Consider a triangle ABC, where AB is the diameter of a circle and C is on the circumference. The Measure menu offers valuable tools for exploring geometry, with commands for perimeter, circumference, area, length, distance, angle, radius, and ratio.
Students may find iteration useful for some investigations, and you might also use it to build sketches that can be used to explore questions such as, “How many diagonals will different polygons have?”, or “How does the area of a polygon change as its number of sides increases?” Measurement
For example, a hexagon can be constructed by rotating a point around a center by 60° repeatedly-you can do that rotation once, then iterate the procedure. The Iterate command can be used to create fractals, as well as a range of other constructions that can be defined in an iterative manner. Predicting and observing the effects of manipulations like these helps students understand geometric properties while also gaining spatial awareness. Now move the whole line of reflection and observe.
Then, drag a point on the original shape and see what happens to the image point. Ask students to draw a shape and reflect across a vertical line. In addition to creating appealing symmetric designs, students can use Sketchpad’s dynamic capabilities to understand the geometric properties of these transformations. Once again, the menu commands help make students aware of the properties that define each transformation-reflections can’t be done without first specifying a line of reflection. The Transform menu allows students to explore reflection, rotation, translation, and dilation. Though often neglected in school curriculum, loci were very important in the study of classical geometry, and locus-based definitions and proofs remain common in advanced geometric contexts. Loci are some of the most powerful mathematical objects in Sketchpad, and let you create geometric objects with any mathematical properties you desire. What do you notice when you draw the diagonals of the kite? When you drag the vertices of the kite, what stays invariant about the diagonals? Are there any counterexamples?Īlso in the Construct menu, you’ll find the Locus command. But more importantly, once a shape such as a kite has been constructed, students can begin to look for invariances and generalizations. The commands in the Construct menu can be used to construct the range of special triangles, quadrilaterals, and other polygons that are studied in geometry. For example, the Construct | Perpendicular Line command will be grayed out until the appropriate foundational objects are selected-a perpendicular line must pass through a given point and be perpendicular to some other line, so a point and a line must be selected. The menu itself helps direct attention to the requirements of such constructions. They can constructing midpoints and intersections, and parallel and perpendicular lines. ConstructionĪfter becoming familiar with Sketchpad’s toolbox, students can use the Construct menu to focus on constructing shapes that involve increasingly sophisticated properties and relationships. Sketchpad will provide instant feedback-students can use the “drag test” to check their own work, rather than relying on you to evaluate it. The resulting triangle has now been constructed to be an isosceles triangle, not just drawn to look like one! Encourage students to always drag their constructions to make sure they behave as they should. Simple drawing is useful in focusing attention on properties, while constructing allows the creation of robust and precise shapes, which can then be used in future geometric explorations.ĭraw a circle, then use the segment tool to construct two radii and the segment that joins the radius points. This simple introduction to Sketchpad helps build a rich conception of the triangle-so that students might not expect them all to have horizontal bases! Dragging allows students to see the continuous variations that are possible in geometric shapes, and to build visual and spatial imagery that will help them become better problem solvers.Īs students attempt to configure their triangles into, say, isosceles ones, you can develop the distinction between drawing and constructing. Use the Straightedge tool to draw a triangle, and then drag one of the vertices of that triangle using the Arrow tool to create an infinite variety of triangles. Starting with the toolbox, you’ll find the Point, Straightedge (line), and Compass (circle) tools-the building blocks of Euclidean geometry, and the starting points for rich explorations in school geometry and beyond.
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