While Kaylee's conjecture may seem obvious to adults and older students, it helps the students make connections between a "square number," such as 5 times 5 equals 25, and a geometric model of a square with 5 rows of 5 square units per row. It also helps the students be more precise and careful as they make observations.

You may notice that this is setting the stage for a far less obvious conjecture that is encountered in middle school, namely, that the difference between two consecutive square numbers, n² and (n + 1)² is 2n + 1. For example, the difference between 19² and 20² is 2 x 19 + 1. By definition, 2n + 1 is an odd number. As you look at the sequence of consecutive squares and compute the differences between each consecutive pair, you always get odd numbers: 1, 3, 5, 7, . . . 39, 41, etc.

Kaylee uses visual representation to explain and justify her reasoning. However, she does not mention the one extra square unit that has to be added in addition to one row and one column.

Because geometric representation of squares was used, the students are able to talk about the number of square units in one row to represent one factor, and the number of rows of square units for the other factor. They were able to continue their discussion by talking about using different colors of square units to help illustrate the difference between a 3-by-3 square and a 4-by-4 square.