All 6 entries tagged Teaching

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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


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


Multiplication of complex numbers


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


de Moivre's theorem and roots of a complex number


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)



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.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

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.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

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 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


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)


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.


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