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MATH-8-Q3-WEEK-1-DESCRIBING-MATHEMATICAL-SYSTEM-AND-ILLUSTRATING-AXIOMATIC-SYSTEM.ppt
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- 2. Activity 1: MULTIPLECHOICE Directions: Read each question carefully and write the letter of the correct answer on a separate sheet of paper. 1. Which has no width and no thickness and can be extended infinitely in opposite directions? A. angle B. line C. plane D. point 2. Which describes a point? A. corner of a room B. cover of a book C. tip of a pen D. top of a box 3. What is a flat surface that extends infinitely in all directions? A. angle B. line C. plane D. point 4. Which is a subset of a line with one endpoint and extends infinitely in one direction? A. angle B. line C. ray D. segment 5 . What term is formed by two noncollinear rays with a common endpoint?
- 3. Activity 2: WordJumble Unscramble the letters using the words from the word bank below. Write them on a sheet of paper. 1.I N L E 6. E T A L U T S O P 2.M X O I A 7. E L G N A 3.T I O N P 8. G E S N T M E 4.E N P L A 9. M E R O E H T 5.Y R A 10. R E U A S E M THEOREM RAY AXIOM MEASURE POSTULATE LINE ANGLE PLANE
- 4. Terms Objects 1. Point 2.Line 3. Plane 4. Line segment 5. Ray 6. Angle Questions: •What objects did you choose in items 1 to 6? •What words can you use to describe the objects you choose for each item? Using the words that you gave in question number 2, form a definition for each term above. Activity 3: Find Me Look at the objects around you and list down one object each that you think best describes or represents a point, line, plane, line segment, ray and angle. Copy the table below and write your answer on it.
- 5. God has givenus bountiful nature with different shapes and measurements that are connected with Geometry
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- 7. Geometry is themeasurement of earthly objects. “geo” which means earth “metrein” which means to measure. But Euclid gave geometry a new definition and developed it into a formal study of statements that involve reasoning. He then defined geometry as the study of a body of logically connected statements. This makes completely clear that in geometry, a statement leads to another statement in an arrangement that is supported by reasons.
- 8. In the mathematicalsystem designed by Euclid, there are four parts namely: 1. Undefined Terms 2. Defined Terms 3. Axioms and postulates 4. Theorems
- 9. The undefined termsin geometry are points, lines and planes which are said to be the building blocks of geometry. How will you describe these undefined terms? Undefined Terms
- 10. Term Description RepresentationNaming Real-life Example Point A point is a figure with no length and width or simply with no dimension. A point is named by a capital letter. So, the point below is read as point B. Tip of a pen
- 11. Term Description RepresentationNaming Real-life Example Line A line has no width and no thickness but it can be extended infinitely in opposite directions. A line is determined by locating two points on the line and it is named with a lower-case letter or two capital letters with a double arrowhead above it. The figure below is read as line BC or line m and denoted as 𝐵𝐶 Crease of a paper
- 12. Term Description RepresentationNaming Real-life Example Plane A plane is a flat surface that extends infinitely in all directions. A plane is named with a capital letter. The plane below is named as plane N. Cover of a book
- 13. Can you citeother real-life examples of points, lines and planes that you see around you?
- 14. In geometry, definedterms are terms that have a proper definition and can be defined by means of other geometrical terms. The basic defined terms in geometry are line segments, rays, opposite rays and angles. Defined Terms
- 15. Term Description RepresentationNaming Real-life Example Line Segment A line segment is formed when two distinct points are connected with a straight line A line segment is determined by locating two points and it is named with a two capital letters. The figure below is read as line segment AB and denoted as or 𝐴B̅ 𝐵𝐴. Ruler
- 16. Term Description RepresentationNaming Real-life Example Ray A ray is part of a line. It extends infinitely in one direction only and stopped by a point on one end. A ray is determined by locating two points and it is named with two capital letters with a single arrowhead above it. The figure below is read as ray AB and denoted as AB. Arrow
- 17. Term Description RepresentationNaming Real-life Example Angles An angle is a figure formed by two rays meeting at a common endpoint. Angles are denoted by a number, vertex and the three capital letters. The angle below can be named as or or 𝑋𝑂𝑌 𝑂 Hands of a clock
- 18. Can you thinkof other real-life examples of line segments, rays, opposite rays and angles around you?
- 19. Here are somecommon terms that you may encounter along with the defined terms.
- 20. Term Description RepresentationExplanation Midpoint of a Segment is a point that divides the segment into two congruent segments In the figure, we let N be a point on 𝐴𝐵 Point N is called the midpoint of ,if 𝐴𝐵 is congruent 𝐴𝑁 to . Congruent 𝑁𝐵 means having equal measure and similar shape. To denote congruency, we used “ ”. Therefore, we ≅ can write 𝐴𝑁 ≅ . 𝑁𝐵
- 21. Term Description RepresentationExplanation Congruent Angles angles having equal measure Since the two right angles have a measure of 90 , ⁰ therefore ∠𝐴𝑋𝐵 ≅ . ∠𝐶𝑌𝐷
- 22. Term Description RepresentationExplanation Bisector of an Angle a ray that divides an angle into two congruent or equal angles If point Y lies in the interior of and ∠𝐵𝐶𝐷 ∠𝐵𝐶𝑌≅∠𝑌 , then 𝐶𝐷 is ∠𝐵𝐶𝐷 bisected by , 𝐶𝑌 and is 𝐶𝑌 called the bisector of . ∠𝐵𝐶𝐷
- 23. Term Description RepresentationExplanation Perpendicular Lines formed when two lines intersect with each other and formed right angles. The perpendicular sign is , it is ⊥ used to show that two lines are perpendicular In the figure, 𝐴𝐶 and intersect at 𝐵𝐷 point F to form right angles. To denote perpendicularity, we used “ ”. Therefore, ⊥ we say that 𝐴𝐶 ⊥ . Since 𝐵𝐷 ∠𝐴𝐹𝐵 and formed a ∠𝐶𝐹 𝐵 linear pair, then they are supplementary. Since perpendicular lines formed four right angles, therefore , , ∠𝐴𝐹 𝐵 ∠𝐶𝐹 𝐵 ∠𝐶𝐹 𝐷 measure 𝑎𝑛𝑑 ∠𝐴𝐹 𝐷 90 . ⁰
- 25. Axioms and Postulates Axiomsand postulates are both statements that are assumed to be true without any proof. Their only difference is that, axioms are used in other areas of mathematics while postulates are widely used in geometry.
- 29. Theorems A theorem isa basic geometric principle which are proven to be true by making connections between accepted definitions, postulates, mathematical operations, and previously proven theorems.
- 34. Generalization: •What is mathematicalsystem? •Enumerate and explain the four mathematical system.
- 36. Complete the statementsbelow by choosing the appropriate word inside the box. The measurement of earthly objects are called (1)_______________. But Euclid gave it a new definition and developed it into a formal study of statements that involve reasoning. He later introduced that mathematical system with four parts. The first one are terms which cannot be defined but can be described, this is called as the (2)_____________. It is composed with the point, line and plane. Second, are terms that have a proper definition and can be defined by means of other geometrical terms, this is called as the (3)____________. It is composed with the line segment, ray, opposite ray and angles. Third, are statements that are assumed to be true without any proof, this is called as the (4)_____________. And last, are basic geometric principle which are proven to be true, which is called as (5)___________.
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- 39. Everything around usinvolves dimensions, space, shapes and all other things related to Geometry. It is very important for us to know how things fit together.
- 40. Activity 2: Sharingis Caring. Students explain their understanding about mathematical system.
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- 42. An axiomatic systemis a way to establish the mathematical truth that flows from a fixed set of assumptions. An axiomatic system is a collection of axioms, or statements about undefined terms. You can build proofs and theorems from axioms. Logical arguments are built from with axioms.
- 43. The following propertiesof an axiomatic system should be considered to establish mathematical truth. Consistency – A statement is said to be consistent if there are no axioms or theorems that contradict each other Independence – An axiom is called independent if it cannot be proved or disproved from the other axioms of the axiomatic system. An axiomatic system is said to be independent if each of its axioms is independent. Completeness – An axiomatic system is called complete if every statement expressible in the terms of the system is either provable or has a provable negation.
- 44. Proof is alogical argument in which each statement is supported/justified by given information, definitions, axioms, postulates, theorems and previously proven statements.
- 45. Example 1 Axiom 1.Every computer set has at least two players. Axiom 2. Every player has at least two computer set. Axiom 3. There exist at least one computer set.
- 46. Explanation: This might describea routine for a shop owner to control activity in a computer shop, but it is also a set of axioms. We have two undefined terms, "computer set" and "player." We have not defined "computer set" or "player," but we can build on those undefined terms to construct various proofs. Let's prove a player exist By the third axiom, a computer set exist. By the first axiom, the existing computer set must have at least one player. Therefore, at least one player for a computer set exist. This limited axiomatic system would be enough to build a network of computers to work in a computer shop.
- 47. Example 2 Axiom 1.Every line is an intersection of two planes. Axiom 2. The plane has at least two lines. Axiom 3. A minimum of one plane exist.
- 48. Explanation Let's prove aplane exist By the third axiom, a plane exist. By the first axiom, the line intersect two planes. By the second axiom, the plane contains at least two lines. Therefore, if two lines intersect then exactly one plane contains both lines. This prove the Theorem that states “ Two intersecting lines determine a plane”.
- 49. Example 3 Axiom 1.There are four real numbers. Axiom 2. The sum of two numbers is equal to the sum of another two numbers. Axiom 3. At least two numbers are equal.
- 50. Explanation By the thirdaxiom, two numbers are equal. By the first axiom, there are four real numbers. By the second axiom, the sum of two numbers is equal to the sum of the other two numbers. Therefore for all real numbers a, b, c and d, if a = b and c = d, then a + c = b + d, this proves the Addition Property of Equality
- 51. Example 4 Axiom 1.There are two triangles. Axiom 2. All angles of the triangle are equal. Axiom 3. A minimum of two angles of each triangle are equal.
- 52. Explanation By the firstaxiom, two triangle exists. By the second axiom, all angles of the triangles are equal . By the third axiom, the triangle contains at least two equal angles. Therefore, if and , then ∠𝐴 ≅ ∠𝐵 ∠𝐵 ≅ ∠𝐶 ∠𝐴 . Thus, Transitive Property of Congruence ≅ ∠𝐶 exists.
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Editor's Notes
- #5 . Shapes like circles, rectangles, squares and triangles are some examples, that can be found in our cars, roads, bridges and buildings or other resources that are available around us. To be able to know, understand and value these, it is important to have basic knowledge and concepts of Geometry.
- #7 It started as a way of calculating land measurement particularly in Egypt and Babylon, WHERE
- #9 There are terms which cannot be defined but can be described. These terms are called Undefined Terms. The undefined terms in geometry are points, lines and planes which are said to be the building blocks of geometry. How will you describe these undefined terms? Examine the following description and real-life examples below.
- #25 Listed below are some of the postulates which are used as guiding rules or assumptions from which other statements on the undefined terms can be derived. Remember that these statements are already accepted as true statements.
- #29 .