This inquiry is suitable for students in years 58 (grades 47). The prompt is adapted from one designed by Theo Giann (a PGCE student at the University of Brighton, UK, during 201112). In his original version, the fourth circle contained a question mark, rather than a hexagon. However, it is possible for students to interpret the question mark as a teacher's closed question and, hence, limit the inquiry to finding the next polygon in the sequence. In its form above, the prompt might lead to more open lines of inquiry, such as the ones below that have been generated by students' questions and comments: names of twodimensional shapes;
 perimeters of polygons;
 construction;
 methods to inscribe shapes in a circle;
 the circumference of a circle;
 regular polygons (including interior and exterior angles); and
 areas of polygons.
New pathway A new pathway for inquiry is suggested by the following questions from a year 7 student in Ollie Rutherford's class at Haverstock School (Camden, UK):  What is the ratio between the number of sides of a polygon and the area of the segments left over?
 Is there a sequence to the ratios?
The answers connect multiple topics in the curriculum: ratio, sequences and expressions for the nth term, area of polygons and a circle, Pythagoras' Theorem, trigonometry (including the cosine rule) and properties of individual shapes (such as the centroid of a triangle) amongst others. Theo tried his prompt with a year 8 class as part of a Masterslevel research project on encouraging autonomy. As this was Theo's first inquiry with an unfamiliar class, he decided not to use the regulatory cards and instead gave the students a limited choice of working in their books, making a poster or completing a worksheet.
Inquiry and misconceptions In his Masterslevel essay, Theo considered the impact of inquiry on one student and raised an important question about misconceptions during inquiry: "One boy, who usually was unmotivated in my lessons, produced a poster with many examples of different inscribed polygons and clear mathematical labelling of each one. He went on to eloquently explain his work to the rest of the class; however, he made an error in his work – he deduced that because he had drawn an irregular inscribed pentagon, all irregular pentagons could be inscribed. I asked him to investigate this statement but he did not see his error and I ran out of time to explain where he had gone wrong. "On the one hand, the student clearly developed his own line of inquiry, explored relevant aspects of the problem and quite frankly worked harder than I had ever seen or ever saw afterwards. The freedom of being able to decide his own workload, his own methods and his own definition of success inspired him to have a really fantastic lesson. On the other hand, his work contained errors which, due to their complex and unique (to him in that lesson) nature, could not be fully addressed – I cannot guarantee that this has not led to misconceptions being formed or even supported. "As a mathematics teacher, I was now faced with a seemingly impossible question – to what extent can mathematical accuracy be discarded for the sake of enthusiasm? No doubt, there is ready at hand an army of purists who would yell from the hilltops “NONE! EVER!” – I myself verge very closely towards that camp – but can we really cast out of hand a method that promotes a child’s enjoyment of mathematics simply because it can lead to unresolved misconceptions? Surely no system can be completely free of that unfortunate phenomenon." Functioning with Geometry and Fractions In this article from the ATM's Mathematics Teaching 207 (March 2008), Derek Ball and Barbara Ball describe the types of students' thinking that arose when considering the prompt above.  Claimevidencereasoning inquiry cycle Courtney Paull (a grade 7 math guide at the International School Manila) posted this picture on twitter. Students created their own claimevidencereasoning cycles from the prompt. The picture shows one claim: "Shapes with sides that are parallel to each other have an even number of sides." Courtney said that she was not sure where the inquiry was going, but the children made "some really awesome claims." After the claim was posted on twitter, one mathematics educator shared this picture as a counterexample to the claim. To maintain the validity of the claim, the wording could be changed to specify 'regular shapes'. Structured and open inquiry These are the questions and observations of Sally Pearson's year 7 mixed attainment class. During the inquiry, students were involved in structured or open inquiries depending on their levels of initiative and independence. Sally reports on the development of the inquiry:
"It was the class's second inquiry so far. I was impressed with the variety of the responses and the way that students were eager not only to provide a new question or comment but also how they were listening so carefully to each other and building on or amending what others were saying. Lots of answers to questions were provided by other students and they were all so desperate to speak we ended up spending about half of the lesson on debates about whether a square is a rectangle, how many lines of symmetry a rectangle has compared to a square, whether there are 360 degrees in a circle and whether a right angle is acute, obtuse or neither. "It was a great start to a unit on angles with a mixed attainment class as some naturally gravitated to exploring angles in triangles, some changed the prompt to see if the sum of the interior angles of any quadrilateral is 360 degrees and some were intrigued by the interior angles in the pentagon. For those that needed it, I closed the inquiry down fairly quickly as they wanted support investigating angles in triangles or polygons with direction from me, but I was interested in the number (about eight out of 30) that were happy to structure their own further inquiry based on changing the prompt with quadrilaterals or extending the pattern they saw to hexagons and beyond." Sally Pearson is a Lead Practitioner in the mathematics department at Brittons Academy (east London, UK). You can follow her on twitter @MissPearson1. Claimsupportquestion inquiry cycle Amelia O'Brien's grade 6 class at the Luanda International School (Angola) asked questions and made observations about the prompt to initiate an inquiry. The students used a cycle of claimsupportquestion to frame their thinking. Amelia reports that "the students were very engaged as they chose their own 'claim'." The cycle led to an advanced mathematical inquiry that included claims such as "irregular polygons cannot fit into circles" and angles in a heptagon sum to 820^{o} (see pictures below). Amelia O'Brien is a grade 6 teacher of the IB's Primary Years Programme and has run training sessions on using Inquiry Maths prompts. You can follow her on twitter @_AmeliaOBrien. Coregulating inquiry Coregulation refers to a process in which the teacher and students direct an inquiry together. In the account that follows, the teacher offered a year 7 (grade 6) mixed ability class the opportunity to decide what to do within a restricted set of options. At the start of the inquiry, the class posed questions and made observations about the prompt (see below). The students focused on the names of the shapes and the interior angles of the polygons, and on the fact that the polygons are inscribed inside circles. As the students read out their questions and observations, the teacher took the opportunity to introduce formal vocabulary, such as ‘segment’, ‘inscribed’ and ‘interior' angle. The teacher then used a protractor on the interactive whiteboard to measure the angles in the triangle and quadrilateral, confirming that they were all, respectively, 60^{o} and 90^{o}. The teacher introduced the six regulatory cards he had chosen for the class (left). In previous inquiries, students had started to choose particular cards automatically instead of thinking afresh about how to regulate their activity. Those cards described how to inquire (for example, "Work with another student"), so now all the cards were about what to do in the inquiry. However, before asking students to select a card, the class discussed what each card might mean in the context of the current inquiry. Under the guidance of the teacher, the students proposed the following: Change the prompt.  Draw the circles inside the shapes.  Decide what the problem is.  Decide what the aim of the inquiry is.  Use a worksheet.  Use a worksheet to practise using a protractor.  Ask a question about the prompt.  Think more to make up another question.  Make up more examples.  Try to draw accurately a heptagon and octagon inscribed inside a circle.  Look for a pattern.  Find the interior angle of a regular pentagon and hexagon and move on to other polygons.  All the students went on to select “Use a worksheet” or “Look for a pattern”. The teacher showed those who chose the latter how to split a polygon into triangles so they were able to calculate the sum of interior angles in multiples of 180^{o} before working out the value of one angle in a regular polygon. Thus, coregulation in this case involved the teacher in designing the regulatory cards and deciding when instruction was appropriate, while students developed the meaning of the cards in the specific context and selected a card that enabled them to learn. You can read more examples of how this inquiry has developed in the classroom in the primary section of the website.
