PYTHAGOREAN THEOREM PROOF GSP5 PROFESSIONAL

The points can be moved, and the properties of a triangle are preserved, for example, that the three angles always add up to 180 degrees. For example, in the video below, a triangle is defined by the lines connecting three points. In addition, a recent review of the research literature supports the use of dynamic representations for conceptual understanding more generally (Roschelle, et al, 2017).Ī dynamic representation is a visual depiction of a mathematical object that changes in time, consistent with the mathematical properties of the object.

PYTHAGOREAN THEOREM PROOF GSP5 PROFESSIONAL

Research with SimCalc has established strong benefits to student conceptual understanding when dynamic representations are integrated into the plan for a curricular lesson sequence, and teachers have received related teacher professional development (Roschelle et al, 2010). In a tool like SimCalc (Hegedus & Roschelle, 2013), students can see how the slope of a line corresponds to a rate of change by seeing how the slope controls the speed of an animated football player who is running. In a dynamic geometry tool like Cabri®, The Geometer’s Sketchpad®, or Geogebra, a student can drag a point in a geometric construction and see how it changes in time - for example, they can see a right triangle not as ONE triangle, but as a family of geometrically similar triangles, all with the same ratio of sides. For example, with a graphing calculator a student can change the numeric value of a slope and see the corresponding change to the slope of a line. Over the past 25 years, learning scientists have established that time-based representations are an important tool to help students make connections and develop conceptual understanding.

Alas, too often slope, similarity and ratio are taught as discrete topics with no connection - and mathematics comes across as a bag of unrelated, incomprehensible procedures and, as a consequence most students cannot develop conceptual understanding. Why?įundamentally, slope is connected to concepts of geometric similarity and ratio - m always comes out the same because it is a measure of the ratio of the sides of similar triangles. The ratio-the slope-will always come out the same. Why does the measure of rise over run always come out the same?Ī student can measure rise over run at any pair of points along a line. We can measure the slope with triangles of “rise over run” drawn anywhere and at any size along a straight line in a graph. To understand mathematics, students need to connect ideas.įor example, the slope of a line is often given as a number - the m in y = mx + b. Jeremy Roschelle, Digital Promise, When integrated with curriculum and pedagogy, visual representations that change in time can improve students’ conceptual understanding of mathematics.

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