Mathematics is the study of patterns and quantities. It is used in many fields and is everywhere in daily life. Some key concepts in mathematics include geometric patterns found in nature, Fibonacci numbers, problem solving techniques, and graph theory. Mathematics uses formal language and symbols to express ideas and relationships precisely. Common tools in mathematics include logic, sets, functions, and statistical analysis of data. Mathematicians develop and apply logical reasoning to understand mathematical concepts and solve problems.

1. MATHEMATICS
IN THE
MODERN WORLD
WORKSHEET
_ Mr. Andrei Irwin Maghirang _
Submitted by:
COMIA, JEMYCAA.
DE TORRES, JANNA KATRINA D.
PEREZ, JUSTIN RED D.
MAGBOJOS, CHARIS LOVE B.
ALCANTARA, ANDREY

2. MODULE ONE
THE NATURE OF MATHEMATICS
1. The following are represent math; except:
a. Art
b. Language
c. Study of patterns
d. Process of thinking
e. None of the above
2. We use Mathematics in our daily tasks and activities. It is an important tool in the field of
humanities, literature, medicine and even in music and arts. Mathematics is __________
a. A tool
b. Everywhere
c. Everything
3. It is a structure, form, or design that is regular, consistent, or recurring. Also they can be found
in nature, in human-made designs, or in abstract ideas.
4. It is a kind of pattern which consists of a series of shapes that are typically repeated.
5. It is a type of symmetry that exists in patterns that we see in nature and in man-made objects.
It acquires symmetries when units are repeated and turn out having identical figures, like the
bees’ honeycomb with hexagonal tiles.
6. It is sometimes called “line symmetry” or “mirror symmetry”, captures symmetries when the
left half of a pattern is the same as the right half.
7. What is an approximation of a golden spiral created by using Fibonacci Numbers?
8. What is Fib(11) + Fib (6) – Fib (9)?
9. If the fibonacci sequence is organized in a way a number can be obtained by adding the two
previous numbers, What is the 9th
and 10th
term by adding the two previous numbers of the
example sequence below?
1,1,2,3,5,8,13,21,____,____,...
10. What is the 26th
Fibonacci number? By using the formula.

3. MODULE TWO
MATHEMATICAL LANGUAGE AND SYMBOLS
1. Twice of a number added by four.
2. Eight times a number is one hundred twenty-eight.
3. Find x and y if ( 2x + 8, y ) = ( x + 12, -14 )
4. If f(x) = x + 10 and g(x) = 4x + 8 , what is (f g)(x)
•
5. It is a set that has no element { }
6. A set that has elements in a given set has no end or not countable {…-1 0 1 2 3…}
7. It is the statement that can be demonstrated to be true by accepted mathematical operations
and arguments . What is it?
8. What is the opposite of the original statements?
9. Who stated in the “Formality of Language: Definition and Measurement”, an expression is
completely formal when it is context-independent and precise (i.e. non-fuzzy), that is, it
represents a clear distinction which is invariant under changes of context. In mathematics, we are
always dealing in a formal way.
10. It is a statement about mathematical objects that are taken seriously as mathematical objects
in their own right.

4. MODULE THREE
PROBLEM SOLVING AND REASONING
1. Every windstorm in this area comes from the north. I can see a big cloud in the distance. A
new windstorm is coming from the north, is it Deductive or Inductive reasoning used to come up
with the conclusion?
2. If a team wins 10 levels of dance competition, then they compete in the finals. If a team will
dance in the finals, then they will travel and move to the Arena. The JAVENS won 10 levels.
CONCLUSION: The JAVENS will travel and move to the Arena.
3. What does the following mean:
∃ an integer k such that n=2k+1
4. The first step of an indirect proof is to?
5. He is known as “The Father of Problem Solving” and one of the foremost recent
mathematicians to make a study of problem solving. He also made fundamental contributions to
combinatorics, number theory, numerical analysis, and probability theory.
6. This part of Polya’s four-step strategy is often overlooked. You must have a clear
understanding of the problem. Are all the words in a problem really understood and clear by the
reader? Does the reader really know what is being asked in a problem on how to find the exact
answer? These questions must be answered on what step ?
7. It is the ability to understand and think with complex concepts that, while real, are not tied to
concrete experiences, objects, people, or situations.
8. Consider the figure below.
8th
layer
How many triangles are there on the 8th layer?
a. 15
b. 17
c. 14
d. 16

5. 9. In a house there are 8 people and you order only one pizza, if you cut the pizza in half it’s only
2pcs, and if you cut again in the half it’s only 4pcs. How many cuts do you need to make to get
8pcs slice of pizza? 🍕
10. To complete the table, the sum is 12
2
8 0
1 6

6. MODULE FOUR
MATHEMATICAL SYSTEM
I. Calculate using the 12 hour clock.
1. 4 + 10 = ?
2. 4 + 26 = ?
3. 4 + 20 = ?
4. 4 + 23 = ?
5. 4 + 11 = ?
6. This statement a ≡ b (mod n) is called?
7. Find the positive value of x such the 71 ≡ x ( mod 8 ).
8. What is the time 100 hours after 7 a.m.?
9. ______________ is a set with one or more binary operations defined on it.
10. What is (a b) (c d)?
∗ • ∗
a b c d
a b a c d
b c b a d
c d c b a
d a d c b

7. MODULE FIVE
DATA MANAGEMENT
1. A numerical value that describes some characteristics of a sample or population is called a…?
2. This word means the middle of the data . What is it?
3. What is a data entry that is far removed from the other entries in the data set?
4. It is where a numerical value is obtained that is considered typical of the quantitative variable.
5. Mark operates Technology Titans, a Web site service that employs 8 people. Find the mean
age of his workers if the ages of the employees are as follows: 55, 63, 34, 59, 29, 46, 51, 41
6. Find the median of the following data: 7, 9, 3, 4, 11, 1, 8, 6, 1, 4
7. You begin to observe the color of clothing your employees wear. Your goal is to find out what
color is worn most frequently so that you can offer company shirts to your employees.
Monday: Red, Blue, Black, Pink, Green, and Blue
Tuesday: Green, Blue, Pink, White, Blue, and Blue
Wednesday: Orange, White, White, Blue, Blue, and Red
Thursday: Brown, Black, Brown, Blue, White, and Blue
Friday: Blue, Black, Blue, Red, Red, and Pink
What is the mode of the colors above?
8. What is the statistical measure that expresses the extent to which two variables are linearly
related?
9. It is another way to visually show the behavior of data. To create a graph, distribution of
scores must be organized?
10. Mean?
150 125 150 99 115
125 175 135 41 485

8. MODULE SIX
LOGIC
A. IDENTIFICATION
1. It is a table that shows the truth value of a compound statement for all possible truth
values of its simple statement.
2. A statement whose truth value is always true regardless of the truth value of its
individual simple statement.
3. Any variable, such as statement variable or an input signal, that can take one of only
two values is called______.
B. DIRECTION: Write each symbolic statement as an English sentence. Use p, q and r as
defined below.
p: Kim Yohan is a dancer
q: Kim Yohan is a hockey player
r: Kim Yohan is a singer
4. (p ∧ q) v r
5. (r ∧ q→ p)
6. ~p → (p v r)
7. It is used in many electrical appliances, as well as in telephone equipment and computers.
8. It states that the negation of the conjunction statement is equivalent to the disjunction of the
negation of each simple statement. And, the negation of the disjunctive statement is equivalent to
the conjunction of the negation of each simple statement.
9. According to Immanuel bant (1780), Logic is a science of the necessary Laws of thought
without which no employment of the understanding and the reason takes place. Logic can be
applied to many disciplines. TRUE or FALSE
10. DR. Morgan Law states that the negation of the conjunction statement is equivalent to the
disjunction of the negation of each Simple statement. And the negation of the disjunction

9. statement is equivalent to the conjunction of the negation of each simple statement. TRUE or
FALSE

10. MODULE SEVEN
MATHEMATICAL OF GRAPHS
DIRECTION: Write MATH if the statement is correct and change the bold word if the statement
is false.​
1. A graph can be represented by a diagram in the plane where vertices are usually
represented by small circles (open or solid) and whose edges by line segment or curve
between the two points in the plane corresponding to the appropriate vertices.
2. A graph without loops and multiple edges is called a simple graph.
3. Two edges with the same end points are said to be adjacent edges.
4. Graph middle is a graph representing certain situations.
5. A subgraph can be obtained by removing vertices or edges from a graph.
6. A run in a graph is a sequence of vertices such that consecutive vertices in the sequence
are adjacent.
7. A connected graph is eulerian if and only if every vertex of the graph is of even degree.
8. A closed walk such that no edge is repeated is called a circuit. A closed walk that repeats
no vertex is called a cycle.
9. ___________ is a graph representing certain situations, and there are many real life
situations that can be modeled by graphs.
10. How Many known theorems in graph coloring?
a. 1 b. 2
c. 3 d. 4

11. ANSWER KEY
MODULE ONE
1. E. NONE OF THE ABOVE
2. B. EVERYWHERE
3. PATTERN
4. GEOMETRIC PATTERN
5. TRANSLATIONS
6. REFLECTION SYMMETRY
7. FIBONACCI SPIRAL
8. Fib(11) + Fib (6) – Fib (9)?
= 89 + 8 - 34
= 63
Solution:
𝐹𝑖𝑏 11
( ) =
1.618
( )
11
− 1−1.618
( )
11
5 =89
𝐹𝑖𝑏 6
( ) =
1.618
( )
6
− 1−1.618
( )
6
5 =8
𝐹𝑖𝑏 9
( ) =
1.618
( )
9
− 1−1.618
( )
9
5 =34
9. 34 and 55
10. 121,326.81
Solutions: 𝑋 𝑛
=
φ
𝑛
− 1 − φ
( )
𝑛
5
𝑋 26
=
(1.618)
26
− 1 − 1.618
( )
26
5
𝑋 26
=
217,294.79 − 0.000003679
5
𝑋 26
=
217,294.79
5
𝑋 26
= 121, 326. 81

12. MODULE TWO
1. 2x + 4
2. 8(x) = 128
3. x = 4 and y = -4
Solution: Since (2x + 8, y) = (x + 12, -4)
2x + 8 = x + 12
2x - x = 12 - 8
x = 4
Solving for x, we got x = 4 ang obviously y = -4
4. 4x2
+ 48x + 80
Solution: ( f g ) (x) = f (x) g (x)
• •
= (x + 10 )(4x + 8)
= 4x2
+ 48x + 80
5. EMPTY SET or NULL SET
6. INFINITE SET
7. THEOREM
8. NEGATIONS
9. BY HEYLIGHEN F. AND DEWAELE J-M
10. MATHEMATICAL LOGIC

13. MODULE THREE
1. INDUCTIVE REASONING
2. DEDUCTIVE REASONING
3. n is ODD
4. Assume the negation of what you are trying to proof
5. GEORGE POLYA
6. 1ST STEP / STEP 1. UNDERSTAND THE PROBLEM
7. ABSTRACT REASONING
8. A. 15
9. 4 Cut to get 8 pcs of Pizza
10. All sides Is equal to 12
3 2 7
8 4 0
1 6 5

14. MODULE FOUR
1. 2
2. 6
3. 0
4. 3
5. 3
6. CONGRUENCE
7. 7
Solution: 71 ≡ x ( mod 8 )
64 ≡ 0 ( mod 8 )
64 + 7 ≡ 0 + 7 ( mod 8 )
71 ≡ 7 ( mod 8 )
So x = 7
8. 11 a. m.
Solution: 7 + 100 ≡ x ( mod 12)
107 - x ≡ 2n
107 -x is a multiple of 12
Therefore, the least positive integer of x must 11.
Since 107 - 11 = 96 is the nearest multiple of 11 less than 107
So the time is 100 hours after 7 o’clock a.m. is 11 a.m.
9. MATHEMATICAL SYSTEM
10. B.
Solution: (a b) = a
∗
(c d) = a
∗
(a b) (c d) = b
∗ • ∗

15. MODULE FIVE
1. DESCRIPTIVE STATISTICS
2. MEDIAN
3. OUTLIER
4. Measures of Central Tendency
5. The mean age of all 8 employees is 47.25 years, or 47 years and 3 months.
SOLUTION: Step 1: Add the numbers to determine the total age of the workers.
55 + 63 + 34 + 59 + 29 + 46 + 51 + 41 = 378
Step 2: Divide the total by the number of months.
6. The median is 5.
SOLUTION: Step 1: Organize the data, or arrange the numbers from smallest to largest.
1, 1, 3, 4, 4, 6, 7, 8, 9, 11
Step 2: Since the number of data values is even, the median will be the
mean value of the numbers found before and after the position.
Step 3: The number found before the 5.5 position is 4 and the number
found after the 5.5 position is 6. Now, you need to find the mean value.
1, 1, 3, 4, 4, 6, 7, 8, 9, 11
7. The mode is blue. The color blue was worn 11 times during the week. All other
colors were worn with much less frequency in comparison to the color blue.
8. CORRELATION
9. GRAPHICAL REPRESENTATION

16. 10. 160
Solution:
MODULE SIX
A.
1. TRUTH TABLE
2. TAUTOLOGY
3. BOOLEAN VARIABLE
B.
4. Kim Yohan is a dancer and Kim Yohan is a hockey player, or Kim Yohan is a
singer.
5. If Kim Yohan is a singer and Kim Yohan is a hockey player, then Kim Yohan is a
dancer.
6. If Kim Yohan is not a dancer, then Kim Yohan is a dancer or Kim Yohan is a
singer.
7. SWITCHING NETWORKS
8. DE MORGAN’S LAW
9. FALSE (1785)
10. TRUE

17. MODULE SEVEN
1. MATH
2. MATH
3. FALSE (PARALLEL EDGES)
4. FALSE (GRAPH MODEL)
5. MATH
6. FALSE (WALK)
7. MATH
8. MATH
9. GRAPH MODEL
10. C. 3