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TOK in Math

While Grade 10 students do not have Theory of Knowledge (TOK) yet in their course of study, it is a good practice to give them the opportunity to think about the world they live in, even in Mathematics.

I only have 9 students in class, they were divided into four groups, each with a different question to answer; and each student submits an individual paper on their thoughts about the question prompts.

Here are excerpts from the essays submitted.

QUESTION: Do we discover the mathematical basis of the world or we impose mathematical structures onto the world?

Platonism, nominalism, and realism, all present convincing arguments to whether Mathematics was discovered or imposed by us onto the world. In my opinion, math has always been present in nature was simply discovered by humans. The concepts and theories of Mathematics present today were all synthesized man, but would not have been derived without the presence of mathematics in the universe. Simply put, math is innate amongst everyone, and can be found wherever we go.Chloe Uy

QUESTION: Is Mathematics a formal language?

Math was not created to express our frustrations when we cannot solve a problem or to tell the world how hot it is on a specific day, but what it can do is show statistics of temperatures before and how many students find it difficult. In relation to the definition of what a Language is, Math does communicate something and is most likely understood because of its universality in syntax and orthography which makes it conform to Natural Languages. The fact that teachers are able to check each and every equation in tests means that there is something to be understood with the particular symbols the student wrote. The astronomer, mathematician, and philosopher, Galileo Galilei once said “The laws of nature are written in the language of mathematics”, with this Mathematics is something understood as a formal medium and is used to explain what can’t be explained in most times.Karrel May Sinfuego

QUESTION: Is gambling an application of Mathematics?

There is math in gambling, but it is still the gambler’s choice to take advantage of it to get a mathematically better chance at winning. Roulette, blackjack, betting and etc. all involve math mainly statistics so mathematicians have a mathematical advantage over gamblers who play by luck; but no matter how much math you put in it there is never a 100% chance of winning because the casino too has mathematicians working to give the casino the advantage; so it’s a tug of war between them.Edgar Sia III

QUESTION: Is zero nothing?

I personally agree that zero is not equal to nothing. If zero was truly equal to nothing, then we would have no need for the number itself. The numeral zero serves a purpose, and has a definite usage in modern numbers, and so therefore, I believe that zeros are not the same as nothing, and instead, one of the most complex numbers along with infinity.Grant Park

QUESTION: Do we discover the mathematical basis of the world or we impose mathematical structures onto the world?

What I think is that Math itself is around everywhere in nature, the world, and it is just waiting to be discovered by humans or any other animals which are capable to do so. To put support to it I would give the different civilizations using different types of numerical system. Asian civilization, European civilization and Arabic civilization, despite being located far from each other they all still have math, which indicates that math is not actually imposed by humans onto the world but is discovered. Mathematical basis can be discovered just by observing the world, for example, Isaac Newton discovered the law of universal gravitation just sitting under a tree and get hit by a dropping apple, Rene Descartes discovered the Cartesian coordinate system just by staring at his checkered roof with a bug stuck to it. In conclusion, math is actually around us among nature and humans impose the language to the math in order to use it, so we discover the mathematical basis of the world.John Yun

QUESTION: Is Mathematics a formal language?

Mathematics is much more like a formal system than it is a mere formal language. This is because, in addition to an alphabet, strings, and formation rules, mathematics requires several inference rules and axioms in order to make sense. Although the many symbols used in the representation of inference rules may seem confusing, we were actually introduced to the idea of mathematics being a formal system ever since we began learning the subject. The idea behind the inference rules of mathematics is logic, and we were taught at a young age about the logic behind mathematics.Pamela Esguerra

QUESTION: Is gambling an application of Mathematics?

Gambling is undoubtedly an application of mathematics. It involves a fair amount of mathematics. One of the fundamental concepts applied in gambling is of probability, which in itself includes other important sub-concepts such as independent and dependent events, expected value and house edge. Being able to apply these concepts of mathematics aids a player to have an advantage over the other players. Although applying these concepts may not inform us when exactly an event will occur, it does tell us about the likelihood of an event occurring and this is  advantageous in any gambling scenario to make well informed betting decisions. Nonetheless, mathematics can always be used in gambling, and that for a fact, is always true.Nandika Gupta

QUESTION: Is zero nothing?

Zero is not nothing, it’s a state. It will always have a context; the scale on a thermometer when measured in Celsius reads a zero as the point of phase change from ice to water and water to ice; it is not reading that means there is no temperature. If I had a box with ten shiny marbles, then I remove 9 marbles, the box has 1 marble left, but I have the 9 marbles in my hand. If I remove the last marble, the box has zero marbles in it, but I have all ten of the marbles. So there are never zero marbles except in the context of the box.Benjamin Antonio

QUESTION: Do we discover the mathematical basis of the world or we impose mathematical structures onto the world?

I believe that simple mathematics is innate; it is up to us to invent the rest. The complex formulas and other mathematical theories are all pieces of a puzzle which we made in order to finish it. I believe the Platonist point of view restrains us from achieving more in the field of mathematics.  When waiting to be spoon fed by nature, it restricts us from a greater freedom of thought, as if we simply discover, then we cannot be more daring, ask deeper questions, and be motivated to create further change. While these point of views do not affect our calculations, they will define the discoveries in the future of mathematics.Isara Suwansilp

Which question provokes you to answer? Comment and make your thoughts known.

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IB World Mag asks about “Productive Failure”

I follow the official twitter account of the IB World magazine, @IBWorldmag mainly because they share valuable insights into the world of IB. From time to time, they also throw in very interesting questions to its followers.

One question tweeted last November 2 was: Can a “productive failure” method provide students with a deeper learning experience? Here’s my answer.

I completely agree that this provides a deeper learning experience in Math.

My students are challenged on a regular basis with problems that require critical thinking. This way, they strive to solve by understanding the math processes involved. They are trained to be THINKERS and RISK TAKERS. Whether they are abe to solve it or not is NOT as important as having a deeper understanding of mathematics; hence, a more meaningful learning experience.

A nurturing yet disciplined learning atmosphere must be present so failure is not deemed as a source of embarrassment leading to a loss of confidence. In my class, students are always reminded that what is important is that we learn from our mistakes; that next time we are faced with similar problems, we are able to solve them with confidence.

I will attribute the success of my students in challenging them regularly, while ensuring that I will be with them each time they fail, not to show them the way, but to help them remember that they have solved problems they thought they never could, but they have.

My Grade 9 have been with me for 3 straight years, and I am proud to say that from a dismal performance in international standardized measure of academic progress, 8 of 11 students are now 94% above the rest of the world, with four earning a percentile rank of 99%, one 98%, one 96% and one 95%. Wonder if they were ever as good 3 years ago? Definitely not. Our turning point was when I poured out my frustration that they could not even solve word problems involving simple arithmetic. It worked because since then, they have journeyed with me for three years in mathematics, and they are loving every bit of challenge I give them.

On another batch, half of Grade 10 have been with me for 3 straight years too, and I am proud to say that all of them are now 91% above the rest of the world, with three earning a percentile rank of 99%, one 98%, one 97%, one 96% and one 95%. Again, were they this smart three years ago? They will honestly disagree.

Learning is an experience both the class and the teacher should enjoy together. I have asked for my class’ trust that whenever I give them challenging questions I do not intend them to feel bad about themselves, but that I like them to explore different ways of solvig. Ultimately, I want them to learn how to persevere, which is something they will use when faced with challenges. Their success is due largely to the trust they have given me, and thus simply giving hard questions in class will not make them geniuses. It is with failing productively with them and succeeding together in learning that make everything more meaningful.

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Binomial or Poisson? Their Difference Simplified

A number of students get confused when to use Binomial and Poisson. For guidance, here is a simplified difference between the two.

  • An exact probability of an event happening is given or implied in the question, and you are asked to find the probability of this event happening k times out of n. Use binomial distribution.
  • A mean or average probability of an event happening per unit time/per page or for any given interval is given, and you are asked to find the probability of n events happening in a given time/number of pages or for any number in an interval. Use poisson distribution.

Try these examples for starters:

  1. Cars have been observed to pass a given point on a backroad at a rate of 0.5 cars per hour. Find the probability that no cars pass this point in a two hour period.
  2. A manufacturer finds that 30% of the items produced from one of the assembly lines are defective. During a floor inspection, the manufacturer selects 6 items from this assembly line. Find the probability that the manufacturer finds two defectives.
  3. Sophie has 10 pots labelled one to ten. Each pot, and its contents, is identical in every way. Sophie plants a seed in each pot such that each seed has a germinating probability of 0.8. What is the probability that more than eight seeds do germinate?
  4. A radioactive source emits particles at an average rate of one every 12 seconds. Find the probability that at most 5 particles are emitted in one minute.

ANSWERS: (1) poisson; 0.3679 (2) binomial; 0.3241 (3) binomial; 0.3758 (4) poisson; 0.6159

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Circle Theorems

An inscribed angle of a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle. The following theorems involve inscribed angles:

  • If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc.
  • If inscribed angles of a circle or congruent circles intercept the same arc or congruent arcs, then the angles are congruent.
  • If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.

A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Three important theorems involving tangents are the following:

  • In a plane, if a line is a tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
  • In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent of the circle.
  • If two segments from the same exterior point are tangent to a circle, then they are congruent.

A line that intersects a circle in exactly two points is called a secant of the circle. You can find the measures of angles formed by secants and tangents by using the following theorems.

  • If a secant angle has its vertex inside a circle, then its degree measure is one-half the sum of the degree measures of the arcs intercepted by the angle and its vertical angle.
  • If a secant angle has its vertex outside a circle, then its degree measure is one-half the difference of the degree measures of the intercepted arcs.

For secant-tangent angles, we use the following theorems:

  • If a secant-tangent angle has its vertex outside a circle, then its degree measure is one-half the difference of the degree measures of the intercepted arcs.
  • If a secant-tangent angle has its vertex on a circle, then its degree measure is one-half the degree measure of the intercepted arc.
  • The degree measure of a tangent-tangent angle is one-half the difference of the degree measures of the intercepted arcs.

For segment measures, we use the following theorems:

  • If two chords of a circle intersect, then the products of the measures of the segments of the chords are equal.
  • If two secant segments are drawn to a circle from an exterior point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.
  • If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.
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Permutations vs Combinations

Permutation is the total number of ways of arranging n objects, while combination is the total number of ways a set can be selected from n objects. If arrangement is to permutation, then selection is to combination. As such, order matters in permutation while order does not matter in combination.

Because enumeration is such an important part in finding probabilities, a sound knowledge of permutations and combinations can help ease the workload involved.

Here is an example from the Math HL 3rd Ed of Cirrito:

Light bulbs are sold in packs of 10. A quality inspector selects two bulbs at random without replacement. If both bulbs are defective the pack is rejected. If neither are defective the pack is accepted. If one of the bulbs is defective the inspector selects two more from the bulbs remaining in the pack and rejects the pack if one or both are defective. What are the chances that a pack containing 4 defective bulbs will in fact be accepted?

The scenario may be best understood using a tree diagram, while the counting process used is combination (not permutation).

Try working out the problem until you arrive at the correct answer which is 11/21.

Enjoy solving!

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Key Features of Quadratic Functions

Quadratic functions are functions of the form f(x) = ax² + bx + c, where a, b and c are real numbers. Its graph is called a parabola.

The lowest (minimum) or the highest (maximum) value of a quadratic function is found in its vertex. It is the point where the graph “turns”.

The value of a affects the concavity of the parabola. If a > 0, parabola is concave up; hence, the vertex is a minimum. If a < 0, parabola is concave down; hence, the vertex is a maximum.

The imaginary vertical line that passes through the vertex is called the axis of symmetry. Every parabola is symmetrical about its axis of symmetry.

As in all other functions, the point where the graph crosses the y-axis is called the y-intercept, and the point where the graph crosses the x-axis is called the x-intercept. Furthermore, the x-intercepts correspond to the roots of the equation y = 0; and the y-intercept corresponds to the value of c.

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How complex are Complex Numbers?

Complex Numbers are any number of the form a + bi where a and b are real and equals square root of (-1).

Notice that all real numbers are complex numbers where = 0. A complex number is purely imaginary if = 0.

Two complex numbers are equal when their corresponding parts  are equal. That means, a + bi = c + di      if and only if     a = c and b = d.

Complex numbers a + bi  and a - bi  are conjugates.

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How Does One Succeed in Math?

There is no single formula in achieving success in Math. There are however tried and tested strategies that make one do well in Math. Let me give four of the not-so-new-yet-we-ignore tactics to achieve that elusive A in Math.

  1. ASK. When something bothers you during the lecture or an activity, do not hesitate to throw your question. Letting it pass in the hope of getting the answer eventually will not help. Confusion will only compound in your head; and before you know it, everything doesn’t make sense at all. So before leaving the classroom, make sure you understand everything that the teacher has said. Make sure you achieve the aim(s) set at the beginning of the period.
  2. DYH. Do your homework. Homework is given to reinforce the skill we learn in class. Doing something at home gives us the opportunity to explore ways of solving a single problem, and finding out which one suits us. It is a time for discovery without the pressure of finishing within a 50-minute period.
  3. REVIEW. If you have heard of the cliché practice makes it perfect, then you should have figured by now that the pronoun it usually refers to MATH. While practicing doesn’t guarantee a perfect score, it certainly improves your skill and retains the processes you undertake for each type of problem. If you stumble upon a question you cannot answer, refer to number 1: ASK.
  4. RELAX. Learning is best achieved with a healthy mind, heart and soul. Spend time with your family and friends during weekends and special occasions. Engage yourself in sports and activities you like. Everybody needs a break; thus the 7th habit of effective people: sharpening the saw.

Surely these are no special tips, and these we have heard many times from our Math teachers. Let us aim for this school year that we follow these game plans. After all, these strategies work NOT only for Math, but for other subjects as well. =)

Let me know what you think makes one succeed in Math. Comment below.