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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) 2016 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). It is recommended to spend a week observing lessons in a secondary school. 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. Also, make yourself aware of the changes in the national curriculum by looking at the AQA and EdExcel websites as well as the revision materials 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. 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. You can borrow a graphics calculator form the learning centre and use it during the time of your course. 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. 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: [email protected] 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 2 1 x 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. 1-3x+4x2-4x3+4x4......, 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:RR g:RR 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:RR p : x 2x - 3 q:RR 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:RR g:RR f : x 3x - 2 g:xx+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:RR g:RR 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:RR g:RR f : x 7x - 3 g : x x2 + 2 f:RR 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 = 3x + 1/ 3x (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 B2 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 A1 and B 1 and verify that AB 1 B 1A1. 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) . 3 If X 2 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 b. x z 4 y z 5 17. Given that x y z 7 2x y z 11 y z 1 c. x y z 3 x 2y 3z 8 2x y z 9 2x y z 11 d. x 2y z 12 5 x y 3z 14 3 4 A and aI bA cA2 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 9 6 1 1 6 1 b) c) No inverse d) e) 2. a) 140 2 3 5 2 21 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 a n 0 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 4 20 8 14 1 1 6 b) 9 14. a) 0 1 0 3 c) 0 1 4 61 12 1 11 62 21 23 6 4 8 5 8 6 7 20 0 4 53 3 34 1 1 1 0 8 c) 15. a) 1 3 1 b) 4 12 8 14 15 44 230 10 15 5 17 11 1 52 42 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
