Summer Induction Work

Transcription

1 A-level Maths Summer Induction Work Deadline: Monday 11th September The Hazeley Academy

2 Mathematics A Level Maths Edecel: Pure Maths, Statistics and Mechanics Objective: To reinforce key GCSE skills in order to provide a firm foundation for starting the A-Level Course. Tasks Complete the Keeping Your Maths Brain booklet. You should give yourself a minimum of one hour each week to complete this. Further study may be necessary on some topics if you are not yet confident. Use the information to refresh your memory on each topic. Try all of the questions before marking them, using the solutions at the back. You may like to use other revision techniques to remember methods. The deadline for completing and handing in this homework booklet is 11 th September and you will be tested on these topics in your net lesson. Each topic will contribute equally to the final grade and this assessment will go towards your PR1 (Progress Review 1). The grade thresholds for the test are as shown in the table below Grade A B C D E U Grade threshold 80% 70% 60% 50% 40% Below 40% Those getting below grade A will be given more targeted support and then asked to resit the test. PLC Your mini personalised learning checklist is attached to the summer work you might like to RAG (Red/Amber/Green) as you go Resources/Research Use the notes provided in the booklet along with resources such as (login: hazeley, password: angle), Mr Hegarty Maths and Mathswatch clips to support your learning. Wider Reading If you are interested in some wider reading, the following books may be of interest: Fermat s Last Theorem Simon Singh Ale s Adventures in Numberland Ale Bellos The Code Book Simon Singh Why Do Buses Come In Threes? Rob Eastaway Submission Date 11 th September 017

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4 Hazeley School Page Welcome to your 7 week guide to keeping your maths brain. In this booklet you will find 7 sections, one to complete per week between now and September. Each section covers a topic from your Higher GCSE course that will be encountered again in your Year 1 course. By completing the eercises gradually through your break you should find that these techniques are still fresh when you return to start in Year 1. The deadline for completion is 11 th September. Each section starts with a review of the key techniques required for the topic. It includes worked eamples,, and hint bubbles,, to help you when you are working through the eercises. If you are still unsure about a topic or technique, on the contents page you will also find key words that should help you to find the relevant topics and worksheets on Also, you can find more marked questions on the the relevant activity numbers are also on the contents page. Remember to check your answers as you go and highlight any areas that you think you need more help with when you come back in September. Good Luck

5 Hazeley School Page Linear Equations Page 4: Page 5: Eercise 1 Question no Mymaths Equations 5 Equations Equations 4 Equations 5 Studymaths Algebra: & 7 Algebra: Algebra: 0 & 1 Page 6: Page 7: Eercise Quadratic Equations Question no Quadratic Quadratic Quadratic Completing the Quadratic Mymaths Equations 1 Equations Equations 1 Square Formula Studymaths Algebra: 4 Algebra: Algebra: 1 Page 8: Page 9: Eercise Simultaneous Equations Question no. 1 Mymaths Simultaneous Equations, Simultaneous Equations Negative Studymaths Algebra: 5 Algebra: 6 Inequalities Quadratic Simultaneous Equations Page 10: Page 11: Eercise 4 Question no. 1 Mymaths Inequations, Negative Inequalities Quadratic Inequalities Page 1: Page 1: Eercise 5 Algebraic Fractions Question no. 1 4 Cancelling Algebraic Adding Algebraic Multiplying Algebraic Mymaths Fractions Fractions Fractions Indices Page 14: Page 15: Eercise 6 Question no. 1 Mymaths Indices 1, Indices Indices Studymaths Algebra: 18 Page 16: Page 17: Eercise 7 Trigonometry Question no. 1 Mymaths Trigonometry Missing Angles Sine Rule Trigonometry Missing Sides Cosine Rule Sides, Cosine Rule Angles Studymaths Geometry: 15 Page 18: Answers Quadratic Equations - Fractions

6 Hazeley School Page 4 An equation that only includes variables with a power of 1 is a linear equation. All linear equations will have graphs that form a straight line. Solving Linear Equations: Always ensure that you show your steps when solving any equations. Eample 1: Eample : Eample : Solve + 7 = 4 Solve = 6 = 1 = = = = 1 Solve + 4 = = = 5 = Minimum methods acceptable: As e.g. with no arrows To help decide the order: imagine secret brackets around the numerator of the fraction. (5 ) 7 Minimum methods acceptable: As e.g. with no arrows Collect terms first. Move the SMALLEST amount of s to the biggest Minimum methods acceptable: As e.g. with no arrows

7 Hazeley School Page 5 Eercise 1: 1) Solve the following equations. Write your solution as a fraction when required. a) 11q 17 = 60 b) 6p + 10 = 5 c) 8y 1 = 0 d) 4 + 5n = 64 e) 1 b = 7 f) 0 c = 8 ) Solve these equations involving divisions. Write your solution as a fraction when required. a) 5 4 b) 10 4 c) d) 6 4 ) Solve these equations involving unknowns on both sides. Write your solution as a fraction when required. a) 5m + 6 = m + 1 b) p + 4 = p c) 5q 4 = q d) 7 = 5 4) Solve these equations involving brackets. Write your solution as a fraction when required. a) e + (e + 1) = e b) 5(f + 6) = 5f c) (g + 1) + (g 1) = d) 5h (h 1) = 9 5) Solve these more comple equations. Write your solution as a fraction when required. 1 a) 7 6 b) 0 y 1 c) d) 4 e) 7 f) Now check your work, answers on page 18.

8 Hazeley School Page 6 An equation that only includes variables with a power of is a quadratic equation. All quadratic equations will have graphs that form or shaped parabola. Factorising Quadratic Epressions: Eample 1: a) Factorise b) Factorise 49 c) Factorise Product = -4 Sum = +5 Product = -49 Sum = +0 Product = -1 Sum = = -7 7 = 0-4 = = - Separate : = +5 Factorise pairs: ( ) +1( ) Solution: ( )( + 8) Solution: ( 7)( + 7) Solution: ( + 1)( ) Eample : Hence solve a) = 0 b) 49 = 0 c) 6 = 0 ( )( + 8) = 0 ( 7)( + 7) = 0 ( + 1)( ) = 0 So = 0 or + 8 = 0 So 7 = 0 or + 7 = 0 So + 1 = 0 or = 0 = or = -8 = 7 or = -7 = -½ or = ⅔ Completing the Square: Eample : Write in the form ( + a) + b = 0 and hence solve = 0 Compare ( + a) = + a + a with the 1 st two terms of the equation + 6 a = 6 Remove etra a term Include rd term of original a = Substitute into required form: ( + ) 16 = 0 equation Simplify: ( + ) 5 = 0 (Completed Square Form) Rearrange to solve: ( + ) = 5 ( + ) = ±5 = - ± 5 = or = -8

9 Hazeley School Page 7 Using the Quadratic Formula: Eample 4: Use the quadratic formula to solve 7 + = 0 Substitute values from a + b + c = 0 into = = = = 6 b 5 6 b 4ac a = = or 6 REMEMBER A quadratic equation MUST equal ZERO before you solve it. Eercise : 1) Factorise the following quadratic epressions: a) b) c) 15 ) Factorise completely: d) 5y y e) f) y y 1 g) 11 h) 4 81 a) b) 5 + c) d) 6 1 ) Solve the following equations by first factorising them: a) y y = 0 b) y y 4 = 0 c) = 0 d) + = 5 e) y + 5y = 4) Solve the following equations by first completing the square, give answers to 1 dp: a) = 0 b) 8 = 0 c) + 4 = 0 5) Solve the following equations using the quadratic formula, give answers to 1 dp: a) = 0 b) = 0 c) = 0 d) 4 = e) 1 = 8 + Now check your work, answers on page 18.

10 Hazeley School Page 8 Problems that involve more than one equation and more than one unknown that are to be solved at the same time with the same values are known as Simultaneous Equations. Linear Simultaneous Equations, Elimination Method: Eample 1: Unknowns have the SAME SIGN SUBTRACT DIFFERENT SIGNS ADD Solve the following pair of equations simultaneously: + y = y = y = y = - The aim is to get the same amount of one unknown by multiplying one or both equations by a constant 17y = 119 y = 7 Substitute into to find : + 14 = 9 = 5 Check results in = 0 1 = -1 Linear Simultaneous Equations, Substitution Method: Eample : Solve the following pair of equations simultaneously: + y = 5 + y = 40 y = 5 + ( ) = = 40 = 4 Substitute into to find y: y = 4 = 10 The aim is to get one of the unknowns on its own and then substitute it into the other equation

11 Hazeley School Page 9 Quadratic Simultaneous Equations: Eample : See Factorising and Solving. Page 4 Solve the following equations simultaneously: y = + y = 1 The aim is to substitute the linear equation into the quadratic equation. + = 1 0 = 0 = ( )( + 1) = or = -1 Substitute into to find y: =, y = 8 = -1, y = 0 Eercise : 1) Solve the following simultaneous equations using the elimination method: a) + 5y = y = 0 b) a + b = 9 4a + b = 1 c) y = -4 + y = 9 d) 5 7y = 7 4y = 16 ) Solve the following simultaneous equations using the substitution method: a) + y = 5 + y = 5 b) y = + y = 10 c) a + 4b = 6 8b a = - d) = 4 + z 6 5z = 18 ) Solve the following linear and quadratic simultaneous equations: a) y = y = + 4 b) y = 7 8 y = + 7 c) y = y = 8 d) y = 9 4 y = Now check your work, answers on page 18.

12 Hazeley School Page 10 Equations involving greater than >, less than <, greater than or equal to or less than or equal to are called Inequalities. Inequalities have a set of solutions. Linear Inequalities: Eample 1: Solve the following inequalities: a) + 7 < 1 b) < < 8-1 See Solving Linear Equations. Page REMEMBER Change the direction of the inequality when you or by a negative number. Quadratic Inequalities: Eample : Solve the following inequality: < 0 See Solving Quadratic equations. Page 4 ( + )( + 5) < 0 so = 0 when = - or = -5 Now consider the graph of y = So the graph has y < 0 when is between -5 and - Eample : -5 - Solution: -5 < < - Solve the following inequality: 4 5 > 0 ( 5)( + 1) > 0 so 4 5 = 0 when = -1 or = 5 Now consider the graph of y = So the graph has y > 0 when is less than -1 or greater than 5 Solution: < -1 or > 5 (Ensure you write the solutions separately in this case)

13 Hazeley School Page 11 Eercise 4: 1) Solve the following inequalities: a) > 10 b) + 1 < 0 c) d) 5 < + 1 e) + 1 < + 5 f) ( + 1) 7 g) ( 1) < (1 ) h) 4 ) Solve the following quadratic inequalities: a) < 0 b) 8 9 > 0 c) d) < 0 e) 4 10 > 0 f) Now check your work, answers on page 18.

14 Hazeley School Page 1 Any fraction that involves an unknown is an Algebraic Fraction. You may be asked to simplify epressions or solve equations involving algebraic fractions. Simplifying Algebraic Fractions, Adding & Subtracting: Eample 1: a) Simplify the following algebraic fractions: 5 b) REMEMBER 8 1 Only add the numerators. c) REMEMBER Before you can add or subtract fractions you must have the same denominators Simplifying Algebraic Fractions, Multiplying & Dividing: Eample : Simplify the following algebraic fractions: REMEMBER Always factorise before you multiply or divide. a) b) REMEMBER Cancel whole brackets from any numerator & denominator. REMEMBER Never Divide: Flip the second fraction over & multiply.

15 Hazeley School Page 1 Solving Equations involving Algebraic Fractions: Eample : Solve the following equations: 6 8 a) 5 4 b) When you have a single fraction on one side multiply up to remove all fractions (using quadratic = -0.7 or = formula) Eercise 5: 1) Simplify the following algebraic fractions: 7a b 5ab 5ab 15a 10a 18a ab 6a a) b) c) d) e) f) 4ab 8a ab g) h) i) ) Write the following epressions as a single fraction: a) b) c) d) 4 5 e) f) ) Write the following epressions as a single fraction: a) b) 4) Solve the following equations giving answers to two decimal places where necessary: b) a) 4 1 c) Now check your work, answers on page 19.

16 Hazeley School Page 14 An Inde is also known as a power. The plural of inde is Indices. Simplifying Indices: You can only simplify indices when the bases are the same. The Rules: Base Inde 5 = = 7 = = = 5 = 5 = ( ) 5 = = ++++ = 10 = = 1 9 ½ = 9 =, 8 ⅔ = 8 = = = = 5 15 y a y b = y a+b y a y b = y a b (y a ) b = y a b y 0 = 1 y n / m y n = 1 y n = ( m y) n Eample 1: Simplify the following epressions: a) c 4 c 7 b) p 4 p 6 c) (r 4 ) 6 Eample : = c 4+7 = c 11 = p 4 (-6) = p 10 = r 4 6 = r 4 Evaluate the following: a) 4 ½ b) (6 ½ ) 6 ½ c) 49 d).5 ½ = 1 1 = 4 = 6 6 ½ = 49 = 6 + ½ = 6 = 6 = 7 =

17 Hazeley School Page 15 Eercise 6: 1) Simplify the following epressions: a) 4 b) m m c) y ½ y ½ d) w 7 w e) (k ½ ) 6 f) ( ) g) h) () () ) Evaluate the following quantities: a) 100 b) (5 4 ) ½ c) 81 ¼ 16 ¼ d) 0 01 ½ e) 0 04 ½ f) g) h) i) Now check your work, answers on page 19.

18 Hazeley School Page 16 Trigonometry is the use of Sine, Cosine and Tangent to calculate sides and angles in triangles. Right-angled Triangles: Sin(a) = opposite hypotenuse Eample 1: Tan(a) = opposite adjacent Cos(a) = adjacent hypotenuse opposite opposite the angle a hypotenuse opposite the right angle a adjacent net to the angle a Find the sides and angles indicated: a) b) c) o 58 o 6 9 o a) b) c) Opp Hyp 5 5 o Adj sin(5) 5 5sin(5) 87 Adj 6 Hyp 58 o Opp 6 cos(58) 6 cos(58) 11 Opp 6 Hyp o 9 Adj 6 tan( ) tan 9 7 o Non right-angled Triangles: Label angles as capital letters, sides as lower case. The sides are labelled opposite their corresponding angles. Sine Rule: a sin A b c sin B sinc Cosine Rule: a b c b c a c a b bccos A accosb abcosc b C A a c B Good Habits: Write a list of the angles and sides you know. Which Rule? If you have a pair use the Sine Rule.

19 Hazeley School Page 17 Eample : Find the sides and angles indicated: a) A b) A c) A d) C b 46 o 85 o 5 c A = 85 a = A = 68 a = A = 76 a = 15 A = a = 4 B = b = B = b = 7 B = b = 1 B = b = 10 C = 46 c = 5 C = c = 9 C = c = C = c = 9 Pair so sine rule: No pair so cosine rule: Pair so sine rule: No pair so cosine rule: cos sin85 sin46 88 sin76 sin sin85 sin76 sin sin cos sin76 sin cos sin 078 Eercise 7: a B C b 7 68 o a 9 c B 1 C b 76 o a c o B C b 10 A 4 9 a c o 7cos o B 9cos 4 1) Find the missing sides and angles: a) b) c ) d) 7 7 o 6 o e) f) g) h) 4 o 8 7 ) Find the missing sides and angles: 9 o 17 4 o 4 51 o o 1 o 7 a) b) 7 c) 51 o 9 6 o 44 o o 95 o d) e) 8 f) 115 o 4 6 o o 7 11 o 8 7 o 9 Now check your work, answers on page 19.

20 Hazeley School Page 18 Eercise 1: 1) a) q = 7 b) p = 7 c) y = ⅛ d) n = 1 e) b = 6 f) c = 4 ) a) = 18 b) = 8 c) = 45 d) = 10 ) a) m = b) p = -7 c) q = 1⅙ d) = 4) a) e = -1½ b) f = 1 c) g = ¾ d) h = 18 5) a) = b) y = ⅕ c) = d) = e) = ⅗ f) = -4 Eercise : 1) ) a) ( 1)( 4) b) ( + 5)( + ) c) ( 5)( + ) d) ( + )( + ) e) ( 1)( 10) f) ( + )( 4) g) ( 11)( + 11) h) ( 9)( + 9) a) (4 + )( ) b) ( 1)( 1) c) (5 + )( + 5) d) ( + 1)( 1) ) a) y(y ) = 0 y = 0 or y = b) (y 4) (y + 1) = 0 y = 4 or y = -1 c) ( + 1)( 7) = 0 = -½ or = ⅓ d) ( + 7)( 5) = 0 = -7 or = 5 e) (y 1) (y + ) = 0 y = ⅓ or y = - 4) 5) a) ( + 5) = 0 = -0 or = -9 7 b) ( 4) 18 = 0 = -0 or = 8 5 c) ( + ) = 0 4 = 1 or = -4 a) = 7 or = 0 b) = 4 8 or = 0 c) = -0 1 or = - 1 d) = 1 or = -0 7 e) = 7 9 or = 0 1 Eercise : 1) a) =, y = 4 b) a =, b = 1 c) =, y = d) = 4, y = -1 ) a) =, y = 1 b) =, y = 1 c) a = 5, b = ¼ d) = ½, z = - ) a) = 4, y = 8 = -1, y = b) =, y = 1 = 5, y = 7 c) =, y = 7 = 5, y = 17 d) = ½, y = ½ = 4, y = Eercise 4: 1) a) > 1 b) < -1 c) 5 d) < ¼ e) < 4 f) -9 g) < 1 h) 1 ) a) -4 < < - b) < -1, > 9 c) -1 1 d) ½ < < ⅚ e) < -1¼, > f) = 7

21 Hazeley School Page 19 Eercise 5: 1) a) b) c) d) e) f) g) h) i) a 5b b a 6 b a b a b ) ) 4) a) b) c) d) e) f) a) b) a) = -⅔ or = 1 b) = -½ or = c) = 0 or = Eercise 6: 1) a) 7 b) m c) y d) w -5 Eercise 7: e) k f) 6 g) 6 4 h) ) a) 1000 b) 1 5 c) 1 5 d) e) f) 1 5 g) 10 h) 64 i) 1 1) a) = 18 b) = 4 09 c) = 60 d) = 6 17 e) = 60 o f) = 8 1 o g) = 10 9 h) = 54 o ) a) = 10 b) = 4 86 c) = d) = 7 o e) = 94 1 o f) =4 6 o

22 Hazeley School Page 0 Mini Personalised Learning checklist Topic Linear equations Quadratic Equations Simultaneous Equations Inequalities Algebraic Fractions Indices Trigonometry RAG