| Chords in a Circle | Students construct chords in a circle and manipulate them to discover various properties. |
| Dynamic Proof | Students create a sketch in which a dramatic animation shows how the sum of the areas of two squares is equal to the area of the third square. |
| Exterior Angles in a Polygon | Students construct a pentagon with its exterior angles, measure the angles and find the sum. They then dilate the pentagon to look at the problem, and understand their result, in a new way. |
| Leonardo da Vinci's Proof | Students use reflection and rotation to build a sketch that demonstrates a famous proof by Leonardo. |
| Medians in a Triangle | Students discover various properties of the three medians of a triangle. |
| Midpoint Quadrilaterals | Students construct the midpoints of the sides of a quadrilateral, connect them to form the midpoint quadrilateral, and then discover interesting properties of the midpoint quadrilateral. |
| Properties of Parallel Lines | Students discover relationships among the angles formed when parallel lines are intersected by a transversal. |
| Reflections Over Two Intersecting Lines | Students explore what happens when they reflect a figure over a line and then reflect the image over a second line that intersects the first. |
| A Right Triangle with Squares | Students make a square tool and use the tool to investigate the Pythagorean Theorem and its converse. |
| GSP Resource Files (Right Click to Download) |
||
| Areas: Proof as Verification | Students form a conjecture about the ratio of two areas in a quadrilateral, and use Sketchpad to search for a proof. In the process they verify or disprove their results for two different kinds of quadrilaterals. |
Areas [gsp] Areas 2 [gsp] |
| The Congruent Triangles Construction: Ellipses | A pair of congruent triangles holds the key to this unusual Sketchpad ellipse construction. |
Congruent Triangles [gsp] Gears [gsp] |
| Distances in an Equilateral Triangle: Proof as Explanation | Students explore the sum of certain distances in an equilateral triangle, and develop a proof to explain their observations. | Distances [gsp] |
| The Folded Rectangle Construction: Parabolas | With just a blank sheet of paper and a single point, students can fold a genuine parabola. Students model the technique with Sketchpad to reveal the underlying mathematics. |
Folded Rectangle [gsp] Conic Connection [gsp] Tangent Circle [gsp] Headlights [gsp] |
| Kite Midpoints: Proof as Discovery | Students investigate and prove a property of kites, and use the proof to make a discovery about other quadrilaterals. | Kite [gsp] |
| Polygon and Circle Areas | Students manipulate the number of sides in a regular polygon, develop a formula relating the area and perimeter of the polygon, and use their result to develop a formula relating the circumference and area of a circle. | To A Circle [gsp] |
| Triangle Midpoints: Proof as Systematization | Students use Sketchpad to prove that the midpoints of two sides of a triangle form a segment that is parallel to the third side and half its length. This relationship was demonstrated in a previous activity, but not proved. | Triangle Midpoints [gsp] |
| Visual Demo of Pythagorean Theorem | Students do a visual demonstration of the Pythagorean theorem based on Euclid’s proof. By shearing the squares on the sides of a right triangle, they create congruent shapes without changing the areas of their original squares. | Shear Pythagoras [gsp] |