Download the file: TactileFunctions_Linear_Cubic.gsp

The constructions in this sketch are complicated by two primary constraints. First, I wanted the function to pass through the points in order by horizontal position, rather than by a rigid sequence such as point A, then point B, then point C, etc. This gives us a curve which always passes the vertical line test. The implementation of this accounts for much of the mess when you tell Sketchpad to Show All Hidden. The second constraint is that I wanted the piecewise curves to be single function objects, rather than merely patched together constellations of multiple function curves, as previous versions of this construction have included. This accounts for some of the columns of functions and calculations.

The primary objective of this sketch is to play with the idea of a tactile function. In most of our education, functions are defined symbolically and the curve is treated as a dependent object which a clever person might manipulate by correctly anticipating the effects of changes to the symbolic definition. In such a setting, you can only make a curve look the way you want if you are deeply familiar with the quantities in the symbolic definition that generates the curve. Dynamic geometry permits us an escape from that. The functions in this sketch are tactile, we can define them visually and graphically by pushing them back and forth. The defining is done with the physical manipulation of a curve, and we are not concerned about the symbolic definition that might describe what we have done. This puts us in the driver's seat. Math ought to be something we make, not something that happens to us.

Use the hide/show buttons to conceal the anti-derivative function. Project the sketch on a whiteboard or use a tablet to try to draw the anti-derivative function. Show the anti-derivative curve and see how close you were able to get.

Hide the anti-derivative function. Draw any function curve over the graph. Move the function so that its antiderivative will be as close as possible to the curve you have drawn. Reveal and compare.

Change your perspective to consider the tactile function as a derivative (of the anti-derivative function) Then the derivative function is the second derivative, and you can explore concavity.

**Construction notes:**The constructions in this sketch are complicated by two primary constraints. First, I wanted the function to pass through the points in order by horizontal position, rather than by a rigid sequence such as point A, then point B, then point C, etc. This gives us a curve which always passes the vertical line test. The implementation of this accounts for much of the mess when you tell Sketchpad to Show All Hidden. The second constraint is that I wanted the piecewise curves to be single function objects, rather than merely patched together constellations of multiple function curves, as previous versions of this construction have included. This accounts for some of the columns of functions and calculations.

The primary objective of this sketch is to play with the idea of a tactile function. In most of our education, functions are defined symbolically and the curve is treated as a dependent object which a clever person might manipulate by correctly anticipating the effects of changes to the symbolic definition. In such a setting, you can only make a curve look the way you want if you are deeply familiar with the quantities in the symbolic definition that generates the curve. Dynamic geometry permits us an escape from that. The functions in this sketch are tactile, we can define them visually and graphically by pushing them back and forth. The defining is done with the physical manipulation of a curve, and we are not concerned about the symbolic definition that might describe what we have done. This puts us in the driver's seat. Math ought to be something we make, not something that happens to us.

Exploration ideas:Exploration ideas:

Use the hide/show buttons to conceal the anti-derivative function. Project the sketch on a whiteboard or use a tablet to try to draw the anti-derivative function. Show the anti-derivative curve and see how close you were able to get.

Hide the anti-derivative function. Draw any function curve over the graph. Move the function so that its antiderivative will be as close as possible to the curve you have drawn. Reveal and compare.

Change your perspective to consider the tactile function as a derivative (of the anti-derivative function) Then the derivative function is the second derivative, and you can explore concavity.

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