In geometry, area is the two-dimensional space or region occupied by a closed figure, while perimeter is the distance around a closed figure i.e. the length of the boundary. For example, the area can be used to calculate the size of the carpet to cover the whole floor of a room. Perimeter can be used to calculate the length of fence required to surround a yard or garden. Two shapes may have the same perimeter, but different areas or may have the same area, but different perimeters.
The first recorded use of areas and perimeter was in ancient Babylon, where they used it to measure the amount of land that was owned by different people for taxation purposes. Later in the early 287BC the great mathematician Archimedes from Sicily, Greece, discovered the area and the perimeter of the circle and the relationship between spheres. Archimedes was probably not the first to realise the fact. However he was, as far as we know, the first to prove it formally. (Heather Hasan, 2006). He also gave the earliest known proofs for the surface area and volume of the sphere. Archimedes provides us with what is probably the first mathematically rigorous range for pi (as opposed to a practical approximation), correctly stating that it lies between 223/71 and 22/7. The latter is often called the "Archimedean value" of pi, but this approximation was in use long before his time, and continues to be used today. Despite widespread belief to the contrary, only an estimate, since pi is an irrational number and cannot be expressed exactly as the ratio of two whole numbers.
Key Mathematical Ideas
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Area and perimeter are being utilised in every day school life. They were first built upon in key stage 2 (levels 3-5), where children would be using squared paper to draw shapes on and calculate the perimeter from counting the number of squares in the shape. Then they are taught to multiply the number of squares on the length and the width to work out the area. Area is only taught to only basic shapes like squares and triangles. Then it gets more complicated to see that in key stage 3, students are trying to gain levels between 5 to 7 by deducing and using the formula of length multiplied by width to find the area of a rectangle. For triangles, they are performing half the base length multiplied by the height. For levels 3-5, the heights are normally given and the students just work out the area by substituting the base length and height values in. However preferably for levels 5-7, the student would need to work out the value of the height, and then perform the area. Mainly calculating the perimeter and area of rectangular shapes are used, as well as calculating the perimeter.
Figure : Rectangle divided into 2
The area of a rectangle is base multiplied by the height. If the rectangle is divided into two parts diagonally as shown in the figure 1, it forms two triangles. This is how the area of the triangle is deduced.
The more complex problems come into mind when students have to derive and use a formula for the area of a right-angled triangle, deduce for formulas for the area of a parallelogram, triangle and trapezium. They also calculate the areas of triangles, parallelograms, and trapeziums. According to the information provided by the UK Metric Association It is important in this day and age to use standard metric units, suggesting appropriate units and methods to estimate or measure length and area. (Using 2D representations to visualise 3D shapes and deduce some of their properties. In the modern technology world, there are pieces of software that give an idea of what a shape will look like and also give a prediction of the area or perimeter when the dimensions are given.
Bruner (2004 : 4) explores how the usage of the formula for area is an opportunity "for learners to construct new knowledge and new meaning from authentic experiences." He progressed three stages to show "enactive, iconic and symbolic" which are not age specific. The enactive stage is when the student are able to do a physical activity much better than explaining the same task that has been finished. The iconic stage shows when students are represented with new information, it is helpful for them to visualise the concepts that are being taught. However in the symbolic stage, arbitrary words and mathematical symbols are mostly used. His theory can be applied in area and perimeter.
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Area and Perimeter have links to different topics covered in the National Curriculum, and measurements and scales are one of the mathematical connections branched out around area and perimeter. It's a good continuation especially when students have just learnt the formulas and know how to work the area of a triangle. The next bit of extension work could be to give them a real life example:
Example 1) Sally lives very near to her friends in Newham, and from her home at the point A, she drives 10km to her friend Sarah's house (point B). Half an hour later, she drives 9km to her second friend Johnny's house (point C) and drive 11 km to come back home.
Sally comes home and decides to draw on a map, which places she went to and saw she made a triangle. She now wants to know what is the distance she has travelled around, and the area she has covered in Newham.
A 10 KM
Figure : Triangles
Using the Figure 2 above, year 7 students were asked to calculate the distance Sarah has travelled. They managed to add up the lengths of 10, 9 and 11 making 20 km. They understood the concept of perimeter, but when they saw they had use the formula to calculate the area of a triangle, they were confused. Either they would multiply the three lengths or they would assume as the formulas as a half, they would multiply only two lengths.
Teaching real life examples are a great advantage to students so that they can see how area and perimeter on simple scale drawings are calculated. This topic can be further studied using maps. This example above is designed for key stage 3 students. The following example is more complicated and advanced and would be intended for key stage 4 students:
Figure : Triangle 2
The same question can apply, and the same answer would apply to calculate the perimeter. However when calculating the area of this triangle, students would assume the height is 8km, but to make it more complex questions wouldn't hint 8km and make students calculate the height.
Area and perimeter is mainly applied in practical work and the appropriate method to progress those ideas is from realistic circumstances and problem- solving situations. It is frequently unreasonable to consider measuring skills in the format of using a pen and paper - even fundamental environments for measurement require the three dimensional and the tactile element that conveys measurement activity to life. Never the less pen and paper answers to very detailed tasks are aimed to discover concept progression and achievement of specific calculating skills. It is generally known that children have difficulties with area and perimeter, and it is more than just a question of memorizing. Problems arise when there is purely confusion over the terms 'perimeter' and 'area' when students are examined to match shapes with the same perimeter. The Mathematics Assessment for Learning and Teaching (MaLT) has given a project database show that 36% of 11 year olds make an error by matching the area. Another problem occurs when calculating the perimeter of shapes with diagonal sides. Again MaLT has researched saying 13% of 11 year olds regarded the diagonal of a unit square in place of the same length as the side of the square. As students increase their age by three years, it is assumed that they can work out the area of rectangles utilizing a formula, but again most students would probably use the formula of the perimeter rather than the formula of the area when searching a missing measurement. Many students could possibly calculate this question:
Work out the distance a footballer ran around the football pitch of 60m long and 90m wide.
However some would calculate the area and a few would easily add 60 and 90 together. As students start to work out the units of area, conversion between square units turns out to be very problematic. I have experienced that most of 13 year old students could work out the area in square metres using a calculator measuring 210 mm by 295 mm (A4 sheet of paper), the rest made errors in decimals. In this situation, it seems that the errors are showing that their idea of measurement does not recognise the significance of recognizing the unit of measure. (Julie Ryan & Julian Williams, 2007)
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An idea gives pupils with opportunities a way to progress their understanding of the area of a rectangle away from the basic formulaic idea of 'Length multiplied by the Width'. The activity is to search all the rectangles on a square grid having an area of 20cm2 with the aim of that the corners of the rectangle lie always on a point. There are three simple solutions like the dimensions: 1 by 20, 2 by 10 and a 4 by 5. There are more areas, for example: 72cm2, 90cm2 and 100cm2, which will help pupils to think about where each of the areas has a minimum of at least nine answers. There could a whole class activity where students could find all the possible answers from 1 to 100, or doing this activity in small groups of pupils could be given ten or more different numbers between 1 and 100 to search for all the possible answers. Another idea is for pupils to explore triangles with a constant area and a constant one side length. The purpose is to assist the pupils to construct a comprehension of the formula for the area of a triangle: A = Â½ bh or A = bh/2. On a 1 cm square paper, asking pupils to draw many triangles that are non-congruent where all the triangles have the same area of 6cm2 and a constant length of 4cm. Using A4 piece of paper there will be more than a few triangles that fits the criteria. Students will easily find the three triangles: an isosceles, a right angled triangle and a scalene triangle. Once a lot of triangles are collected have been found, these can be made in to a display by sticking cut outs on a big sheet of paper with each triangle positioned therefore the constant length (in this example: 4 cm) turns to be the base of each triangle. This display visuals that all the triangles have the same perpendicular height of 3cm. Altering the constant lengths to a different measure e.g: 3 cm and maintaining the constant area as 6 cm2, then another display can be made and the perpendicular height will always remain to be 4 cm. Students can discover what happens for triangles with different area and base lengths that are constant. (Mike Ollerton, 2007)
Advantages of Using Computer Software
Using Excel to Find the Area and Perimeter of a rectangle or square
(2*(A3*C3))+(2*(C3*B3))+(2*(A3*B3)) Eq Used to find the surface area
((4*A5)+(4*B5)+(4*C5)) Equation used to find the perimeter
The advantages of using an Excel spreadsheet is that it saves time and is much more accurate than hand calculation. Another advantage is that it can be shared within a team or students, this can save time to do calculations that require calculating the area or the perimeter of something. The use of Microsoft Excel for the calculation of area and the perimeter is better because multiple calculation can be calculated at one time. For example; calculating area of ten rectangles at one time. It also provides a good platform for comparison of different rectangles or any shape that we are dealing with.
Part 2: Developing, analysing and evaluating approaches to teaching
During my PGCE placement, I have utilized an extensive variety of expertise knowledge, assessment, teaching and learning approaches from Part 1. I believe that this is essential in terms of offering the most excellent teaching to students. Having repetitive, predictable and non-contemporary approaches are to compromise the usefulness of teaching and reduce the chance of accomplishing the learning objectives.
It is vitally important to understand that the students under the guidance of the teacher will not frequently learn at the same speed as the other students. Likewise, the students will not comprehend or uniformly understand mathematical ideas. For that reason, it is significant to adapt my methods to support each individual as best as possible. By implementing my method of teaching I would not only provide with an a explanation that suits most of the students as well as make it more comfortable for me to teach, so that I will be able answer the questions that students asks, and many examples are explained as follows:
During my first placement of my PGCE course, I attempted to build on or challenge/test students in my lessons to take full responsibility as much as possible for their own performance by making them involve in projects and tasks which permit them to plan, produce and either self-assess or peer assess mutually. This approach not only provides assistance from a well-built support group, and also it lowers the fear of students who may sense isolation and hesitant when doing challenges in Mathematics. Mathematics is like learning a new language the more we practice we get better at it. So by giving more exercises in a way that it is more interactive and fun which involves all of the pupils attention, will not only give way to the pupils understanding the topic more, but also provide with an attraction towards the class.
A lot of lesson planning went towards developing the perception of community and teamwork within the classes. At the beginning of each teaching topic, I summarized what my expectations were in terms of approach, behavior and outcomes. At this initial period, I did not want to consider talking about the qualitative outlooks, as I was being afraid of intimidating the students who were less certain; instead I summarized the outlook of how the mathematics task would be represented as and what the evaluation was to be at the end of the lesson. I was ascertained to distinguish that the challenges I showed to the pupils would not be simply and academic beside the point.
I planned two lessons concentrating on area and perimeter at the same time. As part of my first lesson which is shown here detailed focuses the way the actual application of an example in the classroom might progress in surprising techniques and to specify how significant pedagogical subject knowledge is in managing with this.
I started by studying the concept of area where I stressed on the definition of Area - "Area measures the space inside a shape, so the number of squares inside the shape" (Colin Foster, 2003). I asked the pupils to annotate a rectangle with an area of 20cm2 in their books on squared paper and to cut the rectangle out. My choice of this task was called a "reversed" open task, was by most suitably given to the pupils who had worked with area previously, especially the formula of area for the rectangles. Before long after the instruction the resulting conversation occurred in the middle of a student and me (Miss Arun):
S: Can I do a square?
Miss Arun: Is a square a rectangle? â€¦ What's a rectangle?
S: Two parallel lines
Miss Arun: Good but two sets of parallel lines â€¦ and what else does a rectangle have?
S: Four right angles.
Miss Arun: Better â€¦(I point to the square the student has drawn) so is that a rectangle?
Miss Arun: Right! â€¦ But has this got an area of 20?
S thinks: Ermâ€¦ no.
Miss Arun nods: Okay! â€¦ (and leaves student to think).
It is not distinct whether my original choice of 20 was made with any understanding of geometrical suggestions however the smoothness where I moved from area dimension to three-dimensional problems showed with positive consideration to the properties of geometry and yet again needed perceptive approach to the pedagogical subject knowledge of both the dimension and three dimensional areas. I also showed the useful practise of questioning to stimulate understanding from the pupil. Following this I talked to my class about the properties of rectangles.
Next I invited a pupil to come to the front of the class with her cut out rectangle of dimension of 5 by 4, and proved that it has an area of 20cm2. The student led the class by discussing how multiplying the length and the width together is similar to counting squares and therefore calculates the area. I then used the student's cut out rectangle as an example to emphasise the connection between the theoretical definition of area and the calculation. I carried on as I knew the students needed to recognise that the area formula "L x W" is only applied to particular shapes.
Miss Arun: When [the first student] mentioned that is how you find the area of a shape, is she completely right?
(Another student answers)
S: That's what you do with a 2D shape.
Miss Arun: Yes, 2D shape like this shape â€¦ what kind of shapes would it not work for?
(More students answer)
S: A circle.
I questioned more as it provokes out that the formula "L x W" only works for rectangles.
A pupil then implied on a rectangle with dimensions of 10 by 2 as another example with an area of 20cm2, therefore at this point I made sure that all the pupils had picked either dimensions of 4 by 5 or 10 by 2. The pupils were asked for more possibilities and they proposed the initial examples however turned at 90Â°, simultaneously with dimension of 1 by 20 - this dimension had not been mentioned before.
On the interactive whiteboard using Smart Notebook, I showed rectangles with numbers on the sides. I asked the pupils to search for a pattern in the rectangles displayed, and then interestingly the pupils discussed the factors of 20. I continued to talk about the topic:
Miss Arun: Can you find any more numbers that give an area of 20?
I waited with an attitude of ambiguity. The students gave me no response.
Miss Arun: No? How do we know that there isn't any more numbers giving an area of 20?
S: You could put half by 40.
Miss Arun: Oh! You have now gone into decimals. We are going to have many factors of 20 with decimal numbers in, won't we?
I was aiming only on whole numbers and, consequently some disagreement about stressing the factors of 20. I knew the pupil's unexpected response was important and to what level of the curriculum it would relate. My open choice of questioning allowed this extension to start, though it was not my intention nevertheless, I decided not to engage in this part, although it might have been a beneficial use of the 20cm2 example because I just wanted to go on to examples that were different. As an alternative I utilised the 20cm2 example to aim on the pursuit for all the factors of 20. This study of the 20cm2 example took the first fifteen minutes of the lesson. I then had pupils redo the exploration for rectangles with a different area of 16cm2. I used this example to focus the procedure of finding factors, and to emphasise "a square is a rectangle".
I reminded on what they had studied on, and then told the pupils about the perimeter and how to calculate it for rectangles. I helped the class to calculate the perimeter of rectangles that are of different dimensions but of the same area of 16cm2 and showed that shapes may have the same area however they do not always have the same perimeter. I returned to the examples with the area of 20cm2 and the perimeter for each rectangle was worked out to focus on the aspects in perimeter.
My last and final task for the lesson was planned for the pupils to work in pairs to search for lots of shapes and not only rectangles, but limit to contiguous squares having an area of 12 cm2 and establish the perimeters. I showed a spider diagram of the centre part saying "What is the connection between perimeter and area?" on the board and let pupils to talk and explore this activity for roughly five minutes. They were to jot down what they had in mind.
I went around the classroom and interfered with their work to assist them to develop approaches to work in a systematic way and teach them to note the perimeters of each shape. Five minutes later, I discussed that there were "many" possible shapes. As a class we discussed at the work from a group and then questioned pupils to concentrate on discovering a shape of biggest perimeter and a shape of smallest perimeter. The one hour lesson finished with a last minute quick discussion of the students' outcomes, which highlighted the shapes of small perimeters were more compact. Also shifting one of the squares on a shape not including altering the number of contiguous edges will not change the perimeter.
My conclusion was that the area and perimeter can have similar or different numbers; two shapes can have different perimeters but have the same area.
These learning objectives were accomplished through three examples.
Reflection on my practice at my teaching placement was, without a doubt, the most essential method I had acquired during my PGCE course. Reflection on my practice at my teaching placement was, without a doubt, the most essential method I had acquired during my PGCE course. This utilized entirely to the assessment and evaluation of my own work and to the students. By initializing a scheme of checking and reviewing, in the perspective of the national curriculum, students are encouraged to be analytical and perceptive in the middle of a task of their individual work. The value of initializing this scheme is important. The students are not only given confidence to break down and reassess smaller chunks of their work but they can amend mistakes or have the tendency to tangent or set off activities in which, revising for a considerable examination, this activity could be particularly detrimental to the management of their time.
I have experienced the benefits by making use of a mixture teaching approaches and strategies being both the practitioner in addition to the students: engaging students in the creation and evaluation of activities, a combined ownership is began and a common sense of principle is produced.
It must be ensured that I must have a deep theoretical understanding of area and perimeter, and show rich pedagogical subject knowledge for teaching.
Given the important examples to the learning and teaching procedure, never-the-less time must be crucially spent utilising this perception to an exploration of examples and also the pedagogical propositions. I have learnt how to alter examples to make them more theoretically difficult or simpler, to create counter examples or to show a different approach. I used my opportunity from my placement to involve with examples, to trial them with students and to learn how to successfully adapt them to assemble needs of different kinds.
In each and every one of the parts in area and perimeter, it must be necessary to have an in depth discussion of the mathematical connections in addition to the mathematical topics and also finding how an example illustrates these connections. As a final point, there needs to be a discussion of how to apply the examples in the classroom, therefore the examples develop achieving, educational objects that show the preferred general principle. If these points were lacking, the chances for learning presented by examples may possibly go unconvinced.