TEACHER: Oh... so if I draw my best square on the board... one thing we
know about a square is that it has...
ALL[Kendrick loudest]: Four equal sides.
TEACHER: Very good! You’re right Kendrick, 4 equal sides... right? If I
draw a 1-by-1 array, what do I get?
ALL: It’s going to be 1 square...
TEACHER: Oh! So then 1 1 1, which makes 1...
ALL: A square number!The array representation seemed to help further their understanding, so I
decided the students were ready to revisit some of their initial confusions. It is
important to figure out what representations are meaningful to the students, as
opposed to my conducting a “lesson” on arrays.
I was also pleased that by seeing the arrays, Kendrick was able to offer an
observation about the attributes of a square. Once he made that connection to
arrays, I wanted to return to his original assertion about square numbers being big.
Even though he was incorrect, I wanted to validate his willingness to take a risk
and to use this discussion to clarify other students’ ideas. I said, “So let’s talk about
Kendrick’s idea about all square numbers having to be big numbers... what do
you think, Kendrick? Do you still think the same? Because 1 is not a big number.”
I decided to start from a small number and build it, hoping Kendrick would see
the number pattern.
TEACHER: So what about a 2-by-2 array... will I get a big number?
CHELSEA: That’s going to be 4!
TEACHER: So if I write 2 2, that will give me 4, which is...
ALL: A square number!
TEACHER: Oh, yeah, because it makes a square! OK... since you know this,
you can tell me what the next square number will be!
ALL: Three!
ALL:[except Chelsea] No! Nine!
CHELSEA: Oh, yeah!
TEACHER: So the next square number is 9?
MALIA: Because it will be a 3-by-3 array...
TEACHER: OK. And the next square number will be...
MALIA: Sixteen!
TEACHER: And how did you know that?
MALIA: Because 4 4 is 16.
SHARONDA: Yeah, because 4 4 is 16, and if you put the dimensions in and
you count the squares in the array you’ll get 16, see... 4 4 is 8 and then
8 8 is 16!Talking About Square Numbers