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Questioning techniques to challenge all learners at A level Mohammed Basharat Nick Thorpe 1730-1830 Thursday 25th June 2015 While we are waiting…. Change one coefficient of the quadratic 𝑦 = 𝑥 2 + 6𝑥 + 8 so that the graph touches the 𝑥-axis Watson and Mason ( 1998) 1. Change one aspect of …. so that … Change one aspect of……so that….. Change one coefficient of the quadratic 𝑦 = 𝑥 2 + 6𝑥 + 8 so that the graph touches the 𝑥-axis Change one aspect of 1 (2𝑥 0 Change one aspect of 2 −1 𝒓 = 3 + 𝑡 1 so the line 1 2 intersects one of the axes. + 3)𝑑𝑥 = 0 so it is true. Time for you to… Think about these questions and/or try and design your own question using this question stem. GeoGebra input: (x - a)² + (y - b)² = r² Change 1 (2𝑥 0 GeoGebra input: + 3)𝑑𝑥 = 0 so it is true. Integral[a x + b, c, d] Change one aspect of 2 −1 𝒓 = 3 + 𝑡 1 so the line 1 2 intersects one of the axes. GeoGebra input (3D view): Line[(a, b, c), (p, q, r)] Time for you to… Think about these questions and/or try and design your own question using this question stem. 2. Give me an example of…… GeoGebra input: ax+by+c=0 Give me an example of… …a question involving simultaneous equations with solution 𝑥 = 2, 𝑦 = −3 …a cubic curve with no stationary points …a transformation so the image of 𝑦 = 𝑥 − 1 2 passes through (2,4) …a function with (natural domain) 𝑥 > 0 and range 𝑦 ≥ 0 3. What is the same and what is different about …. The actions of comparing and contrasting force us to think about the properties of mathematical objects and students may make connections they might not normally consider. They are prompted into mathematical thinking processes such as generalising, conjecturing about what stays the same and what can change (called ‘variance’ and ‘invariance’), and then verifying these conjectures. What is the same, and what is different about… …𝑦 = f 𝑥 , 𝑦 = f( 𝑥 ) and 𝑦 = f(𝑥) …𝑥𝑛+1 = 𝑥𝑛 +1 , 𝑥1 2 = 5 and 𝑦𝑛+1 = 𝑥𝑛 +2 , 𝑦1 3 =5 …the roots of f 𝑥 = 0 and f 𝑥 + 2 = 0 …summing an arithmetic series and the area of a triangle. Preparing the learning Environment “Everyone gets stuck. It cannot be avoided, and it should not be hidden. It is an honourable and positive state, from which much can be learned.”