GB2248715A - A teaching aid to illustrate mathematical principles - Google Patents

Abstract

A pivotally-mounted transparent ruler is movable over a three-hundred and sixty degree transparent protractor marked with various geometric figures and lines that enable the nature and varying relationships of different trigonometric and other functions to be illustrated, and that also can be used to demonstrate a number of mathematical principles and relationships employed in other curriculum areas, such as physics and geography. <IMAGE>

Description

A TEACHING AID TO ILLUSTRATE DYEAXICALLY SOME BASIC MATHEMATICAL PRINCIPLES Mathematical teaching aids are relatively uncommon and the principles illustrated by the present device are currently demonstable only statically (and hence confusingly) by means of partial and sequential diagrams in books or on blackboards, or dynamically by expensive computer-generated models. It is well-known to teachers of mathematics that the average pupil tends to learn only mechanically how to calculate trigonometical and geometrical functions, rather than understanding the principles that underlie these functions and their varying inter-relationships with other mathematical functions.These principles can only be properly grasped as particular cases of more general and varying relationships, and a knowledge of these relationships is essential to proceeeding to higher mathematics, particulary where the differential and integral calculus is concerned. Whereas the mathematical principles underlying the device described below have of course been known for many centuries, and the basic construction of a ruler moving over a three-hundred and sixty degree protractor incorporated in a number of instruments such as the sextant, these have not so far been applied to the development of a simple and inexpensive teaching device that can illustrate the dynamic generation of basic trigonometric functions and geometric relationships.The instrument described has been developed as a teaching aid for pupils up to GCSE level mathematics, but has a number of applications in other currciculum areas, including geography and physics.

The instrument consists of a transparent linear-ruled surface symetrically positioned above, and secured to, the centre of a transparent and specially-marked three hundred and sixty-degree protractor such that the ruler can be freely-rotated by hand. As the linear rule rotates and moves over colour-coded triangles and other markings on the protactor the nature, variations, and inter-relationships of certain geometric and trigonometric functions may be simply illustrated and rough but instructive calculations made.

One particular form of the device is descibed below with reference to the accompanying diagram, which illustrates a plan of the basic instrument with appropriate markings. The diagram is of a device some twenty-two and a half centimetres in diameter, but without the grid markings which are drawn on some versions.

It should be noted that the various markings on the instrument itseif are colour-coded to facilitate concentration on particular areas and features of the protractor. The diagram shows the ruler at an angle of fourty-five degrees to the horizontal, where, for example, the tangent of the angle theta 1 is equal to 1.

The protractor is divided into four quadrants. In the upper left quadrant a triangle is marked out having an angle theta 1 at the centre of the protractor. Starting with the ruler on the horizontal W/E axis, it may be shown how the tangent, sine and cosine of the angle varies in a regular sequence with the angle as the ruler is rotated in a clockwise direction. Rough measurements of the trigonometric ratios may be made on the scales provided, their regular variations noted, and compared with accurate figures in standard tables. Thus for example it may be shown how the tangent of an angle increases from 0 to 1, and then towards infinity.The right-hand upper quadrant demonstrates how, with the hypotenuse of fixed length (now of course a radius) the sines and cosines of an angle vary as functions of the opposite side (y) and the adjacent side (x) respectively, and thus how sine and cosine waves are generated For example, by rotating the ruler from a horizontal position in an anti-clockwise direction it may be shown how the sine of an angle varies from 0 to 1 in the upper right-hand quadrant, then from 1 to O in the upper-left quadrant, then from 0 to -1 in the bottom-left quadrant, and finally from -1 to O in the bottom-right quadrant. It is thus easily seen how, with time, the upper semi-circle is the postive part of the sine wave and the lower semi-circle the negative part of the since wave.It is evident from the additional markings on the principal N/S and W/E axes that, moving in an anti-clockwise direction from E, the four quadrants contain positive and negative values of x and y as follows: (+x, +y); (-x, +y); (-x, -y); (+x,-y).

The four principal axes are thus also Cartesian co-ordinates that can be used to illustrate the principles underlying the graphical representation of equations, an understanding of which is fundamental to grasping the principles of the infinitessimal calculus. One version of this device (not illustrated) also has a grid marked out over the protractor surface, thus allowing points to be plotted and their relationship to angles to be understood.

The upper- and lower-left quadrants illustrate the nature of corresponding, alternate, vertically-opposite, supplementary and complementary angles, and contain the standard proof of the theorem that the sum of the angles of a triangle is equal to one hundred and eighty degrees. The two parallel lines on either side of the main N/S axis demonstrate the nature of transversals and how the various angles formed by intersecting parallel lines are related to each other in value and vary in a regular fashion as the ruler is rotated from a N/S position in either direction. The instrument also permits a number of other mathematical functions such as radian measure to be illustrated and calculated, and has many uses in the teaching of the dynamic geometry of circles, triangles and other figures. The instrument also permits compass bearings and angles of elevation and depression to be illustrated and calculated, and scale drawings to be made.

Claims (1)

1. A mathematical teaching device consisting of a transparent linear-ruled surface symetrically positioned above, and secured to, the centre of a transparent and specially-marked three-hundred and sixty-degree protractor such that the ruler can be freely-rotated by hand. As the linear rule rotates and moves over colour-coded triangles and other markings on the protactor the nature and variations of certain geometric, trigonometric and other mathematical functions may be simply illustrated and rough but instructive calculations made.