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What Materials Can You Use to Teach Fractions at the Abstract Level?

Mathematics is abstract by nature as values, symbols and expressions are distinct from the real-world things these expressions are intended to represent. Teaching and learning math isn't dependent on objects; students can interact exclusively with symbols. Bringing real-world quantities and relationships to the student, called "manipulatives" by Teacher Vision, illuminates the abstract representations of math. These objects may give the student an easier time assembling concepts -- particularly spacial relationships.
  1. Rulers

    • While some students will gravitate towards geometric forms and other linear or sequential concepts, a ruler is usually helpful to all students at first. It simply provides a visual aid and an index to help conceptualize relationships. It doesn't even have to be a sequential or numbered ruler with quantities getting higher or lower like a gas gauge; it can simply be an incremental index as a visual aid.

    Liquid and Containers

    • Liquids can be conceptually helpful for students, especially in conjunction with a variety of containers. Containers can be many shapes -- in common geometric form from cylinders to boxes. Some of the containers can be indexed, such as a measuring cup. Differentials in relationships can be deduced when liquid is split between an indexed measuring cup and a nonindexed container. It can also be illuminating to use some indexed containers that have a continuous shape, such as a test tube, where one linear inch of additional liquid is equal to the previous inch, and some that indexed containers that taper so linear measures and volumetric measures differ.

    Sand

    • Sand can be a useful manipulative as it behaves differently than water. For example, sand can be spread out on a table top and compared to a volume of sand in a container. This physical difference can illustrate concepts in ratios that is difficult with water, which must be contained more tidily or will dissipate.

    Geometric Shapes

    • Geometric shapes such as purpose-built teacher's aids or building blocks can help demonstrate increments in three dimensions. It can be advantageous to begin using same-shaped blocks to represent single quantities. The student doesn't need to jump ahead to more complex relationships and has a simple, tangible representation of basic fractions. Though shapes can be used to illustrated more advanced concepts too, such as the relationship of two equal volumes in varied shapes.

    Money

    • Money can be a helpful aid in that it is already assigned increments students are familiar with. With other manipulatives, the a whole unit is arbitrary. A brick, for example, can be one-quarter of four bricks or one-eighth of eight bricks, but a quarter of a dollar is always a quarter of a dollar.

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