Mathematics - Schudio · Pre-enrolment task for 2014 entry Mathematics Why do I need to complete a...

Huddersfield New College, New Hey Road, Huddersfield, HD3 4GL Telephone: 01484 652341 email: [email protected] www.huddnewcoll.ac.uk

Pre-enrolment task for 2014 entry

Mathematics

Why do I need to complete a pre-enrolment task? To be successful on AS Maths it is vital that you are able to use higher level GCSE topics as soon as you start the course – they are assumed knowledge. You must be able to manipulate algebra; solve linear and quadratic equations; know the shape of basic graphs; understand indices and calculate fractions without a calculator. We have brought together topics which we feel are most needed in the first few weeks of your course. We have provided instruction, worked examples and plenty of exercises. Also, lessons and assessments are set from the

‘mymaths’ website for you to work through.

You should aim to complete the booklet and ‘mymaths’ task for enrolment. You can bring this with you to enrolment and also to your first lesson.

How will I be given feedback on how well I have done?

The booklet contains answers so you can check your answers as you go along, making sure you understand. You can contact staff via e-mail for help. At the end of your first week in college you will sit an initial diagnostic test. The results of this test, together with your ‘mymaths’ assessments and progress with the booklet will allow us to determine which course is best for you.

Task In Detail

1. The Bridging Booklet

A 23 page booklet providing information about the maths course offered at New College and covering the

following topics:

Indices, surds, Factorising, solving quadratic equations, graphing quadratic functions, solving equations,

challenging common mathematical concepts.

2. My Maths tasks – lessons and homework. Remember to ‘check out’.

When should I hand it in?


AS Mathematics – BRIDGING MATERIALS Contents

Page

1. Contents

2. Introductory letter

3. A guide to Maths Courses at HNC

4. Indices

5. Fractional Indices

6. Negative Indices

7. Two special powers

8. Surds

9. Surds - Exercise F

10. Quadratics

11. Difference of two squares and factorising

12. Solving quadratic equations

13. Solving quadratic equations - Exercise K

14. Graphs of quadratic functions

15. Graphs of quadratic functions – Exercise M

16. Common Misconceptions

17. Equations

18. Enhancement Questions

19. Answer Sheet 1

20. Answer sheet 2


HUDDERSFIELD NEW COLLEGE AS Mathematics – BRIDGING MATERIALS

Welcome to Maths at Huddersfield New College! The aim of this pack is to aid your preparation for an Advanced Mathematics course and make the transition from GCSE to AS level as smooth as possible. It contains some of the GCSE Higher tier material that is required before you begin your AS course. We want you to be successful and we will do all we can to be supportive but the hard work has to come from you. You MUST work through this booklet over the summer and bring your completed booklet with you to enrolment. You will be tested on what you have learned at the end of your first week in college. Please read ‘ The Guide to Level 3 Mathematics courses’ on the next two pages it provides information about which Maths course would be the most appropriate for you. A decision will need to be made about which course will best fit with your intended career path, your other subjects and your ability. Only those students who demonstrate a sound grasp of mathematical skills and show the necessary commitment will be considered for the traditional AS Mathematics course. You have also been provided with an individual log-in to the ‘mymaths’ website (please refer to the mymaths letter). Many of the lessons will help refresh your maths skills and knowledge and you may find these useful. We have also set up a number of tasks for you to complete - make sure you ‘check out’ because we will be monitoring how successful you are! Ensure you complete these tasks to the best of your ability – persevere with them until you get a green light! Should you encounter any problems with the work you can e-mail [email protected] and we will get back to you as soon as possible. ( Don’t worry if you are unable to access the internet from home – let us know and we will arrange times over the first week in September for you to come in and complete the on-line tasks then.) Good Luck with the work. We hope you enjoy your summer break and we look forward to meeting you at enrolment. The Maths Team (Any queries please contact Mark Webber, Head of Maths email: [email protected])

mailto:[email protected]


A Guide to Level 3 Mathematics Courses at Huddersfield New College

The maths department offers a broad range of courses designed to match the abilities and ambitions of students. Experience shows that enjoyment of the subject and eventual exam success depends to a large extent on appropriate enrolment. The information below outlines the content of the courses and gives some indication of their level of difficulty and the suitability for combination with other subjects.

A Level Mathematics (AQA) Minimum GCSE entry B.

This is the traditional advanced level course. It is available as an AS level which is often continued to a full A Level. As a demanding subject, it is particularly appropriate for mathematically able students who show both interest and commitment. Algebraic competency is very important, since 2 of the 3 modules studied for AS rely heavily on this skill. A high level at GCSE is not always an automatic indicator of this, and even students achieving an A at GCSE may need to work especially hard to succeed. A Level Maths combines well with most other subjects, particularly academic ones. A Level Use of Maths (AQA)

Minimum GCSE Higher-level grade C.

Graphical Calculator is essential.

This course is designed to apply Mathematical skills in real-life, applied contexts. AS students study 3 modules: USE1 (Algebra & Graphs) and 2 from Data Analysis, Discrete Maths and Dynamics. The last 3 are FSMQs, which, if passed, count as qualifications in their own right, gaining UCAS points. For all modules there is a strong emphasis on relating the theory covered to real life situations. This makes the course especially suitable for those who like to see the practical relevance of what they learn or who might be studying the subject to support their other choices. The subject content is generally covered in a slightly more straightforward way than for Mathematics, which makes the course suitable for a wide range of students. For many degree courses (including, in some cases, Engineering and Chemistry, but not mathematics), most universities are perfectly happy to accept Use of Maths. The department can offer more advice on this issue.

A Level Statistics (AQA) Minimum GCSE Higher-level grade C.

Graphical Calculator essential

This course has a strong tradition at A Level. Algebraic ability is less important than for maths, and for this reason the course is probably more accessible to a broader range of students, but it is in no way an easy option. Statistics is about applying techniques to describe, predict and understand numerical behaviour in a wide variety of real life situations. Successful students are those able to work methodically and accurately and who can relate their results to the context of the situation being studied. Statistics may be taken alongside A Level Maths and it combines very well with subjects that include some element of data analysis, such as Psychology, Biology, Sociology, Economics, Geography and Business Studies.

A Level Further Maths (AQA) Minimum GCSE entry A, plus a strong GCSE profile.

This must be studied alongside A Level Mathematics. As, quite possibly, the hardest A-Level, it is suitable only for those students with a passion and flair for the subject. Typically, students on the course will have achieved very high grades in all their subjects at GCSE, simply because they will not have found them at all challenging! Further maths is available at both AS and A Level. Occasionally, students take up AS Level on their second year. Further Maths is an excellent choice for those contemplating Maths or related courses at university and it combines well with a wide range of subjects.


Level 3 FSMQ (AQA) Data Analysis, Decision Maths, Dynamics or Personal Finance.

Minimum GCSE Higher-level grade C.

Graphical Calculator is essential for Data Analysis

Graphical calculators will be available for purchase or rental from the Maths Department.

A free standing Maths Qualification is offered as an enrichment option. This course will involve a weekly lesson throughout a year. It would be suitable for students with a grade C or above in GCSE maths who wish to extend their subject knowledge, perhaps to support their subject choices such as Psychology, Sociology, Biology, Physics, Geography, IT, Design and Business Studies. FSMQ gain UCAS points as follows:

Grade A B C D E

UCAS Points 20 17 13 10 7


AS Mathematics – BRIDGING MATERIALS Indices

An index is another name for power. The index is the number of times a (base) number is multiplied by itself. index or power

aⁿ

base E.g 53 = 5 x 5 x 5 = 125 The following rules only apply when multiplying or dividing powers of the same number or variable (same base). Rule for multiplying numbers in index form (add the powers):

1. 34 x 35 = 3(4+5) = 39

2. 104 x 10-2 = 10(4+-2) = 102

3. am x an = a(m+n)

Rule for dividing numbers in index form (subtract the powers):

1. 45 ÷ 42 = 4(5-2) = 43

2. 10-2 ÷ 10-4 = 10(-2--4) = 102

3. am ÷ an = a(m-n)

Rule for raising a power term to a further power(multiply the powers):

1. (62)4 = 62x4 = 68

2. (7-2)4 = 7-2x4 = 7-8

3. (am)n = amxn

Question – What are the rules for adding and subtracting numbers in index form?


AS Mathematics – BRIDGING MATERIALS Exercise A

Write as a single power:

1. 56 x 5-3 =

2. 4 x 42 =

3. 64 ÷ 6-2 =

4. 6-3 ÷ 64 =

5. (42)3 =

6. (43)-2 = 7. (4-2)-3 =

8. (47)0 =

9. 42 x 25 =

10. (22)3 x 25 =

Simplify the following expressions:

1. 6a3 ÷ 2a2 =

2. 2a2b3 x 4a3b =

3. 6a4 = 3a

4. 9a4b3 =

3ab

5. Can you simplify this? 2a2bc2 x 6abc3 4ab2c


AS Mathematics – BRIDGING MATERIALS Fractional indices

am/n m is the power and n is the root

When m = 1

i.e Indices of the form n

1 - this means ‘the nth root of’.

Examples

Exercise B (non-calculator) Evaluate:


AS Mathematics – BRIDGING MATERIALS Fractional Indices

Indices of the form am/n : the nth root of a raised to the power of m

Examples:

Exercise C Evaluate

1. 1252/3 = 4. (27p3) 2/3 2. 163/2 = 5. (x4 y10)3/2

3. 2434/5 =

Rewrite in index form:


AS Mathematics – BRIDGING MATERIALS Negative indices

Consider the index rule for dividing: 52 = 5 2 – 3 = 5 -1

53

Numerically, 52 ÷ 53 = 25 ÷ 125 = ⅕

Therefore, 5-1 = ⅕

i.e The negative sign means ‘one over’ (reciprocal)

Exercise D. Write each of these in fraction form.

1) 10-5

2) 8-2

3) T -1

4) 4q-4


AS Mathematics – BRIDGING MATERIALS Two special powers

A number raised to the power 1 stays the same number. e.g 51 = 5 A number raised to the power 0 is always equal to 1. e.g 40 = 1 Because:

43 ÷ 43 = 43-3 = 40 and 64÷64 = 1 Exercise E. Simplify the following


AS Mathematics – BRIDGING MATERIALS Surds

Surds are roots of numbers: i.e. Rules for multiplying numbers in surd form.

Rules for dividing numbers in surd form.

In maths, we would prefer to write the square root of 300 in surd form rather than as a decimal because it is EXACT.

√300 =√(100 x3) =√100√3 = 10√3 Similarly,

√50 = √(25x2) = √25√2 = 5√2

√80 = √(16 x5) = √16√5 = 4√5


AS Mathematics – BRIDGING MATERIALS Surds - Exercise F (non-calculator)

Simplify each expression. Leave your answer in surd form.

8) √75

9) √700 10) √128

Can you do these?


AS Mathematics – BRIDGING MATERIALS Quadratics

1. Quadratic Expansion

Expression such as 5p(2p-3) and (3y + 2)(4y – 5) can be expanded (multiplied together) to give quadratic expressions. 5p(2p – 3) = 10p2 – 15p and (3y + 2)(4y -5) = 12y2 -7y -10 There are many methods used for expanding such expressions as (t + 5)(3t – 4) but the rule is to multiply everything in one bracket by everything in the other bracket. You may have used FOIL (First, Outer, Inner, Last) to help you with this. Example: Expand (x + 3)(x + 4)

F: x × x = x² O: x × 4 = 4x I: 3 × x = 3x L: 3 × 4 = 12 Simplify: x² + 4x + 3x + 12 = x² + 7x + 12

Example: Expand (2t + 3)(3t - 1) 6t² - 2t + 9t – 3 = 6t² + 7t – 3 Exercise G. Expand the following expressions:

1. 2x( x- 1)

2. 3x (5 – 2x)

3. (x + 3)(x + 2)

4. (x – 3)(x + 4)

5. (3x – 2)(2x + 5)

6. (2 + x)(1 + 3x)

7. (7x – 1)(7x + 1)

8. (5x + 4)(5x – 4)


AS Mathematics – BRIDGING MATERIALS The Difference of Two Squares & Factorising

A quadratic expression with only two terms, both of which are perfect squares separated by a minus sign, is called the difference of two squares. Eg. x² - 9, x² - 25, x² - 100, etc. Recognise the pattern as x² minus a square number n² Its factors are (x + n)(x – n)

Factorise x² - 36. Recognise this as x² minus 6², this then becomes (x + 6)(x – 6)

Similarly, recognise 9x² - 100 as 3x squared minus ten squared which will factorise to (3x + 10)(3x – 10) Exercise H. Factorise the following expressions.

1. x² - 9 2 k² – 100 3. 16 x² - 9 4. 16y² - 25 x²

Factorising ax² + bx + c = 0

Factorise 3x² + 8x + 4 Both signs are positive, so both bracket signs must be positive 3 has only 3 × 1 as factors, the brackets must start (3x + )(x + ) Factors of 4 are 4 × 1 and 2 × 2 Find which pair of factors of 4 combines with 3 & 1 to give 8; We see that the combinations 3 × 2 and 1 × 2 adds to 8 So, the complete factorisation becomes (3x + 2)(x + 2) Note : You can check your solutions by expanding brackets.

Exercise I. Factorise the following expressions.

(1) x2 + 6x + 8 (5) 3x2 + 3x - 36 (9) 4x2 - 25 (2) z2 + 10z + 16 (6) 5x2 - 9x - 2 (10) 5x2 + 10x (3) y2 + 8y + 7 (7) 3y2 - 14y + 8 (11) ¼x2 – 9 (4) 2x² + 5x + 2 (8) 7x² + 8x + 1 (12) 3t²- 16t – 12

3 4

2

1 1 2


AS Mathematics – BRIDGING MATERIALS Solving Quadratic Equations

There are three main methods used to solve quadratic equations algebraically:

1. By factorising. 2. Using the quadratic formula (at AS this needs to be memorised) 3. By completing the square.

1.By Factorisation. Example. Solve 6x2 + 15x = 9 firstly set the equation to zero If 6x2 + 15x - 9 = 0 now factorise

(6x - 3) (x + 3) = 0

6x - 3 = 0 or x + 3 = 0

either x = ½ or x = -3 Exercise J. Solve the following quadratic equations: (1) x2 - 6x + 5 = 0 (6) 13x2 = 11 – 2x (2) x2 - 4x = 5 (7) 4x2 - 4x = 35 (3) 4x2 + 4x - 35 = 0 (8) 7x2 + 5x + 4 = 2x2 - x + 3 (4) 4x2 - 25 = 0 (9) 4x2 + 2x - 25 = 2x (5) 9x2 - 24x - 9 = 0 (10) 2x2 - x = 6 2. Using the quadratic formula. Many quadratic equations cannot be solved by factorisation. One way of solving this type of equation is to use this formula: The solution of the equation ax² + bx + c = 0 is given by Example: Solve the following equation using the quadratic formula. (Give your answer correct to 2 decimal places.) 2x² + x – 8 = 0 (note: a = 2, b = 1, c = -8) Substituting into the formula gives

4

06.81

4

651

4

)824(11

x

Hence, x = 1.77 and -2.27

2a

4acbbx

2


AS Mathematics – BRIDGING MATERIALS Solving Quadratic Equations – Exercise K

Solve the following equations, giving your answers to 2 decimal places.

1. x² - x – 10 = 0

2. 4x² + 9x + 3 = 0

3. 6x²+ 22x + 19 = 0 4. 4x² - 9x + 4 = 0

3.Completing the Square Example: i) Rewrite x² + 4x – 7 in the form (x + a) ² - b. Solution: Since x² + 4x = (x + 2)² - 4, So, we have x² + 4x – 7 = (x + 2)² - 4 – 7 = (x + 2)² - 11 (check this answer by expanding)

ii) Hence, solve the equation x² + 4x – 7 = 0, giving your answer to 2 decimal places. Solution: When (x + 2)² - 11 = 0 then (x + 2)² = 11 x + 2 = ±√11 x = -2 ±√11 = 1.32 or -5.32 Exercise L Solve the following equations by completing the square.(Leave your answers in surd form).

1. x² + 4x – 1 = 0 2. x² + 14x – 5 = 0 3. x² - 6x + 3 = 0 4. x² + 6x + 7 = 0


AS Mathematics – BRIDGING MATERIALS Graphs of Quadratic Functions

A quadratic function is usually written in the form ax2 + bx + c where a, b and c are constants. All quadratics are parabolas and look like:

NOTICE – the graphs are symmetrical and have one turning point (a maximum or a minimum)

We can find the point of intersection with the y-axis by substituting x=0 into the equation. For the equation y = x2 + 2x – 8 at x = 0 we obtain y = -8.

We can find any points of intersection with the x-axis by substituting y = 0 into the equation. For the equation y = x2 + 2x – 8 at y = 0 we obtain

0 = x2 + 2x – 8 which can be factorised 0 = (x + 4)(x – 2) So x = -4 and x = 2.

To find the coordinates of the minimum (or maximum) point of the graph you need to write the equation in completed square form. y = x2 + 2x – 8 = (x + 1)2 – 1 – 8 = (x + 1)2 – 9 When (x +1) = 0 , y has its lowest value of -9

The graph of (x + 1)2 – 9 has minimum point (-1, 9) From this information we can complete our sketch.


AS Mathematics – BRIDGING MATERIALS Graphs of Quadratic Functions – Exercise M

Sketch the following curves, showing where they intersect the axes, and the minimum point. 1. y = x2 + 5x + 6 2. y = x2 + x – 2 3. y = 2x 2 + 2x - 4


AS Mathematics – BRIDGING MATERIALS Common Misconceptions

A misconception is a false view of how things are. In mathematics, some things that look sensible are, in fact, completely wrong. It is important that you are aware of common mistakes some students make and don’t make them yourself! A good way to check is to substitute values. For example, to show that: . 1 + . 1 is NOT equal to . 1 . x y x + y substitute x = 1 and y = 1 .

Then 21

1

1

1 and

2

1

11

1

NOT the same.

Exercise N. Are the following true or false?

1. 3(x - y) = 3x - 3y

2. (x + y) 2 = x2 + y2

3. (a + b) + 5 = a+5 + b+5

4. xxx

x

x

x 11

11

5. xx

x

x

x

x

1

11

6. qpqp

7. qppq

8. sin (A + B) = sin A + sin B

9. (6t)2 = 6t2

10. 2x-3 = . 1 . 2x3 (Did you get 3 true? – if not, take another look…....)


AS Mathematics – BRIDGING MATERIALS Equations

Exercise O Using all your algebraic skills solve the following equations:


AS Mathematics – BRIDGING MATERIALS Enhancement Questions

The following questions are taken from recent UK Senior Mathematics Challenge papers. They are non-calculator questions and are designed to get you thinking mathematically. Some are tough but don’t be put off, persevere and play!

1. Which of the following is not a multiple of 15? A 135 B 315 C 555 D 785 E 915

2. What is the value of 16 – 25 + 34 – 43 + 52 – 61 ? A 1 B 2 C 3 D 4 E 5

3. Consider all three-digit numbers formed by using different digits from 0, 1, 2 ,3 and 5. How many of these numbers are divisible by 6? A 4 B 7 C 10 D 15 E 20

4. Suppose that x – 1 = y – 1 and x ≠ y. What is the value of xy ? x y A 4 B 1 C -1 D -4 E more information is needed

5. For how many integers n is n also an integer ? 100 – n A 1 B 6 C 10 D 18 E 100

6. In this subtraction the values of P, Q, R and S are digits. 8 Q 0 S What is the value of P + Q + R + S ? - P 0 R 2 2 0 0 8

A 12 B 14 C 16 D 18 E 20 7. Positive integers m and n are such that 2m + 2n = 1280.

What is the value of m + n? A 14 B 16 C 18 D 32 E 640

8. The numbers x, y and z satisfy the following equations:

x + y + 2z = 850, x + 2y + z = 950 , x + y + 2z = 1200. What is their mean? A 250 B 1000 C 750 D 1000 E more information is needed


AS Mathematics – BRIDGING MATERIALS Answer Sheet 1

Exercise A: pg 6 Exercise E: pg 10

1. 53 1. 3

2 43 2. 1

3. 66 3. 1/8

4. 6-7 4. y16

5. 46 5. 5/2

6. 4-6 6. 27/64

7. 46

8. 1 Exercise F: pg 12

9. 29 1. √6

10. 211 2. 6

3. 3

Simplify: pg 6 4. 2

1. 3a 5. 8√5

2. 8a5b4 6. 20√6

3. 2a3 7. 144

4. 3a3b2 8. 5√3

5. 3a2c4 9. 10√7

10. 8√2

Simplify: pg 12

Exercise B: pg 7 1. 6

1. 3 2. a/c

2. 3

3. 5 Exercise G: pg 13

4. 10 1. 2x2 – 2x

5. 5/6 2. 15x – 6x2

6. 3/5 3. X2 + 5x + 6

4. X2 + x – 12

Exercise C: pg 8 5. 6x2 + 11x – 10

1. 25 6. 2 + 7x + 3x2

2. 64 7. 49x2 – 1

3. 81 8. 25X2 - 16

4. 9p2

5. X6y15 Exercise H: pg 14

1. (x + 3)(x - 3)

Rewrite: pg 8 2. (k + 10)(k - 10)

1. t2/3 3. (4x + 3)(4x -3)

2. M3/4 4. (4y + 5x)(4y – 5x)

Exercise D: pg 9 Exercise I: pg 14

1. 1/105 = 1/100000 1. (x + 4)(x + 2)

2. 1/64 2. (z + 8)(z + 2)

3. 1/t 3. (y + 7)(y + 1)

4. 4/q4 4. (2x + 1)(x + 2)

5. 3/4x5 5. (3x – 9)((x + 4)

6. (5x + 1)(x – 2)


AS Mathematics – BRIDGING MATERIALS Answer Sheet 2

Exercise I: pg 14 continued Exercise N: pg 19

7. (3y – 2)(y – 4) 1. T

8. (7x + 1)(x + 1) 2. F

9. (2x + 5)(2x – 5) 3. F

10. 5x(x + 2) 4. T

11. (½x – 3)(½x + 3) 5. F

12. (3t + 2)(t – 6) 6. F

7. T

Exercise J: pg 15 8. F

1. x = 5 and x = 1 9. F

2. x = 5 and x = -1 10. F

3. x = 2.5 and x = -3.5

4. x = 2.5 and x = -2.5 Exercise O: pg 20

5. x = 3 and x = -1/3 1. x = -2 and x = -6

6. x = 11/13 and x = -1 2. x = 4.5

7. x = 3.5 and x = -2.5 3. x = 8.5

8. x = -1/5 and x = -1 4. x = 34

9. x = 2.5 and x = -2.5 5. x = -6 2/3

10. x = 2 and x =-1.5 6. x = ±5

7. x = 5 ± 2√10

Exercise K: pg 16 8. x = 25

1. x = 3.70 and x = -2.70 9. x = 1/16

2. x = -0.41 and x = -1.84

3. x = -1.39 and x = -2.27

4. x = 1.64 and x = 0.61

Enhancements questions:

Exercise L: pg 16 To be discussed...........

1. x = -2 ± √5

2. x = -7 ± 3√6

3. x = 3 ± √6

4. x = -3 ± √2

Exercise M: pg 18

Mathematics - Schudio · Pre-enrolment task for 2014 entry Mathematics Why do I need to complete a...

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