Archive for the ‘Mathematics’ category

Psychology by the numbers

September 29, 2013

Psychology would seem, at first look, to be one of the sciences with the least involvement with mathematics. After all, psychology is all about the mind, all about people – what does mathematics have to do with that?

Well, as it turns out, a lot! The main reason is that mathematics is of use, is that psychology has one aspect that most people don’t consider. Most think of psychology as relating to a single person, but there is one other consideration. We also care about how groups respond, and how the individual compares to the group.

To this end, we need a simple way to describe groups, so we can compare them to each other and to the individual. The best way to do this reliably is to use mathematics – and we are interested in two particular concepts – the idea of what the average value of a group is, and the idea of how much the group is distributed.

In this post, I will cover the basic mathematics you will need to describe these concepts – which you will need for your Empirical Research Activity (ERA) report that is due at the end of term 4. The mathematics is *not* difficult, and you will be able to the majority of it on your computer, by using microsoft excel (or a similar piece of software).

Get some rhythm – Some LOGARITHM!

July 8, 2013

Logarithms are a frustrating function – the inverse of exponential functions, they are infuriatingly irritating to deal with.  But logarithms are a critical part of applied mathematics, and have been a critical tool in the mathematical kit since the dawn of mathematics (Which was when Ugg the caveman wondered how many rocks he had).

Logarithms are what you get when you find what power you must raise ten to to get another number.

The most fascinating part of history is the role of logarithms in the development of arithmetic methods. The simplest aspect is that it is simpler (and less error prone) to add instead of multiplying, so if you find the logarithm of two different numbers, the product of those numbers is equal to the sum of the logarithms. This meant, that in the age before calculators or computers, the process of multiplication was sped up and simplified by using a table of logarithms to do all calculations.

But logarithms are more than just an outdated way of speeding up arithmetic calculations – they are an important tool in graphing. By using a logarithmic scale on one or more axes, you can produce graphs that cover a broad range of numbers by compressing the axis. The graph to the left is an example of this – in one simple picture it presents the entirety of the universe – if this were linear scale, either the small items would be invisible, or the the graph would have to be so large that it is useless.

April 21, 2013

So, we’ve gone through two and a bit chapters of Mathematical methods so far, and you are beginning to think that you’ve done the worst of it – it can’t get that much harder, can it? Really?

Well, welcome to cubics, quartics and higher order polynomials – they are the big brothers and sisters to the humble quadratic, and they’d like to have a word with you about beating up on their sibling, the poor little quadratic – something about “completing your square”

On a more serious note, if you are feeling alright about algebra so far, the next section isn’t any harder – just a few more ideas – nothing new or harder than we’ve done so far, just more. Just like when we started quadratics, the first thing we do with cubics and quartics  is to identify the common pattern, and be able to factorise them directly from recognising that pattern.

Once we can do that, we will look at how you can start to graph these functions from their intercepts – but that means we need a way of finding all the intercepts. We already knew from our studies of the Null Factor Law (NFL) that the factors of a quadratic equation can give the intercept. The Remainder theorem (right now, that irritating chorus should be going through your head) allows us to find the factors by finding values for a where f(a) = 0 for a given f(x). Combining all this knowledge gives us the ability to sketch almost any graph!

February 24, 2013

Welcome to Mathematical Methods – a subject which will prepare you for using mathematics to analyse the world and introduce the concept of developing theoretical models to investigate potential results of decisions. It is also a subject that will require you to work hard to master the skills and techniques so that you can use them effectively.

We have already completed the simplest work of the year, revising skills which you should be familiar with from previous years of mathematics – simple linear equations, systems of functions and basic coordinate geometry. Now, we start to go further, by investigating non-linear relationships – that means all lines that can be drawn on a graph that are not straight. The simplest of these are the quadratics – which have the shape of a smile (or a frown), just like the one in graph just above.

You will have studied this last year, and you may have become quite good at factorising and sketching quadratics – which is good. If you are not so confident, you will have time to revise, but you will have to put in additional work. We will be going a lot further with quadratics over the next month, and you will need to be prepared!

Physics Codes

June 7, 2012

Well, no class today for you – but there are still things to do. First of all, at the end of this post is the information I told you that you might want. Of course, being a physics teacher I can’t make it easy for you – I have to teach you something interesting.

Encryption is a fun mathematical process  – but it also has some very useful applications – like putting information in public view, but knowing that only those who are supposed to be able to read it can, because they have the key.

I have used a free online encrypt/decrypt tool which you can also use the link is: http://www.tools4noobs.com/online_tools/decrypt/, and I have used Blowfish/CBC/Base64. The key is our where we have our classes at Croydon…

Tree Diagrams

August 27, 2011

Tree diagrams can be a helpful way of organizing outcomes in order to identify probabilities. For example, if we have a box with two red, two green and two white balls in it, and we choose two balls without looking, what is the probability of getting two balls of the same color?

P(samecolor) = P(RR or GG or WW)

We use the tree diagram to the left to help us identify the possible combinations of outcomes. Here we see that there are nine possible outcomes, listed to the right of the tree diagram. This number is the size of the sample space for this two state experiment, and will be in the denominator of each of our probabilities.

Each of these possible nine outcomes has a probability of 1/9, which we can find using the multiplication rule P(RR or GG or WW) = 3/9.

Stick or Switch?

August 26, 2011

During a certain game show, contestants are shown three closed doors. One of the doors has a big prize behind it, and the other two have junk behind them. The contestants are asked to pick a door, which remains closed to them. Then the game show host, Monty, opens one of the other two doors and reveals the contents to the contestant. Monty always chooses a door with a gag gift behind it. The contestants are then given the option to stick with their original choice or to switch to the other unopened door.

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