7 Geometrical Reasoning
Task 7.4.3 Mean Geometry
Interpret Figure 7.4b as a sequence of moves to construct a square of equal area to a given rectangle. Provide reasons to justify each step.
Comment
Note that the length of the side of the square, being the square root of the product of the lengths of the sides of the original rectangle, is also their geometric mean.
Denoting the sides of the original rectangle by a and b, the circle has radius (a + b)/2 (the arithmetic mean). Using similar triangles and Pythagoras’ theorem shows the vertical side of the square to have the desired length.
Figure 7.4b
Indeed, any number formed by using addition, multiplication, division and square roots (of positive quantities) can be constructed as a segment.This turns out to be the set of numbers that can be constructed using just an unmarked straight edge and a pair of compasses.
Considering areas, even addition is not at all clear. Euclid did know that any polygonal region can be decomposed into triangles. He also proved theorems that show how a triangle can be converted to a rectangle of the same area, and then how a rectangle can be converted to a square of the same area. Pythagoras’s theorem comes to the rescue, because it shows how to add areas of squares. The final square can then be recomposed as a triangle or some other shape by reversing the opera- tions that led to the two squares in the first place. Euclid did not include anything about multiplying areas, presumably because there is no physical object whose meas- ure requires four dimensions.
Geometry offers an entry to ways of constructing chains of reasoning and tying these arguments back to real pictures in the mind. One way of working with geome- try is to start with the geometry and stay with it, but there is a second aspect; geometry is a way into number. In no way should we underestimate the power of being able to think with an unstructured number line. Nor should we underestimate the way a visual starting point can captivate and intrigue. Diophantus, a mathemati- cian who lived and worked in Alexandria about 300 years after Euclid, produced a number of books, some of which survived the upheavals of the next thousand years. Diophantus posed several ‘whole number’ problems. Copies of his books were still circulating amongst mathematicians a thousand years later. The famous remark by Fermat that he had proved that An + Bn= Cn only has whole number A, B and C
solutions when n = 2 was written in the margin of his copy of Diophantus alongside ‘this margin is too small to hold the proof ’.This problem stimulated amateur mathe- maticians for the next 200 years and was only solved by Andrew Wiles at the end of the twentieth century. Perhaps one lesson to take from this story is that whole number starting points can sometimes make hard problems accessible.
7.5 PEDAGOGIC PERSPECTIVES
The pedagogic approach in this chapter has been to stick with just a few results, but to use them to explore the nature of reasoning and proof. This is more in line with the approach to teaching geometry in Japan and Hungary than it is with that com- monly found in England.The curriculum specifies a variety of results, or ‘geometrical facts’, whereas what is most important is that learners realise the fact that there are geometrical facts, rather than internalising specific facts, learning to reason, through shifting attention from specific relationships to properties, and then to reasoning solely on the basis of agreed or accepted properties.Throughout what matters most is 130 BLOCK 2
Reflection 7.4
Review mentally the different roles that Pythagoras’ theorem plays in mathematics.
Review the features that different proofs of the same theorem might offer, and how these might affect a choice of what to offer learners.
that learners’ powers are engaged and developed, and that they encounter persistent mathematical themes such as invariance within change. The pedagogic task is to take a number of important mathematical processes and themes, and then select those items of content to work with in depth that will allow pupils to get a feel that they have ownership of working and thinking visually.
Perhaps the most important power of all is the power to imagine, and to express what is imagined in gestures, movements, diagrams, words and symbols.You have fre- quently been invited to imagine. Certainly both before and after using dynamic geometry software it is important to imagine what is possible, and to re-imagine what happened. When working with others it is valuable to try to imagine what they are describing and to learn to describe accurately what you are imagining. This is how the power to imagine is developed.
Asked to imagine a square, attention may be attracted to many different features: vertices, sides, angles, parallelism, equality of length, equality of angles, but also colour, thickness, orientation and taste. Some aspects are useful mathematically, others are not; some aspects are to be discerned, others are relationships to recognise; others are properties that are being exemplified.The strategy of getting learners to ‘say what you see’ and to negotiate these different features is an effective way of directing attention to mathematically significant features and aspects, and of directing attention to recog- nising relationships and then perceiving properties.
Many pupils find geometry, in the early stages, more accessible than algebra. This accessibility is tied in to being more comfortable in handling materials, reading and working with pictures and figures than manipulating abstract symbols. But pedagogi- cally a teacher needs to move pupils on from enjoying and being comfortable to becoming involved and effective as a geometer. A key to becoming sharper in work- ing with images is contained in the phrase: say what you see – watch what you do.
The gap between working on your own with course materials and working as a teacher with pupils is very sharp here. A pedagogic strategy with a group of pupils, is to ask individuals to ‘say what they see’.There are further strategies, like insisting that what is seen is described in words and not by approaching the white board and waving one’s arms about. The teacher’s job in setting up this situation is to provide the scaffolding for discussing what is seen, but also to fade this scaffolding as the dis- cussion progresses. However, the teacher needs to be watchful to ensure that the removal of the scaffolding does not result in a deterioration of the quality of the geo- metric seeing and reasoning.
A second attribute of proof in geometry is concerned with developing chains of reasoning. Learners need to experience reasoning (the giving of reasons for why assertions are true), and to try to provide reasons for themselves, convincing them- selves, then a friend, then a sceptic.This was why sometimes you were given proofs to follow and sometimes invited to reason for yourself.
This chapter sits within measurement because every thing discussed either involves some sort of measurement (as in lengths, angles, and radii) or plays an important role in measurement. Most of the work required you to reason and not measure. This is central to classical Euclidean geometry.Aristarchos, in 250 BCE, used Euclidean geom-
etry to measure the diameters of the sun and the moon and to estimate the relative distances of the two from the earth. His observations and results led him to postulate that the sun was the centre of our universe and not the earth. This story is a good GEOMETRICAL REASONING 131
example of the interplay between geometrical deduction and producing a measure- ment in a case where it is not possible to lay down a ruler and read off a result. 132 BLOCK 2
Reflection 7.5
Next time you see some mathematics that you find off-putting, try approaching it using ‘say what you see’ and notice the effect on you.
8
Visualising
This chapter develops links between the square of a number, the area of a square, the path traced by a point that moves equidistant from a fixed point and a fixed line, a quad- ratic expression such as x2– 1 or (x + 2)(x – 1) and the graph of a quadratic expression.
The unifying theme of this chapter is how geometrical thinking can be developed to assist in the visualisation of mathematical ideas and, in particular, developing the concept of locus to gain an understanding of graphs.
An important step was made by René Descartes developing the notion of co- ordinates in the seventeenth century. Before Descartes, mathematicians thought of curves as defined by intrinsic properties: a circle is the set of point a fixed distance from a fixed point, a parabola is the set of points equal distance from a fixed point and a fixed line, and so on. Descartes wrote about a fevered dream in which he realised that instead of studying curves by their intrinsic properties, it would be possible to use the power of algebra by specifying curves as an equation relative to a co-ordinate system.This insight transformed the way curves were studied.
This chapter makes links between what we have come to think of as distinct branches of mathematics and shows how these branches of mathematics have applica- tions in other areas of study.
8.1 DYNAMIC GRAPHS
When you hear or see the word graph, what comes to mind? Most likely what the word conjures up is a static picture of co-ordinate axes and a curve. But the static is merely a snapshot from a dynamic ‘film’.This is brought out in the idea of a locus.
Locus
The term locus comes from the Latin for place, and means the set of points that satisfy some condition.