Problem
Counting right triangles in a figure
A square is cut into the seven tangram pieces. Every side must lie on a drawn tangram line. Joined pieces count only when they form a single triangle. Count every right triangle you can find.
Reasoning · Grade 3-1 Counting Large and Small Shapes
Your answer
Watch out Joining pieces seems to keep making triangles, but most close into a square or a pentagon instead.
How to solve
Strategy Make a Systematic List — I count in two organized groups: first the single pieces that are already right triangles, then the right triangles formed by joining pieces. To find the combinations safely I look at the big square's structure - the two large triangles join into one big right triangle, and all five smaller pieces join into the matching big right triangle. Imagining (or sliding) the pieces helps me see which unions actually make a triangle and which make a square or a slanted shape instead.
1STEP 1
Count the single pieces that are right triangles
Five pieces are already right triangles: 5.
2 + 1 + 2 = 5
A right triangle is a triangle with one square corner; just sort the seven pieces into 'triangle' and 'not a triangle' and the triangles are five.
3.G.A.1Make A Systematic List2STEP 2
Join the two large triangles
The two large ones join into half the square as one triangle.
1 new triangle
Two equal right triangles put together along their long side make one larger right triangle - the same way two set squares make a bigger one.
K.G.B.6Create A Physical Representation3STEP 3
Join the five smaller pieces into the other half
The other five together make a triangle of the same size.
1 new triangle
The square is split by the diagonal into two equal right-triangle halves, so the leftover five pieces must close up into a right triangle that matches the large pair.
K.G.B.6Identify Subproblems4STEP 4
Check other ways of joining pieces
No other join closes into a triangle — none more.
A piece-union is only a triangle if it has exactly three straight sides; checking the shapes shows the only triangle-shaped unions are the two big halves.
3.G.A.1Make A Systematic List5STEP 5
Add the two groups
Altogether 5 + 2 = 7.
5 + 2 = 7
The singles and the combinations are different triangles, so adding the two counts gives every right triangle exactly once.
2.OA.B.2Make A Systematic ListThe single pieces and the joined pieces are different triangles, so adding the two counts finds every right triangle exactly once.
Why?
A triangle either is one of the drawn pieces or is made of several of them, and it can never be both at once.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
Why?
A joined triangle is exactly the pieces inside it put together with no gap and no overlap, which is what makes it a genuinely new shape to count.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Answer
7 triangles
5 + 2 = 7
Seven is reasonable: there are 5 triangular pieces, and the only larger triangles you can actually trace are the two halves of the big square, giving 5 + 2 = 7. Every counted shape has three sides and a square corner, and the square and parallelogram pieces were correctly left out because they are not triangles.
Takeaway
Count the triangle pieces first (5), then look for bigger triangles you can build by joining pieces (2) - that's all 7!
- Count the single pieces that are right triangles
- Join the two large triangles
- Join the five smaller pieces into the other half
- Check other ways of joining pieces
- Add the two groups