Proving Pythagoras’ Theorem

Hopefully everyone reading this can state Pythagoras' Theorem by heart - but, just in case: "Given a right-angled triangle, the square of the length of the hypotenuse is the sum of the squares of the two shorter sides."  Stated more simply, and with the aid of a diagram:   c2 = a2 + b2     But how can this result be proved? It's no good drawing lots of right-angled triangles and measuring their sides - firstly, because it's impossible to measure accurately; and secondly, showing that something is true ...

Understanding significant figures

I'll start with a question - to how many significant figures is the number 500? What does "significant" mean? Consider the number 315 - which digit can you change which would have the least effect on the size of the number? Clearly the 5: increase or decrease it by 1, and 315 changes by 1. But move the 3 up or down 1, and the number will change by 100. So, the 3 is more "significant" than the 5 - and the further ...

A Human Binary Counter

At the heart of every computer, however complex, are binary numbers - that is, numbers formed only of 0's and 1's. This is because it is easy to represent just two digits electronically: a switch can be on or off; a current can flow one way or the other; a pulse can change a stored digit from one state to the other. How does the binary system work?  When we count up to 9 in the decimal system, we've run out ...

Last Call for Revision

I have just returned from teaching on an OSC revision course at the Anglo-American school in Moscow, and was impressed by the focus of the students there towards revision, and how well they were able to sustain that focus over a fairly gruelling week. By the time this blog is published I hope to have seen some of you at the Oxford revision courses as well; my aim, and that of all the teachers on OSC revision courses, is to ...

Puzzle time again

Last year I posed three mathematical puzzles in a blog. This time there's just one, but don't just read it and then look at the answer - it's a really good puzzle to spend some time trying to solve; most people give up, and yet when you see the solution, it's surprisingly straightforward. And please note: there are no catches, no tricks, no sneaky wordplay; the puzzle is exactly as I present it. Four people are being chased by a dragon. ...

Understanding logarithms

A popular topic on OSC revision courses is logarithms. I usually start by asking who knows what a logarithm is, to be met by most of the class staring at their fingernails! To fully comprehend the laws of logarithms, and how to use them, it really helps to start with a clear understanding of the basics: and as long as you can cope with indices (powers), logs follow closely behind. A logarithm is really just a power. If I asked you: ...

Bubble sort

Suppose you want to buy a new phone accessory online. You go to a search site and it finds 100 retailers stocking the accessory, and offers to sort them by price. Or you've got a list of 1000 names on a spreadsheet which you want to sort alphabetically. Have you ever wondered how a computer carries out a sort? Such a problem can be solved using an algorithm which is simply a set of simple, unambiguous instructions which will lead to a solution. ...

The Hailstone Sequence

Follow this rule to generate a sequence of integers:    Choose any integer n as the first term of the sequence     If n is even, divide it by 2 to get the next term. If n is odd, multiply it by 3, add 1, and then divide by 2 to get the next term. In other words, if uk is even, uk+1 = uk/2; if uk is odd, uk+1 = (3uk + 1)/2. For example, if u1 = 14, subsequent terms are 7, 11, 17, 26, 13, 20, 10, 5, ...

Try these New Year Resolutions…

The five mathematical resolutions I have suggested below are, unlike some New Year Resolutions, very achievable! But you do have to stick with them if you are to get the benefit. 1.   I will know my times tables up to 12 by the end of January. Does that sound a bit beneath you? It isn't - many students find themselves doing questions slower that they should - or getting wrong answers - because their recall of simple multiplications is poor. Don't ...

Using the TI-84 to solve equations

By Wednesday, November 30, 2016 , , , , 0

You must have a graphic display calculator (GDC). For HL and SL exams the GDC can only be used in Paper 2, but you will require it for both papers if you are following the Studies course. But don't wait until the exams are close before practising the GDC functionality: you need to do this throughout the course so that you can use it to maximum effect in the exams. Whatever level you are taking, it is almost inevitable that you ...