I draw several rectangular, circular, or hexagonal fraction models on the board. About half of them are correctly labeled and drawn. The rest of them have uneven pieces or incorrect labels. I ask students to look at the models and think about what they see. What fraction is represented by each model? How do they know? (The most commonly forgotten piece in their explanation is that the pieces must be equal). Then I call on them to share what they see and if they don't catch the errors I call on other students and ask guiding questions. I work to guide them to discovering the errors on their own.
"What do you notice about the fourths in this model as compared to the fourths in this model?"
I tell students that today we will match the more traditional fraction models (parts of a rectangle or other shape) with what we have been representing on the number lines.
As a class, we draw number line models for halves, quarters, eighths, thirds, sixths, and tenths. I may add on more if the class is both interested and able.
Then, after talking to them about the inherent flaws of the shape models (see Reflection), we carefully draw circular or rectangular fraction models that match the number line models.
Unit Fractions Number Line 2 and Shape Models.
"What is something you observed today about fractions? Specifically, what did you notice about the denominator (number of equal pieces it takes to make one whole) and its relationship to the size of the pieces?"
Here are some of their observations. As you will see, some students understand the role of the denominator and can express that understanding. Some students understand what the denominator represents but then make a fallacious conclusion (that everything is divided in half when it gets smaller - I know why he said this!).
This student is clearly developing understanding but doesn't yet have the precision to express what he understands. I really admire this student's ability to manipulate the vocabulary he has to express what he doesn't yet have the words for - yes, he needs to work on precision of language, but from talking with this child, I know he understands and his mind is far ahead of his academic English vocabulary. He will get there, and when he does, there will be no stopping him!
Finally, this student expressed a partial understanding. He understands the result of dividing by halves but didn't use any vocabulary about how this process is represented by the denominator.