BSc (Hons) Mathematics
Secondary Education with QTS*
A Guide to Starting at Edge Hill University
PRE-PROGRAMME INFORMATION
Appendix: Mathematics
specialism information
Direct entry to Year 2 (Level 5)
2015 entry
(* Indicates our recommendation for QTS at the end of your course. Full QTS is only
confirmed on successful completion of your first induction year in teaching)
All mathematics trainees should try to do some background reading on the teaching
of mathematics in Schools. A recommended text is Learning to Teach Mathematics in
the Secondary School: A Companion to School Experience (Learning to Teach
Subjects in the Secondary School Series) by Sue Johnston-Wilder, Peter JohnstonWilder, David Pimm and Clare Lee (11 Aug 2010).
There is also a wealth of general information on training to teach that can be found on
the Teaching Agency website http://www.education.gov.uk/get-into-teaching/
The National Centre for Excellence in the Teaching of Mathematics – NCETM, have
a large number of interesting resources. During the course you will complete a subject
knowledge audit and file for mathematics taught up to GCSE and A level so you might
find the NCETM’s self evaluation tools useful. These are found at
https://www.ncetm.org.uk under the Personal Learning tab. Trainees on the three year
course will have completed an audit against GCSE requirements in their first year.
If you want to brush up on GCSE mathematics topics, try looking at
www.bbc.co.uk/schools/gcsebitesize/ and http://www.mymaths.co.uk Inexpensive
revision materials are available from The Mathematics Co-ordination Group
www.cgpbooks.co.uk. Keep any revision work that you complete, it can be added to
your subject knowledge file.
Part of the course involves the study of mathematics at degree level. We call this
subject study. Edge Hill supports the use of technology in teaching and learning
mathematics so you will certainly need access to a scientific calculator with statistical
functions, but access to a graphical calculator is preferable. If you wish to purchase a
graphical calculator, you should remember that our current regulations do not allow
calculators that have built in computer algebra systems to be used in examinations.
The subject study element of the first year course covers the key areas of pure
mathematics relating to algebraic structure, matrices and complex numbers. Trainees
have also learnt about mathematical reasoning, proof and logic, and the principles that
underpin the work that you have seen around functions and graphs. Applications of
mathematics form a strand that continues throughout the course and, in the first year
involves learning about modelling with mechanics. These will be built on in the two
years of your course so you will need a firm foundation in these areas from your
previous studies.
A condition of entry into year two of the degree course is that you must have completed
observational placements in schools amounting to 40 days in a secondary setting and
5 days in a primary setting.
To help you to prepare, please find below a set of pre-course material for mathematics.
This contains a great deal of work, some of which may be familiar to you linking to
some elements of the first year course. However, some of it may be completely new.
Please feel free to select the exercises which are of the most value to you personally.
Support for doing these exercises can be found in an Advanced Level
mathematics/further mathematics textbook, although you may need to look in different
texts for the pure mathematics and the applications to thoroughly cover the first year
material.
Suggested further reading:
Boaler, J. (2009). The elephant in the classroom: Helping children learn and love
maths. Souvenir Press.
Chambers, P., & Timlin, R. (2013). Teaching Mathematics in the Secondary School.
Sage.
Richard Bronson (1 Mar 2011) Schaum's Outline of Matrix Operations
Smedley, R. and Wiseman, G. (1998) Introduction to Pure Mathematics Oxford
University Press
Seymour Lipschutz and Marc Lipson (Oct 2009) Schaum's Outline of Discrete
Mathematics, Revised Third Edition (Schaum's Outline Series)
Jefferson B. and Beadsworth T. (2000) Introducing Mechanics Oxford University
Press
Ghada Nakhla
Secondary Undergraduate Mathematics Course Leader
e-mail address:
ghada.nakhla@edgehill.ac.uk
Tel number: 01695 657380
Pure
Exercise 1 The Binomial Theorem
1. Expand, using the Binomial Theorem, up to and including the term
in x4 :
(i) (1 + 2x)-3
(ii) (2 - x)-2
(iii) (1 - 3x)1/2
2. Use the Binomial Theorem to expand (1-x)2 and (1+x)-1. Hence, find the
1  x  2
expansion of
up to and including the term in x4, and hence obtain, without
1  x 
a calculator, the value of 0.992/1.01 to 4 decimal places.
Answers:
27 3
405 4
x  128
x
1. 1-6x+24x2-80x3+240x4, 41  41 x  163 x 2  81 x 3  645 x 4 , 1  23 x  98 x 2  16
2
3
4
2. 1-3x+4x -4x +4x ......, 0.9704
___________________________________________________________________
Exercise 2 Trigonometry
1. Complete the following table:
Angle in degrees
Angle in radians
270
3/4
/18
315
-210
-11/12
2. Draw the graphs of y = sin and y = cos taking values of  from
-3 to +3
3. Given tan = sin/cos, what happens to tan when  = /2, 3/2,
5/2, -/2, -3/2, -5/2, ----- ?
Draw a graph of y = tan taking values of  from -3 to +3.
4. Draw up a table in surd form for the three trigonometric functions
when  = 0, /6, /3, /4 and /2
Without using a calculator, but using the sin(A+B) etc formulae, and the above
results
find the values of:
(i) sin(2/3) (ii) cos(5/6)
(iii) sin(7/6)
(iv) cos(5/3) (v) cos(7/3)
(vi) cos(7/6)
_________________________________________________________________
Exercise 3 Functions
1. Functions f and g are defined as follows:
f:RR
g:RR
f : x  2x - 3
g : x  x2 + 1
(a) Write down the values of (i) f(2)
(iv) f(-3) (v) g(7)
(ii) g(2) (iii) f(5)
(vi) g(-3)
(b) Write down expressions for (i) fog(x) (ii) gof(x) (iii) fof(x)
(iv) gog(x)
(c) Write down the values of (i) gof(2) (ii) fof(2) (iii) gog(2)
(iv) fog(2) (v) fog(-3) (vi) gof(-5)
2. Functions h and k are defined as follows:
h : [1,2]  [2,4]
k : [3,5]  [0,6]
h : x  x2 + 2
k : x  (2x - 5)
Only one of the two functions hok(x) and koh(x) can be formed.
(a) For the one that can be formed, state the rule, the domain and the
codomain.
(b) Explain why the other function cannot be formed.
3. Functions p and q are defined as follows:
p:RR
p : x  2x - 3
q:RR
q : x  (x + 3)/2
Find expressions for poq(x) and qop(x).
4. Let s and q be two functions mapping R to R defined by:
s(x) = x and q(x) = x + 1
Express the following in terms of s and q.
(a) h(x) = x + 1
(b) j(x) = (x + 1)
(c) k(x) = (x + 1)
(d) l(x) = x + 2
(e) m(x) = (x + 2)
(f) n(x) = x
Answers:
1. (a) 1,5,7,-9,50,10 (b) 2x2-1, 4x2-12x+10, 4x-9, x4+2x2+2
2. koh=(2x2-1)
3. x, x
4. qs, sq, sqs, qq, sqq, ss.
(c) 2,-1,26,7,17,170
___________________________________________________
Exercise 4 Inverse Functions
1. Explain what you understand by the terms injective, surjective and
bijective.
2. Find the inverse function, f -1(x), in each case, by replacing f(x)
with y and then making x the subject of the formula.
(i) f(x) = (5x-3)/2 (ii) f(x) = 3(x - 5)/4
(iii) f(x) = 1/(x-3)
(iv) f(x) = (x+1)/(x+2)
3. The functions f and g are defined by:
f:RR
g:RR
f : x  3x - 2
g:xx+4
-1
-1
(i) Find f (x) and g (x) (ii) Find gof(x)
(iii) Show that (g o f)-1 = f -1 o g-1
4. The functions f and g are defined by
f:RR
g:RR
f : x  3x + 4
g : x  7 - 6x
Find the rule for each of the following functions:
(i) f -1 (ii) g-1
(iii) g-1o f -1 (iv) f o g (v) (fog)-1
5. One of the functions below is one-one and the other is not. Which is
which? Sketch graphs to support your answer.
f:RR
g:RR
f : x  7x - 3
g : x  x2 + 2
f:RR
f : x  2x + 3
Give the rule and domain of f -1 .
Draw f(x), f -1(x) and y = x on the same axes and hence verify that the graph of f
the reflection of the graph of f in the line y = x.
6.
-1
is
Answers:
2. (2x+3)/5, 4x/3 + 5, 3 + 1/x, 1/(1-x) - 2
3. (x+2)/3, x-4, 3x+2
4. (x-4)/3, (7-x)/6, (25-x)/18, 25-18x, (25-x)/18
5. f is 1-1
6. (x-3)/2
__________________________________________________________________
Exercise 5 Differentiation
1. Differentiate the following functions:
(a) f(x) = x6 (b) f(x) = 3x4 - 6x3 + 2x + 1 (c) f(x) = 3sinx + 2cosx
(d) f(x) = x (e) f(x) = 1/x
(h) y = 3x2/3 - 4x -1
(f) y = 3x + 1/ 3x
(g) y = (2x + 1)2
(i) y = tanx - 3cosx (j) y = 7secx - 0.25sinx
2. Find the gradients of the following curves at the points indicated
(a) y = x2 - 4
at x = 0
(b) y = 1/x3
at x = 1
3. Find the co-ordinates of the points on the following curves at which the
gradient is zero.
(a) y = 6x2 - 2x + 1
(c) y = 1 + 4x - x3/3
(b) y = x3 - 6x2 + 9x + 1
Answers:
2. (a) 0 (b) -3
3. (1/6,5/6), (1,5) and (3,1), (2,19/3) and (-2,-13/3)
__________________________________________________________________
Exercise 6
1. Determine the intervals on which each of the following function is increasing
and those on which it is decreasing. Find the co-ordinates of the stationary
points distinguishing between maxima and minima.
Sketch the curves.
(a) y = x2 - 6x + 8 (b) y = 6 + x - 3x2
(d) y = 3sinx - cosx
(0 < x < 2)
(c) y = 2x3 - 9x2 - 60x + 7
2. Determine the co-ordinates of the stationary points of the following
curves distinguishing between maxima, minima and points of inflection.
Hence sketch the curves.
(a) y = x3 - 6x2 + 12x - 7
(b) y = 3x4 - 28x3 + 96x2 - 144x + 80
3. A thin rectangular piece of metal is 2m long and 50cm wide. Equal
squares of side xcm are cut from each corner and the sides then
turned up to form a box. Find the value of x for which the volume
is a maximum.
Answers:
1. min (3,-1), max (1/6,73/12), max (-2,75) and min (5,-268), max (2/3,2) and min
(5/3,-2)
2. (a) pt of inflection (2,1), (b) pt of inflection (2,0), min (3,-1)
3. x = 11.62cm
Exercise 7 Further Differentiation
1. Differentiate the following functions with respect to x:
(a) y = e5x
(b) y = e-7x
(c) y = 7ex/2
(d) y = (2/3)e-cosx
(e) y = 1/ex
(f) y = e(1-x)
2. Differentiate the following functions with respect to x:
(a) y = 6lnx (b) y = ln8x (c) y = (1/4)ln2x (d) y = ln(x4 + 2)
(e) y = -(3/5) ln(3x2 - 2x + 1) (f) y = ln(56x)
Answers:
1. 5e5x , -7e-7x , 3.5ex/2 , (2/3)sinx e-cosx , -1/ex , -e(1-x)/(2(1-x))
2. 6/x, 1/x, 1/(4x), 4x3/(x4+2), -6(3x-1)/(5(3x2-2x+1)), 1/x
_________________________________________________________________
Exercise 8 Integration
1. Integrate the following functions:
(a) (3x2 + 1) dx
(b) (1/x1/3) dx
(c) (x2 - 1/x2 ) dx
2. Find the area between the curve y = (x + 2)2, the x-axis and the two ordinates
x = 2 and x = 3
3. Evaluate the following integrals:
2
(a)
2
 3x  1 dx (b)
1

8
1
1
dx (c)
x 1/ 3
[Refer to Q1]

3
2
x2 
1
dx
x2
4. Use partial fractions to integrate, and hence evaluate:
(a)
3
1
2
x ( x  3)

2
dx
(b)

2
1
3x  4
dx
x ( x  1)
Answers:
1. x3+x+c, (3/2)x2/3+c, x3/3 + 1/x + c.
2. 61/3
3. 8, 9/2, 37/6
4. 0.163, 2.367
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
Matrices
1. Given that
 4 2
  1 1
  3 0
A
, B  
 and C  
,
  1 0
 6 2
  1 4
find
a. AB
b. BA
c. A(BC)
d. (AB)C
Comment on your results.
e. A2  B 2  C 2
f.  A  3I 
2
g. 4I  7 A  3C
2. Find the inverse of each of the following matrices. If the inverse does not exist
explain why not.
b.  3 2 


  1 1
a.  2 1


 5 3
c.  3  6  d. 7  3  1 e. 21  1






9  2 6 
4 
 2
2 6 
3. Given that
 3 1
1 2 
A
 and B  

 5 3
 1  4
a. Show that AB  BA  A  B .
b. Find A1 and B 1 and verify that  AB
1
 B 1A1.
4. Given that
 a a
c
X 
 (a, b  0) and Y  
 b 0
d
c
 (c, d  0).
0
Show that X and Y commute if and only if ad  bc  0.
5. Given that
1 0 
 2  1
A
 and B  

 3  1
5 0 
show that
 A  B 2  A2  2AB  B2 . Explain why this is the case.
a a2 
 (a  0) , find A2 and A3 . Write down, in terms of n, an
6. Given that A  
0 a 
expression for A n .
7. By writing each of the following sets of simultaneous equations in the form Ap  q
, where p and q are column vectors, solve for x and y:
a. 3 x  2y  15
2x  5 y  28
b.
x  5 y  18
 7 x  4 y  30
c. 3 x  21 y  16
 51 x  3 y  5
8. Illustrate by drawings, starting with a unit square, and describe the effects of the
following pairs of transformations given by the matrices below:
 1  2
 1 0
 followed by 
 ;
a) 
0 1 
0 3
 1 0
 1  2
 ;
 followed by 
b) 
0 1 
0 3
5 0
 0 .6  0 .8 
 followed by 
 ;
c) 
0 5
 0 .8 0 .6 
1 1 
 1  2
 followed by  1 31  .
d) 
 3 6 
2 6
In each case, write the matrix for the single transformation that results from
combining the pairs above.
 0.6  0.8 
 .
9. Describe fully the transformation whose matrix is R  
 0.8 0.6 
1 0 
 .
A matrix X is given by the relationship XM  R . Where M  
 0  1
Describe fully the transformations represented by M and X and evaluate X.
10. Given that
 cos 
A
  sin 
sin  
 cos 
 and B  
cos  
  sin 
sin  
,
cos  
show that AB can be expressed in the same form as A and B (Hint: you need results
from A level so ask if not sure) .
 23
If X   1
 2
1
2
3
2

 , find the least positive integer n such that X n  I.

11. Calculate the determinant of each of the following matrices:
a.  2 0 1


 1  1 0


 0 1 2
b.   3 1 0


 2  1 1


1 1
 1
c.  0
1 3


 5  1 1


  2 2 0
d.   1 2  1


 3 1 0


 4 1 2
12. Given that
1 1 1
1 2 3  0, find the value of the constant k.
1 3 k
13. Solve the equation
1
2
x
1  x 3  1  0.
0
2 1
14. Find the inverse of each of the following matrices:
a.   4 3  1


 6 11 0 


 1 2 0
b.  2 3  5


1 1 1 


0 4 0
c.  3 2 1 


 3 0  2 .


1 7 1 
15. Given that
 0 5 1
  2 1 0




A   1  2 1 and B   3  2 4 ,




5 0
 3 4 2
 1
find
a) A 1
b) B 1
c)  A  B .
1
16. Solve each of the following systems of simultaneous equations by expressing
each in the form Mx  c, where x and c are column vectors:
a. 2x  y  z  7
x z  4
y  z  5
17. Given that
b.
x y z 7
2x  y  z  11
y z 1
c.
x y z  3
x  2y  3z  8
2x  y  z  9
d.
2x  y  z  11
x  2y  z  12
5 x  y  3z  14
 3  4
A
 and aI  bA  cA 2  0,
 2  1
where a, b and c are non-zero constants, show that a  5c  0. Hence write down a
set of values for a, b and c such that aI  bA  cA2  0.
 1 x
18. Let G be the set of matrices of the form 
 , x  R. Show that
 0 1
a. G forms a group under matrix multiplication.
b. G does not form a group under matrix addition.
(In both cases you may assume that the operation is associative)
 a b
19. Let G be the set of matrices of the form 
 , a, b  R. Show that
 2b a
a. G forms a group under matrix addition.
b. G does not form a group under matrix multiplication.
(In both cases you may assume that the operation is associative)
20. Find the eigenvalues and eigenvectors of each of the following matrices:
a.  1 1


  2 4
b.  1 2


 3 2
c.  2  1 6 


 3  3 27


1 1 7 
d.  2 2 1


 2 5 2


 3 6 4
In cases (a) and (b) show that the matrix satisfies its own charactersitic equation.
Hence, find the inverse of each of the matrices (a) and (b).
Similarly, together with the Cayley-Hamilton Theorem find the inverse of each of the
matrices (c) and (d).
Matrices:
Outline answers:
8 8 
  5  2    32 32    32 32 
 b) 
 c) 
 d) 

1. a) 
 1  1
 22 12    2  4    2  4 
Answers show that matrix multiplication is not commutative.
12 9    1  4 
  33  14 
 f) 
 g) 

e) 
7 
16 
 2  8  2
 4
 3  1
  1 2
1 6
9  6 1
 b) 
 c) No inverse d)

 e) 
2. a) 
140   2 3 
5  2
 5 2 
  1 3
1  3  1
1   4  2
1   1  1
 B 1  
 ( AB) 1  

3. A 1  
4  5 3 
2   1  1
4  1  1
4. Discuss with tutor.
5. Discuss with tutor.
 a n na n 1 

6. A n  
n 
0
a


7. a) x  1 y  6 b) x  2 y  4 c ) x  5 y  2
1  2
1  6
 3  4
0 0
 b) 
 c) 
 d) 

8. a) 
0 3 
0 3 
4 3 
0 0
9. R=rotation 53.1o
M= reflection in x axis
X =reflection in x axis followed by rotation 53.1o
10. n  12
11. a) -3 b) 5 c) 22 d) -13
12. k  5
1 10
13. x 
2
2  11
5
 4
0
 14
  4  20 8 




1
1
6  b)
0
3  c)
9 
14. a)  0  1
1 4
 0
61
12 
  1  11 62 




 21  23  6 
  4  8 5
 8  6 7 
  20 0 4 
  53  3


1
1 
1 
0 8  c)
15. a)
 1  3 1  b)
 4
 12  8
15 
44 
230 


 10 15  5 
 17 11 1 
 52 42
1

1
2
34 

14 
 16 
 1 
 6
  2
 5 
 
 
 
 
16. a)   2 
b)  1 
c)  1 
d)   2 
 3 
0
 4 
 3 
 
 
 
 
17. a  5t b  2t c  t for any value of t.
18. b) not closed
19. b) If a 2  2b 2 , matrix has no inverse.
 1  1
1  4  1
.
20. a)
eigenvalues 2, 3 eigenvectors t   s  inverse 
6  2 1 
 1  2 
b)
 2    1
1  2 2 

.
eigenvalues 4, -1 eigenvectors t   s  inverse
4  3  1
3  1 
 3   1  1
     
c) eigenvalues 1, 2, 3 eigenvectors t  9  s 6  r  5  inverse
 1  1   1 
     
6 1  9 

1
 6 8  36  .
6

0 1  3 
  3   1   1
     
d) eigenvalues 1, 1, 9 eigenvectors t  1  s 0  r  2  inverse
 1   1  3 
     
 8  2  1

1
  2 8  2 .
9

 3  6 6 