Active Learning in FP1
Aims
To identify three topics which would benefit from a new
approach because
– Students find them hard (Proof by Induction)
– Students find them strange to start with (Matrices)
The Activities
Algebra-Free Proof by Induction
Defining Matrices through transformation
Competitive Team Graph Sketching
– Students lack discipline in them (Graph Sketching)
To try out three activities that actively engage the learners
Proof by Induction
Students struggle with two aspects of this:
– the concept
– the algebra
Let’s separate these aspects and nail the concept without
the algebra
Three problems with no or minimal algebra…
Proof by toppling dominoes (aka Proof by Induction)
In fact, we normally reverse
steps 1 and 2, which would
be no good for the
dominoes, but makes for a
neater proof…
Proof by induction
Step 1
Check it is true for the first
one
Step 2
Show that if the result is true
for any individual number,
then it is also true for the
next one
Result
It is true for all integers
bigger than the first one
Proof by toppling dominoes (aka Proof by Induction)
Toppling dominoes
Step 1
Set up the dominoes so that
each domino will knock the
next one over
Step 2
Knock the first domino
Result
All the dominoes fall over
Proof by induction
Step 1
Show that if the result is true
for any individual number,
then it is also true for the
next one
Step 2
Check it is true for the first
one
Result
It is true for all integers
bigger than the first one
Algebra-free proof by induction
Prove that every 2n x 2n grid can be covered with Lshaped 3 tiles so that 1 square is left uncovered and this
can be in any desired position.
Every road in a particular country is one-way. Every pair
of cities is connected by exactly 1 direct road. Prove by
induction that there exists a city which can be reached
from every other city either directly or via at most one
other city.
Let’s start with the main application
Introducing Matrices
Most courses introduce matrices as a way of recording
information.
They begin with addition, subtraction and multiplication by
a scalar
Multiplication of matrices is often contrived, or worse,
simply defined.
We define matrices as a way of recording a linear
transformation – perhaps referring to the need to be able
to program a computer to do it.
We define matrix multiplication to be that required for
transformations to work (the MEI text does this bit).
Initial work involves lots of drawing and sketching
Students’ first reaction to matrices is generally ‘What’s the
point?’
Reflection in the y-axis
We can see that x → -x & y → y.
or
x’ = -x
y’ = y
So we can now write these
equations as a pair of
simultaneous equations as
multiples of x and y.
x’ = -1x + 0y
y’ = 0x + 1y
Finally we can summarise the
equations coefficients by using
matrix notation.
Reflection in the x-axis
What happens to x and y?
Write this as a pair of
simultaneous equations.
(2,-1)
(-2,1)
 −1 0


 0 1
Try some more transformations
Reflection in the line y = x
Reflection in the line y = -x
Enlargement, scale factor 2, centre O
Rotation 90°anticlockwise
(2,1)
(2,1)
Now write this information in a
matrix
A different way of thinking about
transformation matrices
Look at our general simultaneous equations:
x‘ = ax + cy
y‘ = bx + dy
with matrix:
a c 


b d 
What happens if we transform the points (1,0) and (0,1)?
Where next?
Try this worksheet with students in pairs or threes.
Still thinking in terms of
So if we find the images of the points (1,0) and
(0,1), we get the matrix of the transformation
straight away.
either writing the transformations as linear
transformations and using the matrix as a shorthand,
or considering what happens to (1, 0) and (0, 1)
And then?
Introduce multiplication in terms of applying the
transformation to a set of points.
Move on to combinations of transformations.
And finally formalise algebra of matrices.
Competitive Team Graph Sketching
Students are good at identifying vertical asymptotes and
intersections with axes.
They tend to have a sloppy approach to completing the
rest of the sketch, however carefully we teach them a
step-by-step method.
Competitive Team Graph Sketching
Competitive Graph Sketching requires each team
member to have responsibility for one part of the sketch.
Rotating the roles will give everyone experience of each
requirement.
All members of the group are fully involved.
Weaker students can be helped to complete their task by
stronger ones, but the team will only improve as a whole
as each member gets better at his/her task, so that they
can all work at the same time.
Create mixed ability teams to make the competition fair.
The Roles
Points:
Vertical asymptotes and intersections
Above:
Behaviour approaching vertical asymptotes
from above
Below:
Behaviour approaching vertical asymptotes
from below
Infinity:
Behaviour tending towards positive and
negative infinity
The Competition
Each member of the team does their bit on paper – aim to
look at limits algebraically, but allow weaker students to
work numerically. (All students should check on a
calculator.)
The Points Person draws and labels the axes after he/she
has worked out the asympotes, then draws in the
asymptotes and marks and labels the intersections.
The rest of the team add in their bits, check it works and
join it all up.
The first team to finish get a small prize.