Mark Greenaway, an Advanced Skills Teacher in the UK, used the prompt above. Appearing to show two pairs of parallel lines, it suggests an inquiry about the angle properties of those lines. However, students used at least five areas of mathematics to attempt to interpret the prompt (illustration below): 

• Parallel and perpendicular lines;

• Coordinates on a graph, gradients and equations of straight lines, and equations of circles;

• Areas of shapes;

• Problem-solving involving the radius of a circle (requiring the use of Pythagoras' Theorem in one way to solve the problem); and

• Vectors.

Indeed, if this prompt had been designed to 'target' angles, then it is clear the inquiry teacher has failed by making the prompt open to multiple interpretations. Interestingly, Mark called this a "straight lines inquiry" and, in providing such an ambiguous prompt, ran it as an open inquiry. (For a description of the levels of openness in inquiry mathematics lessons, read Levels of Inquiry Maths).

One alternative is to use a guided inquiry, which might start with either of the prompts above. Marking an angle with a red dot or adding a statement (and indicating the parallel lines with arrows) has, in the past, steered students towards posing questions and making comments about angle properties. Often, they speculate about the size of other angles by measuring or 'by eye'. The teacher can intervene further by guiding students towards a consideration of how many independent angles are required to find all the angles in the diagram. The answer in this case is two (see diagram below). The students would, as normal, decide upon the direction of the inquiry from there.

Two independent angles are required to find all the angles in the diagram. Given one red angle, all the other red angles are known. Similarly, given any green angle afterwards, all the other angles are either known or could be calculated.

Another alternative is a structured inquiry, which is similar to an investigation in mathematics classrooms. (For a comparison of inquiries and investigations, click here). The teacher structures the inquiry by posing the question: How many independent angles would you need to find all the angles in the diagram? Students would be directed to draw diagrams with straight lines of which none, some or all will form pairs of parallel lines. They would then work out the number of independent angles required and tabulate the results. The final step would be for each student to induce (or 'discover') an answer - in this case, the formula giving the relationship between the number of lines (l), the number of pairs of parallel lines (p), and the number of independent angles (i).