1 Express each of the following percentages as fractions:

• (a) 40%

• (b) 8%

• (c) 70%

• (d)
Author(s): The Open University

In a recipe the quantity of each ingredient needed depends upon the number of portions. As the number of portions increases, the quantity required increases. The quantity per portion is the same. This is called direct proportion. The quantity is said to be directly proportional to the number of portions. If 2 potatoes are required for one portion, 4 will be required for two portions etc. A useful method for direct proportion problems is to find the quantity for one and multiply by the
Author(s): The Open University

Activity 20

Convert each of the following to percentages. Round off the percentages to whole numbers.

• (a)

• (i) 0.8

• (ii) 0.
Author(s): The Open University

Given any number, you now know how to find its square. But, given the squared number, how do you find the original number?

Example 3

After studying this course, you should be able to:

• understand and use the basic terms for the description of the motion of particles: position, velocity and acceleration

• understand, use and differentiate vector functions

• understand the fundamental laws of Newtonian mechanics

• solve mechanics problems in one dimension by drawing a sketch, choosing a suitable x-axis and origin, drawing a force diagram, applying Newtonâ€™s second law, tak
Author(s): The Open University

After studying this course, you should be able to:

• appreciate the concept of force, and understand and model forces such as weight, tension and friction

• model objects as particles or as rigid bodies, and the forces that act on an object in equilibrium

• use model strings, rods, pulleys and pivots in modelling systems involving forces

• understand and use torques

• model and solve a variety of problems involving systems in equ
Author(s): The Open University

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

Course image: Mathias Appel in Flickr made available under Creative Commons Public Domain 1.0 Licence.

Don't miss out:

Author(s): The Open University

Exercise 1

A vector a has magnitude |a|Â =Â 7 and direction Î¸Â =Â âˆ’70Â°. Calculate the component form of a, giving the components correct to two decimal places.

Author(s): The Open University

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
Author(s): The Open University

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)
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 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
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 Î¸ measured anticlockwise from the positive x-axis, then cosÂ Î¸ is the x-coordinate of P and sinÂ Î¸ is the y-coordinate of P (
Author(s): The Open University

After studying this course, 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

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

Course image: fdecomite in Flickr made available under Creative Commons Attribution 2.0 Licence.

Don't miss out:

If reading this text has inspired you to learn m
Author(s): The Open University

Hart, K., Kerslake, D., Brown, M., Ruddock, G., Kuchemann, D. and McCartney, M. (eds) (1981) Children's Understanding of Mathematics 11-16, London, John Murray.
McCague, W. (2003) 'A mathematical look at a medieval cathedral', Math Horizons, April, pp.11-15 and p.31. See also http://www.maa.org.
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Author(s): The Open University

The content acknowledged below is Proprietary (see and conditions made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence) and used under licence.

Course image: rod
Author(s): The Open University

The doâ€“talkâ€“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