Mathematics subject knowledge
enhancement course
Developing Subject Knowledge
- Quadratics
Don’t hesitate to mail (C.Bokhove@soton.ac.uk) or tweet me
(@cb1601ej)
A BIT MORE ABOUT ME
2
Candidates should be able to:
3
Quadratics
- factorising
x 2  14 x  24  0
( x  2)( x  12)  0
x  2,12
4 x 2  12 x  5  0
(2 x  5)( 2 x  1)  0
(2 x  5)  0
or
(2 x  1)  0
5 1
x ,
2 2
4
Useful representations
x 2  14 x  24  0
a
b
sum
prod
1
24
25
24
2
12
14
24
3
8
11
24
4
6
10
24
• …become tables
5
Quadratics
– completing the square
x  14 x  24  0
4 x 2  12 x  5  0
( x  7)  49  24  0
12 x 5
4( x 
 )0
4
4
( x  7)  25  0
4( x 2  3x  1.25)  0
2
2
2
( x  7) 2  25
( x  7)  5
x  2or 12
2
4[( x  1.5) 2  2.25  1.25)]  0
4[( x  1.5) 2  1]  0
4( x  1.5) 2  4  0
x  2.5or 0.5
6
Quadratics
– graphical representation
x 2  14 x  24  0
4 x 2  12 x  5  0
( x  2)( x  12)  0
(2 x  5)( 2 x  1)  0
( x  7) 2  25  0
4( x  1.5) 2  4  0
7
Activity 1 – Linking forms
Match the correct factorised form,
completed square form, intercepts, and
turning points to each of the seven
graphs
See Blackboard
8
Quadratics
- the quadratic formula
9
Activity 2 – Proof sorter
See http://nrich.maths.org/1394
and Blackboard
10
The discriminant
b 2  4ac
The discriminant is the part of the quadratic formula which lies
under the square root sign.
Investigate the value of the discriminant and the graph of each of
the following:
2 x 2  3x  5  0
2x2  4x  5  0
2x2  4x  2  0
What generalisations can you make?
11
Quadratics
The discriminant
b  4ac  0
2
2 real roots
b 2  4ac  0
1(repeated) root
b 2  4ac  0
no real roots
12
Quadratics
- simultaneous equations
Simultaneous equations can be used to find the point (if any) of
intersection of a curve and a straight line.
The most common method is to rearrange one of the equations,
substitute it into the second, and solve the resulting equation formed.
Solve the simultaneous equations
y  2x 1
Show that the same result can be achieved by two
different methods
y  x2  x  3
Thinking of a graphical solution, and giving
examples, explain how many roots would be
possible for sets of such simultaneous equations.
13
Quadratics
- other functions
Sometimes equations which are not quadratic may be changed into
quadratic equations by making a suitable substitution.
Solve the equation:
Let x
 t2
t 4  13t 2  36  0
the equation can now be written as
x 2  13 x  36  0
( x  4)( x  9)  0
x  4or 9
t 2  4or 9
t  2or  3
14
Quadratics
- other functions
How could you solve:
t  6t
If we rearrange we can see:
t  t 6  0
x2  t
x2  x  6  0
( x  3)( x  2)  0
It is always important to check solutions to
such equations. Replacing t with x2 has
created additional roots which do not satisfy
the original equation.
x  3,2
x
t
t  9,4
t4
15
The parabella
http://nrich.maths.org/785
16
17
What now?
Additional readings:
Look at the Peter Powers Party activity on Blackboard
Read the article from Mathematics Teacher for a very
interesting ‘collaborative planning’ session by three teachers.
But now, Activity 3:
Test your skills by making the quadratics practice exercises
(on blackboard). Self-check with answers, and ask questions.
(Every 10 minutes or so I’ll try to wrap up, and provide
explanations if necessary)
18
Understanding and skills
Activity 4
Log in to Integral and make a start on your subject
knowledge by working through the Additional maths
materials (Quadratics). Once you are happy with the
content of each section you can take the online multiple
choice test.
http://integralmaths.org/
(Another useful resource is on
http://www.fi.uu.nl/dwo/soton/ . You can register,
login, look at instruction but above all, practice, now
with the opportunity to enter in-between steps.)
Activity 5
Make a start on Developing Deep
understanding in Algebra to hand in to me
on 19th November.
19
Further resources
Further resources can be found at:
http://www.fi.uu.nl/dwo/soton/
http://www.purplemath.com/modules/quadform.htm
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algeb
ra/col_alg_tut17_quad.htm
http://www.youtube.com/watch?v=4dxF1V52glg
20