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Negative numbers inquiry

This prompt, which would normally be introduced to students in parts, was inspired by a discussion of extending patterns in Raffaella Borasi's Learning Mathematics Through Inquiry (1992). Borasi writes that the approach suggested by the prompt "relies on the discovery of patterns among already established results and assumes that these patterns will continue to hold in moving into the new expanded system" (pp. 59-60). 
  
She then invites us to consider the following multiplication sequence:
3 x 4 = 12
3 x 3 = 9
3 x 2 = 6
3 x 1 = 3
3 x 0 = ?
3 x (-1) = ?
3 x (-2) = ?
Students can derive the remaining values in the sequence (on the right) by continuing the pattern. Subtracting three from the product in the line above gives:
3 x 1 = 3
3 x 0 = 3 - 3 = 0
3 x (-1) = 0 - 3 = (-3)
3 x (-2) = (-3) - 3 = (-6)
Borasi concludes, "While we may all be aware that patterns can occasionally be deceptive, they nevertheless provide another valuable heuristic to guide the extension of a known operation to a wider domain" (p. 60).
Extending patterns and structural reasoning
The prompt invites students to extend a sequence of operations in order to derive rules about adding and subtracting (and multiplying) negative numbers. Indeed, extending a sequence and spotting the pattern is a requisite to ‘discover’ the rule. Consequently, there is a danger that the inquiry will restrict students to inductive thinking that involves them in comparing and describing the sequences. In such a circumstance, the teacher must encourage students towards structural reasoning that explains rather than just describes. To this end, the prompt below has proved effective.
  
  
Classroom inquiry
This picture, which was posted on social media, shows how the inquiry started in a PYP classroom. Once students have identified the pattern, they can create their own examples before trying (under the direction of the teacher) to explain their observations using number lines.