Ohio MAA Talk, Au 21

Brad Findell, Oct 15, 2021

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

1.

Exploring Span

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

  • 1. Span 1
  • 2. Span 2
  • 3. Span 3
  • 4. Span 4

1.1.

Span 1

Is [math]v[/math] in the span of [math]w_1[/math] and [math]w_2[/math]?[br][br]Vary the parameters [math]a[/math] and [math]b[/math] to change the position of the linear combination [math]aw_1 + bw_2[/math]. Can you hit [math]v[/math]?[br][br]To try another, set the vectors [math]w_1[/math] and [math]w_2[/math] and the target vector [math]v[/math]. 
– Brad Findell

1.2.

–

1.3.

–

1.4.

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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.

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3.

Exploring Eigenvectors

Find eigenvectors and eigenvalues for given matrices.

  • 1. Eigenvectors 1
  • 2. Eigenvectors 2
  • 3. Eigenvectors 3
  • 4. Eigenvectors 4
  • 5. Eigenvectors 5
  • 6. Eigenvectors 6

3.1.

Eigenvectors 1

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

3.2.

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3.3.

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3.4.

–

3.5.

–

3.6.

–

4.

Congruence via Transformations

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

  • 1. Transformation 1
  • 2. Transformation 2
  • 3. Transformation 3
  • 4. Transformation 4

4.1.

Transformation 1

One of the pairs of figures below shows a [b]translation[/b], and the other pair does not. To identify which is which, draw segments between each point and its image. Use those segments to explain your reasoning.
– Brad Findell

4.2.

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4.3.

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4.4.

–

5.

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

5.1.

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

5.2.

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6.

Other possibilities


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