Off-the-shelf Masterclass: Sierpinski’s triangle

Explore Sierpinski’s triangle, a beautiful mathematical shape which appears in surprising places. Create the shape in different ways and investigate some of its properties. See how Sierpiński’s triangle links to chaos and find out how simple rules can lead to amazing outcomes.

Sierpiński's triangle drawn on the ground with chalk
Image credit: Melissa Youngern via flickr

About this Masterclass

Sierpinski’s Triangle (properly spelt Sierpiński) is a beautiful mathematical object, and one of a special type of objects called fractals. Fractals are made up from simple rules but appear to be very complex and have lots of amazing properties, on top of being stunning to look at. Sierpinski’s Triangle is even more special than most as it crops up in some surprising places.

The aim of the Masterclass is for students to investigate this amazing and very visual shape. They will explore how to construct it in a number of different ways, from playing 'the chaos game' to looking at how it links to another very interesting mathematical object called Pascal’s triangle. These activities will allow them to explore how something random can end up having very organised behaviour, and how areas such as combinatorics (the number of ways of choosing different options) and simple addition rules can produce some surprising behaviour. Many of the results are counter-intuitive and will develop their questioning skills and pattern spotting. They will find out how simple things can have underlying complexity. If there is time, they can also start to explore the properties of fractals and think more about these beautiful mathematical objects.

Length: 2 hours (see the session leader notes for ideas on how to shorten the Masterclass).

Resources needed

  • 'Sierpinski's Triangle' presentation (download from "Supporting documents" below)
  • Student worksheets (download below)
  • Supporting (helper/session leader notes (download from "Supporting documents" below)
  • Paper and pencils
  • Coloured pencils
  • OHP transparencies (pre-printed with Worksheet 1)
  • Six-sided dice
  • Dry-wipe markers
  • 30cm rulers
  • Isometric paper (or worksheet 7)
  • Internet access to run Geogebra applets (more details in session leader notes and the presentation)
  • OHP (overhead projector) or backlit visualiser (you must be able to shine a light through several transparencies)
  • Computer and data projector
  • Further details of quantities etc. can be found in Session Leader Notes

Worksheets

01 Chaos game transparency worksheet 02 Fractal properties worksheet 03a Small Pascal's triangle worksheet (9 rows x 2 portrait) 03b Small Pascal's triangle worksheet 3A (9 rows x 1 landscape) 04 Addition rules worksheet 05 Large Pascal's triangle worksheet (32 rows) 06 Pascal's triangle colouring worksheet 07 Isometric paper

Solutions

01 Fractal properties worksheet solutions 02 Addition rules worksheet solutions 03 Completed Pascal's triangle worksheet

Supporting documents

Session leader notes Helper notes Additional information Session script Sierpinski's Triangle presentation

Support for this Masterclass

All supporting notes, worksheets and solutions can be downloaded from this page along with the Powerpoint presentation.

The Session Leader Notes offer full notes on the activities and a detailed list of resources, including quantities needed. 

The Session Helper Sheet can be distributed amongst helpers so they can better support the students.

Ask the Ri

You can collect any student questions and email them to the Ri Masterclass team at masterclasses@ri.ac.uk - we will endeavour to answer them before the next Masterclass in your series.

Masterclass series support

For more information on running a Masterclass series and the support, templates and other resources available to you, please get in touch via masterclasses@ri.ac.uk.

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