Amanda Klahn's grade 4 class came up with the wonderfully mathematical observations and questions above There are a variety of rich lines of inquiry that the pupils could follow.

The inquiry was the first time the class used the regulatory cards. Amanda restricted the pupils' choice to six cards. She reports that the cards worked well, making the pupils more aware of the direction the session was taking and stopping them from defaulting to 'ask the teacher'. As the pupils become more experienced with inquiry, Amanda hopes that the cards "will help them know where to take their learning without relying on me."

These are the questions and observations of Amanda Klahn's grade 4 PYP class at the Western Academy of Beijing the year after the inquiry reported on above. The students are thinking about the properties of the inequality and whether the solutions are in proportion. They also propose to change the prompt (including the digits and the operation) and then consider whether the change continues to fulfill the inequality in the prompt or, indeed, another condition set by the pupils. For example, one asks "Can we change the order so it is true for 'less than'?" Amanda reports that the prompt generated great excitement in the class with a pupil exclaiming that, "My brain is bursting with ideas."

At the time of carrying out the inquiries, Amanda was an IB PYP teacher at the Western Academy of Beijing (China). She writes a blog about classroom inquiries called Doing Maths: From Worksheets to Wonderings.

These are the initial responses of a year 7 class at Haverstock School (Camden, UK). Students describe the meaning of the prompt and verify that the inequality is correct. They suggest using other digits and reversing the inequality. That the class had been involved in inquiries before is evident in the students' selection of a regulatory card. Asked to suggest the direction of the inquiry, many pairs of students chose two complementary cards (as indicated by the colour of the ticks in the picture below).

As the inquiry went into the second lesson, students began to think about reversing the inequality. Many thought that four digits arranged __ __ x __ __ would always give a greater product then the same digits arranged __ x __ __ __. The first example supported the conjecture: 59 x 87 > 5 x 987. However, this was quickly followed by a counter-example: 17 x 42 < 1 x 742. The examples led students to explore how the size and order of the digits affected the inequality.

The comments from a year 7 class show the students' attempts to make sense of the prompt, either by commenting on a feature of the prompt they notice or, in one case, by trying to make a conjecture about the general relationships. Other than asking about the calculator button, the students did not spontaneously ask questions about the mathematics underlying the prompt.

The final question was particularly novel. The student who asked the question went on to find these inequalities:

215 x 34 > 215 x 3 x 4 > 2 x 1 x 53 x 4

215 + 34 > 215 + 3 + 4 > 2 + 1 + 53 + 4

In the picture (right), which shows the result of one open inquiry, the connections between the pupil's steps in reasoning are clearly presented. The inquiry combines procedural steps to ascertain if the inequality is true and observations based on a conceptual understanding.