In the following subsections, we apply the vector ideas introduced so far to displacements and velocities. The examples will feature directions referred to points of the compass, known as bearings.

The direction of Leeds relative to Bristol can be described as ‘15° to the East of due North’, or N 15° E. This is an instance of a bearing. Directions on the ground are typically given like this, in terms of the directions North (N), South (S), East (E)
Author(s): The Open University

The following activity illustrates how the conversion processes outlined in the preceding sections may come in useful. If two vectors are given in geometric form, and their sum is sought in the same form, one approach is to convert each of the vectors into component form, add their corresponding components, and then convert the sum back to geometric form.

Author(s): The Open University

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ Theorem: the magnitude of a v
Author(s): The Open University

In some applications of vectors there is a need to move backwards and forwards between geometric form and component form; we deal here with how to achieve this.

To start with, we recall definitions of cosine and sine. If P is a point on the unit circle, and the line segment OP makes an angle θ measured anticlockwise from the positive x-axis, then cos θ is the x-coordinate of P and sin θ is the y-coordinate of P (
Author(s): The Open University

On completion of this unit you should be able to:

• convert a vector from geometric form (in terms of magnitude and direction) to component form;

• convert a vector from component form to geometric form;

• understand the use of bearings to describe direction;

• understand the difference between velocity and speed;

• find resultant displacements and velocities in geometric form, via the use of components.

Author(s): The Open University

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.

## Activity 3 Constrained numbers

W
Author(s): The Open University

This unit focuses on your initial encounters with research. It invites you to think about how perceptions of mathematics have influenced you in your prior learning, your teaching and the attitudes of learners.

Author(s): The Open University

The content acknowledged below is Proprietary (see made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence  and conditions) and is

All other materials included in this unit are derived from content originated at the Open University.

1. Joi
Author(s): The Open University

In this unit you have been introduced to the difference between mathematical content and processes. You have worked on the do–talk–record (DTR) framework for learning mathematics.

Author(s): The Open University

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
Author(s): The Open University

## Task 10 The Möbius band

Take a long thin strip of paper (preferably squared or graph paper) about 30 cm by 3 cm. Give one end a half twist and then tape it together. This is a Möbius band as shown in
Author(s): The Open University

Assuming that both the content of mathematics and the processes need to be included in programmes and curricula, the problem becomes one of how a suitable curriculum can be structured. One possibility is to construct a very specific curriculum with clearly defined objectives for both content and processes separately, and possibly with suggested learning activities. However, content and process are two complementary ways of viewing the subject.

An alternative is to see the curriculum in
Author(s): The Open University

You will come to this unit with many memories of mathematics, both as a teacher and a learner. It may help if you start by recalling memories of learning mathematics and making a record of them in your notebook.

When you work on a task, get into the habit of having your notebook to hand to record your thinking. Use the notebook in any way that helps you to think about the work you have done. Some people find it helpful to divide a page into two columns using the left-hand side to record
Author(s): The Open University

This unit is aimed at teachers who wish to review how they go about the practice of teaching maths, those who are considering becoming maths teachers, or those who are studying maths courses and would like to understand more about the teaching process.

This unit is from our archive and is an adapted extract from Teaching mathematical thinking at Key Stage 3 (ME624) which is no longer taught by The Open University. If you want to study formally with us, you may wish to explore other cour
Author(s): The Open University

Except for third party materials and otherwise stated (see terms and conditions), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

The following files accompany the exercise in Section 4.2

Clicking on 'View document' below opens an extract from Section 4.2 of the unit (PDF, 1.7 MB) which accompanies the audio clips, also below. Listen to each of them in turn with the extracted pages open (you may like to print them out). Work on the problems at the appropriate places – you'll find the answers at the foot of this page.

After studying this unit you should:

• be able to perform basic algebraic manipulation with complex numbers;

• understand the geometric interpretation of complex numbers;

• know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.

Author(s): The Open University

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
Author(s): The Open University

Online bookshops and some of the major search engines offer ‘Alerts’ services. These work by allowing you to set up a profile once you have registered on their site, and when there are items meeting your criteria you receive an email. The good thing about alerts is that you don’t have to do anything once you have set up your profile. The downside, particularly with alerts services from the search engines, is that given the extent to which internet traffic is on the increase whether new
Author(s): The Open University

The process of keeping up-to-date in your chosen subject area is useful for your studies and afterwards, for your own personal satisfaction, or perhaps in your career as part of your continuing professional development.

There are a great many tools available that make it quite easy to keep yourself up to date. You can set them up so that the information comes to you, rather than you having to go out on the web looking for it. Over the next few pages, you will be experimenting with some
Author(s): The Open University