Prompts are mathematical statements, equations or diagrams stripped back to the bare minimum, while simultaneously loaded with the potential for exploration. In short, a prompt should have "less to it and more in it" as one teacher has said. (1) A prompt must promote curiosity and questioning in students of the sort "that can't be right" or "I've noticed ...". Prompts should be engaging and ripe for speculation or conjecture. (2) A prompt must be aimed at students' developing mathematical knowledge, challenging them to decide whether new concepts are required to understand it fully. It must be accessible, yet should stand just beyond the recognition of a class. It should not be designed to intimidate a class; rather, students must feel confident enough to be able to manipulate and change the prompt. (3) A prompt must be open enough to offer students the opportunity to regulate their own activity. Ideally, it will offer a number of pathways and incorporate different areas of the curriculum at both abstract and concrete levels. (4) A prompt should provide opportunities for different forms of thinking, including induction (exploration and generalisation) and deduction (logical reasoning and proof).
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Five steps to creating an inquiry prompt Three maths teachers were involved in creating the inquiry prompt that appears at the end of this sequence. Each step shows the prompt becoming progressively more open  literally being stripped back to the essence of the mathematical diagram. Sometimes it is difficult for just one teacher to see the full potential in a prompt, and successive iterations might take the involvement of other inquiry teachers.
Step 1 The diagram was devised by Kier Tipple, a maths teacher in Brighton (UK). Defining A, B, and C as the areas of the three regions, he posed the problem: "At what distance do you need to set the radii apart to have the three sections equal in area?" Kier introduced the problem at a meal for maths educators, drawing the diagram on the back of a train ticket.Step 2 The diagram remains the same, but the problem has gone. This gets nearer an inquiry prompt because students are expected to pose the questions. However, in retaining the equation A = B = C, it is clear what is on the mind of the teacher: the prompt relates to three areas.
Step 3 This version hints at the original problem by including the centres of the circles. However, the suggestion of finding their areas has gone and the inquiry is potentially more open for it.
Step 4 In the fourth version, we return to the concept of area. The shading of two of the regions suggests they have something in common, but it is not obvious what that might be. For the first time, the prompt could be interpreted as a Venn diagram.
Step 5 The final prompt is stripped back to the minimum  two overlapping circles. This is the most open, offering a number of paths of exploration  including Venn diagrams, labelling parts of a circle, areas and circumferences of circles, and trigonometry and Pythagoras' Theorem (if we go back to the original problem).
Colm Sweet (a maths teacher in Horsham, UK)  who was responsible for steps 3, 4, and 5 above  has since developed the prompt further by combining it with triangles and squares to create the overlapping shapes prompt.
