Six Triangles

Noticing and using mathematical structure
Grade level
Use an App
pattern shapes
Pattern Shapes

How many different shapes can you make with triangles?

six green triangles

Using 6 same-sized green triangles, how many different designs can you make?

Each design must include all 6 triangles without gaps. Each triangle must match at least 1 other triangle along a side. 

The sides of the triangles should look like this when they match: 

Two green triangles sharing one edge
How could you get started?
  • Tap the green triangle six times to add them to the board.
  • Rotate and join the triangles together.  
  • What letters can you make with the triangles?
Ready to explore more?
  • How many other designs can you make? 
  • Can you rotate or move the triangles to make a new design? 
For Teachers: More about this activity

This task asks students to compose and decompose shapes and to apply spatial reasoning to consider different arrangements of the same six shapes. Students may engage in a range of geometry skills as they work on this task, including recognizing and naming shapes; using small shapes to compose larger shapes; and generalizing that shapes remain the same, regardless of their arrangement in space (e.g., a triangle is still a triangle even when rotated). 

When using the Pattern Shapes app, students might try to fit all the triangles together to compose a recognizable shape, like a hexagon. They may move one triangle at a time to create new figures (as shown here) or they may rotate and rearrange the triangles. They may resize the triangles to fit multiple arrangements on a single screen. They can share and save their arrangements. Encourage students to keep track of the shapes and the number of shapes they have made. They can do this within the app (as shown here) using the drawing or text tool.

Students may attempt to make only recognizable shapes, such as a hexagon or a parallelogram. Invite them to use the triangles to create new shapes. To push their thinking further, invite them to consider: Is there a set number of shapes that can be made with 6 triangles? You might also ask students: Is every shape you’ve created unique? This may push them to consider whether they’ve created a new shape or if a shape they’ve created has been rotated in space, essentially repeating a previous shape.