Maths Item of the Month Archive 2007

December 2007

A mathematical Christmas message from MEI

Christmas message

Answer: compliments of the season to you (complements of the C's on 2u)


November 2007

Finding the tangent to a parabola geometrically

To find the tangent to a parabola at a point P:

Draw a vertical line through P.
Draw a horizontal line through Q, the vertex of the parabola.
R is the intersection of these two lines.
M is the mid-point of Q and R.
MP is the tangent to the parabola at P.

Can you prove that MP is the tangent to the parabola?


Solution


October 2007

An implicit equation of a line?

The graph of x3 + 3xy + y3 = 1 is shown:

Can you prove that the graph is a straight line?

Are there any points that satisfy the equation which aren't on the line?

Item submitted by Dr Jeremy D. King, Tonbridge School


Solution



September 2007

Cats

Bill's cat has 8 kittens: 3 female and 5 male. Alex is going to have two of them. Alex decides that she must have both kittens of the same gender. One kitten is chosen at random then the other will be chosen at random from those of the same gender.

What is the probability that she gets two female kittens?

Two possible answers:

  1. There is a 3/8 probability that the first kitten is female. If the first kitten is female then so is the second kitten.

    The probability that she gets two female kittens is 3/8.

  2. Suppose the female kittens are A, B, C and the male ones are D, E, F, G, H. The two kittens she gets (in order) could be:
    AB, AC, BA, BC, CA, CB, DE, DF, DG, DH, ED, EF, EG, EH, FD, FE, FG, FH, GD, GE, GF, GH, HD, HE, HF, HG.

    The probability that she gets two female kittens is
    6/26 = 3/13 .

Both these answers cannot be correct. Which one is wrong and why?


Solution


August 2007

Equilateral triangle in a rectangle

The diagram shows an equilateral triangle in a rectangle. The two shapes share a corner and the other corners of the triangle lie on the edges of the rectangle.

Prove that the area of the green triangle is equal to the sum of the areas of the blue and red triangles. What is the most elegant proof of this fact?


Solution


July 2007

Motivating proof by asking impossible questions

If you ask students to find an example of something which is impossible their natural reaction is likely to be to ask "Why?". To be fully satisfied that there are no such example most students will want to see a proof.

e.g. (1) "Find a Pythagorean triple where two of the numbers are even and one is odd."

Many mathematical ideas requiring proof can rephrased in such a way.

e.g. (2) "Show that any graph (in discrete mathematics) has an even number of odd nodes" can be rephrased as "Find a graph with an odd number of odd nodes".

e.g. (3) "Show that the difference between the squares of two consecutive numbers is always odd" can be rephrased as "Find two consecutive numbers such that the difference between their squares is even".

It is important not to tell the students that the question is impossible for the question "Why?" to arise naturally.


June 2007

Circles with integer co-ordinates

The circle x2 + y2 = 52 has 12 points with integer co-ordinates, as does the circle x2 + y2 = 132.

To find a circle with more than 12 points with integer co-ordinates you could multiply 52 and 132 to obtain x2 + y2 = 652 or you could multiply 5 and 13 to obtain x2 + y2 = 65 (as 65 can be written as the sum of two distinct squares in two different ways).

Does this result generalise: can the product of the largest values in two Pythagorean triples always be written as the sum of two distinct squares in two different ways?


Solution


May 2007

MEI Branches

MEI has a number of branches around the country. These are local groupings of MEI teachers who meet two or three times a year.

Branch meetings are intended to give teachers an opportunity to discuss issues and help them in their delivery of MEI courses. They are a valuable form of professional development and often include sessions from MEI professional officers as well as discussion of recent examination papers.

Branches are a vital communication link within MEI. There is an annual Branch Chairmen's conference for Chairmen to receive information and for MEI to hear what is going on in the Branches and the concerns of their members.


April 2007

Zeckendorf representations

The first few Fibonacci numbers: F1=1, F2=1, F3=2, F4=3, F5=5, F6=8, F7=13, ...
1, 1, 2, 3, 5, 8, 13, 21, ?

Zeckendorf representations: Let the notation (anan-1...a4a3a2)F where each ar = 0 or 1, represent the number anFn+an-1Fn-1+...+a4F4+a3F3+a2F2

Such a Fibonacci representation is called a Zeckendorf representation.

For example, 17=13+3+1 and also 17=8+5+3+1. Therefore there are two Zeckendorf representations of 17: 17 = 11101F = 100101F

Theorem (Edouard Zeckendorf 1972):
Every positive integer n has a unique Zeckendorf representation with no consecutive 1s.

13

8

5

3

2

1

1

0

1

=4

1

0

0

0

=5

1

0

0

1

=6

1

0

1

0

=7

1

0

0

0

0

=8

1

0

0

0

1

=9

1

0

0

1

0

=10

1

0

1

0

0

=11

1

0

1

0

1

=12

1

0

0

0

0

0

=13

1

0

0

0

0

1

=14

1

0

0

0

1

0

=15

1

0

0

1

0

0

=16

1

0

0

1

0

1

=17

1

0

1

0

0

0

=18

1

0

1

0

0

1

=19

1

0

1

0

1

0

=20

For more information see the Zeckendorf Wikipedia page.


March 2007

Past papers

Did you know that all of the past papers on this site are provided with the markscheme and the examiner's report as a single file? This is to make it easy for teachers to see the standard expected of their students and identify common mistakes made in examinations.


February 2007

Possibly the best counter-example in the world!

Hypothesis: An irrational number to the power of an irrational number cannot be rational.

Disproof

First of all consider the number x = .

Is x rational or irrational?

This is a very difficult question to answer but we do know it's one or the other!

So we have two cases to consider:

If x is rational then we have found our counter-example because we know is irrational and so x = is an example of an irrational to the power of an irrational with a rational answer.

If x is irrational then think about the number . By our assumption (that x is irrational) then this is an irrational to the power of an irrational. But we know the value of :

and this is clearly a rational number.

Therefore one of and must be an example of ?irrational to the power of irrational equals rational?, but we don't know which one and nor do we need to know, our counter-example is in there!

(In fact it was proved as recently as 1930 that is irrational and so it is the latter case which is the counter-example.)


January 2007

Sums of cubes


Solution