Try some yourself Which of these triangles are similar? 2.7 Rotational symmetry There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim. Look at the trim on the left. It does not have line symmetry but 1.4 Parallel lines Two straight lines that do not intersect, no matter how far they are extended, are said to be parallel. Arrows are used to indicate parallel lines. Try some yourself Calculate all the angles at the centres of these objects. 4.2 A better solution Here is an improved solution which shows working. Suppose you plan to redecorate your bathroom. The end wall has the following shape, with dimensions as shown on the diagram. The quality of the plasterwork is not good and you are considering tiling the wall. 4.3 ‘Find’ questions Suppose you have decided to tile the wall using square tiles of side 10 cm. You are proposing to use the tiles across the full 5 metre width of the wall up to a height of 1.8 m. Find the number of boxes of tiles that you will require to cover the wall if the tiles a 4.3 Section summary The modulus function provides us with a measure of distance that turns the set of complex numbers into a metric space in much the same way as does the modulus function defined on R. From the point of view of analysis the importance of this is that we can talk of the closeness of two complex numbers. We can then define the limit of a sequence of complex numbers in a way which is almost identical to the definition of the limit of a real sequence. Another analogue of real analysis arises 4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem You will no doubt recall that in real analysis extensive use is made of the modulus function 3.3 Section summary In this section we have seen a correspondence between complex numbers and points in the plane using Cartesian coordinates; the real part of the complex number is represented on the real axis (“horizontal”) and the imaginary part on the imaginary axis (“vertical”). We can also use polar coordinates (r,θ) in which case r, the modulus of a non-zero complex number z is positive and θ is an argument of z, defined only to within an additive intege 3.2 Relationship between complex numbers and points in the plane We have seen in Section 2.2 that the complex number system is obtained by defining arithmetic operations on the set R × R. We also know that elements of R × R can be represented as points in a plane. It seems reasonable to ask what insight can be obtained by representing complex numbers as Keep on learning There are more than 800 courses on OpenLearn for you to 6 Curves from parameters In Section 5 we show how functions may be used to sketch curves in the plane, even when these curves are not necessarily the graphs of functions. Click the link below to open Section 5 (8 pages, 151KB). 1 Overview A fundamental concept in mathematics is that of a function. Consider, for example, the function f defined by
This is an example of a real function, because it associates with a given real number 3.3.1 Increasing by a percentage Our everyday experience of percentages includes percentage increases (like VAT at For example, £8 plus 2.1 Introduction Proportion is another way of expressing notions of part and whole. You might say that the proportion of village inhabitants who are children is a quarter, or that the proportion of fruit juice in the punch is two thirds, or that the proportion of sand in the concrete is three quarters. All these examples involve the fractions 1.6 Do, talk and record triad The do–talk–record triad (DTR) is a description of what is likely to take place in collaborative mathematics classrooms. It is concerned with observable events, and with the learner rather than the teacher, though many teaching insights flow from it. Although the order of the triad suggests that it should be followed in a particular sequence, this is not necessarily the case. Sometimes talking comes before doing or recording before talking. It also takes time for a learner to move 1.2.1 To know or to do? The so-called ‘content/process’ debate in mathematics involves discussion of the relative importance of content and process in mathematics. It originated as part of a discussion about the nature of mathematics, particularly of school mathematics, and of the purposes for which mathematics is learned. Identifying content and process in mathematics draws attention to the idea that mathematics is a human activity. As a teacher of mathematics in the UK, you are faced with a national curr 5 Approaches to problem solving You should not expect always to be able to read a problem and then just write down the answer. When you are faced with a written mathematical question or problem to solve, read it carefully. It is important that you get to grips with the question in two ways: first, that you absorb the information given; and second, that you find out what the question is really asking. Your solution will link the two. This method can be summarised by the following questions. World in transition: Migration and Trade World in transition: Managing Resources
Question 1
Question 1
Example 13
Example 14
. It gives us a way of measuring the “closeness” of two numbers, which we exploit in writing expressi
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%, or a service charge of 15%) and percentage decreases (such as a discount of 15%).
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