Mathematical System.pptx

Grade 8 Mathematics- Mathematical System

1. Mathematical system

2. Pre-test 2 1. What do you call the statements that are assumed to be true and do not need proof? A. axioms B. defined terms C. theorems D. undefined terms

3. Pre-test 3 2. There are four parts of the Mathematical system. Which of the following is not a part of the mathematical system? A. corollary B. theorems C. axioms or postulate D. defined and undefined terms

4. Pre-test 4 3. Which of the following statements is true about axioms or postulates? A. These are concepts that need to be defined. B. These are statements accepted after it is proved deductively. C. These are concepts that can be defined using the undefined terms. D. These are statements assumed to be true and need no further proof.

5. Pre-test 5 4. Which of the following statements is true about a postulate? A. It is never accepted to be true. B. It is accepted as true without a formal proof. C. It is only accepted as true after being formally proven. D. It is usually not obvious as true, so it must be proven.

6. Pre-test 6 5. Which of the following best describes a theorem? A. It is the same thing as a postulate. B. It is a statement that is accepted as true without a formal proof. C. It is a statement that is impossible to be proven by mathematical reasoning. D. It is a statement that has been formally proven using mathematical reasoning.

7. Pre-test 7 6. Which of the following is NOT a property of Mathematical system? A. conjecture C. postulates B. define terms D. theorems

8. Pre-test 8 7. Which of the following statements is true about defined terms? A. These are concepts that need to be defined. B. These are statements accepted after it is proved deductively. C. These are concepts that can be defined using the undefined terms. D. These are statements assumed to be true and need no further proof.

9. Pre-test 9 8. Which of the following illustrates reflexive property? A. (𝐴𝐵 + 𝐵𝐶) + 𝐶𝐷 = (𝐴𝐵 + 𝐵𝐶) + 𝐶𝐷 B. (𝐴𝐵 + 𝐵𝐶) + 𝐶𝐷 = (𝐵𝐶 + 𝐴𝐵) + 𝐶𝐷 C. (𝐴𝐵 + 𝐵𝐶) + 𝐶𝐷 = 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐷 D. (𝐴𝐵 + 𝐵𝐶) + 𝐶𝐷 = 𝐶𝐷 + (𝐴𝐵 + 𝐵𝐶)

10. Pre-test 10 9. Which of the following illustrates distributive property of equality? A. 3 + 4 = 4(3) + 3(4) B.5(2) + 6(7) = 5(7) + 6(2) C.8(𝑥 + 𝑦 + 𝑧) = 8𝑥 + 8𝑦 + 8𝑧 D.−8(𝑥 + 𝑦 + 𝑧) = −8𝑥 + 𝑧 + 𝑦

11. Pre-test 11 10. Which of the following is a commutative axiom? A. 2 + 3 = 2 + 3 C. 2 ∙ 3 = 2 ∙ 3 B. 2 + 3 = 3 + 2 D. (2 + 3) = 2(3) + 3(2)

12. Objectives: 12 Define each part of the mathematical system; a. Undefined terms b. Defined terms c. Postulates d. Theorems; 2. Illustrate how the four parts of mathematical system related to one another.

13. MATHEMATICAL SYSTEM 13 UNDEFINED TERMS DEFINED TERMS AXIOMS/ POSTULATES THEOREMS in geometry, we come across with terms which cannot be precisely defined . In modern mathematics. We do accept certain undefined terms by description. Unlike undefined terms (which do not have formal definition), these have a formal definition . They are used to define even more terms. A statement which is accepted as true without proof . These statements can be used as reasons in proving some mathematical statements. a statement that can be proven . Once a theorem is proven, it can also be used as a reason in proving other statements.

14. MATHEMATICAL SYSTEM 14 UNDEFINED TERMS DEFINED TERMS AXIOMS/ POSTULATES THEOREMS Point Line Plane Collinear Points Coplanar Points Subsets of a Line Axioms Postulates Theorems Line Line Segment Two points determine a line The shortest segment from a point not on a line to the line is the perpendicular segment

15. UNDEFINED TERMS 15 Point Line Plane DEFINED TERMS Collinear Points Coplanar Points Subsets of a Line Noncollinear and Noncoplanar Points

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20. Directions: Identify whether each of the following represents a point, line or a plane. Stars in the sky Curtain rod Edge of a ruler Cartolina A knot on a piece of thread A clothesline 20 7. Top of a box 8. Page of a book 9. A magic wand 10. Button 11. Mole 12. Handkerchief Point Line Line Plane Point Line Plane Plane Line Point Point Plane

21. Point 21

22. Point 22 A point is a position in space. It has only location but no dimension; length, width, thickness and does not occupy an area. It is named using a CAPITAL LETTER and it can be modeled by a dot. A B C D All other geometric figures are made up of a collection of points.

23. Line 23 A straight, continuous arrangement of infinitely many points. Its length is infinite. It extends infinitely in two directions. And it has no thickness. The arrowhead symbolizes infinity Lines are named by a single lower case script letter or by any two points on a line

24. 24 Line d line d d A B line AB line BA AB BA T wo points determine a line

25. Plane A flat surface that extends along its length and width. It is like an “infinite sheet of paper”. It has length and width but no thickness It is named by a single script CAPITAL LETTER or by any three points in the plane which are not on the same line. 25

26. Plane A flat surface that extends along its length and width. It is like an “infinite sheet of paper”. It has length and width but no thickness It is named by a single script CAPITAL LETTER or by any three points in the plane which are not on the same line. 26 D plane D L I E F plane LIF plane IFE plane LEF plane EFI plane LIFE At least three noncollinear points determine a plane.

27. Defined Terms 27 Col line ar Points Non-col line ar Points A B C Points A, B, and C are colinear points A B C Points A, B, and C are non-colinear points are points that lie on the same line are points that do not lie on the same line

28. 28 Defined Terms Co plana r Points Non-co plana r Points A B C Points A, B, and C are coplanar points B C A Points A, B, and C are non-coplanar points are points that lie in the same plane are points that do not lie in the same plane

29. Subsets of a Line LINE SEGMENT 29 A B A B A B B A C A line segment is a part of a line consisting two endpoints and all the points in between. A line segment may be called: AB or BA Its endpoints are A and B. RAY A ray is a part of a line with one endpoint and extending in only one direction. A ray is named with its endpoint first, followed by another point on the ray. The ray can be named AB read as “ray AB” OPPOSITE RAY are rays with a common endpoint but extending in opposite directions. Common Endpoint: B Opposite rays: BA and BC B is between points A and C

30. 30 Let’s Try! 1. Name all the points. A, W, X, Y and Z 2. Name all the lines using a script letter. line a line b line c line d 3. What is the other name for line a? line WZ or line ZW 4. Using a script letter, name a plane form by the four lines. plane L 5. Name the plane in three different ways using three points. plane WXY plane XYZ plane WZY W X Z Y L A c d a b

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32. 32 Assignment