Homecoming 2014: Great Scots
John Napier > Numeracy and Mathematics > First, second and third level
Key theme: John Napier: Great Scottish Mathematician
(multiplication, decimal fractions and logarithms)
Introduction
This series of learning experiences at first, second and third level, offer activities to support learners in
developing an understanding of multiplication, decimal fractions and powers and roots. The
suggested learning experiences can provide a foundation for exploring contexts for learning in the number
and number processes, powers and roots and fractions decimal fractions and percentages
experiences and outcomes.
John Napier was a 16th century Scottish mathematician born in Edinburgh in 1555 and educated at St
Andrews University in Fife. He is recognised as a Great Scot for his contribution to mathematics as we know
it today. These contributions included the creation of mathematical instruments such as Napier’s Bones
for multiplication and the introduction of decimal notation for fractions.
John Napier is most recognised for the invention of logarithms. He had a great interest in astronomy which
led to his contribution to mathematics. However, Napier was not just a star gazer; he was involved in
research that required lengthy and time consuming calculations of very large numbers. Once the idea came
to him that there might be a better and simpler way to perform large number calculations, Napier
focused on the issue and spent twenty years perfecting his idea. The result of this work is what we now call
logarithms.
"Seeing there is nothing that is so troublesome to mathematical practice.... than the multiplications,
divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are...
subject to many slippery errors, I began therefore to consider [how] I might remove those hindrances."
Source: A Description of the Wonderful Canon of Logarithms by John Napier
www.educationscotland.gov.uk/studyingscotland
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Image source:
http://commons.wikimedia.org/wiki/File:John_Napier.JPG
Homecoming 2014: Great Scots
John Napier > Numeracy and Mathematics > First, second and third level
Prior learning
Pupils would benefit from:
First level – familiarity with a variety
of methods for single digit
multiplications.
Second level – confidence with
fractions and recognising that there
are numbers between whole
numbers.
Capabilities
Successful Learners -.apply their
understanding in unfamiliar contexts.
Confident Individuals – explain
their mathematical thinking to others.
Responsible Citizens – understand
the contribution John Napier made to
number.
Effective Contributors – solve
mathematical problems by applying
their knowledge of concepts
Social Studies experiences and outcomes
First Level: Napier’s Bones
I can use addition, subtraction, multiplication and division when solving problems, making
best use of the mental strategies and written skills I have developed. MNU 1-03a
By exploring places, investigating artefacts and locating them in time, I have developed an
awareness of the ways we remember and preserve Scotland’s history. SOC 1-02a
Second Level: Napier’s decimal point
I have extended the range of whole numbers I can work with and having explored how
decimal fractions are constructed, can explain the link between a digit, its place and its
value. MNU 2-02a.
To extend my mental map and sense of place, I can interpret information from different
types of maps and am beginning to locate key features within Scotland, UK, Europe or the
wider world. SOC 2-14a
Through exploration and discussion, I can understand that food practices and preferences
are influenced by factors such as food sources, finance, culture and religion. HWB 2-34a
Third Level: Napier’s logarithms – understanding powers
Having explored the notation and vocabulary associated with whole number powers and
the advantages of writing numbers in this form, I can evaluate powers of whole numbers
mentally or using technology.
MTH 3-06a
Pupils may have prior experience of second level experience and outcome: By observing
and researching features of our solar system, I can use simple models* to communicate
my understanding of size, scale, time and relative motion within it.
*Extend this at third level by applying knowledge of indices to make mathematically accurate scale
models. SCN 2-06a
By applying my knowledge and skills of science and mathematics, I can engineer 3D
objects which demonstrate strengthening, energy transfer and movement.
TCH 3-12a
www.educationscotland.gov.uk/studyingscotland
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Interdisciplinary
opportunities
Social Studies:
Investigate life in Scotland
at the time of John Napier
and consider the impact of
the mathematical
discoveries on wider
society. Apply knowledge
of decimal notation to
scale drawing.
Technologies: Apply
knowledge of powers in
measurement contexts
including area and volume
calculations.
Science: Apply
mathematical notation to
demonstrate
understanding of scientific
concepts such as cell
division.
Higher Order Thinking
Learners can express the
depth of their
understanding of the
concepts explored through
model making,
externalising their
mathematical thinking.
Homecoming 2014: Great Scots
John Napier > Numeracy and Mathematics > First level
Learning experience at First level: Napier’s bones (multiplication)
Introduction
‘Napier’s Bones’ investigates the
multiplication aspect of the number and
number processes experiences and
outcomes by asking learners to investigate
different methods for multiplying.
Learners would benefit from having spent
time learning their multiplication facts (at
least 2, 3 and 5 to make this accessible)
prior to embarking on this lesson.
Stimulus
Napier’s Bones
John Napier was a Scottish mathematician
who developed a clever calculating tool for
finding the answers to challenging
multiplication questions. This ingenious
tool for multiplying became known as
‘Napier’s Bones’ and was revealed to the
world in 1617.
Possible learning opportunities/tasks
Possible evidence
Learners could:
 Discuss the different methods used for multiplying
and/or the way they remember their multiplication
tables and create a short video to demonstrate
these methods (to their peers to ensure a real
purpose).
 Use arrays to consolidate understanding of
multiplication
 Create their own set of Napier’s Bones using lolly
sticks or cardboard/paper
 Use area models to introduce multiplying a single
digit by a 2 digit number – this could be
differentiated e.g. 2 x 12 for less confident learners
and 4 x 27 for more confident learners)
 Be the teacher! Learners create word problems
involving multiplying, swap their word problems with
another group and mark the answers using
Napier’s Bones and check their answer using one
other method that they know.
SAY - explain their methods for multiplication
WRITE – multiplication problems
MAKE – Napier’s bones /videos
DO – present videos to younger pupils and design activities
based on this
Useful resources
Nrich activity exploring Napier’s Bones
http://nrich.maths.org/1132
Key Learning
Flip cameras or equivalent
Learners can:
Scrap paper/maths jotters
 demonstrate understanding of
Mini white boards & pens
multiplication
Manipulatives e.g. Cuisenaire rods, multilink, base ten
 multiply a variety of numbers using a
materials
range of strategies.
Lolly sticks/cardboard/paper
Lesson plan: Napier’s bones
http://nzmaths.co.nz/resource/napiers-bones
Demonstration – how Napier’s Bones work
http://ww2.gannon.edu/cetl/napier/
www.educationscotland.gov.uk/studyingscotland
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Reflecting on learning
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Can learners explain their methods for multiplying
confidently and competently?
When creating Napier’s rods, can learners remember their
multiplication table facts when they are creating Napier’s
rods and if they can’t, do they have efficient strategies for
working out the stations?
Are pupils using known strategies for estimating their
answers prior to calculating 2 digit numbers multiplied by
single digit numbers?
Are word problems, written by the learners, of an
appropriate context for the numbers involved.
Taking it further
Numeracy and Mathematics - Learners could look consider
other ancient methods for calculations including the abacus.
Learners could make their own abacus using beads and string.
Social studies – Learners could investigate what was happening
in Scotland at the time of John Napier: Mary Queen of Scots was
on the throne, Union of the Crowns
http://www.educationscotland.gov.uk/scotlandshistory/timeline.asp
Homecoming 2014: Great Scots
John Napier > Numeracy and Mathematics > Second level
Learning experience at Second Level – Napier’s Decimal Point (decimalisation)
Introduction
Possible learning opportunities/tasks
Possible evidence
‘Napier’s Decimal Point’ involves learners
exploring the depth of their understanding
of decimals. Learners will have the
opportunity to investigate the significance
of the decimal point which will support
their understanding in number and
number processes, measurement and
fractions, decimal fractions and
percentages experiences and outcomes.
Pupils could:
WRITE – create mind map to demonstrate knowledge of the
importance of decimals notation in everyday applications.
DO – take photographs of their decimal models
SAY – justify why they have chosen the materials they have to make
their model
MAKE – decimal models to demonstrate understanding
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Investigate other learners’ understanding of decimals
using examples of other pupils’ work (see graphic
discussion tool ‘Decimals – What do you think?’).

Make 3D decimal models using a variety of materials

Use Cuisennaire or base ten materials to create
decimal models (linking fractions to decimals)
Stimulus
Napier was responsible for advancing the
notion of the decimal fraction by
introducing the use of the decimal point.
His suggestion that a simple point could
be used to separate whole number and
fractional parts of a number soon became
accepted practice throughout Great
Britain.
Key learning
Learners can:
 understand and explain what a decimal
point is and why it is used
 name a variety of contexts where the
decimal point would be important
Work together to create a mind map showing all the
contexts in which decimal numbers are used in
everyday life. (e.g. Money, measurements, time) This
video may be useful to assess what they have found
or to stimulate conversation:
http://www.bbc.co.uk/skillswise/topic/decimals
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Make 2D decimal models using squared paper
(understanding of place value of decimals)
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Graphic discussion tool ‘Decimals – What do you
think?’ for assessing pupils understanding of
decimals (which can be adapted to meet your
learners’ needs)
Large sheets of paper
Selection of materials for making decimal models:
see attached - Decimal models guidance sheet
3D and 2D decimal models examples
Squared paper
www.educationscotland.gov.uk/studyingscotland
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Can pupils think of at least 3 different contexts where decimals
are used in everyday applications?
Do pupils’ models demonstrate an understanding of the
magnitude of number? (the quantity or size of the materials used
in the models reduces or increases depending on the position –
see 3D and 2D decimal models examples for further guidance)
Are pupils able to demonstrate their understanding of the link
between fractions and decimal fraction notation using a model?
Taking it further
Useful resources
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Reflecting on learning
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Numeracy and Mathematics – use decimal notation in applications
such as recording measurements – converting between units of
measure e.g. millimetres to metres
Social subjects – measure distances on a map and calculate actual
distances by applying scale. Record lengths using decimal notation.
Health and wellbeing –plan a shopping list for a recipe. Identify the
quantities of milk, flour etc. required and converting units of measure,
in order to decide how many bags/bottles etc. are required.
Homecoming 2014: Great Scots
John Napier > Numeracy and Mathematics > Third level
Learning experience at Third Level – Napier’s logarithms (Understanding powers and roots)
Introduction
Possible learning opportunities/tasks
Pupils could:
‘Napier’s logarithms’ can develop understanding of
powers, indices and roots. Learners will have
the opportunity to experiment with visual
representations of familiar powers before moving
on to explore indices in a broader sense.
‘Taking it further’ provides ideas for learners to
apply their understanding in context.
Stimulus
Napier realised that all numbers can be expressed
in what is now called exponential form, meaning 8
can be written as 23, 16 as 24 and so on. What
make logarithms so useful is the fact that the
operations of multiplication and division are
reduced to simple addition and subtraction.
The relevance and history of Napier’s work could
be explored using this short video clip:
http://ed.ted.com/lessons/how-does-math-guideour-ships-at-sea-george-christoph
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Discuss the meaning of square and cubic numbers in
familiar, real life applications.
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Use concrete materials to explore square and cube
numbers and square roots – squared paper, peg boards,
geoboards with elastic bands or dotted paper Base Ten
Dienes Blocks could be used or Visualising Squares and
square roots interactive tool .
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Explore the effect of indices through problem solving
investigations.
Learners could be encouraged to first look for a pattern and
then apply a rule, using index notation to record the
relationships.
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Sissa’s reward http://nrich.maths.org/1163
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BBC class clips: A geometric sequence problem
Multiplying Bacteria Problem
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Play Indices, Powers and Roots game
Useful resources
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Key learning:
Learners can:
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represent multiplication as a power
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understand and use square and cube roots.
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www.educationscotland.gov.uk/studyingscotland
National STEM centre: SMILE resource contains two
packs of games, investigations, worksheets and practical
activities supporting the teaching and learning of powers
and roots.
http://www.nationalstemcentre.org.uk/elibrary/resource/78
55/powers-and-roots
The history of place values and index notation: BBC
class clips clip 13585
http://www.bbc.co.uk/learningzone/clips/the-history-ofplace-values-and-index-notation/13585.html
Indices explained: BBC class clips
http://www.bbc.co.uk/learningzone/clips/bitesize-mathsindices/13681.html
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Possible evidence
DO – show their working when carrying out investigations which
exemplify what mathematics is taking place when indices are used.
MAKE –. Models to represent square numbers, cubic numbers and
roots.
SAY – explain why indices are useful in real life.
WRITE – index notation to represent the solution to a word problem
Reflecting on learning
Are learners:
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able to suggest at least one real life application for indices,
powers or roots?
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demonstrating confidence when writing the mathematical
notation which represents the models they have made?
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able to show their working in a systematic way which naturally
demonstrates how indices could be used as shortened notation
for their working?
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able to identify pairs when playing Indices, Powers and Roots
game?
Taking it further
Numeracy and Mathematics Learners could go on to explore
indices in algebra to assess deeper understanding.
Science Make scaled down models of the solar system using
indices to inform their scale.
Technologies Learners could investigate packaging to explore
cubic numbers further http://nzmaths.co.nz/resource/boxing