Mathematics Activity

Sheet
Quarter 3–Week 1 & 2
❖ Describing a Mathematical System
❖ Illustrating the Need for an Axiomatic
Structure of a Mathematical System in General
and in Geometry in Particular

REGION VI – WESTERN VISAYAS

 Quarter 3, Week 1, 2

LEARNING ACTIVITY SHEET NO. 1 – 2

MATHEMATICS 8 ACTIVITY SHEET
Describes a mathematical system. Illustrates the Need for an Axiomatic Structure of
a Mathematical System in General and in Geometry in Particular.

I. Learning Competency

➢ Describes a mathematical system. (M8GE-IIIa-1)

➢ Illustrates the Need for an Axiomatic Structure of a Mathematical
System in General and in Geometry in Particular. (M8GE-IIIa-c-1)

II. Background Information for Learners / Educational Sites

https://faculty.uml.edu/klevasseur/ads/s-math-systems
https://www.sunysuffolk.edu/explore-academics/faculty-and-staff/faculty-
websites/jean-nicolas-
pestieau/documents/MAT102_math_systems_and_groups.pdf
http://www.solano.edu/academic_success_center/forms/math/Basic%20Numb
er%20Properties.pdf
INTRODUCTION
This lesson will focus on describing a mathematical system and how to
illustrate the need for an axiomatic structure of a mathematical system in general,
and in Geometry in particular: (a) defined terms; (b) undefined terms; (c) postulates;
and (d) theorems.

PART 1
A structure formed from one or more sets of undefined objects, various concepts
which may or may not be defined, and a set of axioms relating these objects and concepts is
called mathematical system. It consists of the following:
1. A set or universe, U.
2. Definitions: sentences that explain the meaning of concepts that relate to the
universe.
3. Axioms and Postulates: Axiom is an assertion about the properties of the universe
and rules for creating and justifying more assertions. These rules always include the
system of logic that we have developed to this point. Postulate is a statement that is
accepted without proof.
4. Theorems: True propositions derived from the axioms of a mathematical system.
Properties of Mathematical System
1. Closure. S is closed under the operation (). If for any two elements a, b in S, the
element (a)(b) is also in S. For example, the naturals are closed under addition since
adding any two naturals always results in another, bigger, natural (e.g. 2 + 6 = 8).

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 2. Identity Element. S has an identity element i. If for any element a in S, a(i) = (i) a = a.
Example: a. 3(1) = (1)3 = 3, 1 is the identity element.
b. 3 + 0 = 3, 0 is the identity element.
3. Inverse Element. If an element b in S is such that a(b) = (b)a = i, then this element is
called the inverse of a and (a, b) is called an inverse pair.
1
Example: a. 3 + (−3) = 0 b. 3(3) = 1
4. Associative Property. If for any three elements a, b, c in S, (a)(b) ∗ c = a ∗ (b ∗ c), the
operation ∗ is said to be associative.
5. Example: a. (2 + 3) + 5 = 2 + (3 + 5) b. (2 x 3) 5 = 2 (3 x 5)
6. Commutative Property. If for any two elements a, b in S, a ∗ b = b ∗ a, the operation
∗ is said to be commutative.
7. Example: a. 3 + 4 = 4 + 3 b. 3 (4) = (4) 3
8. Distributive Property. The sum of two numbers times a third number is equal to the
sum of each addend time the third number.

PART 2
Defined and Undefined Terms

A point suggests an exact location in space. It has no dimension. It is named by using a

capital letter.
A line is a one –dimensional figure that contains set of points arranged in a row which
extended infinitely in both directions. Two points determine a line. That is, two
distinct points are contained by exactly one line. A line can be named using lower
case letter or any two points on the line.
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. To name the plane, we use a lower-case letter or three points
on the plane.
Collinear points are points on the same line.
Noncollinear points are points not on the same line.
Coplanar points are points on the same plane.
Noncoplanar points are points not on the same plane.
Example 1: (a) The tip of a pencil represents the idea of a point.
(b) The telephone wires represent the idea of a line.
(c) The surface of the page of a notebook represents the idea of a plane.
Example 2: Consider the given box below. E
(a) A and B are collinear points.
F
(b) D and G are noncollinear points. D C
(c) A and C are coplanar points. G
A postulate is a statement that is accepted without proof. A B
A theorem is a statement accepted after it is proved deductively.

Axioms of Equality
• Reflexive Property of Equality
For all real numbers p, p = p.

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 • Symmetric Property of Equality
For all real numbers p and q, if p = q, then q = p.
• Transitive Property of Equality
For all real numbers p, q, and r, if p = q and q = r, then p = r.
• Substitution Property of Equality
For all real numbers p and q, if p = q, then q can be substituted for p in any
expression.

Properties of Equality
• Addition Property of Equality
For all real numbers p, q, and r, if p = q, then p + r = q + r.
• Multiplication Property of Equality
For all real numbers p,q and r, if p=q, them pr = qr.

Definitions, Postulates, and Theorems on Points, Lines, Angles, and Angle Pairs
• Definition of a Midpoint
If points P, Q, and R are collinear (P–Q–R) and Q is the midpoint of ̅̅̅̅
𝑃𝑅 , then
̅̅̅̅ ≅ 𝑄𝑅
𝑃𝑄 ̅̅̅̅ .
• Definition of an Angle Bisector
̅̅̅̅ bisects ∠PQR, then ∠𝑃𝑄𝑆 ≅ ∠𝑆𝑄𝑅.
If 𝑄𝑆
• Segment Addition Postulate
If points P, Q, and R are collinear (P–Q–R) and Q is between points P and R, then
̅̅̅̅ + 𝑄𝑅
𝑃𝑄 ̅̅̅̅ = ̅̅̅̅
𝑃𝑅.
• Angle Addition Postulate
If point S lies in the interior of ∠𝑃𝑄𝑅, then 𝑚∠𝑃𝑄𝑆 + 𝑚∠𝑆𝑄𝑅 = 𝑚∠𝑃𝑄𝑅.
• Definition of Supplementary Angles
Two angles are supplementary if the sum of their measures is 180 0.
• Definition of Complementary Angles
Two angles are complementary if the sum of their measures is 90 0.
• Definition of Linear Pair
Linear pair is a pair of adjacent angles formed by two intersecting lines
• Linear Pair Theorem
If two angles form a linear pair, then they are supplementary.
• Definition of Vertical Angles
Vertical angles refer to two non-adjacent angles formed by two intersecting lines.
• Vertical Angles Theorem
Vertical angles are congruent.

III. Practice!
Exercise 1
Direction: Write TRUE if the statement is correct and FALSE if not.
_______________ 1. Mathematical system is a structure formed from undefined objects only.
_______________ 2. Theorems are assertions about the properties of the universe and rules for
creating and justifying more assertions.
_______________ 3. A mathematical system consists of a set elements.

GRADE 8 MATHEMATICS INSTRUCTIONAL PACKET QUARTER 3 WEEK 1-2 Page 4

 _______________ 4. Descriptions explain the meaning of concepts that relate to something.
_______________ 5. If an element b in S is such that a ∗ b = b ∗ a = i, then this element is called
the inverse of a and (a, b) is called a reverse pair.
_______________ 6. x ∗ y = y ∗ x is an example of commutative property.
_______________ 7. The additive inverse of 7 is -7.
_______________ 8. Commutative property is also true to subtraction.
_______________ 9. For any three elements x, y, z in S, (x ∗ y) ∗ z = x ∗ (y ∗ z), the operation ∗ is
said to be associative.
_______________ 10. 5 + 0 = 5. 0 is the identity element.

Exercise 2
Directions: State the property of mathematical system shown below.
1. 9 + 0 = 9 6. 8 x 1/8 = 1
2. 6 x 9 = 9 x 6 7. 3 (2n – 1) = 6n – 3
3. m ( n + o ) = mn + mo 8. 11 + 2x = 2x + 11
4. 1 + (-1) = 0 9. 12 x 1 = 12
5. d + ( e + f ) = (d + e ) + f 10. (3x + y) + 5 = 3x + (y + 5)

Exercise 3
Directions: These are some of the objects around us that could represent a point, a line, or a plane.
Classify each object and place it in tis corresponding column in the table.
Tip of a needle A grain of rice cover of your book
the wall of a room meter stick laser
the string on a guitar tip of a ballpen intersection of a side
wall
the floor of your and ceiling
a clothesline
bedroom
intersection of the
top of the table a star in the sky front wall, a side wall
and ceiling

Objects that could Objects

Objects
thatthat
could
could Objects
Objects
that could
that could
represent a point represent
represent
a line
a line represent
represent
a plane
a plane

Exercise 4
Direction: Write TRUE if the statement is correct and FALSE if the statement is wrong. Use the
figure below for the given item. E F
1. Points A, B, C, D are collinear.
2. Points A, D, F are noncollinear A B
3. Points B, F and G are on the same line
G
4. Points G, C, D are not on the same line.
5. Points A, E, F are coplanar.
C D
6. Points A, F, G are not coplanar.
7. Points A, B, D , E are on the same plane.
8. Points A, B, F, E are coplanar.
9. Points A. B, D are collinear and coplanar.
10. Points B, F, C are collinear and coplanar.

IV.
Direction: Match the statements from column A to its corresponding terms in column B.

GRADE 8 MATHEMATICS INSTRUCTIONAL PACKET QUARTER 3 WEEK 1-2 Page 5

 Column A Column B
1. If two angles form a linear pair, A. Point
then they are supplementary.
2. Vertical angles are congruent. B. Angle Addition Postulate
3. If point S lies in the interior of ∠PQR, C. Line
then 𝑚∠𝑃𝑄𝑆 + 𝑚∠𝑆𝑄𝑅 = 𝑚∠𝑃𝑄𝑅.
4. If points P, Q, and R are collinear D. Segment Addition Postulate
(P–Q–R) and Q is between points
P and R, then ̅̅̅̅
𝑃𝑄 + ̅̅̅̅
𝑄𝑅 = ̅̅̅̅
𝑃𝑅 . E. Linear Pair Theorem
̅̅̅̅
5. If 𝑄𝑆 bisects ∠𝑃𝑄𝑅, F. Definition of an Angle Bisector
then ∠PQS ≅ ∠SQR
6. It suggests an exact location G. Vertical Angles Theorem
in space.
7. They are points found on the
same line. H. Coplanar points
8. It is a set of points in an I. plane
endless flat surface.
9. They are points found on
the same plane.
10. It is a one –dimensional figure
that contains set of points arranged J. Collinear points
in a row which extended
infinitely in both directions.

ANSWER KEY:

Exercise 4 Exercise 3

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