Introduction to Euclid's Geometry
| Philosophy of Mathematics |
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Additional resources
OER
Non-OER
- Web resources
- Video on angles - http://study.com/academy/lesson/types-of-angles-vertical-corresponding-alternate-interior-others.html
- Additional information on axioms and postulates
- To learn types of angles click here
- The following videos provide an introduction to axioms, postulates and lines
- Books and journals
- Textbooks:
- Syllabus documents (CBSE, ICSE, IGCSE etc)
Learning Objectives
Teaching Outlines
Concept 1 - Introduction to geometry
One interesting question about the assumptions for Euclid's system of geometry is the difference between the "axioms" and the "postulates." "Axiom" is from Greek axíôma, "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply postulated, e.g. "let" this be true. There need not even be a claim to truth, just the notion that we are going to do it this way and see what happens. Euclid's postulates, indeed, could be thought of as those assumptions that were necessary and sufficient to derive truths of geometry, of some of which we might otherwise already be intuitively persuaded. As first principles of geometry, however, both axioms and postulates, on Aristotle's understanding, would have to be self-evident. This never seemed entirely quite right, at least for the Fifth Postulate -- hence many centuries of trying to derive it as a Theorem. In the modern practice, as in Hilbert's geometry, the first principles of any formal deductive system are "axioms," regardless of what we think about their truth -- which in many cases has been a purely conventionalistic attitude. Given Kant's view of geometry, however, the Euclidean distinction could be restored: "axioms" would be analytic propositions, and "postulates" synthetic. Whether any of Euclid's original axioms are analytic is a good question.
It is useful to discuss with students about Euclid and his great contribution to Mathematics. The below two statements helps to understand and prove the theorems in geometry. Also, through a combination of activities, help the students understand results in the nature of axioms and postulates.
- Certain statements which are valid in all branches of mathematics whose validity is taken for granted without seeking mathematical proofs is called axioms
- Some statement which are taken for granted in a particular branches of mathematics is called postulates.
Concept 2 Axioms and postulates
Concept 2 - Euclid's Axioms and Postulates
- First Axiom: Things which are equal to the same thing are also equal to one another.
- Second Axiom: If equals are added to equals, the whole are equal.
- Third Axiom: If equals be subtracted from equals, the remainders are equal.
- Fourth Axiom: Things which coincide with one another are equal to one another.
- Fifth Axiom: The whole is greater than the part.
- First Postulate: To draw a line from any point to any point.
- Second Postulate: To produce a finite straight line continuously in a straight line.
- Third Postulate: To describe a circle with any center and distance.
- Fourth Postulate: That all right angles are equal to one another.
- Fifth Postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side of which are the angles less than the two right angles.
Activities
Activity No #
Estimated Time Materials/ Resources needed Prerequisites/Instructions, if any Multimedia resources Website interactives/ links/ / Geogebra Applets Process/ Developmental Questions Evaluation Question Corner
Activity No #
Estimated Time Materials/ Resources needed Prerequisites/Instructions, if any Multimedia resources Website interactives/ links/ / Geogebra Applets Process/ Developmental Questions Evaluation Question Corner
Learning objectives
Notes for teachers
The teacher can talk about Euclid and his contributions to geometry, Euclid's Elements (is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid).
Geometric basic facts taken for granted are called axioms. These are universally accepted and do not need any proofs. These statements are the basis to understand and prove higher geometrical theorems.
Euclid's Axioms (or Common Notions)
- Axiom 1 - Things which equal the same thing also equal one another.
- Axiom 2 - If equals are added to equals, then the wholes are equal.
- Axiom 3 - If equals are subtracted from equals, then the remainders are equal.
- Axiom 4 - Things which coincide with one another equal one another.
- Axiom 5 - The whole is greater than the part.
Activity No # 1. Euclid's Axiom #1
Estimated Time:45 minutes
Materials/ Resources needed
Laptop, geogebra, projector and a pointer.
*Prerequisites/Instructions, if any
Multimedia resources : Laptop
Website interactives/ links/ / Geogebra Applets
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Process Developmental Questions
Evaluation
Ask students to give some more examples to reiterate the axioms.
- Given <P = <Q and <Q = <R, according to which axiom of Euclid, the relation between <P and <R is established ?
- If a + b = 8cm, Is it true to say that a + b + y = 8 + y ?
Question Corner
- What was the name of the book written by Euclid ? How many chapters did it have ?
- If AB = 4cm, CD = 8cm and PQ = two times AB. Are CD and PQ equal ? Which axiom is used for proving this ?
Concept # 4. What is a theorem ?
Learning objectives
Notes for teachers
Activity No #
Estimated Time Materials/ Resources needed Prerequisites/Instructions, if any Multimedia resources Website interactives/ links/ / Geogebra Applets
Process/ Developmental Questions Evaluation Question Corner
Hints for difficult problems
Project Ideas
Math Fun
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