Stretching question

Sunday 18 June 2006 at 2:35 pm | In Articles | 13 Comments

A level maths questions these days vary little from year to year, so working through past papers is the best way to revise. Occasionally though, the examiners ask for something slightly different at the end of a question to separate the best students from the good ones.

This happened in this week’s Further Maths paper. It’s not hard, but the last part of the question may not have been seen by students before and so requires a little thought on their part.
The question started off with

    M=\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \vspace{-2mm} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}
\end{pmatrix}

then asked for the transformation that M represents, to calculate M^2 and M^4 and then, the ‘stretcher’ for three marks, calculate M^{2006}.

It will be fascinating to see the examiners’ report on how students coped with this non-standard question.

13 Comments »

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  1. Being an American, I’m not sure what a “Further Maths paper” is, or how much time students have to work on it, but I’d be willing to bet that if it’s longer than a few minutes that at least one person attempted to perform 2005 multiplications.

    True, there was a time when I thought that any sane person wouldn’t come up with such a thing. That was before I was shown a geometry exam by one of my teachers in which all the lengths and angle measurements appeared to have been rounded to the nearest multiple of 5.

    Apparently the determined but not-so-studious student had constructed a makeshift ruler and protractor out of scratch paper, constructed large versions of the figures, and then measured the requested angles and lengths directly rather than calculating them.

    Comment by Daniel McLaury — Monday 26 June 2006 12:53 pm #

  2. This particular exam is Further Maths AS Paper 1 and last year’s paper can be found here.

    It’s a 1.5 hour paper taken by 17 year-olds who want to extend their knowledge of mathematics beyond that taken by the normal A level student.

    Since there are 75 marks for the complete paper, the 3-mark stretcher should take about 3.5 minutes. It will be fascinating to see if the exam board reports in the autumn that any student answered the question the ‘long’ way. :-?

    Comment by Steve — Monday 26 June 2006 5:34 pm #

  3. While we’re at it, what are ‘A’ and ‘O’ levels? Any time I hear anything about the British educational system I have no idea what they’re talking about.

    Comment by Daniel McLaury — Monday 3 July 2006 10:17 pm #

  4. The British system keeps changing so it’s hard for us to keep track let alone people in other countries :)

    ‘O’ level (Ordinary level) was a national exam series taken by some students at the age of 16 until the mid-eighties when it was replaced by GCSE, which is now taken by almost all 16-year olds. Two years later, academic students take ‘A’ level (Advanced level) which are entry qualifications for University. (There are hundreds of other post-16 qualifications for vocational courses, but let’s keep things simple.)

    In 2000-01, AS (Advanced Subsidiary) levels were introduced which are taken by 17 year olds and count as half an A level.

    Just to complicate matters, this is the system for England & Wales; Scotland has its own system.

    The British Council in the US says:
    The American concept of a school transcript is unfamiliar in the UK. Schools in the UK do not generally rank pupils within their year; currently, the principal standards are the GCSE, SCE and AS and A-Level examination results.

    There is no official method of equating British and American educational qualifications. The educational systems are entirely different and attempts to compare them must be done on a strictly provisional basis. However, general experience seems to indicate that 5 GCSE’s or the Scottish Certificate of Education-Standard Grade passes are considered to weigh closest to the American high school diploma.

    British students tend to specialise much earlier than US students so they may only study 3 subjects (eg Mathematics, Physics, Chemistry) by the time they reach A level. However, taking into account that English & Welsh students don’t go to university until the year in which they are 19, I believe that the mathematics studied in both countries is not too dissimilar.

    Comment by Steve — Monday 3 July 2006 10:49 pm #

  5. hi
    i have a question. how can i write mathematical test like you

    Comment by Smile of Math — Monday 3 July 2006 11:44 pm #

  6. LatexRender or LatexRender for WordPress

    Comment by Steve — Tuesday 4 July 2006 9:26 am #

  7. a problem
    prove that if 0<a^2+b^2-abc \leq c then
    a^2+b^2-abc is a perfect number

    Comment by mahyar — Monday 10 July 2006 1:41 pm #

  8. i need further maths question to solve

    Comment by rasheed — Wednesday 19 July 2006 5:24 am #

  9. Here’s a question… if you were to attempt this question by repeated multiplication, what is the minimum number of multiplications that would be required?

    Comment by Nick — Tuesday 25 July 2006 7:41 pm #

  10. Great question Nick. You may want to take a look in here:

    http://en.wikipedia.org/wiki/Exponentiation

    Comment by filipe — Friday 28 July 2006 1:06 am #

  11. …and in here.

    http://en.wikipedia.org/wiki/Addition-chain_exponentiation

    Comment by filipe — Friday 28 July 2006 1:07 am #

  12. helo mi name is brett i love maths so much some times i cry when i carnt do maths cause my mum has put a limit on how much maths i do now

    Comment by brett sherwood — Sunday 12 November 2006 8:34 pm #

  13. i do love to study mathematics but due o my health situation i can’t continue so my mum has advise i only do reseach to upgrade myself so as to better undersand my programing .

    Comment by roland — Wednesday 12 September 2007 2:47 pm #

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