EdSurge Fusion 2018: GeoGebra Resource Demonstrations

Tim Brzezinski, Sep 10, 2018

Table of Contents

  1. Geometry
    1. Vertical Angles Exploration (1)
    2. Area of a Triangle (Discovery)
    3. Exploring Polygon Angles: Triangle through Octagon
  2. Algebra
    1. Special Pair of Lines in the Coordinate Plane
    2. Perpendicular Lines in the CP: Quick Investigation
  3. Precalculus & Trigonometry
    1. Half-Life Action (3)!!!
    2. Conic Section Explorations
  4. Calculus
    1. Derivative Function: Without Words
    2. Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions

1.

Geometry

[b][color=#1551b5]Resources that Illustrate how GeoGebra can foster DISCOVERY LEARNING & facilitate MEANINGFUL REMEDIATION in Geometry:[/color][/b] 1) Quick ACTIVE Discovery that Vertical Angles are always congruent (equal in measure) 2) Resource quickly helps students DISCOVER (or RE-DISCOVER) that the area of a triangle = 1/2 the area of a parallelogram that has the same base and height as the triangle. Proof that the quadrilateral formed IS a parallelogram is also included. 3) Resource helps students powerfully discover the sum of the measures of the interior angles of any n-gon = 180(n - 2) degrees. Also helps students readily discover that the exterior angles of any convex polygon always sum to 360 degrees.

  • 1. Vertical Angles Exploration (1)
  • 2. Area of a Triangle (Discovery)
  • 3. Exploring Polygon Angles: Triangle through Octagon

1.1.

Vertical Angles Exploration (1)

Interact with this app for a few minutes. Be sure to drag the LARGE POINT around and be sure to move the 2 sliders you see.
Notice anything interesting? Describe as best as you can.
What happens if you make the size of the [b][color=#ff00ff]angle[/color][/b] BIGGER or smaller? [br][br][b][color=#ff00ff](You can change the size of the manila angle by using the manila-colored slider). [/color][/b]Does your response to the question above change? Why? Why not? Describe.
– Tim Brzezinski

1.2.

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1.3.

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2.

Algebra

[color=#1551b5][b]Resources that Illustrate how GeoGebra can foster DISCOVERY LEARNING & facilitate MEANINGFUL REMEDIATION in Algebra:[/b][/color] 1) Activity for students to DISCOVER (or RE-DISCOVER) that if 2 parallel lines are drawn in the coordinate plane, then they have equal slopes. Note: Adjusting the "Tilt" slider causes the slope triangle to [color=#c51414]turn red[/color] if slope becomes negative. 2) Quick illustrator for students to quickly DISCOVER (or RE-DISCOVER) that (perpendicular lines graphed in the coordinate plane have slopes that are opposite reciprocals.

  • 1. Special Pair of Lines in the Coordinate Plane
  • 2. Perpendicular Lines in the CP: Quick Investigation

2.1.

Special Pair of Lines in the Coordinate Plane

In the applet below, feel free to move [b]ANY LARGE POINT[/b] anywhere you'd like [b]AT ANY TIME. [/b][br][br]Interact with this app for a few minutes. Be sure to tamper with all sliders and be sure to move the LARGE POINTS around. After doing so, please reflect on these dynamics by answering the few questions that follow.
1.
Press the reset button. Slide the long black slider slowly (once again) and stop it at the exact moment the [b][color=#ff00ff]2nd pink angle[/color][/b] reaches the 2nd line. [br][br]What does the action of the [b][color=#ff00ff]pink angles[/color][/b] imply about both black lines? [br]How/why do you know this to be true?
2.
Now resume sliding the slider where you left off in question (2) above. [br][br][b]What does the motion of the 2nd right triangle imply about both black lines? [/b]
3.
What causes the right triangle (that appears) to [b][color=#0000ff]turn blue[/color][/b]? What causes it to [b][color=#ff0000]turn red[/color][/b]? Please describe as best as you can.
4.
Write a conditional (a statement of the form "If _____, then _____") that describes the phenomena through which you've just dynamically interacted.
5.
If we formed the converse ("flip") of the conditional ("If _____, then _____") statement you wrote for (4), would that statement always be true? That is, if you swap both hypothesis and conclusion of the statement you wrote for (4), would this statement be true? Why? Why not? Explain.
– Tim Brzezinski

2.2.

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3.

Precalculus & Trigonometry

[color=#1551b5][b]Resources that Illustrate how GeoGebra can foster DISCOVERY LEARNING & facilitate MEANINGFUL REMEDIATION in PreCalculus & Trigonometry:[/b][/color] 1) Dynamic and modifiable resource that illustrates the HALF-LIFE of a substance is independent of BOTH the initial amount of substance you have AND the amount of substance you have after any time. 2) Conic Section Explorations: This resource helps students explore how the cross section (intersection) of a plane and cone is a CONIC SECTION. Illustrations in GeoGebra's 3D Graphing Calculator and GeoGebra Augmented Reality are provided.

  • 1. Half-Life Action (3)!!!
  • 2. Conic Section Explorations

3.1.

Half-Life Action (3)!!!

[b]Recall the half-life of a substance is defined to be the amount of time it takes for half of ANY AMOUNT of that substance to decay. [br][br][/b]This applet illustrates that [color=#1e84cc][b]the half-life of an substance[/b][/color] is INDEPENDENT OF (i.e. DOES NOT DEPEND UPON) [color=#980000][b]the INITIAL AMOUNT of SUBSTANCE[/b][/color]. Thus, [color=#e06666][b]at any point in time[/b][/color], the [color=#1e84cc][b]half-life of a substance always remains invariant. [/b][/color][br][br]To illustrate this, [color=#e06666][b]move the pink point anywhere you'd like[/b][/color] and then slide the black slider. [br][br]
Quick Demo: 1:05 sec to END. (BGM: Simeon Smith)
– Tim Brzezinski

3.2.

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4.

Calculus

[color=#1551b5][b]Resources that Illustrate how GeoGebra can foster DISCOVERY LEARNING & facilitate MEANINGFUL REMEDIATION in Calculus:[/b][/color] 1) This dynamic & modifiable GeoGebra resource powerfully illustrates the concept of derivative of a function (at x = a certain input) in a GRAPHICAL (non-algebraic) setting. 2) This GeoGebra resource allows students to create a custom-made solid of revolution formed by rotating the area between 2 (modifiable) functions about any specified horizontal or vertical line.

  • 1. Derivative Function: Without Words
  • 2. Calculus: Surface of Revolution formed by Rotating Area Between 2 Functions

4.1.

Derivative Function: Without Words

In the applet below, a function [i]f [/i]is shown (black). [br]Its derivative function, f', is shown in gray. [br][br]Feel free to input any function you'd like. [br][br]Slide the slider to see the physical meaning of the derivative of a function (at a certain input value: x-coordinate of white point) within this graphical context. [br][br][i]How can we sketch the graph of the derivative of a function? Explain. [/i]
Quick Demo (BGM: Andy Hunter)
– Tim Brzezinski

4.2.

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