GeoGebra 3D with AR (Google): Explorations & Lesson Ideas

Tim Brzezinski, Oct 16, 2018

This book contains an introduction, lesson ideas, and various means through which math teachers can use GeoGebra Augmented Reality to create dynamic, student-centered, exploratory learning environments. This volume will continue to grow over time.

Table of Contents

  1. Getting Started (Intro)
    1. GeoGebra 3D and Google Play Services for AR: A Beautiful Integration
    2. GeoGebra 3D with AR (Android): Quick Setup Instructions
    3. Quick Tutorial: How to Use GGB AR on Devices with ARCore by Google Installed on Them
    4. GeoGebra 3D: From Computer to Phone/Tablet to Augmented Reality in < 2 MIN
    5. How to Explore Any Public GeoGebra Resource in Augmented Reality
  2. Demos With Resource Links
    1. Geometry Demos with Resource Links: GeoGebra 3D Calculator with AR (Android)
    2. Algebra & Precalculus Demos with Resource Links: GeoGebra 3D with AR (Android)
    3. Calculus Demos with Resource Links: GeoGebra 3D with AR (Android)
  3. Algebra & Geometry
    1. Surface Area: Intuitive Introduction
    2. Volume: Intuitive Introduction
    3. Diagonal Hint for a Rectangular Prism
    4. Surface of Revolution: GeoGebra Augmented Reality Template
    5. Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
    6. Build Your Own Right Triangular Prism (V2)!
    7. Unwrapping a Cylinder: REVAMPED!
    8. Cone Anatomy
    9. Square Pyramid: Underlying Anatomy
    10. Sphere as a LOCUS: Quick Exploration and Prompt
    11. Sphere + Plane = ?
    12. Cube: Exploration Template
    13. Tetrahedron: Exploration Template
    14. Octahedron: Exploration Template
    15. Dodecahedron: Exploration Template
    16. Icosahedron: Exploration Template
    17. 3D Abe: Silhouette = Surface of REV Cross Section
  4. Precalculus
    1. Conic Sections: Introduction
    2. Parabola: Cause and Effect
    3. Campbell's Test: Maximizing Volume
    4. Varignon 3D Action: REVAMPED!
    5. Sphere Investigation
    6. How to Create a 3D Surface of Revolution by Rotating ANY 2D Figure about ANY LINE
  5. Calculus
    1. Surface of Revolution: 3D Augmented Reality Template
    2. Rotating Function Graphs about the xAxis within GeoGebra AR
    3. Disc Method: REVAMPED!
    4. Disc Action!!!
    5. Washer Method: REVAMPED!
    6. Cylindrical Shells: REVAMPED!
    7. Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions
    8. Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VA)
    9. Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VB)
    10. Modifiable Solid: Equilateral Triangle Cross Sections Parallel to yAxis
    11. Modifiable Solid: Isosceles Triangle (Modifiable) Cross Sections Parallel to yAxis
    12. Modifiable Solid: Square Cross Sections Parallel to yAxis
    13. Custom Solid: Changeable Rhombus Cross Section Parallel to yAxis
    14. Modifiable Solid: Regular Pentagon Cross Sections Parallel to yAxis
    15. Modifiable Solid: Regular Hexagon Cross Sections Parallel to yAxis
    16. Modifiable Solid: Regular Heptagon Cross Sections Parallel to yAxis
    17. Modifiable Solid: Regular Octagonal Cross Sections Parallel to yAxis
    18. Modifiable Solid: Regular Nonagon Cross Sections Parallel to yAxis
    19. Modifiable Solid: Regular Decagon Cross Sections Parallel to yAxis
    20. Modifiable Solid: Semicircular Cross Sections Parallel to yAxis
    21. Custom Solid: Elliptical Cross Sections Parallel to yAxis
    22. Solids with Various Cross Sections: AR Templates
    23. Arc Length to Surface of Revolution: Calculus
    24. Partial Derivatives: Graphical Illustrator
    25. Plotting Points in 3D: AR Template
    26. Cylindrical Coordinates: Dynamic Illustrator
    27. Spherical Coordinates: Dynamic Illustrator
    28. Circular Paraboloid of Revolution: 3D Augmented Reality Template
    29. Special Hyperboloid of 1 Sheet as a Locus
    30. Special Hyperboloid of 2 Sheets as a Locus
    31. Ellipsoid: Quick Exploration Template
    32. Special Spheroid Action: Locus Illustration
    33. 3 Tori Morphing
    34. Surface of Rev with Helical Cross Section
  6. 3D + AR Modeling Challenges
    1. 3D AR Modeling Challenge 1
  7. Solar System Demos
    1. Season Changes: Sun-Earth Demo
  8. Miscellaneous
    1. 3 Golden Rectangles Surprise!
    2. Dodecahedral Surprise!
    3. Holiday Fun!
    4. Seashell Surface
    5. Dandelin Sphere Action!
    6. Shortest Path Between 2 Points on a Sphere
    7. Modifiable Torus

1.

Getting Started (Intro)

  • 1. GeoGebra 3D and Google Play Services for AR: A Beautiful Integration
  • 2. GeoGebra 3D with AR (Android): Quick Setup Instructions
  • 3. Quick Tutorial: How to Use GGB AR on Devices with ARCore by Google Installed on Them
  • 4. GeoGebra 3D: From Computer to Phone/Tablet to Augmented Reality in < 2 MIN
  • 5. How to Explore Any Public GeoGebra Resource in Augmented Reality

1.1.

GeoGebra 3D and Google Play Services for AR: A Beautiful Integration

[size=100][size=150][b][color=#1e84cc]GeoGebra 3D[/color][/b] empowers students to mathematically BUILD 3D MODELS of the world around them. [br][color=#9900ff][b][br]Google Play Services for AR[/b] [/color]empowers students to virtually TEST the accuracy of their models and any changes students make to IMPROVE them! [/size][/size]
MODELING A BOWL
MODELING A FENCE POST
MODELING A CONE (TO SCALE)
MODELING 2 CONES (TO SCALE)
CUP & STRAW
LIGHT FIXTURE MODELING
GLASS MODELING
[size=150]In addition to providing students with powerful mathematical modeling tools, [b][color=#1e84cc]GeoGebra 3D [/color][/b]with Augmented Reality (powered by [b][color=#9900ff]ARCore by Google[/color][/b]) also helps students explore (and go inside) 3D surfaces to help, at times, provide a deeper conceptual understanding of concepts that they often learn in an abstract manner. [/size]
TORUS MORPHING: PLAYGROUND SLIDE OR DONUT?
CIRCULAR HELIX EXPLORATION & SLINKY MODELING
– Tim Brzezinski

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2.

Demos With Resource Links

This chapter contains quick screencasts of how GeoGebra 3D with Augmented Reality (powered by ARCore by Google) can be used to foster active student engagement & conceptual understanding the mathematics classroom. Links to resources (located in the next few chapters) are provided.

  • 1. Geometry Demos with Resource Links: GeoGebra 3D Calculator with AR (Android)
  • 2. Algebra & Precalculus Demos with Resource Links: GeoGebra 3D with AR (Android)
  • 3. Calculus Demos with Resource Links: GeoGebra 3D with AR (Android)

2.1.

Geometry Demos with Resource Links: GeoGebra 3D Calculator with AR (Android)

Below is a list of quick screencasts that illustrate how GeoGebra 3D with Augmented Reality (AR) can provide both teachers and students an ACTIVE, HANDS-ON approach to exploring some geometry concepts.[br][br][b]Enjoy! [/b][br][br]NOTE: [br]Augmented Reality powered by [b][color=#9900ff][url=https://play.google.com/store/apps/details?id=com.google.ar.core]Google Play Services for AR[/url][/color][/b].
EXPLORING SURFACE AREA OF A RECTANGULAR PRISM
Link to the GeoGebra resource shown above can be found [color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/fmbmkpj7][b]here[/b][/url][/color].
EXPLORING VOLUME
Link to the GeoGebra resource shown above can be found [url=https://www.geogebra.org/m/qbxbcmqw#material/dp6ghmvv][b][color=#0000ff]here[/color][/b][/url].
FINDING THE DIAGONAL OF A RECTANGULAR PRISM: BIG HINT
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/NNUNEtR8]here[/url][/color][/b].
MODIFIABLE TRIANGULAR PRISM STUDENTS CAN EXPLORE
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/j6xVky5C]here[/url][/color][/b].
UNWRAPPING A CYLINDER
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/rCxXxFhE]here[/url][/color][/b].
ANATOMY OF A RIGHT CIRCULAR CONE: QUICK EXPLORATION
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/j9yyv3md]here[/url][/color][/b].
OCTAHEDRON NET
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/s7u5unpk]here[/url][/color][/b].
EXPLORING THE SPHERE AS A LOCUS
Link to the GeoGebra resource shown above can be found [b][color=#0000ff][url=https://www.geogebra.org/m/qbxbcmqw#material/g7qmdmqw]here[/url][/color][/b].
– Tim Brzezinski

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3.

Algebra & Geometry

  • 1. Surface Area: Intuitive Introduction
  • 2. Volume: Intuitive Introduction
  • 3. Diagonal Hint for a Rectangular Prism
  • 4. Surface of Revolution: GeoGebra Augmented Reality Template
  • 5. Rotating 2D Graphs about Lines to Create 3D Surfaces of Revolution in GeoGebra AR
  • 6. Build Your Own Right Triangular Prism (V2)!
  • 7. Unwrapping a Cylinder: REVAMPED!
  • 8. Cone Anatomy
  • 9. Square Pyramid: Underlying Anatomy
  • 10. Sphere as a LOCUS: Quick Exploration and Prompt
  • 11. Sphere + Plane = ?
  • 12. Cube: Exploration Template
  • 13. Tetrahedron: Exploration Template
  • 14. Octahedron: Exploration Template
  • 15. Dodecahedron: Exploration Template
  • 16. Icosahedron: Exploration Template
  • 17. 3D Abe: Silhouette = Surface of REV Cross Section

3.1.

Surface Area: Intuitive Introduction

TEACHERS:
For an introductory class activity related to this, [url=https://www.geogebra.org/m/mgwejudc]click here[/url].
– Tim Brzezinski

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4.

Precalculus

  • 1. Conic Sections: Introduction
  • 2. Parabola: Cause and Effect
  • 3. Campbell's Test: Maximizing Volume
  • 4. Varignon 3D Action: REVAMPED!
  • 5. Sphere Investigation
  • 6. How to Create a 3D Surface of Revolution by Rotating ANY 2D Figure about ANY LINE

4.1.

Conic Sections: Introduction

TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to MENU, OPEN. Under SEARCH, type [b]rascu98w[/b].[br] Note these characters can be found in the URL above.
STUDENTS:
Once you're all set up, please answer the questions located below this applet.
1.
[color=#bf9000][b]Note the equation of the plane is z = some constant. [/b][/color][color=#bf9000][b]Change the equation of this yellow plane to z = 2. [br]Then change it to z = 1. [br]Then change it to z = 4. [/b] [/color][br][br]How would you describe the intersection of this [b][color=#bf9000]plane[/color] [/b]and [color=#1e84cc][b]double-napped cone[/b][/color]?
2.
Change the [color=#bf9000][b]equation of the plane[/b] [/color]to [math]z=x+2[/math] . How would you describe the intersection of this [color=#bf9000][b]plane[/b][/color] and [color=#1e84cc][b]double-napped cone[/b][/color] now?
3.
Change the [color=#bf9000][b]equation of the plane[/b] [/color]to [math]z=0.5x+2[/math] . How would you describe the intersection of this [color=#bf9000][b]plane[/b][/color] and [color=#1e84cc][b]double-napped cone[/b][/color] now?
4.
Change the [color=#bf9000][b]equation of the plane[/b] [/color]to [math]z=4x+2[/math] . How would you describe the intersection of this [color=#bf9000][b]plane[/b][/color] and [color=#1e84cc][b]double-napped cone[/b][/color] now?
– Tim Brzezinski

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5.

Calculus

  • 1. Surface of Revolution: 3D Augmented Reality Template
  • 2. Rotating Function Graphs about the xAxis within GeoGebra AR
  • 3. Disc Method: REVAMPED!
  • 4. Disc Action!!!
  • 5. Washer Method: REVAMPED!
  • 6. Cylindrical Shells: REVAMPED!
  • 7. Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions
  • 8. Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VA)
  • 9. Modifiable Solid: Isosceles Right Triangle Cross Sections Parallel to yAxis (VB)
  • 10. Modifiable Solid: Equilateral Triangle Cross Sections Parallel to yAxis
  • 11. Modifiable Solid: Isosceles Triangle (Modifiable) Cross Sections Parallel to yAxis
  • 12. Modifiable Solid: Square Cross Sections Parallel to yAxis
  • 13. Custom Solid: Changeable Rhombus Cross Section Parallel to yAxis
  • 14. Modifiable Solid: Regular Pentagon Cross Sections Parallel to yAxis
  • 15. Modifiable Solid: Regular Hexagon Cross Sections Parallel to yAxis
  • 16. Modifiable Solid: Regular Heptagon Cross Sections Parallel to yAxis
  • 17. Modifiable Solid: Regular Octagonal Cross Sections Parallel to yAxis
  • 18. Modifiable Solid: Regular Nonagon Cross Sections Parallel to yAxis
  • 19. Modifiable Solid: Regular Decagon Cross Sections Parallel to yAxis
  • 20. Modifiable Solid: Semicircular Cross Sections Parallel to yAxis
  • 21. Custom Solid: Elliptical Cross Sections Parallel to yAxis
  • 22. Solids with Various Cross Sections: AR Templates
  • 23. Arc Length to Surface of Revolution: Calculus
  • 24. Partial Derivatives: Graphical Illustrator
  • 25. Plotting Points in 3D: AR Template
  • 26. Cylindrical Coordinates: Dynamic Illustrator
  • 27. Spherical Coordinates: Dynamic Illustrator
  • 28. Circular Paraboloid of Revolution: 3D Augmented Reality Template
  • 29. Special Hyperboloid of 1 Sheet as a Locus
  • 30. Special Hyperboloid of 2 Sheets as a Locus
  • 31. Ellipsoid: Quick Exploration Template
  • 32. Special Spheroid Action: Locus Illustration
  • 33. 3 Tori Morphing
  • 34. Surface of Rev with Helical Cross Section

5.1.

Surface of Revolution: 3D Augmented Reality Template

Here, you can use this template to create a function ([b]f[/b]), and rotate it about the xAxis (by using the [b]n[/b] slider). The [b]Filling[/b] slider adjusts the opacity of this surface. [br][br]Note the [b][color=#e06666]surface of revolution[/color][/b] formed. [br][br]You can change the phrase "xAxis" to "yAxis" to rotate the graph of function [b]f [/b]about the yAxis instead.
TO EXPLORE IN AUGMENTED REALITY:
1) To explore in Augmented Reality, open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]qgcz82ae[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) Position what you see accordingly on your device. Press the AR button in the lower right corner. [br] Follow the directions that appear.
– Tim Brzezinski

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6.

3D + AR Modeling Challenges

  • 1. 3D AR Modeling Challenge 1

6.1.

3D AR Modeling Challenge 1

STUDENTS:
The goal of this first modeling challenge is to build this trash can (shown in this screencast below) using the tools of GeoGebra's 3D Graphing Calculator. If your device has [b][color=#9900ff]ARCore by Google[/color][/b] installed on it, you can then explore this model within GeoGebra Augmented Reality! [br][br][b]Question to consider: [/b][br]What 2 solids compose this trash can model? [b][br][br]Modeling Clues:[br][/b]For this particular trash can, the height of the [b][color=#1e84cc]blue surface[/color][/b] is three times the radius of the [b][color=#a64d79]red/pink surface[/color][/b].
Quick Demo
– Tim Brzezinski

7.

Solar System Demos

  • 1. Season Changes: Sun-Earth Demo

7.1.

Season Changes: Sun-Earth Demo

Even though the [b][color=#bf9000]Sun[/color][/b] and [b]Earth [/b]are not constructed to scale (nor is the distance between the centers of these two masses) , the [b]Earth[/b] (shown here as [b][color=#1e84cc]northern hemisphere[/color][/b] and [b][color=#ff00ff]southern hemisphere[/color][/b]) does have an axis that is tilted approximately 23.5 degrees about its center with respect to the line that is perpendicular to the plane containing its orbit.
1.
For the hemisphere in which you live, what Sun-Earth event does [b][color=#ff7700]point A[/color][/b] represent? [br]What does [b][color=#ff7700]point B[/color][/b] represent? What about [color=#ff7700][b]point C[/b][/color]? What about [b][color=#ff7700]point D[/color][/b]? [br][br]Give reasons for your responses.
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device. [br][br]2) Go to the MENU (horizontal bars) in the upper left corner. Select OPEN. [br] In the Search GeoGebra Resources input box, type [b]jdkh6cz8[/b][br] (Note this is the resource ID = last 8 digits of the URL for this resource.)[br][br]3) The buttons will not appear in the 3D app. However, the sliders SPIN and SPEED will. [br] (Simply alter these and nothing else.)
– Tim Brzezinski

8.

Miscellaneous

  • 1. 3 Golden Rectangles Surprise!
  • 2. Dodecahedral Surprise!
  • 3. Holiday Fun!
  • 4. Seashell Surface
  • 5. Dandelin Sphere Action!
  • 6. Shortest Path Between 2 Points on a Sphere
  • 7. Modifiable Torus

8.1.

3 Golden Rectangles Surprise!

Creation of this applet was inspired by a [url=https://twitter.com/pickover/status/902195365407059969]tweet[/url] from [url=https://twitter.com/pickover]Cliff Pickover[/url]. [br][br]Slide the slider [b]very slowly[/b]. [br][br][b]Enjoy! [br][br][color=#1e84cc]To explore in Augmented Reality[/color][color=#1e84cc], see the directions below the applet. [/color][/b]
TO EXPLORE IN AUGMENTED REALITY:
1) Open up GeoGebra 3D app on your device.[br][br]2) Go to MENU (upper left corner). [br] Go to OPEN. Under [b]Search[/b], type [b]YGu6kVpR[/b]. [br] Find the slider named [b]b[/b] and slide it slowly.
– Tim Brzezinski

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Information