Reasoning · Grade 3-1 Filling and Joining Shapes

Problem

Count distinct shapes from joined pieces

One square, plus two right triangles from cutting another square along its diagonal. All three pieces join edge to edge into one connected shape. Short edges only meet short edges, and a long slanted edge only meets another. Count the distinct shapes, treating turns and flips as the same.
Your answer
How to solve
Strategy Make a Systematic List — Every finished shape covers the same area (one square plus two half-squares = 2 square units), so none can have a gap or overlap. I sort the search into two clear groups: first glue the two triangles to each other and then add the square, or attach each triangle straight onto a side of the square. Cutting paper pieces and sliding them lets me flip each picture to catch any repeats, so I list every shape exactly once.
1STEP 1

Group A: glue the two triangles to each other first, then add the square

The two triangles can join in three ways.

1 + 1/2 + 1/2 = 2
2STEP 2

Group B: attach each triangle onto a side of the square by a leg

Adding the square gives a rectangle, a big triangle and a parallelogram — 3.

3STEP 3

Group B results

Attaching them separately adds a trapezoid, a pentagon and hexagons — 5 more.

2 + 3 = 5
4STEP 4

Remove repeats and count

None coincide under turns or flips: 8 in all.

4 + 4 = 8
Answer
8 shapes
4 + 4 = 8
Every finished shape has area equal to one square plus two half-squares, which is 2 square units, matching the three pieces exactly, so none has a gap or overlap. Grouping by the number of sides gives four 4-sided shapes (rectangle, big right triangle, parallelogram, trapezoid) and four shapes with a dent (one 5-sided, three 6-sided). An exhaustive paper check finds no ninth shape and no duplicate among the eight.
Takeaway

Group the cases (triangles glued first, or each on a square side), draw them all, then flip to remove copies: simple Grade 1 shape-building finds all 8 shapes (4 with straight sides and 4 with a dent)!

  • Group A: glue the two triangles to each other first, then add the square
  • Group B: attach each triangle onto a side of the square by a leg
  • Group B results
  • Remove repeats and count