Daniel Walker (a secondary school maths teacher) devised the prompt. In his original design, the prompt was y = ax + b and y = bx + a. As Daniel explains, this would have involved students working at an abstract level immediately: "If the students solve these simultaneously either using graphs or algebra, they'll find the solution is (1, a + b)." However, Daniel had a change of mind: "It's just occurred to me that if I use numbers and maybe
throw in an added twist like a + b = 1, then I give more scope for the students to generalise for themselves. Starting with, for example, y = 3x - 2 and y = -2x + 3 will give a solution (1, 1), but will allow them to investigate the effect of varying the relationship between a and b." The final version of the prompt encourages students to explore by changing the gradient and constant. They can then make and test conjectures and generalisations before ending the inquiry with an algebraic proof.
The solution (1, a + b) for the general case can be arrived at in various ways. On this sheet there are three methods. Asking students to discuss the methods can lead to claims about which one is best and what 'best' means in this context.
Daniel Walker is a teacher of mathematics at North Bridge House Canonbury (London, UK).
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