A swimming elephant can breathe by holding the end of its trunk out of the water, but if it tried to find its food under the surface, like the desman, it would have to hold its breath. Neither the mammalian lung nor the skin can extract enough oxygen from water to sustain life, so aquatic mammals must come to the surface at intervals to breathe; and all of them – pinnipeds, sirenians and cetaceans – drown if they are prevented from doing so for prolonged periods.

Lungs form 7% of th

Mammals share a number of biological characteristics that mark them out as members of the class Mammalia. Many of these are adaptations to a life on land. For example:

Mammals give birth to young at a relatively advanced stage of development and feed their young on milk.

Most mammals have hair, or fur, covering part or all of the body.

Mammals have a high metabolic rate and maintain a relatively high and constant body temp

7.2.1 The equilibrium constant

An expression for the equilibrium constant of a reaction can be put together from the concentrations of the reactants and products at equilibrium. A concentration of a reactant or product is represented by enclosing its chemical formula in square brackets. Thus, the concentration of NO(g) is written [NO(g)].

To write down the equilibrium constant of a reaction, we start with the concentrations of the products. Each one is raised to the power of the number that precedes it in the reactio

6.3.1 Refinements and difficulties

In Section 6.2, we said that inter-axis repulsions vary in the order:

non-bonded pair–non-bonded pair > non-bonded pair–bond pair > bond pair–bond pair

There is evidence for this in the inter-bond angles in molecules. For example, in wat

3.2 The electronic configurations of atoms

The quantum theory of the atom tells us that we cannot say exactly where an electron in an atom will be at any particular moment; we can speak only of the *probability* of finding an electron at a particular point. So the precise orbits shown in the Rutherford model of Figure 1 misrepresent the arrangement of electrons about

Having discussed *n*th roots, we are now in a position to define the expression *a*^{x}, where *a* is positive and *x* is a rational power (or exponent).

## Definition

If *a* > 0, *m* Author(s):

This unit explores reasons for studying mathematics, practical applications of mathematical ideas and aims to help you to recognize mathematics when you come across it. It introduces the you to the graphics calculator, and takes you through a series of exercises from the *Calculator Book*, *Tapping into Mathematics With the TI-83 Graphics Calculator*. The unit ends by asking you to reflect on the process of studying mathematics.

**In order to complete this unit you will need**

**In Section 4 we prove that some of the properties of the groups appearing earlier in the unit are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse.**

**Click 'View document' below to open Section 4 (9 pages, 237KB).**

**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 'View document' below to open Section 5 (8 pages, 151KB).**

**3.4 Drawing and interpreting graphs **

**A graph shows the relationship between two quantities. These quantities may be very different: for instance, the price of coffee in relation to different years, or the braking distance of a car in relation to different speeds, or the height of a child at different ages. Because the quantities are different, there is no need to have equal scales on the graph, and it is often impractical to do so. However, it is essential that the scales are shown on the axes: they should indicate exactly what **

**After studying this unit you should:**

**be able to solve homogeneous second-order equations;****know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;****know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.**

**First-order differential equations **

**This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.**

**This unit is an adapted extract from the course Mathematical methods and **

*In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in Section 2, in which you will also work with bearings.*

*This unit is an adapted extract from the Open University course *

*It is crucial to remember that you are a learner of mathematics as well as a teacher. In this unit you will be asked to undertake some mathematical tasks. The aim of these tasks is not to improve your mathematics, but to give you experience of doing mathematics for yourself—experience that you can reflect upon subsequently. The reflection is used to develop your awareness of the ways that learners deal with mathematical tasks, and how learners' mathematical thinking is influenced by the way*

*Working in mathematics education involves a sense of both past and future, and how the two combine to influence the present. It may seem that, because the past has already happened, it cannot be altered; however, you can alter how you perceive the past, and what lessons you take from it. Each of us has a personal past in mathematics education—the particular events of our personal lives, who taught us, where, what and how they taught us, and what we took from the experiences. Each of us also*

*After studying this unit you should:*

*reflect in depth on aspects of mathematics learning, whether you are directly concerned with mathematics teaching or simply interested in issues of mathematics education;**examine established views about existing practice in a critical way and engage with research evidence on mathematics and learning.*

*Mathematical processes are different from content in that they overarch the subject and are not thought of as hierarchical. A list of processes could contain:*

problem-solving (including investigating);

mathematical modelling;

reasoning;

communicating;

making connections (including applying mathematics); and

using tools.

*Each of the six processes liste*

*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*

* Technorati reports that over 100 000 new ‘blogs’ are created each day. Because these online diaries offer instant publishing opportunities, you potentially have access to a wealth of knowledge from commentators and experts (if they blog) in a wide range of fields. Most internet searches will turn up results from blogs, but there are some blog-specific search engines such as: Blogdigger*

*Encyclopedias can be useful reference texts to use to start your research. There are some available online, such as Wikipedia, which is a freely available collaborative encyclopedia.*

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