When Euclid wrote about ratios of lengths and areas and similarity, without algebra, theorems were dependent on spatial representations. Five diagrams, all related to each other, appear in his text in various places, so I designed a ‘Match the theorem’ task in the manner of Malcolm Swan’s tasks. There isn’t a one-to-one correspondence and you may have to draw some new diagrams (typical of Malcolm’s card matching tasks). I have used language from the versions of Euclid by Todhunter and Heath, but have also elaborated in my own language – thinking sometimes that I can do a better job (arrogant!). The phrase ‘about the diameter’ can be deciphered from the text and diagrams.
The diagrams could also be used for: ‘say what you see, and can derive, in this diagram’.
Match these diagrams to the theorems below:
The complements of the parallelograms which are about the diagonal of any parallelogram are equal to one another. (This means that if you draw the diagonal of any parallelogram, and split the original parallelogram into two similar ones along the diagonal with some bits left over, the bits are equal in area.)
If a straight line be divided into any two parts, the square on the whole line is equal to the squares on the two parts, together with twice the rectangle contained by the two parts.
If a straight line be divided into any two parts, the square on the whole line, and on one of the parts, are equal to twice the rectangle contained by the whole and the part, together with the square on the other part.
If a straight line be bisected, and then extended (produced) to any point, the rectangle contained by the whole line thus produced, and the part of it produced, together with the square on half the original line bisected, is equal to the square on the straight line which is made up of the half and the part produced.
If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square on the other part, is equal to the square on the straight line which is made up of the whole and that part.
If a straight line be divided into two equal parts and also into two unequal parts, the rectangle contained by the unequal parts, together with the square on the line between the points of the section, is equal to the square on half the line.
Equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and conversely parallelograms which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another (n.b. ‘equal’ here means ‘equal area’).
Parallelograms which are equiangular to one another have to one another the ratio which is compounded of the ratios of their sides.
Parallelograms about the diameter (this means diagonal) of any parallelogram are similar to the whole parallelogram and to one another.
If two similar parallelograms have a common angle, and be similarly situated, they are about the same diameter (that means diagonal).
You can imagine that I was delighted to find the following ‘Further exploration’ suggestion in Jemma Sherwood’s new book: How to enhance your mathematics subject knowledge.
‘Take a look into Descartes’ work on Euclid’s Book VI Proposition 12 to see how a multiplicative product can be constructed from straight lines.’
So this is good news for my sanity – two other people, Descartes and Jemma Sherwood, also think that Euclid could still be a source of some searching ideas about number and spatial relationships.
I don’t want my blogs to turn into advertisements for OUP books but I could not resist writing about Jemma’s book. I won’t say more about this particular Proposition, but I will say more about her book. It is bursting with pedagogic content knowledge – that is, the background understanding and linkages that make school mathematics into a connected, meaningful network rather than a list of things needed for ‘the next test’. Each chunk presents a core mathematical idea through exposition and tasks, with varied representations and critical observations. She has a knack for putting into text pedagogic wisdom that I have only ever heard in teacher talk. For example, I don’t know where I first thought about the difficulty of moving learners away from images and models that turn out to be limited but were useful at the time – probably in a discussion at an ATM session. Jemma puts several of these into text, for example:
‘… once the abstract idea of moving numbers, unknowns, or even entire expressions over the equals sign and inverting, or doing the same thing to both sides to balance the equation, is mastered, then the image of the flowchart, function machine or balance becomes redundant, and this redundancy is what we guide our students towards.’
You can probably see the danger here: we have all encountered teaching that goes straight for the abstract ‘change the side, change the sign’ but Jemma is not advocating that. Throughout the book she describes journeys through numerical and spatial meaning, representations, the meaning of the mathematical actions we do in particular representations, and the need to – eventually – wean students away from depending on a limited image. As well as this she contextualises core mathematical ideas into both the history and advanced issues of the subject expressed in accessible ways that could be adapted for school use.
When I look at any maths text intended for school level I look at two areas: proportionality and functions. This is because they are crucial in later understanding and often introduced in over-simplified, fragmented ways (and sometimes using wrong mathematics!) Ratio, scaling and proportion get twenty pages of Jemma’s book spanning counted bead patterns, rational number, rates of change, gradients, formulation and a segue into continued fractions.
Back to Euclid, this time Book V, for some insights about ratio:
‘Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.’
Translated into Anne Watson’s language this means that to say that is the same ratio as implies that, for some real numbers p and q, if we multiply the numerators by p and the denominators by q, we get three possible relationships. Either:
pa > qb and pc > qd simultaneously
pa = qb and pc = qd simultaneously
pa < qb and pc < qd simultaneously.
In other words, we are comparing to and hence to and this tells us something about equal and unequal ratios, with multiplicative relationships as the focus of attention. Note how hard it is to read this, even when you are very confident and fluent with proportionality.
These relationships are taken for granted in our modern use of proportionality with the benefit of algebra and graphs to demonstrate it. Jemma says, in this context: ‘One of our responsibilities is to make explicit the things we take for granted’. It isn’t a surprise to find that her twenty pages do not start with this approach, but instead she offers a sequence of ways to represent the family of multiplicative relationships involved in understanding proportionality, usually using scaling as the basic idea. I wish I had started my teaching with this book. You can view it on OUP’s site.
Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications. Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.