Depth · one concept from first principles · step 2 of 4
Using matchsticks of equal length, the shape shown on the right was built. How many equilateral triangles of all sizes can be found in this shape in total?
Reading the rows top to bottom, the smallest upward triangles number 1 + 2 + 3 = 6.
Grade 4 students can identify the unit equilateral triangles and add row by row.
4.G.A.2Make A Systematic ListDownward-pointing ones sit between the upward triangles: 1 in row 2, 2 in row 3, giving 1 + 2 = 3.
Separating downward triangles as their own subproblem makes them easy to spot and not forget.
4.G.A.2Identify SubproblemsA size-2 upward triangle covers 4 unit triangles; the grid fits 3 of them (no size-2 downward fits).
Checking each size as a separate subproblem catches the larger triangles the eye tends to skip.
4.G.A.2Identify SubproblemsThe whole figure is 1 size-3 triangle. Total: 6 + 3 + 3 + 1 = 13.
Organizing by size shows a clear pattern and lets the subtotals simply add to the answer.
4.G.A.2Look For A PatternThe total number of equilateral triangles in the figure equals the sum of the counts from the separate size-and-orientation groups.
The groups together hold every equilateral triangle in the figure, and no triangle sits inside two of them at once.
Each triangle has one definite size and points one definite way, so listing it under its own size and direction records it exactly once, with nothing left over.
Once the groups have their separate counts of 6, 3, 3, and 1, finding how many there are altogether is just adding those four counts.
Merging four separate piles into one total lets you push any two of them together first without changing the final sum.
This only needs Grade 4 shape sense and tidy counting by size you already know!
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The same shape shows up later on the AMC, as Casework by Position / Vertex Type, Lattice / Grid Pattern from Small Cases, Overlap / Complement Counting, Small-Domain Constrained Search. Why the shapes repeat →