Problem
Count shapes from points on open lines
Two parallel lines carry 3 points each. Three corners on one line make no triangle at all. Find how many triangles the six points can make.
Reasoning · Grade 2-1 Connecting Points to Make Shapes
Your answer
Watch out The 20 ways of picking 3 from 6 overcount — three points on one line lie flat and make no triangle.
How to solve
Strategy Make a Systematic List — A triangle needs corners from both lines (three corners on one line would be flat), so we split into two simple cases: 2 corners on top + 1 on bottom, and 1 on top + 2 on bottom. Counting each case and adding is systematic and complete. A diagram helps see why all-same-line choices fail.
1STEP 1
See why corners must come from both lines
A triangle must reach across both lines, so there are just two cases.
Three dots in a straight row cannot bend into a triangle, so a triangle must reach across to the other line.
2.G.A.1Draw A Diagram2STEP 2
Case A: 2 corners on top, 1 corner on bottom
Two from the top and one from the bottom: 3 × 3 = 9.
3 × 3 = 9
Each top pair is the triangle's base and each bottom point is its tip, so multiplying the choices counts them all.
3.OA.A.3Make A Systematic ListChoosing a pair of top points for the base and any bottom point for the tip counts all of Case A by multiplying the two choices.
Why?
The base choice and the tip choice do not restrict each other, so every base can be joined to every tip.
🧱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.
Why?
A triangle has either two corners on top or two on the bottom, never both at once, so the two cases can be counted separately and added.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
3STEP 3
Case B: 1 corner on top, 2 corners on bottom
One from the top and two from the bottom gives the same 9.
3 × 3 = 9
This is the mirror of Case A - a bottom base with a top tip - so it also has 9 triangles.
3.OA.A.3Make A Systematic List4STEP 4
Add the cases
The cases never overlap, so 9 + 9 = 18.
9 + 9 = 18
Splitting into separate, non-overlapping cases lets us just add the counts for the total.
2.OA.B.2Identify SubproblemsAnswer
18 triangles
9 + 9 = 18
Choosing any 3 of 6 points gives 20 groups; only the 2 flat groups (the 3 top points, the 3 bottom points) are not triangles, and 20 - 2 = 18 matches the casework, so 18 is correct.
Takeaway
A triangle has to reach across both lines, so just count the two cases (2-and-1, then 1-and-2) and add - that is all Grade 3 multiplication!
- See why corners must come from both lines
- Case A: 2 corners on top, 1 corner on bottom
- Case B: 1 corner on top, 2 corners on bottom
- Add the cases