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we bring forth to you the HISTORY of circles (:
The circle has been known since before the beginning of recorded history. It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus.
Early science, particularly geometry and Astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.
Some highlights in the history of the circle are:
* 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of π.
* 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
* 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.
Euclid
~325 BC - 265 BC
The Elements
Many geometrical axioms and propositions
Wrote all this in 300 BC!
Java applets: Euclid’s Elements, Book III, Proposition 1
SOURCE:www.wikipedia.com
*Circle & God:
Early science,particularly geometry and astronomy/astrology,was connected to the divine for most medieval scholars.The compass in the13th century manuscript is a symbol of God's act of creation,as many believed that there was something intrinsically "divine" or "perfect" that could be fund in circles.
In ancient Rome,circles were worshipped as they were thought to be divime and holy
There should be no need for introduction again here. So well lets get to the main aim of this blog: for you all to better understand the properties of circles.
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Paper
PROPERTIES OF CIRCLES
angle properties of circles:
(1) Angle at Centre,
(2) Angle in Semicircle,
(3) Angles in Same Segment,
(4) Angles in Opposite Segments.
symmetrical properties of circles:
(1) Straight line perpendicular to chord
(2) equal chords equidistant from centre
angle in opposite segments of a circle
tangents in a circle:
(1) Tangent perpendicular to radius
(2) Tangent from an external point
Entries
Stories of C-I-R-C-L-E-S //
Blogged on : Monday, July 20, 2009
Blogged at : 7:32 AM
The entire unit, “geometrical properties of circles”, can be broken down into two specific subtopics, namely “symmetrical properties of circles” and “angle properties of circles”. The corresponding syllabus requirements from each subtopic are listed below:
A. Symmetrical properties of circles (under section 22, Symmetry): Students should be able to use the following symmetry properties of circles: 1. Equal chords are equidistant from the centre; 2. The perpendicular bisector of a chord passes through the centre; 3. Tangents from an external point are equal in length.
B. Angle properties of circles (under section 23, Angle): Students should be able to calculate unknown angles and solve problems (including problems leading to some notion of proof) using the following geometrical properties: 1. Angle in a semi-circle; 2. Angle between tangent and radius of a circle; 3. Angle at the centre of a circle is twice the angle at the circumference; 4. Angles in the same segment are equal; 5. Angles in opposite segments are supplementary.
The unit will be covered in three double-period lessons. The first lesson, “symmetrical properties of chords”, will cover points 1 and 2 under subtopic A. This lesson will be covered in greater detail in this lesson plan.
The second lesson (double-period), “symmetry and angle properties of tangents”, will cover point 3 from subtopic A as well as points 1 and 2 from subtopic B. This is because point 3 from subtopic A deals with tangents to the circle, the angle properties of which are dealt with in point 2 of subtopic B. Points 1 and 2 from subtopic B join well together because both are properties relating right angles and circles.
The final lesson (double-period), “interior angle properties of circles”, of this unit will cover points 3, 4 and 5 from subtopic B. Students will be taught the various relations between angles inside the circle, such as angle at the centre of a circle vs. angle at circumference, angles in the same segment, and angles in opposite segment. This will wrap up the entire unit on “geometrical properties of circles”. Learning Theories Co-operative Learning and Social Constructivism Vygotsky introduced the concept of a Zone of Proximal Development (ZPD) in learners, where scaffolding can make the difference between what a learner can do by himself (unassisted performance) or should be able to achieve with guidance (assisted performance). In this lesson, students are able to solve simple geometric problems without assistance, whereas some scaffolding from the teacher will increase the range of problems they can solve. Furthermore, scaffolding also includes group and co-operative learning, a concept that can be drawn upon for better results in the Mathematics classroom. Students will be asked to work in pairs to come up with ideas for a problem they are posed with at the beginning of class, and the teacher will utilise a “think-pair-share” classroom model to encourage discussion and sharing of ideas. Guided Discovery Bruner introduced the guided discovery approach in the 1970s, where “pupils try to discover mathematics on their own by working through various activities.” In this lesson, students will be made to discover various properties on their own, albeit with guidance from the teacher, through a series of investigative enquiry tasks in the first worksheet. This way, students can discover things for themselves, and would end up more motivated to learn. Stages of Cognitive Development Piaget, in his work in the 1960s, put forward his organisation of cognitive development into a series of stages. Many students in our secondary schools have been shown to be in the concrete operational stage, hence supporting the greater usage of concrete material for learning. As such, this lesson will begin with students getting to “play” with circles and geometrical instruments, before moving on to other equally-concrete investigative acts in the first worksheet. The use of concrete objects will help solidify the foundation of these abstract properties in the students’ minds. Van Hiele Theory Pierre and Dina van Hiele postulated in the 1960s that students learnt geometrical concepts within a certain hierarchy of levels. The levels are recognition at level 1, analysis, ordering, deduction, and finally rigour at level 5. The van Hiele theory suggests that students can only progress from one level to the next, without skipping levels of understanding. In this topic, the lesson is structured to accommodate for the van Hiele theory. To begin with, students are asked to recognise the general appearance of the properties, i.e., the circles, chords and perpendicular bisectors that they will be dealing with. After the initial recognition stage, students will be brought up to the analysis and ordering stages through the use of the investigative worksheet. Finally, as the students are not required to produce rigorous mathematical proofs, an informal proof of each property will be presented to satisfy students at the deductive stage. Teaching Approaches Some of the teaching approaches employed here are commonly used in teaching geometry (as seen in the “Teaching of Geometry” section in Teaching Secondary Mathematics). In this lesson, deductive reasoning will be utilised, asking students to “deduce, from previously known information” such as perpendicular bisectors and chords, the method to obtain the centre of a circle without folding. The approach of inductive reasoning and conjecturing, wherein the teacher would have the student make “conscious guesses of generalisations by observing instantiations or analogies”, is reflected in an investigate-style worksheet that asks students to measure angles and induce relations and properties. Finally, some problem-based teaching will be used to engage the students. When beginning the class, students will be challenged to find the centre of a circle with restrictions on what they can do (e.g. no folding of the circle), and with limited tools. This will drum up greater interest in the topic among the students. Learning Difficulties This topic can be difficult for students to visualise. The very properties that the students have to learn seem very complex when spelt out in words, and this does not help the students visualise what they have to learn. This is addressed in many parts of the lesson – the lesson begins by letting the students use their geometrical instruments to find the centre of a cut-out circle, and the student continues to investigate the properties of these cut-out circles as the lesson moves on. Various visual aids are utilised to ensure the students understand what the geometrical terms are referring to, such as slides, OHTs, and one self-constructed Teacher’s Aid that shows why equal chords are equidistant from the centre of a circle. A further misconception could arise on the definition of “equal chords”, and this will be addressed as necessary when the term is introduced.
Blogged on : Saturday, July 18, 2009
Blogged at : 4:58 AM
Did you know:
Properties of circle can be divided mainly into the following two parts
- Symmetrical properties of circle
- Angle properties of circle
Symmetrical properties of circles
Chords of a circle
Chord is a straight line that touches the circumference of the circle at any two points.
- Perpendicular bisector of a chord (perpendicular bisector of chord)
The perpendicular bisector of a chord of a circle passes through the centre of circle.
- Perpendicular from the centre of a circle to a chord bisects the chord (perpendicular from centre bisects chord)
- Equal chords of a circle are equidistant from the centre of a circle (Equal chords, equidistant from centre)
- Tangents to a circle
Tangent at a Point
Tangent from anExternal Point
(1) Tangent at a point
A tangent of a circle is perpendicular to the radius of the circle drawn from the point of contact.
(2) Tangent from an external point
It there are two tangents from an external point to a circle,
-the length of the tangents are equal
-the line joining the external point and the centre bisects the angle between the tangents
Angle properties of circles
Arc and segments of a circle
<(in our notes given by Mrs. Sim, on pg 1, we highlighted the arc to show big and small arch plus shade to show the segments, it look something like the one I drew.)>
- Angles in a circle
(1) Angle at centre = 2 × angle at circumference subtended by an arc (angle at centre = 2 x angle at circumference)
(2) An angle in the semicircle is a right angle (angle in a semicircle)
(3) Angles in the same segment are equal (angles in the same segment)
- Angles in the opposite segments (angles in opp. segments)
Here we will display questions for you to try out on your own. GD LUCK(:
SOURCE:www.google.com
In the diagram, the points A, B, C, D and E lie on the circumference of the two circles such that AEDC forms a cyclic quadrilateral and BCD is a straight line. The centre of the smaller circle is marked as O. Given that AB = BC, AE = CE, angle AED = 65° and angle ACE = 70°, find
In the diagram, AB is the diameter of the circle with centre O. CE and BE are tangents to the circle. The tangent to the circle at point B meets OC produced at D. Given that angle CBE = 28° and that the radius of the circle is 5cm, find,
(i) angle BEC
(ii) angle CAO
(iii) length of BC
(iv) angle CDE
ANSWERS:
(i) angle ACB = angle AED = 65° (ext. angle of cyclic quad.)
(ii) angle ABC = 180° - 2 x 65° (sum of triangle)
= 50°
(iii) angle AOC = 50° x 2 (angle at centre= 2 angle at circumference)
= 100°
angle OAC = (180° - 100°)
= 40°
(iv) angle AEC = 180° - 2 x 70°
= 40°
angle ADC = 40° ( angles in the same segment)
(i) BEC is an isos. triangle
angle BEC = 180° - 2 x 28°
= 124°
(ii) angle CAD = angle CBE = 28° (alt. segment theorem)
(iii) triangle ACB is a rt. triangle and AB = 10cm
sin 28° = BC/10
BC = 10 sin 28°
= 4.69 cm (3s.f.)
(iv) angle COB = 28° x 2 (angle at centre= 2 angle at circumference)
= 56°
angle OBD = 90°
angle CDE = 180° - 56° - 90°
= 34°
There is a time for everything, this time, its the time to share.
Chats
REFLECTIONS of members:
Keith:
I feel that this project is really useful as it helps us gain a deeper understanding into the “properties of circles” topic. Having gained this knowledge I think that I will be better at answering mathematical questions concerning this topic. This project also helps to improve the teamwork, communication and understanding between us, as teammates, in order to complete the project smoothly, we need to learn to cooperate with our classmates.
Shu Yi:In my opinion, the properties of the circles is just basically an aid to help us to find the angles in the circle. There are many different types of properties for us to memorise in order for us to find the angles in the circle easily and in the shortest amount of time. Some of the properties of circles include:
-Perpendicular from centre bisects chord.
-Equal chords, equidistant from centre.
-Angle at centre is equals to two times the angle at the circumference.
-Angle in a semicircle.
-Angles in the same segment.
-Angles in the opposite segments.
-Tangent bisects radius.
-Tangent from an external point.
There are so many properties such that it can drive us crazy! But as long as we can memorise them and use them at the right time, this topic isn’t hard to learn at all.
There are also many other points and factors that are very important and are related to the properties as mentioned above.
Circumference- It is basically the sides of the circle, meaning the line segment.
Tangent- A tangent is which is in such a way that it is perpendicular to the centre of the circle when a line is drawn from it.
Chord- A chord is basically a line segment where the two ends of it touches the circumference of the circle.
These points are the basics before we can learn the properties of the circles. Basically, in my eyes, the properties of the circles don’t seem to be the properties of it, but the properties of the angles in it. All the methods used to find the properties of the circles are meant to find the angles in the circle compound.
Arffah:
I feel that through this project I can have a better understanding of the properties of circles and in the same time have fun designing and making our blog attractive. I was also able to learn about moral values such as the importance of having teamwork whereby each member fulfils his/her role and contributes to the work. I also realized that it is important to hear out everyone’s opinion on a matter and decide on the best solution to ensure the project is a success. Learning seems so much easier and I feel that I am able to digest what I put up on the blog more easily.
Tian Cong:
Through this project, I managed to have a much better understanding of the properties of circles. Having this project not only let me have a better understanding of this topic, but also allows me to share this knowledge i have with the rest of my friends. Doing this project in blog way allow us to have fun, because we get to design our blog the way we want. On the other hand, doing project in the form of blog is much more efficient as it allow us as students to save some meeting time, especially we only have a short period to complete this project. I learn that it is very important for each member to complete their assigned task well and on time, so that the leader would have a much easier time adding up all the works from the group members. Hopefully, we get to do our projects in the form of a blog or webpage form more often in the future.
Afiqah:
this project give us a better understanding on the properties of angles and how to solve the circles problems because we have to personally do a research about it. Like others, we did not gain in terms of practice of moral values but also extra knowledge that we gotten from the research. We are glad to be given the opportunities to share our knowledge.
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