Chemical elements contain atoms of the same atomic number. But most materials consist of chemical compounds. These are a combination of the atoms of two or more chemical elements. Such combinations often occur in simple numerical ratios. Thus, when sodium metal (Figure 2b) and chlorine gas (
Brazil is undergoing what is considered its worst economic crisis in seventy years, and there is usually no agreement when it comes to the causes of this situation. President Rousseff and the Labor Party say that it was the corollary of the “International Crisis,” a ghost of the 2008 depression created in their minds. The reality, however, is different. Since expresident Lula Da Silva of the Labor Party entered office in 2003, the government has clung to the typical Keynesian pro
This unit lays the foundations of the subject of mechanics. Mechanics is concerned with how and why objects stay put, and how and why they move. In particular, this unit â€“ Modelling static problems â€“ considers why objects stay put.
Please note that this unit assumes you have a good working knowledge of vectors.
This is an adapted extract from the Open University course Author(s):
The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.
All materials included in this unit are derived from content originated at the Open University.
This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.
This is an adapted extract from the Open University course Mathematical methods and models (MST209)
This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.
This unit is an adapted extract from the course Mathematical methods and models (MST209
1 Firstorder differential equations
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.
Click 'View document' to open the workbook (PDF, 1.6 MB).
All materials included in this unit are derived from content originated at the Open University.
Exercise 1
A vector a has magnitude aÂ =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.
1.4.2 Displacements and bearings
The displacement from a point P to a point Q is the change of position between the two points, as described by the displacement vector
If P and Q represent places on the
ground, then it is natural to use a bearing to describe the
direct
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 â€˜15Â° to the East of due Northâ€™,
or NÂ 15Â°Â E. This is an instance of a
bearing. Directions on the ground are typically given like
this, in terms of the directions NorthÂ (N),
SouthÂ (S), EastÂ (E)
1.3: Summing vectors given in geometric form
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.
1.2: Converting to geometric form
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 aÂ =Â a _{1} iÂ +Â a _{2} j, what are its magnitude a and direction Î¸?
The first part of this question is dealt with using
Pythagorasâ€™ Theorem: the magnitude of a v
1.1: Converting to component form
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 Î¸ measured anticlockwise
from the positive xaxis, then
cosÂ Î¸ is the xcoordinate of
P and sinÂ Î¸ is the
ycoordinate of P (
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 SectionÂ 2, in which you will also work with bearings.
This unit is an adapted extract from the Open University course
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.
Activity 3 Constrained numbers
W
Task 10 The MÃ¶bius 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 MÃ¶bius band as shown in
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:

problemsolving (including investigating);

mathematical modelling;

reasoning;

communicating;

making connections (including applying mathematics); and

using tools.
Each of the six processes liste
1.1 Experiences of learning mathematics
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 lefthand side to record
After studying this unit, you should:
understand some current issues in mathematics education, such as the relationship of mathematics content to mathematics processes.
understand a variety of approaches to the teaching of mathematics such as 'dotalkrecord'
be able to approach mathematical problems and tasks in a flexible way.