Maths Investigations using Penrose, the Mathematical Cat: Investigating Polygons

polygons-homeschool maths

I have written before about our change to a living maths approach using Life of Fred as our backbone, and I think I might have mentioned wanting to spend some of our time investigating a certain aspect of numbers in greater depth.  I have a few maths based books, but Penrose the cat is by far my most favourite maths teacher 🙂  For our first maths investigation I chose polygons.  The chapter is entitled ‘Penrose discovers mathematical stars’.

Firstly, I jotted down a few things I wanted the girls to investigate together on my white board.  However,  we took lots of rabbit trails, and really did not stick to my aforementioned plans at all!

polygons-homeschool maths

Meeting Penrose, the Mathematical Cat

Next, I read the chapter The Adventures of Penrose the Mathematical Cat, where Penrose (the cat) discovers the beauty of polygons and their polygram (star) making skills:

polygons - homeschool maths

What is a Polygon?

Afterwards, I made sure each of the girls fully understood what a polygon actually was.  It is a shape which:

  • Is flat (2 dimensional)
  • Has 3 or more sides
  • Is a closed shape

Irregular and Regular Polygons

In order to show the girls the difference between regular and irregular polygons, I photocopied a table with both.  This was a useful illustration even though we would only be focusing on the regular polygons.  They made a notebook page with the table and an envelope of the polygons we would be learning about:


They had some fun creating their own irregular polygons and soon learnt that whether regular or irregular, a pentagon always has five sides and five corners.  I gave them a sheet of examples and they had to say whether they thought they were a polygon or not, and if not, why not.  They then cut them out, and using their knowledge of sets, created three sets (Regular Polygon, Irregular Polygon, Non Polygon) and stuck them in their notebook:


Naming Polygons

I photocopied a set of polygons, cut them out to keep in my maths games file.  Using these, I cut out the coloured polygons with five, six, seven and eight sides, explaining that each one had a name based on the number of sides (ie octagon had eight sides just like an octopus had eight legs).  I gave them cut outs of the names and they tried to match the name to the polygon:

I had them watch the following videos about polygons, including a rather cool rap song:

Polygons and Peg-boards

I got out the peg boards and elastic bands and had the girls try to recreate each shape.  This was a great activity because they really had to pay attention to the sides, points and their position to each other:

B6 found this hard and I needed to help her but in the end she was doing them herself, understanding how to ‘read’ each polygon:

I let them take a Polaroid photo of each shape to pop in their note book.  The end results were beautiful.

  • A Pentagon:
  • A Hexagon:
  • A Heptagon:
  • An Octagon:

Polygons and their Stars

I then asked the girls to create some mathematical stars inside the polygons.  They found this really hard:

Soooo, I decided to come at it a different way.  I gathered marshmallows and cocktail sticks and the girls re-made the polygons using these:

A8 found this very easy:

And happily helped her sister, who did not:

Once they had done one of each type (see example above), they attempted once more to make mathematical stars.  This time they found it much easier.  A8 figured out very quickly that if she used one extra marshmallow in the middle the star came together very easily:

It was simple to see that the mathematical stars within a polygon contained the same number of points to the sides (or in deed corners) of that same polygon.

How Can You Tell How Many Points a Polygon’s Star Will Have?

So they had found out the answer to the main question posted on their white board at the beginning of this study – that a polygon’s star will have the same number of points as the number of sides of the polygon.  Yay!  So that was it?  Well, no.  We hadn’t finished Life of Fred – Apples, and I only wanted to cover one investigation per book.

Doodling Mathematical Stars

Vihart’s YouTube offered some much needed inspiration, and whilst I did have the girls watch the video, it was clearly lost on them.  But me, not so much.  In fact it fascinated me so much I felt the peculiar urge to watch it again and again until I felt I understood it well enough to simplify.  Watch it yourself.  This woman is beyond good at explaining her thought process, and she puts mathematical investigation right within the grasp of a mediocre maths person (ie me!):

As I was watching the video for the umpteenth time (it takes a while for anything to sink in….must be age), I had the girls play about with this funky game of making stars on the computer:

This reminded me of something from my childhood.  Y’know, those spiradoodles, where you place a circle thing in a larger ring and draw….I was sure we had one somewhere, and I thought they would be a great introduction to drawing many pointed stars, as well as being a great fun drawing tool as well:

Obviously the video above, whilst excellent, was a little too advanced but it did make me think a return to studying polygons and stars was in order once we had finished multiplications and factorisation.  Fun, fun, fun 🙂

Finding the Polygons

I found a hexagon shape which had been divided into lots of sections.  I photocopied a few and the girls took it in turns to find all the triangles, the quadrilaterals, the pentagons…..basically any polygon they could find.  We used one sheet per type of polygon, and stuck them in to their notebooks.  Can you believe they found over 25 each (I did help B6 just a little), and I am certain there were more to find, only we had run out of time:

This is one of three posts I am writing to showcase our living maths/non text book maths.  The girls have learnt heaps.  And I have not even had one complaint about maths since we began learning this way.  The icon below will take you to the Life of Fred: Apples post:


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