One of the great joys in Maths is exploring something seemingly very simple and finding layers upon layers of complexity, and connections with other areas of Maths which at first sight appear to be totally separate. I'm sure you have come across Pascal's triangle; here it is: To create each new row, start and finish with 1, and then each number in between is formed by adding the two numbers immediately above. Pattern 1: One of the most obvious patterns is the ...

## Prepare Your Body and Your Mind for the Exams

This will probably be my last blog before you dive into the exams in May. I hope your revision has gone well: there's still time to go over more past papers to become as familiar as possible with the way the examiners ask questions. And don't forget that, in Maths exams, there's only one mark for the correct answer – all the other marks are for working and intermediate answers. So train yourself to write as much as you can ...

## The Wine Glass and the Water Glass Puzzle

This is an old one but is a wonderful example of a puzzle with an unexpected, counter-intuitive solution. 'Two 50ml glasses are filled to the brim, one with wine and the other with water. A teaspoon (5ml) of water is transferred to the wine glass and thoroughly mixed in. Then a teaspoon of the mixture is transferred back to the water glass. Question: is there now more water in the wine, or wine in the water?' The more you think about it, ...

## Understanding Keywords in Mathematics Exams

Whether you are approaching your final exams, or you're at the stage of mid-course exams, you're probably beginning to put in some practice of past paper questions. Sometimes it isn't the maths which is hard, but understanding the language of the question; and each question will contain certain keywords which tell you how you should be tackling the question, and what you should be writing down. I can't emphasise enough how important it is to understand what, in the context ...

## Understanding the Chain Rule

Most students are first introduced to the chain rule when shown how to differentiate a function such as y = (3x - 2)5. The problem is that is tempting to try and fit all chain rule differentiations into that format, for example trying to differentiate e3x - 2 in the same way. What is the chain rule? It's a calculus formula with a wide range of uses, just one of which is differentiating a 'function of a function.' Quite simply, differentiation concerns the rate ...

## Straight Line Curves

It's a really relaxing bit of recreational mathematics (albeit with some serious theory behind it), creating beautiful curves just by drawing straight lines. I can only give a glimpse in this blog, but the possibilities are endless. If you're tempted to have a go, one word of caution: be as accurate as you can since the final result can be disappointing if you are careless, or work too fast. My first example is probably about the simplest you can do. I've ...

## Great Mathematicians 5: Al-Khwarizmi

Not heard of him? Born in 780 in what was then Persia, he became one of the learned men of the House of Wisdom in Baghdad. He lived to the age of 70 and, because of the breadth of his work in mathematics and the sciences, he must have been an amazing person to know. The House of Wisdom acquired and translated scientific and philosophical treatises, mainly from Greek, as well as publishing original research. Al-Khwarizmi's first major publication was The ...

## Some Mathematical Tips for the New Year

Happy New Year everybody! At this time last year I suggested some new year resolutions to help you in your studies. They are of course valid at all times, and if you didn't see them, then have a look now. This time I've got some tips which you might want to note down; they cover specific areas of the syllabus, although they aren't applicable to all of HL, SL, and Studies. Degrees or Radians? Well, it depends on what the question is asking you ...

## Concerning Leap Years

Who Introduced the Leap Year? Julius Ceasar, believe it or not. The Roman calendar had got seriously out of date with the solar calendar, even though Roman officials were supposed to introduce an extra month every so often. So Caesar consulted his top astronomers who worked out that the solar year—the time it takes for the Earth to do exactly one rotation around the sun—was 365.25 days. So, by maintaining a calendar with 365 days, and adding one extra day every ...

## Drawing Graphs on the TI-84

A previous post on solving equations with the TI-84 has proved very popular, so here are some tips and tricks when using the GDC for drawing graphs and associated functionality. Pre-set scales It's easy enough to type in an equation and then press the graph button. However, the most important thing to do which helps make sense of any graph is to set up the correct window: the minimum and maximum value of the displayed coordinates, and the associated scale marks. The ...