# Negative numbers inquiry

# The prompt

**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.

# Classroom inquiry

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.