Monthly
Maths
I s s u e
With days to go
until the Diamond
Jubilee
celebrations, we
have links to some
mathematics
resources written
especially for the
occasion, courtesy
of TES
Resources.
On the following
pages are other
resources linked
to the theme of the
number 60 and to
diamonds for you
to follow up the
event after the
bank holiday, and
to explore some
interesting and
enriching
mathematics.
MEI will be
celebrating its
50th anniversary
in 2013, so if you
have any
interesting ideas
for ways to help us
to celebrate, or
mathematics
activities and
resources around
the number 50,
please email us.
We will find
rewards (sorry, no
diamonds or gold!)
for the best ideas!
www.mei.org.uk
1 7
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be registered
(free of charge) to download resources.
Diamond Jubilee
Pythagoras (by adc199)
A KS3/4 resource using
Pythagoras to calculate
the length and cost of
bunting required for a
Diamond Jubilee street party. Read more
How to draw the Union
Flag accurately (by
J u n e
2 0 1 2
Diamond Jubilee
Mathematical Mysteries
(by laura.reeshughes)
This KS3/4 resource
consists of three
mathematical mysteries
which are based around
some of the celebrations taking place, but
which could also be used in line with your
curriculum . The first mystery requires the
use of bearings and constructions, the
second probability and proportion and the
third distance, time and ratio. Read more
sbe1978)
This KS3 activity works
as an aptitude test to see how well
students follow instructions involving
measurement. Read more
Maths - the Queen’s Problem
(by Jonny Griffiths)
A chess-based problem with
many permutations! For
example, how many queens
can they place on a 60 by 60
chessboard so that no queen
can be taken by any other? Read more
Diamond Jubilee Problem Solving
(by Andrew Chadwick)
KS2/3 lesson starters e.g.
If A=1, B=2, C=3 … Z=26,
the word DIAMOND has a
total of 60
(4 + 9 + 1 + 13 + 15 + 14 + 4).
What other words can you find with a
total of 60? Read more
More about the number
60 on Page 2
Diamond Jubilee Top Trumps
(by laura.reeshughes)
A KS3/4 activity using a
set of top trumps cards to
commemorate the
Diamond Jubilee,
featuring well known royal
faces. The questions are
all based around the
number 60, and involve
fractions, BIDMAS and some decimals.
Read more
Diamond Jubilee
Arithmetic Game
(by Owen Elton)
A KS3 activity based on
the popular "24 Game".
The resource file contains
a set of cards where, instead of 24, a goal
of 60 is set. Each card contains four
single-digit numbers which must be used
exactly once to reach the target.
Read more
MEI Maths Item
of the Month
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Why is 60 an
interesting number?
Class
investigation
Introduce and
discuss different
types of number.
The concepts can
vary depending on
the abilities of your
class.
This PowerPoint
by davecavill and
this worksheet by
goldson1 on TES
Resources may
help to introduce/
revise some of the
concepts.
Make cards, each
having a different
type of number,
along with its
definition on the
back, and ask
students to use
that criteria to
investigate its
relationship to the
number 60, e.g.
even number
square number
cube number
triangular
number
factor
multiple
prime number
prime factor
Opposite are other
number types that
you might discuss
with the class.
60...
60...
is the smallest common multiple
is adjacent to 2 prime numbers:
(number divisible) of numbers 1 to 6
is a composite number (a number that
can be divided evenly by numbers other
than 1 or itself)
e.g. 60 can be factored as 2 x 30
is a composite number with 12
positive divisors:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
is a highly composite number (a
positive integer with more divisors than
any smaller positive integer)
is one of the 6 numbers that are
divisors of every highly composite
number higher than itself:
1, 2, 6, 12, 60, 2520
has 11 proper divisors:
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30
is an abundant number (a number that
is less than the sum of all of its proper
divisors)
1 + 2 + 3 + 4 + 5 + 6 +10 +12 +15 + 20 + 30
= 108
is a unitary perfect number (a number
made by the sum of its unitary divisors those having no common factor other
than 1 - excluding itself)
1, 3, 4, 5, 12, 15, 20
is the sum of consecutive odd
numbers:
5 + 7 + 9 + 11 + 13 + 15
is the sum of consecutive even
numbers:
8 + 10 + 12 + 14 + 16
is the product of Pythagorean triples:
3x4x5
59 and 61
is the sum of consecutive primes:
11 + 13 + 17 + 19
is a Goldbach number (the sum of two
odd prime numbers - see this TI
investigation):
e.g. 23 + 37
is the sum of twin primes:
29 + 31
is the smallest number which is the
sum of two odd primes in 6 different
ways:
29 + 31
23 + 37
19 + 41
17 + 43
13 + 47
7 + 53
is the difference of squares:
82 - 22
162 - 142
is the difference of powers with the
same base:
26 - 22
can be made using ‘four fours’:
4x4x4–4
44 + 4 x 4
44/4 - 4
is a Harshad number in base 10 (an
integer that is divisible by the sum of its
digits when written in that base)
is the number base (Sexagesimal)
passed down to the ancient Babylonians
by the Sumerians and is still used for
measuring time, angles, and geographic
coordinates (see Page 3)
Clock geometry
Time for debate
Why do we have
60 seconds in a
minute, 60
minutes in an
hour, etc?
The system was
set up by the
Babylonians who
used a number
system with a
base of 60,
possibly because
it has many
factors.
Why are they
called seconds?
The word
"seconds" come
from the second
sexagesimal place
in the base 60
expansion.
How can we use
our hands and
fingers to count
to 60?
We do not have
60 fingers to count
on - surely ancient
civilisations didn’t?
So how did they
do it? Read more
Time to ban the
digital watch…
from the
classroom at
least? The
Dozenal Society of
Great Britain
extols the merits
of base twelve.
Read more
KS2/3 Class activity:
Using clocks to add fractions
The aim is to be able to add fractions with
unlike denominators. One way to get
students to see that they can list several
names for one fraction is by exploring
what we could name each section of a
clock face with the hands
placed on different
numbers. Draw a clock
with hands of equal
length, with one hand at
12 and the other at 2.
Ask students what fraction they could
describe this section as. Most students
first see this as 2 out of 12 hours, and
they write the fraction 2/12. Many also
see that this could be 10 out of 60
minutes, or 10/60, an equivalent fraction.
Point out that neither of these fractions is
stated in its lowest terms and ask them
what this might be. Some will realise that
2/12 can be stated as 1/6 by dividing the
fraction by 2/2; likewise 10/60 can be
stated as 1/6. It still has the value 10/60
not because "you can drop the zeros",
but because 10/60 divided by 10/10 = 1/6
Now draw another clock
with one hand on the 12
and one on the 8. This
shows 8/12 or 40/60.
Again, neither fraction is
in lowest terms: one could
divide 40/60 by 10/10 to get 4/6, then
divide 4/6 by 2/2 to get 2/3. Most students
will now see that 8/12 is the same as 2/3.
Now move onto a clock
illustrating the addition of
1/4 + 2/3. Some students
may not understand how
these fractions can be
added together as they
have unlike denominators and thus are
not like fractions.
By finding the lowest common
denominator for both fractions they can
be added together. 1/4 is seen to be
equivalent to 3/12 and 2/3 was proven to
be equivalent to 8/12. So, we get 3/12 +
8/12 = 11/12.
This TenMarks video shows
why 60 having so many
factors makes a clock a good device for
adding/subtracting fractions with different
denominators. View here and here.
Colin Billett has
produced a PowerPoint display exploring
the methods for the four rules with
fractions. It is shared here.
KS3 Bitesize has interactive
activities and videos that introduce
equivalent fractions, mixed numbers and
improper fractions, and practise
operations using fractions. View here
GCSE Bitesize has useful
and entertaining videos that develop the
theme of adding, subtracting, multiplying
and dividing fractions with unlike
denominators. View here and here
Clockwatching
When do the
hands of the
clock line up?
Mike Rosulek,
assistant
professor of
computer
science at the
University of
Montana, has come up some fascinating
observations. Read more
The mathematics of
diamonds
More gems
Round Brilliant Cut diamonds
The South Africa
necklace was
given to the young
Princess Elizabeth
on her 21st
birthday in 1947,
with round brilliant
cut diamonds
breathing vigorous
life into these
lustrous stones.
The symmetry of diamonds is an
important aspect of its cut, along with its
polish. The symmetry refers to alignment
of the facets. With poor symmetry, light
can be misdirected as it enters and exits
the diamond. It is the cut and symmetry
of a diamond that determines how light is
bounced from facet to facet, resulting in
the wonderful sparkle diamonds are so
loved for.
'Martian Pink'
Diamond sold for
£11.1m on May
29, 2012.
Named after the
red dust on Mars
in 1976 for the
successful US
landing of the
Viking 1 mission, a
rare 12.04-carat
diamond sold at
auction after six
minutes of
frenzied bidding in
Hong Kong.
The modern round brilliant has 57 facets
(polished faces), counting 33 on the
crown (the top half), and 24 on the
pavilion (the lower half). The girdle is the
thin middle part. The function of the
crown is to diffuse light into various
colours and the pavilion's function to
reflect
light back
through
the top of
the
diamond.
The round brilliant cut diamond is
believed to be the ideal shape in terms of
brilliance (white light reflected up
through the top of the diamond), fire
(coloured light reflected from within a
diamond) and sparkle (combination of
fire and brilliance).
Although no single inventor has officially
been credited with the invention of the
Round Brilliant Cut, many sources credit
an 18th Century Venetian cutter named
Vincenzio Perruzzi.
The Russian mathematical genius Marcel
Tolkowsky, a member of a large and
powerful diamond family, subsequently
calculated the cuts necessary to create
the ideal diamond shape.
In 1919, as part of his thesis in
mathematics, Tolkowsky considered
variables such as the index of refraction
and covalent bond angles to describe
what has become known as the Round
Brilliant Cut. He focused on diamonds
with a pavilion angle of 40° 45'. He wrote
that this angle "gives the most vivid fire
and the greatest brilliancy, and that
although a greater angle would give
better reflection, this would not
compensate for the loss due to the
corresponding reduction in dispersion."
Tolkowsky developed a geometric model
of the crown, using a knife-edge girdle for
simplicity. This geometric model let him
take a given pavilion angle and calculate
the best crown angle and table size for
that pavilion angle. This geometric model
can be used to calculate the crown
angles and table sizes that correspond to
other pavilion angles.
See Folds website for "Diamond CrossSection" software for calculating the
proportions of Tolkowsky's diamonds.
You can read Tolkowsky's book here and
play with his mathematic model thanks to
Mathematician Jasper Paulsen.