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Division inquiry

Amanda James
, a UK secondary school maths teacher, contacted the Inquiry Maths website to ask for a prompt related to the concept of division. She was working on a project about transition between primary and secondary schools with colleagues in both sectors. The project studied the methods students use to solve division problems.
A preliminary prompt (below) might lead to a short initial inquiry to develop the meaning of division and introduce the concept of commutativity. Students might struggle with the right-hand side of the inequality, but it could lead to discussions of how whole numbers can be partitioned. It could also be used to tackle the issue of remainders, which might have relevance in some problems involving actual items, but can be an obstacle to greater conceptual understanding in secondary school. The prompt also starts to correct (the surprisingly high number of) students who continue to say through secondary school "5 divide 40" when they mean to divide 40 by five.
The main inquiry starts with a compare and contrast prompt (at the top of the page) through which students could develop an understanding of the difference between multiplication and division. What is happening? Why doesn't doubling and halving work for both? What would we have to do to both amounts to turn the "is not equal" sign into an "equals"? How many ways are there to do that? Can we make up more examples like this? How would we explain why one is double and halve and the other is double and double? What would happen if we halve and halve in both equations? During the initial phase of students' questions and comments, the teacher is advised to emphasise that students' calculations are permissible. In this way, students can demonstrate their methods for multiplication and division, giving the teacher valuable information about the current level of knowledge in the class.

Task sheet (These tasks were devised as part of a structured inquiry.)

You can read more examples of how this inquiry has developed in the classroom on the primary section of the website.
Questions and observations about the prompt
These questions and comments on the prompt come from year 7 classes. The top two are from mixed ability classes and the bottom one comes from a middle ability class. The students in the mixed ability classes developed the inquiry by choosing from four regulatory cards:
  • Ask the teacher to explain
  • Ask the teacher for a worksheet on multiplication or division
  • Make up more examples
  • Change the prompt.
You can read more examples of how this inquiry has developed in the classroom in the primary section of the website.