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Mathematics subject knowledge enhancement courseDeveloping Subject Knowledge
- Quadratics
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Candidates should be able to:
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Quadratics - factorising
024142 xx
12,2 x
0)12)(2( xx05124 2 xx0)12)(52( xx
0)12(
0)52(
x
or
x
2
1,2
5x
Useful representations
• …become tables
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a b sum prod
1 24 25 24
2 12 14 24
3 8 11 24
4 6 10 24
024142 xx
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Quadratics – completing the square
024142 xx 05124 2 xx
02449)7( 2 x
025)7( 2 x
25)7( 2 x
5)7( x
122 orx
0)4
5
4
12(4 2
xx
0)25.13(4 2 xx
0)]25.125.2)5.1[(4 2 x
0]1)5.1[(4 2 x
04)5.1(4 2 x5.05.2 orx
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Quadratics – graphical representation
024142 xx
0)12)(2( xx
025)7( 2 x
05124 2 xx
0)12)(52( xx
04)5.1(4 2 x
Activity 1 – Linking forms
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Match the correct factorised form, completed square form, intercepts, and turning points to each of the seven graphs
See Blackboard
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Quadratics- the quadratic formula
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See http://nrich.maths.org/1394 and Blackboard
Activity 2 – Proof sorter
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The discriminant
0242 2 xx
acb 42
0532 2 xx
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:
What generalisations can you make?
0542 2 xx
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Quadratics The discriminant
042 acb042 acb
2 real roots
042 acb
no real roots1(repeated) root
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Quadratics- simultaneous equations
32 xxy
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
Show that the same result can be achieved by two different methods
Thinking of a graphical solution, and giving examples, explain how many roots would be possible for sets of such simultaneous equations.
12 xy
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Quadratics- other functions
03613 24 tt2tx
Sometimes equations which are not quadratic may be changed into quadratic equations by making a suitable substitution.
Solve the equation:
Let the equation can now be written as
32
94
94
0)9)(4(
03613
2
2
ort
ort
orx
xx
xx
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Quadratics- other functions
tt 6How could you solve:
If we rearrange we can see:
4
4,9
2,3
0)2)(3(
06
06
2
2
t
t
tx
x
xx
xx
tx
tt
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.
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The parabella
http://nrich.maths.org/785
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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)
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Understanding and skillsActivity 4Log 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 5Make a start on Developing Deep understanding in Algebra to hand in to me on 19th November.
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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_algebra/col_alg_tut17_quad.htm
http://www.youtube.com/watch?v=4dxF1V52glg
