We first meet geometry through shapes and their properties. The activities in this category touch upon many aspects of shape.

Geometry and spatial sense are vast; developing deep understanding takes years and encompasses many subfields. The mathematics here spans a range as well, but by no means "covers" geometry in grades K–8.

Symmetry, coordinates, and proportion are the core ideas of the following three activities, which approach shape and properties of shape from different directions. Take a look at the background to each activity for additional ideas and connections.

• In Quilts you see a block design and have to reproduce it using different colored triangles. Think symmetrically!

• Clues in the Taxicab Treasure Hunt lead you through a grid of city blocks as you search for a store of riches. How many clues do you need?

• From Corner to Corner supplies you with a ruler to measure the side of a square and its diagonal. Eventually, you develop a rule of thumb for estimating the length of a diagonal given the length of a side.

Our first school activities about shape may have been about identifying different figures. We were asked to color the triangles red and the circles blue. Later, especially using concrete manipulatives, we explored how shapes relate to one another: Can you make a square out of triangles? Can you cover the floor with hexagons? Soon we learned about measurement and began to work with area, perimeter, and volume.

We used symmetry to help make figures and describe them elegantly. By the end of elementary school, our developing proportional reasoning let us make scale drawings—and that led to our understanding about similarity.

Around the same time, we began to identify locations with coordinates—which leads us to analytic geometry, where we use coordinates to help study shape. (In the Taxicab Treasure Hunt activity, you use the coordinates of city blocks.)

Such activities across the grades traditionally come together when we are asked to reason soundly about the properties of shapes—usually in the geometry class that follows algebra.

The new NCTM Standards discuss geometry in Chapter 3: Geometry.

Mathematics instructional programs should include attention to geometry and spatial sense so that all students

analyze characteristics and properties of two- and three-dimensional geometric objects;

select and use different representational systems, including coordinate geometry and graph theory;

recognize the usefulness of transformations and symmetry in analyzing mathematical situations;

use visualization and spatial reasoning to solve problems both within and outside of mathematics.