Portuguese Man o’ War puzzles are a new Mini-Mathematical Universe. Click here to see step by step instructions on its implementation in the classroom. Many suitable Mini-Mathematical Universes already exist. The picture above is created by asking how many ways an integer can be expressed as a sum*. To present this to the class I would just show them the empty rectangles. For example, the last rectangle would be an empty 15 x 7 rectangle. They can ask me to fill in the leftmost or rightmost column of any rectangle, or they can take a guess at what one of these columns looks like.

Mini-Mathematical Universes should complement and not replace the traditional way of teaching the Scientific Method through an immersion in the natural world.Students still need to struggle with collecting data from imprecise instruments. Students still need to feel the goo.

*Mathematicians can view a sequence linked to this exploration on the Online Encyclopedia of Integer Sequences here.

**Other people are realizing the strength of this style of activity totally independently from me. See here – a brilliant way to introduce the Euclidian Algorithm by Fawn Nguyen.

This way of looking it as investigating mini mathematical universes is a great way of framing a challenge.

Your first activity here would work well with Cuisenaire rods, either real or in an online environment:

https://nrich.maths.org/content/id/4348/cuisenaire.swf