Monthly

Monthly
Maths
I s s u e
English Heritage
has a useful
document
‘Prehistoric
Henges and
Circles’ that
includes
illustrations of the
different types and
a map showing
their distribution.
There is also
extensive page
by page
information about
individual henges
in England,
archaeological
research reports
and mapping
projects.
The English
Heritage Avebury
Teacher’s kit has
extensive
resources to
support a visit to
Avebury henge
and surrounding
area in Wiltshire.
Also on the
English Heritage
site is information
about Education
at Stonehenge,
as well as a
Stonehenge
Teacher’s Kit.
2 8
Henges
A henge is basically a simple bank and
ditch enclosing an area of land. The bank
is outside the ditch, so it would not have
been a defensive enclosure, but was
more likely to have been a form of
religious and ceremonial gathering place.
The oldest henge was built about 3300
BC. A henge should not be confused with
a stone circle within it, as henges and
stone circles can exist together or
separately.
The largest henges enclose up to 12
hectares (1 hectare = 10000 square
metres). The three largest stone circles in
Britain (Avebury, the Great Circle at
Stanton Drew stone circles and the Ring
of Brodgar) are each in a henge. The
Stone-Circles website estimates there
to be around 1300 stone circles in Great
Britain, with the number of identified
round barrows currently standing at over
10000 and many new rock-art sites being
discovered every year.
Examples of henges without significant
internal monuments are the three henges
of Thornborough Henges, unique
because of their immense size (each 240
metres in diameter, 732 megalithic yards
in circumference) and because of their
relationship
to each other
- the entire
monument
covers a mile
of North
Yorkshire.
Click here for the MEI
Maths Item of the Month
www.mei.org.uk
J u n e
2 0 1 3
The archaeological term ‘henge’ is
actually a back-formation from the name
Stonehenge (coined in Anglo-Saxon
times – meaning ‘the hanging stones’),
but Stonehenge, which represents the
culmination of stone circle engineering, is
not at all a typical henge. It is atypical in
that the ditch is outside the main
earthwork bank.
Stonehenge was built around 3100 BC,
comprising a ditch, bank, and the Aubrey
holes (round pits in the chalk, about one
metre wide and deep, with steep sides
and flat bottoms). The famous standing
stones were added at later dates. There
is a diagram and map of the surviving
stones on the Stonehenge website.
Investigations
Archaeologist Mike Pitt shares evidence
that some henges may be oval rather
than circular. The Henges Engineering
in Prehistory site presents a new henge
theory. All the stones that make up
Stonehenge were transported over
varying terrain from up to 30 miles away.
Engineering Timelines demonstrates
how this might have been done.
A new MEI teaching resource is at the end of this
bulletin. Click here to download it from our website.
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Classroom activities
Summer solstice
Every year around
the 21st of June,
the sun rises
directly over
Stonehenge’s Heel
Stone, a sarsen
(sandstone block)
which marks the
avenue
surrounding the
inner stone circle,
as seen from the
centre of the
monument through
the horseshoe
arrangement. As
such, rays shine
directly into the
site’s core.
Stonehenge's
orientation in
relation to the
rising and setting
sun has always
been one of its
most remarkable
features.
Did its builders
come from a sunworshipping
culture or were its
circle and banks
part of a huge
astronomical
calendar?
Read Bruce
Bedlam’s
theories.
Build Your Own Stonehenge
Sunrise Sunset
The website Scientific Explorations
with Paul Doherty has some interesting
practical activities that explore the
solstice and equinox using a light bulb
and paper plates, and challenges
students to design a structure with angles
incorporated into the structure that point
to solstice sunrises and sunsets in their
chosen location. Mathematical tools are
provided, including a Solstice Calculator
and a Lunar Standstill Calculator.
Paper Plate
Education offers
an activity with
the aim of
developing an
awareness of the
horizon making a circle around the
observer. This activity involves drawing
a local horizon around the perimeter of
the eponymous paper plate*. Over
several months, track the changing
position of the sunrise or sunset against
the local background. The
accompanying video provides a short
introduction to this investigation. There
are other relevant activities on this
website, including:
Analemmatic
sundials:
How to build one and why they work
Chris Sangwin and Chris Budd explain
how to build this kind of
horizontal sundial in
which the shadowcasting object is vertical,
and is moved depending
on the declination of the
sun on a given day.
The Stonehenge
Problem
The nrich website gives suggestions for
practical investigations into how the
massive stones were transported some
distance to the site by water and land:
Moving Stonehenge: looks at the fluid
mechanic questions that are raised by the
Stonehenge ‘Bluestone’
Stonehenge: aims to develop a better
understanding of relative velocity through
a simple experiment with pencils and a
book
Stonehenge is Going Nowhere:
extension of the Stonehenge problem
Altitude of the Noon Sun: observing,
collecting data, and discovering the
pathway of the sun. Video available.
Altitude of the Noon Sun II:
investigating how the altitude of the noon
sun varies by season and by location.
Altitude of the Noon Sun III: using an
Altitude/Latitude finder to understand,
explain and demonstrate how the sky
changes when one changes latitude
*Note: We’ve looked
for the foam scallopedged plates
recommended for the
paper plate activities.
It is possible to buy them in UK from
Mashers disposable goods online store.
A new MEI KS4 teaching
resource follows. It can be
downloaded in its original format from the
Monthly Maths page of our website.
What is a day?
The time it takes
for the earth to
revolve once on its
axis.
A day or not a day?
The time it takes
for the earth to
revolve once on its
axis.
Which is 24
hours…
A day or not a day?
Actually, it’s not.
A day or not a day?
The earth moves
round the sun…
Everyone
knows that!
A day or not a day?
…and when you
face the sun, it’s
daytime…
…when you don’t,
it’s dark!
A day or not a day?
Let’s assume you’re right, that it takes
24 hours for the Earth to make one full
turn on its axis…
A day or not a day?
• Now imagine yourself at midday, looking out at
the sun…
A day or not a day?
• One full turn later, it’s midday again.
Where are you?
A day or not a day?
• Still facing the sun…
A day or not a day?
• Several full turn later, it’s midday again.
Where are you?
A day or not a day?
• Still facing the sun?
A day or not a day?
• Five months and lots of full turns later, midday
again… where are you?
A day or not a day?
• Now we’re really in trouble! Facing away from
the sun so definitely dark at midday.
A day or not a day?
A day is 24 hours, but
that’s not how long it
takes for the earth to
revolve on its axis
So what are you
saying? A day
isn’t 24 hours?
A day or not a day?
So how long
does it take?
A day or not a day?
How would you work
it out?
What do you know?
Stonehenge
• No-one actually knows for certain what
Stonehenge – or other Henges - were built for,
but they have properties which suggest that they
had religious purposes – and some people still
use them as places of worship.
• Many have special alignments with the sun on
certain days of the year.
• At Stonehenge the sun aligns with the Heel
stone on June 21st – the Summer Solstice.
Constructing a Henge
• Many Henges are not actually circular but seem
to have 3 centres.
• Look at the shape of Avebury henge
• One theory is that a single centre would mean
that centre point would ‘have too much power’.
• …but unless someone is very, very old, they
don’t really know for sure!
Constructing a Henge
• How can we construct a shape which is ‘round’
but not circular and has 3 centres?
• Look at the diagram on the next slide.
• Can you work out which centres and lengths
have been used to construct the shape?
Constructing a Henge
Constructing a Henge
• Firstly construct the Henge using the pencil and
paper method.
• How does the shape change when you place the
3 centres differently?
• You might then try constructing it outside or by
using Dynamic Geometry Software (DGS)
• Using DGS will enable you to more easily
explore how the shape changes.
Teacher notes: A day or not a day?
In this activity students will use basic arithmetic, but with a complex problem.
They will need to convert measures and work with time.
It is suitable for all students who like a challenge!
•
•
•
Show slides 1 to 16
Slide 17 How can we work out how long it takes the Earth to revolve once
on its axis. Ask students to discuss this problem in pairs. They need to
arrive at the following:
– It takes 365 ¼ days for the Earth to orbit the sun
– Every 24hours the Earth has to turn a little bit extra to remain facing the
sun at midday.
There are then 2 ways to calculate the time it takes for the Earth to turn
once:
Teacher notes: A day or not a day?
1. During the year, the Earth actually completes one extra turn.
– Minutes in a year = 365.25 x 24 x 60 = 525 960
– Turns in a year = 366.25
– Minutes per turn = 525 960 ÷ 366.25 = 1436.068 = 23 hours 56 mins 4
seconds
2. Thinking about how far the earth has to turn in each 24 hours, it’s a full turn
plus 1/365.25 of a turn which is 1.00274 turns.
– Seconds in a day are: 24x 60 x 60 = 86400
– Second per turn are: 86400÷ 1.00274 = 86164
– This is 23 hours 56 minutes and 4 seconds
•
•
•
Students might be able to work out what they need to do for the calculation,
but may need support in carrying it out.
Working with a partner will help.
Ask students to feedback their solutions to the problem
Teacher notes: Constructing a Henge
This activity gives students the chance to experiment with geometry using
pencil and paper methods. It is suitable for students of all abilities.
It is also possible to create an outdoor version, which could be photographed
from above. Students can then explore ‘roundness’ (see Monthly Maths April
2013) and check to see how round the shapes they’ve created are.
Using Dynamic Geometry Software will make it easier it explore how the shape
changes as the centres are moved.
Teacher notes: Constructing a Henge
•
•
•
•
•
•
Show the students slide 20
Show students the diagram on slide 22.
Ask them to discuss in pairs how the coloured shape might have been
constructed from points A, B, C and D.
Copies of the slide could be printed out for students to explore and test out
their ideas.
If they have some ideas then let them use paper, pencil, compass and ruler
to have a go at constructing one for themselves.
If they are struggling to work out how it has been created then hand out the
instruction sheets, allowing them to read and make sense of the instructions
for themselves.
Constructing a Henge: Pencil and Paper Method
You will need:
 Pencil
 Paper
 Compasses
 Ruler
Choose 3 points, label them A, B and C.
Draw 3 lines joining the points and extending beyond them in each direction.
Place a point D on AB, beyond B as shown.
Use centre B and radius BD to draw the red arc – stopping when it reaches the line BC.
Label where they meet as E.
Use centre C and radius CE to draw the orange arc – stopping when it reaches the line AC.
Label where they meet as F.
Use centre A and radius AF to draw the yellow arc – stopping when it reaches the line AB.
Label where they meet as G.
Use centre B and radius BG to draw the green arc – stopping when it reaches the line BC.
Label where they meet as H.
Use centre C and radius CH to draw the blue arc – stopping when it reaches the line AC.
Label where they meet as J.
Use centre A and radius AJ to draw the blue arc – stopping when it reaches the line AB.
This should meet at D.
What happens to the shape when you place the 3 centres differently?
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Constructing an ancient circle: Outside
You will need:
 3 people
 String
 Playground Chalk
 More people or cones or other markers
(the PE department might be able to help)
Make sure that you are familiar with how a henge is constructed on paper.
Place a person at A, B and C.
Use string and playground chalk to draw the lines AB, BC and AC. Pulling the string taut
and lining it up so it just touches a pair of people at a time.
Decide on a point D on AB, beyond B as shown.
Use centre B and radius BD to construct the red arc – stopping when it reaches the line BC.
You can either use playground chalk to sketch this arc out as the string moves, or place
students or cones along the arc.
Label where they meet as E and continue forming arcs as described on the pencil and paper
construction method.
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