Teacher resources and professional development across the curriculum

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RepresentationSession 05 Overviewtab atab bTab ctab dtab eReference
Part C

Defining Representation
  The Representation Standard | Connections | Structure | Additional Points | Summary | Your Journal

 
 

Students and teachers make the most effective use of representation when they are open to a variety of forms and are regularly watching for interconnections among representations. When students see a concept depicted in several forms, that concept is strengthened. For example, the foundational concept of slope can be seen as a regular rate of change in the graph of a line. It can also be related to a constant value in a related equation. And it can be related to a rate; for example, "3 dollars/pound" in a situation and in a corresponding table of values. Such basic understandings support students as they move on to study more complex concepts.


Table of Values

x = number of pounds


y = cost in dollars


The total cost of nuts that cost $3 per pound is found by multiplying the number of pounds by 3 dollars per pound.


Equation and Graph of y = 3x

Note that these ways of representing the situation may also prompt insights. For instance, here we have used a continuous graph, although in practical terms, when purchasing something, we usually divide down only as far as pennies. In a classroom context, this observation could introduce discussion on continuous versus discrete phenomena.


The slope of the graph can be related to the constant term in the equation and to the rate in the verbal statement. It can also be related to the change in total cost when one more pound is purchased, as is easily seen in the table of values. All of these connections deepen understanding of slope beyond "rise over run" to include the fundamental underlying idea of constant rate of change.


One point to remember is that every representation highlights some mathematical features and conceals others, prompting different kinds of communication with different results. For example, the slope can be directly interpreted from the table of values and from the graph, although it is implied in the equation and verbal statement.


A conscious effort must be made by the teacher to highlight commonalities between representations. This can be done through choice of problems, questions, and representations. The goal should be for students to move flexibly among representations, including from concrete to abstract, among abstract, and from abstract to more concrete representations.

Next  How structure is revealed through representation

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