Stretching Students:
Mechanics in Further Maths
Abstract
• This session will give ideas and
suggestions to stretch the more able
students in mechanics – this will include
the use of extended contextualised
examples.
• This session will be of interest to all those
teaching further mathematics mechanics
units.
Stephen Lee
It should be said that…
1. STEP questions
…No two groups of students are alike
• Useful information/links at:
www.maths.cam.ac.uk/undergrad/admissions/step/
• Need to consider each situation on its
own merits
• Consider having a range of different ‘exercises’
at your disposal
• Hopefully this session will provide you with some
ideas which you can relate to your own cases
• Past papers and solutions:
STEP questions
Reflective Learning (using STEP)
• Should be considered for use, even if
students are not sitting the STEP exams
Provide students with:
• a (suitable) question
• A mark scheme
And/or
• Worked Solution
• Examiners report/comments
• See mechanics examples used for tutoring
students on a distance learning STEP
course
www.admissionstests.cambridgeassessment.org.uk/adt/step/Test+Preparation
1
Reflective Learning
Ask students to undertake the following:
• Read through the whole question, to get a ‘feel’
for it
• Then, before attempting the question, try and
determine what they think the solution strategy
will be and what the key things to note (e.g.
information given and/or information needed to
be obtained) are in each part of the question
• Attempt to solve the question
Reflective Learning
• Read through the mark scheme/example
solution/examiners report
• Consider additional questions about what you have just
undertaken.
– Did the solution strategy that you noted turn out to be
appropriate or did it prove that you needed to revise your
strategy part way through?
• If so, was this unavoidable, i.e. is it actually feasible to prepare the
perfect strategy before you begin the question?
– Was the information you identified as being key BEFORE
starting the question necessary and sufficient or was it inevitable
that further key information not initially considered would arise
whilst undertaking the question?
2. Mechanics in Action
Mechanics in Action
• 1990s publication of above name to
accompany a ‘Leeds Mechanics Kit’
• Contained 53 worksheets, with additional
notes
• Many relevant activities that can be used
to stretch students
• See example handout
‘A stop-go phenomenon’
(possible issue: can be difficult to get hold of this
publication)
• Others include:
– Over hang (sheet 12)
– High road and low road (sheet 26)
– The superball as a deadly weapon (sheet 53)
3. Gems questions
4. Contextualised examples
• Each of last two conferences there has been a
session on ‘6 gems in mechanics’
• Obviously very useful to have questions
that are set in context
• These are excellent questions/ideas for students
to look at and consider
• See handout (or MEI website)
• A suggestion may be to consider the use
of undergraduate books/materials
– See handout from ‘Engineering Mathematics
through Applications’ Kuldeep Singh
2
5. Simulations/Investigations
Real Life Golf Problem
• A staple diet of stretching students is open
ended problems
• 19th Hole at Legends Golf Course has been
designed to involve teeing off from a cliff onto a
green in the shape of South Africa at the foot of
the cliff
• Ideal situation to consider by using modelling
from Mechanics units
• Could use an interactive Geogebra file and
additional exploratory/investigatory worksheet(s)
• Students could investigate real values (or
teachers could provide them…)
• Well designed simulations and
investigations can often lead to students
develop their own thinking of how to look
at a topic/idea
Simulations/Investigations
Session re-cap
• Additional ‘soft’ skills can also be developed
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• Creating posters
• Presentations based around an experiment
• Exemplars document
(Mechanics in the real world?)
STEP questions
Mechanics in Action
‘Gem’ questions
Contextualised examples
Simulations/investigations
• Participants’ experiences
3
MEI Conference, July 2007
Six gems in mechanics
I have selected some pieces of work that have intrinsic interest and are nice
demonstrations of the application of a principle or technique. Some of the solutions
give insight into the solution of whole classes of problems including the identification
of critical considerations.
M1
Suppose that a number of fireworks are fired from ground level and travel
over horizontal ground and that air resistance is negligible. No matter what
their initial speeds and angles of projection those that reach their highest
points at the same time do so at the same height. [If the fireworks turned into
flares when they reached their highest points you would see a rising horizontal
plane of light.]
M1
In projectile motion, consider the two trajectories that intersect at each point
inside the parabola of safety. Must one of the angles of projection be greater
than 45° and the other less than 45°? What can you say about whether the
projectile is rising or falling on each of the trajectories?
M1
Using a force triangle to investigate the equilibrium of an object suspended by
two light strings.
M2
Both an empty and a full can of lager have their centre of mass
(approximately) in the middle. When you drink a little lager, the centre of
mass clearly moves towards the base. When is it at its lowest point?
M2
Suppose that a particle of mass m moving in a vertical plane slides down a
uniformly rough slope. Suppose that the vertical displacement downwards is
H, the horizontal distance travelled is L and the coefficient of friction is If
the path of the particle is any smooth curve and the particle is at all times in
contact with the slope, the KE gained by the particle is mg ( H   L) .
M3
A particle is held in equilibrium by two light strings. What happens to the
tension in one string when the other is cut?
David Holland
1
MEI Conference , July 2007
MEI Conference, July 2008
Six more gems in mechanics
I have selected some more examples that have intrinsic interest and are nice demonstrations
of the application of a principle or technique.
1
Two particles are projected at the same time and the only force acting on them is that
of gravity. Particle A has initial position r0 and velocity u; particle B has initial
position R0 and velocity U; all units are SI. The particles meet no obstructions.
What is the condition that the particles collide with each other?
M1
2
In many self-service cafes, plates are stored ready for use in a stack where the top
plate is at counter level however many plates are in the stack. This may be achieved
very simply by suspending the plates in a rack supported by suitable elastic strings, as
shown in the diagram. How does it work?
M3
3
Two objects, A and B, slide on a uniform plane inclined at an angle to the
horizontal. They are joined together by a light, rigid coupling that is parallel to the
plane. Object A has mass mA and resistance to motion FA; object B has mass mB and
resistance to motion FB. When the objects are being pulled up the plane by a force of
magnitude P, the force in the coupling is T. If FA and FB are independent of  and
the value of g, then so is the equation connecting P and T.
M1
4
A fly of mass m stands on the edge of a circular disc of mass M. The disc is on a
smooth horizontal table. Initially the disc is at rest. Describe what happens to the
disc and the fly relative to the ground as the fly walks across a diameter of the disc at
a constant speed u relative to the disc.
M2
5
A particle moves on a rectangular horizontal table (not necessarily smooth) that has
smooth raised edges. If the coefficient of restitution between the particle and each
of the edges is the same then after contact with two adjacent edges the particle is
travelling parallel to its direction before its first contact.
M2
6
DO is a chord of a circle in a vertical plane with O at the lowest point. A smooth
ring slides on a thin straight rod joining D and O and is at rest when released from D.
The time taken for the ring to slide from D to O is independent of the position of D on
the circle.
In an alternative formulation, O is the highest point of the circle and the particle
starts from rest and travels down a chord OD.
M1
[If from the highest or lowest point in a vertical circle there be drawn any inclined
planes meeting the circumference the times of descent along these chords are each
equal to the other. Galileo Galilei: Dialogues Concerning Two Sciences; Day 3,
Theorem VI, Proposition VI (1635)]
David Holland
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version 3; MEI Conference 2008