INFORMATION SHEET GE4-MMW-1.1.

1
“Mathematics in Our World”

This module discusses about …” what is physical is subject to the laws of mathematics, and what is
spiritual to the laws of God, and the laws of Mathematics are but the expression of the thoughts of
God.” – Thomas Hill.

SELF-CHECK GE4-MMW-1.1.1

A. Fill in the blanks with the correct answer.

___________1. is a recursive sequence, generated by adding the two previous numbers, the first two
numbers of the sequence being 0 and 1.
___________2. is an interdisciplinary theory stating that within the apparent randomness of complex
systems, there are underlying patterns, constant feedback loops, repetition, self-similarity, fractals and
self-organization.
___________3. is a never-ending pattern.
___________4. often represented by the Greek letter φ.
___________5. is the science that deals with the logic of shape, quantity and arrangement.

SELF-CHECK ANSWER KEY GE4-MMW-1.1.1

1. Fibonacci Sequence
2. Chaos
3. Fractal
4. Golden Ratio
5. Mathematics
INFORMATION SHEET GE4-MMW-1.2.1
“Mathematics in Our World”

Mathematics is a tool. Play with it any way you want and see if you can make something. Don’t
worry if you break the tool, we’ll rebuild it, together.

Today, we’ll be talking about the essence of mathematics and how it shapes the world around us. The
intention behind this post is to show the beauty of math to people, how it governs nature without most
of us even noticing it.

One of the things about Mathematics that we love the most is its uncanny ability to reveal hidden
beautiful patterns in our everyday life, the nature around us. These patterns can be sequential, spatial,
temporal, and even linguistic. There are connections between things that don’t seem connected, but can
be observed with the intellect of math. One beautiful example is — fireflies flashing in unison and a
pattern that can be solved mathematically.

In short, we can say mathematics is the science of patterns.

Talking more about patterns, let’s have a glimpse of “Chaos Theory, which is a hot topic among many
mathematicians. ‘Chaos’ is an interdisciplinary theory stating that within the apparent randomness of
complex systems, there are underlying patterns, constant feedback loops, repetition, self-similarity,
fractals and self-organization.
One of the simplest examples to understand is the ‘Butterfly Effect’ that describes how a small change
in one state of a deterministic nonlinear system can result in large differences in a later state, e.g. a
butterfly flapping its wings in Brazil can cause a tornado in Texas. Such kind of phenomena are often
described by fractal mathematics. Chaos theory and chaotic models have applications in many areas
including geology, economics, biology, meteorology etc., and can help demystify the huge dynamic
complex systems.

What is a fractal? — A fractal is a never-ending pattern. They are the images of dynamic systems — the
pictures of ‘Chaos’. Geometrically, they exist in between our familiar dimensions, nature is full of fractals,
for instance: trees, mountains, seashells, clouds, ferns, even human body! These things look very complex
and non-mathematical. Now, think what it took to produce what you see. You’ll realize it takes endless
repetition and that gives rise to one of the defining characteristics of a fractal, a self-similarity. Fractals
have vast applications in astronomy, fluid mechanics, image compression etc. as they hold the key to
describe the real world better than traditional science.

Let’s have a closer look at some of the real-world fractal examples around us.

1. Fern — As you look deeper and deeper, you see a never-

ending repetitive pattern.

2. Koch Snowflake — A beautiful example of a

fractal with infinite perimeter but finite area.
The idea is, make an equilateral triangle. Now
make another equilateral triangle above the
previous one, but in opposite direction. You’ll
see small equilateral triangles on the boundary.
Keep doing the same for them, and keep doing,
keep doing…
When you keep doing it, soon after some
depth, you’ll start seeing the resemblance of
pattern with a snowflake.

3. Fractal Antenna — Above example of ‘Koch snowflake’

shows a fractal of perimeter increasing infinitely while
it’s area can be bounded. Using such property, fractal
antenna was invented, using a self-similar deign to
maximize the length of material that can receive much
weaker signals and transmit signals over long distance
without compromising the area and volume taken by
the antenna due to its fractal nature. This is very
compact and have useful applications in cellular
telephone and microwave communications.
 So, you see, there are examples around us shaped by mathematics with hidden patterns, without us
even knowing about it. That’s the beauty of math.

Now let’s have a look at one of the famous mathematical number sequences, the ‘Fibonacci
Sequence’. The Fibonacci sequence is a recursive sequence, generated by adding the two previous
numbers, the first two numbers of the sequence being 0 and 1.

So, Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 …

An interesting fact is that the number of petals on a flower always turns out to be a Fibonacci number.
Statistically, this sequence appears a lot in botany. Another example is if you look at the bottom of pine
cone, and count clockwise and anti-clockwise number of spirals, they turn out to be adjacent fibonacci
numbers (see image below).

Let’s have a look at a property of Fibonacci numbers. I’m going to write continuous sums of squared
Fibonacci numbers.
Squared Fibonacci Sequence: 0, 1, 1, 4, 9, 25, 64, …
Continuous sums:
0=0x1
0+1=1x1
0+1+1=1x2
0+1+1+4=2x3
0+1+1+4+9=3x5
0 + 1 + 1 + 4 + 9 + 25 = 5 x 8 … and so on. (You see every time product of the sum is two consecutive
Fibonacci numbers)

Well, there’s a mathematical explanation for the pattern we see above. Suppose you have squares
of sides representing Fibonacci numbers, and assemble them in the way shown below. The above
pattern is nothing but area of the rectangle formed by joining the squares (continued Fibonacci
squares sum).
The figure on the right is called the Fibonacci Spiral

Eye of hurricane
Fibonacci spiral recurs throughout the nature — in
the seed heads of sunflower, the petals of a rose,
the eye of the hurricane, the curve of a wave, even
the spiral of galaxies!
It seems that when we keep comparing ratios of
two consecutive Fibonacci numbers, as we move
further in the sequence, the ratio approaches a
value of 1.618034… which is called Φ or better, the
golden ratio. This ratio has a beauty of special kind
and is important to us. Why? — The golden ratio
appears everywhere — DNA, human body, eye of
hurricane etc — it appears in various structures of
nature. This occurrence of Φ in various aspects of nature, gives rise to the question that ‘Was our
universe intelligently designed, or is it just a cosmic coincidence?’

The Golden Ratio

The golden ratio (often represented by the Greek letter φ) is directly tied to a numerical pattern
known as the Fibonacci sequence, which is a list composed of numbers that are the sum of the previous
two numbers in the sequence. Often referred to as the natural numbering system of the cosmos, the
Fibonacci sequence starts out simply (0+1= 1, 1+1=2, 1+2=3, 2+3=5, 3+5=8...), but before long, you'll find
yourself adding up numbers in the thousands and millions (10946+17711=28657, 17711+28657=46368,
28657+46368=75025...) and it just keeps going on forever like that.
When a Fibonacci number is divided by the Fibonacci number that came before it, it approaches the
golden ratio, which is an irrational number that starts out as 1.6180339887... and, once again, goes on
forever.
When the golden ratio is applied as a growth factor (as seen below), you get a type of logarithmic spiral
known as a golden spiral.
Learn more about the Fibonacci sequence and natural spirals in this fascinating video series by
mathematician Vi Hart, who talks fast, but she's interesting and will remind you of the way your brain
once hopped from subject to subject: https://youtu.be/ahXIMUkSXX0

Golder Ratio manifest in nature

Chameleon tails
A chameleon tail is famous for its tight spiraling shape. (Photo: Ryan M. Bolton/Shutterstock)
Seashells
A seashell is one of the most well-known examples of the golden ratio spiral in nature. (Photo:
Tramont_ana/Shutterstock)

Fern fiddleheads
The curled-up fronds of a young fern are known as fiddleheads. (Photo: Zamada/Shutterstock)

Ocean waves
Despite their tumultuous nature, ocean waves are another example of the golden ratio manifesting in
nature. (Photo: irabel8/Shutterstock)

Golder Ratio manifest in Arts

Leonardo da Vinci
The Golden Section was used extensively by Leonardo Da Vinci.

Michelangelo
In Michelangelo’s painting of “The Creation of Adam” on the ceiling of the Sistine Chapel, look at the
section of the painting bounded by God and Adam.
Golder Ratio manifest in Architecture
Ancient Greek architecture used the Golden Ratio to determine pleasing dimensional relationships
between the width of a building and its height, the size of the portico and even the position of the
columns supporting the structure. The final result is a building that feels entirely in proportion.

Ancient Greek
architecture uses the
Golden Ratio to
determine pleasing
dimensions

Ration in animals:

The eyes, beak, wing and key body markings of the penguin all fall at golden sections of its height.
 Application of mathematics in the World

All the key facial features of the tiger fall at golden sections of the lines defining the length and
width of its face.

The body of knowledge and practice known as mathematics is derived from the contributions of
thinkers throughout the ages and across the globe. It gives us a way to understand patterns, to quantify
relationships, and to predict the future. Math helps us understand the world — and we use the world to
understand math.

The world is interconnected. Everyday math shows these connections and possibilities. The
earlier young learners can put these skills to practice, the more likely we will remain an innovation
society and economy.

Algebra can explain how quickly water becomes contaminated and how many people in a third-
world country drinking that water might become sickened on a yearly basis. A study of geometry can
explain the science behind architecture throughout the world. Statistics and probability can estimate
death tolls from earthquakes, conflicts and other calamities around the world. It can also predict profits,
how ideas spread, and how previously endangered animals might repopulate. Math is a powerful tool
for global understanding and communication. Using it, students can make sense of the world and solve
complex and real problems. Rethinking math in a global context offers students a twist on the typical
content that makes the math itself more applicable and meaningful for students.

For students to function in a global context, math content needs to help them get to global
competence, which is understanding different perspectives and world conditions, recognizing that issues
are interconnected across the globe, as well as communicating and acting in appropriate ways. In math,
this means reconsidering the typical content in atypical ways, and showing students how the world
consists of situations, events and phenomena that can be sorted out using the right math tools.
INFORMATION SHEET GE4-MMW-2.1.1
“Set and Venn Diagrams”

A Venn diagram, also called primary diagram, set diagram or logic diagram, is a diagram that shows
all possible logical relations between a finite collection of different sets. ... A Venn diagram consists of
multiple overlapping closed curves, usually circles, each representing a set.

INFORMATION SHEET GE4-MMW-1.2.1

“Set and Venn Diagrams”

SETS

A SET is a collection of things.

For example, the items you wear is a set: these include
hat, shirt, jacket, pants, and so on.

You write sets inside curly brackets like this:

{hat, shirt, jacket, pants, ...}

You can also have sets of numbers:

 Set of whole numbers: {0, 1, 2, 3, ...}
  Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

Ten Best Friends

You could have a set made up of your ten best friends:

{alex, blair, casey, drew, erin, francis, glen, hunter, ira, jade}
Each friend is an "element" (or "member") of the set. It is normal to use lowercase letters for them.

Now let's say that alex, casey, drew and hunter play Soccer:

Soccer = {alex, casey, drew, hunter}

(It says the Set "Soccer" is made up of the elements alex, casey, drew
and hunter.)

And casey, drew and jade play Tennis:

Tennis = {casey, drew, jade}

We can put their names in two separate circles:

UNION
You can now list your friends that play Soccer OR Tennis.

This is called a "Union" of sets and has the special symbol ∪:

Soccer ∪ Tennis = {alex, casey, drew, hunter, jade}

Not everyone is in that set ... only your friends that play Soccer or Tennis (or both).

In other words, we combine the elements of the two sets.

We can show that in a "Venn Diagram":

A Venn Diagram is clever because it shows lots of information:

 Do you see that alex, casey, drew and hunter are in the "Soccer" set?
 And that casey, drew and jade are in the "Tennis" set?
 And here is the clever thing: casey and drew are in BOTH sets!

All that in one small diagram.

INTERSECTION
"Intersection" is when you must be in BOTH sets.

In our case that means they play both Soccer AND Tennis ... which is casey and drew.
The special symbol for Intersection is an upside down "U" like this: ∩
And this is how we write it:

Soccer ∩ Tennis = {casey, drew}

In a Venn Diagram:

Which Way Does That "U" Go?

union symbol looks like cup
Think of them as "cups": ∪ holds more water than ∩, right?
So, Union ∪ is the one with more elements than Intersection ∩

DIFFERENCE

You can also "subtract" one set from another.

For example, taking Soccer and subtracting Tennis means people that play Soccer but NOT Tennis ...
which is alex and hunter.

And this is how we write it:

Soccer − Tennis = {alex, hunter}
In a Venn Diagram:
Summary So Far
 ∪ is Union: is in either set or both sets
 ∩ is Intersection: only in both sets
 - is Difference: in one set but not the other

THREE SETS
You can also use Venn Diagrams for 3 sets.

Let us say the third set is "Volleyball", which drew, glen and jade play:

Volleyball = {drew, glen, jade}

But let's be more "mathematical" and use a Capital Letter for each set:

 S means the set of Soccer players

 T means the set of Tennis players
 V means the set of Volleyball players

The Venn Diagram is now like this:

You can see (for example) that:

 drew plays Soccer, Tennis and Volleyball

 jade plays Tennis and Volleyball
  alex and hunter play Soccer, but don't play Tennis or Volleyball
 no-one plays only Tennis

We can now have some fun with Unions and Intersections ...

And how about this ...

 take the previous set S ∩ V
 then subtract T:

Hey, there is nothing there!

That is OK, it is just the "Empty Set". It is still a set, so we use the curly brackets with nothing inside: {}
The EMPTY SET has no elements: {}

UNIVERSAL SET
The Universal Set is the set that has everything. Well, not exactly everything. Everything that we are
interested in now.
Sadly, the symbol is the letter "U" ... which is easy to confuse with the ∪ for Union. You just have to be
careful, OK?
In our case the Universal Set is our Ten Best Friends.

U = {alex, blair, casey, drew, erin, francis,

glen, hunter, ira, jade}
We can show the Universal Set in a Venn Diagram by putting a
box around the whole thing:

Now you can see ALL your ten best friends, neatly sorted into
what sport they play (or not!).
And then we can do interesting things like take the whole set
and subtract the ones who play Soccer:

We write it this way:

U − S = {blair, erin, francis, glen, ira, jade}

Which says "The Universal Set minus the Soccer Set is the Set {blair, erin, francis, glen, ira, jade}"

In other words, "everyone who does not play Soccer".

COMPLEMENT
And there is a special way of saying "everything that is not",
and it is called "complement".

We show it by writing a little "C" like this: Sc

Which means "everything that is NOT in S", like this:

Summary
 ∪ is Union: is in either set or both sets
 ∩ is Intersection: only in both sets
 - is Difference: in one set but not the other
 Ac is the Complement of A: everything that is not in A
 Empty Set: the set with no elements. Shown by {}
 Universal Set: all things we are interested in

WRITTEN WORK GE4-MMW-2.1.1

Instruction:
Evaluate the set and Venn Diagram Below. Write your solution on a bondpaper and BOX
your final answer.
1. If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∪ B?
2. If A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, what is A ∩ B?
3. If X = {a, e, i, o, u} and Y = {a, b, c, d, e}, then what is Y - X ?

4.
From the above Venn diagram, what is the set U - T?
5. The Universal Set = { x ∈Z | -4 ≤ x < 4} and A = {0}.What is the complement of
A?

6. If
P = The set of whole numbers less than 5
Q = The set of even numbers greater than 3 but less than 9
R = The set of factors of 6
Then what is (P ∩ Q) ∪ (Q ∩ R)?
7. If
P = The set of whole numbers less than 5
Q = The set of even numbers greater than 3 but less than 9
R = The set of factors of 6
Then what is (P ∪ Q) ∩ (Q ∪ R)?

8.

From the above Venn diagram, what is the set (S ∩ T) ∪ V?

9. If
A is the set of factors of 15,
B is the set of prime numbers less than 10
C is the set of even numbers less than 9,
 then what is

10. If A, B, and C are any three sets, then which of the following is equal to

Hint: Use Venn diagrams to help you answer this question.

INFORMATION SHEET GE4-MMW-3.1.1

“Word Problems on Sets”

In this module we will continue the discussion in the previews lesson that focus in real life problem that
can be solve using sets.

INFORMATION SHEET GE4-MMW-3.1.1

“Word Problems on Sets”

Word problems aren’t just on school tests. You solve word problems every day in your work or
even while you’re just out and about. these steps make solving word problems easier than you think.

SIMPLE STEPS FOR SOLVING WORD PROBLEMS:

 Read the problem.

Begin by reading the problem carefully. Don’t jump to any conclusions about the answer
until you understand the problem.

 Identify and list the facts.

Look at all the information given in the story problem and make a list of what you know.

 Figure out exactly what the problem is asking for.

Know what you’re trying to find. The problem often states the required answer, but
sometimes you have to ferret it out from the information given. If an important fact isn’t
there, you can often convert some piece of the given information.

 Eliminate excess information.

The problem may include facts that don’t help you find the solution, so clear them out
of the way.

 Pay attention to units of measurement.

For example, if dimensions are given in inches, but the answer must be in square feet,
you need to convert units. You can use a table of conversions, an online calculator, or
your own memory to get the conversion factors.

 Draw a diagram.
Sometimes a diagram helps you visualize the problem (but not always).

 Find or develop a formula.

When you see the math that needs to be done, you probably know a common math
formula to use for the computation. You will find that you use some special formulas
again and again.
  Consult a reference.
If you’re stuck, look for a reference of some kind, such as a conversion chart or even a
blog where someone has encountered the same problem you’re having (although as
always, consider Internet sources carefully).

 Do the math and check your answer.

After you have an answer, be sure to test it. If the result is shockingly high or low, verify
whether you made a mistake.

DIFFERENT NOTATIONS IN SETS

What are the different notations in sets?

To learn about sets we shall use some accepted notations for the familiar sets of numbers.
Some of the different notations used in sets are:

∈ Belongs to
∉ Does not belongs to
: or | Such that
∅ Null set or empty set
n(A) Cardinal number of the set A
∪ Union of two sets
∩ Intersection of two sets
N Set of natural numbers = {1, 2, 3, ……}
W Set of whole numbers = {0, 1, 2, 3, ………}
I or Z Set of integers = {………, -2, -1, 0, 1, 2, ………}
Z+ Set of all positive integers
Q Set of all rational numbers
Q+ Set of all positive rational numbers
R Set of all real numbers
R+ Set of all positive real numbers
C Set of all complex numbers

These are the different notations in sets generally required while solving various types of
problems on sets.
Note:
 The pair of curly braces { } denotes a set. The elements of set are written inside a pair of
curly braces separated by commas.
 The set is always represented by a capital letter such as; A, B, C, …….. .
 If the elements of the sets are alphabets then these elements are written in small
letters.
 The elements of a set may be written in any order.
 The elements of a set must not be repeated.
 The Greek letter Epsilon ‘∈’ is used for the words ‘belongs to’, ‘is an element of’, etc.
 Therefore, x ∈ A will be read as ‘x belongs to set A’ or ‘x is an element of the set A'.
 The symbol ‘∉’ stands for ‘does not belongs to’ also for ‘is not an element of’.
 Therefore, x ∉ A will read as ‘x does not belongs to set A’ or ‘x is not an element of the
set A'.
WORD PROBLEMS ON SETS
 Word problems on sets are solved here to get the basic ideas how to use the properties of union
and intersection of sets.

Addition Theorem on Sets

Theorem 1 :
n(A ∪ B) = n(A) + n(B) - n(A ∩ B).

Solved basic word problems on sets:

1. Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩ B).

Solution:
Using the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
then n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

= 20 + 28 - 36

= 48 - 36

= 12

2. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B).

Solution:
Using the formula n(A∪B) = n(A - B) + n(A ∩ B) + n(B - A)

70 = 18 + 25 + n(B - A)
70 = 43 + n(B - A)

n(B - A) = 70 - 43
n(B - A) = 27

Now n(B) = n(A ∩ B) + n(B - A)

= 25 + 27
= 52

Different types on word problems on sets:

3. In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at least
one of the two drinks. How many like both coffee and tea?

Solution:

Let A = Set of people who like cold drinks.

B = Set of people who like hot drinks.

Given

(A ∪ B) = 60 n(A) = 27 n(B) = 42 then;

n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

= 27 + 42 - 60
= 69 - 60 = 9
=9

Therefore, 9 people like both tea and coffee.

4. There are 35 students in art class and 57 students in dance class. Find the number of students
who are either in art class or in dance class.

• When two classes meet at different hours and 12 students are enrolled in both activities.

• When two classes meet at the same hour.

Solution:
n(A) = 35, n(B) = 57, n(A ∩ B) = 12

(Let A be the set of students in art class.

B be the set of students in dance class.)

When 2 classes meet at different hours n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

= 35 + 57 - 12
= 92 - 12
= 80
When two classes meet at the same hour, A∩B = ∅ n (A ∪ B) = n(A) + n(B) - n(A ∩ B)
= n(A) + n(B)
= 35 + 57
= 92

Further concept to solve word problems on sets:

5. In a group of 100 persons, 72 people can speak English and 43 can speak French. How many can
speak English only? How many can speak French only and how many can speak both English and
French?

Solution:

Let A be the set of people who speak English.

B be the set of people who speak French.
A - B be the set of people who speak English and not French.
B - A be the set of people who speak French and not English.
A ∩ B be the set of people who speak both French and English.

Given,
n(A) = 72 n(B) = 43 n(A ∪ B) = 100
Now, n(A ∩ B) = n(A) + n(B) - n(A ∪ B)

= 72 + 43 - 100
= 115 - 100
= 15

Therefore, Number of persons who speak both French and English = 15

n(A) = n(A - B) + n(A ∩ B)

⇒ n(A - B) = n(A) - n(A ∩ B)
= 72 - 15
= 57

and n(B - A) = n(B) - n(A ∩ B)

= 43 - 15
= 28

Therefore, Number of people speaking English >

Number of people speaking French >

WORD PROBLEMS ON SETS with VENN DIAGRAM

In a survey of university students, 64 had taken mathematics course, 94 had taken chemistry
course, 58 had taken physics course, 28 had taken mathematics and physics, 26 had taken mathematics
and chemistry, 22 had taken chemistry and physics course, and 14 had taken all the three courses. Find
how many had taken one course only.

Solution :

Step 1 :Let M, C, P represent sets of students who had taken mathematics, chemistry and
physics respectively

Step 2 : From the given information, we have

n(M) = 64, n(C) = 94, n(P) = 58,

n(M∩P) = 28, n(M∩C) = 26, n(C∩P) = 22
n(M∩C∩P) = 14

Step 3 : From the basic stuff, we have No. of students who had taken only Math

= n(M) - [n(M∩P) + n(M∩C) - n(M∩C∩P)]

= 64 - [28 + 26 - 14]
= 64 - 40
= 24

Step 4 : No. of students who had taken only Chemistry:

 = n(C) - [n(M∩C) + n(C∩P) - n(M∩C∩P)]
= 94 - [26+22-14]
= 94 - 34
= 60

Step 5 : No. of students who had taken only Physics :

= n(P) - [n(M∩P) + n(C∩P) - n(M∩C∩P)]

= 58 - [28 + 22 - 14]
= 58 - 36
= 22

Step 6 : Total no. of students who had taken only one course :

= 24 + 60 + 22
= 106
Hence, the total number of students who had taken only one course is 106.

ALTERNATIVE METHOD (USING VENN DIAGRAM)

SAME PROBLEM ABOVE

Step 1 : Venn diagram related to the information given in the question:

Step 2 : From the venn diagram above, we have

 No. of students who had taken only math = 24

 No. of students who had taken only chemistry = 60
 No. of students who had taken only physics = 22

Step 3 : Total no. of students who had taken only one course :

= 24 + 60 + 22
= 106

Hence, the total number of students who had taken only one course is 106.
Summary

SIMPLE STEPS FOR SOLVING WORD PROBLEMS:

 Read the problem.

 understand the problem.
 Identify and list the facts.
 Figure out exactly what the problem is asking for.
 Eliminate excess information.
 Pay attention to units of measurement.
 Draw a diagram.
 Find or develop a formula.
 Consult a reference.

WRITTEN WORK GE4-MMW-3.1.1

Instruction:
Evaluate the problem using ALTERNATIVE METHOD (USING VENN DIAGRAM) Below. Write your
solution on a bondpaper and BOX your final answer.

1. Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩
B).
2. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B).
3. In a group of 60 people, 27 like cold drinks and 42 like hot drinks and each person likes at
least one of the two drinks. How many like both coffee and tea?

PRECAUTIONS:
 Do not just copy all your output from the internet.
 Use citation and credit to the owner if necessary.
ASSESSMENT METHOD: WRITTEN WORK CRITERIA CHECKLIST

INFORMATION SHEET GE4-MMW-4.1.1

“Languages of Functions and Relations”

An ordered pair is a set of inputs and outputs and represents a relationship between the two values.
A relation is a set of inputs and outputs, and a function is a relation with one output for each input

INFORMATION SHEET GE4-MMW-4.1.1

“Languages of Functions and Relations”

WHAT IS FUNCTION
Some relationships make sense and others don’t. Functions are relationships that make sense. All
functions are relations, but not all relations are functions.

It is like a machine that has an input and an output.

And the output is related somehow to the input.
 "f(x) = ... " is the classic way of writing a function.
And there are other ways, as you will see!

INPUT, RELATIONSHIP, OUTPUT

We will see many ways to think about functions, but there are always three main parts:

 The input
 The relationship
 The output

Some Examples of Functions

 x2 (squaring) is a function
 x3+1 is also a function
 Sine, Cosine and Tangent are functions used in trigonometry
 and there are lots more!

NAMES
First, it is useful to give a
function a name.
The most common name is "f",
but we can have other names
like "g" ... or even
"marmalade" if we want.
But let's use "f":

what goes into the function is put inside parentheses () after the name of the function:
So f(x) shows us the function is called "f", and "x" goes in

And we usually see what a function does with the input:

f(x) = x2 shows us that function "f" takes "x" and squares it.
The "x" is Just a Place-Holder!
Don't get too concerned about "x",
it is just there to show us where
the input goes and what happens
to it.
It could be anything!

Sometimes There is No Function

Name

Sometimes a function has no name, and we see something like:

y = x2

But there is still:

 an input (x)
 a relationship (squaring)
 and an output (y)

RELATING
we said that a function was like a machine. But a function doesn't really have belts or cogs or
any moving parts - and it doesn't actually destroy what we put into it!
A function relates an input to an output.
Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16
What Types of Things Do Functions Process?
"Numbers" seems an obvious answer, but ...
... which numbers?

For example, the tree-height function h(age) = age×20 makes no sense for an age less than zero.
... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So, we need something more powerful, and that is where SETS come in:

A set is a collection of things.

Here are some examples:

 Set of even numbers: {..., -4, -2, 0, 2, 4, ...}

 Set of clothes: {"hat","shirt",...}
 Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}
 Positive multiples of 3 that are less than 10: {3, 6, 9}

Each individual thing in the set (such as "4" or "hat") is called a member, or element.
So, a function takes elements of a set, and gives back elements of a set.

A FUNCTION IS SPECIAL
But a function has special rules:
 It must work for every possible input value
 And it has only one relationship for each input value
This can be said in one definition:

The Two Important Things!

1.
"...each element..." means that every element 2.
in X is related to some element in Y.
"...exactly one..." means that a function is single
We say that the function covers X (relates every valued. It will not give back 2 or more results
element of it). for the same input.

(But some elements of Y might not be related to So "f(2) = 7 or 9" is not right!
at all, which is fine.)
 When a relationship does not follow those two rules then it is not a function ... it is still a
relationship, just not a function.

Vertical Line Test

On a graph, the idea of single valued means
that no vertical line ever crosses more than one
value.
If it crosses more than
once it is still a valid
curve, but is not a
function.

Infinitely Many
My examples have just a few values, but
functions usually work on sets with
infinitely many elements.

Domain, Codomain and Range

In our examples

 the set "X" is called the Domain,

  the set "Y" is called the Codomain, and
 the set of elements that get pointed to in Y (the actual values produced by the function) is
called the Range.

Functions have been used in mathematics for a very long time, and lots of different names and ways of
writing functions have come about.

Here are some common terms you should get familiar with:
Example: z = 2u3:
 "u" could be called the "independent variable"
 "z" could be called the "dependent variable" (it depends on the value of u)
Example: f(4) = 16:
 "4" could be called the "argument"
 "16" could be called the "value of the function"
Example: h(year) = 20 × year:

 h() is the function

 "year" could be called the
"argument", or the
"variable"
 a fixed value like "20" can
be called a parameter

Another example:
1. Write an equation to represent the function from the following table of values:
ANSWER: The y value is always 2 times the x value. So, the equation is y = 2x.

2. Which one of the following relations is not a function?

ANSWER: In B the numbers 1 and 2 are both related to more than one number, so
this cannot be a function.

3. A = {-3, -2, -1, 0, 1, 2, 3}

f is a function from A to the set of whole numbers as defined in the
following table: What is the domain of f?

ANSWER:
The domain is the set of values of x = A = {-3, -
2, -1, 0, 1, 2, 3}.

4. Which one of these graphs does not illustrate a function?

 ANSWER: In C the points on the vertical line all have the same x coordinate, so one value of x is
related to more than one value of y.
C does not illustrate a function.

5. The function f is defined on the real numbers by f(x) = 2 + x − x2: What is the value of f(-3)?
ANSWER:
f(x) = 2 + x − x2
So f(-3) = 2 + (-3) − (-3)2 = 2 − 3 - (+9) = -10

6. g(n) = 3n + 2
f(n) = 2n2+ 5
Find g( f(2))
ANSWER:
f(2)= 2(2)2+5
=2(4)+5
=8+5
=13

g(13)= 3(13) + 2
= 39 + 2
= 41

Summary:
 a function relates inputs to outputs
 a function takes elements from a set (the domain) and relates them to elements in a set (the
codomain).
 all the outputs (the actual values related to) are together called the range
 a function is a special type of relation
 every element in the domain is included, and
 any input produces only one output (not this or that)

WRITTEN WORK GE4-MMW-4.1.1

Instruction:
Perform the indicated operation.
Write your solution on a bondpaper and BOX your final answer.

1. The function f is defined on the real numbers by f(x) = 2 + x − x2: What is the value of f(-3)?
2. The function f is defined on the real numbers by f(x) = 2 + x − x2: What is the value of f(2)?
3. The function f is defined on the real numbers by f(x) = 2 + x2 − x2: What is the value of f(3)?
 4. The function f is defined on the real numbers by f(x) = 2x − x: What is the value of f(5)?
5. The function f is defined on the real numbers by f(x) = x3 + 3: What is the value of f(2)?

INFORMATION SHEET GE4-MMW-5.1.1

“Function and Relation: Applications in Real Life Problem”

In this module we will continue the discussion in the previews lesson that focus in real life problem
that can be solve using functions and relations

INFORMATION SHEET GE4-MMW-5.1.1

“Function and Relation: Applications in Real Life Problem”

TRANSLATING KEY WORDS AND PHRASES INTO ALGEBRAIC EXPRESSIONS

The table below lists some key words and phrases that are used to describe common mathematical
operations. To write algebraic expressions and equations, assign a variable to represent the unknown
number. In the table below, the letter “x” is used to represent the unknown. In translation problems,
the words sum, total, difference, product and quotient imply at least two parts – use parentheses when
a sum or difference is multiplied. For example, the phrase "the sum of three times a number and five"
translates to "3x + 5," while the phrase "three times the sum of a number and five" translates to "3(x +
5)."
 Variable and Verbal Expressions

Write each as an algebraic expression.

Write each as a verbal expression.

Evaluate each expression.

Strategies in solving problems:
I. Translate the problem into mathematical models by visualizing the situation given.
II. Write the equation by assigning a variable for the unknown.
III. Substitute the given value, if necessary.
IV. Solve the problem through algebraic method

Example 1:

Jane: What a nice Watch! It’s so pretty.

Mary: you like it? I bought it When I Was still in Manila.
Jane: Really? how much was it?
Mary: if you multiply the price by 4, and to the product add 70, and from it sum subtract 50, then equal
to 220 dollars.
Given:
 Let the price of the Watch be represented by x; x
 The price is to be multiplied by 4; 4x
 To the product, 70 is to be added; 4x + 70
 From this, 50 is to be subtracted; 4x + 70 - 50
 The preceding terms are said to be equal to 220; 4x + 70 – 50 = 220

Solution:
4x+70-50 = 220
4x=220+50-70
4x/4=200/4
X=50

Therefore, the cost of the Watch is 50 dollars.

Example 2:

If a farmer Want to cultivate a farm field owned by NDD Foods Inc. at Pampanga on time, he
must cultivate 120 hectares a day. For technical reasoning he cultivated only 85 hectares a day, hence
he had to cultivate 2 more days than he planned and he still has 40 hectares left.

a. What is the area of the farm field and;

b. how many days the farmer planned to Work initially?

Given:
 X = no. of days in the initial plan
 120x = hectare of the Whole field
 85(x+2) = cultivated hectares
 40 = hectares uncultivated

Solution:

120x=85(x+2)+40
120x-85x=170+40
35x/35 = 210/35
X=6

Therefore, the farmer planned to have the Work done in 6 days, and the area of the farm field
is 120x60 = 720 hectares.

Example 3:

The time it takes a mother to make dinner is represented by the function t(x) = 20 + 2x, where x
is the number of people at the dinner and t(x) is the number of minutes to make dinner.

Find the time it takes a mother to make dinner if 8 people come to dinner.

Given:
 x: Number of People at the dinner
 t(x): number of minutes to make dinner
 t(x) = 20 + 2x: time it takes a mother to make dinner

Solution:
x = 8 people come to dinner
t(x) = 20 + 2x
t(8) = 20 + 2(8)
t(8) = 20 + 16
t(8) = 36

Therefore, mother can make dinner for 8 people in just 36 minutes.

Example 4:
two planes, which are 2400 miles apart, fly toward each other. Their speed differs 60 miles per
hour. They pass each other after 5 hours.
a. Find their speed.

Given:
Plane Time x Speed Distance
st
1 5 hours x 5x
2nd 5 hours x+60 5(x+60)
 Distance to two planes: 2400 miles apart
Solution:

5x + 5(x+60)=2400
5x + 5x + 300 = 2400
10x = 2400 – 300
10x = 2100
10x = 2100/10
X = 210 mph

Therefore, the first plane’s speed is 210mph

The second plane’s speed is 210 + 60 =270 mph

Summary:
a word problem is a mathematical exercise where significant background information on the
problem is presented in ordinary language rather than in mathematical notation.

INFORMATION SHEET GE4-MMW-6.1.1

“Linear Functions”

The real questions are not how we got the data, but how do we interpret this data. Everyone should
easily spot that the Quantity for Supply and Demand are the same at a price of P1.40 per gallon. Why is
this important and what would happen if we set our price lower or higher than this amount?

Price (per gallon) Quantity demanded (millions of gallons) Quantity supplied (millions of gallons)

P1.00 800 500

P1.20 700 550

P1.40 600 600

P1.60 550 640

P1.80 500 680

P2.00 460 700

P2.20 420 720

INFORMATION SHEET GE4-MMW-6.1.1

“Linear Functions”

Learning Outcomes

 Given sufficient information about a line, find the slope and write the equation of a line
 Given an equation of line, identify the slope, find the intercepts, and graph the line
 Find the equation of a line approximating data using technology to compute a linear regression.
 Model cost, revenue, profit, supply, and demand using linear functions.
 Find the intersection of lines, and use that to determine the break-even point for cost/revenue
functions, and equilibrium price for supply/demand functions.
 Solve linear inequalities and show the solution graphically and symbolically

Linear Functions

As you hop into a taxicab in Makati, the meter

will immediately read P3.30; this is the “drop” charge
made when the taximeter is activated. After that initial
fee, the taximeter will add P2.40 for each mile the taxi
drives.

In this scenario, the total taxi fare depends

upon the number of miles ridden in the taxi, and we
can ask whether it is possible to model this type of
scenario with a function. Using descriptive variables,
we choose m for miles and C for Cost in Peso as a
function of miles: C(m).

We know for certain that C(0) = 3.30, since the P3.30 drop charge is assessed regardless of how
many miles are driven. Since P2.40 is added for each mile driven, then

C(1) = 3.30 + 2.40 = 5.70

If we then drove a second mile, another P2.40 would be added to the cost:

C(2) = 3.30 + 2.40 + 2.40 = 3.30 + 2.40(2) = 10.50

If we drove a third mile, another P2.40 would be added to the cost:

C(3) = 3.30 + 2.40 + 2.40 + 2.40 = 3.30 + 2.40(3) = 10.50

From this we might observe the pattern, and conclude that if

m miles are driven, because we start with a P3.30 drop fee and then for each mile increase, we add
P2.40.

It is good to verify that the units make sense in this equation. The P3.30 drop charge is measured in
Peso; the P2.40 charge is measured in Peso per mile.
When Peso per mile are multiplied by a number of miles, the result is a number of Peso, matching the
units on the 3.30, and matching the desired units for the C function.

Notice this equation C(m) = 3.30 + 2.40m consisted of two quantities. The first is the fixed P3.30 charge
which does not change based on the value of the input. The second is the P2.40 Peso per mile value,
which is a rate of change. In the equation this rate of change is multiplied by the input value.

Looking at this same problem in table format we can also see the cost changes by P2.40 for every 1-mile
increase.

m 0 1 2

C(m) 3.30 5.70 8.10

It is important here to note that in this equation, the rate of change is constant; over any
interval, the rate of change is the same.

Graphing this equation, we see the shape is a line, which is how

these functions get their name: linear functions

When the number of miles is zero the cost is P3.30, giving the
point (0, 3.30) on the graph. This is the vertical or C(m) intercept. The
graph is increasing in a straight line from left to right because for each
mile the cost goes up by P2.40; this rate remains consistent.

In this example you have seen the taxicab cost modeled in words, an equation, a table and in
graphical form. Whenever possible, ensure that you can link these four representations together to
continually build your skills. It is important to note that you will not always be able to find all 4
representations for a problem and so being able to work with all 4 forms is very important.

LINEAR FUNCTION

A linear function is a function whose graph produces a line. Linear functions can always be
written in the form f(x) = b + mx or f(x) = mx + b; they’re equivalent, where b is the initial or starting
value of the function (when input, x = 0), and m is the constant rate of change of the function.

Many people like to write linear functions in the form f(x) = b + mx because it corresponds to the way
we tend to speak: “The output starts at b and increases at a rate of m.”

For this reason alone, we will use the form f(x) = b + mx for many of the examples, but
remember they are equivalent and can be written correctly both ways.

SLOPE AND INCREASING/DECREASING

m is the constant rate of change of the function (also called slope). The slope determines if the function
is an increasing function or a decreasing function.

f(x) = b + mx is an increasing function if m > 0

f(x) = b + mx is a decreasing function if m < 0

If m = 0, the rate of change zero, and the function f(x) = b + 0x = b is just a horizontal line passing
through the point (0, b), neither increasing nor decreasing.

Example 1
Marcus currently owns 200 songs in his iTunes collection. Every month, he adds 15 new songs.
Write a formula for the number of songs, N, in his iTunes collection as a function of the number of
months, m. How many songs will he own in a year?

The initial value for this function is 200, since he currently owns 200 songs, so N(0) = 200. The
number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. With
this information, we can write the formula:

N(m) = 200 + 15m

N(m) is an increasing linear function.

With this formula we can predict how many songs he will have in 1 year (12 months):

N(12) = 200 + 15(12) = 200 + 180 = 380. Marcus will have 380 songs in 12 months.

Example 2
If you earn P30,000 per year and you spend P29,000 per year write an equation for the amount
of money you save after x years, if you start with nothing.

“The most important thing, spend less than you earn!”

y=30,000x-29000x

Calculating Rate of Change

Given two values for the input, x1, and x2 two corresponding values for the output, y1, or y2 a set of
points, (x1, y1) and, (x2, y2) if we wish to find a linear function that contains both points we can calculate
the rate of change,

Rate of change of a linear function is also called the slope of the line.

Slope of a line is a ratio of the rise over run. Where rise is the difference of the y values and run is the
difference of the x values.

Note in function notation, y1 = f(x1) and y2 = f(x2), so we could equivalently write

Example 1
 The population of a city increased from 23,400 to 27,800 between 2002 and 2006. Find the rate
of change of the population during this time span.

The rate of change will relate the change in population to the change in time. The population
increased by 27800 – 23400 = 4400 people over the 4-year time interval. To find the rate of change, the
number of people per year the population changed by:

Notice that we knew the population was increasing, so we would expect our value for m to be positive.
This is a quick way to check to see if your value is reasonable.

Example 2
The pressure, P, in pounds per square inch (PSI) on a diver depends upon their depth below the
water surface, d, in feet, following the equation P(d) = 14.696 + 0.434d. Interpret the components of this
function.

The rate of change, or slope, 0.434 would have units

This tells us the pressure on the diver increases by 0.434 PSI for each foot their depth increases.

The initial value, 14.696, will have the same units as the output, so this tells us that at a depth of 0 feet,
the pressure on the diver will be 14.696 PSI.

Example 3
If f(x) is a linear function, f(3) = –2, and f(8) = 1, find the rate of change.

f(3) = –2 tells us that the input 3 corresponds with the output –2, and f(8) = 1 tells us that the input 8
corresponds with the output 1. To find the rate of change, we divide the change in output by the change
in input:

If desired we could also write this as m = 0.6

Note that it is not important which pair of values comes first in the subtractions so long as the first
output value used corresponds with the first input value used.

Example 4
Given the two points (2, 3) and (0, 4), find the rate of change. Is this function increasing or decreasing?

Numerator: 4-3=1
Denominator: 0-2=-2
Rate of Change: 1/-2=-1/2 (decreasing)

We can now find the rate of change given two input-output pairs, and can write an equation for a linear
function once we have the rate of change and initial value. If we have two input-output pairs and they
do not include the initial value of the function, then we will have to solve for it.
Example 5
Write an equation for the linear function graphed to the right.

Looking at the graph, we might notice that it passes through the points
(0, 7) and (4, 4). From the first value, we know the initial value of the
function is b = 7, so in this case we will only need to calculate the rate
of change:

This allows us to write the equation:

Lesson activity: Try it on your own.

Given the table below write a linear equation that represents the table values

w, number of weeks 0 2 4 6

P(w), number of rats 1000 1080 1160 1240

Important Topics of This Lesson

 Definition of Modeling
 Definition of a linear function
 Structure of a linear function
 Increasing & Decreasing functions
 Finding the vertical intercept (0, b)
 Finding the slope/rate of change, m
 Interpreting linear functions

WRITTEN WORK GE4-MMW-6.1.1

Instruction:
Solve the problem below:
Note: Write your solution on a bondpaper and write an explanation on how you get your final
answer.

1. The balance in your college payment account,

G, is a function of the number of quarters, x, you attend. Interpret the function G(x) =
15000 – 5000x in words. How many quarters of college can you pay for until this
account is empty?
2. If you earn P40,000 per month and you spend P19,000 per month write an
equation for the amount of money you save after x months, if you start with
nothing.

Note for Modular Distance Learning – Printed: Write your Name, Course & Section,
Subject, Module number and Subject Teacher on your bondpaper.