MEI Conference 2016mei.org.uk/files/conference16/SIMONC-H1-PDF.pdfEquipment: - a number of metre...
Transcript of MEI Conference 2016mei.org.uk/files/conference16/SIMONC-H1-PDF.pdfEquipment: - a number of metre...
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MEI Conference 2016
Preparing to teach moments
Simon Clay
mailto:[email protected]
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2 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0
Is it possible for this to happen? Teacher notes
From Twitter via @MaxCRoser
Points to consider:
You might want to encourage students to consider:
- which areas of mechanics are applicable in this situation
- any measurements they require such as the mass of car, mass of ‘average’ person, dimensions of
car, position of centre of mass of car, etc
- any assumptions they need to make to model the situation
Extension:
The report suggests that the car was moving when the people jumped onto the bonnet. Model this
situation. Does this change your conclusions in any way?
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3 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0
Is it possible for this to happen? Student sheet
From Twitter via @MaxCRoser
Extension:
The report suggests that the car was moving when the people jumped onto the bonnet. Model this
situation. Does this change your conclusions in any way?
Task:
Decide what would need to happen in order for the above situation to be possible.
Make sure that you can justify your model and any measurements or assumptions
you have used.
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Designing a ‘Teeter-Totter’
Teacher notes
This task is aimed at reinforcing students’ knowledge of the theory of moments and help to develop
their ability to apply it. It is designed to be open-ended with plenty of scope for allowing students to
generate a suitable solution. There is deliberately very little guidance on the student sheet below as
this places students in a position where they have to make decisions and identify assumptions they
are making.
Students could be given the prompt below as a handout or simply shown the photo with a class
discussion to help direct their initial thinking. However, be aware that this can stymie creative
solutions!
The apparatus is called a ‘Teeter-Totter’ although this could well not be its ‘official’ name!
Encourage students to give as much detail as possible in their design and justify any decisions they
make mathematically. Some prompt questions you may wish to use during the activity could be:
- What do you want the outcome to be for a person using this equipment?
- Who is the apparatus suitable for?
- How does the equipment behave for a small child? Teenager? Adult? Does this matter?
- How far does a child with a mass of 25 𝑘𝑔 have to walk along the beam before it pivots?
What about a person with mass 10 𝑠𝑡𝑜𝑛𝑒?
- What type of wood should be used? Why?
- What assumptions have you made in your model?
- What are the dimensions of the ‘Teeter-Totter’?
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Designing a ‘Teeter-Totter’
The photo below shows a piece of apparatus from a ‘Commando-style’ obstacle course, called here a
‘Teeter-Totter’. It consists of a log that is hinged on a pivot so as you walk up the beam from the
ground it gets to a point where it then tips and you can walk off the other end. (Note that you can see
a second one in the background.)
Task:
Produce a suitable design for a ‘Teeter-Totter’ justifying the decisions
you have made.
Indicate any measurements that would be required to produce your
‘Teeter-Totter’ and identify any modelling assumptions you have
made.
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6 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0
Investigating Moments - Teacher notes
Equipment: - a selection of metre rulers
- selection of masses, including 10 𝑔 masses totalling 200 𝑔
- Blu-tac
- Play-Doh
- Optional: some lengths of string for attaching rulers together
These investigations work well with students working in pairs. The above equipment needs to be
available for each pair. Following the initial experiment there are a series of prompt questions to
encourage students to think more deeply about what they have discovered. Students should also be
encouraged to ask their own questions and explore these.
Practical notes:
- A student can use their finger to balance the ruler but for increased accuracy they may wish to use
some sort of wedge or block instead. An effective alternative is to use the edge of a table.
- Masses can be attached to the rulers by using Blu-tac.
- Rulers can be attached to each through the use of Blu-tac or by tying a piece of string around them.
Investigation 1: Balancing a ruler
This investigation could be used with students who are yet to meet the theory of moments and be a
way in to exploring and generating the theory. It provides an opportunity for students to think through
the assumptions they make during the modelling process (for example, the beam is uniform, the mass
is a particle, etc.) and also confront some of the issues of experimental approaches such as the
accuracy of measurements.
You may need to encourage students to try several numerical cases in order to build up a data set.
Plotting a graph of distance of support from the end of the beam, 𝑑, against the total of the masses
supported by the beam, 𝑚, leads to some interesting models being produced.
Investigation 2: Beam balance
This investigation is split into two complementary parts. The first provides students with an
opportunity to apply their knowledge of the theory of moments and explore various solutions for a
specific problem as well as a solution for general masses. The second part offers the opportunity for
students to begin to use the theory to problem solve.
Adapted from activities of the same name in ‘Mechanics in Action’ (1990)
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7 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0
Investigating Moments - Student sheet
Investigation 1: Balancing a ruler
Equipment: - a number of metre rulers
- at least five 10 𝑔 masses
- Blu-tac
1. A ruler will usually balance at its midpoint. Check that this is true for your ruler.
2. Attach a 10 𝑔 mass to the end of the ruler. Where does the ruler balance now?
Q: Can you predict where the ruler will balance when 20 𝑔 is placed on the end of it? What about
when using 30 𝑔? Continue to investigate until you have a conjecture that you can test with a
prediction.
Q: Can you explain what is happening?
Some further points to explore:
What happens when you have a number of rulers stuck together?
What happens when you place masses at various points along the length of the ruler?
d
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8 of 8 SC 07/06/16 ‘Preparing to teach moments’ v1.0
Investigation 2: Beam balance
Equipment: - a metre rule
- 40 𝑔, 70 𝑔 and 90 𝑔 masses
- Blu-tac
- a small and a large lump of Play-Doh or similar
Part A:
Place the 40 𝑔, 70 𝑔 and 90 𝑔 on a beam (metre ruler)
Q: Can you do this and make the beam balance?
Q: Is there more than one way of doing this? How many ways can you find?
Q: What about if the masses were 𝑥, 𝑦 and 𝑧?
Part B:
Using five 10 𝑔 masses and the beam can you find the masses of the small and large lumps of Play-
Doh?
Q: How large a mass can you measure?
Q: How small a mass can you measure?
Q: How accurate can you be in these measurements?
Some further points to explore:
Q: Can you find the mass of the beam using the equipment above?
Q: Can you place two 10 𝑔 masses on adjacent numbers on one side of the metre rule and then
balance these using three 10 𝑔 masses on the other side?
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Preparing to teach Moments
Simon Clay
mailto:[email protected]
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Session description This session is one of four designed to help teachers prepare for teaching
mechanics topics in the new A level from 2017.
"Preparing to teach moments" will cover some of the basic subject content,
links to GCSE and other A level topics, as well as exploring ideas and
approaches for teaching moments. The session will demonstrate how
simple practical classroom activities can provide a stimulus for students to
develop their understanding of moments. We will also consider how we can
develop models of situations involving moments.
The session is suitable for teachers who have not previously taught any
mechanics. It will also be suitable for teachers who have previously taught
M1 modules, but for whom moments has not been included within their
current M1 specification.
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What will happen and why?
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The particle model When a system of forces acts on a particle, the particle may be in one of the following states
- static equilibrium
- constant velocity
- accelerating in the direction of the resultant force
Typically in early study of Mechanics students have seen situations where bodies may be successfully modelled by single particles:
- a book at rest on a slope
- a book sliding down a slope
- a car and trailer accelerating along a straight road
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Moving on from the particle model
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The table could slide or topple
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Maximising turning effect of a force
q
F
d
A simple experiment with your students
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The moment of a force
O d
F
The turning effect of a force is called the moment of a force.
The moment of a force F about an axis through O
perpendicular to the plane containing O and the line of
action of F is F x d, where d is the perpendicular distance
from O to the line of action of F.
- Moment has sense, usually
described as clockwise or anti-
clockwise and is signed positive or
negative according to the
convention adopted for that problem.
- The SI unit of moment is the N m.
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‘Weighing’ your piece of wood
Equipment: A piece of wood
A 100 g mass
A ruler
Task: Calculate the mass of the piece of wood
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What examples of moments can
you think of?
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Alternative bathroom scales?
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Reformed A level Content
www.gov.uk/government/publications/gce-as-
and-a-level-mathematics
https://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematicshttps://www.gov.uk/government/publications/gce-as-and-a-level-mathematics
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Simplifications
At A level, the following simplifications are made:
• Only static equilibrium situations are considered
• The moments are easy to calculate, mostly with forces
that are parallel
• We do not usually mention that the moment is about an
axis perpendicular to the plane through a point, say A,
we just say the moment about A.
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Modelling assumptions
• The body is rigid - it is not deformed by the forces that
act on it
• The axis is fixed
• The body is free to rotate about the axis - for instance,
there is no frictional or other force impeding rotation
about a hinge
• The moment of the whole weight may be found by taking
its line of action to be through the centre of mass
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Conditions for static equilibrium
For the equilibrium of a body it is necessary and sufficient
that the resultant of all the external forces is zero and that
the total moment of these forces is zero about any axis.
In the case of coplanar forces it is sufficient for the
equilibrium of a body that the resultant of all the external
forces is zero and that any point can be found about which
these forces have zero moment.
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Solving problems
To meet the conditions for equilibrium we have to resolve in two directions (we choose) to establish a zero resultant force and take moments about one point (we choose) to establish that the forces have zero moment about this point.
As you might suppose, a wise choice of the directions and point can make the solution easier.
Not every problem requires two resolutions and moments taken
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AQA A level Mathematics (2017)
draft sample paper 2, Q12
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AQA A level Mathematics (2017)
draft sample paper 2, Q12
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Calculating moments
But finding the perpendicular distance onto the line of
action of a force is not always quite so straightforward!
Example: Suppose that the force of 10 N acts through P
which is 4 m from O and that the force is at an angle of 60°
with OP.
O 4 m
P
60°
10 N
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Calculating moments
10 cos 60 N O
4 m
P
10 N
60°
10 sin 60 N
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Experiments - Investigating
moments
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http://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.html
Interactive simulation
http://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.htmlhttp://phet.colorado.edu/sims/html/balancing-act/latest/balancing-act_en.html
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Experiments - The Ladder
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‘Teeter-totter’ task Produce a suitable design for a ‘Teeter-Totter’ justifying the decisions
you have made.
Indicate any measurements that would be required to produce your
‘Teeter-Totter’ and identify any modelling assumptions you have made.
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The car & the quayside
Via Twitter
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Problem: Two supports
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OCR A level Mathematics (2017)
draft sample paper 1, Q15
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OCR A level Mathematics (2017)
draft sample paper 1, Q15
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‘Rowing and the Same-Sum Problem Have Their Moments’
by John D. Barrow
Solving the boat ‘wiggle’
problem
http://arxiv.org/pdf/0911.3551v3.pdf
Rowing moments
http://arxiv.org/pdf/0911.3551v3.pdf
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About MEI
• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students
and work with industry to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources
SessionH_Moments_Delegate_handout.pdf (p.1-8)SessionH_PreparingToTeachMoments_webversion.pdf (p.9-39)