Ohio MAA Talk, Sp 22

Brad Findell, Mar 25, 2022

GeoGebra classroom allows students to work simultaneously on the same “lesson,” providing the instructor simultaneous views of all the students’ screens. Instructors can then share individual students sketches and interact with them during whole-class discussions. This session will share some particularly fruitful lessons from an undergraduate Linear Algebra course, along with other lessons designed for undergraduate Geometry courses for teachers. [list] [*]After the presenter presses "Create Lesson", participants may engage [b]as students[/b] in these activities. A GeoGebra account is not required. [*]But if you sign up for a GeoGebra account, you will be able to save your work to return to it later, "follow" activity writers, and identify "favorite" activities. [*] These are works in progress. Please send comments to findell.2@osu.edu. [/list]

Table of Contents

  1. Exploring Span
    1. Span 2
  2. Exploring Linearity
    1. Linearity 1
    2. Linearity 2
  3. Exploring Eigenvectors
    1. Eigenvectors 3
  4. Congruence via Transformations
    1. Transformation 4
  5. Quadrilateral Diagonals
    1. Rectangle Diagonals
  6. Point to line or angle
    1. Point to Line
    2. Point to Side of Angle

1.

Exploring Span

Exploring the span of a set of vectors in 2D and 3D.

  • 1. Span 2

1.1.

Span 2

Is [math]v[/math] in the span of [math]w_1[/math] and [math]w_2[/math]?[br][br]Set the vectors [math]w_1[/math] and [math]w_2[/math] and the target vector [math]v[/math].  Vary the parameters [math]a[/math] and [math]b[/math] (or the vector [math]u[/math]) to change the position of the linear combination [math]aw_1 + bw_2[/math]. Can you hit [math]v[/math]?
– Brad Findell

2.

Exploring Linearity

Demonstrating linearity of matrix transformations.

  • 1. Linearity 1
  • 2. Linearity 2

2.1.

Linearity 1

Move [math]u[/math] and [math]v[/math] to observe linearity. Explain what the sketch demonstrates. (Edit the entries of [math]M[/math] to try different transformations.)
What does the sketch demonstrate algebraically? Geometrically?
– Brad Findell

2.2.

–

3.

Exploring Eigenvectors

Find eigenvectors and eigenvalues for given matrices.

  • 1. Eigenvectors 3

3.1.

Eigenvectors 3

Move vector v to determine eigenvectors and eigenvalues of M.
[size=85][size=100]An [i]eigenvector[/i] is a vector that is mapped onto a multiple of itself. That multiple is called an [i]eigenvalue[/i].[/size][/size]
– Brad Findell

4.

Congruence via Transformations

Explores the basic rigid motions: translations (slides), reflections (flips), and rotations (turns).

  • 1. Transformation 4

4.1.

Transformation 4

Two figures are said to be [b]congruent[/b] if there is a sequence of basic rigid motions that take one figure onto the other.[list=1][*]Specify a sequence of two or three basic rigid motions that takes one F onto the other. Illustrate intermediate images. Explain your reasoning.[/*][/list][i]Note that your toolbar now includes the basic rigid motions. To use one of them, choose the transformation you want (e.g., reflection), select the object to be transformed, and then select object(s) to specify the transformation (e.g., for a reflection, a line).[/i] [br][br]Also: The F is not selectable in the image, so use the [b]polyline[/b] or [b]polygon[/b] tool (on the line menu) to create your own version.
– Brad Findell

5.

Quadrilateral Diagonals

  • 1. Rectangle Diagonals

5.1.

Rectangle Diagonals

Determine condition(s) on the diagonals that will guarantee a rectangle.
What condition(s) [b]on the diagonals[/b] will guarantee a rectangle?
– Brad Findell

6.

Point to line or angle

Find and construct the distance from a point to a line.

  • 1. Point to Line
  • 2. Point to Side of Angle

6.1.

Point to Line

Find the distance from [math]C[/math] to line [math]f[/math].
Can you construct a segment whose length will give the distance exactly?
– Brad Findell

6.2.

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