply the shortcut to 5 dots
- Apply the shortcut to 6 dots
- Check that the three answers fit together