Reasoning abstractly and quantitatively, one of the Standards for Mathematical Practice in the Common Core State Standards for Mathematics, includes making sense of symbols.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Illuminations offers three applets that highlight this important mathematical practice by offering multiple representations for equality: Pan Balance–Shapes, Pan Balance–Numbers, and Pan Balance–Expressions. The applets visually display shapes, numbers, and graphical representations of expressions as ‘weights’ in a ‘pan balance’ with the goal of literally balancing them. This analogy helps students make connections and develop the facility to flexibly transition between the many concrete and abstract natures of representing quantities. The progression of applets offers an opportunity for students, even those at the elementary level, to explore the concept of variable by uncovering the quantities (weights) that result in the two pans (opposite sides of an equation) balancing (achieving equality).
Read success stories about how parents, teachers, and mentors all found value in using the applets for teaching and learning mathematical concepts to students of all ages, one who refers to himself as just as much the student here as the teacher!
Are you new to teaching? Or looking for new resources to teach about symbols? Here are a few ready-to-use lesson plans for each grade level that complement the pan balance applets. Find even more resources aligned to the Common Core Standards and related to this topic in the Making Sense of Symbols document.