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Coefficients inquiry

If the coefficients and constant of a quadratic equation are all odd, then the solutions can never be integers.
  
The prompt could be used after students have learnt about solving quadratic equations by factorisation through inquiring into the quadratic equations prompt. The coefficients prompt is an opportunity for the teacher to introduce the quadratic formula, particularly when students discover that they cannot solve an equation with the method they have learnt previously. Indeed, students might select a regulatory card to request an explanation of or instruction in an alternative method.
   
In the initial phase of the inquiry, students' questions and observations will be aimed at understanding the prompt and creating equations whose properties meet its conditions. In the general form of a quadratic equation ax2
+ bx + c = 0, the
coefficients (a and b) and constant (c) are all odd.
  
The teacher could structure a phase of exploration by providing groups of equations to solve. In group A (see the table below) a = 1. The equations are deliberately similar to those students might have seen when factorising in order to increase the potential for cognitive conflict in their thinking. Group B uses the same odd numbers for coefficients and constant, but changes the operations. If students make up more examples of their own, they can generalise about solutions to this type of group. In group C, the coefficients form a linear sequence, which is a constraint students might use to create their own equations.
The inquiry ends with the proof that the prompt is true 
(see right above), which is accessible to students in upper secondary school.

Resources 


Notice: Please be aware that, to the knowledge of the authors, this prompt has not, as yet, been used in a classroom. If you do use it with a class, we would very much appreciate hearing about the students' questions and observations and how the inquiry developed. Please contact Inquiry Maths here.
Proof by contradiction