1
Water → Focus on Mathematics
Mathematical Modelling Learning Journey
Introduction
This mathematics learning journey, aimed at fourth level, looks at the learning context of flooding and makes use of
v
mathematical
modelling to inform thinking in relation to questions such as:
o How might you use mathematics to model the mechanisms that produce flooding?
o How might you calculate the risk of flooding in a particular area?
o If you had to report to the media to the people about taking action against flooding, how confident could you be
about your figures (and recommendations)?
o If you were to recommend flood defences, how could you justify the expense? Would the cost of defence be less
than the cost of repair? How might you estimate these costs?
This learning context would provide rich opportunities for planning for interdisciplinary learning with departments such
as technologies and geography.
An overview planning sheet is shown on the next page which gives a plan for learning and teaching starting from the
experiences and outcomes. Learning intentions, success criteria and possible evidence of learning which could be
gathered are shown. The pages that follow the overview give examples of learning experiences which would address
the learning intentions. They illustrate opportunities for learners to develop knowledge and understanding and to
apply relevant skills.
Mathematics experiences and
outcomes
Using proportion, I can calculate the change in one
quantity caused by a change in a related quantity
and solve real-life problems. MNU 4-08a
Having explored how real-life situations can be
modelled by number patterns, I can establish a
number sequence to represent a physical or pictorial
pattern, determine a general formula to describe the
sequence, then use it to make evaluations and solve
related problems. MTH 4-13a
Responsibility of all areas which
could be addressed in this learning
journey:
By considering the type of text I am creating, I can
independently select ideas and relevant information
for different purposes, and organise essential
information or ideas and any supporting detail in a
logical order. I can use suitable vocabulary to
communicate effectively with my audience.
LIT 4-26a
Interdisciplinary learning planning
opportunities
An engineering challenge related to flooding could
be incorporated into planning for interdisciplinary
learning between departments relating to a scenario
where a local authority wants to consider introducing
a scheme to encourage the use of electric transport.
Possible links
Technologies TCH 4-14b, TCH 4-12b
2
Water → Focus on Mathematics
Mathematical Modelling Learning Journey
Experiences and
outcomes
Using proportion, I can calculate the
change in one quantity caused by a
change in a related quantity and solve
real-life problems.
MNU 4-08a
Learning intentions
Investigate and evaluate a simple
‘analogy’ model of a flooding scenario
Having explored how real-life situations
can be modelled by number patterns, I
can establish a number sequence to
represent a physical or pictorial pattern,
determine a general formula to
describe the sequence, then use it to
make evaluations and solve related
problems.
MTH 4-13a
I can explore a number of analogies
between everyday situations and the
problem of flooding such as
an overflowing bath, crowd control,
motorway management, and relate
these to the problem of flooding
I can use mathematics confidently to
model situations
I can recognise some of the limitations
of my model
Possible evidence
Evidence of learning which could be
gathered could include:
o their practice familiarising
themselves with each model
examined
o the numeracy associated with
particular examples
o algebraic approaches to
generalising the situation to be
analysed
Reporting back in a range of ways on
the pros and cons of each model and
some description of a solution to
flooding based on them
Prior knowledge
o Work with others to explore, and
present findings on, how
mathematics impacts on the world
and the important part it has played.
o Explore number sequences, and
apply rules to extend pattern.
o Apply number facts to solve
problems algebraically.
o Use representations of 3D objects
and a range of methods to interpret
them.
Success criteria
Use data to evaluate the risk of
flooding and possible costs associated
with the clean up
Compile a report outlining some
possible solutions to flooding
I can identify some of the factors which
affect flooding and can begin to use
these to measure the risk of flooding
I can estimate costs associated with
cleaning up damage caused by
flooding, given data to work with, for
example, population of an area
I can use technology to produce a
report of my findings including tables,
graphs and charts
I can use the results of my
investigations to inform
recommendations in my report
Data gathered with annotations on
what it may show
Costings
Recordings of group discussions
A report, supported by facts and
figures, charts and tables, on a
possible particular local problem, with a
measure of the risk (for example, once
every 10 years) and of the costing of
some solutions, a costing of possible
damage repair; comparison between
the two measures; a recommendation
for action
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Water → Focus on Mathematics
Mathematical Modelling Learning Journey
Learning experience
Possible tasks
Possible evidence
Introduction
Discuss the scenario of a bath left unattended due
to unforeseen circumstance, for example, taps
turned on but plug not yet in place when the phone
rings.
Evidence of learning which could be gathered could
include:
o their practice familiarising themselves with each
model examined
o the numeracy associated with particular examples
o algebraic approaches to generalising the
situation to be analysed
Explain to the learners that they are going to be
using a number of analogies to help them explore
flooding and that as they explore each of these
they will consider:
o how the model can be used to help them
understand some of the factors involved in
flooding
o possible solutions
o limitation the simplified model/analogy has
In the possible tasks suggested on this page, the
model of the overflowing bath is explored.
Teacher’s notes with ideas relating to this
learning journey can be found in supplementary
resources on the STEM Central website.
Learning intention
Investigate and evaluate a simple ‘analogy’ model
of a flooding scenario.
What information is needed to analyse the
situation?
Things to be elicited from discussion could include:
o how long it would take to fill the bath using the
hot tap only with the plug in
o how long it would take using the cold tap only
with the plug in
o how long it would take to empty the bath with
the plug out and the taps off
Discuss rates of flow and how they could be
combined.
Some problems could be given to the learners to
work with, to help them become comfortable with
the combination of rates of flow.
Discuss either a time limit for which the state of
affairs has to be endured or a fixed volume of
water passing through the system.
o their discussions on the possible limitations of
the model
Taking it further
Learners could be given problems to work on to help
them explore:
o the critical time for which the situation can be
endured (without overflow)
o the volume passing through the system in this
critical time
o the rate of flow with which any overflow pipe has
to cope
Discuss other possible solutions that would:
o put off the critical time
o manage any excess water
Ask learners to consider how this model could help
them understand flooding. A particular flooding
scenario could be given and the learners could
discuss what is similar and different about the
different situations. Where is the model useful? What
could it be used for? Which factors that could be
involved in a flooding situation does the bath model
not cover?
4
Water → Focus on Mathematics
Mathematical Modelling Learning Journey
Learning experiences
experiences: Modelling –
crush
Possible
Possibletasks/challenges
tasks/challenges
Taking it further
Introduction
Discuss the phenomenon of the unexplained traffic
Introduction
Learners
Discusscould
the phenomenon
be introduced
with
to the problem
learners of
using
flooding
questions
withtoa stimulate
discussion
discussion
of the problem.
such as:
Learners
Which
could
of you
work
hasinencountered
groups to discuss
it (as passengers
some information
onobviously)?
the 'Glen cinema disaster' and ‘Ibrox disaster' to
help illustrate the importance of getting good
solutions
Discuss:to this problem.
the parameters – speed
They
be asked
consider whetherinthey
have
thecould
Highway
Code to
recommendations
relation
met
with
a
similar
scenario
in
their
life?
Elicit
the
to: (a) stopping distances; (b) safe following
idea
that ‘things have been improved’. Note that
distances.
these both occurred as people left the venue.
Ask the learners
Learners
could to
bereflect
givenon
particular
questionsscenarios
like:
to
explore the problem further, for example, the
headmaster
o
How does wants
this relate
to put
to the
a display
problem
cabinet
of flooding?
in the
atrium
o
Whatwith
are athe
'footprint'
similarities?
of 8 m2. Advise him based
onWhat
o
calculations.
are the differences?
o What solutions are transferable and how would
Ask
they
themanifest
learners themselves?
to reflect on questions like:
o How could we estimate the data corresponding
o to
How
thedoes
inflow
this
and
relate
the outflow
to the problem
of cars?of flooding?
o What are the similarities?
o What are the differences?
o What solutions are transferable and how would
they manifest themselves?
o How would we gather data corresponding to the
inflow and the outflow of people?
o Given that the atrium corresponds to a flood
plain, what does the display cabinet represent?
o How might we figure out the size of a managed
flood-plain?
jam:
travelling
on the an
motorway
70 mph
we get
Discuss
the problem
architectathas
in designing
caught
in
a
traffic
jam
doing
1
mph;
after
about
a public building, a football stadium or any place20
minutes
wetocome
theaend
of volume
the jam,ofnever
which has
cope to
with
large
people in
having
come
across
a
reason
for
it (likecrushes.
roadworks
a short time while avoiding dangerous
or
a crashreal
or an
'on-ramp').
Consider
lifeoverloaded
instances of
disasters where not
getting this right has partially contributed to the
Ask the learners to calculate the answer to
disaster, for example, Glen cinema disaster, Ibrox
questions such as: (1) over what distance does the
disaster. Draw attention to the atrium of most
jam exist? (2) if drivers are acting responsibly what
public buildings. Consider the scenario of 1000
is the density of traffic at 70, 60, 50, 40, etc and 0
students in the playground, the bell rings, all enter
mph. (3) how might we calculate the rate of flow of
through doors into atrium, all leave atrium by
traffic going into an area or coming out of the area
corridors and enter classrooms; this is to take no
(4) how fast is the area filling up (5) when it fills we
more than 5 minutes.
have a jam – how long will this take for a range of
flow rates?
Learning intention
Investigate intention
and evaluate a simple ‘analogy’ model
Learning
of a flooding scenario.
Investigate and evaluate a simple ‘analogy’ model
of a flooding scenario.
Give an the
exercise
converting
thesetofigures
to 'traffic
Consider
situation
with regards
a school.
densities'.
Make up some figures, rates of flow of people
through a door, and ask the learners to explore the
Collect
data
on of
thefactors
Internet
or give
as a data
problem.
What
sorts
would
theyit need
to
v the
sheet.
take
into consideration? They could discuss factors
the time it takes to get people into class with
Draw a graph
known
stopping
1/2/3/4/5/6
doorsofopen
into
atrium. distances and
interpolate to find others (an able learner might
Learners
encouraged
questioning
find the should
functionberelating
speedthrough
to stopping
todistance).
evaluate the solution in each case, for example,
45 minutes is too long, or with all doors open people
are
flowing
the atrium
thanofthey
can get
Use
this tointo
calculate
carsfaster
per mile
motorway
out
–
dangerous!
...
or
the
atrium
is
filling
up
lane for different speeds; use this to get the rate of
because
peopleatare
flowingspeeds.
in faster
than out;aitrate
will
flow of traffic
different
Assuming
take
x time
to fill
byawhich
timeofall
1000 at
willwhat
be inrate
–
in of
70 mph
and
rate out
1 mph,
safe!
is the lane filling? Make a table of filling rate at
different
speedstoand
a graph
to use possible
for
Ask
the learners
discuss
in groups
interpolation.
Discuss
ways
of
avoiding
tailbacks.
practical experiments/methods of collecting
the
relevant data required to solve a particular situation
in their school.
5
Water → Focus on Mathematics
Mathematical Modelling Learning Journey
Learning experiences
Possible tasks
Introduction
Discussion should take place of earlier lessons
and what flood defences are suggested by them.
Based on the various analogous problems, we can
establish possible solutions to the flooding
problem. Identify each and give examples.
We need to ask ourselves of each: (1) are they
effective; (2) are they cost-effective.
Learning intention
Compile a report outlining some possible solutions
to flooding.
o the bath: (1) create an overflow; (2) build up the
sides
o crush zone: (1) reserve a flood-plane which
cannot be built on; (2) control inflow; (3) keep
surface permeable
o motorway: (1) reduce speed of entry into area;
(2) deepen the channel
In Groups, learners could pick a solution and
explore things like: (1) examples of its use; (2) an
estimate of its cost and how long it is expected to
last and thus its average cost per year; (3) how
much value of real estate is saved per year;
(4) report on the cost-effectiveness of the method;
(5) is cost the only thing we should care about?
Ask learners to consider: could we build houses
which were either flood-proof or could rise above
any flood (float) or permanently be on stilts. What
would be the pros and cons of such an approach?
Possible evidence
A report, supported by facts and figures, charts
and tables, on a possible particular local problem,
with a measure of the risk (for example, once
every 10 years) and of the costing of some
solutions; a costing of possible damage repair;
comparison between the two measures; a
recommendation for action.
Responsibility to all
This learning experience would provide a strong
opportunity to address an aspect of responsibility
of all such as:
By considering the type of text I am creating, I can
independently select ideas and relevant
information for different purposes, and organise
essential information or ideas and any supporting
detail in a logical order. I can use suitable
vocabulary to communicate effectively with my
audience.
LIT 4-26a
Taking it further
Learners could now be given a scenario where
they are asked to use the results of their
investigation to write a report for a local authority
which is creating a flood prevention plan. This
would provide rich opportunities for planning for
interdisciplinary learning with departments such as
geography and technologies.