Defining Connections
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Connections between representations can lead to the development of a robust understanding of new mathematical ideas and to an inquisitive, confident outlook on the part of students. Each type of representation has a specific focus. When they are related to one another, a multi-dimensional grasp of a concept is possible.

Before studying the volume of rectangular prisms and cylinders, a fifth-grade teacher presented a science investigation involving a variety of rates to her students. As one element of the investigation, each group took a turn filling a large glass container with water from the sink that flowed at a constant, slow rate. They used a stopwatch and metric ruler to collect data. Here is one group's container and the graph they created:

Not only does this activity demonstrate connections between science activities and mathematical concepts, but it is also rich in connections that can lead to a robust understanding of several important mathematical ideas. Students are actively involved in collecting values of variables that are literally varying before their eyes in a physical representation. They see that the height of the water is a function of the amount of time that has passed.

The data table shows the relationship between each number of seconds of filling and its related height of water. These ordered pairs are connected to the points on the coordinate graph. The entire graph then represents the overall relationship between time and height of water for this container.

The class went on to compare the graphs for different containers to investigate the relationship between container shape and the rate at which the water level rises at different points. The teacher can also build on this investigation to move to the development of formulas for containers that have parallel sides. Students will be able to understand the role of the height of a container in a volume formula and will be better prepared to understand why volume is sometimes explained through the use of diagrams that show multiple cross-sections that are one unit high.

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