2013 Conference Proceedings - University of Nevada, Las Vegas

t-charts made it easier for students to identify patterns because it “isolates pertinent informationfrom everything else in the word problem.” Since students tended to look for patterns in onevariable and developing recursive generalizations, teachers used t-charts to encourage students tolook for patterns between the input and output variables and come up with algebraic equations.In Figure 3, a student used an input–output table to explore relationships between independentand dependent variable for the pentagon table task. The exploration led to a conclusion that forthe pentagon task, multiplying the number of pentagons by three and adding two to the productgives the number of people that can sit around a train of pentagon tables.Figure 1. Student’s Input-Output Table for the Pentagon Table TaskTable 2 is a series of a student’s work that illustrates the teacher’s observation in Episode 1.After a student explored patterns with geometric representations of the square table task, hewrote a recursive rule. When he continued his explorations with a t-chart, he related theindependent and dependent variable and wrote a correct explicit rule. This student went back tothe geometric representation to explain that his explicit rule is valid because the number of seatswould be equal to two times the length of the train table plus the two ends. This example showsstudent’s algebraic reasoning through connecting ideas from different representations.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 184