From 2D to 3D Modeling: It's Easier Than You Think!
- Tim Brzezinski
For students that have studied various classes of functions (linear, quadratic, square root, trigonometric), YOU CAN engage them in 3D modeling activities (& challenges) within GeoGebra Augmented Reality! In this screencast below, note the two surface equations and . If we were to replace z with y, we would have the equations of the top and bottom halves (respectively) of 2 semicircles with radius = 4 units we would typically have students graph in the coordinate plane. Yet instead - here, we write z as a function of x and restrict the domain of this surface to be , or . More info can be found below the screencast.
Quick (Silent) Demo
Yet where did this domain restriction come from? That part is easy. Before building the model, we measured the radius of this cylindrical coffee maker. Here, r = 10.5 cm and the height of this cylinder = 4.6 cm. Since we chose 4 UNITS in GeoGebra Augmented Reality to represent 10.5 cm = radius, we need to determine how many units represent the height of this cylinder. Thus, . And upon solving, we get ? = 1.75 units. Since we chose the plane y = 0 to split this cylindrical lateral area in half, this surface needs to extend 1.75u / 2 = 0.876 units in both the positive y-direction and negative y-direction. Hence, the need for the domain restriction . And this simple level of proportional reasoning is a task that MANY STUDENTS can do by the time they study various classes of functions in high school. Building 3D mathematical models of real-world objects IS a task MANY STUDENTS CAN DO after studying various classes of functions. So why restrict modeling to only be within the (2D) coordinate plane?