Building large models to show equivalence, an exploration, Clip 2 of 4: A doubling conjecture

In the second clip, David and Meredith worked on building models to represent their solution to the problem: Which is larger, two thirds or three quarters, and by how much. David first built two models to show halves and quarters, and researcher Carolyn Maher pointed out that the problem was asking to find the difference between two thirds and three fourths. The researcher suggested that David listen to Meredith’s solution, which she had shared with the classroom teacher. Meredith had built a twenty-four centimeter-long model using a blue, black, and brown train. She lined up four dark green rods, three brown rods, and four red rods. She reasoned that the reds could be called twelfths and that the difference between two thirds and three fourths was one twelfth. As she spoke, she lined two white rods against one red rod in her model. She then said that it could be bigger by two twenty-fourths. The researcher then asked Meredith to explain her second model to David. Before letting her do so, she told David and Meredith that their next challenge would be to think about what a third model would look like, and what they would call each of the rods in that model, even if they didn’t build the model. Meredith then showed David that in her twelve centimeter-long model, the difference between the two fractions was one twelfth. Later, Meredith re-explained her two models to the researcher, who encouraged them to think about the challenge that she had posed David and Meredith thought about what a third model would look like. David conjectured, and Meredith rationalized that in a third model that was twice the size of her first one, the red rods would be called one twenty-fourth, and that the white rods would be called one forty-eighth. However, he incorrectly reasoned that the light green rod would be called one twelfth. This error was only peripheral to the central line of reasoning. David then repeated his conjecture to the researcher, who called Alan and Erik over to David’s desk and asked him to repeat his conjecture in their presence. David explained that originally the whites were one twenty-fourth and the reds were one twelfth. But if it were doubled the reds would be one twenty-fourth, the whites would be one forty-eighth, and the light green would be one twelfth. The researcher suggested that David, Meredith, Alan, and Erik combine supplies and try to build a model to test David’s conjecture on the floor at the front of the room. Erik then told the researcher about Andrew’s model and his way of finding thirds and fourths, and joined the others on the floor.Transcript and student work are also available.Robert B. Davis Institute for Learning. (1993). Building large models to show equivalence, an exploration, Clip 2 of 4: A doubling conjecture [video]. Retrieved fro