Students can improve a variety of basic math and problem-solving skills by exploring geometry. Teachers and parents often share the belief that basic arithmetic relates most to the real world, but as the Annenberg Learner website states, "The world is built of shape and space, and geometry is its mathematics." Since geometry lends itself to hands-on experiences, it is also an effective way to reach learners who have difficulty engaging in a traditional classroom.

## Lines and Angles

Students can explore straight lines that are horizonal, vertical, diagonal, parallel, bisecting, adjacent, opposite and perpendicular, as well as the angles that form when two straight lines meet. Students should recognize and form angles that are small and large, and then acute, right and obtuse. To engage different types of learners, students should go beyond worksheets and practice making lines and angles with popsicle sticks, pipe cleaners and their own arms and legs. Simple games can be modified to interest students ("Simon Says make a small acute angle").

## Two- and Three-Dimensional Shapes

Students can identify and sort figures by number and type of lines and angles before adding new concepts such as lines of symmetry, edges, faces and vertices. Students should be encouraged to build large shapes using smaller shapes. Paper cutouts, pattern blocks, geoboards, toothpicks and marshmallows are inexpensive materials. Students should also be challenged to locate two- and three-dimensional shapes in the everyday world.

## Similarity and Congruence

Students should differentiate between lines, angles and shapes that are similar (sharing many properties) and congruent (sharing all properties, including size). Keep in mind that location and orientation are not properties that make shapes different or the same -- for example, some students may not recognize a square that has been rotated to look like a diamond. This type of thinking will help students look at other types of problems from different perspectives.

## Spatial Reasoning

Students can improve their spatial awareness by problem solving. Introductory activities can include creating or using isometric drawings, or constructing nets for three-dimensional figures. Students can begin to identify "hidden" parts of a drawing or figure or imagine and draw the figure from a different perspective. Dot paper, geometric solids and linking cubes can be used for these types of activities. Tangram puzzles can also help students improve spatial reasoning skills.

## Transformations

Geometric shapes can be moved in predictable ways. Students can learn to perform transformations and recognize when a shape has undergone a translation (slide), rotation (turn) or reflection (flip). As students become proficient, they should develop the ability to imagine the result of a transformation before checking with hands-on materials. Visualizing a problem, creating a sketch or using a grid are skills that easily transfer to other subjects in school, as well as real life.

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