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July 08, 2011

Stationary points of a polynomial

This was a post-PGCE entry! It was a presentation of the properties of a polynomial up to 4th degree. The characteristics of the stationary points are determined by the derivatives of the polynomial.


This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com



June 25, 2011

An algebraic question

I once set a question inspired by the Corandic that I came across in one of the training sessions in my first placement. Corandic is an emurient grof with many fribs, ...

Corandc granks an increasing number starps at each storp. The number of starps at each storp increases according to an arithmetic sequence. There are 45 starps at 7th storp and 63 starps at 10th storp. Find the how many starps there are in 1st storp and 5th storp.


June 24, 2011

Complex number mathematics

I happened to have taught some complex number mathematics to students studying the International Baccalaureate Higher Level Mathematics. The students need to produce some graphical presentation using dynamic geometry software. Below are some the resources that I produced.

These pages illustrate the geometrical transformation of complex numbers in the complex(Argand) plane. Addition performs translation. Multiplication performs rotation, dilation or contraction. The complex number which involves in the additon or multiplication operation is the parameter of the transformation. It is instructive for students to identify the different transformation effects that can be achieved by different operations and parameters. Useful and interesting exercises are reflection, rotation by a right angle clockwise and anti-clockwise, dilation and contraction by a scale factor.

de Moivre' theorem is illustrated using of sequence of (green) vectors to indicate the results of the multiplications of a number complex W by itself many times from 2 to n. It is very important for students to verify algebraically that the effect the multiplication of two complex numbers on the arguments is their addition.

(cos A + i sin A)(cos B + i sin B) = cos(A+B) + i sin(A+B)

Thus the de Moivre's theorem can be proved by the method of induction. On the "roots" page, W which is the nth root of Z, can be found by adjusting it until W^n = Z. Notices that there n different ways to work. Thus actions illustrate the fact that there are n roots to the nth root of a complex number. Hide and show the root (orange) vectors to test and check the answers. The n roots of the nth root of a complex number form a regular polygon with n sides. It is instructive for students to construct a regular polygon using GeoGebra to verify the results.

Addition of complex numbers


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Multiplication of complex numbers


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de Moivre's theorem and roots of a complex number


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April 10, 2011

Teaching geometric optics of mirrors and lenses

Geometric optics

GeoGebra is very useful for illustrating the principles of geometric optics. The GeoGebra construction sequence of the ray diagram differ very little from textbook description. The only difference is that the GeoGebra page is interactive and works for other scenarios. The GeoGebra page allows the investigator to explore the effect on the image when the position of the object or focal length is varied. There is no need to reconstruction another ray diagram for every variation. One construction covers all scenarios! This is powerful demonstration of the generality of construction principle.

The constructions are fairly straightforward. The pages below include a construction step toolbar. Press the play button to playback the construction sequence. There are three different ways to use the GeoGebra pages for teaching. (1) The pages can be used as presentations in the lesson. (2) Students can use the pages as simulations of optics experiments. (3) The pages are benchmarks for what could be emulated by the GeoGebra. Students could construct similar pages as projects.

Concave mirror

Convex mirror

mirror_concave.png convex mirror



Converging lens

Diverging lens

converging lens diverging lens

The pages (click to open a separate page for the actual GeoGebra pages) were created for demonstration in a physics lesson and for students to investigate the properties of the mirrors and lenses. In addition to the mirrors and lenses, the following pages of the parabola and prism were created to illustrate the design principles of the optical elements. The mirrors are based on the mathematics of parabola. The lenses are based on the physics of the refraction described by Snell's law.

Parabola

This dynamic geometry page shows that a parabola is a locus of a point P which is equidistant to the focus F and a point G on the directrix d. PG is parallel to the x-axis. The lengths of PF and PG are equal, therefore the FGP is an isosceles triangle. b, the perpendicular bisector FG is the tangent to the locus. This properties is important in Physics. A ray incident in parallel to the x-axis of the parabolic mirror is reflected to to the focus F. The reverse is also true because a ray is reversible.

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In this construction, F' is a reflection of F about y-axis. Therefore, the parabola will always passes through the origin. Move F on the x-axis will change the curvature of the parabola.

Prism

This simulates the refraction of light in a glass prism using Snell's law. The refractive index is adjustable and glass prism can be transformed into a trapezoidal or triangular prism. The page is capable of producing two totally internal reflections inside the prism. The page aims to illustrate the concept that the cross-section of a converging or diverging lens resembles a glass prism. Focusing the light at the focus is the result of the application of Snell's law.

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It was a "Tour de force" of GeoGebra skill to produce the page. There are 105 construction steps. There is a maximum of two refraction and reflection points on each side of the trapezium. Thus, the physics simulated by the page is correct up to a limit. The page should not be used by students unsupervised and teachers should explore many different scenarios to prepare for any surprise. Snell's law is simulated using a string of commands "Rotate[R_1, If[abs(n sin(Angle[R', R, R_2])) ≤ 1, -asin(n sin(Angle[R', R, R_2])), Angle[R', R, R_2] - π], R]" for the refraction of light at the point R. R_1 and R_2 are points at the end of the interfacial normal at R. Each refraction/reflection point can accept an incident ray from another reflection points.


March 31, 2011

Teaching Chinese Remainder Theorem

Introduction

This teaching idea was a spin-off of the subject session on "Cross-cultural mathematics". I based my material largely on the lecture notes on a course of Cryptography by Leonid Reyzin of the Computer Science Department at Boston University. The Chinese remainder theorem can be stated in different levels of algebraic abstraction. We can introduce the Chinese remainder theorem not only for cultural curiosity, but also for its mathematical contents and practical applications. Proving the Chinese remainder theorem in full generality is normally investigated in the abstract algebra course at university. In the following questions and examples, I try to make the topic accessible to school level students. In fact, solving problems using the Chinese remainder theorem were examination questions for scholars in ancient China! Nowadays, the Chinese remainder theorem type of questions appear in Maths Challenges for schools!

Examples and questions

Select two numbers such that their only common divisor is 1. Technically the numbers are known as mutual primes, coprimes or relative primes. Consider 4 and 5, for example, and investigate the following table


0 1 2 3 4
0 0 16 12 8 4
1 5 1 17 13 9
2 10 6 2 18 14
3 15 11 7 3 19
  • How the table was generated? Answer
  • What other properties do the table exhibit? Answer

Some open questions

  • Choose another pair of mutual primes and show that the rules are still applicable.
  • Choose a pair of non-mutual primes, how do they affect the rules?
  • Choose a pair of mutual primes p and q such that p and q are greater than 100. Find a method to obtain n if the a and b are the remainders when n is divided by p and q.
  • Choose 3 numbers p, q and r such that they are mutually primes. How to solve the problem of finding n if a, b and c are the remainders when n is divided by p, q and r respectively?

Some specific questions

  1. What is the remainder of (17 + 15) when divided by 20?
  2. What is the remainder of (17 x 15) when divided by 20?
  3. How can the table above help to work out the answer of the previous questions? Answer
  4. What is the remainder of (17 x 17) when divided by 20? Answer
  5. Find the number(s) k such that (k x k) when divided by 20 is 9? Answer

An examination question in ancient China

A general counted the number of soldiers in his army. 2, 3 and 2 soldiers stood out of the files if they were ranked in groups of 3, 5 and 7. How many soldiers were there?

On MacTutor at University St Andrews, there is a good reference to the origin of the Chinese remainder theorem.

Application of the Chinese remainder theorem in the RSA algorithm

The RSA algorithm is an encryption and decryption technique used in many privacy applications and electronic commerce.


March 13, 2011

Teaching energy, power and efficiency in physics

workdone

The picture shows a snap-shot of a dynamic work sheet to support classroom discussion on the topic of workdone, power and efficiency in a Physics lesson. The animals pull along some loads for some distances. Students are encourage to discuss the merit of which animal did most work, which was most powerful and which worked most efficiently. They are encourage to give reasons to support their assertions. Pressing the star buttons at the bottom of screen will reveal the answers graphically and numerically.

The performances of the animals are determined anew randomly every time the program is restarted. The size of the animals is not an indicator. However, the size of the food is directly proportional to the energy it contains. Energy which is the product of force and displacement is presented by the area of a rectangle. Power is how much the energy is output per unit time. Efficiency relates the energy output to the how much food the animal has eaten. Scratch was used to create the work sheet. It is quite popular and is quite easy to use.


February 06, 2011

Teaching the concept of limit in calculus

Limit


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This entry was another spin-off from the session of mathematics education for post-16. The topic was teaching calculus. I chose to introduce differentiation from the concept of limit using Geogebra. The dynamic geometry software is meant to illustrate the existence of limit for the gradient of a function. The points A and B on the curve y=xn in the Geogebra page can be made arbitrarily close but never coincide. As separation of AB in x tends to 0, the gradient of the curve tends to a limit. I would begin the introduction of limit with Zeno's paradoxes. When the students had comprehended the concept of limit, the derivative of y=xn can be algebraically derived from the binomial expansion of (x+dx)n. The zoom and unzoom actions of the page were achieved using the javascript extension of Geogebra.


December 22, 2010

Shopping worksheet

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I used this page for teaching "divide in ratio" and "an introduction to algebraic expressions". The "divide in ratio" was introduced using the multi-buy discount offer. The question was how many items to pay if the shop offers a discount of "buy N get M free". The emphasis was on how the sum was worked out. The students were encouraged to explain their methods of working out the division. Their tasks were not about working out the sum, but to write the expression of the sum. The worksheet prints the expression for the total. The objective was to show the commutative law of addition and subtraction and the distributive law of multiplication using a shopping list. The amount of money to pay is independent of the order by which the items and the loyalty discount were counted.



Simple harmonic motion

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I constructed the simple harmonic motion page for a physics lesson. I had this dynamic image in my mind when I learnt the topic years ago! However, I could only stare at the equations and mentally register the interdependencies of different terms. I suppose that most students would have done the similar mental and manual exercise: plot a few different versions of the oscillation corresponding to different sets of controlling parameters. It was the process of training to be a mathematician or physicist. Now, it is a wonderful experience to see the instantaneous response of the changes graphically.

"The simple harmonic motion with damping" page is not a simulation. Actually, the analytic solution to the second order linear differential equation was entered into the page directly. Geogebra simply plots the graph. The spring was a locus of a point on a small circle rolling up the y-axis! I certainly have done a lot of thinking on the topic of "constructions and loci".

The pages that I created can also be viewed and downloaded from http://sites.google.com/site/huivictorc/geogebra.


Projectile explorer

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The projectile explorer was prepared for a mechanics lesson. The construction is general enough such that the page could be adapted into a Geogebra tool. Then two or more projectiles could be put together in a single page to enable the exploration of interception.



December 08, 2010

Kites and bisectors

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Angle bisector as a locus

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Spirograph

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I created the spirograph to illustrate the concept of locus. I later realised that it only works if the locus completes the circuit in 2π. Otherwise, the angle returns discontinuously to 0 when it reaches 2π. I could circumvent the limitation by using a slider to make the angle runs for an interval continuousely beyond 2π. However, introducing the slider will disable the manual movement of the pen that I consider an advantage for pedagogical purpose for my students. Later, I found a very good spirograph construction by Mike May at the Department of Mathematics and Computer Science of Saint Louis University. I found the similarities of the two constructions very remarkable!

It was a great opportunity for me to meet Michael Borcherds, the lead developer of Geogebra, at a workshop during my teaching placement. He went through the construction techniques of the roots of complex numbers, conformal mapping, transformations using matrices, cobweb techniques, etc. I thank Michael for showing me how to put a button to switch the trace on and off in my spirograph and to construct the vector decomposition such that the components and the resultant vectors are all interdependent.  The vector decomposition technique was used in the constructing the projectile explorer.

Constructing the "trace-on/off" button

Using the spirograph as an example. It shows a button "Show Trace" (Boolean a) to control the page to show the trace of the locus of the point E. It is done by creating another point TRACE which is defined to be E. The trace is set to on for the point TRACE. However, TRACE is shown on the condition that the boolean value of a is true. Therefore boolean a controls the switching the trace on and off.

Constructing the inter-dependent vector and vector components

The projectile explorer shows the initial velocity of a projectile. The velocity is resolved into its horizontal and vertical components. Adjusting any one of resulting vector and its horizontal and vertical components automatically updates the two remaining vector. Four points P0, pxv0, pv0 and pvy0 define a rectangle whose coordinates are (B2,C2), (D2+B2,C2), (D2+B2,E2+C2) and (B2,E2+C2) respectively. The values (B2,C2) are coordinates of the position P0 projectile at time t0. (C2,D2) are the horizontal and vertical components of the velocity vector v0. A2, B2, C2 and D2 link to the cells on the spreadsheet. Therefore, moving the points on the graph page update the values in the corresponding cells.


October 28, 2010

Cloud chamber experiment

A very successful experiment demonstrated the alpha particle radiation using a cloud chamber in the Westwood Teaching Building! The radioactive source was placed at the head of a bended pin which was pierced into a cork. The photograph on the left showed the cloud chamber apparatus. The photograph on the right showed four bright spots. The brightest spot where the tracks of the condensation originated was the pin head holding the radioactive source. The remaining three spots were multiple reflections of the pin head of the chamber and my camera (see my gallery for more pictures).

a cloud chamber apparatus
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