KS4: Algebra and Vectors

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1 Page1 KS4: Algebra and Vectors

2 Page2 Learning Objectives: During this theme, students will develop their understanding of Key Stage 4 algebra and vectors requiring algebraic manipulation. It will build on from knowledge gained from KS3 algebra, in order to develop students fluency in the topic. KS4 National Curriculum Algebra In addition to consolidating subject content from key stage 3, pupils should be taught to: simplify and manipulate algebraic expressions (including those involving surds {and algebraic fractions}) by: - factorising quadratic expressions of the form x 2 + bx + c, including the difference of two squares; {factorising quadratic expressions of the form ax 2 + bx + c} - simplifying expressions involving sums, products and powers, including the laws of indices know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments {and proofs} where appropriate, interpret simple expressions as functions with inputs and outputs; {interpret the reverse process as the inverse function ; interpret the succession of two functions as a composite function } use the form to identify parallel {and perpendicular} lines; find the equation of the line through two given points, or through one point with a given gradient identify and interpret roots, intercepts and turning points of quadratic functions graphically; deduce roots algebraically {and turning points by completing the square} recognise, sketch and interpret graphs of linear functions, quadratic functions, simple cubic functions, the reciprocal function with x 0, {the exponential function y = k x for positive values of k, and the trigonometric functions (with arguments in degrees) y = sin x, y = cos x and y = tan x for angles of any size} {sketch translations and reflections of the graph of a given function} plot and interpret graphs (including reciprocal graphs {and exponential graphs}) and graphs of non-standard functions in real contexts, to find approximate solutions to problems such as simple kinematic problems involving distance, speed and acceleration {calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, velocity-time graphs and graphs in financial contexts} {recognise and use the equation of a circle with centre at the origin; find the equation of a tangent to a circle at a given point} solve quadratic equations {including those that require rearrangement} algebraically by factorising, {by completing the square and by using the quadratic formula}; find approximate solutions using a graph solve two simultaneous equations in two variables (linear/linear {or linear/quadratic}) algebraically; find approximate solutions using a graph {find approximate solutions to equations numerically using iteration} translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or two simultaneous equations), solve the equation(s) and interpret the solution

3 Page3 solve linear inequalities in one {or two} variable{s}, {and quadratic inequalities in one variable}; represent the solution set on a number line, {using set notation and on a graph} recognise and use sequences of triangular, square and cube numbers, simple arithmetic progressions, Fibonacci type sequences, quadratic sequences, and simple geometric progressions (r n where n is an integer, and r is a positive rational number {or a surd}) {and other sequences} deduce expressions to calculate the n th term of linear {and quadratic} sequences Geometry and measures In addition to consolidating subject content from key stage 3, pupils should be taught to: describe translations as 2D vectors apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors; {use vectors to construct geometric arguments and proofs}.

4 Page4 Introductory week: (KS4 Algebra Pre-requisites) Task 1: Simplify the following: 1) 5a x 4b = 2) 6c x 2d = 3) 2e x 8f = 4) 6m x 3m = 5) 4g x 3h = 6) 7i x 2k = 7) 5p x 5q = 8) 10n x 7n = Extension: 1) 3c x 4g x 2k = 2) 2y x 2h x 6p = 3) 5d x w x 6w =

5 Page5 Task 2: Algebraic Simplification

6 Page6 Task 3: Algebraic Simplification

7 Page7 Task 4: Expanding and Simplifying Brackets

8 Page8 Task 5: Expanding and Simplifying Brackets

9 Page9 Task 6: Worded Questions 1. Mr Smith owns minibuses and coaches. Each minibus has 12 seats. (a) Write an expression, in terms of m, for the number of seats in m minibuses. Each coach has 48 seats (b) Write an expression, in terms of m and c, for the number of seats in m minibuses and c coaches. 2. Lisa packs pencils in boxes. She packs 12 pencils in each box. Lisa packs x boxes of pencils. (a) Write an expression, in terms of x, for the number of pencils Lisa packs. Lisa also packs pens in boxes.... She packs 10 pens into each box. Lisa packs y boxes of pens. (b) Write down an expression, in terms of x and y, for the total number of pens and pencils Lisa packs. 3. Jennifer made x cakes. She put 4 sweets on top of each cake. (a) Write down an expression, in terms of x, for the number of sweets she used.

10 Page10.. Paul made 3 more cakes than Jennifer. (b) Write down an expression, in terms of x, for the number of cakes Paul made... Paul also put 4 sweets on each of his cakes. (c) Write down an expression, in terms of x, for the number of sweets Paul used Martin cleaned his swimming pool. He hired a cleaning machine to do this job. The cost of hiring the cleaning machine was for the first day, then for each extra day. Martin s total cost of hiring the machine was For how many days did Martin hire the machine?... days 5. Andrew, Brenda and Callum each collect football stickers. Andrew has x stickers. Brenda has three times as many stickers as Andrew. (a) Write down an expression for the number of stickers that Brenda has.. (1) Callum has 9 stickers less than Andrew. (b) Write down an expression for the number of stickers that Callum has..

11 Page11 Section A: Simplify and manipulate algebraic expressions (Rearranging difficult formulae) Starter: Task 1:

12 Page12 Task 2: Make the letter in brackets the subject of the formula 1. Make the subject 2. Make the subject 3. Make b the subject 4. V 2 = u 2 + av 2 Make v the subject 5. Make r the subject 6. Make y the subject 7. Make a the subject 8. Make r the subject 9. Make r the subject 10. Make x the subject Task 3: Rearrange for the subject of the formula

13 Page Task 4:

14 Page14 Task 5: 1. Rearrange a(q c) = d to make q the subject. 2. (a) Make n the subject of the formula m = 5n 21 (b) Make p the subject of the formula 4(p 2q) = 3p P = πr + 2r + 2a Make r the subject of the formula 4. Make a the subject of the formula 2(3a c) = 5c Make m the subject of the formula 2(2p + m) = 3 5m

15 Page15 6. Make x the subject of 5(x 3) = y(4 3x) 7. When you are h feet above sea level, you can see d miles to the horizon, where Make h the subject of the formula 8. Rearrange the formula to make t the subject 9. Make b the subject of the formula 10. Rearrange the formula to make a the subject 11. Make x the subject of the formula 12. Rearrange to make u the subject of the formula. Give your answer in its simplest form.

16 Page16 Task 6: Forming equations

17 Page17 Section B: Solve quadratics by Factorisation; solving quadratics using the Formula Task 1: Solve via factorisation method 1. (i) Factorise x 2 4x 45 (ii) Solve the equation x 2 4x 45 = 0 2. (i) Factorise x 2 7x + 12 (ii) Solve the equation x 2 7x + 12 = 0 3. (a) Factorise x 2 3x 18 (b) Solve x 2 3x 18 = 0 x = or x = 4. (a) Factorise x 2 + 6x + 8 (b) Solve x 2 + 6x + 8 = 0 x = or x = 5. (a) Factorise x 2 x 56 (b) Solve x 2 x 56= 0 x = or x = 6. (i) Factorise x 2 + 9x + 20 (ii) Solve the equation x 2 + 9x + 20 = 0 7. (i) Factorise x 2 12x + 35 (ii) Solve the equation x 2 12x + 35 = 0 8. (i) Factorise x 2 x 72 (ii) Solve the equation x 2 x 72 = 0 9. (a) Factorise x 2 15x + 56 (b) Solve x 2 15x + 56 = 0 x = or x = 10. (a) Factorise x 2 + 9x + 18 (b) Solve x 2 + 9x + 18 = 0 x = or x =

18 Page (a) Show that x 2 x 56 = 0 (b) (i) Solve the equation x 2 x 56 = 0 (ii) Hence find the length of the shortest side of the trapezium Task 2: Solve the following using the Quadratic equation formula a) 2x + x - 8 = 0 b) 3x + 5x + 1 = 0 c) x - x 10 = 0 d) 5x + 2x - 1 = 0 e) 7x + 12x + 2 = 0 f) 3x + 11x + 9 = 0 g) 4x + 9x + 3 = 0 h) 6x + 22x + 19 = 0 i) x + 3x 6 = 0 j) 3x - 7x + 1 = 0 k) 2x + 11x + 4 = 0 l) 4x + 5x - 3 = 0 m) 4x - 9x + 4 = 0 n) 7x + 3x - 2 = 0

19 Page19 o) 5x - 10x + 1 = 0 Extensions: 1) Mavis is solving a quadratic equation using the quadratic formula. She correctly substitutes in values for a, b and c to get: What is the equation Mavis is trying to solve? 2) Junior uses the quadratic formula to solve: Sienna uses factorisation to solve: They both find something unusual in their solutions. Explain what this is Task 3: Use the quadratic formula to solve the equations below. Leave your answers to 3sf where necessary. 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) Extension: For each of the equations below substitute the values and solve them. Note down how many solutions they have. Which bit of the formula affects the number of solutions the equation has? Can you make a generalised statement about this? a) b) c) d) e) f) Task 4: 1. x 2 +4x+3=0 2. x 2 +6x-16=0

20 Page20 Task 5: 3. x 2-10x+21=0 4. x 2 +6x+2=0 5. 3x 2-6x- 30= 0 1. Solve x 2 2x 1 = 0 Give your solutions correct to 2 decimal places. 2. Solve x 2 + 3x 5 = 0 Give your solutions correct to 4 significant figures. 3. Solve 3x 2 + 7x 13 = 0 Give your solutions correct to 2 decimal places. 4. Solve the equation 2x 2 + 6x 95 = 0 Give your solutions correct to 3 significant figures. 5. The diagram below shows a hexagon. All the corners are right angles. All measurements are given in centimetres. The area of the shape is 25 cm 2. (a) Show that 6x x 39 = 0

21 Page21 Task 6: Using Quadratic formula 1. Solve 3x 2 + 7x 13 = 0 Give your solutions correct to 2 decimal places x =... or x = Solve the equation 2x 2 + 6x 95 = 0 Give your solutions correct to 3 significant figures. x =... or x = Solve x 2 + 3x - 5 = 0 Give your solutions correct to 4 significant figures. 4. Solve this quadratic equation. x 2 5x 8 = 0 Give your answers correct to 3 significant figures. 5. (a) Solve x 2 2x 1 = 0 Give your solutions correct to 2 decimal places. (b) Write down the solutions, correct to 2 decimal places, of 3x 2 6x 3 = 0 6. (a) Solve x 2 + x + 11 = 14 Give your solutions correct to 3 significant figures. y = x 2 + x + 11 The value of y is a prime number when x = 0, 1, 2 and 3. The following statement is not true. y = x 2 + x + 11 is always a prime number when x is an integer (b) Show that the statement is not true

22 Page22 (a) Show that 2x 2 + 6x 95 = 0 (b) Solve the equation 2x 2 + 6x 95 = 0 Give your solutions correct to 3 significant figures. 8. (a) Show that 6x x 39 = 0 (b) (i) Solve the equation 6x x 39 = 0 (ii) Hence work out the length of the longest side of the shape.

23 Page23 (a) Show that 9x 2 17x 85 = 0 (b) (i) Solve 9x 2 17x 85 = 0 Give your solutions correct to 3 significant figures. (ii) Hence, work out the length of the shortest side of the 6-sided shape.

24 Page24 Section C: The difference of 2 squares; solving quadratics by Completing the Square Task 1: Solve by completing the square

25 Page25 Task 2:

26 Page26 Task 3: Write an equivalent expression in the form (x ± a) 2 - b (a) x 2 + 4x + 3 (b) x 2 + 8x -13 (c) x 2 + 6x + 5 (d) x 2-4x + 5 (e) x 2-8x + 9 Task 4: Solve the following equations using completing the square leaving your answer in the form of a square root. (a) x 2 + 6x -1= 0 (b) x 2 + 4x + 3= 0 (c) x 2 + 8x +13 = 0 (d) x 2 + 2x - 3= 0 (e) x 2-4x - 6= 0 (f) x 2-10x -10 = 0 Task 5: Complete the square on the following (Note some answers involve fractions!!) (a) x 2 + 3x - 2 (b) x 2-7x +12 (c) x 2-3x - 5 (d) 2x 2-16x + 24 (e) 3x 2 + 6x -15 (f) 2x 2 + 6x + 5

27 Page27 Section D: Simultaneous equations using a Quadratics Starter: Simple Simultaneous equations 1. Solve the simultaneous equations 3x + 2y = 4 4x + 5y = Solve the equations 3x + 5y = 19 4x 2y = Solve the simultaneous equations 3x + 4y = 200 2x + 3y = Solve the simultaneous equations 5x + 2y = 11 4x 3y = Solve the simultaneous equations 4x 3y = 11 10x + 2y = 1 6. Solve the simultaneous equations 3x + 7y = 26 4x + 5y = Solve the simultaneous equations 6x 2y = 33 4x + 3y = 9 Task 1:

28 Page28 Task 2: 1. 2.

29 Page Extension:

30 Page30 Task 3: Exam style questions 1. By eliminating y, find the solutions to the simultaneous equations x 2 + y 2 = 25 y = x 7 x =... y =... or x =... y =... (Total 6 marks) 2. Bill said that the line y = 6 cuts the curve x 2 + y 2 = 25 at two points. (a) By eliminating y show that Bill is incorrect. (2) (b) By eliminating y, find the solutions to the simultaneous equations x 2 + y 2 = 25 y = 2x 2 x =... y =... or x =... y =... (6) 3. Solve the simultaneous equations x 2 + y 2 = 29 y x = 3 (Total 7 marks)

31 Page31 Section E: Algebraic Fractions Task 1: Simplifying Algebraic fractions

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33 Page33 Task 2:

34 Page34 Task 3:

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36 Page36 Task 4: Multiply/Divide Algebraic fractions

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38 Page38 Section F: Vectors Pre-requisites: manipulating algebraic fractions, Pythagoras theorem, and ratio knowledge (to work out vector magnitude and manipulate data provided) Task 1: Column Vectors d) Task 2:

39 Page39 Task 3:

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41 Page41 Task 4:

42 Page42 Section G: Supplementary Questions (Mock Exam Paper style) 1. (a) Solve the equation x 1 3x 1 = (4) (b) Solve the equation 3 5 = 1 x 1 3x (4)

43 Page43 2. Solve the equation 3x 1 2 2x 1 = (4) 3. Algebraic fractions... (Total 3 marks) (Total 3 marks) (Total 3 marks) 6.

44 Page44... (Total 3 marks) (Total 3 marks)... (Total 3 marks) (Total 3 marks) (Total 3 marks) (Total 3 marks)

45 Page (a) The triangle has angles x, 2x and 84 as shown. Find the value of x. 84 Not drawn accurately x 2 x Answer... degrees (3) 12. Here are four expressions. n n2 3 n n (a) If n = 3, which expression has the greatest value? Show your working. (b) Answer... If n = 0.3, which expression has the greatest value? Show your working. (2)......

46 Page Answer... (2) (Total 4 marks) 13. This shape is made up of rectangles. x Not to scale 4y 3x y (a) Write down an expression, in terms of x and y, for the perimeter of the shape Answer... (2) (b) If x = 2 cm and y = 5 cm, find the area of the shape Answer... cm 2 (2) (Total 4 marks)

47 Page x x z 2x y (a) (i) Write down and simplify an expression, in terms of x, for the area of the triangle x... (2) (ii) Write down an expression, in terms of x, y and z, for the total area of the triangle and the two rectangles (1) (b) (i) Factorise your answer to part (a)(ii) (ii)... Calculate the total area of the triangle and two rectangles, if x = 5 cm and x + y + z = 28 cm (2) (1)