Solve quadratic equations by factorising Grades 6/7 Quadratic Equations Solve quadratic equations by factorising If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

Lesson Plan Lesson Overview Progression of Learning Objective(s) Expand brackets, including multiplication of binomial expressions. Grade 6/7 Prior Knowledge Expanding binomial expressions; linear equations Duration 60 minutes (variable). Resources Slides 11 onwards are printable versions of some of the earlier slides. Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) Starter; Recap of linear equations needed later for factors in quadratics. Teacher led activity. Eight equations of the general form ax + b = s. IF A STUDENT IS CORRECT, CLICK ON ANY EQUATION TO SHOW THE ANSWER AND TURN SECTOR GREEN; IF AN ANSWER IS INCORRECT, CLICK ON A BLANK AREA OF COLOUR IN THE SECTOR TO TURN IT AMBER. THE RED CIRCLE IS TIMING DOWN ONE MINUTE FROM THE TIME AT WHICH THE START BUTTON IS PRESSED A printable sheet (SLIDE 11) is provided if more practice is needed. 10 Factorising Introductory “investigation”; pair the eight factors with the expressions. This can be used to emphasise the role of negative signs (note the x2 ± 3x − 18 expressions). PRINT SLIDE 12 Differentiated practice at factorising. Later questions use common factors and difference of two squares. PRINT SLIDE 13 15 Solving quadratics by factorising “I have two numbers multiplied together…”. Then to differentiated sheet of factorising questions and follow up sheet of solving by factorising. PRINT SLIDE 14 15-20 Reasoning. Four differentiated reasoning problems that need solution by factorisation. PRINT SLIDE 15 Plenary: Error spotting in answers. You’re the examiner. Question from SAM, with two incorrect answers. Click on slide brings up MS so students can work out how many marks each is worth and gain insight into what examiners are looking for in their answers and how they are marked. 5 Next Steps Other methods of solving quadratic equations. Assessment PLC/Reformed Specification/Target 6/Algebra/Quadratic Equations (factorisation)

You have one minute from the moment you press start 2x − 1 = 0 2x + 1 = 3 x = 1 2 x = 1 3x + 1 = 1 3x + 2 = 0 START x = 0 x = − 2 3 2x + 1 = 9 5x − 1 = 8 x = −2 1 2 x = 1 4 5 You have one minute from the moment you press start 4x = 6 3x = 1 x = 1 1 2 x = 1 3 Click the start button to begin the timer and reveal eight equations. Click the equation to reveal the correct answer; click the colour in the sector to try again

Key Vocabulary Expand Factor/factorise Simplify Solve/solution

Quadratic Equations; factorising Pair the factors with the expression x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) ( x + 4 ) ( x − 4 ) ( x − 6 ) ( x + 2 ) ( x − 3 ) ( x + 5 ) x2 + 3x − 18 x2 − 3x − 18

Quadratic Equations; factorising Factorising All but two of these can be factorised; express them using brackets – and how about explaining why the other two don’t work? SILVER x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 x2 + 10x + 25 x2 + 3x x2 + 16 x2 − 16 x2 − x Fancy a challenge? All but one of these can be factorised… GOLD 2x2 + 7x + 3 3x2 + x − 2 3x2 + x 4x2 − 25 9x2 + 16 4x2 − 8x − 5 Click on a box to see if it will factorise. If it turns red, it won’t. If it turns green, you know what to do…

Quadratic Equations; factorising I have two numbers in these sacks. When I multiply them together, the result is 0. What can you tell me about the two numbers? × = 0 What can you tell me about a and b? ab = 0 What can you tell me about x? (x − 2)(x − 4) = 0 What can you tell me about x? x2 + 7x + 12 = 0

Solve these quadratic equations by factorising Quadratic Equations; factorising Solve these quadratic equations by factorising BRONZE SILVER GOLD x² + 7x + 10 = 0 x² + 6x + 8 = 0 x² + 3x + 2 = 0 x² + 9x + 20 = 0 x² − 9x + 20 = 0 x² − 6x + 5 = 0 x² + 2x − 24 = 0 x² − x − 30 = 0 3x² + 7x + 2 = 0 2x² + 9x + 9 = 0 2x² + 3x + 1 = 0 3x² + 8x + 5 = 0 2x² − 9x + 9 = 0 2x² − 9x + 10 = 0 5x² + 26x + 5 = 0 4x² + 4x + 1 = 0 4x² + 13x + 3 = 0 4x² − 4x + 3 = 0 x² − 3x = 0 2x² + 8x = 0 x² − 9 = 0 4x² − 25 = 0 2x² + 3x = 9 3x² − x = 4 x = −2 or x = −5 x = −2 or x = −4 x = −2 or x = −1 x = −4 or x = −5 x = 4 or x = 5 x = 1 or x = 5 x = −6 or x = 4 x = 6 or x = −5 x = −2 or x = −⅓ x = −3 or x = −1½ x = −½ or x = −1 x = −1 or x = −1⅔ x = 3 or x = 1½ x = 2½ or x = 2 x = −5 or x = ⅕ x = ½ x = −¼ or x = −3 x = 1½ or x = ½ x = 3 or x = 0 x = −4 or x = 0 x = 3 or x = −3 x = 2½ or x = −2½ x = −3 or x = 1½ x = −1 or x = 1⅓ Click on “Bronze”, “Silver” or “Gold” to see answers for that section

Quadratic Equations; factorising Reasoning BRONZE SILVER The area of the square is the same as the area of the rectangle. Find x. Solve the equation x3 + 2x2 − 8x = 0. x 2 x + 4 x SILVER GOLD The area of the rectangle is 140cm2. Find x. Solve the equation (x + 2)(x − 7) = 52. (x + 4)cm2 (x + 6)cm2

Quadratic Equations; factorising You’re the examiner. Look at these answers. Is either of them completely correct? How many marks (out of 3) would you award to each one? (x − 4)(x − 5) = 0 (x + 4)(x − 5) = 0 x = 4 or x = 5 x = 4 or x = −5 WJEC Sample examination question and mark scheme

2x − 1 = 0 2x + 1 = 3 4x + 3 = 0 2x + 7 = 10 3x + 1 = 1 3x + 2 = 0 3x − 1 = 5 3x + 2 = 0 2x + 1 = 9 5x − 1 = 8 5x + 1 = 11 2x − 1 = 10 4x = 6 3x = 1 2x = 7 7x = 3 If these were the answers, what were the questions? 2x − 9 = 0 2x + 1 = 3 This is the set of questions from the starter slide x = 2 x = − 3 4 4x + 3 = 7 4x + 3 = 2 x = 0 x = 4 2 3 3x + 1 = 0 6x − 5 = 7 x = 1 2 x = −3 5x = −2 2x = −5 x = −3 1 4 x = −1 1 2

Quadratic Equations x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) Factorising Pair the factors with the expression x2 + 6x + 8 x2 + x − 20 ( x + 6 ) ( x + 3 ) ( x + 4 ) ( x − 4 ) ( x − 6 ) ( x + 2 ) ( x − 3 ) ( x + 5 ) x2 + 3x − 18 x2 − 3x − 18

Quadratic Equations x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 Factorising All but two of these can be factorised; express them using brackets – and how about explaining why the other two don’t work? SILVER x2 + 7x + 10 x2 + 4x + 6 x2 + 3x − 18 x2 + x − 42 x2 + 10x + 25 x2 + 3x x2 + 16 x2 − 16 x2 − x Fancy a challenge? All but one of these can be factorised… GOLD 2x2 + 7x + 3 3x2 + x − 2 3x2 + x 4x2 − 25 9x2 + 16 4x2 − 8x − 5 On the screen, the ones that factorise change colour and become green; the ones that don’t become red. If you have a coloured crayon…

Solve these quadratic equations by factorising Quadratic Equations; factorising Solve these quadratic equations by factorising BRONZE SILVER GOLD x² + 7x + 10 = 0 x² + 6x + 8 = 0 x² + 3x + 2 = 0 x² + 9x + 20 = 0 x² − 9x + 20 = 0 x² − 6x + 5 = 0 x² + 2x − 24 = 0 x² − x − 30 = 0 3x² + 7x + 2 = 0 2x² + 9x + 9 = 0 2x² + 3x + 1 = 0 3x² + 8x + 5 = 0 2x² − 9x + 9 = 0 2x² − 9x + 10 = 0 5x² + 26x + 5 = 0 4x² + 4x + 1 = 0 4x² + 13x + 3 = 0 4x² − 4x + 3 = 0 x² − 3x = 0 2x² + 8x = 0 x² − 9 = 0 4x² − 25 = 0 2x² + 3x = 9 3x² − x = 4

Quadratic Equations; factorising Reasoning BRONZE SILVER The area of the square is the same as the area of the rectangle. Find x. Solve the equation x3 + 2x2 − 8x = 0. x 2 x + 4 x SILVER GOLD The area of the rectangle is 140cm2. Find x. Solve the equation (x + 2)(x − 7) = 52. (x + 4)cm2 (x + 6)cm2