Investigating Quadrilaterals: Angles and Tessellations

The first thing I wanted to do was have a look at angles.  Although T11 has done lots of work with them, the girls have not.  I showed them how to measure angles using a protractor.  They practiced measuring the angles in different shapes.  Once they were really comfortable with the basics, I asked them how they might go about finding out how much all the angles in a quadrilateral added up to, without the use of a protractor.  Could they take a guess?

T11 knew the answer, but the girls took a guess using what they knew of squares, rectangles and right angles to make a hypothesis.  L11 thought they all would have angles which added up to 360 whereas C11 hypothesised that they all would have angles that added to 360 apart from the deltoids.  Having put forward their hypotheses they began to attempt to prove or disprove it.  This is such a great way to work because there is no right or wrong answer, only a postulation of what they think might happen.  Whether they prove or disprove it, the learning that is done is second to none and therefore all three children come out feeling victorious rather than just my strong maths student:

I had given them a huge supply of paint chips, which were all, rather usefully, rectangular.  All the different quadrilaterals could easily be made from them.  The children found the angles in each quadrilateral added up to 360, with all of the children choosing to use four of the same quadrilateral to demonstrate this:

I asked how it could be proven using just one quadrilateral.  They tried different things until they discovered that by labelling the four angles and cutting the quadrilateral shape into four they were able to place the four angles together to make a full 360 degree angle:

And finally all three children measured the angles in each quadrilateral using a protractor, showing once and for all that the interior angles in all quadrilaterals add up to 360 (give or take for inaccuracies due to the small size of the shapes).  Here are their very roughly tabulated results:

Next I taught the children a little about tessellations.  A tessellation is a pattern made of identical shapes.  These shapes must fit together without any gaps and they should not overlap.

I introduced the children to tessellations using video:

We chatted about tessellations in nature such as tortoise shells, honey comb and fish scales and then I sent them around the house to find some examples.  They came up with brick work, fire guard, puzzle pieces, window segments, tiled floor, graph paper…

I had each of them try some tessellation fun on this  website whilst the others played around with some actual plastic shapes.  This was just play and investigation, with no other goal than simply having fun:

When I felt they had all got the hang of the idea of a tessellated shape I set them an investigation: ‘Do all the quadrilaterals that you have learnt about tessellate?’  I let them decide how they would investigate this and asked them to prove or demonstrate their findings.  Each child thought that all the quadrilaterals would tessellate, the reason being that the angles, when tessellated, would add up to 360.  They demonstrated their findings in the following ways:

They did really well, and I could tell they all really understood that which I was trying to teach.  However, I wanted to show them a way to prove that all quadrilaterals tessellated.  This was a proof I had read about in one of the books in the photo below.  I cut up three lots of 12 pieces of paper and asked the children to draw any 4 sided shape on the top of their pile.  They cut all 12 pieces in this drawn shape, until 12 congruent four sided shapes were obtained:

I asked the children to create a tessellation with their pieces of paper, thus proving that no matter the shape, so long as it has four sides it can be tessellated.  Here are their designs:

To finish up I had them watch this short clip:

I then had them read the tessellation chapters of the following books which basically summed up all they had discovered for themselves:

This was a great lesson in the properties of quadrilaterals, angles, protractor use and of course tessellations.  It was useful to be able to introduce the idea of tessellating shapes prior to our study on Islamic geometric art, when we would be looking at this idea in much greater depth.