Students commented on the connection between the areas using the large squares as a unit of measure and the areas using the small squares. 

At the end of the lesson, the students had found the areas of the rectangle and triangle and had started to make suggestions for changing their dimensions in order to make the areas equal. 

We also had estimates for the area of the circle (using the large squares) of between 30.5 and 33.5. Students reasoned that the small squares would give a more accurate estimation because there were more whole squares to count.

Inquiry lesson 3

The final lesson addressed the two remaining points from the students' initial questions and comments. The first question was about whether other shapes could have the same area. 

Students drew a rhombus, parallelogram and regular trapezium, trying to create the three shapes with equal areas. They checked the areas by using strategies to count squares.

On completing the task, some students attempted to make one cut to each of the three shapes so that they could rearrange the two pieces to make a rectangle.

The second question involved the perimeters of the shapes: Is it possible for the perimeters of a rectangle, triangle and circle to be the same?

After I explained why C = πd, students either practised finding the lengths of circumferences or tried to establish if the shapes with the same areas (introduced at the end of lesson 2) had the same perimeters. 

The class decided that if the areas of a rectangle, a triangle and a circle are equal, it would be unlikely they would also have the same perimeter. Many in the class wanted to go further and say it was impossible, but no student could establish a solid reason why this might be so.

The picture shows the initial responses to the prompt from Amanda Klahn's grade 4 (year 5) PYP class at the Western Academy of Beijing (China). The question about how to work out the area of the triangle and circle at the bottom demonstrates that pupils have identified the new concepts and procedures they need to make progress in the inquiry. The ability to recognise missing knowledge is a key part to developing as a self-regulated learner. During the inquiry, Amanda described how the class explored π before returning to the prompt, adding "I love how students are led to explore new concepts!"

The design of Inquiry Maths prompts at just above the current understanding of the class encourages students to 'reach up' for new knowledge. The inquiry process then involves using the new concepts in the service of the pupils' inquiry. This makes the concepts relevant at the point of meeting them and connected to other concepts that form part of the inquiry.

These are the questions of a year 8 class. The picture was sent to the website by Eduardo Abend who used Inquiry Maths prompts in the classroom and wrote a project about inquiry learning while training  to be a secondary school teacher at London Metropolitan University. Eduardo reports on the inquiry: "It was the class's first Inquiry Maths lesson, so I made it quite structured. I had the plan of covering the area of a circle, moving from what the class already knew about calculating a circle's circumference to the new knowledge. The class was really receptive to the new style of teaching and we managed to do a practical activity to discover the area of a circle. We cut circles into small sectors and arranged them into a form as close to a rectangle as possible. From there it was a matter of using previous knowledge to find out the area of the circle."