Fill the Flower

Explaining and justifying
Noticing and using mathematical structure
  • Geometry
Grade level
Use an App
pattern shapes
Pattern Shapes

How can you put together shapes to fill a larger shape?

pattern blocks arranged in the shape of a flower

How can you fill the flower with pattern shapes without any of the same shapes sharing a side?

  • The first 4 shapes have been placed for you.
  • If you use 2 of the same shape, make sure that their sides don’t touch each other. It’s okay if their vertices (corners) touch each other.
  • If you get stuck, remove some of the shapes and keep trying.
How could you get started?
  • Which pattern shapes could you place next to the green triangle? What about next to the red trapezoid?
  • How can turning a pattern shape help it fit?
  • Which shapes do you think will be more useful? Which shapes won’t be useful?
Ready to explore more?

Here are some other puzzles you can solve with the Pattern Shapes app. (For these puzzles, it’s okay to have matching shapes touch each other.)

  • Try filling this turtle with exactly 12 pattern shapes.
  • Try filling this fox with only 2 kinds of pattern shapes.
For Teachers: More about this activity

This open-ended activity asks students to fill a flower picture with pattern shapes without any of the same shapes sharing sides. Solving this problem requires students to put together shapes to compose larger shapes, rotate and slide shapes, apply problem-solving skills, and use spatial reasoning related to the sides and corners of two-dimensional shapes. This activity also prepares them for more sophisticated work with the angles of two-dimensional shapes in later grades.

There are multiple strategies that students may use as they fill the flower with shapes. Many will place pattern shapes one at a time into the flower, being careful not to let any of the same shapes touch. If they get stuck, they may remove one or more shapes and replace them with other shapes. This adds an element of perseverance to the task. Alternatively, students may first fill the flower shapes by letting matching shapes share sides, then remove the matching shapes that share slides and replace them with other shapes. As students engage with this activity, they may make generalizations about which pattern shapes are more useful than others for this problem, recognizing that the vertices of some shapes are problematic for further placements. Many students will correctly conclude that the red trapezoid, blue rhombus, green triangle, and yellow hexagon are the only possible shapes that can be used. These are the only shapes with sides and vertices that line up well against each other and the outline of the flower.

The Pattern Shapes app is the featured app for this problem, and there are numerous ways to place the pattern shapes successfully within the flower.

  • Here is an example of a student who is on their way to successfully filling the flower with pattern shapes. Here is another example, with a different placement of shapes.
  • While it is possible to resize the pattern shapes in this app and successfully fill the flower, we recommend that students keep the default size (large) of the shapes as they work.

Encourage students to use math words to explain their thinking. As they work, model the use of geometric shape names. For example, instead of referring to “the yellow shape,” call it “the hexagon.” Ask questions such as these to extend students’ thinking:

  • Which pattern shapes can be swapped for a pair or group of smaller pattern shapes? How could swapping pattern shapes like this help you find more than one solution?
  • Do you see a group of 3 pattern shapes that form a hexagon? How could repeating that group of shapes help you fill the flower? How can turning (rotating) that group of shapes help you find a solution?
  • Are you using some pattern shapes more than others to fill the flower? Are there some shapes that aren’t usable? Why do you think that happens?

If you’re interested in using the Pattern Shapes app to create customized puzzles for your students, you can make your own black outlines by putting together pattern shapes to form a larger shape, then selecting the dashed-line icon at the bottom of the screen.