M-ZONE WHAT IS MATH?

Is math a language or

a tool? I say its Both! - Durwin Santos

This month's issue is all about learning math as a language and how it is being used as a tool! Keep an eye out for QR CODES for free posters and answer sheets!

Volume 11, November 21

CONTENT p. 3 - p. 6

| (Intro) Math as a LANGUAGE

p. 7 - p. 8 |Math Sentence & Expressions p. 9 - p, 14 | Truth Tables p. 15 - p, 18| Truth Tables (Activity) p. 20- p, 26| Deductive Vs Inductive Reasoning p. 27- p, 32|(Intro) Math as a TOOL

nt e m e Stat

n o i s s e Expr 1

MATH

NOTE LIST TO READ LIST: Content List Math as a Language Math as a Tool

important reminders

p. 33- p, 38| Measures of Central Tendency p. 38- p, 44| Measures of Dispersion p. 45- p, 46| Why do we Learn Statistics? p. 47- p, 54| BEST-FIT LINE p. 55- p, 58| Coefficient Correlation p. 59- p, 64| Ending Chapter

2

icebreaker! do you really

know math? 1. Who is the father

of mathematics?

2. Is Pi a rational or

irrational number?

3. Is Zillion a real

number?

3

SCAN THE QR

CODE TO GET

THE ANSWERS! Don't cheat! answer it first

4

Time to recall the first time you've heard of math,the initial

language we learned from it was NUMBERS.

MATH AS A: LANGUAGE

5

Try to recall a memory in your elementary

years when math was first introduced to

the class. Initially they introduce the

concept of numbers first before they

move on to the operations of math.

Now that we're older, its time for us to

understand math in a different

perspective. We will tackle how math is

considered a language and a tool in our

lives today.

6

Expressions versus

SENTENCES

Like the English language,

math also has a different

way of determining

equations!

To understand the concept of math

having its own language, let's try

learning simple relations of math to

the English language

7

EXPRESSION A mathematical expression is an arrangement of symbols that are

placed accordingly to their functions. They represent in a correct

order that is expressable and represents the object of interest.

The symbols can designate numbers, variables, operations,

functions, brackets, punctuations, and groupings to help determine

the order of operations

SENTENCE A mathematical sentence makes a statement about TWO

expressions, either using numbers. variables or a

combination of both

In a math sentence, we use symbols like equals, greater

than, or less than to separate two represent expressions in

one equation.

In an expression, you are

not able to determine

whether the statement is

true or false

21 + 7 (23 x 75) / 2

EXPRESSION

As for a sentence, you

are able to determine

whether the statement is

true or false

x + 34 -2 x 12 = 98 SENTENCE

8

TRUTH TABLES Try to remember an event in your life

where you wanted to find out if a

statement was true or not. Most of the

time we ask our own elders if something

was true or false

For this lesson, lets try to figure out if a

statement is true by learning the truth

table. First we will tackle the difference of a

simple and a compound proposition, then

we will continue to discuss the kinds of

compound propositions.

SCAN TO GET A

FREE POSTER! 9

simple proposition A simple proposition conveys a single

thought or idea. In terms of the English

Language, it composes a single

sentence.

compound proposition A compound proposition combines TWO

simple propositions by using

connectives such as: if, and, then, and if and only if.

LETS TAKE A CLOSER LOOK!

TRUTH TABLE

10

CONJUNCTION A conjunction is formed when two simple

propositions are joined using the connective

and. Example: My mom is a doctor and my dad is a chef. In symbolic form, we write it as p q The propositions p and q are called conjuncts A conjunction is true when both conjuncts are true.

Otherwise, its value is false. Observe the conjunction truth table below:

11

DISJUNCTION A disjunction is formed when two simple

propositions are joined using the

connective or.

alone or I wait for

Example: Either I go home my mom In symbolic form, we write it as p v q The propositions p or q are called disjuncts. A disjunction is false when both disjuncts are false.

In short, a disjunction is true when at least one of the

disjuncts is true. Observe the disjunction truth table below:

12

CONDITIONAL The proposition p is called the antecedent while

the proposition q is called the consequent A conditional proposition takes the form: if p, then q Example:If my homework is due on Monday,

then I should work on it by Sunday In symbolic form, we write it as p q Another term for a conditional proposition is

implication A conditional is false when the antecedent is true and

the consequent is false. Otherwise, it is true. Observe the conditional truth table below:

13

BICONDITIONAL A biconditional proposition has the form of: p if and only if q. Example: I will attend the Taylor Swift concert if

and only if I'm going with my bestfriend. In symbolic form, we write it as p q Observe the biconditional truth table below:

Now you have reached the end of this lesson, try to form your own compound propostions with a friend!

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POP QUIZ! Take out one whole sheet of paper, time to

apply what you just learned!

Consider the following propositions:

P: there are no classes tomorrow q: I will sleep until afternoon

Construct a: Conjunction, Disjunction,

Conditional, and Biconditional statements out

of the following propositions 15

Use this space to write down your answers, don't forget to check your spelling!

16

SCAN THE QR CODE FOR

THE RIGHT ANSWER!

NO CHEATING! ANSWER IT FIRST 17

Find any mistakes? write down your notes here!

18

INDUCTIVE VS DEDUCTIVE

19

20

Have you ever encountered an

event or situation and you try to

make sense of it by making

conclusions?

Like from the picture before this

page we see a man wondering why

his fish is gone but he sees his cat

sitting down on the other side of

the table too...

21

The language of Inductive and

Deductive reasoning helps us

analyze situations in order for us to determine whether the

evidences stand with the truth of

our statements.

(also I ate the fish)

22

DEDUCTIVE

REASONING

It is a method of reasoning that

starts with general

observations, then ends with a

specific conclusion.

23

indUCTIVE

REASONING

Inductive reasoning on the

other hand, starts by looking

at specific observations, then

makes a generalized

conclusion.

24

Observe the illustrations shown below to understand an example of how we use Deductive Reasoning in real life:

Point A: Work ends at 4pm Point B: Your device states that its already 4pm Conclusion: Therefore your work shift is over.

M P 0 0 4:

25



Observe the illustrations shown below to understand an example of how we use Inductive Reasoning in real life:

You check the pantry room to see what's for lunch, and all the dishes are red. You assume that all red-colored dishes are spicy. Therefore you concluded that all meals are spicy today.

26

MATH AS

27

A TOOL

28

The world as we know it has revolved around human

development throughout the years. And for us humans to evolve and learn more, we use our math tools, such

as statistics and data.

29

We will be focusing on the concept of math used as a tool for us humans to be able to collect, compute, and analyze data.

30

INTRODUCING: WHAT IS DATA? 1

2

3

4

5

6

7

8

9

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Look at the dots in front of you and treat them as

your data set. How many dots are there? What is the

average of this data? What is the center value of this

set of data? Can all of these questions be answered

with this set of dots? Yes. Because data is what you

need to be able to answer all these questions.

31

HOW AND WHY DO

WE USE DATA?

Data is essential for multiple reasons, but the main reason is that we use data to measure the effectiveness or importance of a given category. And how we use data can be through different measurements such as measures of central tendency and measures of dispersion.

32

MEASURES OF CENTRAL TENDENCY Let's learn each

measurement used in

a central tendency

33

MEAN MEDIAN MODE

MEAN The mean is the sum of the total values of a dataset, then divided by the number of observations. This measurement aims to determine the AVERAGE value of the dataset.

MEDIAN The median is a measurement used to determine the CENTRAL VALUE of a set of data. It is usually found by looking for the middle value of a dataset.

MODE The mode determines what value APPEARS THE MOST in a dataset. In some cases, there are datasets that have no presence of the mode.

34

LET'S TRY IT OUT! 1

2

3

4

5

6

7

8

9

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

Let's pertain to the dataset given to us earlier.

We can observe the following:

The number of values in this data set is: n =24 Let's try to find the mean, median, and

mode of this dataset.

35

Finding the Mean: X

Sum of the dataset's values ( x) = Number of observatio ns (n)

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 9...... + 24 = 299 = 299 / 24 X = 12.4583

1. first, we calculate the sum of the values. In this

dataset, the sum is 299. 2. The next step is to divide the sum by the total

number of observations or the dataset, which is 24 3. After dividing the sum of values to the total number

of observations of the dataset, we then mark this as

the mean. you may also round up the mean to 12.46

36

Finding the Median: 1

2

3

4

5

6

7

8

9

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

x = 12.5

s

12 + 13 = 25/2 =

Observe the rules in finding the median: If n is odd: we look for the middle number of the data in array If n is even: we calculate for the average of the two middle

values of the data in array in this case, since n = 24, we see that the middle value

of the dataset is 12 and 13. We then solve for the 37

average to determine the median

Finding the Mode: 1

2

3

4

5

6

7

8

9

9

11

12

13

14

15

16

17

18

19

20

21

22

23

24

In finding the mode, we just look for the number

value that appears the most in a dataset.

Therefore, the value nine is considered our mode

since it appeared twice on our dataset.

x=9 38

MEASURES OF

DISPERSION 8

39

45

56

In measures of dispersion, its purpose is to discover how spread out the data values is on the number line. Earlier, we tried to solve for the typical or average values. This time, we're looking for their varieties. We measure them through: range, standard deviation, and variance. 39

Why do we use it?

It may not seem obvious, but measures of dispersion are used in many occasions or situations. For example, let's say we want to compare the runs scored by two batters in the last ten matches: Batsman A: 25, 20, 45, 93, 8, 14, 32, 87, 72, 4 Batsman B: 33, 50, 47, 38, 45, 40, 36, 48, 37, 26

average, you can see the

While they may have the same difference if we calculate their measures of dispersion. (try to solve (B) by yourself by the end o f this lesson!) reference: https://www.brainkart.com/article/Measures-of-Dispersion_39436/ 40

RANGE When computing for the

range, you arrange the

numbers in highest to

lowest values first, then

subtract the highest value

from the lowest value.

PERTAIN TO BATSMAN A DATASET

Highest Value = 93 Lowest Value = 4 Range = HV - LV

Range: 93 - 4 = 89

VARIANCE Variance is a statistical measurement of the spread

between numbers in a data set. Here is the formula:

(for population data)

(for sample data)

STANDARD DEVIATION Standard deviation gives a clear idea about how far the

values are spreading or deviating from the mean. Here is the formula:

41

LET'S SOLVE FOR THE VARIANCE Xi



( xi -

)

( xi -

)

25

40

-15

225

20

40

-20

400

45

40

5

25

93

40

53

2809

8

40

-32

1024

14

40

-26

676

32

40

-8

64

87

40

47

2209

72

40

32

1024

4

40

-36

1296

sum:

400

400/1

0 = 40

sum = 0

sum

=9572

2

To make

your

calculations

clearer, it's

best advice

to make a

table of the

variables

you need to

solve for the

variance.

want to see

what the

answer for

the variance

is? check the

QR CODE!

42

LET'S SOLVE FOR THE STANDARD DEVIATION In solving for the standard deviation, you

calculate the square root of the variance. Population variance = 957.2 Sample variance = 1063.56

Now that we have the variance, we can find

the standard deviation for both population and

sample data: Population SD = 957.2 = Sample SD = 1063.56 =

43

30.94 32.61

Now it's your turn to try! Now that we calculated the measures of

dispersion of Batsman A, it's time for you to

try it on your own! Use the formulas we used

earlier. And to end our lesson here's a free

poster for you to put up in your room!

^ SCAN AND DOWNLOAD ^ 44

why do we learn

statistics?

With everything we've learned today, we can

conclude that numbers can be analyzed in

many different ways and have multiple

interpretations. With statistics, we can

understand data at a new level and in the

most accurate way possible! 45

Like in the business industry, people usually

hold meetings to discuss their statistical

analysis on a specific category, whether it

may be about: sales, marketing strategies,

budget planning or inventory reports!

46

BEST-FIT LINE

Okay, hear me out. I know I said that was the last

lesson, but I figured I needed to give you one last

lesson that can help you find out how strong the

relationships between two variables are! Aren't

you curious to see the connection between the

temperature at a rock concert and the number of

people attending the concert? Let's find out! 47

Suppose that you're an event organizer for a

rock concert. You heard someone from the

crowd complain about the heat. You're unsure

if the heat can cause a negative effect on the

audience, so to find out, we use line

regressions to determine their relationship.

48

Temperature (x)

No. of People (y)

25

300

32

285

30

290

36

180

35

195

37

100

This table is a six days report of the temperature degrees and the number of people arriving at the concert. The formula that we'll be using for this will be: Method of Least Squares

y = mx + b

The least-squares method is used to predict the behavior of the dependent variable to the independent variable.

49

We'll be using the

following formulas to

get the variables needed:

50

(

x

y

xy

x

25

300

7500

625

32

285

9120

1024

30

290

8700

900

36

180

6480

1296

35

195

6825

1225

37

100

3700

1369

y =

1,350

xy =

42,325

x =

6,439

x= 195 x) = 38,025 2

2

2

The first step is to make a table of the

variables we need. Make sure to have all the

variables complete, and always calculate

the sum of each column.

Now that everything is complete, we can

move on to the next step!

51

m=

6 (42,325) - (195)(1,350) 6(6439) - 38,025 m=

253,950 - 263,250 38,634 - 38,025 -9,300 m= 609 m = -15.2709

Here is a step-by-step solution to finding the (m). It's best to do it this way so that you may trace back where you got your answers. 52

b=

6439(1350) - (195)(42,325)

6(6439) - 38,025 b=

8,692,650 - 8,253,375 38,634 - 38,025 439,275 b= 609 b = 721.3054

Here is a step-by-step solution to finding the (b). now that we both have the y, m, and b variables, we can use the least of method squares formula.

y = (-15.27)x + 721.31 53

GRAPH OF THE DATA

In front of you is the graph of the relationship between temperature and the people attending the concert. We can see through these lines that there is a relationship between these two variables. Now, the question here is HOW STRONG is the connection? The correlation coefficient is then used to determine or measure if there is a solid or weak relationship.

54

CORRELATION

COEFFICIENT (FORMULA)

If you look closely and observe the correlation

coefficient formula, some of these variables

were already given earlier. We only have to add

one variable, which is (y 2). We only have to add

another column to the table, and then we can

start solving!

55

(

x

y

xy

x

y

25

300

7500

625

900,000

32

285

9120

1024

81,225

30

290

8700

900

84,100

36

180

6480

1296

32,400

35

195

6825

1225

38,025

37

100

3700

1369

10,000

xy =

42,325

2

2

x= 195 x) = 38,025

y = 1,350

2

(

y) = 1,822,500 2

2

x =

6,439

2

y =

335,750

Now that are variables are complete, we

can finally work on solving for the

correlation coefficient! Let's have YOU try

it out! Get a whole sheet of paper and

show your solutions! 56

let's solve for it! Here's the formula for the correlation coefficient!

Scan the QR code to see the

answer and solutions! (answer it yourself first, no

cheating!) 57

CORRELATION

COEFFICIENT Now, if you already solved for the

correlation coefficient (r), then the

answer you got is supposed to be:

r = -0.86 Now to analyze if the relationship is strong or

not, take a look at the following conditions: If (r) is > 0.7 , then the relationship is strong If (r) is < - 0.7, then the relationship is strong. If (r) is -1 or 1, then its a perfect linear relationship If (r) is 0, then there is no perfect linear relationship

Given the conditions, we can conclude that there

is a minimal negative relationship between the x

(temperature) and y (audience).

58

Now that we have an

answer, I guess there is

not much to worry about

for the rock concert's

success!

59

What better way to end

this lesson than enjoying

some tunes? Have a break

and listen to this song.

Good job you!

60

MATH AS

Now that We've seen how math hope that our lessons can someh on math being more than just add dividing n

Math is everything around us! W problems as long as we know ho takeaway is that math is not onl language that we must continue always count on! So what are you you've learned from our sessio

61

A WHOLE

h is both a language and a tool, I

how give you a bigger perspective

ding, subtracting, multiplying, and

numbers.

We can always find solutions to

ow to solve them! So the main

ly homework in class but also a

to learn and a tool that we can

u waiting for? Time to use what

on! Take Care, dear student!

62

Special thanks to: Sir Durwin Santos For the lessons you taught us this semester.

63

Take care, students!

64

Designed and Made By: Patricia Marie Zotomayor Enderun Colleges | Bachelor of Science in Business Administatrion | Major in Marketing Management

Creator's Note

This magazine is dedicated to all the students that are eager to see math in a different light. Hope you enjoy my twists on the topics and don't forget to check on the QR Codes for posters and answer sheets!