INforM – Interactive Notebooks for Mathematics
Book 17 – Radioactive Dating
Radioactive dating
This is the third of four `thematic’ Notebooks whose aim is to present some
mathematics teaching ideas based on realistic problems and modelling which link to
other subjects and which can be pursued in an open-ended and problem-solving manner.
The activities are related to work in History and Science. The mathematics covers
sequences, linear functions, quadratic functions and logarithms – so different pages may
be omitted and the resulting Notebook saved for a particular year group.
Organisation of the materials
The Smart Notebook file is saved as `Book 17 Carbon dating.notebook’.
It consists of 16 pages of which the first is the title page, shown above.
There are 13 pages to support the activity and its extension. Page 15 is a blank page.
Page 16 contains teacher notes which are amplified here.
The first activity
The title page shows images of
King Arthur and his castle at
Camelot. Page 2 shows
images of the great round table
which hangs in Winchester.
There is an opportunity for a
(brief) discussion about what
the students know about the
Arthurian legend in general
and the Knights of the Round
Table in particular.
INforM
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December 2006
Page 3 just helps set the
problem in context – it is
a real problem which
was solved by ground
breaking work in British
archaeological research.
The radiocarbon dating
technique was invented
in the US by Willard
Frank Libby in the late
1940’s for which he
received the Nobel Prize
for Chemistry in 1960.
Page 4 provides the
scientific detail needed
to begin the
investigation. Needless
to say it is a somewhat
simplified version of
what actually took place,
but the basic techniques
are realistic and sound.
In practice, there are
methods for detecting,
and correcting, any
`wobbles’ in the natural
occurrence of 14C in any
particular period.
Page 5 shows an
accurate decay graph
drawn in TI InterActive!
of the function y = ax
where a = 0.51/5730 
0.999879 . The
questions are designed to
give practice in reading
off from the graph
(interpolating) and in the
inverse process. Lines
can be sketched with a
pen or drawn with a tool.
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Page 6 shows just a small
portion of the decay graph,
which is more or less linear.
If you interpolate an age of
1500 years it corresponds to
about an 83% concentration.
Inverse interpolation gives
an age of about 800 years.
Page 7 shows the start of a
trial and improvement
method of solving
0.5 = a5730 without
introducing index notation
for nth roots. It can be
carried out using scientific
or graphical calculators.
We need a value for a just a
little smaller than 0.999 so
this is a good exercise in
place value.
Page 8 shows a display
taken from the TI SmartView
software emulator of a TI-84
graphical calculator. If you
have a copy it would be an
ideal tool to use on the
SMARTBoard. Here the
notation for nth root as ^(1/n)
is introduced. Now the
solution is given by using
trial and improvement on the
equation 0.91 = 0.999879x.
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Page 9 shows an approach to
the graphing and inverse
interpolation methods using
the Graph, Trace and Table
features of a graphical
calculator.
Page 10 show what an
excellent fit can be obtained
from a linear function for
ages between 0 and 1000
years. The coloured
rectangles can be dragged
away to reveal each of the
answers. Here is a nice
example of the use of
algebraic manipulation to
find the inverse of a linear
function and then using
substitution to complete the
solution of an equation.
Page 11 shows a painting of
John Napier taken from the
History of Mathematics
project web-site at St.
Andrews University:
http://www-history.mcs.standrews.ac.uk/Mathematicia
ns/Napier.html . There is
also a biography of him
there. So this is to help
prompt some collaborative
project work in maths and
history about great
mathematicians and the
times they were living and
working in.
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December 2006
Advanced students may enjoy making their own searches for information about
mathematical topics such as natural logarithms. Two very useful sources are Eric
Weisstein’s Mathworld and Wikipedia:
http://mathworld.wolfram.com/NaturalLogarithm.html
http://en.wikipedia.org/wiki/Natural_logarithm .
Page 12 introduces the
very important issue of
accuracy when expressing
a result – and introduces
the idea of error bounds,
which are very important
in practice. Of course the
margin of error here just
relates to the accuracy
with which the
concentration was
measured, and not to that
of the assumption that 14C
concentration stayed
constant.
Page 13 relates the
mathematics back to the
original problem of the
Round Table and provides
a lead to a different
technique used by
archaeologists to date
wooden material. The
king in question would
appear to be Henry III
(1216-1272) who was born
in Winchester.
We have rather assumed that all the wood in the table is the same age. Of
course, in practice, as pieces of wood rot, or are otherwise damaged, they would have
been likely to be repaired with contemporary wood – so it was important that many of
the pieces of wood used in the table had their concentration measured, and only the
oldest ones used for the analysis. A good case in point is Nelson’s flagship, HMS
Victory, kept in dry dock in the Portsmouth naval base. This is constantly being
maintained with old timbers being replaced – so it is debatable whether any part of her
actual belongs to the original ship!
This is the important issue in the controversy over the dating of the tiny
swatches of fabric from the shroud in Turin cathedral. Independent radio carbon dating
at the Universities of Oxford and Arizona showed that fabric to be mediaeval – but the
counter argument says that these could have come from a repaired patch of the shroud,
and that the main fabric is much older. See for example:
http://www.shroud.com/nature.htm
http://news.bbc.co.uk/1/hi/sci/tech/4210369.stm
http://www.mcri.org/Shroud.html .
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The opening is also given to explore the relation between the archaeological technique
of dating using radioactive decay and the forensic technique of using Newton’s law of
cooling to date the time of (recent) death. There are web-sites with mathematical
activities based on TV programs about forensic analysis of crimes and `scene of crime
investigations’.
http://www.tiforensics.com/
http://education.ti.com/educationportal/sites/US/nonProductSingle/activitybook_forensics.html
http://www.weallusematheveryday.com/tools/waumed/home.htm
Page 14 provides the links to the
software file included in the
Attachments. They each allow
you to slide a point to change
the value of the Age and to read
off the corresponding value of
the % concentration, and vice
versa. A new feature being
exploited here is that Cabri has
now provided a way to create
interactive diagrams on a webpage using Cabri II Plus.
Page 16 has the Teacher notes.
The Attachments tab also has
files in Excel and TI InterActive!
which can be opened using the
links on this page. There is also
a link to the CabriLog website
from where the free `plug-ins’
needed for the html documents
can be downloaded.
The whole Notebook is intended as an exemplar of how information can be gathered
and presented around a theme. Such a presentation can be developed by a teacher, by
an individual student or a group of students.
Reference
Biddle, M.: 2000, King Arthur’s Round Table. Woodbridge: Boydell Press.
INforM
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December 2006