Geometry from the Point of View of Measurements - Essay Example

About geometric thinking Geometric thinking or rather geometry is all about finding measurements. It has been applied in almost every sphere of life by people from every walk of life, right from farmers who staking out land by the Nile before and after floods, by astronomers making calculation on the movement of planets and stars in the skies, to philosophers as the basis for their deductions. If you look at geometry from the point of view of measurements, two types of measures come into focus namely practical and pure. Practical measures deals with concepts such as length, angle, perimeter surface area and volume, while pure measure deal with co-ordinates, and the use of trigonometry as a way to measure angled and triangles. A look into any three mathematical tasks, which was undertaken for study Task 1: Measurement of Angles The measurement of angles is vital to understanding the measurement of triangle, a calculation which is oft used in many areas ranging from astronomy to building constructions. Angles however are very difficult to calculate by way of comparison. In Figure 1.1 (see reference section) all the angles represent the same rotation. Therefore, in order to understand how the measurement takes place, we have looked into the following example for guidance. In the above Figure 1.2, if angle A has to be measured, a circle must be drawn. This circle (radius 1), should be centered at the vertex, using the length of the arc of the circle as the measure for the angle. This type of angle measurement is known as the radian measure of an angle. If we look into the arc length, an arc length of 1 gives an angle of about 57.3°. Thus, it can be seen that radian values for commonly used angles are not whole numbers, but when a full circle is drawn, with a radius of 1 the arc length would be 2PI which is considered as the measure of a full turn of the circle or the equivalent of 360° in degrees. Task 2: Measuring the volume of solids The volume of solids can be determined by making use of other solids whose volumes are already known. So it is possible to find out the volume of solids by the process of decomposition and comparison of the solid figure of which measurement has to be taken. If we want to measure the volume of a tetrahedron (which is a three-dimensional form of a triangle) then the volume is calculate s the (area of base)*(height)/3. Similarly the area of a triangle is half the area of the surrounding rectangle. The volume of a prism or cylinder is (area of base) *(height). If this principal of arriving at measurement by decomposition and comparison has to be better understood, then what is required is an example to demonstrate such logical analysis. In the Figure 1.3, we consider the shaded part of the cube as the surface of a square-based pyramid. Similarly, we should consider another identical pyramid in the figure, with the side HGCD as its base, and E as its apex. Now what is remaining is what is left, which is a square based pyramid identical to the other two. This leads to the conclusion that the volume of the original pyramid is of the volume of the cube carried out through this process of decomposition and comparison. Task 3: Pythagoras theorem Pythagoras theorem states that the length of the right angled side of a right-angled triangle is the sum of the square of the side opposites the right angle of the triangle. In understanding this theorem, we looked into one of the proofs of this theorem, which is derived from the concept of tiling. There are three ways in which the tiles can be arranged. One way of arrangement is putting rows or columns of the same type as seen in Figure 1.4 . The second way is by arranging blocks of one-size tiles bordered by infinite rows and columns of the other as seen in Figure 1.5. The third way is arranging running diagonals with the small tiles as spacers as seen in Figure 1.6. Pythagoras’ theorem brings about a connection between the lengths of the sides in a right-angled triangle, but as it connects the square of the two sides of a triangle to the third side, the theorem concerns itself with area. In this context it is connected to the concept of tiling.. To understand this better lets look at this Figure 1.7. In this figure we must first concentrate on the two-sized tiling of the plane and then concentrate on the larger tiles, which are tilted. Further concentrating on the tilted tiles as tiles derived from the two-sized tiling, we can consider them as tilted tiles, which are in the form of squares. Now if we look into one of the tilted squares, we will se that it consists of pieces of the tiling. By rearranging the pieces mentally, we will find the area of the tilted square in terms of the areas of the tiles. This concept can be applied to all the tiles in the figure. Task 4: Understanding about Locus The term Locus, is used to denote a set of points, which satisfy some condition. If we take the example of two straight lines, with a point that moves in such a way, that it is at the same shortest distance, from each straight line, we will be asking ourselves the question as to what its locus is. Similarly, if convert these straight lines into perpendicular axes and focus attention on some point equidistant from the two axes, we will want to know about what relationship there exists between the co-ordinates of the point?. If one co-ordinate is the falls on the x-axis and the other co-ordinates on the y-axis, the question is what path will the co-ordinates follow?. The answers to all these question is the locus, because it is the set of points, requires to make such measurements. Understanding the structure of topic studied An understanding oft the structure of a topic through a study of the above mentioned four tasks is as follows: The measurement of angles by way of comparisons is difficult, because of their nature and so it is important to use a specific form of measure or a formula to ascertain their measure. This concept can be understood using similar examples. If we were to hold a ruler up horizontally and try to estimate how many copies of the ruler is required to make the height of a wall in the room, we will not be able to make a definite or accurate measure visually. If we were to take a book and try to give some estimate on the thickness of the book, we will only be arriving at a relative measure and not the exact measure. Another common misconception is that if the beginnings and endings of a pair of segments are aligned vertically, we assume then the segments are the same length: This is due to horizontal displacement rather than to the segment itself. Obliqueness is a factor that causes lengths to appear greater length even though all the segments on the right are the same length. All these reasons further re-instates the need for formulas to bring correct measures. The process of measuring solid can be done by the process of decomposition and comparison as a solid can be understood to be made up of smaller solids. This concept bring forth the need to focus and defocus my attention, and be able to ignore, discern and recognize relationships and such abilities enable me to understand how such measurements, which, in short, is what geometric thinking is all about. The Pythagoras theorem is particularly interesting and one theory used to prove it has been studied here, which show that the theorem can be proved by it connection to the concept of tiling. Connecting the proof of a theorem to other related concepts, helps one to develop a logical sense of analysis and skills for derivation of proofs from existing concepts, which is what Pythagoras emphasized through his mathematical concepts. Thus, understanding one ways of proving this theory (as studied here) requires use of the imagination, which can be developed only with practice. Such practice helps to work our minds on the details of what we see, so that we can find relationships amongst features that we discern and it is such abilities that help us to relate the theory to the concept of tiling. The locus is a set of point, which satisfy a particular condition and this has been understood by it varied applications as in the questions posed passage on locus. Experience with the study of works of Pythagoras and Archimedes Pythagoras theorem is one of the most interesting theorems presented in the field of mathematics. Great historian E. Bell described Pythagoras as mystic, mathematician, one-tenth of him genius, nine-tenths sheer fudge’. He is the first European who understood that that proof of a theory must arise from agreed properties and explicit assumptions. The Pythagoras theorem can be proved in many ways, which is why the theories of this mathematician are the most interesting to study. One way of proof has already been put forth; the second one is being discussed from the point of my experience with this leaner of geometry. In the Figure 1.8 below, we should imagine that the hole is a square. If we calculate and compare areas of the large square, which are four copies of the triangle, by calculating and comparing areas of the large square, four copies of the triangle, and the hole, we will come to understand the theorem by ways of the process of comparison of figures. Another important learner who has influenced my learning of geometric thinking is Archimedes, through his work with areas, and shapes. In Figure 2.1, he determined that the area of the circle lies between the areas of the outer and inner square, and that the length of the perimeter also lies between the perimeters of the two squares. Similarly, the area and perimeter of the circle lies between the corresponding measures of the inner and outer hexagons in part. By calculating the areas of polygons with very large numbers of sides, up to 96, he was able to derive at the area and perimeter of a circle. He stated that the ratio of the perimeter of a circle to its diameter was between 3 and 3 .This ratio is denoted as pi. He discovered that the area of a circle was a fixed ratio times the radius squared and that this ratio was also pi. All these are very important forms of measurements which are used in every aspect of life. Select a page concerning geometry or geometric thinking from classroom materials that either you have used or you have seen being used. Practical measures always make use of conventional units. These conventional units such as meters, from which there has been further derivatives in the form of kilometers, millimeters, light years and nanometers. Since measures are carried out through a process of comparison with a standard measure, there is the possibility of errors. So, any measurement consists of three parts, which are the number of units, the units involved, and an error interval within which the quantity is confidently placed. It is only when pure measures are made that two lengths can be compared exactly in an ideal figure, with no room for errors. Taking the topic of co-ordinates on a plane as a point of measure, co-ordinates are discussed in algebra and in arithmetic they are determined as positions on a number line. If we imagine a situation wherein we are in a boat, and find ourselves within sight of land, what would be the different ways by which we could locate our position on a chart? The many ways could be any of the following: Looking into the distances from two identifiable landmarks on the shore. Finding out our position by means of latitude and longitude measures. Calculating the distance from nearest point on the shore and from some other point. Calculating the angle between two landmarks and distance from one of them. Finding the angle between one landmark and two others. Whatever, be the form of measurement we take, if we made a small extra measure, this would be useful as a check on accuracy of our measurement. Ultimately, what all these forms of measurement indicate is that there are different co-ordinate systems on a surface. For example, with a range-finder and compass, you can use a polar co-ordinate system, which has an origin (pole), and measures position by distance to the pole and angle from a fixed direction through the pole (for example, north). On the other hand a bipolar co-ordinate system, sues distances from two fixed points (poles), while a Cartesian co-ordinate system uses two fixed axes, and measures distances parallel to those axes (usually perpendicular). This come back to the original point that pairs of co-ordinates are used on maps (with references such as F7, and as (x, y) in algebra and geometry) because they, uniquely define positions, but do this in different ways. In order to understand this let’s take an example by looking at Figure 1.9. First we draw a triangle and label the vertices of the triangle with the co-ordinates (0, 0), (1, 0) and (x, y) and then draw a square outwards on each side of the triangle. We then join each square to its neighbor by joining to the nearest vertices. If we wanted to find the co-ordinate of the other vertices of each of the squares we would have to consider the sides of the triangle. All of the triangles have a side on the square. We have to use these sides as the base. Furthermore all of them have the same height, which means that all three have the same area. In order to deal with the fourth triangle, we need to imagine that there is another pair of axes with their origin at the point (x,y) and one axis along the side of the original triangle to (1,0). Now, the fourth triangle and two others all have the same length base (the sides of another square), and the same height and so must have equal areas. Therefore, the areas of all four triangles must be equal. In this way co-ordinate can be used as a way of measure of areas of a triangle. If we consider a house, which is a three-dimensional object, if we can consider it as an object made up of number of two-dimensional faces. One part of the house can be considered as that of a cube, the roof can be considered to consist of a triangle and a polygon. Of we wanted to find out the total area of the house we would have to calculate the area of the triangle, cube and the polygon for this purpose. If we wanted to find out the shortest distance in the house, would have to transform its floor surface into a net of joined planar regions that are cut out, folded up and glued along edges to make the ‘solid’ shape in order to find this shortest path. Lets take the Figure1.10 as an example. If this represented the floor surface of the house, then there would be three possible ways of showing the shortest path, which is depicted by the nets drawn in this figure. References Askew, M., Brown, M., Rhodes, V., Johnson, D. and William D. (1997) Effective Teachers of Numeracy. London, Kings College. ATM mats (webref) available from ATM Derby, http://www.atm.org.uk (accessed April 2005). Cundy, H. and Rollett, A. (1989) Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Publishers Cuoco, A., Goldenberg, E. P. and Mark, J. (1996) ‘Habits of mind: an organizing principle for mathematics curricula’, Journal of Mathematical Behaviour, 15: 375–402. Eagle, R. (1989) ‘Centres of a triangle and a central dilemma’, Mathematics Teaching, 126: 9–11. Ellis, H.F. (1981) A.J. Wentworth, B.A. London: Arrow Books. Hart K. et al, (eds) (1981) Children's Understanding of Mathematics: 11–16. London: John Murray. Hewitt, D. (1996) ‘Mathematical Fluency: The nature of practice and the role of subordination’. For the Learning of Mathematics, 16(2), pp. 28–35. Polya, G. (1957) How to Solve It: A New Aspect of Mathematical Method. (2nd edn.). Cambridge, MA: Princeton University Press. Pythagoras (webref) http://www.sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html (accessed April 2005). Figures Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 1.7 Figure 1.8 Figure 1.9 Figure 2.0 Figure 2.1 Read More