© ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions

FUNCTIONING WITH GEOMETRY AND FRACTIONS Derek Ball and Barbara Ball describe the thinking that has arisen from looking at pictures of polygons within polygons.

Figure 1

Over the past few months, we have worked on the image in figure 1 with a number of groups. Mostly these have been Y8 students in masterclasses, but occasionally they have been older or younger, and sometimes they have been adults. We have usually worked with one projected image with the whole group, and participants have not written or drawn during the discussion. In particular, they have not used isometric paper to recreate the pictures. We have been surprised at how ably some groups have

worked with this image and how other groups seem to have been unable to work with it at all. Here we develop our thesis that working on this image is about functioning mathematically and share some of our experiences. Our starting point is most frequently to show the image and ask the group what they see. Some groups seem able to talk about this, while others seem unable or unwilling to respond to such an open invitation.

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© ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions

•

Functioning mathematically means being prepared to and able to talk about a situation you are facing in order to make more sense of it. Having decided that we are looking at four regular hexagons and that in some of them midpoints of sides are involved, we focus on what fraction of the area of each hexagon is coloured blue. We have had very mixed responses when we ask the group which hexagon they think it would be easiest to start with. • Functioning mathematically means being able to think about where might be a good place to start. Most groups who are prepared to think about where to start suggest the bottom right hexagon. We prompt some groups to start with the bottom right hexagon. We ask what fraction of the area of this hexagon is blue. With just about every group we have worked with someone has suggested . • Functioning mathematically means having hunches about how to proceed in making sense of a situation. With some groups,  is suggested because participants have some idea about why  might be the correct fraction. They are prepared to offer explanations, the most common of which is that the grey triangles can be folded on to the blue triangle and will fit exactly. On one occasion we were offered a quite different, but thoroughly correct explanation (figure 2). With other groups,  seems to be offered because it is a familiar fraction and it looks about right and participants seem to have no clear notion that such an answer could be explained in some way. When pressed to give an explanation, some groups have offered what we – and presumably the participants also – see as ludicrous explanations: the triangle has half the area of the hexagon because the triangle has three sides and the

hexagon has six sides; or because the angle of a triangle is 60° and the angle of a hexagon is 120°. • Functioning mathematically means being able to give yourself convincing reasons for your hunches. If groups have not offered an explanation for the fraction being , folding in the grey triangles might be suggested. With all groups, we would ask how they know that the grey triangles fold exactly onto the blue triangle. This might sometimes involve a discussion of the interior angle of a regular hexagon. It seems clear to us that anyone is greatly helped in their understanding of the geometry of a regular hexagon if they can perceive it as composed of six equilateral triangles. This idea can readily be introduced to children from the age of four (or younger), and yet seems quite unfamiliar to some groups of Y8 students. On the other hand, some participants become highly involved in the justification process. We remember one Y8 student who was clear that the isosceles triangles when folded in exactly filled the angles of the equilateral triangle, but agonised for some time about whether they met in the middle or not. • Functioning mathematically means seeing the need to justify your reasons for believing something is true. Attention is then directed – preferably by members of the group, and, failing that, by us – to the top left hexagon. What fraction of this hexagon is blue? We are sometimes offered  by participants who are unwilling to explain their hunch; perhaps we are being offered any fraction that is bigger than . Sometimes we are offered , the correct answer, on the same basis, we think. We frequently find we need to offer an extra image (figure 3) to help participants find the fraction of this hexagon that is blue. Appreciating the relevance of this image is greatly aided by seeing a regular hexagon composed

Figure 2

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MATHEMATICS TEACHING INCORPORATING MICROMATH 207 / MARCH 2008

© ATM 2008 ● No reproduction except for legitimate academic purposes ● [email protected] for permissions

Figure 3

of six equilateral triangles. Some groups are far more adept than others at seeing the connection between this image and the top left hexagon and also seeing the connection between this image and the bottom right hexagon – the grey triangle is one of three that can be folded in to cover the blue triangle. Having agreed that the blue triangle has an area three times the grey triangle, if the group is reluctant to proceed we ask what fraction of the kite is blue. We are surprised when this is found to be a very difficult question by groups of Y8 students attending a masterclass. • Functioning mathematically means being able to use ratio and proportion to help make sense of situations. This involves being able to manipulate simple fraction concepts that involve a straightforward change of viewpoint. A debate can be had about which hexagon to work on next. Let us suppose we work on the top right hexagon. Hunches about the fraction of the area coloured blue can again sometimes be wild guesswork. Being able to use the results for the first two hexagons to deduce the fraction for this hexagon requires insight that only a few of the Y8 students we have worked with have had unaided. • Functioning mathematically means making connections between situations and is a skill that we think needs to be developed much more than at present. When given the hint that the inner hexagon in the top left picture can be added into the top right picture, participants vary greatly in their ability to deduce the fraction. This may be partly lack of practice in putting together a short chain of reasoning. • Functioning mathematically means being able to deduce a result from results previously obtained. Perhaps another difficulty some participants have in deducing that the fraction of the area

coloured blue in the top right hexagon is  of  =  is the difficulty of seeing the relevance of this fraction statement, or if they do see its relevance of being able to operate with it in a situation that is not a textbook exercise on fractions. This is even more starkly the case when considering the bottom left hexagon. Reaching the point of seeing that  of  of the hexagon is blue does not mean that you can necessarily express this as a simple fraction. Not one student in one large group of apparently quite able Y8 students could complete this calculation, and they demonstrated basic lack of understanding of equivalent fractions, for example, in a practical situation. Our conclusion is that students’ understanding of the manipulation of fractions is often limited to the performance of textbook exercises and is rarely developed in activities requiring the solution of problems. • Functioning mathematically means operating with fractions in problem-solving situations. We like using these images with groups, because they require the ability to solve problems involving geometry and fractions, where there is – we presuppose – a limited amount of knowledge required and where the emphasis is clearly on using skills and understanding that might sometimes be developed only through routine drill and practice. Little geometry is required and – we used to think – little understanding of fractions. Perhaps we are wrong. Equally importantly, we like using these images with groups because we believe that the mathematical thinking involved is that required by anyone who uses mathematics to help them function intelligently in their world. Why not try using this image with your students? Derek and Barbara Ball are retired but run masterclasses.

Note The image is from Task Maths Interactive: Mathematical Art, £95 from Cambridge Hitachi. This CD of images and a drawing program, written by Derek and Barbara, provides plenty of opportunities for functioning mathematically. Go to www.cambridge-hitachi.com/taskmaths for more information.

Barbara and Derek are currently working with Mike Ollerton on Functioning Mathematically, a collection of mathematical activities to be published by ATM in time for Easter conference 2008. Functioning Mathematically will be available as both a book and as a download. The idea for this book originated in a session run by Mike Ollerton at ATM Easter conference 2007, where participants worked on problems similar to those in this new publication. www.atm.org.uk/buyonline/

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