Discovering equivalent fractions and introducing fraction notation, Clip 5 of 5: Compare one half and two thirds, establishing equivalence

In the fifth clip, researcher Carolyn Maher called the class together. At the overhead, Erin, Jackie, and Jessica had built the six and twelve centimeter models. The researcher asked the class to provide the solution to the problem, and they answered in unison that two thirds was larger than one half by one sixth. The researcher then asked if anyone had built a model that gave another solution. Meredith indicated that she had, and she was asked by the researcher to tell the class what she found. Meredith lined twelve white rods against the model using the orange and red train. Before she explained what she had done. Michael began to shake his head in disapproval. The researcher asked Michael what was wrong. Michael said he didn’t think they should use a white rod but rather a red rod to show the difference between two thirds and one half. The researcher asked the class what rod they used to represent one sixth and they replied that they had used the red rod. The researcher pointed out that Meredith had showed it to be bigger by two whites. Michael replied that the two whites together would be one sixth”. Erik then offered his view on the matter. He said that the two whites together could be called two twelfths and if combined they would be called one sixth. The researcher pointed out that Meredith was calling it two twelfths as well as one sixth. Meredith said, “There's two answers." Michael and Erik responded that they were the same answer and Erik elaborated on that explaining that Meredith had taken a red and divided it into half with each half being called one twelfth. The researcher then began to record the students’ arguments using mathematical notation at the OHP. She wrote 1R=2W. Erik then continued his direct argument, saying, And since she's calling a white rod one twelfth and the other white rod one twelfth and the red rod is really one sixth. But, when she calls them two twelfths, the two twelfths are actually just two white rods put together to equal a red, so it should be really, it's really one sixth. The researcher then recorded 1/12 + 1/12 = 2/12, and Erik pointed out that it’s also one sixth. The researcher then said and wrote that Erik had also said that 1/2 of 1/6 = 1/12, and that 1/6 = 2/12. Erik agreed. Brian then added to the argument that there were two answers. Jessica then continued Erik’s argument, saying, that two whites would equal one red. The class agreed that all the statements that the researcher had recorded were true.Transcript and student work are also available.Robert B. Davis Institute for Learning. (1993). Discovering equivalent fractions and introducing fraction notation, Clip 5 of 5: Compare one half and two thirds, establishing equivalence [video]. Retrieved fro