NATIONAL QUALIFICATIONS CURRICULUM SUPPORT
Mathematics
Numeracy
Advice and Guidance for
Practitioners
[NATIONAL 5]
This advice and guidance has been produced to support the profession with the delivery of
courses which are either new or which have aspects of significant change within the new
national qualifications (NQ) framework.
The advice and guidance provides suggestions on approaches to learning and teaching.
Practitioners are encouraged to draw on the materials for their own part of their continuing
professional development in introducing new national qualifications in ways that match the
needs of learners.
Practitioners should also refer to the course and unit specifications and support notes which
have been issued by the Scottish Qualifications Authority.
http://www.sqa.org.uk/sqa/34714.html
Acknowledgement
© Crown copyright 2012. You may re-use this information (excluding logos) free of charge in
any format or medium, under the terms of the Open Government Licence. To view this licence,
visit http://www.nationalarchives.gov.uk/doc/open-government-licence/ or e-mail:
psi@nationalarchives.gsi.gov.uk.
Where we have identified any third party copyright information you will need to obtain
permission from the copyright holders concerned.
Any enquiries regarding this document/publication should be sent to us at
enquiries@educationscotland.gov.uk.
This document is also available from our website at www.educationscotland.gov.uk.
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ADVICE AND GUIDANCE FOR PRACTITIONERS
Advice and guidance for practitioners
Numeracy is a skill for life, learning and work. Having well -developed
numeracy skills allows people to be more confident in social settings and
enhances enjoyment in a large number of leisure activit ies. For these and
many other reasons, practitioners have important parts to play in enhancing
the numeracy skills of all learners.
A numerate person will have acquired and developed fundamental skills and
be able to carry out number processes, but beyond this being numerate also
allows us to access and interpret information, identify possibilities, weigh up
different options and decide on which option is most appropriate.
This advice and guidance promotes and supports effective learning and
teaching of numeracy to stimulate the interest of learners and promote
creativity.
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ADVICE AND GUIDANCE FOR PRACTITIONERS
Area of numeracy
Selecting and using appropriate
numerical notation and units
Learning and teaching approaches
Applying is the ability to use existing
information to solve a problem in a different
context, and to plan, organise and complete a
task.
Learners could be encouraged to think about
how they are going to tackle problems, decide
which skills to use and then carry out the
calculations in order to complete the task. To
determine a learner’s level of understanding,
learners could be encouraged to show and
explain their thinking.
Selecting and carrying out
calculations involving whole
numbers, fractions, decimals,
percentages, a simple formula,
ratio and proportion
Analysing and evaluating is the ability to
identify and weigh up the features of a
situation or issue and to use your judgement of
them in coming to a conclusion. It includes
reviewing and considering any potential
solutions.
Practitioners are encouraged to promote
learning and understanding through interactive
activities using a variety of real-life contexts.
Practitioners have the opportunity to adopt a
range of resources, particularly ICT and other
media.
Investigative work using the internet, Google
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Exemplification
Learners’ skills should be developed so that they can:
 use knowledge of rounding and estimation
 use mental strategies to carry out calculations
efficiently, such as number facts, doubling,
approximations and times tables
 understand the interrelationships between fractions,
decimal fractions and percentages to choose an
efficient route to a solution.
Calculating speed, distance and time of journeys using
Google maps. 1
Calculating cost of journeys, exploring different time
zones, measuring the area and perimeter of buildings and
compound shapes using Google Earth 2 ruler tool, 3
examining temperature differences between countries
using Google Earth layers. 4
Teachers’ TV hotel activity – numerical calculations
within the context of running a large hotel . 5
TES also provides a range of activities for ratio, direct
and indirect proportion.
National Centre for Excellence in the Teaching of
Mathematics (NCETM): Maths in work videos
illustrating numeracy in real-life contexts. 6
Improving learning in mathematics standards unit to
promote deeper understanding of fractions, decimals and
ADVICE AND GUIDANCE FOR PRACTITIONERS
Maps and Google Earth.
Interdisciplinary activities which involve the
selection and use of a range of numerical
processes such as calculating food miles, with
links to developing global citizenship.
Exploring changes in temperatures to examine
global warming along with social studies.
Learners will be able to apply their knowledge
of temperature change to analyse and evaluate
the impact of rising sea levels and other
climate change issues.
A range of other opportunities for numeracy in
real-life contexts exists across the skills for
work (SfW) courses. Centres are encouraged
to explore these opportunities to enhance the
learners’ experiences of using numeracy
across disciplines.
percentages, including percentage increase and
decrease. 7
Skilled to go web resources for utility bill calculations
and purchasing goods activities include compound
percentage calculations in real-life contexts. 8
Maths case studies involving selecting and carrying out
calculations to promote deeper understanding and
reasoning skills. 9 Investigative learning opportunities for
volume calculations and scale drawing.
Games-based short-response questions available on
Manga High, 10 including adding and subtracting simple
fractions and calculating fractions of quantities.
Contextualised short- and extended-response
questions using worksheets and textbooks may
be used to reinforce key concepts.
Other National Qualification (NQ) courses (eg
Accounting) may provide opportunities to
develop skills in number processes, money,
time and information handling.
Reading measurements using a
Practitioners are encouraged to promote
learning and understanding through interactive
Using an appropriate mathematical instrument where the
major divisions are marked, learners should be able to
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ADVICE AND GUIDANCE FOR PRACTITIONERS
scale on an instrument
activities using a variety of real-life contexts.
Practitioners may explore opportunities for
interdisciplinary learning activities, including
measuring. For example, real-life contexts
exist within the SfW courses, including
Construction Crafts, Energy, Engineering
Skills and Hospitality which require learners
to undertake a range of different measuring
activities from scale drawings to weighing
ingredients.
measure to the nearest marked minor unnumbered
division, eg measuring millimetres and millilitres using
appropriate scales.
Use of a planning tool from a DIY retailer will help to
make this relevant to learners. Opportunities to include
scale drawing, area, perimeter, volume calculations and
measurements.
TES provide a number of useful activities, including the
My room activity, 11 providing lesson plans and materials
to deliver a series of lessons involving measure.
An opportunity for interdisciplinary learning that may be
explored is the SfW Energy: Domestic Wind Turbines
Systems unit. This unit provides learners with the
opportunity to design a tail vane for a wind -turbine. This
practical exercise provides a real-life context for
learning and could be supplemented to include further
measuring activities and calculations to meet the criteria
set out above.
Additional opportunities for real-life contexts include
the SfW Hospitality and Construction Craft courses.
Practical cookery skills for the Hospitality Industry unit
require learners to weigh and measure ingredients
accurately.
Similarly, the Bench Joinery unit within SfW
Construction Craft requires learners to measure
accurately.
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ADVICE AND GUIDANCE FOR PRACTITIONERS
These practical activities could be supplemented to
include measuring to the nearest marked unnumbered
division on a scale.
Justifying decisions using the
results of calculations
Case studies, project work and collaborative
learning opportunities to demonstrate
knowledge and understanding of money/time
and measurement.
The SfW Financial Services: Personal Finance
Awareness unit requires learners to create
examples of three real-life events that would
require them to borrow using a range of
borrowing products, ie mortgages, loans etc.
Practitioners may wish to use this opportunity
to examine the percentage calculations
involved in comparing and contrasting
different financial products.
Calculations may include compound percentages and
percentage increase/decrease to compare and contrast
different financial products.
Financial education materials are available from pfeg 12
to answer questions including:
1.
How will I manage my money
2.
What financial risks are out there?
3.
How can I protect myself against these risks?
4.
Is all debt ‘bad debt’?
5.
How can numeracy skills help me deal with money
at work?
Renewable energy projects on STEM central 13 provide a
number of learning opportunities involving a number of
calculations, including measure and money. Explore
local issues, eg building a bypass.
The local environment provides ample contexts for
learning that is meaningful and relevant.
Extracting and interpreting data
from at least three different
graphical forms
Opportunities for practitioners and learners to
explore, research and investigate data and its
representation, including tabular, charts, maps,
dot plots and graphs. Higher-order thinking
can be developed through high-level
Improving learning in mathematics standards units – cooperative learning activities, including interpreting bar
graphs, pie charts and box plots. 14
Sources for investigative work could include
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ADVICE AND GUIDANCE FOR PRACTITIONERS
questioning.
Centres delivering SfW courses may wish to
explore opportunities to develop this area of
numeracy within the existing courses. The
SfW Energy: Conventional Technologies and
the Grid unit requires learners to investigate
different sources of energy and organise their
information and data, producing clear
summaries. This data and information could be
extracted from appropriate graphs and charts
as detailed above.
newspapers, magazines, The Guardian data site and TV
advertisements.
Opportunities for interdisciplinary learning in this area
exist across a range of different subjects, including:
- National 5 Geography – ability to use a range of
numerical and graphical sources of evidence
- National 5 Modern Studies – ability to use a range of
sources of information, which are mostly
straightforward but may have some complex features,
to detect and explain bias, exaggeration and
selectivity in the use of fact
Making and justifying decisions
using evidence from the
interpretation of data from
graphical forms
Promoting an awareness of how data,
information and statistics are portrayed in the
media and using this to make informed
decisions and justify those decisions.
- National 5 Biology – demonstrate analysing and
evaluating information, drawing conclusions, giving
explanations and making predictions in the context of
life on Earth.
Making and justifying decisions
based on probability
Opportunity for case studies and project work
using a range of real-life data sources.
Improving learning in mathematics – evaluating
probability statements. 15
Case studies examining health and safety statistics and
crime rates in specific postcode areas. Sources include
the Scottish neighbourhood statistics website 16 and
gapminder website. 17
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Notes
When evaluating bias, the following suggestions may be considered:
 Which supermarket is best, based on a survey taken outside a particular
supermarket?
 Is weekend homework valuable? Significantly more pupils asked than
teachers.
 The government based their decision on a survey taken in a specific
demographic.
Long whiskers on a boxplot may be due to a single extreme value – the
maximum or minimum – or a large proportion of the upper/lower quarter of
data. More advanced exploratory data analysis makes use of upper and lower
fences (Q 3 + 1.5 × IQR and Q 1 – 1.5 × IQR). Hence, each datum outside the
fences is denoted by an asterisk and the whisker ends at the
minimum/maximum value inside the fences. An outlier is defined as anything
outside the fences.
Learners will be well practised on interpolation: the estimation of one
variable given the other when the values are within the range of the data
given. They will probably not be so familiar with extrapolation, ie extending
the line of best fit to estimate values outside the range of the given data. This
will enable them to make predictions, but care should be taken when
extrapolating. Learners should understand that a line of best fit is based on
data in the given range and may well not apply to values outside the range.
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ADVICE AND GUIDANCE FOR PRACTITIONERS
http://maps.google.co.uk/maps?hl=en&tab=ll – Google Maps website.
http://www.google.co.uk/intl/en_uk/earth/index.html – Google Earth
website.
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http://support.google.com/earth/bin/answer.py?hl=en&answer=148134 –
Google Earth, measuring distances and area.
4
http://support.google.com/earth/bin/answer.py?hl=en&answer=181050 –
Google Earth, weather.
5
http://www.tes.co.uk/teaching-resource/Hotel-6049007/ – Mathematical
sequences from behind the scenes at a London hotel.
6
https://www.ncetm.org.uk/resources/11329 – National Centre for Excellence
in the Teaching of Mathematics.
1
2
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http://tlp.excellencegateway.org.uk/resource/su_mat_5822/screens/math_004_
002_005/page.html – Excellence Gateway, improving learning in
mathematics.
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http://www.oft.gov.uk/about-the-oft/partnership-working/partnershipworking-info/consumer-education/resources/sthome – Skilled to Go, learning
everyday consumer skills.
9
http://www.bowlandmaths.org.uk/index.htm – Bowland Maths, an
imaginative resource for teaching mathematics.
10
http://www.mangahigh.com/en_gb/ – Manga High, a free games-based
maths teaching resource.
11
http://www.tes.co.uk/teaching-resource/My-Bedroom-6019967/ – A TES
resource based around ‘My Bedroom’.
12
http://www.pfeg.org/ – Personal Finance Education Group.
13
http://www.ltscotland.org.uk/stemcentral/index.asp – STEM Central,
engineering through sciences, technologies and maths .
14
http://tlp.excellencegateway.org.uk/resource/su_mat_5822/screens/math_004_
005_005/page.html – Excellence Gateway, improving learning in mathematics
(interpreting bar graphs, pie charts and box plots) .
15
http://tlp.excellencegateway.org.uk/resource/su_mat_5822/screens/math_004_
005_005/page.html – Excellence Gateway, improving learning in mathematics
(evaluating probability statements).
16
http://www.sns.gov.uk/Reports/Report.aspx?ReportId=2&AreaTypeId=LA&A
reaId=400 – Scottish Neighbourhood Statistics, the Scottish Government’s
ongoing programme to improve the availability, consistency and accessibility
of small area statistics in Scotland.
17
http://www.gapminder.org – Gapminder, for a fact-based world view.
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