Learning outcomes By the end of this unit you should be able to: explain what is meant by a symmetry of a plane figure; specify symmetries of a bounded plane figure as rotations or reflections; describe some properties of the set of symmetries of a plane figure; explain the difference between direct and indirect symmetries; use a two-line symbol to represent a symmetry; describe geometrically th
4 Two identities Section 4 introduces some important mathematical theorems. Click 'View document' below to open Section 4 (7 pages, 237KB). Learning outcomes By the end of this unit you should be able to: Section 1: Sets use set notation; determine whether two given sets are equal and whether one given set is a subset of another; find the union, intersection and difference of two given sets. Section 2: Functions determine the image of a given function; determine whether a given function is one-one 2.1 Introduction In this section we shall define the complex number system as the set R × R (the Cartesian product of the set of reals, R, with itself) with suitable addition and multiplication operations. We shall define the real and imaginary parts of a complex number and compare the properties of the complex number system with those of the real number system, particularly from the point of view of analysis. 5 Hyperbolic functions In Section 4 we introduce the hyperbolic functions sinh, cosh and tanh, which are constructed from exponential functions. These hyperbolic functions share some of the properties of the trigonometric functions but, as you will see, their graphs are very different. Click 'View document' below to open Section 4 (5 pages, 104KB). 3.1 Expressing numbers in scientific notation Earlier you looked at place values for numbers, and why they were called powers of ten. 2.4.1 Try some yourself 1 Write the following as a number to a single power: (a) 26 ÷ 22 (b) 1010 ÷ 107 (c) 78 ÷ 74 2.1 The impact of a power Here is a tale based on an ancient Eastern legend, which gives an idea of the impact of raising a number to a power. A long time ago there lived a very rich king whose son's life was saved by a poor old beggar woman. The king was naturally very grateful to the woman, so he offered to 2.2 Tables and percentages Tables often give information in percentages. The table below indicates how the size of households in Great Britain changed over a period of nearly 30 years. 1 Modelling static problems The main teaching text of this unit is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Once you have completed the workbook and exercises return to this page and watch the video below, ‘The arch never sleeps’, which discusses a practical application of some of the ideas in workbook. Click 'View document' to open the workbook (PDF, 0.8 3 Work on your own mathematics Two activities are given below. You are asked to work on them in turn and to record not only your working, but observations on what you notice about your emotions as you work through step by step. W 1.2.2 Content School mathematics curricula often focus on lists of content objectives in areas like number, arithmetic, statistics, measurement, geometry, trigonometry, and algebra. A typical list of content objectives might contain over one hundred objectives to be introduced or revisited and learned each year. These can be seen as hierarchical in nature but many textbooks do not attempt to organise the objectives in ways that enable the bigger underpinning ideas to become apparent to the pupils. In addit 1.4.4 O is for Objectivity One of the characteristics of ‘good’ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective. This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may be 5 Conclusion – new ways of looking at the world There is a variety of new approaches or terms that are interlinked, and have been prominent throughout this book. All of them have played a part in this book's journey through the scientific, political, philosophical and social implications of climate change. Governance of climate change is about: decision making under uncertainty; understanding and representing vulnerability even when vulnerabilities are difficult to assess or unknowable; and making every aspect of human 4.4 Signing everyone up to sustainability The proposers of step-by-step progress towards sustainability would include in their plans many of the ideas proposed in the previous two subsections. However, what distinguishes this group is that they stand in the middle of the scale between faith in unfettered business voluntarism and a conviction that radical transformations are required. Their incrementalism is reflected in the kinds of pragmatic solutions they propose; their radicalism shows in the way they think about new roles and pro 3.2.2 Good green governance in five easy steps It would be a serious error to imagine that ‘government’ has evaporated: it still shapes many aspects of our lives from beginning to end (welfare, taxation, transport and, of course, the recording of births and deaths). Governments are the central negotiators of environmental-change policies at international level, and of their implementation at national and local level. Nevertheless, for many areas of life, governance is undeniably a better description both of new processes that are alre Learning outcomes By the end of this unit you will: Have gained an understanding of the four dimensions of globalisation in relation to climate change; Be able to distinguish between the three approaches to achieve sustainability; Know the difference between ‘government’ and ‘governance’; Identify what makes ecological citizenship distinctive; Understand how the medium of the web can aid transitions to sustainability. 3.2 The past temperature of the planet Measuring the concentration of lead in the ice is called a direct measurement: the ice sample is melted and the water produced contains a very small but readily measured quantity of lead dust. A very accurate set of scales would be needed to measure it, but it is a directly measured quantity. There are also many indirect measurements that can be made using proxy data. The concept for using proxies is both simple and brilliant: one measured property allows inference about other states o 6.3 The role of active citizens and communities Few people agree that individuals should take the main responsibility for tackling environmental issues. For example, in a 2007 poll of over 2000 UK citizens, 70% agreed that the government should take a lead in combating climate change, even if it means using the law to change people's behaviour. However, over 60% disagreed that there was nothing they could do to avert climate change and over half agreed that they would do more if others did more too, although 40% thought that recycling was 6.2 The role of individuals and households You've been considering how to reduce your own carbon footprint to help tackle the worst effects of climate and other environmental changes. To that extent, ‘I’ as an individual consumer has a role to play. But unless you live alone, you share your household with other people, a group that could be called ‘we’. Everyone in the household may have similar views on living lightly. But, even within a household, there may be different views and priorities about what, if anything, sho
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Example 6
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Activity 3 Constrained numbers
