Problem
Count all shapes on a dot grid
Six dots form a triangle pattern: 1 on top, 2 in the middle, 3 along the bottom. Three corners on one straight line make no triangle. Count every triangle, large and small, these dots can form.
Reasoning · Grade 2-1 Connecting Points to Make Shapes
Your answer
Watch out Only the bottom row looks flat, but each slanted edge also carries three dots — three flat picks in all.
How to solve
Strategy Change Focus / Count the Complement — Instead of hunting triangles directly, we count every way to pick 3 of the 6 dots, then subtract the few flat (straight-line) picks. Listing the straight lines that hold 3 dots is short and sure, so the complement method is fastest and safe from missing triangles.
1STEP 1
Count every way to pick 3 dots
There are 20 ways to pick 3 of the 6 dots.
(6 × 5 × 4)/(3 × 2 × 1) = 20
Every triangle is some choice of 3 dots, so first we count all choices of 3, then weed out the bad ones.
3.OA.A.3Make A Systematic List2STEP 2
Find the picks that do NOT make a triangle
Three dots line up on the bottom row and on each slanted edge: 3 flat picks.
Three dots in a straight row stay flat and never bend into a triangle, so these are the only picks to remove.
2.G.A.1Draw A Diagram3STEP 3
Subtract the flat picks
Removing the flat ones leaves 20 − 3 = 17.
20 - 3 = 17
All choices minus the flat ones leaves exactly the choices that bend into triangles.
2.OA.B.2Change Focus Count The ComplementTaking the flat picks away from all the picks of three dots leaves exactly the picks that make a triangle.
Why?
Every pick of three dots is either flat or a triangle and never both, so the two kinds together make up all the picks.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Knowing the whole and the unwanted part, subtracting returns the wanted part, because taking away undoes putting together.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
4STEP 4
Sense-check with sizes
Only 5 are the neat ones — the other 12 are slanted triangles.
Seeing the 5 obvious shapes inside the 17 confirms we did not overcount, while remembering the slanted ones confirms we did not undercount.
2.G.A.1Draw A DiagramAnswer
17 triangles
20 − 3 = 17
The total picks (20) is clearly bigger than the triangle count, and removing exactly 3 straight-line picks leaves 17, which is sensible for 6 dots. Counting the neat shapes (5) plus the slanted ones agrees with 17.
Takeaway
Count ALL ways to pick 3 dots, then cross out the few that fall in a straight line - what is left are your triangles!
- Count every way to pick 3 dots
- Find the picks that do NOT make a triangle
- Subtract the flat picks
- Sense-check with sizes