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Rhombus Hexagons

How it works

If you are given an outline of a hexagon, can you completely cover it with pattern block rhombi? (This is called tiling the hexagon with rhombi.) If so, how many rhombi does it take? What orientations do these rhombi have? How many rhombi of each orientation are there?

In this activity, students explore the different ways to tile different hexagons with rhombi. In doing this, they may notice that when they look at their tilings, they see something... interesting that helps them to think about what's going on.

Intro handout
Irregular Hexagons handout
Hexagon Side Length Sequence Recording handout
Cube Stacks handout
Rhombus Hexagon and Cube Stacks Recording handouts

Why we like this activity:

  • It’s fun! The tilings are really visually appealing.

  • It helps to develop spatial reasoning.

  • It helps to develop combinatorial reasoning (reasoning about discrete features (like the number of rhombi of each orientation in a tiling) as opposed to continuous ones (like the exact shape of the rhombi and hexagons)).

  • It requires students to engage in mathematical habits of mind:

    • Finding similarities and differences between different tilings.

    • Making observations about the correspondence between rhombus hexagons and 3D pictures of cube stacks.

    • Making observations / finding similarities and differences / looking for patterns / making and testing predictions / understanding and explaining when trying to figure out which hexagons can and can't be tiled by rhombi.

    • Making observations / finding similarities and differences / looking for patterns / making and testing predictions / understanding and explaining when trying to figure out how many rhombi (and how many of each possible orientation) it takes to tile a given hexagon.

    • Making observations / finding similarities and differences / looking for patterns / making and testing predictions / understanding and explaining when trying to figure out which rhombus hexagons can and can't be translated into 3D pictures of cube stacks, and vice-versa.

  • It has a low floor and a high ceiling: It's easy for students to get started making tilings, but there are lots of deep and challenging questions to explore!

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