8
MATHEMATICS
FOURTH QUARTER
LEARNING ACTIVITY SHEETS
Republic of the Philippines
Department of Education
REGION II – CAGAYAN VALLEY
COPYRIGHT PAGE
Learning Activity Sheet in MATHEMATICS
(Grade 8)
Copyright © 2020
DEPARTMENT OF EDUCATION
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Content Editor
Language Editor
Focal Persons
: RICHARD S. CABALZA, MICHAEL M. ACUPAN,
ARLON T. MACARUBBO, JANSTEN B. MAPATAC,
TUGUEGARAO CITY SCIENCE HIGH SCHOOL
: JOAQUINA L. BIRUNG, MARLO T. MELAD, NOLI B. ABRIGO Jr., PhD
ENRIQUE GARCIA, JACKILYN ALAMBRA, MAI RANI ZIPAGAN
: ISAGANI DURUIN, PhD, JESSICA T. CASTANEDA, PhD
: NOLI B. ABRIGO, PhD
JESSICA T. CASTANEDA, PhD
ISAGANI R. DURUIN, PhD
RIZALINO G. CARONAN
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_____________________________________________
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ii
Table of Contents
Compentency
Page
number
Illustrates theorems on triangle inequalities (Exterior
Angle Inequality Theorem, Triangle Inequality
Theorem, Hinge Theorem).
.....................
1
Applies theorems on triangle inequalities.
.....................
18
Proves inequalities in a triangle.
.....................
27
Proves properties of parallel lines cut by a
transversal.
.....................
39
Determines the conditions under which lines and
segments are parallel or perpendicular.
.....................
45
Illustrates an experiment, outcome, sample space
and event.
.....................
55
Counts the number of occurrences of an outcome in
an experiment: (a) table; (b) tree diagram; (c)
systematic listing; and (d) fundamental counting
principle.
....................
61
Finds the probability of a simple event.
....................
68
Illustrates an experimental probability and a
theoretical probability.
....................
74
Solves problems involving probabilities of simple
events.
....................
83
Note: Practice Personal Hygiene protocols at all times.
iii
1
MATHEMATICS 8
Name of Learner: _________________________________
Section: _________________________________________
Grade Level: __________
Date: ___________
LEARNING ACTIVITY SHEETS
TRIANGLE INEQUALITY
Background Information for Learners
The Triangle Inequality Theorem states that the sum of the lengths of any
two sides of a triangle must be greater than the measure of the length third side.
Meaning, if the sum of the two sides is less than or equal to the measure of a third side,
then the sides cannot form a triangle.
Remember:
▪ If a, b, and c are the sides of a triangle then, a + b > c.
▪ The sum of two remote angles is equal to its exterior angle.
▪ Six exterior angles are formed when all the sides of the triangle are extended.
▪ The Hinge Theorem is sometimes called the “Alligator Theorem” because if
you consider the sides as the (fixed length) jaws of an alligator the wider it
opens its mouth, the bigger the prey it can fit.
Learning Competency:
Illustrate theorems on triangle inequalities (Exterior Angle Inequality Theorem,
Triangle Inequality Theorem, Hinge Theorem). M8GE-IVa-1
Think about this:
Father will put up a small sturdy overhang cabinet on a wall for his carpentry tools.
For assistance, he instructed his son to cut pieces of wood for triangular braces. He gave the
lengths as follows 6, 8, 10, and 6, 8, 16 all in inches. Using the cut woods with their indicated
lengths, can the son form a triangle by joining the endpoints of the wood?
Note: Practice Personal Hygiene protocols at all times.
2
Reused picture
ACTIVITY 01
unFIT and RIGHT
Materials
At least 10 bamboo sticks
Notebook (for recording)
Ballpen
Directions:
1. Copy the table below in your notebook.
2. Cut the bamboo sticks into the desired lengths. (The lengths of the sticks must be in
inches)
3. Form a triangle using the sticks by connecting their endpoints
4. Under the unFit and Right on the table, write R if the sticks can form a triangle U if
the sticks cannot form a triangle.
.
No.
1.
2.
3.
4.
5.
6.
7.
Length of the sticks (in inches)
l
a
s
6
8
10
5
2
6
4
4
4
3
2
6
6
8
16
4
4
8
6
8
9
unFit and Right
For all the sets of lengths given in the table, perform the three cases below:
Case # 1. l + a, compare the sum of the lengths of the two sides with the length of side s
Case # 2. l + s, compare the sum of the lengths of the two sides with the length of side a
Case # 3. a + s, compare the sum of the lengths of the two sides with the length of side l
Note: Practice Personal Hygiene protocols at all times.
3
Let’s try # 1: Refer to the table
Given: l = 6
a=8
s = 10
a=8
l=6
Case # 1. 6 + 8=14, the sum of the lengths of l and a is greater than the length of side s
which is 10. (Take note: l + a > s)
6 + 8 > 10
Case # 2. 6+10=16, the sum of the lengths of l and s is greater than the length of side a
which is 8. (Take note: l + s > a)
6 + 10 > 8
Case # 3. 8+10=18, the sum of the lengths of a and s is greater than the length of side l
which is 6. (Take note: a + s > l)
8 + 10 > 6
Do the same with the remaining items
Guide Questions
1. Do the bamboo sticks form a triangle or not?
2. How many sets of sticks were able to form a triangle?
3. How many sets did not form a triangle?
4. Were you able to form a triangle in # 1?
5. What pattern did you observe with the sets of sticks that form and do not form a
triangle?
TO SUM UP:
Supply the blanks below with word/s to complete the thought of the statements.
1. A straight line parallel to the third stick is formed if the sum of the lengths
of two sticks is ____________ to the length of the third stick.
2. A triangle cannot be formed if the sum of the lengths of the two sticks is
________________ the length of the third stick.
3. A triangle can be formed if the sum of the lengths of the two sticks is ___________
the length of the third stick.
4. If the sum of the lengths of two sticks is ________________ the length of
the third stick, then a triangle cannot be formed.
5. If the sum of the lengths of two sticks is__________________ the third stick, then
a triangle can be formed.
Items 1 and 5 in the table answer the questions in the “Think about this” portion.
Note: Practice Personal Hygiene protocols at all times.
4
KEY POINTS:
Triangle Inequality Theorem
▪
▪
For any triangle, the sum of the lengths of any two sides must be greater
than the length of the remaining side.
It is not possible to construct a triangle from three line segments if any of
them is longer than the sum of the other two.
The Converse
A triangle cannot be constructed from three line segments if any of them is longer
than the sum of the other two.
ACTIVITY 02
FINDING THE POSSIBILITY
Draw a
triangle
whose two
sides are 10
and 15,
respectively.
Huh! how about the
third side? Isn’t it that
the third side should not
just be any number?
What
length
should I
use for
the third
side?
The situation above requires the knowledge about the possible lengths of the third side of
a triangle given its two sides.
Let us see how to determine all the possible lengths (h) of the third side
Note: Practice Personal Hygiene protocols at all times.
5
CASE # 1
10 + 15 > h
25 > h
valid statement
CASE # 2
10 + h > 15
h > 15 – 10
h>5
valid statement
Let us combine the two valid statements
25 > h
or
h < 25
and h > 5
or
5<h
Putting the two statements together, we have:
5 < h < 25
Algebraically we have:
15 – 10 < h < 15 + 10
difference of
the two sides
CASE # 3
15 + h > 10
h > 10 - 15
h>- 5
Not valid, because
a negative integer
cannot be a length
to any linear
distance.
sum of the
two sides
Therefore,
▪ the possible lengths of the third side are lengths between 5 and 25 units
exclusive or 5 < h < 25
▪ the possible set of lengths are { 6, 7, 8, 9, ...24}
▪ the number of possible lengths is 24 – 5 = 19
Key Points:
Given three sides a, b and c, where c is the third side.
The possible lengths of the third side c of the triangle can be determined by:
b–a<c<b+a
FOLLOW UP EXERCISES:
A. Determine the possible lengths of the third side of the triangle given two sides; and
B. List down the possible lengths in set notation form
1. 4 in, 8 in
2. 7 in, 12 in
3. 5 in, 14 in
4. 22 in, 30 in
5. The three sides of a triangle are a, b and c where a>b. If 12 < c < 22, what is the
length of each a and b?
6. The lengths of the sides of a triangle are 16 – k, 16, and 16 + k. What is the range of
the possible values of k? Create a table of the possible integer lengths of the
sides of the triangle if 16 – k is always the shortest length?
Note: Practice Personal Hygiene protocols at all times.
6
ACTIVITY 03
TRACKING THE OPPOSITES
Materials
Protractor
Ruler
Ballpen
Notebook (for recording)
Directions:
1. Copy the table below in your notebook.
2. Use protractor to measure the angles.
3. Measure the lengths of the sides opposite the angles.
4. Write the measures of the indicated parts in the table in your notebook.
Name of s
ABC
DEF
GHI
Degree measures of
the Angles
A
B
D
E
G
H
Lengths of sides (in cm)
Opposite the Angles
BC
AC
EF
DF
HI
GI
GUIDE QUESTIONS:
1. In each triangle, what is the opposite side of the largest angle?
▪ The opposite side of the largest angle is the________________ side.
2. In each triangle, what is the opposite angle of the shortest side?
▪ The opposite angle of the shortest side is the ________________ angle.
3. What is the relationship between the longer side of a triangle and the measure of the
angle opposite it?
4. Without a protractor, is it possible to determine the measure of the third angle in each
triangle in the table? How?
Note: Practice Personal Hygiene protocols at all times.
7
5. Given LMS, where L = 1000 ; M = 500
a. What is the S?
b. What is the longest side?
c. Arrange the sides in ascending order
Key Points:
Angle side relationships in Triangles
If two sides of a triangle are not congruent, then the largest angle is opposite the
longest side. (Angle-Side Relationship Theorem)
If two angles of a triangle are not congruent, then the longest side is opposite the
largest angle. ( Side-Angle Relationship Theorem)
Note: Practice Personal Hygiene protocols at all times.
8
EXTERIOR ANGLE INEQUALITY
THEOREM
ACTIVITY 04
WORD TREASURE
Directions: Fill in the crossword puzzle with the words missing from the sentences below.
Match the number of the sentence to the boxes placed across or down the grid. If
filled out correctly, the words will fit neatly into the puzzle.
Name: ___________________________________ Date: ______________
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Note: Practice Personal Hygiene protocols at all times.
9
Across
6. A 3-sided closed polygon
Down
1. A triangle with an angle whose measure
is between 90 degrees and 180 degrees
7. Angles that do not share a vertex or 2. Two angles with a common vertex and a
corner of a triangle with the exterior
common side
angle
3. An angle formed between one side and
9. The three angles inside the triangle
an extended side of a triangle
11. Two rays with a common endpoint 4. The verbal phrase of the inequality
12. Two angles whose non-adjacent
symbol >
sides form a straight line
5. A mathematical statement that is
13. Set of points that can be extended
accepted with a proof
infinitely
8. A triangle with two congruent sides
14. The verbal phrase of the symbol < 10. A pair of opposite angles formed by two
intersecting lines
WHO AM I ?
ACTIVITY 05
Directions: Classify the different parts of the triangle inequality below as exterior angle,
interior angle, an angle adjacent to an interior angle, and remote angles of an
angle.
B C
A 1
2
I
3
H
Exterior
Angles
D E
F
Interior
Angles
G
Adjacent angles (an
exterior angle
adjacent to an interior
angle)
A and ___
D and ___
G and ___
C and ___
F and ___
I and ___
Note: Practice Personal Hygiene protocols at all times.
Remote angles of the
given angle
A
D
G
C
F
I
10
TRISOME ANGLING
ACTIVITY 06
Materials
Protractor
Ruler
Ballpen
Notebook (for recording)
Directions:
1. Copy the table below in your notebook.
2. Use protractor to measure the angles.
3. Write the measures of the indicated parts in the table in your notebook.
T
C
AS
G
ET
A
M
H
U
L
Angle measures
Triangle
ATM
LAC
TUG
Exterior
Angle
H =
S =
E =
Remote Interior s
A =
L =
U =
T =
C =
G =
GUIDE QUESTIONS:
1. Are all outer angles considered remote exterior angles?
2. How many exterior angles can be formed when all the sides of the triangle are
extended?
3. What is exterior angle in Triangle Inequality?
4. What is the relationship of an exterior angle with its two remote angles?
5. In LAC, how big is S?
Note: Practice Personal Hygiene protocols at all times.
11
ACTIVITY 07
FIND THE MISSING PART
A. Solve for x and determine the measure of the missing angle/s. (For items 1-6)
B. Answer the following.
7. True or False: The remote angles of an exterior angle are supplementary.
8. In ABC, AB = 14, AC = 8.
Q
a. What is the largest integral possible length of BC?
1
b. Arrange the angles in ascending order if BC has the largest integral value.
9. Use the figure at the right to answer the questions below.
a. Which angle/s is/are less than 2?
b. If m P = 40° and mPQR = 50°, what is m2?
2
10. If m P = 50° and m2 = 100°, what kind of triangle P
R
is PQR?
Note: Practice Personal Hygiene protocols at all times.
12
HINGE THEOREM
THE ALLIGATOR THEOREM
ACTIVITY 08
Directions: Fill in the blank with the appropriate word to complete the statement.
1. The triangles.
8
8
8
P
P
8
P
8
8
P
8
8
8
P
8
8
P
8
▪ As the angle increases, the length of the opposite side ________________.
2. The opening of the door.
Hinge
Door frame
Hinge
Door frame
Kove and Shak went into the dugout after the scrimmage game. Kove, a medium
size player went ahead of Shak who is a lot bigger, and he opened the door which is
just enough for him to get inside. Now, what should Shak do to enter the dugout like
Kove?
▪
Shak should open the door a bit wider to make the length of the
opening ________enough for him to get through the door.
3. The opening of the alligator’s mouth.
Reused picture
▪
The bigger the prey the __________the mouth of the alligator should open.
Note: Practice Personal Hygiene protocols at all times.
13
For items 4 – 7, use <, > or = to compare the two parts of the two triangles.
(Not drawn to scale)
4.
5.
15
16
B___E
BC___FE
6.
7.
GJ___JI
2___1
8. Two navigating ships bound to the same port came from a short trip as
shown in the diagram below. The distances that the ships travel and the distances
back to the port form two triangles. The triangles have two congruent sides with
lengths of 3 miles and 1.8 miles. Which of the two ships is closer to the port?
Ship 2
1.8 mi
3 mi
0
35
Port
450
3 mi
1.8 mi
Ship 1
KEY POINTS:
Hinge Theorem:
▪
If two triangles have two congruent sides, then the triangle that has the larger
included angle has the longer third side.
Converse:
▪
If two triangles have two congruent sides, then the triangle that has the longer
third side has the larger included angle.
Note: Practice Personal Hygiene protocols at all times.
14
Rubrics for Scoring
Level 1 (0 pt.)
Amount of
Learner shows
Work
no attempt to
really do any of
the problems,
no answer
given.
Understanding
Learner shows
no
understanding
of using triangle
inequality to
solve real life
problems
involving
triangles
Level 2 (1 pt.)
Learner only
shows answer.
Level 3 (2 pts.)
Learner only
shows answers
but showed
partial work.
Level 4 (3 pts.)
Learner
completed each
step and gave
complete
answers.
Learner shows
limited
understanding
of using triangle
inequality to
solve real life
problems
involving
triangles
Learner shows
partial
understanding
of using triangle
inequality to
solve real life
problems
involving
triangles
Learner shows
thorough
understanding
of using
triangle
inequality to
solve real life
problems
involving
triangles
Reflection:
What have you learned about Triangle Inequality?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
________________________
ANSWERS
Activity 1: unFIT and RIGHT
1. R
2. R
3. R
4. U
5. U
6. U
7. R
GUIDE QUESTIONS
1. Not all. Depends on the lengths of the sticks.
2. 4
3. 3
4. Yes
Note: Practice Personal Hygiene protocols at all times.
15
5. If the sum of the lengths of two sticks is greater than the third stick, then a triangle
can be formed. Otherwise, no triangle is formed.
TO SUM UP
1. equal
2. less than
3. greater than
4. less than
5, greater than
Activity 2: FINDING THE POSSIBILITY
Exercises
1. 4 < s < 12, s = {5,6,7…11}
2. 5 < s < 19, s = {6,7,8…18}
3. 9 < s < 19, s = {10,11,12…18}
4. 8 < s < 52, s = { 9,10,11…51}
5. a = 5, b = 17
6. 0 < k < 8
Table
0<k<8
1
2
3
4
5
6
7
16 - k
15
14
13
12
11
10
9
16
16
16
16
16
16
16
16
16 + k
17
18
19
20
21
22
23
Activity 3: TRACKING THE OPPOSITES
GUIDE QUESTIONS
1. longest side
2. smallest angle
3. The angle opposite the longer side is the larger angle.
4. Yes, subtract the sum of the two angles from 1800.
5. a. 300
b. SM
c. LM, LS, SM
Activity 4: WORD TREASURE
exterior angle
adjacent angles
linear pair
vertical
greater than
remote
lesser than
interior
Note: Practice Personal Hygiene protocols at all times.
triangle
isosceles
angle
obtuse
line
theorem
16
Activity 5: WHO AM I?
Exterior
Angles
Interior
Angles
A
D
G
C
F
I
1
2
3
Adjacent angles (an
exterior angle
adjacent to an interior
angle)
A and 1
D and 2
G and 3
C and 1
F and 2
I and 3
Remote angles of the
given angle
A
D
G
C
F
I
2 and 3
1 and 3
1 and 2
2 and 3
1 and 3
1 and 2
Activity 6 : TRISOME ANGLING
GUIDE QUESTION
1. No
2. 6
3. An angle formed between one side and an extended side of a triangle.
4. The exterior angle is equal to the sum of its two remote angles.
5. Based on the figure, there is no algebraic computation for the measure of S,
but surely S is greater than both the measures of  L and C.
Activity 7 : FIND THE MISSING
Note: MA denotes the third missing angle.
1. x = 790, MA= 500
2. x = 550, MA = 550
3. x = 1200, MA = 600
4. x = 1000, MA = 300
5. x = 12, 1= 4x + 5 = 530
2 = 6x +8 = 800
6. x = 52, 1 = x – 3 = 490
2 = 2x – 7 = 970
7. FALSE
8. a. 21
b. B, C, A
9. a. P and PQR
b. 900
10. Isosceles 
MA = 1000
MA = 830
Activity 8: HINGE THEOREM or THE ALLIGATOR THEOREM
1. increases
2. longer/larger
3. wider
4. <
5. >
Note: Practice Personal Hygiene protocols at all times.
17
6. <
7. >
8. ship 2
References
Grade 8 Learner’s Module (Module 6), pp. 390-409. Fourth Year
Triangle Trigon
https://www.google.com/search?source=hp&ei=eG80X7X8PIqg0gTi7ILIAg&q=hinge+theorem
+or+alligator+theorem&oq=hinge+theorem+or+alligator+theorem&gs_lcp=CgZwc3ktYWIQAzo
OCAAQ6gIQtAIQmgEQ5QI6BQgAELEDOgIIADoICAAQsQMQgwE6BggAEBYQHjoFCCEQoAE6
Trianhttps://study.com/academy/lesson/comparing-triangles-with-the-hinge-theorem.html
Thttps://www.expii.com/t/sss-inequality-theorem-converse-of-hinge-theorem-1006odule 2 (L
Prepared by:
ARLON T. MACARUBBO
Teacher III, TCSHS
Note: Practice Personal Hygiene protocols at all times.
18
MATHEMATICS 8
Name: _____________________
Date: ______________________
Grade Level: ____
Score: _________
Learning Activity Sheet
TRIANGLE INEQUALITIES
Background Information for Learners
Most of us start our day with the sandwiches which are triangular. Our mothers make a
sandwich in triangular shape because it looks more appetizing and because of the triangular
forms, the sandwiches come in handy.
By using the Triangle Inequality Theorem, an engineer can find a reasonable range of
values for any unknown distance. This can be extremely beneficial when trying to find a rough
estimate of the amount of material needed to build a structure with undetermined lengths.
The concepts and skills you will apply from this activity is on the axiomatic
development of triangle inequalities. This will improve your attention to details, shape, and
your deductive thinking, enhance your reasoning skills, and polish your mathematical
reasoning.
In this activity sheet, you will discover more useful facts about angles and sides of a
triangle.
Learning Competency: The learner applies theorems on triangle inequalities.
(M8GE-IVb-1)
Note: Practice Personal Hygiene protocols at all times.
19
Hands-On Activity 1 WHEN CAN YOU SAY “ITS ENOUGH”
Segments that Form Triangles
Materials: plastic straws, barbecue sticks or hard broom sticks, scissors, ruler
Steps:
1. Cut out plastic straws, barbecue sticks or hard broom sticks with these lengths:
21 cm
18 cm
15 cm
12 cm
6 cm
2. Select three strips randomly. Check if you can use them to form a triangle. Record the
lengths of the strips and whether or not they make a triangle.
3. Repeat STEP 2 several times. Write your findings on a table. (Note: The first one in the
table is done for you.)
Measures of three strips
Do the strips form a triangle? (YES/NO)
21, 18, 15
YES
4. Make a conclusion on the relationship among the side lengths of any triangle.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
__________________
Note: Practice Personal Hygiene protocols at all times.
20
HANDS-ON ACTIVITY 2 “OPPOSE ME”
Which angle is largest?
Materials: ruler, protractor
Procedure:
1. Measure each angle in the two triangles.
2. Measure the three sides of each triangle.
Note: Practice Personal Hygiene protocols at all times.
21
3. Arrange the angles of each triangle from the largest to smallest and its sides from the
longest to shortest. In the triangle, which side is opposite the largest angle? Which side
is opposite the smallest angle?
COV Largest angle ______
Longest side ______
______
Shortest side ______
______
Smallest angle ______
VIR Largest angle
______
______
Smallest angle______
Longest side ______
______
Shortest side ______
4. Make a conclusion as to where the largest and smallest angles in a triangle are, in
relation to the longest and shortest sides.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
____________
Rubrics for Scoring
CRITERIA
Amount of
Work
Mathematical
Reasoning
Level 1
(0 pt.)
Learner
showed no
attempt to
do any of
the
problems
and no
answer was
given.
Learner
showed no
explanation
of the
concept.
Level 2
(1 pt.)
Learner
only
showed
answer.
Level 3
(2 pts.)
Learner
only
showed
answers
but only of
partial
work.
Level 4
(3 pts.)
Learner
completed
each step
and gave
partial
answer.
Learner
showed
explanation
with
illogical
reasoning.
Learner
showed
explanation
with gaps
in
reasoning.
Learner
shows
Note: Practice Personal Hygiene protocols at all times.
Level 5
RATING
(4 pts.)
Learner
completed
each step and
gave
complete
answers.
Learner
showed
explanation explanation
with
with
thorough
substantial
reasoning
reasoning and
insightful
justifications.
OVERALL
RATING
22
ACTIVITY 1
For items 1 – 5, refer to the figure below.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Name all the exterior angles: _________________________________________
Name all the interior angles: _________________________________________
What are the remote interior angles to 2? ______________________________
What is the adjacent interior angle to 1? _______________________________
If m4 = 85 and m3 = 117, find the following:
a. m1 = ___
c. m5 = ___ e. m7 = ___ g. m9= ___
b. m2 = ___
d. m6 = ___ f. m8 = ___ h. m10= ___
In JHS , JH = 7, HS = 11, and JS = 8. Name the angles of JHS from the smallest
to largest._________________________________________________
In SDO , SO = 9, DO = 7, and SD = 5. Name the angles of SDO from the largest to
smallest. _________________________________________________
In REY , mR = 29, mE = 92, and mY = 59. Name the sides of REY from the
shortest to longest. ______________________________________________
In NOL , mN = 18, mO = 58, and mL = 104. Name the sides of NOL from the
longest to shortest. ______________________________________________
Note: Practice Personal Hygiene protocols at all times.
23
10. List the numbered angles in the figure below from the smallest to the
largest.__________________________________________________________
ACTIVITY 2
A. Directions: Determine whether the following sets of numbers could represent the lengths of
the sides of a triangle. Show possible illustrations.
1. 8, 15, 9
2. 3, 5, 9
3. 5, 18, 11
4. 14, 6, 20
5. 20, 37, 39
Note: Practice Personal Hygiene protocols at all times.
24
B. Directions: Fill in the blanks with,, or =. Refer to the figure at the right.
6. m1 _____ m2 + m3
7. m3 _____ m1
8. m1 _____ m2
9. AH ______ AC
10. CH ______ AH
Reflection
Life is replete with inequalities. In terms of material wealth, opportunities, intelligence,
and even physical beauty, people are not equal. Yet, in the eyes of God, we are all equal. What
matters is what we do with the talents that he has given us. What talents do you have? In what
ways do you develop your talents and share them with others?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________
Note: Practice Personal Hygiene protocols at all times.
25
Answer Key
Hands-on Activity 1 WHEN CAN YOU SAY “IT’S ENOUGH
3.
Measures of three strips
Do the strips form a triangle?
21,18,15
YES
21,15,12
YES
21,12,6
NO
18,15,12
YES
18,12,6
NO
15,12,6
YES
15,6,21
NO
15,6,18
YES
21,18,6
YES
6,18,15
YES
(NOTE: order of measures of three strips may vary.)
4. The sum of the lengths of any two sides of a triangle is greater than the length of
the third side.
Hands-on Activity 2 OPPOSE ME
1. m1 = 72.9; m2 = 57.34; m3 = 49.36
m4 = 56.31; m5 = 43.15; m6 = 80.54
2. COV : CO = 6.71; OV = 7.62; CV = 6.08
VIR : VI = 6.08; IV = 7.21; RV = 5
3.
COV Largest angle C
V
Smallest angle O
VIR
Longest side OV
CO
Shortest side CV
Largest angle R
Longest side IV
V
IR
Smallest angle I
Shortest side RV
4. In a triangle, if one side is longer than the other side, the angle opposite the longer
side is the larger side.
In a triangle, if one angle is larger than the other angle, the side opposite the larger
angle is the longer side.
Activity 1
1. 1, 2, 3, 7, 8, 9, 10
2. 4, 5, 6
Note: Practice Personal Hygiene protocols at all times.
26
3. 5, 6
4. 5
5. a. 117
c. 63
b. 95
d. 32
6. S, H, J
7. D, S, O
8. EY, ER, RY
9. NO, LN, LO
10. 1, 3, 2
e. 63
f. 148
g. 148
h. 32
Activity 2
A. 1.YES; 8 + 15  9, 15 + 9  8, 8 + 9  15
2. NO; 3 + 5  9
3. NO; 5 + 11  18
4. NO; 14 + 6 = 20
5. YES; 20 + 37 39, 37 + 39 20, 20 + 39  37
B. 6. 
7. 
8. 
9. 
10. 
References
“Sandwich Cartoon Transparent & PNG Clipart Free Download.” YAWD, yawebdesign.com/explore/sandwich-cartoon-png/. Accessed 4 June 2020.
“Circus, Tent, Big Top, Show, Stripes, Carnival - Circus Cartoon Png, Transparent Png Vhv.” Vhv.Rs,
www.vhv.rs/viewpic/wimTRo_circus-tent-big-top-show-stripescarnival-circus/. Accessed 4 June 2020.
NIVERA, GLADYS. GRADE 8 MATHEMATICS PATTERNS AND PRACTICALITIES.
Antonio Arnaiz cor. Chino Roces Avenues, Makati City, SalesianaBOOKS by DON
BOSCO PRESS, 2014.
Prepared by:
RICHARD S. CABALZA
Teacher III
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
27
MATHEMATICS 8
Name of Learner: _________________________________
Section: _________________________________________
Grade Level: __________
Date: ________________
LEARNING ACTIVITY SHEET
Proving Inequalities in A triangle
Background Information for Learners:
This Learning Activity Sheet is a teacher-made instructional material designed for Individual
Self-Directed Learning, which aims to guide students in their study of Proving Inequalities in
a Triangle. It is a reinforcement if not a substitute to the Learner’s Material which is previously
used by students before the COVID-19 Pandemic. This is a simplified learning material in
mathematics which covers one of the identified Most Essential Learning Competency in Grade
8 Mathematics which is “Proving Inequalities in A triangle”.
This topic requires your background knowledge on axioms, proofs, conjectures, theorems, and
mathematical generalizations. It is
The following are the definitions of the words previously stated which are lifted from esources:
Axiom: a statement accepted as true as the basis for argument or inference
Proof: A rigorous mathematical argument which unequivocally demonstrates the truth of a
given proposition
Conjecture: A proposition which is consistent with known data, but has neither been verified
nor shown to be false. It is synonymous with hypothesis.
Theorem: A statement that can be demonstrated to be true by accepted mathematical operations
and arguments.
Generalization: Making mathematical conclusions based on definitions, axioms and proved
theorems.
A mastery of the Axioms of Equality, Properties of Equality, Theorems on Triangles
and Triangle Congruence, which are explained in the previous lessons will also give you a head
start in understanding the lesson presented in this LAS.
Since the lesson is on Proving Inequalities in a Triangle, we therefore review the
different properties of inequality presented in the Learner’s Material.
Properties of Inequality
• For all real numbers 𝑝 and 𝑞 where 𝑝 > 0, 𝑞 > 0
If 𝑝 > 𝑞 then 𝑞 < 𝑝
If 𝑝 < 𝑞 then 𝑞 > 𝑝
• For all real numbers 𝑝, 𝑞, 𝑟 and 𝑠, if 𝑝 > 𝑞 and 𝑟 ≥ 𝑠, then 𝑝 + 𝑟 > 𝑞 + 𝑠
• For all real numbers 𝑝, 𝑞, 𝑟 if 𝑝 > 𝑞 and 𝑟 > 0, then 𝑝𝑟 > 𝑞𝑟.
• For all real numbers 𝑝, 𝑞, 𝑟 if 𝑝 > 𝑞 and 𝑞 > 𝑟, then 𝑝 > 𝑟.
• For all real numbers 𝑝, 𝑞, 𝑟 if 𝑝 = 𝑞 + 𝑟 and 𝑟 > 0, then 𝑝 > 𝑞.
Note: Practice Personal Hygiene protocols at all times.
28
Property of Inequality Used in Geometry
P
Q
R
Q is between P and R
If ̅̅̅̅
𝑃𝑅 ≅ ̅̅̅̅
𝑃𝑄 + ̅̅̅̅
𝑄𝑅 then 𝑃𝑅 > 𝑃𝑄 and 𝑃𝑅 > 𝑄𝑅
P
1
2
Q
R
∠1and ∠𝟐 are adjacent angles
m∠PQR>m∠1 and 𝑚∠PQR>m∠2
Things to consider in writing proofs:
1. Illustrating the problem as it is being stated and described.
2. Labeling properly the drawn figure.
3. Writing down logically the steps. Normally what is being stated first are the given and the
final step is to write the statement and reason of what you need to prove.
How indirect proofs are written?
1. Negating the statement to be proven.
2. Reason out logically until the contradiction of a known fact is reached.
3. Assumptions written must be false, thus the statement to be proven must be true
The Two-Column Proof Way of Proving Theorems in Mathematics:
At this point, the Two-Column Proof Way of proving theorems will be utilized. A two-column
proof consists of a list of statements, and the reasons why those statements are true. The
statements are in the left column and the reasons are in the right column. The statements consist
of steps toward solving the problem.
Note: Practice Personal Hygiene protocols at all times.
29
In the previous lesson, several theorems have been verified though observations,
constructions and computations. To note some, the following Inequalities in One Triangle
and Two Triangles are being presented to you.
In the succeeding part of this LAS, you will be asked to show the proofs of these theorems
using the two-column proof. In proving theorems using the two-column proof, observation
skills, deductive reasoning and logical proving are needed. The need to determine the
appropriate statement and its corresponding reason is necessary. Knowing the right statement
and the corresponding reason will provide you a hint on the succeeding statements and reasons
that will eventually lead you to the proof of the theorem you need to prove in the very
beginning.
The proof of the first theorem is done for you.
Inequalities in One Triangle
Triangle Inequality Theorem 1 (𝑺𝒔 → 𝑨𝒂)
If one side of a triangle is longer than a second side, then the angle opposite the first side is
larger than the angle opposite the second side.
Triangle Inequality Theorem 2 (𝑨𝒂 → 𝑺𝒔)
If one angle of a triangle is larger than a second angle, then the side opposite the first
angle is longer than the side opposite the second angle.
Triangle Inequality Theorem 3 (𝑺𝟏 + 𝑺𝟐 > 𝑺𝟑 )
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either
remote interior angle.
Inequalities in Two Triangles
Hinge Theorem or SAS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the
included angle of the first triangle is greater than the included angle of the second, then the
third side of the first triangle is longer than the third side of the second.
Converse of the Hinge Theorem or SSS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third
side of the first triangle is longer than the third side of the second, Then the included angle of
the first triangle is larger than the included angle of the second.
Note: Practice Personal Hygiene protocols at all times.
30
Consider the succeeding example. In the table below you are given a sample of a two-column
proof. Study the example and consider the explanation why the statements and the
corresponding reasons are sequentially written in that logical manner.
Triangle Inequality Theorem 1 (𝑺𝒔 → 𝑨𝒂)
If one side of a triangle is longer than a second side, then the angle opposite the first side is
larger than the angle opposite the second side.
B
Given: 𝑨𝑪 > 𝑨𝑩, 𝑩𝑨 = 𝑨𝑿
Prove: 𝒎∠𝑩 > 𝒎∠𝑪
x
C
Statements
Reasons
1. 𝐴𝐶 > 𝐴𝐵
Given
2. 𝐵𝐴 = 𝐴𝑋
Given
3. ∆𝐴𝐵𝑋 is isosceles
Definition
Triangle
4. 𝑚∠𝐴𝑋𝐵 = 𝑚∠𝐴𝐵𝑋
A
Note that these statements are
given in the problem.
of
an
Isosceles
Base Angles of Isosceles
Triangles are Congruent
5. 𝑚∠𝐴𝑋𝐵 = 𝑚∠𝐶 +
𝑚∠𝑋𝐵𝐶
Exterior Angle Theorem
6.
𝑚∠𝐴𝐵𝑋 = 𝑚∠𝐶 +
𝑚∠𝑋𝐵𝐶
Substitution
7.𝑚∠𝐵 = 𝑚∠𝐴𝐵𝑋 +
𝑚∠𝑋𝐵𝐶
Angle Addition Postulate
8. 𝑚∠𝐵 = 𝑚∠𝐶 +
𝑚∠𝑋𝐵𝐶 + 𝑚∠𝑋𝐵𝐶
Substitution
9. 𝑚∠𝐵 > 𝑚∠𝐶
Definition of >
Isosceles triangles are
triangles with two equal sides
Always, the base angles of
isosceles triangles are
congruent.
The measure of an Exterior
angle is equal to the sum of the
measures of the remote interior
angles
Study steps 4 and 5 to come
up with this reason.
Measures of Adjacent Angles
can be added to form one
larger angle.
Consider steps 6 and 7 in this
part of the proof.
The addition of the two angles
in step 8 makes angle B
greater than angle C.
Note: Practice Personal Hygiene protocols at all times.
31
Learning Competency:
The learner proves inequalities in a triangle (M8GE-IVc-1)
Directions:
Study the remaining triangle inequality theorems. The activities that follow allow you
to think on the right statements and reasons that support the logical flow of proving a particular
inequality theorem. In some of the examples, hints will be provided for you.
Activity 1
Instructions: Apply the indirect way of proving theorems in proving Triangle Inequality
Theorem 2. Note: (The example presented here is directly lifted from the learner’s module.)
Triangle Inequality Theorem 2 (𝑨𝒂 → 𝑺𝒔)
If one angle of a triangle is larger than a second angle, then the side opposite the first
angle is longer than the side opposite the second angle.
L
Given:∆𝐿𝑀𝑁; ∠𝐿 > ∠𝑁
Prove: ̅̅̅̅̅
𝑀𝑁 > ̅̅̅̅
LM
M
N
Indirect Proof:
Assume: ̅̅̅̅̅
𝑀𝑁 ≯ ̅̅̅̅
LM
Statements
1. ̅̅̅̅̅
𝑀𝑁 = ̅̅̅̅̅
LM or ̅̅̅̅̅
𝑀𝑁 < ̅̅̅̅̅
LM
2. Considering 𝑀𝑁 ≅ LM: If 𝑀𝑁 ≅ LM then
Reasons
1. Assumption that ̅̅̅̅̅
𝑀𝑁 ≯ ̅̅̅̅̅
LM
2. Definition of
Consequently, what can you say about ∠𝐿 and
∠𝑁
of isosceles triangles are congruent
The assumption that ̅̅̅̅̅
𝑀𝑁 ≅ ̅̅̅̅̅
LM is
True
False
The conclusion that ∠𝐿 ≅ ∠𝑁
the given that ∠𝐿 > ∠𝑁.
̅̅̅̅̅:
3. Considering ̅̅̅̅̅
𝑀𝑁 < LM
̅̅̅̅̅
̅̅̅̅̅
If 𝑀𝑁 < LM then
3. Base angles of isosceles triangles are congruent
The assumption that ̅̅̅̅̅
𝑀𝑁 < ̅̅̅̅̅
LM is
True
False
4. Therefore, ̅̅̅̅̅
𝑀𝑁 > ̅̅̅̅̅
LM must be
True
False
Note: Practice Personal Hygiene protocols at all times.
The conclusion that ∠𝐿 < ∠𝑁 contradicts the given
that
4. The
that ̅̅̅̅̅
𝑀𝑁 ≯ ̅̅̅̅̅
LM contradicts the known fact
that∠𝐿 > ∠𝑁
32
Activity 2
Instructions: Use the two-column proof in proving the Triangle Inequality Theorem 3.
Complete the table by adding the missing statement or reason.
Triangle Inequality Theorem 3 (𝑺𝟏 + 𝑺𝟐 > 𝑺𝟑 )
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
T
5
V
4
2
1
6
3
S
E
̅̅̅̅ bisects ̅̅̅̅̅̅̅
Given: ∆𝑆𝐸𝑇,𝑉𝐸
∠𝑆𝐸𝑇
Statements
Reasons
̅̅̅̅ bisects ̅̅̅̅̅̅̅̅
1. ∆𝑆𝐸𝑇,𝑉𝐸
∠𝑆𝐸𝑇
1._______________________________
2. ____________________________
2. An angle bisector divides an angle into two
congruent parts
3. 𝑚∠4 = 𝑚∠1 + 𝑚∠3
3. ______________________________
4. 𝑚_______ > 𝑚∠3
4. Angle Addition Postulate
5. 𝑚∠4 > 𝑚∠6
5. _____________________________
6. 𝐸𝑇 > 𝑇𝑉
6.______________________________
Activity 3
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either
remote interior angle.
Instructions: Use the illustration below and the guide questions to provide a step-by-step twocolumn-proof solution to the Exterior Angle Inequality Theorem.
Given:∆𝐿𝑀𝑁 with exterior angle ∠𝐿𝑁𝑃
Prove: ∠𝐿𝑁𝑃 > ∠𝑀𝐿𝑁
L
Note: Practice Personal Hygiene protocols at all times.
33
P
N
M
To aid you in your proof Let us first Construct the following
R
L
Q
3
4
1 N
M
2
P
̅̅̅̅ ≅ ̅̅̅̅
1. Midpoint Q on ̅̅̅̅
𝐿𝑁 such that 𝐿𝑄
𝑁𝑄
̅̅̅̅̅ through Q such that 𝑀𝑄
̅̅̅̅̅ ≅ 𝑄𝑅
̅̅̅̅
2. 𝑀𝑅
Guide Questions:
1.
2.
3.
4.
5.
6.
7.
What is usually the first statement and reason in writing a two column proof for a theorem ?
What relationship exists between ∠3 and ∠4?
How are triangles ∆𝐿𝑄𝑀 and ∆𝑁𝑄𝑅 related? What postulate supports your answer?
How can CPCTC be applied in this part of the proof?
What angles to be added to form ∠𝐿𝑁𝑃? What postulate supports answer?
How can Inequality be applied in this part?
Hoc can Substitution Property of Inequality be applied in the final step of the proof?
Activity 4
Hinge Theorem or SAS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the
included angle of the first triangle is greater than the included angle of the second, then the
third side of the first triangle is longer than the third side of the second.
Instructions:
Instructions: Supply the missing reasons for the proof of the Hinge Theorem
A
Given: 𝐴𝐵 ≅ 𝐴𝐷
B
2
1
Prove: 𝐸𝐵 > 𝐸𝐷
E
D
Note: Practice Personal Hygiene protocols at all times.
34
Statements
Reasons
1. 𝐴𝐵 ≅ 𝐴𝐷
1.________________________________
2. 𝐴𝐸 ≅ 𝐴𝐸
2.________________________________
3. ∠𝐴 ≅ ∠1 + ∠2
3.________________________________
4. ∠𝐴 > ∠1
4.________________________________
5. 𝐸𝐵 > 𝐸𝐷
5.________________________________
Activity 5
Converse of the Hinge Theorem or SSS Inequality Theorem
If two sides of one triangle are congruent to two sides of another triangle, but the third
side of the first triangle is longer than the third side of the second, Then the included angle of
the first triangle is larger than the included angle of the second.
Instructions: Show the proof the of the Converse of the Hinge Theorem. Use any book as your
refererence or you may opt to search the net for possible solutions. Revise how it is being
presented in the book or the internet and place your solution in the table that follows. Include
also in your answer sheet your source/s.
Given:
Illustration:
Note: Practice Personal Hygiene protocols at all times.
35
Statements:
Reasons:
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Note: Add rows to the table if needed.
Note: Practice Personal Hygiene protocols at all times.
36
Reflection:
1. What particular part of the lesson is difficult for you? How did you address the difficulty
you have encountered in this lesson?
2. What are the things that you learned in this LAS? How can you apply the things you
learned in this LAS in the remaining lessons in your Subject ?
Note: Practice Personal Hygiene protocols at all times.
37
References:
1.
2.
3.
4.
5.
Mathematics Learners Material for Grade 8
https:// www.dummies.com/education/math/geometry/proofs
https://mathworld.wolfram.com
Merriam Webster Dictionary
https://www.onlinemathlearning.com/exterior-angle-theorem.html
Answer Key
Answer Key: Activity 1
Statements
1. ̅̅̅̅̅
𝑀𝑁 = ̅̅̅̅̅
LM or ̅̅̅̅̅
𝑀𝑁 < ̅̅̅̅̅
LM
2. Considering 𝑀𝑁 ≅ LM: If 𝑀𝑁 ≅ LM then
Reasons
1. Assumption that ̅̅̅̅̅
𝑀𝑁 ≯ ̅̅̅̅̅
LM
2. Definition of Isosceles Triangle
∆𝐿𝑀𝑁 is an isosceles triangle.
Consequently, ∠𝐿 = ∠𝑁
The assumption that ̅̅̅̅̅
𝑀𝑁 ≅ ̅̅̅̅̅
LM is
False
Base Angles of Isosceles Triangles are congruent.
The conclusion that ∠𝐿 ≅ ∠𝑁 contradicts
the given that ∠𝐿 > ∠𝑁.
3. Considering ̅̅̅̅̅
𝑀𝑁 < ̅̅̅̅̅
LM:
̅̅̅̅̅ then 𝑚∠𝐿 < 𝑚∠𝑁
If ̅̅̅̅̅
𝑀𝑁 < LM
Triangle Inequality Theorem 1
The assumption that ̅̅̅̅̅
𝑀𝑁 < ̅̅̅̅̅
LM is False.
The conclusion that 𝑚∠𝐿 < 𝑚∠𝑁 contradicts the
given that 𝑚∠𝐿 > 𝑚∠𝑁
4. The assumption that ̅̅̅̅̅
𝑀𝑁 ≯ ̅̅̅̅̅
LM contradicts the
known fact that 𝑚∠𝐿 > 𝑚∠𝑁
̅̅̅̅̅ must be True
4. Therefore, ̅̅̅̅̅
𝑀𝑁 > LM
Answer Key. Activity 2
1.
2.
3.
4.
5.
6.
Given
∠3 ≅ ∠6
Exterior Angle Theorem
𝑚∠4
Transitive Property
The longest side is opposite the largest angle
Answer Key: Activity 3
Statements
Reasons
̅̅̅̅̅ ≅ 𝑄𝑅
̅̅̅̅
1. ̅̅̅̅
𝐿𝑄 ≅ ̅̅̅̅
𝑁𝑄 ; 𝑀𝑄
7. ∠3 ≅ ∠4
2. Vertical Angles are congruent
8. ∆𝐿𝑄𝑀 ≅ ∆𝑁𝑄𝑅
3. SAS Triangle Congruent Postulate
9. ∠𝑀𝐿𝑁 ≅ ∠1
4.Corresponding Parts of Congruent Triangles are
Congruent
10. ∠𝐿𝑁𝑃 ≅ ∠1 + ∠2
5.Angle Addition Postulate
11. ∠𝐿𝑁𝑃 > ∠1
6.Property of Inequality
12. ∠𝐿𝑁𝑃 > ∠𝑀𝐿𝑁
7.Substitution Property of Equality
Note: Practice Personal Hygiene protocols at all times.
6. By Construction
38
Answer Key: Activity 4
Statements
Reasons
1. 𝐴𝐵 ≅ 𝐴𝐷
1. Given
2. 𝐴𝐸 ≅ 𝐴𝐸
2. Reflexive
3. ∠𝐴 ≅ ∠1 + ∠2
3. Angle Addition
4. ∠𝐴 > ∠1
4. Definition of Inequality
5. 𝐸𝐵 > 𝐸𝐷
5. The greater the angle of a triangle, the larger is
the opposite side
Answer Key: Activity 5
Solutions may vary depending on the illustration and the process shown by the students.
Prepared by:
MICHAEL M. ACUPAN
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
39
MATHEMATICS 8
Name: _____________________
Date: ______________________
Grade Level: ____
Score: _________
Learning Activity Sheet
PROPERTIES OF PARALLEL LINES CUT BY TRANSVERSAL
Background Information for Learners
Have you watched a ski competition in the television already? In ancient times,
Scandinavians used skis as a way to travel over snow. The modern sport of skiing began in
Norway in the nineteenth century and has been gaining popularity ever since. One of the keys
to the basic downhill movement, the schuss, is keeping the skis parallel.
There are many other applications of parallel lines cut by a transversal line such the
railroads of trains, tallying sheets, and among others. Hence, it is indispensable to add in your
skill set the ability of proving properties of parallel lines cut by a transversal line.
In this Learning Activity Sheet, you will be able to prove the properties of parallel lines
cut by a transversal. The skills that you will be acquiring from the activities prepared will
greatly improve your reasoning and proving skills. It is hoped that as you acquired such skills,
you will be able to utilize them in your day-to-day activities.
Learning Competency: The learner proves properties of parallel lines cut by a transversal.
(M8GE-IVd-1)
Note: Practice Personal Hygiene protocols at all times.
40
HANDS–ON ACTIVITY: “Corresponding Angle Measures”
Materials:
1. Notebook paper
2. Protractor
Steps/Procedures:
3. Straightedge/ruler
4. Colored Pencils
1. Use a pencil and straightedge to darken two lines on a piece of notebook paper. Use
your straightedge/ruler to draw transversal t.
t
2. Label each angle.
t
1 2
3 4
5 6
7 8
3. Use your protractor to measure each of the four pairs of corresponding angles.
4. Make a conjuncture about the corresponding angles formed by two parallel lines cut by
a transversal.
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
5. What appears to be true about alternate interior angles? Consecutive interior angles?
Alternate exterior angles?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Note: Practice Personal Hygiene protocols at all times.
41
Rubrics for Scoring
CRITERIA
Level 1
(0 pt.)
Amount of
Learner
Work
showed no
attempt to
do any of
the
problems
and no
answer was
given.
Mathemati
Learner
cal
showed no
Reasoning explanation
of the
concept.
Level 2
(1 pt.)
Learner
only
showed
answer.
Level 3
(2 pts.)
Learner
only
showed
answers
but only of
partial
work.
Level 4
(3 pts.)
Learner
completed
each step
and gave
partial
answer.
Level 5
(4 pts.)
Learner
completed
each step and
gave
complete
answers.
Learner
Learner
Learner
showed
showed
shows
explanation explanation explanatio
with
with gaps
n with
illogical
in
substantial
reasoning. reasoning. reasoning
Learner
showed
explanation
with
thorough
reasoning
and
insightful
justifications.
OVERALL
RATING
RATING
ACTIVITY
̅̅̅̅ , 𝑚1 = 58, 𝑚2 = 47, and 𝑚3 = 26. Find the measure of each
A. In the figure, ̅̅̅̅
𝐴𝐵ǁ𝐸𝐶
of the following angles.
1.
2.
3.
4.
5.
6.
7
6
8
5
4
9
A
3
6
E
4
8 C
5
7
9
D
Note: Practice Personal Hygiene protocols at all times.
B
2
1
42
̅̅̅̅ , ̅̅̅̅
⃗⃗⃗⃗⃗ bisects EBA, 𝑚8 = 42, and 𝑚3 = 18. Find the
B. In the figure, ̅̅̅̅
𝐵𝐺 ǁ𝐶𝐸
𝐶𝐷 ǁ𝐵𝐺
measure of each of the following angles.
1.
2.
3.
4.
5.
6.
A
7
1
6
5
4
2
8
B
6
7
G
C
5
1
4
2
3
D
E
C. In the figure, 𝑙ǁ𝑚 and 𝑡 is a transversal line. Solve the following problems.
m
l
1 2
8 7
3 4
6 5
t
1. If 𝑚4 = 2x − 25 and 𝑚8 = x + 26, find 𝑚2.
2. If 𝑚6 = 2x + 43 and 𝑚7 = 5x + 11, find 𝑚5. Explain your reasoning.
Reflection
In the context of Plane Geometry, parallel lines never meet. In our lives, there are things
that we fail to meet – or achieve. We feel like we gave it all, but still, it isn’t enough. During
trying times when we do not meet our goals or aspirations in life, how do you cope up with the
challenges offered by the journey of your life? How do you turn failures into inspirations to
continue and battle with the challenges of this life?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Note: Practice Personal Hygiene protocols at all times.
43
Answer Key
Hands-on Activity
4. Make a conjuncture about the corresponding angles formed by two parallel lines
cut by a transversal.
If two parallel lines are cut by a transversal, then each pair of corresponding
angles is congruent.
5. What appears to be true about alternate interior angles? Consecutive interior angles?
Alternate exterior angles?
a. If two parallel lines are cut by a transversal, then each pair of alternate interior
angles is congruent.
b. If two parallel lines are cut by a transversal, then each pair of consecutive
interior angles is supplementary.
c. If two parallel lines are cut by a transversal, then each pair of alternate exterior
angles is congruent.
Activity
A.
1. 58𝑜
2. 75𝑜
3. 73𝑜
B.
1.
2.
3.
4.
5.
6.
42𝑜
42𝑜
96𝑜
42𝑜
42𝑜
120𝑜
Note: Practice Personal Hygiene protocols at all times.
4. 47𝑜
5. 107𝑜
6. 49𝑜
44
C.
1. 𝑚2 = 77
2. 𝑚5 = 101
References
Boyd, Cindy J. 2008. Geometry. New York: Glencoe/McGraw-Hill.
Jurgensen, Ray C, Richard G Brown, and John W Jurgensen. 2000. Geometry. Evanston, Ill.:
McDougal Littell.
"Redirect Notice". 2020. Google.Com. shorturl.at/yzO03
Prepared by:
JANSTEN B. MAPATAC
Teacher III
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
45
MATHEMATICS 8
Name of Learner: _________________________________
Section: _________________________________________
Grade Level: __________
Date: ___________
LEARNING ACTIVITY SHEETS
CONDITIONS UNDER
PARALLELISM and PERPENDICULARITY of LINES
Background Information for Learners
If two non-vertical lines in the same plane never intersect, then they are said to
be Parallel. If a transversal line intersects these two parallel lines, then many
congruent pair of angles are formed. While the two lines that intersect and form four
right angles are said to be Perpendicular. The property of perpendicularity is evident
between two lines which meet at a right angle.
Remember:
The pairs of angles which are formed when a transversal line intersects two
parallel lines are:
▪ Alternate Exterior Angles
▪ Alternate Interior Angles
▪ Corresponding Angles In
▪ Interior angles on the same side of the transversal s on the same side of
▪ Exterior angles on the same side of the transversal the transversal
▪ Interior angles on the same side of the transversal
▪ Interior angles on the same side of the transversal
Learning▪ Competency:
Determine the conditions under which lines and segments are parallel or perpendicular.
M8GE-IVe-1
Think about this:
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46
For aesthetic and attraction purposes, a footbridge was built with an inclination towards
the exit point. The bridge as shown in the figure below goes up from 12 meters and 15 meters
posts respectively, from the ground and 20 meters apart. What are the measures of the interior
angles of the bridge?
B
Z
R
I
20 m
15 m
12 m
WORD TREASURE
ACTIVITY 01
Directions:
Search for the mathematical terms in the word puzzle and define each of the terms. Write the
word and its definition in your notebook
L
A
S
R
E
V
S
N
A
R
T
R
R
N
P
P
E
R
P
E
N
D
I
C
U
L
A
R
Y
A
S
L
A
N
C
Y
D
R
F
E
N
J
C
L
R
M
I
N
T
O
R
G
Z
O
G
D
O
T
T
A
E
N
A
M
R
A
B
P
L
I
M
E
N
O
L
R
E
L
G
R
T
T
E
S
P
R
E
S
E
L
O
A
P
L
E
N
S
O
L
N
C
E
W
I
E
E
R
O
J
S
E
K
E
A
A
O
N
T
K
L
Note: Practice Personal Hygiene protocols at all times.
H
L
C
E
P
M
M
T
J
Z
N
V
S
X
B
T
U
N
Z
O
E
E
D
G
C
G
H
N
Q
E
R
I
E
C
N
L
A
C
I
T
R
E
V
S
G
L
I
N
T
D
P
N
W
G
D
U
Q
W
S
K
H
O
A
F
I
P
R
O
I
R
E
T
N
I
V
C
R
L
P
N
U
J
A
I
D
N
S
O
Q
B
Y
C
P
A
G
S
P
O
I
N
T
S
K
D
I
47
CONNECTING PARTS
ACTIVITY 02
Directions: Supply the blanks with appropriate word/s to answer each item.
fig. 1
fig. 2
fig. 3
p
1 2
43
a
l
i
5 6
87
m
g
1 2
6
3
5 4
t
1 2
4 3
y
j
5 6
8 7
GUIDE QUESTIONS:
1. What is a transversal line?
▪ A transversal line is a line that _____________ two or more lines at _____ or
more distinct points.
2. What is the transversal line in:
fig. 1 _____
fig. 2 _____
3. Enumerate the following in fig. 1 and fig. 3
fig. 1
a. Alternate Exterior Angle
__________
__________
fig. 3 _____
fig. 3
a. Alternate Exterior Angle
_________
_________
b. Alternate Interior Angle
__________
__________
b. Alternate Interior Angle
_________
_________
c. Corresponding Angles
__________
__________
__________
__________
c. Corresponding Angles
_________
_________
_________
_________
d. Interior angles on the same
side of the transversal
__________
__________
d. Interior angles on the same
side of the transversal
_________
_________
e. Exterior angles on the same
e. Exterior angles on the same
side of the transversal
side of the transversal
__________
_________
__________
_________
4. How many alternate interior angle does fig. 2 have? Corresponding angle?
5. Compare lines p and a in fig. 1with the lines j and y in fig. 2.
Note: Practice Personal Hygiene protocols at all times.
48
6. In a short bond paper:
▪ Draw two parallel lines and a transversal line similar to fig. 3.
▪ Number the angles similar to fig. 3.
▪ Measure the angles 1 to 8.
▪ Record/write the measures of each pair of angle in your notebook.
Alternate Exterior Angles
1.  ___ =
and  ___ =
2.  ___ =
and  ___ =
a. Conclusion: The measures of alternate exterior angles are___________________.
Alternate Interior Angles
1.  ___ =
and  ___ =
2.  ___ =
and  ___ =
b. Conclusion: The measures of alternate interior angles are___________________.
Corresponding Angles
1.  ___ =
and  ___ =
2.  ___ =
and  ___ =
3.  ___ =
and  ___ =
4.  ___ =
and  ___ =
c. Conclusion: The measures of corresponding angles are___________________.
Interior angles on the same side of the transversal
1.  ___ =
and  ___ =
2.  ___ =
and  ___ =
d. Conclusion: The interior angles on the same side of the transversal
are___________________.
Exterior angles on the same side of the transversal
1.  ___ =
and  ___ =
2.  ___ =
and  ___ =
e. Conclusion: The exterior angles on the same side of the transversal
are___________________.
Let’s try
Given: Lines g and a are parallel lines.
1. Determine the measures of the remaining angles given that 2 = 320.
1 320
4 3
8
5 6
7
g
a
Note: Practice Personal Hygiene protocols at all times.
49
1. 2 and 6 are two corresponding angles, so 6 = 320
2. 6 and 4 are two alternate interior angles, so 4 = 320
3. 4 and 8 are two corresponding angles, so 8 = 320
4. 8 and 1 are exterior angles on the same side of the transversal, so 1 = 1480
5. 1 and 5 are two corresponding angles, so 5 = 1480
6. 5 and 3 are two alternate interior angles, so 3 = 1480
7. 3 and 7 are two corresponding angles, so 7 = 1480
8. Refer to the figure below. Given that line p is parallel to line o,
A. Find the value of x.
B. Determine the measures of the numbered angles.
C. State the conditions of parallelism and perpendicularity of two lines
1 2
4 2x + 25
x + 75 6
8 7
p
o
Solution:
3  5, alternate interior angles are congruent
2x + 25 = x + 75
x = 50
3 = 2(50) + 25 = 125
5 = (50) + 75 = 125
The remaining angles are:
1 = 1250
2 = 550
4 = 550
6 = 550
7 = 1250
The answer to the problem in the “Think about this” portion.
Refer to figure and the given above.
Solution:
B = 2x +18;
Z = 4x
B + Z = 180
2x + 18 + 4x = 180
6x = 180 – 18
6x = 162
x = 27
B = 2x +18 = 720
Z = 4x = 1080
Note: Practice Personal Hygiene protocols at all times.
8 = 550
50
YOU ARE RIGHT !
ACTIVITY 03
Directions: Spot the difference in the given figures below and answer the questions that
follow.
fig. 1
fig. 3
fig. 2
i
r
d
p
n
e
GUIDE QUESTIONS. Refer to the figure above
1. What is common among the 3 figures?
▪ All are ____________________ lines.
2. Which is a different figure among the 3? Why?
▪ The two intersecting lines in figure ____ form _______ angles.
3. Which of the figures show perpendicularity? _________
4. When are two lines perpendicular?
▪ Two intersecting lines are ______________if they form four right angles.
5. TRUE or FALSE: All perpendicular lines are intersecting lines._____________
KEY POINTS:
Conditions that guarantee that two lines are Parallel
1. If 2 coplanar lines are both perpendicular to the same line, then they are parallel.
2. CAP Theorem. If 2 lines have a transversal and a pair of congruent
Corresponding Angles, then the lines are Parallel.
3.AIP Theorem. If 2 lines have a transversal and a pair of congruent Alternate
Interior Angles, then the lines are Parallel.
4. AEP Theorem. If 2 lines have a transversal and a pair of congruent Alternate
Exterior angles, then the lines are Parallel.
5. If 2 lines have a transversal and interior angles on the same side of the
transversal are supplementary, then the lines are parallel.
Conditions that guarantee that two lines are Perpendicular
1. If two lines are perpendicular to each other, then they form four right angles.
2. If the angles in a linear pair are congruent, then the lines containing their sides
are perpendicular.
Note: Practice Personal Hygiene protocols at all times.
51
ACTIVITY 04
FIGURE IT OUT
Solve for x and determine the measure of the missing angle/s. (For items 1-6)
1. Given: 1 = 3x – 10
g
1 2
5 = 2x + 45
4 3
a. Find x
b. 1 =
5 6
a
c. 2 =
8 7
d. 3 =
e. 4 =
f. 5 =
g. 6 =
h. 7 =
i. 8 =
2. Two rods are mounted on the upper and lower part of the wall for home décor. A
connecting rod is placed between the two rods as shown in the figure below. If the rod
connector makes a (x – 30)0 angle with the upper rod and (2x)0 with the lower rod,
what angle measure does the rod make with the lower rod?
(x – 30)0
(2x)0
For items 3-5, refer to the figure below
Given: a //c
a
300 0
70
2
1
3. What is the measure of 1?
4. What is the measure of 3?
5. What is the measure of 4?
Note: Practice Personal Hygiene protocols at all times.
b
3
4
c
52
RUBRIC
CRITERIA
Accuracy
Mathematical
Justification
OUTSTANDING
4
The computations
are accurate and
show a wise use of
geometric
concepts
specifically on
Parallelism and
Perpendicularity of
lines.
Justification is
logically clear,
convincing and
professionally
delivered. The
concepts on
Parallelism and
Perpendicularity of
lines.
SATISFACTORY
3
DEVELOPING
2
BEGINNING
1
The computations
are accurate and
show the use of
geometric concepts
specifically on
Parallelism and
Perpendicularity of
lines.
The computations
are erroneous and
show some use of
concepts on
Parallelism and
Perpendicularity
of lines.
The computations
are erroneous and
do not show the
use of concepts on
Parallelism and
Perpendicularity
of lines.
Justification is not
so clear. Some
ideas are not
connected to each
other. Not all
concepts on
Parallelism and
Perpendicularity
of lines.
Justification is
ambiguous. Only
few concepts on
Parallelism and
Perpendicularity
of lines.
Justification is clear
and convincingly
delivered.
Appropriate
concepts on
Parallelism and
Perpendicularity of
lines.
RATING
Reflection:
What have you learned about Parallel and Perpendicular lines?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
________________________
ANSWERS
Activity 1:WORD TREASURE
adjacent
alternate
angles
complementary
congruent
coplanar
corresponding
exterior
interior
linear
lines
pair
parallel
perpendicular
plane
points
Note: Practice Personal Hygiene protocols at all times.
53
supplementary
theorems
transversal
vertical
Activity 2: CONNECTING PARTS
GUIDE QUESTIONS
1. intersects, two
2. fig. 1. line i
fig. 2. none
fig. 3. line t
3.
Figure 1
Figure 3
a. Alternate Exterior Angles
a. Alternate Exterior Angles
1 and 7
1 and 7
2 and 8
2 and 8
b. Alternate Interior Angles
b. Alternate Interior Angles
3 and 5
3 and 5
4 and 6
4 and 6
c. Corresponding Angles
c. Corresponding Angles
1 and 5
1 and 5
2 and 6
2 and 6
3 and 7
3 and 7
4 and 8
4 and 8
d. Interior Angles on the Same Side of the
d. Interior Angles on the Same Side of the
Transversal
Transversal
3 and 6
3 and 6
4 and 5
4 and 5
e. Exterior Angles on the Same Side of the
e. Exterior Angles on the Same Side of the
Transversal
Transversal
1 and 8
1 and 8
2 and 7
2 and 7
4. none, none
5. Based on the markings, line j and line y are parallel lines while line p and line a are solely
intersecting lines.
6. a. congruent
b. congruent
c. congruent
d. supplementary
e. supplementary
Activity 3: YOU ARE RIGHT
GUIDE QUESTIONS
1. intersecting
2. fig. 3, right
3. fig. 3
4. perpendicular
5. TRUE
Activity 4: FIGURE IT OUT
1. a. x = 550
Note: Practice Personal Hygiene protocols at all times.
54
b. 1 =1550
c. 2 = 250
d. 3 = 1550
e. 4 = 250
f. 5 =1550
g. 6 = 250
h. 7 = 1550
i. 8 = 250
2. 1400
3. 300
4. 400
5. 1400
References
Grade 8 Learner’s Module (Module 6), pp. 441- 457. Fourth Year
Triangle Trigon
https://www.google.com/search?source=hp&ei=eG80X7X8PIqg0gTi7ILIAg&q=hinge+theorem
+or+alligator+theorem&oq=hinge+theorem+or+alligator+theorem&gs_lcp=CgZwc3ktYWIQAzo
OCAAQ6gIQtAIQmgEQ5QI6BQgAELEDOgIIADoICAAQsQMQgwE6BggAEBYQHjoFCCEQoAE6
Trianhttps://study.com/academy/lesson/comparing-triangles-with-the-hinge-theorem.html
Thttps://www.expii.com/t/sss-inequality-theorem-converse-of-hinge-theorem-1006
rigonometry, Mo, Module 2 (L
Prepared by:
ARLON T. MACARUBBO
Teacher III, TCSHS
Note: Practice Personal Hygiene protocols at all times.
55
MATHEMATICS 8
Name: _____________________
Date: ______________________
Grade Level: ____
Score: _________
Learning Activity Sheet
EXPERIMENTS AND SAMPLE SPACE
Background Information for Learners
In the Philippines game show “Wheel of Fortune”, a contestant spins the wheel to
determine his fortune. What is the probability that the contestant will be able to win the jackpot
prize? What is the probability that the contestant will be able to win Php100000?
The possible occurrence of an event can be characterized as impossible, very likely,
unlikely, 50% chance, likely, very likely or certain.
People deal with probability questions on a daily basis. In this activity sheet, you will
attempt to predict the chance that something will or not happen.
Learning Competency: The learner illustrates an experiment, outcome, sample space and
event. (M8GE-IVf-1)
ACTIVITY 1
Directions: Complete the puzzle by reading the clues below.
Note: Practice Personal Hygiene protocols at all times.
56
1
2
3
4
5
6
7
8
9
10
CLUES
Across
Down
3. A process in which an observation is 1. Each possible outcome in the sample space
obtained
2. Measures the chance that an event will
6. The set of all possible outcomes of an occur
experiment
4. The set of some outcomes of an
7. Event that will surely happen
experiment
8. One occurrence of an experiment
5. Probability of an event that will not happen
9. The observed result of an experiment
10. Event that cannot happen
ACTIVITY 2
A. Directions: An event is described in each row. Put a check on the column which best
describes the likelihood of each event.
Impossible
Event
1. A ball is drawn from a box
containing 2 blue balls,3
Note: Practice Personal Hygiene protocols at all times.
Very
Unlikely
Unlikely
50%
chance
Likely
Very
Likely
Certain
57
2.
3.
4.
5.
red and 2 yellow. The
person wants a green ball.
Getting a head when tossing
an unbiased coin.
The date after the 29th is the
30th in a month.
The event that it rains in
summer
The event that you pass all
your subjects if the remarks
of all your grades are
PASSED.
B. Directions: Determine the sample space in each of the following experiments. If the sample
points of the sample space are too many to enumerate, just give the number of sample
points.
NOTE: The set of all possible outcomes is the sample space of the experiment. The sample
space is usually denoted by S and the total number of possible outcomes by n(S).
Example: For each experiment, write the sample space and the total number of possible
outcomes.
a. Rolling a die
S = {1, 2, 3, 4, 5, 6}
n(S) = 6
b. Flipping a coin
S = {Head, Tail)
n(S) = 2
1. Tossing two coins simultaneously
4. Rolling a pair of dice
2. Tossing three coins simultaneously
5. Tossing a coin followed by rolling a die
3. Drawing a card from a standard deck of cards
C. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are written on slips of paper, placed in a box and
thoroughly mixed. One slip of paper is chosen at random. Find the event and number of
sample points if
1. The number drawn is between 1 and 9.
4. The number drawn is a multiple of 4.
2. The number drawn is odd and less than 9.
3. The number drawn is even and greater than 5.
Note: Practice Personal Hygiene protocols at all times.
5. The number drawn is prime.
58
Reflection
Does making it to the top seem like impossible to achieve?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________
Note: Practice Personal Hygiene protocols at all times.
59
Answer Key
Activity 1
S1
P2
A
R
E3
X
P
E
R
I
O
M
N
T
P
B
E4
L
A
V
E
B
S6
A
M
P
L
I
E
S
N
L
C7
E
R
T
P
Z5
A
C
O
A
I
T8
E
I
E
R
N
O
N
R
I
A
L
O9
U
Y
T
C
O
L
E
M
E
S
I10 M
P
O
S
S
I
B
Activity 2
A.
Event
Impossible
Very
Unlikely
1. A ball is drawn from a
box containing 2 blue
balls,3 red and 2 yellow.
The person wants a green
ball.
2. Getting a head when
tossing an unbiased coin.
3. The date after the 29th is
the 30th in a month.
Note: Practice Personal Hygiene protocols at all times.
Unlikely
50%
chance
Likely
Very
Likely
Certain
60
4. When in Tuguegarao,
the event that it rains in
summer
5. The event that you pass
all your subjects if the
remarks of all your grades
are PASSED.
B. 1. {HH, HT, TH, TT}
2. {HHH, HTH, HHT, THH, TTT, THT, TTH, HHT}
3. n(S) = 52
4. n(S) = 36
5. {H , T}
C. 1. {2, 3, 4, 5, 6, 7, 8}; n(S) = 7
2. {1, 3, 5, 7}; n(S) = 4
3. {6, 8}; n(S) = 2
4. {4, 8}; n(S) = 2
5. {1, 3, 5, 7}; n(S) = 4
References
Interactive, Zellion. “Win the Online Contest - Zellion Blog Smartness in Innovation and
Action.” Blog.Zellioninteractive.Com, blog.zellioninteractive.com/win-the-onlinecontest/. Accessed 4 June 2020.
NIVERA, GLADYS. GRADE 8 MATHEMATICS PATTERNS AND PRACTICALITIES.
Antonio Arnaiz cor. Chino Roces Avenues, Makati City, SalesianaBOOKS by
DON BOSCO PRESS, 2014.
CHUA, ARNALDO. 2018 MTAP-DepEd Saturday Program of Excellence in Mathematics 8
Session 7.
Prepared by:
RICHARD S. CABALZA
Teacher III
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
61
MATHEMATICS 8
Name of Learner: _________________________________
Section: _________________________________________
Grade Level: __________
Date: ________________
LEARNING ACTIVITY SHEET
Counting Number of Occurrences of an Outcome In An Experiment
Background Information for Learners:
This Learning Activity Sheet is a teacher-made instructional material designed for
Individual Self-Directed Learning, which aims to guide students in their study of Counting
Number of Occurrences of an outcome in an experiment using a) Tables b)Tree Diagram c)
Systematic Listing and d) Fundamental Counting Principle . It is a reinforcement if not a
substitute to the Learner’s Material which is previously used by students before the COVID19 Pandemic. This is a simplified learning material in mathematics which covers one of the
identified Most Essential Learning Competency in Grade 8 Mathematics as priorly stated.
This topic requires your background knowledge on the definition of experiment,
outcome, sample space and event.
The following are the definitions of the words previously stated which are lifted from
e-sources:
Experiment: Any procedure that can be infinitely repeated and has a well-defined set of
possible outcomes, known as the sample space.
Outcome: A possible result of a probability experiment.
Sample Space: Set of all possible outcomes or results of an experiment.
Event: A set of outcomes of an experiment ( a subset of the sample space).
It is important to know the total number of outcomes in a probability experiment.
Knowing how to properly count the outcomes would certainly guide you to solving probability
problems. In this LAS we will look into the different ways to count the number of outcomes in
an experiment.
Consider the problem below.
“Suppose at a particular restaurant you have three choices for an appetizer (soup, salad, or
brownies) and three choices for a main course (hamburger, sandwich, and spaghetti). If you are
allowed to choose exactly one item from each category for your meal, how many different meal
options do you have”?
Note: Practice Personal Hygiene protocols at all times.
62
Tables
This type of counting technique makes use of rows and columns and counting the number of
inner table cells for the total outcome. To illustrate further, you need to create three columns
for the appetizers and three rows for the main courses.
Hamburger
Sandwich
Spaghetti
Soup
Soup+Hamburger
Soup + Sandwich
Soup+Spaghetti
Salad
Salad + Hamburger
Salad + Sandwich
Salad + Spaghetti
Brownies
Brownies+Hamburger
Brownies+Sandwich
Brownies + Spaghetti
As seen in the table, the combinations of the meals are in the table cells shaded green. Since
we are only looking at the different meal options that we have based on the available appetizer
sand deserts then we can say that there are nine(9) possible options.
Tree Diagram
This method is called tree-diagram because the possible outcomes are counted based on stages
that are branched out. In this solution, we draw first three branches for the appetizers, then for
each appetizer, three branches will be drawn. Look at the illustration below.
Soup
Salad
Brownies
Hamburger
Soup+Hamburger
Sandwich
Soup+Sandwich
Spaghetti
Soup+Spaghetti
Hamburger
Salad+Hamburger
Sandwich
Salad+Sandwich
Spaghetti
Salad+Spaghetti
Hamburger
Brownies+Burger
Sandwich
Brownies+Sandwich
Spaghetti
Brownies+Spaghetti
Counting the branches at the final level, we can conclude that there are nine possible meal
options to choose from.
Note: Practice Personal Hygiene protocols at all times.
63
Systematic Listing
In this technique, meal options are systematically listed without missing any possibility or
listing a possibility more than once. The list, at the end, will be counted properly.
Soup + Hamburger, Soup + Sandwich, Soup + Spaghetti, Salad + Hamburger, Salad +
Sandwich, Salad + Spaghetti, Brownies + Hamburger, Brownies + Sandwich, Brownies +
Spaghetti
As you can see, nine meal options are listed.
Fundamental Counting Principle
The Fundamental Counting Principle (also called the counting rule) is a way to figure out the
number of outcomes in a probability problem. Basically, you multiply the events together to
get the total number of outcomes.
There are three(3) ways to get an appetizer and three(3) ways to get a main course. So applying
the Fundamental Counting Principle, we get 3 times 3, and that is 9.
Learning Competency:
The learner counts the number of occurrences of an outcome in an experiment; a) table
b) tree diagram c) systematic listing d) fundamental counting principle (M8GE-IVc-1)
Directions:
The activities that follow allow you to apply the different ways of counting the number
of occurrences of an outcome of an experiment.
Activity 1
Instructions: Use tables to count the number of outcomes in the following experiment:
1. Count outcomes of drawing two balls in succession and with replacement from a box
containing one red ball, one white ball and one green ball.
2. How many possible outcomes are there in tossing one coin and rolling one die.
Note: Practice Personal Hygiene protocols at all times.
64
Activity 2
Instructions: Use tree diagram in counting the outcomes of the following experiments
1. How many outcomes are there if you are asked to choose a combination of one t-shirt and
one pants in a closet with 4 t-shirts and 2 pairs of pants.
2. How many ways could a person go from Town A to Town B to town C if there are two
roads connecting towns A and B and four roads connecting towns B and C?
Activity 3
Instructions: Use systematic listing in counting the outcomes of the following experiments.
1. How many ways can we select a club president and a secretary if John, Rex ,Shea and Finn
are the people to choose from?
2. Determine the number of ways to mark your answer sheet in a 3-item true or false test.
Activity 4
Instructions: Use Fundamental Counting Principle in solving the problems that follow.
1. How many license-plates with 3 letters followed by 3 digits exist.?
2. How many numbers in the range 1000-9999 have no repeated digits
3. How many license-plates with three letters followed by 3 digits exist if exactly one of the
digits is 1
4. A quiz has 5 multiple-choice questions. Each question has 4 answer choices of which 1 is
correct answer and the other 3 are incorrect. How many ways are there to answer the five
questions?
Reflection:
1. What particular part of the lesson is difficult for you? How did you address the difficulty
you have encountered in this lesson?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
2. What are the things that you learned in this LAS? How can you apply the things you
learned in this LAS in the remaining lessons in your Subject ?
Note: Practice Personal Hygiene protocols at all times.
65
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
References:
1.
2.
3.
4.
5.
https://en.m.wikipedia.org
www.webquest.hawaii.edu
https://faculty.atu.edu
www.nr.edu
Learners’ Material in Mathematics 8
Answer Key
Answer Key: Activity 1: Tables
1.
R
W
G
R
RR
RW
RG
W
WR
WW
WG
G
GR
GW
GG
2.
H
T
1
H1
T1
2
H2
T2
3
H3
T3
4
H4
T4
5
H5
T5
Note: Practice Personal Hygiene protocols at all times.
6
H6
T6
66
Answer Key. Activity 2: Tree Diagram
1.
P1
T1
P2
P1
T2
P2
P5
T3
P6
P7
T4
P8
2.
A
Note: Practice Personal Hygiene protocols at all times.
B
C
67
Answer Key: Activity 3:Systematic Listing
1. J and R, J and S, J and F
R and J, R and S, R and F
S and J, S and R, S and F
F and J, F and R, F and S
2. TTT, TTF, TFF, TFT, FTT, FFF, FFT, FTF
Answer Key: Activity 4: Fundamental Counting Principle
1.
2.
3.
4.
26.26.26.10.10.10 = 17, 576, 000 ways
9.9.8.7 = 4,536 ways
26.26.26.3.9.9 = 4,270,968 ways
4.4.4.4.4 = 1024 ways
Prepared by:
MICHAEL M. ACUPAN
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
68
MATHEMATICS 8
Name: _____________________
Date: ______________________
Grade Level: ____
Score: _________
Learning Activity Sheet
PROBABILITY OF A SIMPLE EVENT
Background Information for Learners
Probability has something to do with a chance. We use it most of time, usually
unconsciously. We don’t perform actual probability problems in our daily life but use
subjective probability to determine the course of our actions or any judgment. Almost
everything around us speaks of probability.
Probability is a mathematical term for the likelihood of any different combination of
outcomes. An application of simple probability is the flipping of coin or throwing a dice. One
has to understand that there is ½ chance of winning in flipping a coin and 1⁄6 chance of
winning in the throwing of dice.
Before planning for an outing or a picnic, we always check the weather forecast on the
television or over the radio. Supposed there is a 20% chance that rain may occur on that day,
then we may be able to alter our decision of whether going or not. In here, meteorologists
utilize a specific tool and technique to predict the weather forecast based from the historical
database of the days which have similar characteristics of temperature, humidity, and pressure.
There are plenty of real-life applications of the probability of a simple event. Hence, it
is vital that you acquire the competency prepared for you in this learning activity sheet.
Learning Competency: The learner finds the probability of a simple event. (M8GE-IVi-1)
Note: Practice Personal Hygiene protocols at all times.
69
HANDS–ON ACTIVITY: “Coin or Die?”
Materials:
1. 5 – peso coin
2. Die
3. Paper
4. Pen
Note: Practice Personal Hygiene protocols at all times.
70
Steps/Procedures:
1. Toss the coin twice. Record the results in this table.
Head (H)
Tail (T)
Result
2. Roll a die six times. Record the results in this table.
Dots
1
2
3
4
Result
5
6
3. Make a conjuncture about the probability of tossing a coin and rolling a die. What is
the probability of getting a tail (or head)? What is the probability of obtaining 1(or 2,
3, 4, 5, 6)?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
4. If simple probability is the possibility that a head is likely to happen in tossing a coin,
or 1 is likely to come out from rolling a die, how do you define simple probability now?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
Note: Practice Personal Hygiene protocols at all times.
71
ACTIVITY 1: Who Wants to be a Millionaire?
Directions: Below are cases where lie five (5) different questions. You can only claim the
million case if you can answer the fifth question successfully. You can only
proceed to the next question after the first box and so on. Start your journey of
being a millionaire now!
1. A class has 11 females and 21 males. What is the
probability of choosing a female as the president
of the class?
2. The number 1 to 10 are written on separate pieces
of paper, folded and put in a box. What is the
probability that a random chosen number is
even?
3. A bag has 3 green, 2 red, 5 purple, 10 white and
5 black marbles. What is the probability of
choosing any colored marble?
4. Find the probability of getting a numbered card
when drawn from a standard deck of 52 cards.
10, 000
50, 000
100, 000
500, 000
5. Two coins are tossed, find the probability that
two heads are obtained.
1, 000, 000
Note: Practice Personal Hygiene protocols at all times.
72
ACTIVITY 2: Wheel of Fortune
Directions: Compute the probability of each event using the wheel of fortune.
1.
2.
3.
4.
5.
What is the probability of obtaining 300?
What is the chance that one will have a free spin?
If spin, what is the probability of landing a thousand?
What is the probability that a player would get a bankrupt?
What is the probability that a player would win 200?
Reflection
Charles Dickens on David Copperfield said that “The most important thing in life is to
stop saying ‘I wish’ and start saying ‘I will’. Consider nothing impossible, then treat
possibilities as probabilities”. In your life as a student, what seemed to be impossible to do and
yet, you were able to accomplish them?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Note: Practice Personal Hygiene protocols at all times.
73
Answer Key
Hands-on Activity
A simple event is an event where all possible outcomes are equally likely to occur.
So the probability of simple events will have all possible outcomes equally likely to happen
or occur.
Activity 1
1. 0.34375 or 34.375% or 11/32
2. 0.5 or 50% or ½
3. 1or 100%
4. 9/13 or 0.6923 or 69.23%
5. ¼ or 0.25 or 25%
Activity 2
1. 1/12 or 0.083 or 8.33%
2. 1/24 or 0.0417 or 4.17%
3. 1/24 or 0.0417 or 4.17%
4. 1/24 or 0.0417 or 4.17%
5. 1/24 or 0.0417 or 4.17%
References
"Cartoon Flipping Coin Stock Illustrations – 45 Cartoon Flipping Coin Stock Illustrations,
Vectors &
Clipart - Dreamstime". 2020. Cartoon Flipping Coin Stock Illustrations – 45 Cartoon
Flipping Coin Stock Illustrations, Vectors & Clipart - Dreamstime.
https://www.dreamstime.com/illustration/cartoon-flipping-coin.html.
"Pin On Funny". 2020. Pinterest. https://www.pinterest.ph/pin/492159065525331805/.
"Probability | Statistics And Probability | Math | Khan Academy". 2020. Khan Academy.
https://www.khanacademy.org/math/statistics-probability/probability-library.
"Recruitment Meets Wheel Of Fortune: G Adventure’S Success Story". 2020. Hcamag.Com.
https://www.hcamag.com/us/news/general/recruitment-meets-wheel-of-fortune-gadventures-success-story/156459.
Prepared by:
JANSTEN B. MAPATAC
Teacher III
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
74
MATHEMATICS 8
Name: _____________________
Date: ______________________
Grade Level: ____
Score: _________
Learning Activity Sheet
THEORETICAL AND EXPERIMENTAL PROBABILITIES
Background Information for Learners
Experimental Probability is the ratio of the number of favorable outcomes and the total
number of possible outcomes obtained in an actual experiment. The probability of an event
may vary from one experiment to the next.
Theoretical probability is the ratio of the number of favorable outcomes and the total
number of possible outcomes. It assumes ideal conditions and is determined through the same
space.
You use probability in daily life to make decisions when you do not know for sure what
the outcome will be. Most of the time, you will not perform actual probability problems, but
you'll use subjective probability to make judgment calls and determine the best course of action.
So far, you have been obtaining theoretical probabilities of events. This activity sheet
will show that the probability of an event in an actual experiment often differs from its
theoretical probability.
Learning Competency: The learner illustrates an experimental probability and a
theoretical probability. (M8GE-IVi-1)
Note: Practice Personal Hygiene protocols at all times.
75
HANDS–ON ACTIVITY “ROLL ME”
Material: Die
Procedures:
a. Roll a die. What is the probability of getting a 3?
b. Do this activity.
Roll a die 20 times. Record the number of times each number appears.
____ time(s)
_____ time(s)
_____time(s)
______ time(s)
_____ time(s)
_____ time(s)
c. Write the experimental probabilities of each event.
____
_____
_____
____
_____
_____
d. Compare the theoretical probability of the event of getting a 5 to its experimental
probability. Are they equal?
e. Roll a die 40 times. Record the number of times each number appears.
____ time(s)
_____ time(s)
_____ time(s)
____ time(s)
_____ time(s)
_____ time(s)
f. Write the experimental probabilities of each event.
____
_____
_____
______
_____
_____
Note: Practice Personal Hygiene protocols at all times.
76
g. Are the experimental probabilities closer to the theoretical probabilities? If you do
the experiment 100 times, do you expect the experimental probabilities to get even
closer to the theoretical probabilities? Why or why not?
_____________________________________________________________________
_____________________________________________________________________
_____________________________________________________________________
_________
Rubrics for Scoring
CRITERIA
Amount of
Work
Mathematical
Reasoning
Level 1
(0 pt.)
Learner
showed no
attempt to
do any of
the
problems
and no
answer
was given.
Learner
showed no
Level 2
(1 pt.)
Learner
only
showed
answer.
Level 3
(2 pts.)
Learner
only
showed
answers
but only
of partial
work.
Learner
showed
Learner
showed
Level 4
(3 pts.)
Learner
completed
each step
and gave
partial
answer.
Learner
showed
explanation explanation explanation explanation
of the
with
with gaps with
concept.
illogical
in
substantial
reasoning. reasoning. reasoning.
Level 5
(4 pts.)
Learner
completed
each step
and gave
complete
answers.
Learner
showed
explanation
with
thorough
reasoning
and
insightful
justifications.
OVERALL
RATING
Note: Practice Personal Hygiene protocols at all times.
RATING
77
ACTIVITY 1
Directions: If the statement is true, write CORONA and if the statement is false, write
VIRUS.
_________ 1. The experimental probability of an event is the same as its theoretical
probability.
_________ 2. The theoretical probability of getting a head when a coin is flipped is ½.
_________ 3. The experimental probability of an event may vary with each experiment.
_________ 4. The theoretical probability of an event is constant.
_________ 5. As an experiment is repeated more number of times, its experimental
probability gets closer to its theoretical probability.
ACTIVITY 2
Directions: Draw a tree diagram and list the possible outcomes for each event.
1. Three coins are tossed.
2. Two dice are rolled.
3. A die is rolled and then a coin is tossed.
Note: Practice Personal Hygiene protocols at all times.
78
4. Suppose that a family has three children. Find all the possible outcomes for the
genders of the children.
Reflection
What statements can you make about yourself that are certain?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
_____________________
Note: Practice Personal Hygiene protocols at all times.
79
Answer Key
Hands-on Activity “Roll Me”
a.
b.
c.
d.
e.
f.
g.
Activity 1
1.
2.
3.
4.
5.
1
6
Answers vary
Answers depend on the result of procedure b.
Answer can be YES or NO
Answers vary
Answers depend on the result of procedure e.
YES. Yes, because the greater number of trials, experimental probabilities get
even closer to the theoretical probabilities.
VIRUS
CORONA
CORONA
CORONA
CORONA
Activity 2
HEAD
HEAD
TAIL
HEAD
1. HEAD
TAIL
TAIL
HEAD
HEAD
TAIL
HEAD
TAIL
TAIL
TAIL
{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Note: Practice Personal Hygiene protocols at all times.
80
1
1
2
2
2.
3
1
4
3
4
4
5
5
6
6
1
1
2
2
3
2
4
3
5
4
5
5
6
6
1
1
2
3
3
4
2
3
6
4
5
6
5
6
(1,1), (1,2 ), (1,3), (1,4 ), (1,5), (1,6 ) 
(2,1), (2,2 ), (2,3), (2,4 ), (2,5), (2,6 )


(3,1), (3,2), (3,3), (3,4 ), (3,5), (3,6 ) 


(4,1), (4,2 ), (4,3), (4,4 ), (4,5), (4,6 )
(5,1), (5,2), (5,3), (5,4 ), (5,5), (5,6 ) 


(6,6), (6,2 ), (6,3), (6,4 ), (6,5), (6,6 )
Note: Practice Personal Hygiene protocols at all times.
81
3.
HEAD
1
HEAD
4
TAIL
TAIL
HEAD
2
HEAD
5
TAIL
TAIL
HEAD
HEAD
3
6
TAIL
TAIL
(1, HEAD), (1, TAIL ), (2, HEAD), (2, TAIL ), (3, HEAD), (3, TAIL ) 


(4, HEAD), (4, TAIL ), (5, HEAD), (5, TAIL ), (6, HEAD ), (6, TAIL )
Note: Practice Personal Hygiene protocols at all times.
82
BOY
BOY
GIRL
BOY
4. BOY
GIRL
GIRL
BOY
BOY
GIRL
BOY
GIRL
GIRL
GIRL
{BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG}
References
“Playing Cards Suits Hand Diamond PNG - Picpng.” n.d. - Picpng - Search All the Free PNGs
and
Make
Your
Own
Transparent
PNG
Images.
https://www.picpng.com/image/playing-cards-suits-hand-diamond-png-77483.
“Man Rolling The Dice/Risk Royalty Free Vector Clip - Cartoon Man Rolling Dice - Free
Transparent
PNG
Download
PNGkey.”
n.d.
PNGkey.Com.
https://www.pngkey.com/detail/u2e6t4y3t4u2r5a9_man-rolling-the-dice-riskroyalty-free-vector/.
NIVERA, GLADYS. GRADE 8 MATHEMATICS PATTERNS AND PRACTICALITIES.
Antonio Arnaiz cor. Chino Roces Avenues, Makati City, SalesianaBOOKS by DON
BOSCO PRESS, 2014.
Prepared by:
RICHARD S. CABALZA
Teacher III
Tuguegarao City Science High School
Note: Practice Personal Hygiene protocols at all times.
83
MATHEMATICS 8
Name of Learner: _________________________________
Section: _________________________________________
Grade Level: __________
Date: ___________
LEARNING ACTIVITY SHEETS
PROBLEMS INVOLVING SIMPLE EVENTS in PROBABILITIES
Background Information for Learners
Probability is the measure of how likely an event to happen. A number is used
to represent the likelihood of an event happening. This number is called the
probability of the event.
The probability of an event is finding the probability of a single event
occurring. When finding the probability of an event occurring, number of favorable
outcomes over the number of total outcomes.
Remember:
Let S denotes the sample space (total number of possible outcomes) or n(S)
and A an event of possible/favourable outcome or n(A). Since all outcomes are
equally likely to occur, then the probability of an event A is:
number of favourable outcomes to A n(A)
total number of possible outcomes n(S)
Probability of A or P(A)
P( A) =
Learning Competency:
Solves problems involving probabilities of simple events. M8GE-IVi-j-1
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84
Yes!, I will
play under
the rain
again
It will probably
rain today
What is the probability
that it will rain today?
Reused picture
ACTIVITY 01
KNOW IT SO WELL
Direction: Read the following events below. In your notebook, write CERTAIN if you think
the event will surely happen and UNCERTAIN if you are not sure that the event will happen.
1. The sun rises in the east.
2. You will lose weight tomorrow.
3. You will be spared with COVID virus.
4. You will win the lotto draw this month.
5. Manny Pacquiao will be next Philippine president.
6. A typhoon will occur anytime of the year.
7. Autumn happens only once a year in the Philippines.
8. The pandemic is detrimental to the national economy.
9. LRT and MRT operations ease the traffic in Metro Manila.
10. The Philippines will be the crowned FIBA champions this year.
ACTIVITY 02
FIGURE IT OUT
Directions: Answer the problems in your notebook
1. In a school’s basketball tournament there are 150 spectators, 48 of which are the school
officials, 30 visitors and the rest are students. If all the spectators are equally likely to
be seated at the front row, what is the probability of:
Note: Practice Personal Hygiene protocols at all times.
85
a. visitor seated at the front row.
b. student seated at the front row.
c. school official occupying the second row if either a guest or a student stays at the
front row.
d. officiating official.
2. What is the probability of the three Balik-Probinsya recipients negative from COVID – 19
viruses after the swab test has been conducted?
3. If the spinner is rotated, what is the probability of:
a. P(5) =
b. P(even number less than 4) =
c. P(odd numbers) =
d. P(less than 5) =
e. P(4 or 8) =
KEY POINTS:
Probability of Simple Events
▪
A simple event is an event where all possible outcomes are equally likely to
occur.
▪
It is the ratio of the number of ways an event can occur to the number of
possible outcomes.
▪
Probabilities expressed in fraction form will have values between zero and one.
▪
One indicates that an event will definitely occur, while zero indicates that an
event will not occur. Likewise, probabilities expressed as percentages
possess values between zero and one hundred percent where
probabilities closer to zero are unlikely to occur and those close to one
hundred percent are more likely to occur.
ACTIVITY 03
SIMPLY YOURS
1. In a computer cafe’ there are 100 customers, 52 of which are college students, 38 are high
school students and the rest are teachers. If every customer is equally likely to leave
then, what is the probability of:
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86
a. teacher leaving ahead.
b. high school student leaving ahead.
c. college student leaving next after either a high school student or a teacher left
ahead.
2. What is the probability of the five Balik-Probinsya recipients POSITIVE from
COVID – 19 viruses after the swab test has been conducted?
3. If the spinner is rotated, what is the probability of:
a. P(M) =
b. P(P or N) =
c. P(vowel) =
d. P(consonant) =
e. P(not I) =
f. P(not P or E) =
4. There are 6 blue marbles, 3 red marbles, 2 green marble, and 1 black marbles in a
bag. Suppose you select one marble at random. Find the probability of:
a. P(blue)
b. P(black)
c. P(not green)
d. P(blue or black)
e. P(not orange)
5. On a standard deck of fifty-two cards, find the probability of:
a. P(spade)
b. P(nonspade)
c. P(red Ace)
d. P(King)
6. When a dice is thrown, what is the probability of:
a. P(4)
b. P(multiple of 2)
c. P(Less than 7)
d. P(Greater than 8)
7. What is the probability that student 5 has LRN that ends with 7?
8. What is the probability that a student has a birthday in February assuming that it is a
leap year?
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87
9. What is the probability that a student has a birthday in March assuming that it is not a
leap year?
10. A party for school government leaders is composed of 8 male and 7 female members.
What is the probability of choosing a male as the president of this party?
Comment: Lessen the activities or examples, since the topic is intended only for
RUBRIC for SCORING
CRITERIA
Accuracy
Mathematical
Justification
OUTSTANDING
4
The computations
are accurate and
show a wise use of
concepts on
Probability
SATISFACTORY
3
The computations
are accurate and
show the use of
concepts on
Probability
DEVELOPING
2
The computations
are erroneous and
show some use of
concepts on
Probability
BEGINNING
1
The computations
are erroneous and
do not show the
use of concepts on
Probability
Justification is
logically clear,
convincing and
professionally
delivered the
concepts on
Probability
Justification is clear
and convincingly
delivered.
Appropriate
concepts on
Probability
Justification is not
so clear. Some
ideas are not
connected to each
other. Not all
concepts on
Probability
Justification is
ambiguous. Only
few concepts on
Probability
RATING
Reflection:
What have you learned about Problems Involving Simple Events in Probability?
______________________________________________________________________
______________________________________________________________________
______________________________________________________________________
________________________
ANSWERS
Activity 1: KNOW IT SO WELL
Know it well
1. Certain
2. Uncertain
3. Uncertain
4. Uncertain
5. Uncertain
6. Certain
7. Uncertain
8. Certain
9. Certain
10. Uncertain
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88
Activity 2: FIGURE IT OUT
1. In a school’s basketball tournament there are 150 spectators, 48 of which are the school
officials, 30 visitors and the rest are students. If all the spectators are equally likely to
be seated at the front row, what is the probability of:
a. visitor seated at the front row.
b. student seated at the front row.
c. school official occupying the second row if either a guest or a student stays at the
front row.
d. officiating official.
Let S be the sample space (total number of possible outcomes) or n(S).
Since all outcomes are equally likely to occur, then the probability of an event A is:
number of favourable outcomes to A n(A)
total number of possible outcomes n(S)
Probability of A or P(A)
P( A) =
a. visitor seated at the front row.
▪ Let A = event where visitor stays at the front row n(A)
▪ n(S) = 150
▪ n(A) = 30
number of favourable outcomes to A n(A)
30
1
=
or
▪ P( A) =
total number of possible outcomes n(S)
150 5
b. student seated at the front row.
▪ Let B = event where a student stays at the front row n(B)
▪ n(B) = 72 ( 150 – 48 – 30 = 72)
number of favourable outcomes to B n(B)
72 36
=
=
▪ P( B) =
total number of possible outcomes n(S)
150 75
c. school official occupying the second row if either a guest or a student stays at the
front row.
▪ If either a visitor or a student stays at the front row, then there would be 149
spectators who will stay behind the first row, 48 of whom are school officials
▪ Let C = event where a school official stays after the visitor or student in the first row.
▪ n(S) = 149
▪ n(C) = 48
number of favourable outcomes to A n(C)
48
=
▪ P(C ) =
or .32
total number of possible outcomes n(S)
149
d. officiating official
▪ Let D = event where an officiating official stays at the first row n(D)
▪ There are no officiating officials among the spectators.
▪ n(D) = {}
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89
▪
P( D) =
0
=0
150
2. What is the probability of the three Balik-Probinsya recipients negative from COVID – 19
viruses after the swab test has been conducted?
Possible Result of Swab Tests
PERSON 1, PERSON 2, PERSON 3
RESULT
1
2
3
4
5
6
7
8
PERSON 1
POSITIVE
POSITIVE
POSITIVE
POSITIVE
negative
negative
negative
negative
PERSON 2
POSITIVE
POSITIVE
negative
negative
POSITIVE
POSITIVE
negative
negative
PERSON 3
POSITIVE
negative
POSITIVE
negative
POSITIVE
negative
POSITIVE
negative
Take note: the total number of outcomes is equal to 8, which is 23.
▪ Swab test results can be POSITIVE or negative – 2 possible results
▪ PERSON 1, PERSON 2, PERSON 3 – the 3 persons who underwent
the Swab test
▪
Let N = event of negative results
1
▪ P(N) =
8
3. If the spinner is rotated, what is the
probability of:
1
a. P(5) =
8
b. P(even number less than 4) =
c. P(odd numbers) =
d. P(less than 5) =
e. P(4 or 8) =
1
8
4 1
or
8 2
4 1
or
8 2
2 1
or
8 4
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90
Activity 3: SIMPLY YOURS
10
1
1. a.
=
100 10
38 19
b.
=
100 50
52
c.
99
1
2.
32
1
3. a.
8
2 1
b. =
8 4
3
c.
8
5
d.
8
7
e.
8
6 3
f. =
8 4
6 1
4. a.
=
12 2
1
b.
12
10 5
c.
=
12 6
7
=
d.
12
12
e.
=1
12
13 1
5. a.
=
52 4
39 3
=
b.
52 4
2
1
=
c.
52 26
4
1
=
d.
52 13
1
6. a.
6
3 1
b. =
6 2
6
c. = 1
6
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91
d.
0
=0
6
1
10
28
14
8.
=
366 183
31
9.
365
8
10.
15
7.
References
Grade 8 Learner’s Module (Module 6), pp. 562-569
https://www.cliffsnotes.com/study-guides/statistics/probability/probability-of-simple-events
https://www.onlinemathlearning.com/probability-problems.html
rigonometry, Mo, Module 2 (L
Prepared by:
ARLON T. MACARUBBO
TCSHS
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