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Usually we cover both subject knowledge and exam technique, although that can change depending on each individual student. Then we go through diagrams, and they ask questions, and we go from there.

Lots of students say that the classes are too big in school, or that they don't have time to ask teachers after lessons. In my tutorials, we take time to explore things in a little in a bit more detail.

I always look up the board my students are taking so the lessons are really relevant. Then we go through past papers or set texts, whatever the student finds helpful.

I use the shared whiteboard. We make diagrams together and label them, and often the student prints it off because they know it's right and they completely understand it.

After tutoring one girl went and told all her friends the new explanation I gave her. And she was so excited about what she wrote in the exam she emailed me immediately afterwards.

There was one girl who had her exam on Monday. She wanted tuition on Friday, Saturday and Sunday beforehand. It was very intense, but she said the exam went well.

To identify a Poisson distribution question, remember that in a Poisson distribution...
Events occur independently of each other
Events occur at a constant rate
Events occur singly (that is, two can't happen at once)
An example could be 'the number of cars passing your house every hour' because the appearance of a car doesn't change the probability of another car passing, they are likely to pass at a constant rate, and two cars cannot pass at the same time.

Answered by Hannah D.

Studies MMath at Warwick

You're right, when you first learn about matrix multiplication it definitely doesn't *feel* like the easiest way of defining a form of multiplication. This is particularly true if you've just learned about adding matrices and multiplying them by scalars, which do feel intuitive - surely it would be simpler to define multiplication by:
| a c | | w y | = | a*w c*y |

| b d | | x z | | b*x d*z | as opposed to the the normal way. On some level it doesn't really matter! Mathematics can be whatever we think of, so if you want matrix multiplication to be like this, the go ahead! You might find some interesting maths if you do this. The fact that it isn't defined like this, however, is probably a good indication that there's a reason for it being defined the normal way. Mathematicians are smart people, after all! The reason for matrix multiplication is because of what a matrix signifies. To a computer scientist, a matrix is just a 2D array of numbers. To a mathematician however, they signify something much greater: matrices represent a type of**transformation, **known as linear transformations because of how they behave. When we perform matrix multiplication, what we are doing from a visual perspective is to ask what happens when one transformation is followed by another. The nature of linear transformations mean that we can write the combination of these matrices as another matrix, and to work out what this overall transformation looks like we follow the rules of matrix multiplication.

| b d | | x z | | b*x d*z | as opposed to the the normal way. On some level it doesn't really matter! Mathematics can be whatever we think of, so if you want matrix multiplication to be like this, the go ahead! You might find some interesting maths if you do this. The fact that it isn't defined like this, however, is probably a good indication that there's a reason for it being defined the normal way. Mathematicians are smart people, after all! The reason for matrix multiplication is because of what a matrix signifies. To a computer scientist, a matrix is just a 2D array of numbers. To a mathematician however, they signify something much greater: matrices represent a type of

Answered by Daniel C.

Studies Physics at Manchester

Simple harmonic motion is a type of motion which occurs when the net force on an object is always acting towards a particular point (e.g. motion on a spring). The force is proportional to the negative of displacement, which can be shown as follows:
**Differentiate twice, with respect to t**
x = asin(ωt) (explanation too complicated for this medium)
dx/dt = ωacos(ωt)
d^2 x/dt^2 = -ω^2 asin(ωt)
**Substitute x into d^2 x/dt^2 and replace d^2 x/dt^2 notation with a:**
a = -ω^2x
**Apply NL2 (F=ma):**
F = -ω^2 mx
From this we can see that Simple Harmonic motion occurs when and only when force is proportional to negative displacement. Good indicators of this scenerio include a clear midpoint of oscillations (e.g. lowest point on a pendulum) and maximum velocity at the midpoint and minimum velocity at extremities (also evident from a pendulum).

7sechx - tanhx = 5
7/coshx - sinhx/coshx = 5
7 - sinhx = 5 cosh x
7 - 0.5(e^{x }- e^{-}^{x}) = 2.5(e^{x }+ e^{-}^{x})
7 -0.5e^{x }+ 0.5e^{-}^{x} = 2.5e^{x }+ 2.5e^{-}^{x}
0 = 3e^{x }- 7 + 2e^{-x}
0 = 3e^{2}^{x }- 7e^{x} + 2
0 = (3e^{x} - 1)(e^{x} -2)
e^{x} = 1/3 e^{x} =2
x= ln 1/3 x=ln 2

Complex numbers were invented by mathematicians to provide solutions to certain kinds of equations. For instance, the equation x^{2}+1=0 doesn't have any solutions in the real numbers, but mathematicians wondered that *if there were solutions* to this equation, what kind of properties would those solutions have? Rearranging this equation in terms of x gives us x^{2}=-1. There are clearly no real solutions to this equation, since there are no real numbers such that when you square it, it gives you -1. Imagining that there is a number that has this property gives us the imaginary unit i, which is defined as i=(-1)^{1/2}. And we can see that by subsituting this in we get i^{2}+1=((-1)^{1/2})^{2}+1=-1+1=0, so i is a solution to x^{2}+1=0.
Consider the quadratic equation x^{2}+2x+2=0. Using the quadratic equation we get the solutions x=-1+(-1)^{1/2}, x=-1-(-1)^{1/2}. Normally we would be stuck and could go no further, but now that we have the imaginary unit we can rewrite these solutions and get x=-1+i, x=-1-i. These new numbers that are combinations of real numbers and multiples of the imaginary unit are whats called complex numbers.

I would begin the session with checking a basic understanding of the methods of long division, as these will not have been taught very recently. Many students will have been using short division, as it is quicker and once this is taught, many students forget the process of long division all together.
Once this has been refreshed, I would move onto explanations of how the process can be applied to algebraic expressions, using examples of simple questions to demonstrate this. I would encourage the student to try some questions in the session with me, and then I would end the session with an exam style question from a textbook or past paper.

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