Negative numbers inquiry
Mathematical inquiry processes: Extend patterns; generate examples; reason. Conceptual field of inquiry: Operations with negative numbers (including multiplication and division).
The prompts, which would normally be introduced to students one after the other, were 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 by continuing the pattern. Subtracting three from the product in the line above gives:
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 prompts invite students to extend a sequence of operations in order to derive rules about the four operations with 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.
This picture, which was posted on twitter, 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.