2.2 Direct proportion 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
3.2.1 Try some yourself Convert each of the following to percentages. Round off the percentages to whole numbers. (a) (i) 0.8 (ii) 0. 1.3 Square roots Given any number, you now know how to find its square. But, given the squared number, how do you find the original number? If the gardener in Author(s): Learning outcomes 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 Newton’s second law, tak Learning outcomes 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 Acknowledgements 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: If reading this text has 1.5 Exercises A vector a has magnitude
|a|Â =Â 7 and direction
θ = −70°.
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 1.4.1 Bearings 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) 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
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 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 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 ( Learning outcomes 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. Acknowledgements 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 References Acknowledgements 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 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 1.5 Studying 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): 1.4.4 O is for Objectivity One of the characteristics of ‘good’ information is that it should be balanced and present both sides of an argument or issue. This way the reader is left to weigh up the evidence and make a decision. In reality, we recognise that no information is truly objective. This means that the onus is on you, the reader, to develop a critical awareness of the positions represented in what you read, and to take account of this when you interpret the information. In some cases, authors may 1.3.1 Introduction You can find a lot of information about the maths and statistics on the internet. To find this information you might choose to use: search engines and subject gateways; books and electronic books; databases; journals; encyclopedias internet resources
Activity 20
Example 3
Exercise 1
Task 10 The Möbius band
