Sharonda had some kernels of understanding; she seemed to understand that you
can square all numbers, but I was not sure she understood that squaring a number
is not the same thing as a square number. I was also not sure she was clear about
the difference between factors and square numbers. I also noted that the students
had mastered some of their multiplication facts, something we could build on.
TEACHER: What do you guys think?
MALIA: I want to say that when you add a number by the same number, you
get a square number.
TEACHER: When you add a number by the...
MALIA: No! When you multiply a number by the same number, I mean.
TEACHER: OK, when you multiply a number by the same number? [Malia
nods.] Why?
MALIA: Because, like, if you do an array,^1 you will have the same amount on
the top and the same amount on the side.
TEACHER: So, if I have an array, and I have the same amount on the top and
the same amount on the side, and you’re talking about the dimensions right?
If all the dimensions are the same, what shape do I end up with?
MALIA: With a square!Here, Malia was able to self-correct. When she first said, “When you add a num-
ber,” she changed to “When you multiply a number by the same number.” I was
pleased to see her being able to reflect on what she said. I restated her comment
using the correct terminology, dimensions, because I want students to use precise
language during our math discussions. Malia was able to recall the visual repre-
sentation of an array to help her think about square numbers. I realized that this
might be helpful for the other members of the group, so I decided to pursue a dis-
cussion of the array representation.
TEACHER: So, why do you think a square number is called a square number?
MALIA: Because when you multiply a number by the same number, you end
up with a square?
CHELSEA: Like 12 times 12 is 144... is it going to become a square?
TEACHER: You tell me... Does a 12-by-12 array look like a square?
MALIA: It will!
CHELSEA: Yeah...
TEACHER: Why? How do you know?
CHELSEA: Because it looks like a square.BUILDINGUNDERSTANDINGTHROUGHTALK(^1) An array is an area model for multiplication that consists of arrangement of objects, pictures, or numbers in
rows and columns.