To continue our investigation, we can also construct two of the interior triangles and measure their areas. With Sketchpad, we can discover that this relationship holds no matter what the size or shape of the triangle. ![]() In a particular configuration, we find that one distance appears to be twice the other. Similarly, the student can now measure two distances from the centroid-to one vertex and to the midpoint on the opposite side-and discover that even when the distances change, there is something constant about the relationship. But with Sketchpad, the student can now transform the result to observe that the statement is true, not just for a single example, but for many differently shaped triangles. On pencil and paper, this would be an observation of only a single case-provided the student had done the construction carefully enough. Students will notice that the third median passes through the intersection of the first two medians. We then use the straightedge again to construct the three medians. ![]() But once they have constructed the first, we allow them to use the Midpoint command to construct the other two midpoints. Because we are very strict, we require students to use the compass to construct this first midpoint. Next we construct the midpoints of one side. The result is not just a single triangle, but by dragging the vertices it can be transformed into any possible triangle-acute, obtuse, isosceles, right, and so forth. We first use the straightedge to construct a triangle. We begin with a blank screen and the tools of Euclidean geometry: the straightedge and compass. The first example comes from Euclidean geometry. When the program is used for demonstration in the classroom, or when it is used by students for direct investigation, the presentation and the examples themselves will be different, and will be adapted to the level and ability of the class. If we had time, we would also show you examples from trigonometry, conic sections, and other areas of mathematics.Īs we look at the examples, please remember that they are chosen to show you the features and benefits of the software. ![]() These examples will come from several different areas of mathematics, including Euclidean geometry, transformational geometry, algebra and the study of functions, graphing of functions in Cartesian and polar coordinates, and calculus (including derivatives, anti-derivatives, and integrals). We would like to show you several examples of the way in which El Geómetra (in English, The Geometer’s Sketchpad) can be used in demonstrations and in student activities in order to develop student understanding that is more significant and longer lasting. Multiplica (Multiplication on the number line)
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