For students that have studied various classes of functions (linear, quadratic, square root, trigonometric), [b][color=#1e84cc]YOU CAN engage them in 3D modeling activities [/color][/b](& challenges) within GeoGebra Augmented Reality! [br][br]In this screencast below, note the two surface equations[br][math]z=\sqrt{16-x^2}[/math] and [math]z=-\sqrt{16-x^2}[/math]. If we were to replace [i]z[/i] with [i]y, [/i]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. [br][br]Yet instead - here, we write [i]z[/i] as a function of [i]x [/i]and restrict the domain of this surface to be [math]\left|y\right|\le0.876[/math], or [math]-0.876\le y\le0.876[/math]. [br][br]More info can be found below the screencast. [br][br]

[b][color=#ff00ff]Yet where did this domain restriction come from? [/color] [/b]That part is easy. Before building the model, we measured the radius of this cylindrical coffee maker. Here, [i]r[/i] = 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. [br][br]Thus, [math]\frac{4u}{10.5cm}=\frac{?u}{4.6cm}[/math]. And upon solving, we get ? = 1.75 units. Since we chose the plane [i]y[/i] = 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 [math]\left|y\right|\le0.876[/math]. [br][br]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. [br][br][color=#1e84cc][i]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? [/i][/color]