Open Middle Calculus: Tangent Line to a Curve

This open-ended calculus activity demonstrates how to construct and verify the tangent line to a curve at (or near) a chosen point, emphasizing the geometric idea behind the derivative as an instantaneous slope. Learners explore the relationship between a curve’s local linear approximation and the resulting tangent direction by iteratively selecting points, forming candidate lines, and checking their intersections with the curve. Key concepts include derivative as slope, tangent-line geometry, and validation of the tangent condition through measured geometric quantities. The GeoGebra build uses commands reflected in the construction summary: Textfield for user input/labels, Intersect to compute intersection points that constrain the construction, Mode to control interaction and object behavior, Length to measure distances for checks or comparisons, and If to switch logic depending on conditions. Objects such as multiple constructed points/lines (e.g., a, b, c, d, t1–t8, g, A, B, l1, l2, h) and text annotations (text1, text2) support an interactive “select—construct—measure—verify” workflow. Suitable for upper-secondary advanced math and first-year calculus, especially for demonstrating the geometric meaning of the derivative and learning tangent construction.