Taster Lesson For
A – Level
Mathematics
j
1

Investigate what happens when
you remove a square from a
rectangle to leave a rectangle
that is in the same proportions
as the original rectangle
Crucifixion of San Sebastian
Some studies of the Acropolis, including
the Parthenon, conclude that many of its
proportions approximate the golden ratio.
The Parthenon's facade as well as elements
of its facade and elsewhere can be
circumscribed by golden rectangles. To the
extent that classical buildings or their
elements are proportioned according to the
golden ratio, this might indicate that their
architects were aware of the golden ratio
and consciously employed it in their
designs. Alternatively, it is possible that
the architects used their own sense of
good proportion, and that this led to some
proportions that closely approximate the
golden ratio. On the other hand, such
retrospective analyses can always be
questioned on the ground that the
investigator chooses the points from which
measurements are made or where to
superimpose golden rectangles, and that
these choices affect the proportions
observed.
Famous Pyramids of Egypt
How good is your maths?
 Can you do better than 8 year old
Fredrick Gauss?

The Question
What is the sum of all of the whole
numbers between 1 and 1,000 inclusive?
 Show all of the necessary calculations
on a single sheet of paper.

The History

Allegedly this activity was set by a teacher as work to
keep the class busy. Unfortunately in 1885, it proved
inadequate. One student, 8 year old Frederick Gauss,
solved the problem in seconds. He later went on to
become a significant mathematician. His biography is
available at:
http://www.geocities.com/RainForest/Vines/2977/gauss/e
nglish.html
Draw any quadrilateral
Mark the midpoints and join
What do you notice?
Investigate.
Y
X
Z
a
b
O
Y
N
P
X
Z
M
Q
a
b
O
Y
N
P
X
Z
M
Q
a
b
O
It is sufficient to show
that MQ = NP
Y
N
P
X
Z
M
Q
a
b
Show that MQ = ½ (b-a)
O
Show XZ = b - a
Show NP = ½ XZ
Hence NP = MQ