My kids can : making math accessible to all learners, K–5

All of the students were able to create an area model for each fraction using
a rectangle that they knew they needed to divide into equal parts. They were able
to explain that the pieces were “equal or fair because everyone is getting the
same” and that “the more parts there were in the whole, the smaller the pieces.”
Although I was pleased with their work using an area model, I knew that this
small group would need additional practice creating a variety of representations,
models, and contexts; telling and being aware of their own stories; and holding
on to mental images. So, I decided to ask them to compare , , and without ac-
tually drawing a model. I was interested in whether they would be able to use the
same reasoning they had applied earlier. The following conversation arose.

KEISHA: is bigger because it has more pieces.

JHALI: Yes, the 6 is bigger.

KEISHA: No! Wait... the 6 has more pieces but they aren’t bigger.

TARA: Yeah, they are skinny, really little compared to the pieces in this one

[pointing to ].

JHALI: But the 6 is bigger.

RON: It is like, remember in class Colton said it is the opposite of the num-

bers on the number line.’ Cause the bigger the numbers, the more, but with

fractions when they are more, they are smaller.

TARA: That is confusing.

TEACHER: I agree. It is like our brain has to say: “Stop! Think about what

the numbers mean for this fraction.” What does the 6 in mean? Remember

what we wrote on the anchor chart.^1

PETE: That is like on the poster (anchor chart) we made that there are peo-

ple sharing a pizza.

As I spent this extra time revisiting concepts we learned in class, I noted that
these students were inconsistent when justifying their thinking. They were not con-
sistent in how they interpreted the denominator. Although they could say “the big-
ger the numbers, the smaller the fractions,” they would sometimes confuse the nu-
merator and denominator. These students were not always aware of how they
learned, so they had difficulty applying solution strategies from one problem to an-
other. My concern about what they knew was justified: their understanding broke
down when they moved away from the rectangular representation. After consider-
ing what I learned about these students, I decided that my next step should be to go
back to comparing two fractions and model how I would do the comparison verbally.

1

6

1

2

1

6

1

6

1

3

1

2

The Pieces Get Skinnier and Skinnier

(^1) I used an anchor chart to summarize our mathematical strategies. The anchor charts that we use in
both literacy and mathematics provide a structure to recapture our ideas after a discussion and give
students a way to see a history of our thinking.