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Other algebra prompts

These prompts have been used, to our knowledge, only rarely or not at all in classrooms. If you do use one with your class, please contact us and let us know your experiences.
Expression for the nth term of a sequence
If an + b is the nth term of a linear sequence with odd numbers only, a is always even and b is always odd.   
Solving quadratic equations
ax2 + bx + c = 0 (a, b, c are integers) always has two solutions for x.
Quadratics with all odd coefficients never have integer solutions.
Algebraic fractions
Devised by Kier Tipple
The prompt can be explored using a graphing package. On 
 for example, you can change the value of a using a sliding scale.
Calculus (differentiation)
Billy Adamson (Head of Mathematics at Thurston Community College, Bury St Edmunds, UK) designed the prompt and posted these pictures on social media of his students' questions and observations.