to be equal). Take apart situations are most naturally formulated with an equation of the form

in which C is known and all the possible combinations of A and B that make the equation true are to be found. There are usually many different As and Bs that make the equation true.


Geometry and measurement provide additional, powerful systems for describing, representing, and understanding the world. Both support many human endeavors, including science, engineering, art, and architecture. Geometry is the study of shapes and space, including two-dimensional (2-D) and three-dimensional (3-D) space. Measurement is about determining the size of shapes, objects, regions, quantities of stuff, or quantifying other attributes. Through their study of geometry and measurement, children can begin to develop ways to mentally structure the spaces and objects around them. In addition, these provide a context for children to further develop their ability to reason mathematically.

Every 3-D object or 2-D shape, even very simple ones, has multiple aspects that can be attended to: the overall shape, the particular parts and features of the object or shape, and the relationships among these parts and with the whole object or shape. In determining the size of a shape or object, one must first decide on which particular aspect or measurable attribute to focus.

Space (both 3-D and 2-D) could be viewed initially as an empty, unstructured whole, but objects that are placed or moved within the space begin to structure it. The beginnings of the Cartesian structure of space, a central idea in mathematics, are seen when square tiles are placed in neat arrays to form larger rectangles and when cubical blocks are stacked and layered to make larger box-shaped structures. These are also examples of composing and decomposing shapes and objects more generally. Composing and decomposing shapes and objects are part of a foundation for later reasoning about fractions and about area and volume.

Viewing or imagining an object from different perspectives in space and moving or imagining how to move an object through space to fit in a particular spot links spatial relations with the parts and features of objects and shapes.

Just as numbers are an abstraction of quantity, the ideal, theoretical shapes (2-D and 3-D) of geometry are an abstraction of their approximate physical versions. The angles in a rectangular piece of paper aren’t exactly right angles, the edges aren’t perfectly straight line segments, and the paper,

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