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At the start of the session, researcher Carolyn Maher asked the students if they remember working on comparing two thirds and three quarters. She mentioned that she had seen students build more than one model, and asked the students how many models they thought were possible to build. Michael conjectured that if one knew the “secret” behind the models, he could build models using all the rods. Amy commented that they had found six models, and the researcher asked them if they thought those were all the models possible. Amy and her partners said that they thought there were more. Meredith then reasoned that if one could divide the white rod one could make more models. The researcher asked her if she meant that more models could be made if there were more rods of different sizes, and she replied that she did. The researcher then asked the class if there was a smallest model that they could build to represent the problem using the rods that were available. Beth and Sarah built a model using a light green and white train, two red rods, and four white rods at the overhead projector (OHP). Erik noted that the light green and white train could be substituted with a purple rod. The researcher asked them if that model could be used the problem under discussion, and Beth noted that the model did not contain thirds. Then, James, Amy and Jackie built a model of an orange and red train, three purple rods, and four green rods at the OHP. James said that the three fourths were larger than two thirds by one twelfth, showing directly that twelve white rods equaled the length of the train. The researcher asked the students if they were convinced by James’ explanation. Jessica said that she didn’t think the model they had built was the smallest one. The researcher asked the three students if there was a smaller one, and commented that if there wasn’t, they should be able to show that. James then restated his argument that the difference was one twelfth. The researcher then asked the students to explain what she meant by a smaller model. Erik answered that it’s smaller in size. The researcher asked the students if they could convince her that the model that James had built was the smallest length possible. Amy, a member of James’ group, offered her justification but did not actually justify the claim made that it was the smallest one possible. Other students also noted that there were other trains that could equal the orange and red train in length. Erik then offered a counterargument to Jessica’s claim that there was a smaller model that could be used to show the difference between two thirds and three fourths. He stated that unless there was a rod smaller than the white rod, one couldn’t make a smaller model The researcher asked if this meant that unless one would use Meredith’s idea of creating new rods that were smaller than the white ones it would not be possible to find a smaller one and Erik agreed.

Robert B. Davis Institute for Learning. (1993). Building large models to show equivalence, a generalization, Clip 1 of 3: Finding the smallest model. [video]. Retrieved from

doi:10.7282/T3NP26DH

The video is protected by copyright. It is available for reviewing and use within the Video Mosaic Collaborative (VMC) portal. Please contact the Robert B. Davis Institute for Learning (RBDIL) for further information about the use of this video.