Download Bridging the gap notes

Aim
 ‘Bridge the gap’
 Allow slower pace
 Allow room for individual
learning
 Allow space for diverse
ways of learning
Algebra can’s and cant’s
Can
Do simple
substitution such as
R = 3t + 6. Find R
when t is 8
Solve simple linear
equations,
e.g. 3a + 5 = 17 or
4(2x + 5) = 44
Understand x2, x3,
etc ,replace values
and find roots
Factorise very
simple expressions
such as 3t + 12
Can’t
Substitution of one
expression in
another, e.g. P =2r,
r = K + 6, Find P in
terms of K
Solve equations with
letter both sides e.g.
5x + 8 = 3x + 24
Cope with (3x)2 or
(x + 3)2
Solve quadratic
equations by
factorisation or
formula,
e.g. x2 – 7x + 12
Solve equations with Solve equations with
one division line
cross-multiplying
such as:
such as
x+4 =2
x+1=x+3
5
6
8
BRIDGING THE GAP
Lecturer
Nonworkshop
students
Workshop
students
The ‘Algebraic Cuboid’
Advantages:
1) Slightly contrived, but shows a possible
practical application
2) Easy to create a gradual approach
3) Visual imagination stimulated
4) Provides a concrete aspect to the
abstraction of an expression
5) The idea of ‘design’ is appealing to
engineers, and creative
6) Link later to calculus-min, max areas,vols
Starting point:-
Initially, work out the total length of edges (T) surface area (A) and
volume (V) of several given cuboids given dimensions.
Build to formulas:
V = LWH,
T = 4L + 4W + 4H,
A = 2LW + 2LH + 2WH
Open questions: e.g. a) How many cuboids can you design from a total
length of wire of 28 cm? Which has the largest volume? b) You have
to design a cuboid box of chocolates with a volume of 180 cm3. How
many cuboids can you find? Which has the smallest surface area?
Then, proceed to algebraic cuboids:
A cuboid is designed with the width twice the length. The
height is 8 cm. Find expressions for:
a) The total length of the edges
b) The volume
c) The surface area
Now, design a cuboid with the above specifications and a volume
of 400 cm3. How large is its surface area?
Comment:
1) A touch of ‘openness’ (they can choose their own variable)
2) Visual and tactile -they can actually touch the box and draw the
faces if they wish
3) They can see the point of allocating letters ( to model that the
length is twice the width)
4) They can see the point of simplifying- to work out the second
part
5) The algebra techniques- they practise collecting like termsi.e. 4(2L) + 4L + 4 x 8 = 12L + 32 for total edge-length
The associative law 2L x L x 8 = 16L2 for volume
Both of the above for surface area
6)More subtly, it helps unconsciously with the substitution
problem (Q1) as stated earlier (i.e. V = LWH, L= L, W = 2L, H =
8 But they are looking at it on the box – so-visually and
tactilely)
The algebraic cuboid idea is very versatile from the tutor’s point of
view. You can adapt it to bring out almost any algebraic technique.
Also, a quick change in the numbers can lead very quickly to other
similar examples of the same type for them to try. In other words, it
is very adaptable to the individual- which is what we need here.
In the following 3 ‘cuboid’ questions, write down what you think are
the algebraic techniques involved and what concepts it helps with.
1) A cuboid is designed with sides in the ratio 3:4:5
a) Find expressions for the total edge-length, surface area and
volume
b) Find the volume if the surface area is restricted to 4606 cm2
2) A square-based cuboid has a height of 8 cm.
a) Find expressions for the total edge-length, surface area and
volume
b) Find the volume if the surface area must be 450 cm2
3) Design a cuboid with length 4 cm more than the width, height 5
cm more than the width and a surface area of 712 cm2
4) A cuboid is designed on the model:
Length: t + 2
Width: t + 2
Height: 4t
Find the dimensions if the surface area is 658 cm2
5) 2 cuboids are made from the same total length of wire. The
first is square based with a height of 6cm, the second has the
same length as the first, but the width is 3 times as large and
the height is 2 cm
Find the dimensions of both cuboids.
6) A measure of the heating cost of a cuboid shaped room is given
as C = 2LW + H2
Suppose the room has width 3 times the length and height
twice the length.. If the cost of heating ,C, as above is 250
find the dimensions of the room.
Difficulties could encounter
Differences in teaching style between
a) Workshop and lecture
b) Workshop and tutorial
c) Workshop and 1-1 sessions
Differences in resources/exercises provided between
a) Workshop and tutorial
b) Workshop and 1-1 sessions
Profile of workshop students
Style of session
 Teach at their level of
understanding
 Teach to the same
point as lecture
 Provide graded
exercises
 Lots of student activity
 Individual help