Revisiting construction of large models to compare fractions, Clip 1 of 5: Which is larger, two thirds or three fourths, and by how much?

In the first of five clips from this classroom session, the researcher, Amy Martino, provided the students with an opportunity to briefly discuss a task that they had been working on during the previous session, which was comparing two thirds and three fourths. Andrew and Jessica described how they had attempted to make large models and use trains for thirds and fourths. The researcher asked the students if they had all recorded their large models, and Andrew said that he had. She then asked the students how many models they thought could be built. Erik conjected that there would be a lot of models and offered as justification the reasoning that he and Alan had used to build models to show halves and fourths during the previous session. Erik attempted to explain the pattern that he had noticed, and then Alan interjected by repeating his argument from the previous session. He said, “We also realized that the bigger, like if you put … four, you couldn't third that unless you made a new rod using two others to be bigger than the orange.” He then re-explained, saying, “Like four oranges you can't third it without making a new rod. But three oranges you could call that a whole and have three more oranges as the thirds.” The researcher asked the class what they thought of Erik’s conjecture that even numbers can be divided into thirds and fourths. Erik then modified his conjecture, saying that most even rods could be used to build models to show thirds and fourths. Michael agreed, but said that he had found that some of the even rods could not be divided into thirds and fourths. The researcher asked Erik what he meant when he called a rod even. Erik replied, “Well, a rod that if you put all of the whites up to it… all the whites real tight, and you determine if you can divide it in half.” Erik reasoned that if the number of white rods that equal the length of the train could be divided in half, the rod could be called an even rod. David added to this definition by explaining that the “white would be one, the reds would be two, so the reds are even, and then the light greens are three… they're odd, and then the purple is even.”Transcript is also available.Robert B. Davis Institute for Learning. (1993). Revisiting construction of large models to compare fractions, Clip 1 of 5: Which is larger, two thirds or three fourths, and by how much? [video]. Retrieved fro