Introduction and

international mindedness

Known as infinitesimal calculus in its early history of calculus, the Newton-Leibnez Formula or the fundamental theorem of calculus shows the relationship between the derivative and integral and was developed in the later 17th century independently by Gottfried Leibniz and Isaac Newton, used for evaluating definite integrals.

In brief, it states that any continuous function over an interval has an antiderivative on that interval. This particularly elegant theorem shows the inverse function relationship of the derivative and the integral function can be used to find the area under the given function. Values input into the Newton-Leibniz formula will give the area under the function in squared units.

Method of exhaustion (Toolkit)

https://drive.google.com/file/d/12C6d48t6oOQAWxRyniZBiH7bIgX0ck6T/view

On the link above will contain a presentation and discussion regarding the method of exhaustion applied in one of the ways in finding the area under the curve, established by Eudoxus. This method is done by inscribing polygons (triangles) between the area of region bounded by a parabola and a straight line.

real-life applications

Throughout the investigation comes to some real-life applications of different concepts used for the method of exhaustion.

Engineering

Calculus could be use for architecture of buildings and bridges, and the optimization in the length of electrical cables in electrical engineering. It can also be used in kinematics where the calculation of speed, direction, and direction are essential such as flight engineering.

Sciences and research

Biology (Medical Science) – Differential calculus used to determine the exact exponential rate of growth in bacterial culture when different variables are applied.

Physics – Use of integration to calculate the Centre of Mass, Centre of Gravity and Mass Moment of Inertia of a sports utility vehicle. Also, calculating velocity and trajectory of an object to be applied in astronomy and electromagnetism.

Chemistry – Determining rate of reaction and information of Radioactive decay reaction.

Statistics and research analysis in businesses

Research analyst and statisticians makes use of calculus to consider the different variables present in businesses and corporations to increase profits by increase efficiency in its processes. Statistics can also be modelled to make best predictions suitable for the company.

IB Learner profiles

Courageous

We are courageous who challenge ourselves into exploring new ideas that leads to investigating theories that are new to us which is the Method of Exhaustion by Eudoxus and also other mathematical theorems that are involved.

INquirers

We nurture our curiosity, developing skills into inquire and research on the investigation. This curiosity is what makes it possible to develop new knowledge and exploring further deep into the topic.

Communicator

We express ourselves confidently and creatively. We collaborate effectively, and be good-listeners to have a good communication in the group that allow us to overcome problems faced in the toolkit. 

Caring

We are empathetic people. We are all committed into investigating this toolkit while generating positive aura so that all members can be motivated and exert all of our capabilities to do a job well done.

knowledgeable

During research, we develop and use conceptual understanding on mathematical concepts and theorems across its range of applications. Engaging on these newly found ideas benefits us into being more knowledgeable in mathematics.

mathematical quote and brain teasers

Introduction

JC1 is coming to an end and this will be the last entry that I will publish in JC1, sed 😦

My journey throughout first year of IB MAAHL has been eventful, fun, stressed, and hardcore 😉 IB have thought me more ways to think and view problems than my previous years of school. Additionally, it also added some wise and extra knowledge on different formulas and mathematical concept, international mindedness, as part of the education. This increases our knowledge and awareness on where and when different mathematical concepts we are using today are invented or discovered. Whereas, a non-ib program does not cause this kind of exploration on the arise of different mathematical concepts, I considered myself lucky, or not…

Coming into my first class of MAAHL was very confusing, as IB set us up in a new math syllabus. It makes us wonder throughout weeks of Term 1, its makes me uncertain as we don’t know what to do yet, so we just study whatever we can. Not long after, mathematics have become a very challenging subject as you often need to critically think when you solve a question.

GROUP PROJECT IN MAAHL

But during the journey, things have become eventful as we started doing our very first video-making group project:

Not long after comes our second project, collaboration with the JC2s, our very first time that we hosted a successful mathematics olympiad event, SISMO.

This huge project will be our biggest step on improving the mathematical department in school. Me being involved in the project make me feel very honored. It ended successfully and this can happened thanks to the perfection of our captains and head of the department for making this possible and a reality. I would also like to credit our seniors for creating this idea in the first place, but sadly not having enough time for them to be involved physically in the day of the olympiad.

We all hope that hosting our 2nd year of SISMO next academic year will be possible with many more improvements in the future to make it more fun and challenging for the competitors!

Look at Mr Kichan’s big smile, makes me so excited. I’m afraid that talking about more pass events will be very emotional, so I’ll stop now. Instead, more of this kinds of memories should be made later on during JC2! I look forward to it very much, not the studies in class but the jokes made! hehe (Calculus)

TOK in maths

Is mathematics discovered or invented?

According to Pythagorians around the 5th century BC, numbers are living entities and universal principles.

Plato argued mathematical concepts were concrete and as real as the universe itself, regardless of our knowledge of them.

Euclid, the Father of Geometry, believed nature itself was the physical manifestation of mathematical laws.

While other mathematicians argues on different opinions among themselves, what’s certain is that mathematical statements don’t exist as they are based on rules that are created by humans, axioms. These axioms created are eligible to an extent in solving further mathematical processes, but we don’t if it is entirely true. The existence of many representatives, symbols, and formulas can be said as false, as mathematicians created this symbols to easily represent meaning when doing mathematics during old times, which is still used by people now.

Questions involving the invention or discoveries of mathematics is very difficult to determine. Debates and arguments between millions of past mathematicians is so deep that results in no conclusion up until the present. The question will remain unanswered and will have no specific answer on the concepts of mathematics.

For more intriguing topics on mathematics is shown below:

International Mindedness

Visual Approach

The Mandelbrot set is generated by what is called iteration, which means to repeat a process over and over again. In mathematics this process is most often the application of a mathematical function. 

The theory of iterated functions is motivated by questions of real life. Modelling the growth of population is an example, and relating it with another variable is possible as well.

Analytical approach

The Bourbaki group was found in Paris, France in 1934 and they are a group of mathematicians originally intended on preparing a new textbook (Elements of Mathematics) about mathematical analysis as their central work. The group tackle mathematics in an analytical approach.

They later on became more ambitious, publishing more textbooks related to mathematics. These books became useful, influencing many people on the knowledge given by the Bourbaki group. Many elements including symbols arise from the past studies of the Bourbaki group and are used until today, like the null set symbol. Books published are also discussed further among them beforehand about abstract algebra, topology, and analysis.

IB Learners Profile

inquirers

We nurture our curiosity, developing skills for inquiry and research. We know how to learn independently and with others. During class discussion with the teacher and even at home where we inquire from our teachers and friends while doing research so we can receive valuable information that may be useful to us.

Knowledgeable

We develop and use conceptual understanding, exploring knowledge across a range of topics and mathematical concepts when we learn in class. We engage with issues and ideas that have important significance.

Thinkers

We use critical and creative thinking skills to analyse and take action on complex problems. This will be my favourite as solving math problems aren’t easy. Literally all of the questions must involve my brain stressing out in order for me to solve the problem correctly.

communicators

We express ourselves confidently and creatively in more than one language and in many ways. We collaborate effectively, listening carefully to the perspectives of other individuals during class discussion and lectures.

principled

We act with integrity and honesty, with a strong sense of fairness and justice, and with respect for the dignity and rights of people everywhere. We take responsibility for our actions and their
consequences. This can refer to SISMO when problems arise in the event. We took care of it responsibly and professionally so that people will view us highly when they compare with other hosted events similar to us.

Open-minded

We seek and evaluate a range of points of view, and we are willing to grow from the experience. Starting in class, we learned to be open-minded to in solving problems as each of us will have different answers. We also evaluate after SISMO so that we can learn from it and grow from the experience that day.

caring

We show empathy, compassion and respect. We have a commitment to service, and we act to make a positive difference in the lives of others and in the world around us. Being part of JC1 MAAHL Fam this past year allow me to grow as an individual to care for each and every one of us so we can also grow together as a group emotionally and academically.

risk-takers

We approach uncertainty with forethought and determination; we work independently and cooperatively to explore new ideas and innovative strategies. We are resourceful and resilient in the face of challenges and change. This will be the very first term in the learner profile I absorbed fully into my mind. Throughout the year, we have been taking a lot of risks to get a better opportunity for us.

balanced

We understand the importance of balancing different aspects of our lives—intellectual, physical, and emotional—to achieve well-being for ourselves and others. We recognize our interdependence with other people.

reflective

One of the important learner’s profile as well is to be reflective and reflect upon the events happened before. This can lead us to grow as an individual and as a group in many different ways so that different and better actions is done to similar problems that will lead to a smaller mistakes and impact, to prevent chaos and misunderstanding.

“We will always have STEM with us. Some things will drop out of the public eye and go away, but there will always be science, engineering, and technology. And there will always, always be mathematics.”

— Katherine Johnson, African-American mathematician

Preparation Day

As the head of liaisons, Aryo and I are in charge of the contacts and keeping in touch with different schools so that the Person in Charge in each school will receive updates and understand what’s going on before the day of the competition. We are to prevent any confusion and misunderstandings as much as possible. I’m also involved in making the test papers together after they were printed, double checking if there are any mistakes in the orientation of pages of the test papers.

On the day before the olympiad, we gather at 5 p.m. to prepare the MPH, awards, name tags, and the test papers for the next day. All of the awards, name tags, and test papers are prepared in the community room. Another set of tags are also taped on top of the table so that participants will know where to sit the next day. The registration area is also set in front of the MPH.

reflection

When we contacted all schools, hoping that all of them will join. Some teacher’s contact are very difficult to get as some are only willing to give their school’s emails. Some of the teacher’s contacts are also difficult to reach as they are not responsive to our texts. Therefore, some schools may not be aware of the presence of this olympiad due to these unresponsive texts of the teachers.

SISMO DAY

TRaffic control:

As early as 8 in the morning, students started coming into the school, registering in front of the MPH. 3 tables are placed in the registration area for each category so I will have to maintain traffic control in the registration area for keeping them in 3 lanes for each category to prevent it to become chaotic.

Invigilating:

During examination hours, I took part in invigilating participants taking the exam in the MPH. As a team, we distribute the examination papers, roam around the MPH to keep an eye and prevent participants from cheating.

Paper checking:

After each round, papers will be collected and committees gather to check all of the examination papers done by the participants. Answers are projected on a screen from the projector so all of us can see while checking.

reflection:

During traffic control, I have no problem in managing participants to be in line for registration. The participants listened very well, preventing the situation to be chaotic. The only problem I can think of during registration is when some of the tags broke so I need someone to go into the Community Room and fix it.

While invigilating, committees walk around the MPH to invigilate. With the help of teachers, we are able to cover all sides and corners of the MPH while some are roaming around to be aware if some participants are cheating. I think that we as committees should also be aware of our actions when we talk to each other or opening our phones around the MPH. Sorry Sir… 🙂

I think that paper checking was the most efficient way of checking them as it was very fast. We can maybe improve in the future if it were to be more organized rather than letting all papers get messy all over the floor.

Aside from that, I also took part in the ice breaker “24 card game” and also helped in teaching several groups on how to play the game. Improvements can be done if more cards and more people who can teach are present so that groups don’t have to wait that long when some other groups are taught by the committees. But as this year being our first official math olympiad, it was an overall success and a good foundation for more improvements and evolution for the olympiad next year.

INternational mindedness

complex numbers

The 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.

Complex numbers can be expressed in 3 forms: Cartesian form, Polar form and Exponential form.

Calculus

Isaac Newton and Gottfried Leibniz independently invented calculus in the mid-17th century. Calculus compose of a mathematical discipline focus on limits, functions, derivatives, integrals, and infinite series. With these theories invented by these mathematicians, calculus are used to a lot of real-life applications.

IB learner profiles

communicators

All of the committees communicate with each other as a team to make the least amount of mistakes as possible. We collaborate effectively, listening carefully to the perspectives of other individuals and groups. Any problem arise by the participants or situation are solved quickly by the good chemistry of our team.

open-minded

We are all aware of any situation and mistakes done, remind each other on what should be done and what shouldn’t be done again. Others reminded are also open-minded and understanding so that no fights or commotion arise during the competition. We also seek and evaluate a range of points of view, and we are willing to grow from the experience.

reflective

When the event finally ended, all of us sit together as a group and reflect back upon the small and major events that happened during the olympiad. We work to understand our strengths and weaknesses in order to support our learning and personal development in the future. This will result improvements to our team for the future that’ll come.

resources

https://www.mytutor.co.uk/answers/10925/A-Level/Further-Mathematics/What-are-the-different-forms-of-complex-numbers-and-how-do-you-convert-between-them/

warning!

This topic may be more interesting than you thought!

What is it?

Theory of Knowledge (TOK) provides an opportunity for students to reflect on the nature of knowledge, and on how we know what we claim to know. It is a mandatory course for all students taking the IB Diploma Program, and this course will twist your mind upside down, back and forth 180° as you go along the 2-year program. Are the axioms in mathematics even true? Or does mathematics don’t even exist at all.

Theory of knowledge 1

What are the platonic solids and why are they an important part of the language of math?

PLatonic solids

Platonic solid, any of the five geometric solids whose faces are all identical , regular polygons meeting at the same three-dimensional angles. Tetrahedron (or triangular pyramid), cube, octahedron, dodecahedron, and icosahedron are all platonic solids, also known as five regular polyhedra.

The tetrahedron, cube, and dodecahedron first arise by Pythagoras, while the octahedron and icosahedron were first discussed by the Athenian mathematician Theaetetus. However, the entire group of regular polyhedra owes its popular name to the great Athenian philosopher Plato.

Cube

Dodecahedron

Icosahedron

Octahedron

Tetrahedron

According to Plato, he equated:

• The tetrahedron to the element ‘fire’ (the only element that gives heat, and is responsible for all transformative processes such as digestion).
• The cube to the element ‘earth’ (the most dense element, the stuff of rock and bone, giving solidity and form).
• The icosahedon to the element ‘water’ (the universal solvent and the ruling archetype for all fluids, including rivers and oceans of course, but also plasma and mucous).
• The octahedron to the element ‘air’ (the element of mobility, carrying ideas, inspiration and disease around the cosmos and our bodies).
• The dodecahedron that corresponds to ‘ether’, or ‘akash’, is commonly called “space,” and that is accurate insofar as it is everywhere around and within us. It is the vacuum; the matrix; the space through which we move; the three-dimensional field in which we arise, survive and transform. It is within the ethereal container that the other four elements dance and intertwine.

Theory of knowledge 2

To what extent do instinct and reason create knowledge?

In TOK, the use of instinct and reason are influenced by emotions, perception, culture, and experience on how people will make decisions in life and in every situation. For instance, people may act and make decisions differently when they act upon instinct or when explaining by using logical reasoning. An example for a real-life situation is when a stove catches on fire. You will instinctively splash water to the stove, while the right thing to do is to cover the fire with a wet cloth to put it out by using reasoning.

Emotions tell you how to immediately react to a situation, whether you are scared, nervous, or excited. This emotional reaction then tells you how to perceive a situation. Culture also plays a huge role in how your instincts cause you to react to a situation. Where you grow up and who you grow up with influence how you react. All three of these categories build up the experience of an individual. This is where instinct transitions into reasoned and logical reactions. The more experience you gain, the more likely you are to react out of reason and less out of instinct. This will let you be more knowledgeable about the particular situation, when you are more experienced that will lead you to use reasoning and instinct together to create the best possible outcome.

Do different geometries [Euclidean and non-Euclidean] refer to or describe different worlds?

Euclidean Geometry

Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid. This type of geometry are taught in schools before non-Euclidean geometry attracted the attention of mathematicians on the second half of the 19th century.

Euclid proposed five unprovable but intuitive principles known variously as postulates or axioms. The axioms are as follows:

• Given two points, there is a straight line that joins them.
• A straight line segment can be prolonged indefinitely.
• A circle can be constructed when a point for its centre and a distance for its radius are given.
• All right angles are congruent.
• If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.

Non-Euclidean Geometry

A non-Euclidean geometry is a rethinking and redescription of the properties of things like points, lines, and other shapes in a non-flat world. Anything that opposes Euclidean geometry.

The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it, whereas non-Euclidean geometry oppose it.

Theory of knowledge 3

Is it ethical that Pythagoras gave his name to a theorem that may not have been his own creation?

pythagoras theorem

The Pythagoras Theorem, named after the Greek mathematician Pythagoras states that: If we let c be the length of the hypotenuse (longest side of the right-angled triangle) and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: a² + b² = c².

“Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries. By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato’s and Aristotle’s mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view.”

By the statement shown above, Pythagoras’ achievements were unclear and inaccurate. Some of his past works remains controversial on some of Pythagoras’ engagements in certain philosophies and theorems. His past works are stated to be doubtful that he had done it by himself, and it is not likely that it is he that proved the theorem. Therefore, it may have not been ethical to be named the “Pythagoras Theorem” as it may be proven by other philosophers and mathematicians other than himself.

USeful Quote

“Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.” 

– Galileo Galilei (1564 – 1642)

Through my entire experience in JC 1, being part of the Math Analysis & Approaches HL class have helped me to grow over this semester. Through the days of being in Mr Kichan’s class, I am more open-minded and exposed into solving different kinds of complicated questions he gave us in class. In this semester, we are able to extend our knowledge about functions and complex numbers.

Below are further investigations of triangle sequences that will make your brain pop! The Koch’s Snowflake and the Sierpinski’s Triangle are further investigated to find out about the sequence after each iteration. Each investigation have their own special shape and features that are proven to be true just by playing around with triangles!

investigation A

What is the koch’s snowflake?

The Koch snowflake is a fractal curve, also known as the Koch island, which was first described by the mathematician Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely.

Perimeter of snowflakes at each iteration:

1st Iteration (Green Snowflake): 81 X 3 = 243 cm

2nd Iteration (Pink/Red Snowflake): 81/3 = 27 cm (length of one side of the small triangle)

27 X 12 sides = 324 cm

3rd Iteration (Blue Snowflake): 27/3 = 9 cm (length of one side of the smallest triangle)

9 X 48 sides = 432 cm

4th Iteration (Orange Snowflake): 9/3 = 3 cm (length of one side of the smallest triangle)

3 X 192 sides = 576 cm

Area of snowflakes at each iteration:

1st Iteration (Green Snowflake): 1/2(81)(81)sin(60) = 2840.996 = 2841 cm^2

As the area of the new equilateral triangle formed is 1/9 of the previous triangle, we can multiply 1/9 X n numbers of small triangles to add into the area of the previous iteration.

2nd Iteration (Pink/Red Snowflake): 2841/9 = 947/3 X 3 small triangles = 947 cm^2

2841 + 947 = 3788 cm^2

3rd Iteration (Blue Snowflake): 947/3 divided by 9 = 947/27 X 12 smallest triangles

= 3788/9 cm^2

= 3788 + 3788/9 = 37880/9 = 4208.888… = 4209 cm^2

4th Iteration (Orange Snowflake): 947/27 divided by 9 = 947/243 X 48 smallest triangles

= 45456/23 cm^2

= 4209 + 45456/23 = 142263/23 = 6185.347… = 6185 cm^2

Table 1

	No. of Sides	Perimeter/cm	Area/cm^2 (Rounded Off)
1st Iteration	3	243	2841
2nd Iteration	12	324	3788
3rd Iteration	48	432	4209
4th Iteration	192	576	6185

Number of Sides- The number of sides in each iteration increases by 4 times from the previous iteration.

Perimeter of the Snowflake- The perimeter of the snowflake is a geometrical sequence between each iteration with a common ratio of 4/3.

Area of the Snowflake- The area of the snowflake increases after each iterations. The area of each new set of triangles after each iteration is 1/9 of the previous area of the triangle, thus making the final area not a whole number.

From the results obtained above, we can deduce to a general formula in finding the the perimeter of the snowflake in any iteration. It is proven that the sequence is a geometrical sequence, 4/3 being its common ratio. We can find the value in any iteration from the geometrical formula, a_n = a₁(r^n – 1), a₁ = 243 and r (common ratio) = 4/3.

But no general formula can be obtained for the area of the snowflake as its values for each iteration is rounded off to a whole number. Which means that it is not a whole number, which may be difficult to form a sequence with these kind of values.

Investigation B

what is the sierpinski’s triangle?

The Sierpinski triangle, also called the Sierpinski gasket or Sierpinski sieve, is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles.

stage 3 of sierpinski’s triangle

Table 2

Stage	0	1	2	3
Number of Green Triangles	1	3	9	27
Length of one Side of one Green Triangle	1	0.5	0.25	0.125
Area of each Green Triangle	1	1/3	1/9	1/27

Number of Green Triangles- The number of green triangle in the sequence can be determined as 3^n, where n is in which stage in the sequence.

Length of one Side of one Green Triangle- The length of one side of one green triangle in the sequence can be determined as 1/2^n, where n is in which stage in the sequence.

Area of each Green Triangle- The area of each green triangle in the sequence can be determined as 1/3^n, where n is in which stage in the sequence.

If the table were to be extended further to stages 4, 5, 6. These values can be found by using the formulas above to find the number of green triangles, the length of one side of one green triangle, and the area of each of green triangle for all the stages you want to find.

From the sets of numbers obtained shown in table above, these numbers can be compared between the number of green triangles and both the length of one side and area of each green triangles. As you can see, the values in the number of green triangles increases. Oppositely to the values of both the length and area as it decreases after each stages. Aside from that, the value of the number of green triangles are exactly inverse proportional to the area of each green triangle.

discussion of tok

How do mathematicians reconcile the fact that some conclusions conflict with
intuition?

Understanding the question, it means on how mathematicians can settle (on a disagreement) the fact that some conclusions in their theories and discoveries conflict with intuition? Intuition is one of the Ways of Knowing in TOK for people. Intuition is defined on people’s hunch or assumption, this idea and conclusion makes no sense explanation in its answer.

As intuition concludes guesses made by mathematicians, the accuracy that it will be the correct answer will not be 100% as mathematics is known to be a type of ‘language’ with exact answers and full accuracy. All mathematicians will always come up with a conclusion that at least includes some intuition, therefore these conclusions need to be backed up with reasoning that will make sense to be called as a fact. But sometimes, other mathematicians deny intuition and try hard to find a solution with solid facts so that their discoveries can be counted as theories that can be used for others in the future.

reflection

During my experience in mentoring Secondary 4 students on their SA 1, they asked me to guide them on their own investigations and a runway graphed into their desmos given by their teacher. They approached me and asked some questions so I told them that they need to circle around the runway before landing on the runway. The lines used in desmos needs to be connected to show the pathway and so that it’s neat.

ib learner profiles

inquirers

We nurture our curiosity, developing skills for inquiry and research. Our curiosity let us find out and explore more about the differences and sequence about each investigations.

knowledgeable

We develop and use conceptual understanding, exploring knowledge across a range of different kinds of mathematical investigations existed. We engage with issues received from these investigations and ideas that are used to explain behind the theory of these investigations.

thinkers

We use critical and creative thinking skills to analyse and take responsible action on complex problems. We exercise initiative in dealing with new problems and discoveries so that we can be exposed to more different kinds of problems and be more prepared.

References

Fung, E. (n.d.). Koch’s Snowflake. Retrieved from http://www.math.ubc.ca/~cass/courses/m308/projects/fung/page.html

Kognity. (n.d.). In Theory of Knowledge.

Weisstein, E. W. (n.d.). Koch Snowflake. Retrieved from Wolfram MathWorld: http://mathworld.wolfram.com/KochSnowflake.html

Weisstein, E. W. (n.d.). Sierpinski Triangle. Retrieved from Wolfram Mathworld: http://mathworld.wolfram.com/SierpinskiSieve.html

Introduction

In mathematics, functions is a common word we use everyday in class. Functions are very important in life as we encounter them everyday. Without functions, there will be no such things as algorithms in physics that will eventually lead to inventions and technologies of the future. A better understanding of functions can also be expressed in graphs to show what they look like.

transformations in functions

3 types:

translation

• Vertical Translation

f(x)+c will cause the function to shift upwards. (by ‘c’ units)

f(x)-c will cause the function to shift downwards. (by ‘c’ units)

• Horizontal Translation

f(x-c) will cause the function to shift to the right. (by ‘c’ units)

f(x+c) will cause the function to shift to the left. (by ‘c’ units)

Reflection

To graph y=-f(x), reflect the graph y=f(x) in the x-axis.

To graph y=f(-x), reflect the graph y=f(x) in the y-axis.

Shrinking & Stretching

• Vertical Shrinking and Stretching

When y=c f(x),

If c>1, the graph will stretch vertically. (by ‘c’ units)

If 0<c<1, the graph will shrink vertically. (by ‘c’ units)

• Horizontal Shrinking and Stretching

When y= f(cx),

If c>1, shrink the graph horizontally by a factor of 1/c.

If 0<c<1, stretch the graph horizontally by a factor of 1/c.

bike introduction

From the translation that I have learned in class, an assignment was given to us given below. We were told to use translations in functions to make a better model of a bike seen below.

While using Geogebra, I found out that the model of the bike between the given and Figure 1 appears to be different as seen below. Therefore, more adjustments in its function must be made to improve the model of the bicycle.

FIGURE 1: MULTIPLE FUNCTIONS GRAPHED TO LOOK LIKE A BIKE

k(x)=0.3*sqrt(x-1.5)+0.76, 1.5<x<3 = f1

v(x)= sqrt(0.25- (x-1)^(2)) = f2

g(x)=0.75, 1.5<x<2.5 = f3

p(x)= -sqrt(0.25- (x-1)^(2)) =f4

j(x)= (x-2.5)^(2)+0.76, 2.34<x<2.66 = f5

t(x)= sqrt(0.25- (x-3)^(2)) = f6

h(x)=-sqrt(0.25- (x-3)^(2)) = f7

r(x)=0, 0.2<x<3 = f8

s(x)=1.5x-1.5, 1<x<1.5 = f9

Due to the Inaccuracy of the bike’s first model, i’ve decided to remodel it by changing some functions and also by adding some more!

By changing :

• f1 into k(x)=0.3*sqrt(x-1.5)+0.76, 1.5<x<1.9
• f8 into r(x)=0, 1<x<2

By adding :

• f10 which is equal to y=-1.5x+4.5, 2.5<x<3
• f11 which is equal to y=1.5x-3, 2<x<2.5
• f12 which is equal to y=0.4, 2.266667<x<2.733333

figure 2: From the new model of the bike created, we can use it to travel around the world! This time, the bike will not break into pieces, not like the first model…

IB Learner profiles

thinkers

We use critical and creative thinking skills to analyse and take responsible action on complex problems. These thinking skills are used to be able to solve and graph different type of functions to be able to make a better model of bike.

Knowledgeable

We develop and use conceptual understanding, exploring knowledge across a range of disciplines. To be able to apply critical thinking, we need to understand the graph beforehand and how it works before improving the first model.

inquirers

We nurture our curiosity, developing skills for inquiry and research. Our curiosity enable us to begin with the thought of mind if graphing functions to look like a bike will be possible or not. This characteristics should be applied alongside a good knowledge and understanding, before critical thinking can be applied.

Complex Numbers

A complex number consists of real and imaginary numbers in the form of a+bi in a Cartesian form, where a is the real part and b is the imaginary part and ‘i’ represents the unit imaginary numbers which is equivalent to the positive square root of -1.

Forms

Cartesian Form

a+bi

Polar form

• r (cosθ + isinθ)
• r cisθ (cis function)

Euler’s formula

• r e^iθ

Note: i² = -1 as square root of -1 squared is -1.

Real life applications

Electricity

Usually denoted by the symbol i, imaginary numbers are denoted by the symbol j in electronics (because i already denotes “current”). Imaginary numbers are particularly applicable in electricity, specifically alternating current (AC) electronics. AC electricity changes between positive and negative in a sine wave. Combining AC currents can be very difficult because they may not match properly on the waves. Using imaginary currents and real numbers helps those working with AC electricity do the calculations and avoid electrocution.

quadratic equations

In quadratic planes, imaginary numbers show up in equations that don’t touch the x axis. Imaginary numbers become particularly useful in advanced calculus.

signal processing

It is useful in cellular technology and wireless technologies, as well as radar and even biology (brain waves). Essentially, if what is being measured relies on a sine or cosine wave, the imaginary number is used.

ib’s learners profile

knowledgeable

I develop and use conceptual understanding, exploring knowledge across different kinds of problems encountered. I engage with issues and ideas to difficult questions about complex numbers.

thinkers

I use creative and critical thinking to solve complex problems. I exercise initiative in encountering problems that contains an element of surprise.

reflective

I thoughtfully consider the way each questions are done properly, other ways to solve it and my experiences. I work to understand my strengths and weaknesses in order to support my learning and personal development.