Technologically Embodied

Geometric Functions

Geometric Functions

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With Technologically Embodied Geometric Functions, students develop conceptual metaphors that directly relate computer-based sensory motor experiences of abstract function concepts. This approach relies on four foundations:

Use the Materials, Support the Work

We’re working hard to create the sketches, worksheets, and support materials for these Web Sketchpad activities. Ours is a volunteer effort on the part of everyone involved, both curriculum developers and field testers, and we make these activities freely available for you to download and use with your own students.

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Enacting Functions from Geometry to Calculus to the Complex Plane

Mathematics Dept. Colloquium at Texas State University (21 October 2016)

Geometry is one of the best opportunities that exists to learn how to mathematize reality….[N]umbers are also a realm open to investigation…but discoveries made by one’s own eyes and hands are more convincing and surprising. (Freudenthal, *Mathematics as an educational task,* p. 407)

In this colloquium Scott Steketee presented an innovative technology-enabled enactivist approach to teaching function concepts. Based on Web Sketchpad and initially developed for secondary students, this visual sensorimotor approach can be extended to more advanced courses, providing students opportunities to enact differentiation, integration, vector operations, operations on complex numbers, and even to build a visual demonstration to determine the value of `e^(i theta)`. In today’s environment college courses increasingly expect students to be active learners, rendering activities such as these particularly useful. Accordingly, the colloquium actively involved participants in creating and manipulating mathematical objects by passing a Bluetooth mouse.

Part 1: Geometry to Algebra

We use the online paper from ICME-13. Each figure in the paper is a fully functional websketch that we used to create and explore mathematical objects, but if you’re reading this online you will likely find it simpler to use the links to access each illustration as a complete activity with a student worksheet and hint videos.

- Open the Abstract section for a brief summary of the theme of this colloquium.
- Open the Geometric Functions section, click Figure 1, and construct and manipulate a function from the Reflect family. Attend to the rate of change of the independent and dependent variables, and describe both their relative speed and relative direction. The full activity is here.
- Open the Connecting Geometric Transformations to Algebra section and click Figure 7. Construct a point on the number line, dilate it by scale factor `s`, and translate the dilated image by vector `v`. Investigate carefully to figure out the algebraic operation that’s accomplished by each of these transformations, and finish by writing the rule of this composed function as an algebraic equation that links the values of the independent and dependent variables. The full activity is here.
- Open the Performance Based Assessment section and click Figure 12. As independent variable `x` moves, you can adjust the function that controls dependent variable `T(D(x))` by adjusting `s` (the scale for dilation) and `v` (the vector for translation). However, there's an extra green variable, `??(x)`, that is constrained by a mystery function. Your job is to figure out the scale factor and vector that control `??(x)` by adjusting the on-screen values of `s` and `v` so that the two dependent variables stay exactly aligned. The activity containing this game is here.

We paused for small-group discussion and rumination about this approach to introducing functions geometrically and then connecting them to algebra: the mathematics, the pedagogy, the technology.

Several small groups reported some item from their discussion back to everyone.

Part 3: On to Calculus and the Complex PlaneWe were a bit rushed here to stay within our allotted time, so we explored only two of the planned four activities below.

[Note: as of this writing, these activities are in draft form. None of them has a worksheet yet, and most don't yet have hint videos.

**The Slope of the Sine Graph**In this investigation students address the question “What’s the slope of the sine graph?” and end up inventing, in their own words, a definition of the derivative that’s equivalent to `lim_(h->0)((f(x+h)-f(x))/h)`.**Multiply Complex Numbers:**In this activity students multiply two complex numbers, `v` and `w`, by expressing `w` as `(x_w + i*y_w)` and then using the distributive property: `v*w = v*(x_w + i*y_w) = v*x_w + v*i*y_w`. This expression can be constructed geometrically by means of dilation, rotation, and translation resulting in a simple and elegant geometric way to express complex multiplication.**Vector Multiplication:**In this activity students construct the dot product and cross product of two vectors. With a slight change in how we represent the cross product, a simple transformation constructs both `a · b` and `a ⨯ b` as the two legs of a triangle.**Euler’s Formula:**In this activity students evaluate Euler's definition of `e = lim_(n->oo)(1 + 1/n)^n`, extend the definition to evaluate `e^x`, and extend it further in the complex plane to evaluate `e^(i theta)`.

We concluded with an opportunity for general discussion, questions, suggestions, and so forth.

I appreciated the attention and the interest of everyone who came. My particular thanks to Professor Yong Yang for setting up this colloquium and making it run so smoothly, to Professor Zhonghong Jiang for intiating the invitation, and to Professors Alejandra Sorto and Alex White for welcoming me and seeing to my needs. —Scott Steketee

RequirementsThese activities require web access using a browser that supports HTML5 and JavaScript. (That means almost any current browser.)

Update History:

26 October 2016: Created this page