Explanation: more Variation

And here is another example of variation theory or at least, I’m going to explain a little bit or say a little bit about a couple of pages from the Max Maths book that is a bit more oriented on geometry. In this case, as you can see, quadrilaterals Quadrilaterals, I think you can even see them used quite a lot also in the literature, in research articles when it’s about variation theory and I think that is because when you’re talking about quadrilaterals or geometric figures then you’re always talking about certain properties. So here for example, on this first page in this section of the Max Maths books, you can see actually that there are some definitions.

Recall the names and properties of these quadrilaterals and you can see that there is some discussion about what a square is and what a rectangle is and what a parallelogram is. And you can see that there are lists of features of these, so that you actually discern the similarities and the differences between these different quadrilaterals. Things like all sides are equal or ‘opposite sides are parallel’ or ‘internal angles are right angles’ etc, etc. So quite a lot of features that you can actually define and you can write down. But of course this doesn’t ensure that you really know how to recognize these quadrilaterals when you see them.

In the course of this section you can see that more and more questions are being

asked: “Look at the quadrilaterals below. They both have the same properties. Can you identify them?”. So you can see two quadrilaterals and they have the same properties, so students are really asked to think about the properties of these quadrilaterals and try to link them

together: are they similar or aren’t they similar? Things like, also things like

a square and a rectangle: is a rectangle a square? Is a square a rectangle? And if you say yes or no, why is that the case or why isn’t that the case? So I think you can already see that by presenting these different definitions, by presenting these different rules and properties of quadrilaterals, you’re already implying some kind of variation between these different figures. And then we go on to the next page where you actually get a kite as well, so not only is it the case that you leave it with a parallelogram and the square and a rectangle, but you actually extend the number of definitions and go to a kite.

And there is a whole building, you could say, of quadrilaterals, that you can build on. So here

a kite is introduced: pairs of adjacent sides are of equal length, a pair of opposite angles are equal. And these things can all be discussed and then another interesting angle on the next page is that you actually start off with a square but then by pushing the sides you can actually change the shape and you get to a rhombus. So here you can really see the relationship between a square and a rhombus and you can actually study what features are the same.

Again all sides are of equal length, which was the case with a square, opposite sides are parallel hmm that’s also something that’s going on, just like a square, but then of course opposite angles are equal hmm yes, that’s also the case in the square, but then of course a square is a special situation where you actually have four right angles. So, again I want to pinpoint and highlight the way

that this is all sequenced: starting off with certain definitions and then extending all of it, highlighting some of the similarities but also highlighting some of the differences, which I think is very essential in variation theory. And then it finishes with the names of the quadrilaterals. And again, this last page

is more like a summary page: you can, you can write in of course the correct answers, so you can think about, well, this is a square, yeah I recognize this. This is a kite. Oh yeah recognize this, this is a rhombus etc etc. And you can see here for example that the way that these figures are positioned is varied. This is a ‘trick’ or this is a very important thing to do, because often students will just assume that the way a kite, for example, is positioned means that a kite always needs to be positioned that way. But actually the orientation of the kite is not so important, what is important are

these properties: does it adhere to the properties of a kite? And one thing you could add here, and that’s not the case in these pages that we’ve chosen as a fragment. What you could also simply do is have a whole row of kites and then have one non-kite, one that looks like a kite but isn’t a kite, or perhaps something that people will think is not a kite, but actually it is. Like for example, as you could say a square is also a rectangle just look at the properties of a rectangle. A square adheres to these properties as well as well.

So you can make sequences of squares, rectangles, kites etc etc, with the only thing varying, is one of those properties. And, well, I’m using the word ‘varying’ already, this I think also cuts to the heart of what variation theory is.