We describe extensions of several key concepts of differential geometry to manifoldswith density, including curvature, the Gauss-Bonnet theorem and formula, geodesics,and constant curvature surfaces.

two main parts, Algebra activities and the Geometry activities. The Algebra activities include activities that encourage students to form conjectures about parallel & perpendicular lines, direct variation, quadratic functions, absolute value functions, and equations of circles.  The Geometry activities include activities that encourage students to form conjectures about vertical angles, triangle inequality, reflections, isosceles triangles, the measure of an exterior angle of a triangle, central angles and arc measures, parallelogram, rectangle, rhombus, midpoint quadrilateral, sum of the angles in a polygon, and right triangle trigonometric ratios.  After each activity is a sample sketch with answers to questions posed to the students in the activity.

This Guide is designed to help you learn about this dynamic math visualization tool in a practical, fun way