Division inquiry 2
The prompt
Mathematical inquiry processes: Verify, explore more cases and reason. Conceptual field of inquiry: Product and quotient of integers; commutativity.
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 (shown below the main prompt) 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 the compare and contrast prompt 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.
Rich inquiry in the primary classroom
When Amanda Klahn used the prompt with her grade 4 (year 5) class, an extremely rich inquiry developed. Amanda structured the inquiry into three steps. Step 1 saw students post questions and comments to a wall on the class blog. The pupils focused on the meaning of the signs and the accuracy of the equations, speculated about or offered explanations for the underlying mathematics and wondered what would happen if they changed the prompt:
I think 42 x 14 is the same as 84 x 7 because if you split 84 you get 42 and if you split 14 you get 7.
Why does it say that 42 x 14 = 84 when the answer is really 588?
I think the answer to 42 x 14 is the same as 84 x 7, and the same with division.
I understand that 42 x 14 = 84 is not what it's trying to say. It's trying to say that 42 x 14 is the same as 84 x 7.
Does it link to doubles?
Now I see! 42 ÷ 84 = 2 and 7 ÷ 14 = 2.
Maybe the equals sign with the line across means does not equal.
I think the equals sign with the cross means the same thing.
Would it work with subtraction and addition?
What if you use decimals?
Step 2 involved students in responding to their peers' questions and comments.
Step 3 was teacher-driven inquiry. Amanda asked: "What could we do to 42 ÷ 14 to change the sign to an equals?"
At the time of the inquiry, Amanda taught at the Western Academy of Beijing (China).
Questions and observations
The questions and observations about the prompt come from two year 7 mixed attainment classes. The students developed the inquiry by choosing from four regulatory cards:
Ask the teacher to explain
Practice a procedure
Make up more examples
Change the prompt.
An alternative prompt
The prompt was devised for upper primary students to develop their knowledge about division. It aims to deepen their conception of a remainder. Do the remainders mean the same in both calculations?
They would in a concrete context. For example, if 42 items are divided among five people, then the two items remaining would be identical to the two items left over when 26 of the same item are divided among three people.
However, this is not the case if we could split the item because the divisors are different.
Imagine the two items are packets of sweets containing 15 sweets. In the first case, each of the five people would receive 12 sweets as their share of the two packets (60 sweets) left over. In the second case, the divisor is smaller and so each person receives 20 sweets.
In order to show the contention in the prompt is false, the teacher should encourage the class to take another step in the chain of reasoning.
In the first case, 12 sweets represents a fifth of the two packets remaining; in the second case, 20 sweets represents a third of the two packets. Thus, the calculations equal 8.2 and 8.333 recurring respectively and are, therefore, unequal.