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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)



June 01, 2011

SCT 6: The Beginning…

What follows is a brief outline of the class that were the focus of my Directed Task.

Vital Statistics:

Class: Top Set Year 7 - Levels ranging from 5< Level <7

Pupils: 30 in total - 15 Boys and 15 Girls

Selected Pupils:

Child x - The highest achiever in the class who is currently working at low/mid level 6. This pupil would welcome the challenge of being stretched to level 7, maybe even level 8 work!

Child y - A middle ability pupil, strong in some areas, weaker in others, working between level 5 and 6.

Child z - A pupil at the lower end of the spectrum. Whilst their mental maths is fairly strong (perhaps explaining why they are in the top set), they seem to struggle with most of the work and are not the most confident mathematcian. I was told they are currently working at level 5, though they often fail to produce work at this level.

The class as a whole are very strong and a pleasure to teach. There are no serious additional educational needs that need to be addressed. A couple of pupils require worksheets on A4 instead of A5, but that is it. Whilst there are many EAL pupils, they are all fairly confident in english.


May 18, 2011

(Wanted) Dead or Alive…?

So it’s been a while since my last blog entry... a very long while. So consider this a quick message to my avid followers that all is well in the world of this trainee teacher. Any rumours of my passing must be forgotten, I may have been absent from the blogging world but I am still very much part of the realm of the living. The only part of me that has died is my social life.

Needless to say the last few weeks (or months!?) have been ridiculously busy, priorities had to be made and unfortunately blogging was never going to be one of them...

I’ve no doubt ‘The Blog’ has its place in the journey we have taken during the course, but in terms of becoming a better teacher, I have found it significantly more beneficial planning and evaluating lessons rather than burning the midnight oil with my lyrical blogging prose.

That said, it seems I will have to submit some mandatory blogs ‘reflecting’ on my teaching particularly my AP2.

They will follow in due course...

Until then.

Adieu.

Jx

Dead or Alive


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 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.


March 07, 2011

I'll Sleep When I'm Dead…

Evening...

So... it's week 2 of that make or break 6 weeks between half-term and Easter. The Masters has been handed in. Pretty much all of the Uni work is done. All that is left is to see out PP2. Simple. What stands in the way? The trifling matter of the 150 lessons I still need to plan. Yes, I said 150.... ish.

The countdown has started.

Do I mean the countdown from 150 lessons to 0? Sort of. Do I mean the countdown to the end of PP2? Sort of.

I mean the countdown of the lessons I need to plan before I can finally go to sleep. Thats right. Sleep.

I'd wager there are not many PGCE students who are too familar with the concept of sleep at the moment... Sure, it will be in the back of their minds, a distant memory... that thing they used to do when it got dark outside and the midnight hour approached. What did it feel like? I can't remember, it was so long ago...

The days are long, the nights are longer. The lesson plans keep on coming. They are relentless. They do not sleep. Not unlike the "Uruk-hai" of Mordor... they keep running all night. Lessons plans need no sleep, they just need to be done. It is the PGCE student who must go without.

PP2. I love it.

For the first time in all of my weeks on PP1 and PP2, last week was the first time I felt like a teacher. The change in attitude, persona, confidence, was all evident. It was noted. It was a good week.

If this is as hard as it is going to get, I think I am over the worst of it. I will survive. Afterall, as Bon Jovi would say:

"I'll sleep when I'm dead"

Jx

I


February 08, 2011

These Days…

Hello Everybody,

Its that time of the month again, time for me to "reflect" on the happenings of my life as a PGCE student... and yet again this will be set against a Bon Jovi soundtrack... This one is called "These Days" (from the 1996 album of the same name).

So what has been happening since last time??? A strong case could be made for "lots has been happening", and an equally convincing case could be made to argue "nothing has been happening"... you decide.

For now I will just focus on the first two serial weeks of PP2 at undisclosed School B.

I'm liking it. A lot. Really I am. Lots and lots. Without having taught any lessons, I already feel a greater part of this school's community than I ever did on PP1 at undisclosed School A. Maybe I'm just more confident in my ability to teach, and therefore embracing the whole experience a little better. Maybe its because it has been made very clear that the support is there if I need it and they really do care about my professional development. Or maybe I'm now in a school I could see myself teaching at. Credit to the SPO... They did alright :-)

Now let us spare a moment to think about our first Masters essay... The rumour mill has been running riot around Westwood; an ill breeze of terror and fear sweeps around the campus like Mary haunting the Celeste.

First I heard 3 out of the first 8 marked were failed. Then it was 25% of all of them. Then I heard 50% have been deemed insufficent to merit a pass... It is only a matter of time before I'm quoted a 113% failure rate... such is the sheer panic that seems to have blinded the mathematical cohort!

I'll be honest. I'd like to pass, I really would. There is nothing I'd like more than to not have to do it again. However, at the end of the day, having to resubmit would not be the end of the world. No tears will be shed. Its only an essay. Its only life. I'd expect a good amount of formtative feedback, then I could improve, resubmit, jobs done. I see no reason to panic.

Results tomorrow??? I thought that was the case. Everyone else tells me it is on thursday...??? Who knows???

Now to re-focus on PP2. I have a really good feeling about my school. It was flagged as a challenging school, but I see no evidence of that. Every school has its challenges, that just makes it normal. It is the pupils that make it special.

It is my job to make them realise: "These Days, the stars aren't out of reach" - (Bon Jovi, 1996)

Jx

These Days


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.


February 03, 2011

trig task

Writing about web page /smithg/entry/enrichment_task/

Writing about an entry you don't have permission to view


January 31, 2011

Sequences and series enrichment pack

Writing about web page /dknowles/entry/enrichment_project_on/

Writing about an entry you don't have permission to view

our group's enrichment project


January 17, 2011

SCT 5: Interactive ICT Resource

SCT 5: Using GeoGebra to investigate the angles in a polygon.

sct_5.docx

dynamic_shapes.ggb

hint_file.ggb


January 13, 2011

Keep The Faith: PP1 Reflections

Evening All,

So I've been asked (repeatedly) to blog about my PP1 experience, with reference to my AP1 report, perhaps regarding my thoughts and opinions, etc...

In keeping with the Bon Jovi motif that is now running through my blog, this ones Keep The Faith.

How did everyone find PP1? I think it was a mixed bag of broken hearts and shattered dreams, a theatre of disillsuion; almost a realisation that maybe there is no hope for the future of this world. Some just called that monday morning.

Oh my god, has mathematics always been such a detested subject? Has it always been the low-point of a childs life? The Albatross around a teacher's neck?

I've heard people refer to teaching as the "best job in the world" (anon et al, 2010), I doubt they were ever maths teachers.

Almost every lesson is a battle, a war of attrition, where moving the trenches of learning forward an inch can be classified a victory. Why do we fight on? What is it that intangible force that drives us forward? The enigma that gets us through the day?

Is it the coffee in the staffroom? The idea that in 20 years lesson planning won't take quite as long? Or maybe it is just the challenge? That feeling that no matter how bad it gets, somebody somewhere will be thankful that you made that difference.

I really enjoyed PP1 at unspecified School A.

Also, the emergence of "Team W" will never be forgotten. Thank you Wanjun ;-)

Maybe not all lessons went the way I wanted, maybe my classroom management needs a lot of work, maybe my lesson plans can best be described as "works in progress", maybe one day I WILL learn all the names of the pupils in my class. But do you know what? I have faith that day will come. I survived PP1, and thats the best I could have hoped for.

My mentor and I both agree that I still have a long way to go, I'm far from the finished article but I am well aware of that, but we also agree that I have the right attributes to become a good teacher. I'm good at the things that come naturally, and I can learn the skills that don't.

My preference for teaching A-Level quickly became apparent, and those lessons were the highlights of my placement. My biggest challenge became my biggest success, so maybe there is hope yet.

In the immortal words of the rock legend himself: "Keep the Faith"

Jx

Keep The Faith


January 12, 2011

Sct4 submission

Writing about web page /jbird/entry/sct4_group_4/

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SCT 4: Group 4's Wall Display

Writing about web page /jbird/entry/sct4_group_4/

Writing about an entry you don't have permission to view


SCT 3 – Group 4's Maths Trail

Writing about web page /dknowles/entry/sct_3_group/

Writing about an entry you don't have permission to view


SCT3 Maths Tail, Group 4

Writing about web page /dknowles/entry/sct_3_group/

Writing about an entry you don't have permission to view

Our Maths Trail, around the Transport Museum


January 11, 2011

Our Wall Display

Writing about web page /jbird/entry/sct4_group_4/

Writing about an entry you don't have permission to view


SCT3 submission

Writing about web page /dknowles/entry/sct_3_group/

Writing about an entry you don't have permission to view


January 10, 2011

Maths Trail Group 4

sct3_final.docx


December 22, 2010

Shopping worksheet

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)

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

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)

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

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)

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.