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Negative numbers

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

Prompt 6 invites students to extend the lists as well in order to derive rules about adding and subtracting negative numbers. Indeed, extending the lists is a requisite to ‘discover’ the rules because they are not evident in the prompt as it was presented to students.

228-229