2013 Conference Proceedings - University of Nevada, Las Vegas

to explain that 2 times t is the same as t + t. When students were having problems multiplyingnumbers (e.g. 100 x 100 for 100 th position of the square number tasks), teachers took up theopportunity to teach multiplication skills involving 10 with exponents of positive integers.Additionally, the square number task created opportunities for fifth grade students to teach ideasof exponents to fourth and third grade students. Classroom vignettes showing connections ofideas across the curriculum will be presented at the conference.Rules for these early algebra activities were 2x + 2 = y for square table task, 3x + 2 = y forpentagon task, x 2 = y for square number task, and 10x + 2 = y where x is the independentvariable (input) and y is the dependent variable (output). As may be noted, these generalizationsrequired some understanding of multiplication. By including a focus on developing other skills inthe school mathematics curriculum, teachers positioned students to be able to make andunderstand algebraic generalizations while simultaneously supporting understanding of othermathematical areas.Connecting multiple representationsStudents were encouraged to use different representations. They often used physicalmanipulatives to model the tasks. For example, they used pattern blocks to explore patterns whenworking on train table tasks and square number tasks. They were encouraged to draw thegeometric representation of the tasks (see Episode 1). According to the teachers’ informalassessment, drawing geometric representations of the tasks helped students to identify what isvarying and what is constant and supported students in making and understanding theirgeneralizations.Episode 1I think it (a pictorial representation) helps them to be able to visualize, for example withthe tables that we have been having with people sitting around (table tasks) . . . it was justeasier for them to visualize it when you know, you had the two on the side and you takethe top number and the bottom number . . . but without that picture it may not have madesense to them.Students were expected to collect data from their explorations and systematically organizethem in input-output charts. As explained in Episode 1, students started exploring patterns withmanipulatives and pictorial representations and moved to exploring patterns using input-outputtables. Figure 1 is an example of a student’s input-output table for the pentagon table task. UsingProceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 183