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doi:10.7282/T3N018PN

The class is investigating the difference between 1/4 and 1/9. In initial discussions, some students agree with Meredith, who says that the difference is 1/5 because 9 – 4 = 5. Researcher Maher indicates to the class that if they apply Meredith’s rule to 1/2 and 1/4, they would get an answer of 1/2, challenging their earlier reasoning that produced an answer of 1/4.

Some students build one model to show ninths and a second model (of a different length) to show fifths. For example, Alan builds one model where the blue rod represents 1 and so the white rod would have the number name of 1/9 and a second model where the purple rod represents 1and so the white rod would have the number name of 1/4. Alan concludes that 1/9 is smaller than 1/4, but he is unable to determine the difference using these models. After he is unsuccessful in his attempt to construct a model that can be divided into ninths and fourths, Alan conjectures that the reason is that it is not possible to build a model when one denominator is odd and the other is even. Researcher Maher reminds him that in previous sessions, he built a model for comparing 1/2 and 2/3; in another previous session, he built a model to compare 3/4 and 2/3.

Meanwhile, James builds a model consisting of 3 orange rods and a dark green rod (36 cm.) that can be partitioned into ninths and fourths. He presents the model to Researcher Davis. First, James claims that the difference is 1/5, illustrating this difference by 5 white rods. Researcher Davis responds by asking James to clarify what number name is assigned to one white rod, and then of two white rods. James responds 1/36 and 2/36, respectively. James realizes that if the difference between 1/4 and 1/9 is represented by 5 white rods, then that difference must be 5/36. James then presents this finding to the class.

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Student model building

Reasoning

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