Engaging weaker students in M1
Session overview
• Common Errors in M1 (David)
• Personal Experiences of Teaching M1 (Sue)
• Learning Resources (Stephen)
• Discussion
David Holland, Stephen Lee, Sue de Pomerai
Picture from bp3.blogger.com
Learning Resources
45 minute Carousel Sessions on Saturday
Learning Resources
Mechanics is one strand
Image from http://www.elated.com/imagekits/carousel-3/
Learning Resources
Learning Resources
• Examples of ‘Flash’ files
• Practical ‘experiments’
– Forces acting on a box modelled as a particle
– ‘Pushing an object’
– Interactive force diagram: Tension/Compression
– Ruler problem
– Pulleys
– Bob on a string
1
Learning Resources
• Geogebra
www.mei.org.uk/ict
Discussion
Discussion
• What are the major problems you face teaching
weak M1 students?
• What works for you?
• What could help you with your teaching?
(use of interactive resources?)
(use of practicals?)
2
Box modelled as a particle
Teacher Notes
A flash resource for use with this topic can be found at
http://www.meiresources.org/flash/mechanics/box_m1.html
Below are some suggestions how you might use this in your teaching.
First of all familiarise yourself with the features:
•
Use the ‘radio’ button to the left to turn the forces ON and OFF (notice
if the forces are ON then their numeric values are displayed below the
box)
•
Drag the white circle, which is attached to the top right hand corner of
the box by a (light inextensible) ‘rope’.
o Notice the ‘Angle of rope’, θ , changes:
ƒ Depending on where it is, it may give the angle ABOVE
or BELOW the horizontal
ƒ Values of θ available are from 61o below the horizontal to
61o above the horizontal
o The value of T also changes when you drag the white circle
(when the forces are turned ON then the other values, of R and
F, change accordingly)
ƒ T can take values between 25 N and 132.9 N (the
smallest is achieved when T is horizontal and at its
shortest length; the largest is when T is at its longest
(diagonal) length above, or below, the horizontal)
A15 minute activity in/for a lesson looking at forces acting on a particle mass:
Initially, start with the forces turned OFF. Have the rope at its maximum
horizontal length.
•
•
On your mini-whiteboards write down one force, in addition to the
tension in the rope, which act upon the particle? Get class to show their
answers.
Collect and review what has been suggested. Which is the most
frequent response from the students?
o If weight and mass are given, spend a moment discussing the
difference (weight is mass x gravity etc)
o If all three (Weight, Friction, Normal Reaction) are not given at
least once between members of the class, try prompting
students on order to obtain them all
•
•
When all three correct forces have been obtained and discussed, ask
students to draw on their whiteboard, where each of these three forces
act (when T is still in the original position of being in its horizontal
position)
Collect responses and ask students to discuss why they put the forces
in the place that they did.
o After a healthy discussion, reveal where the forces are on the
diagram (by clicking forces ON)
With the forces now turned ON, you want to pose some questions about the
forces. On the whole they will focus around the idea of which forces are
‘related’, i.e. which forces cause another forces value to change when it itself
is changed?
•
•
Ask students to think about how T could be varied so that the Normal
Reaction, R, does not change
o The solution is to drag T in a horizontal line, i.e. where θ = 0
o Students should also note that when doing this the ‘vertical’
component of T is 0.
Given the particle is in equilibrium then resolving forces horizontally, it
follows that T – F = 0 (Using Newton’s second law), i.e. T = F
Now considering how T affects both F and R.
• Say that you are going to move T in a vertical (up, down) motion and
ask what will happen to the values of F and R and why.
o The value of F will actually change, as it is equivalent to Tcos θ
when resolving horizontally (and θ is obviously changing)
o It is perhaps most obvious that R will change, but some
discussion of why will be of benefit
ƒ It is interesting to look at the ‘symmetry’ of the R value
ƒ Ask student to resolve vertically when T is above the
horizontal (say θ = 25 degrees) and then to do the same
when T is the same angle below the horizontal.
i.e. R + T sin θ = W and R – T sin θ = W, which leads to
R = W ± T sin θ
ƒ This should allow for the connection of the symmetry
around the W value to be seen
Depending on if the specification which is being followed introduces the
coefficient of friction a discussion can be held on the relationship between F
and R (and hence as W is fixed here, T)
Following this introduction work could then progress to either connected
bodies or bodies on an inclined plane.
Mathematics in Education and Industry
MEI
MEI
Mathematics in Education and Industry
General
ƒ Apart from technical problems with the content of the units, the
work of weaker students suffers particularly from
MEI Conference 2008
Misconceptions and mistakes seen in M1
from weaker students
Mathematics in Education and Industry
MEI
Where are your principles?
Mathematics in Education and Industry
MEI
Common errors: kinematics
ƒ There can be a lack of an attempt to find out what principle
might apply in a situation
ƒ formulae are tried apparently randomly
ƒ numbers given in a question are put into a calculator in systematic
searches for a given answer
ƒ there is little sense of progression
ƒ Despite the poor presentation, the nature of the jottings does
show some knowledge of the principles
ƒ There is an apparent lack of discrimination used in selecting
which principle might apply
Mathematics in Education and Industry
ƒ poor arithmetic and working to too few figures
ƒ the inability to carry out simple algebraic manipulation efficiently and
accurately
ƒ poor notation
ƒ poor presentation
ƒ lack of diagrams or indication of an origin or the positive direction
MEI
Common errors: vectors
ƒ Many mistakes with vectors come from their not being
distinguished from scalars. Notation is important.
ƒ Using column vectors helps to avoid loss of vector form
⎛ x ⎞ ⎛ 1 ⎞ ⎛ −2 ⎞
⎜ y ⎟ = ⎜ 2 ⎟ − ⎜ 5 ⎟ instead of (xi + yj) = ( i + 2 j) − ( −2i + 5 j)
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
ƒ Using zero acceleration formulae inappropriately
ƒ Using constant acceleration inappropriately
ƒ Failure to define an origin and the positive direction leading to
inconsistencies
ƒ Confusion of the meaning of s, u, v, a and t when used repeatedly
in a problem
ƒ encourage students to use working variables
ƒ Confusing x-y and t-y curves
Mathematics in Education and Industry
MEI
Common errors: statics and dynamics
ƒ Failure to identify all the relevant forces
ƒ Failure to produce an adequate diagram
ƒ Know the diagram conventions e.g.
ƒ the use of magnitudes and arrows
ƒ forces with the known same magnitude given the same label
ƒ forces that might be different having different labels
Mathematics in Education and Industry
MEI
MEI
Mathematics in Education and Industry
Common errors: statics
Common errors: dynamics
ƒ Failure to understand what resolution achieves
ƒ False ‘normal reactions’
ƒ Wrongly using N2L as F – mg = ma
ƒ Wrongly applying N2L to connected particles
ƒ e.g. trucks in a train
ƒ e.g. connected particles not moving in the same line
Mathematics in Education and Industry
MEI
Common errors: vectors
ƒ Mistakes with vector equations
ƒ equating vectors to scalars
ƒ not giving the correct signs
e.g. A mass of 20 kg is subject to two forces, F N and 12 j N. Write
down its equation of motion. Students think of the 12 j N as a
resistance and, using Newton’s second law write
F – 12 j = 20 a instead of F + 12 j = 20 a
This is particularly likely if the force is called a resistance
MEI
Mathematics in Education and Industry
Common errors: vectors
ƒ Information about the direction of vectors can lead to
confusion.
Example
Told that the forces F, G and H are in equilibrium, most
students will write down F + G + H = 0.
F
With the diagram,
G
most will write F + G = H.
H
Drawing and labelling diagrams in mechanics
Clear, accurate diagrams are an essential step in the correct analysis of a variety of mechanics
problems. They provide a means of recording the information given, make it possible to define an
origin and a positive direction and frequently reveal aspects of the situation that suggest efficient
methods of solution. Solutions attempted without diagrams frequently lead to inconsistencies in the
analysis. Poor or wrong diagrams can easily lead to false assumptions being made.
Force diagrams
1
Decide on a single particle or a body model according to whether
any moments are involved.
N
T
If no moments are to be considered a single particle model is
always adequate; this may involve representing
W
one body by one particle (e.g. a car),
several bodies each by one particle (e.g. a car and caravan connected by a tow-bar).
The single particle may be placed inside a representation of the object if this aids clarity.
You may always use this model in MEI M1.
T
If moments are involved, use a body model and consider the lines of
action of the forces. Note that R has not been marked as passing
through the centre of mass.
R
F
It often helps clarity slightly to separate items, e.g. the block and the
slope.
W
This model may be necessary in MEI Mn, n >1, but often a single
particle model is adequate.
2
Follow any instructions given in a question. You may be asked to mark in all the forces or only
those with components in a particular direction. If in doubt mark in all the forces acting.
The following are subject to point 2.
3
The diagram should represent all the forces acting at a point, or on a body or system.
Identify that point, body or system and do not include any forces not acting on it.
Example
A particle of mass m is attached to one end of a light string. The other end of the string is
fixed to a ceiling. The particle is in equilibrium. Mark in all the forces acting on the particle.
T
The forces acting on the particle
hanging from a ceiling are shown
on the left. The extra force shown
on the right does not act on the
particle and should not be
included.
T
mg
mg
Diagrams in mechanics
T
1 of 4
It is often better not to indicate an internal force pair on a single diagram. If you have to consider
separately the forces acting on two or more bodies connected or in contact, it is often clearer to
use linked diagrams instead of a single diagram. Clearly label the pairs of forces given by N3L
with equal magnitudes and opposite directions.
4
Put in any forces given in the question, arrowed in the direction of the force and with any given
magnitudes marked. Indicate any angles given.
5
In dynamics problems indicate the direction(s) you are taking to be positive.
Example
A car of mass 950 kg and with a driving force of 500 N is pulling a trailer of mass
500 kg. The coupling between the car and the trailer is light, horizontal and rigid. The resistances
to the motion of the car and the trailer are 100 N and 50 N respectively. Draw a diagram showing
the forces acting on the car and trailer in the direction of motion, including the internal force in
the coupling.
acceleration a
100 N
50 N
trailer 500 kg
car 950 kg
TN
500 N
TN
T N is the internal force in the coupling written as a tension
Note that this is effectively a two particle model using linked diagrams
6
If any of the particles or bodies has a mass, mark and label the corresponding weight. In a body
model the weight must act through the centre of mass.
7
If any surfaces are in contact,
(a)
mark in the normal reaction at 90° to the tangent of contact,
(b)
determine whether or not the surfaces are smooth; if not, indicate a friction
force along the tangent of contact in the direction opposite to the movement of the
body (or opposite to the direction the movement would take place if there were no
friction).
NB1
The friction force opposes movement (i.e. velocity) not acceleration.
NB2
If the direction of the friction force cannot be determined produce diagrams for both cases.
direction of motion
Example A block of mass m is pulled
along a horizontal plane by a string at 30°
to the horizontal.
R
T
30°
F
mg
Diagrams in mechanics
2 of 4
8
If any strings are involved, mark in the force in the line of the string as a tension.
9
If there are any hinges and resolution is required, it is usually better to mark in the reaction force
as two components at right angles instead of a single force at an unknown angle. However, the
reaction at the hinge is better left as a single force if this reduces the problem to one of only three
forces, for which special techniques may be employed.
General points to check.
10
Ensure all forces have labels and arrows; if the direction of a force is known, mark it as being in
that direction. Do use the same label for any forces with known equal magnitudes. Do not use the
same label for forces not known to have the same magnitude. Apply this rule to linked diagrams
as well as single diagrams.
11
Make sure that you have not made any unwarranted assumptions about magnitudes and directions.
In particular
Never mark in friction as taking its limiting value; make this statement in your working only after
checking that it is true. In general F ≤ Fmax = μR .
When dealing with a normal reaction in a body model, consider carefully its line of action; it may
well not pass through the centre of mass.
12
For a statics problem, having completed your force diagram, consider whether a force
polygon will help your analysis.
For a three force statics problem, remember that the forces must be concurrent and consideration
of this fact as well as the triangle of forces will often provide a simple method of solution.
13
We often want to show forces, velocities and accelerations on the same diagram. The best way to
do this without leading to confusion is to use different sorts of arrows. There is no standard
convention but I have used
single arrows for forces
double arrows for acceleration
blocked arrows for velocity
Diagrams in mechanics
3 of 4
Diagrams for kinematics problems
These are often omitted and much confusion follows. It is always necessary to establish the
direction being taken as positive and often necessary to note the position of the origin. A particularly
important application is to projectile motion where the sense of positive and the position of the
origin may well not be clear from the context.
Example
A small stone is projected downwards at 30° to the horizontal from the top of a vertical
cliff that is 80 m above plane horizontal ground. Its initial speed is 20 m s −1 .
Note that with the convention of the diagram, the initial value
of y is 80, the initial vertical component of velocity is negative,
g is in the negative direction and that the stone hits the sea
when y = 0.
30°
20 m s −1
80 m
y
g
Suppose that the origin were taken at the top of the cliff, still
with y upwards. Now the initial value of y is zero and the
stone hits the sea when y = −80.
Suppose now that the positive value of y is downwards with the
origin at the top of the cliff. The initial value of y is zero, the
initial vertical component of velocity is positive, g is in the
positive direction and the stone hits the sea when y = 80.
O
x
Suppose that the positive value of y is downwards with the origin
at the bottom of the cliff.........
A clear diagram defining the origin and positive directions is
vital!
Diagrams for impulse, momentum and impact problems
We are dealing with vector quantities and so once again it is vital that the sense of each such quantity
clear. It is also vital that the direction taken to be positive should be clearly marked.
Example
Two equal spheres A and B, each of mass m are on a smooth horizontal table. Initially,
sphere A is at rest and sphere B is travelling directly towards it with a speed u. The coefficient of
restitution in the subsequent collision is e. Find the impulse acting on A in the collision.
before
u
0
B
A
m
m
+ ve sense
after
vB
Diagrams in mechanics
vA
4 of 4