Chapter 2 - Chapter 2 - The plan in the beginning of the making of the Towel

I figure I should boil down the MM curriculum into its bare essentials to start, and go from there.

The course outline divides the curriculum into 4 'units' - 'Functions and graphs', 'Algebra', 'Calculus' and 'Probability'. What this blog post will do is outline the main ideas that need to be understood in MM, as I see it. I will miss stuff, but they will be minute details on how these things work which will be dealt with later. The big ideas are what I'll deal with here.

I'll mention quickly before this begins that this is not by any means how I will order the textbook, merely the ideas that need to be in there. Also, I will not focus nearly as much on 'that it works' as much as I will on 'how it works'.

Functions and graphs

Familiarity with a few functions - There are very few different types of graphs to be familiar with - They are:

1) Polynomials
2) x^(m/n) (where m,n are integers)
3) e^x and ln(x)
4) sinx, cosx and tanx
5) |x|

That's basically it. Functions that are lumped together are in the form 'if you understand one, you'll understand the others'.

Transformations - Only dilations, reflections and translations in the x and y directions are dealt with in this course. As such, you need only understand what one generalised transformation means, and the rest should fall out. The generalised transformation is af[b(x-h)]+k.

Inverses - What they are and how to find them.

Combining functions - This only really extends as far as adding functions together ('addition of ordinates'), multiplying them together (in other words, use your cas) or putting functions in functions ('fog', or f(g(x)).

Sketching functions - There are 2 things here. One is applying the generalised transformation mentioned above to sketching, the other is knowing what 'base' form looks like and how to deal with that.

Algebra

This is essentially a revision of previous years' algebra. Very little is taught for the first time. That said, there are specific things that people often need help with.

Solving equations - This basically involves factorising/expanding repeatedly until the function is fully factorised, then applying the 'null factor law'. Other than that, understanding the ideas of inverses, basic algebra of indices/logarithms, a basic understanding of the unit circle and a knowledge of 'equating coefficients' is all that is really needed for this.

Generalisation - As a concept, this is (somewhat surprisingly) often misunderstood, and as such I expect I will focus on this throughout the text.

Matrices - Basic matrix algebra, solving systems of linear equations with them (also using parameters)

For what is supposed to be the first half of the year, that's basically it. In each question set, approximately 1/3 of the questions will be for practising/improving use of algebra or graphing. I will post a blog about the structure of question sets in the future. I will also probably be meshing these two units into one big superunit with 4 or 5 chapters (maybe 'functions in general, methods for solving equations, polynomials and functions of the form x^n, powers and logarithms, trigonometry)

Calculus

This is also mainly a revision of year 11, but this often feels new to people, so it will be treated as being taught from scratch. Also, all of the calculus will most likely be taught in one hit of 2 chapters (differentiation, antidifferentiation).

Differentiation -This all really boils down to a few things

 - What differentiation actually is
 - Basic knowledge of how to differentiate the functions mentioned in unit 1 (polynomials, e^x etc.)
 - Chain, product and quotient rules.
 - Applying this knowledge to graphs (stationary points, sketching gradient functions)
 - Linear approximation of derivatives
 - Related rates of change (this is merely a direct application of the chain rule)

Antidifferentiation - Again, not much to know

 - What it is, and how it relates to integration
 - How to antidifferentiate a few basic functions: sinx, cosx, x^n (where n is rational) and e^x
 - Some generalised methods of antidifferentiating
 - Definite integrals
 - How all this culminates in the fundamental theorem of calculus

Probability

Admittedly, this part of the course often seems like something tacked on to the end just to keep the statisticians happy. It's not necessarily like that, but it certainly does need a lot of groundwork to get it up to the same level of understanding as other aspects of the course. Luckily enough, there's not that much to learn. There will probably be 3 chapters (Basic tools of probability, discrete random variables, continuous random variables)

Basic tools of probability - Definition of discrete/continuous random variables, expected value, variance, standard deviation, mean/median/mode, conditional probability

Discrete random variables - Bernoulli sequences, markov chains

Continuous random variables - Using integration for continuous probability distributions, normal distributions

I'll mention here that for most cases involving normal distributions, markov chains and bernoulli sequences, very specific CAS-related knowledge and skills which does not come from the previous 3 units are needed, and as such the CAS will be referenced constantly throughout these probability sections.

But that's it! That's the course! Considering that there are about 30 weeks (not including holidays) in which to learn this, I feel that this gives ample time to learn this material. At uni, all this and more is taught to non-methods students (read: little maths background knowledge) in a semester with no problems, so I don't see why it can't be taught to only slightly younger people in significantly more time.

The last thing that needs to be mentioned is that anyone doing methods needs a basic understanding of how to use their CAS. I'm still unsure of the extent to which I will refer to how to actually use the CAS (certainly my hand is forced when probability is being studied). Anyone with an opinion on it, feel free to comment :)

END CHAPTER 2