MISCONCEPTIONS IN MECHANICS

A document that is under constant updating.The current version   here

velocity and acceleration

We now that acceleration is the rate of change of velocity.

• Constant acceleration 1m/s² means that the change of velocity is +1m/s per sec.
• Constant acceleration -1m/s² means that the change of velocity is -1m/s per sec.

We also know that positive velocity in one-dimensional motions means motion to the positive direction which is usually to the right.
Since velocity and acceleration are both vectors, they have the same direction if they are both positive or both negative. Otherwise, they have opposite directions.

• Let's say that i am moving to the right having velocity 2m/s and positive constant acceleration 1m/s². This means that one second later my velocity will be 2m/s+1m/s= 3m/s. So my motion is accelerated.
  

• Let's say that i am moving to the right having velocity 2m/s and negative constant acceleration -1m/s². This means that one second later my velocity will be 2m/s-1m/s= 1m/s. So we have a decelerated motion.
  
• Let's say that i am moving to the left having velocity -2m/s and positive constant acceleration 1m/². This means that one second later my velocity will be -2m/s+1m/s= -1 m/s. So we have decelerated motion because the magnitude of my velocity has been decreased.
  
• Let's say that i am moving to the left having velocity -2m/s and negative constant acceleration -1m/s². This means that one second later my velocity will be  -2m/s-1m/s= -3 m/s.
  So my motion is accelerated because the magnitude of my velocity has been increased.
  

That's all folks.

Mathematical modelling and deterministic processes.

Laplace who was a great mathematician believed that everything in universe is predictable. 

The other day in class we talked about mathematical modelling which is the process of describing natural processes by mathematical equations. In most of the cases these equations are based on physical laws. 

General relativity for example, which is our current theory of gravity, can be expressed by to one short equation. According to this model:

Everything from how an apple falls to the earth, to how the moon orbits the earth, how

all the planets orbit the sun, how the sun orbits a supermassive blackhole at the center

of our galaxy, how blackholes form and behave, and how the whole universe expands out from the Big Bang. Now that we have this theory, the future is more predictable. I mean, we

can well  predict eclipses thousands of years into the future. 

It seems like everything that happens in universe is a deterministic process which means that given the initial state the outcome is always the same, predictable since we now the rules of the game.

Next : what in maths is the meaning of chaos?

Shannen's spaghetti trigonometry graphs were truly a great inspiration. We had a discussion about how some sinus waves can be listened by the human ear if they are between 16Hz and 20KHz.

In this geogebra file you can listen sinus waves based on 440Hz frequency which is the A tune.
It is quite interesting to notice how frequency not only affects the graph but also the pitch of the sound. The amplitude affects the volume.

 Moreover adding two sinus waves you can have a weird but very interesting effect where the resulting sound 'sounds' somehow trembling. If the two frequencies are close the effect is more intense (we are listening to the so called 'beats'). Thinking about the A level maths, we can use the compound angle formula to prove that the two interfering waves at some time reinforce each other and some other time cancel each other.

Happy by founding out a new (for me) Geogebra command: PlaySound[function,min value, max value] and the ear-saving PlaySound[false]
:) :) :)

Loci on the Argand Plane

Finding loci is a demanding task that requires a robust background in geometry. Even skilful students struggle with it. In one of our last year's FP2 lessons, we were discussing loci on the Argand plane and we found out how challenging it is to understand and construct the loci of the complex numbers z that comply with this :
mod(z-z1)/mod(z-z2)=constant 
or 
arg(z-z1)-arg(z-z2)=constant 
where z1 and z2 are known complex numbers. 

As we all know, you cannot understand a locus until you have constructed it by yourself. Moreover, demonstrations -no matter how meticulously they could have been made by the teachers- fail to really persuade students that what they see is actually what they are looking for. 
We also know that GeoGebra can be used to facilitate understanding in a brilliant way. 

The modulus -argument form of a complex number (the polar form as it is better called) gives us a tremendously convenient way to explain and visualize our loci. We can ask students to use Geogebra to construct complex numbers with the properties that a particular locus requires, place them on the Argand plane and turning the  'trace' on to move them so that they can find out how the loci looks like. We can also ask them to use the Locus inherent command in Geogebra to verify their findings. 
They can get their toe into water and experiment changing the variables’ values  or the  positions of the numbers the construction is based on. Geogebra at this level is as easy to be used by them as any graphic calculator, but far more suitable for this learning goal. Furthermore, it’s worth getting familiarized to this environment as well (so they will not be ‘afraid’ of subjects like ‘fm with technology’) 

Below you can find a link towards a ‘book’ containing  Geogebra files which correspond  to the basic loci problems we meet in FM1 and FM2 under the ‘Complex numbers’ topic.  All of them can easily be made by the students themselves under a really little help. 
https://ggbm.at/n75NZYwC

Why do I want to be a teacher?

"We don't act rightly because we have virtue or excellence, but we rather have those because we have acted rightly.  Excellence, then, is not an act but a habit"
Aristotle.

Teaching has been my occupation for more than twenty years. I want to keep doing this job as it satisfies me, and I find it very fulfilling. When I teach I feel that I play a crucial role in the life of my students. I am responsible for finding a way to help them understand, to gain knowledge, to become better. When I succeed in tuning into my students way of thinking and lead them to form the correct cognitive structures, I feel really great.

I want to keep teaching because I feel renewed when I am with young people who possess pure and plentiful enthusiasm, and who face knowledge as something to be explored. I must admit that their humor and spontaneity makes me forget my concerns which accompany me as an adult. I can remember several instances when I left home pensive, and in a melancholic mood, and then after 10 minutes of teaching I felt optimistic and full of life again.

I insist that in my teaching role I  love coaching my students with their career choices. I am so happy when I am informed about their achievements.  I concede my feeling of pride.

I want to keep teaching because I feel that I have much more to do and to learn. I want to be a part of a great team of teachers. As a private tutor I enjoy face to face interaction with my students, but teaching a class is challenging and rewarding too.

I want to keep teaching as I want to continue improving, as a teacher. I am used to working within strict time restrictions under continuing assessment of my methods and techniques. I have come to like it. However, I feel that there are a lot of steps which I haven’t taken. There exists enough creativity within me which hasn’t come out yet. Now I am mature enough for it to so  in a consistent, and prudent way.

I love mathematics and physics, and I constantly try to pass this love on to my students. This is one more reason for me to keep on teaching. These two sciences are considered difficult, almost inaccessible to many people. My aim is to dissolve this misconception. The most important part of one’s relation to these sciences is the very first experience. A lot of care should be taken during the early school years. Physics and mathematics have their own language, and their own way of being thought about, and acted on. This can be grasped if it is taught properly. We characterize a child as clever, when they understand maths. But clever is not what you are, is what you become. 

Teaching is more challenging now than ever. Students of the 21^st century try multitasking and face multiple stimuli. Their attention is easily distracted.  Today our students live and walk with a mobile phone in their hand. They are all immersed in the internet, and struggle to accommodate all this data in their brains. Our  teaching approach is challenged to accommodate the "gadgets" of our time to an effective learning plan.  

I want to keep teaching to build upon the excessive enthusiasm of the previous decade in new Instructional Computer Technology and participate in forming a better, more effective and up-to-date exploitation of it.

Considering myself as a teacher: I think there is a long path ahead which calls for being walked along. I really want to become a (better) teacher. 

MEI DIFFERENTIAL EQUATIONS:REVISION DOCUMENTS

These documents aim to help the revision of the differential equations as the are taught under the OCR MEI  specification. They are under constant improvement. There are some more which they will uploaded shortly.

A. Modelling with differential equations :construction of models
B. A flowchart for solving second order homogeneous  differential equations with constant coefficients

C. The particular integral: how to find it.
D. Euler's method revision
E. Mind map for solving first order ordinary differential equations
F. Tangent fields and isoclines: what should we be aware of.

the spring-mass-dash-pot system : http://bit.ly/d_h_m