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Mathematics
Quarter 3 - Module 1
Mathematical System
What This Module is About
A typical mathematics system has the following four parts, the undefined terms,
defined terms, axioms and/or postulate, and theorems. Undefined and defined terms are all
about vocabulary, and axioms and/or postulate and theorems are all about principles.
This module will help you understand more on previously learned topics in geometry
specifically on mathematical system. It introduces terms that we’ll be used in order to help us
solve problems and prove scenarios. Axiom and theorem are being described through
illustrative example to widen one’s perspective.
What I Need to Know
At the end of this module, you should be able to:
1. Describe a mathematical system. (M8GE-IIIa-1)
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What I Know
Choose the letter of the correct answer.
1. A point has no dimension.
a. False
b. true
c. maybe
d. cannot be determined
c. dot
d. ball pen
2. Which of the following represents a plane?
a. Pencil
b. blackboard
3. What is the formula of a Pythagorean Theorem?
a. a² + b² = c²
c. a² • b² = c² d. a + b = c²
b. a² - b² = c²
4. It is a subset of a line that has two endpoints.
a. Skew lines
b. Parallel lines
c. ray
d. line segment
5. The following represents a point EXCEPT.
a. Tip of a needle
b. A dot
c. electric wire
d. grain of sugar
6. A statement that can be demonstrated to be true by accepted mathematical
operations and arguments.
a. Axiom
b. Theorem
c. postulate
d. principle
7. Line segment and ray are some of the examples of _______.
a. Defined terms
b. Undefined terms
c. theorem
d. postulate
8. Which of the following represents a surface of a table and a wall?
a. Plane
b. Point
c. line
d. ray
9. “A gentle torturer” is a specific example of?
a. Theorem
b. Postulate
c. Principle of Contradiction
d. Axiom
10. It is a one-dimensional figure which extends endlessly in both directions.
a. Ray
b. Line segment
c. angle
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d. line
Undefined and Defined Terms
Lesson
1
What I Need to Know
In modern mathematics, we accept certain undefined terms. The choice of the
undefined terms is completely arbitrary and generally to facilitate the development of
structure, e.g. point, plane, number, variable, line etc. then, we defined the other terms of the
mathematical system in terms of undefined terms, e.g. angle, line segment, circle, ray etc.
What’s New
Activity 1
Direction: From your learning in geometry in your previous grade, write something about the
following terms:
1. Point
________________________________________________________________
________________________________________________________________
2. Line
________________________________________________________________
________________________________________________________________
3. Plane
___________________________________________________________
___________________________________________________________
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What Is It
Term
Figure
Point
•
Line
• R
• V
A
A line is a set of points arranged in a row.
It is extended endlessly in both directions.
It is a one-dimensional figure.
Two points determine a line. That is, two
distinct points are contained by exactly one
line.
We use a lower case letter or any two points
on the line to name the line.
m
P
Plane
Q
R
Description
A point suggests an exact location in space.
It has no dimension.
We use a capital letter to name a point.
A plane is a set of points in an endless flat
surface.
The following determine a plane:
(a) three non-collinear points;
(b)two intersecting lines;
(c) two parallel lines; or
(d) a line and a point not on the line. We use
a lower case letter or three points on the
plane to name the plane.
Notation
point A
⃡
line m or 𝑅𝑉
plane PQR or
PQR
What’s More
Activity 2
List down 3 objects that could represent each of the following terms below that are
found in your home.
a. a point.
b. a line.
c. a plane.
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What’s New
Activity 3
Direction: Based on what you have learned, try to define the following terms:
1. Angle
2. Line segment
3. Ray
What Is It
An angle is a union of two non-collinear rays with common endpoint. The two non-collinear
rays are the sides of the angle while the common endpoint is the vertex.
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What’s More
Activity 4
Direction: Name at least 2 examples of each of the following terms below from figure 2.
figure 2
Illustrative example from fig. 1:
1. Angle: ______ _______
Angle: LRQS
2. Line segment:_______ _______
̅̅̅̅
Line segment: 𝑄𝑅
3. Ray: ______ _______
Ray: 𝑄𝑆
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Lesson
2
Axioms or Postulate and
Theorems
What’s In
We had just learned some of the undefined and defined terms in geometry in the
previous lesson. Undefined terms and defined terms are purely mental concepts or ideas.
Thus, these undefined terms can only be described. Those terms will help us to understand
more with this new lesson.
What I Need to Know
Early Greeks considered postulates as general truths common to all studies and
axioms as the truths relating to the special study at hand.
A statement that we arrive at by successive application of rule of implication to the
axiom and statements previously arrived is called theorems.
What’s New
Activity 1
Answer these:
1. Why is “0” a natural number?
2. In your own understanding, what does Principle of Contradiction mean?
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What Is It
“0 is a natural number”, is an example of axiom.
Axiom is a concept in logic, a statement which is accepted without question and which does
not require proof.
There are reasons why it has no proof for example:
1. The statement might be obvious. This means most people think it is clearly true.
2. The statement is based on physical laws and can easily be observed.
3. The statement is a proposition.
Principle of Contradiction is an example of an obvious axiom. It says that a statement and its
opposite cannot both be true at the same time and place. Thus, you can have a
contradictory statement.
Example of a contradictory statement: “banal na aso”
“the cleanest mess”
“the sweetest salt”
What’s More
Activity 3
Write at least 5 contradictory statements.
What’s New
Activity 4
Answer.
1. Remember this: a² + b² = c² for a right angled triangle. What is it?
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What Is It
What is a Theorem?
A theorem is a statement that can be demonstrated to be true by accepted
mathematical operations and arguments. In general, a theorem is an embodiment of some
general principle that makes it part of a larger theory.
Example:
Formula:
a² + b² = c²
a = side of right triangle
b = side of right triangle
c = hypotenuse
This theorem states that the area of the triangle whose side is the hypotenuse is
equal to the sum of the areas of the squares on the other two sides.
It is a fundamental relation in Euclidean geometry among the three sides of a right
triangle.
What’s More
Activity 5
List at least 3 theorems you know.
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What I have Learned
Activity 6
Cite the difference between an axiom/postulate and a theorem.
What I can Do
Activity 7
Write an example of an axiom/postulate and a theorem, and then describe each.
Summary
We can use our logic and reasoning skills to develop the mathematical system of
geometry. Begin with undefined terms, which we first describe. The, we used these
undefined terms to formally define terms. Our defined terms are used to write statements
that we do not prove, but instead agree and accept them to be true. These statements are
called postulates or axioms. Our mathematical system grows by using terms, postulates, and
axioms to prove theorems.
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Assessment (Post-Test)
Choose the letter of the correct answer.
1. “The cleanest mess” is an specific example of?
a. Theorem
b. Postulate
c. Principle of Contradiction
d. Axiom
2. What is the formula of a Pythagorean Theorem?
a. a + b = c²
b. a² • b² = c²
c. a² - b² = c²
d. a² + b² = c²
c. line segment
d. ray
c. true
d. false
3. It is a one dimensional figure.
a. line
b. angle
4. A point suggests an exact location in space.
a. cannot be determined
b. a & b
5. A statement that can be demonstrated to be true by accepted mathematical
operations and arguments.
a. Axiom
b. Theorem
c. postulate
d. principle
c. ball pen
d. dot
c. ray
d. line
6. Which of the following represents a plane?
a. wire
b. wall
7. A blackboard and floor represents what?
a. Plane
b. Point
8. It is a subset of a line that has two endpoints.
a. Skew lines
b. Parallel lines
c. ray
d. line segment
c. electric wire
d. edge of a table
9. The following represents a point EXCEPT.
a. tip of a pen
b. a dot
10. Line segment and ray are some of the examples of what?
a. undefined terms
b. defined terms
c. postulate
d. theorem
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