David, Erik, and Meredith Use Reasoning to Resolve Conjectures
PurposesStudent model building; Reasoning; Representation
DescriptionMeredith rebuilds the previous day’s models for comparing 2/3 and 3/4. The longer model is 24 cm long; a train of a blue rod, a black rod, and a brown rod, representing 1. In this model, a dark green rod represents 1/4, a brown rod represents 1/3, a red rod represents 1/12, and a white rod represents 1/24. Meredith shows that the difference between three greens and two browns is one red (1/12) or two whites (2/24). The other model is 12 cm long; it consists of an orange rod plus a red rod. In this model, a light green rod represents 1/4, a purple rod represents 1/3, and a white rod represents 1/12. Meredith shows that in this model, the difference (1/12) is represented by one white rod.
Researcher Maher challenges Meredith, David, and other students to predict what a larger model would look like without building it. David and Meredith conjecture that there could be a larger model in which the white represents 1/48.
Students are asked to work on a model to verify David’s conjecture. Erik builds another model of length 36 cm (3 blue rods plus 9 white rods), which can be used to show thirds and fourths, but which does not model David’s conjecture.
After realizing that they have not built a model for David’s conjecture, students help David build a model in which the white rod represents 1/48. When asked, David says that he is not surprised at this result; that’s what he thought it would be.
