Discovery, Remediation, Differentiation, Assessment

Tim Brzezinski, Dec 1, 2017

This book contains a few GeoGebra worksheets that demonstrate how GeoGebra can be a POWERFUL and EFFECTIVE tool to foster active, student-centered, discovery based learning and meaningful remediation. It also illustrates how teachers can use GeoGebra to EASILY and EFFECTIVELY differentiate their instruction through this platform.

Table of Contents

  1. Active, Student-Centered Discovery
    1. Distance in the Coordinate Plane (With Hints)
    2. Slope: Intuitive Introduction
    3. Alternate Interior Angles: Quick Investigation
    4. Function Transformations
    5. Right Triangle Trigonometry: Intro
    6. Derivative of Sine & Cosine Functions (Quick Investigation)
  2. Introduce Basic Concepts through Dynamic, Meaningful, and Modifiable Means
    1. Angle Bisector Definition (I)
    2. Perpendicular Bisector Definition
    3. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
    4. Angle Orientations (Standard Position)
    5. Coterminal Angles Action!!!
  3. Differentiation for Students with Different Needs
    1. Differentiation: Example 2
    2. Discovery, Remediation, and Differentiation with GeoGebra
    3. Inscribed Angle Theorem: Take 2!
    4. Inscribed Angle Theorem (V1)
    5. Inscribed Angle Theorems: Take 3!
  4. Differentiation: CHALLENGE those students who NEED IT!
    1. YOUR Linear Speed?
    2. Sine & Cosine of a Sum: Discovery
    3. Finsler-Hadwiger Action!!!
    4. Finsler-Hadwiger Exercise (VA)
    5. Finsler-Hadwiger Exercise (VB)
  5. Ongoing Formative Assessment
    1. Slope (Quiz) I
    2. Graphing Linear Equations: Question Generator (V1)
    3. Linear Equation Generator
    4. Perimeter of a Rectangle: Solving Linear Equations
    5. Graphing Linear Inequalities with 2 Variables (Quiz)
    6. Quiz: Writing Exponential Growth & Decay Functions
    7. Writing Equivalent Polar Coordinates: Quiz

1.

Active, Student-Centered Discovery

  • 1. Distance in the Coordinate Plane (With Hints)
  • 2. Slope: Intuitive Introduction
  • 3. Alternate Interior Angles: Quick Investigation
  • 4. Function Transformations
  • 5. Right Triangle Trigonometry: Intro
  • 6. Derivative of Sine & Cosine Functions (Quick Investigation)

1.1.

Distance in the Coordinate Plane (With Hints)

Discovery Lesson (Can be used without interactive figure below)
– Tim Brzezinski

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1.4.

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1.5.

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1.6.

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2.

Introduce Basic Concepts through Dynamic, Meaningful, and Modifiable Means

  • 1. Angle Bisector Definition (I)
  • 2. Perpendicular Bisector Definition
  • 3. True Meaning of Sine, Cosine, Tangent Ratios within Right Triangles
  • 4. Angle Orientations (Standard Position)
  • 5. Coterminal Angles Action!!!

2.1.

Angle Bisector Definition (I)

[color=#000000]In the applet below, the [/color][b][color=#1e84cc]blue ray[/color][/b][color=#000000] is said to be an [/color][color=#1e84cc][b]angle bisector[/b][/color][color=#000000] of [/color][b]angle [i]BAC[/i][/b][color=#000000]. [br][br]The [/color][b]gray slider[/b][color=#000000] adjusts the entire measure of [/color][b]angle [i]BAC[/i][/b][color=#000000]. [br][/color][color=#000000]The [b]black slider[/b] dynamically illustrates what it means for a [/color][color=#1e84cc][b]ray[/b][/color][color=#000000] to [/color][color=#1e84cc][b]bisect[/b][/color][color=#000000] an angle. [br][br][/color][color=#000000]Interact with this applet for a few minutes. [br]Then, answer the questions that follow. [/color]
1.
[color=#000000]From what you've seen, describe what it means for a [/color][color=#1e84cc][b]ray[/b][/color][color=#000000] to [/color][color=#1e84cc][b]bisect[/b][/color][color=#000000] an angle. [br][/color][b][color=#ff00ff]In your description, avoid using the words or phrases "[i]middle", "down-the-middle", "half"[/i]. [/color][/b]
2.
[color=#000000]Use the [b]Point on Object[/b] tool to plot a point F [/color][color=#1e84cc]anywhere on the [b]angle bisector[/b][/color][color=#000000]. [br]Use the [b]Angle[/b] tool to find and display the measure of angle [/color][i]BAF[/i][color=#000000] and [/color][i]CAF[/i][color=#000000]. [br][/color][color=#000000][br]How do these results reflect (i.e. illustrate) your response to (1)? [/color]
– Tim Brzezinski

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2.5.

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3.

Differentiation for Students with Different Needs

  • 1. Differentiation: Example 2
  • 2. Discovery, Remediation, and Differentiation with GeoGebra
  • 3. Inscribed Angle Theorem: Take 2!
  • 4. Inscribed Angle Theorem (V1)
  • 5. Inscribed Angle Theorems: Take 3!

3.1.

Differentiation: Example 2

Example 1
Example 2
– Tim Brzezinski

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3.5.

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4.

Differentiation: CHALLENGE those students who NEED IT!

  • 1. YOUR Linear Speed?
  • 2. Sine & Cosine of a Sum: Discovery
  • 3. Finsler-Hadwiger Action!!!
  • 4. Finsler-Hadwiger Exercise (VA)
  • 5. Finsler-Hadwiger Exercise (VB)

4.1.

YOUR Linear Speed?

[b]Students:[/b][br][br]Use this applet to help you complete the [i][color=#0000ff]How Fast Are You Spinning?[/color] [/i]investigation given to you at the beginning of class. [br][br]Some key questions to consider are listed below the applet.
1.
According to NASA's website, what is the Earth's mean equatorial radius?
2.
According to your answer for (1), what would Earth's circumference be?
3.
What would the linear speed of a person who lives on the Equator be? (Round your answer to the nearest 10 miles per hour).
4.
What is YOUR current latitude? (For Southern Hemisphere users, use a negative number. For Northern Hemisphere users, use a positive number.) .
5.
Use your answer for (3) and your result for (4) to determine YOUR LINEAR SPEED in miles/hr. (Round to the nearest 10 miles per hour). For a hint, interact with the applet above.
– Tim Brzezinski

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4.3.

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4.5.

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5.

Ongoing Formative Assessment

  • 1. Slope (Quiz) I
  • 2. Graphing Linear Equations: Question Generator (V1)
  • 3. Linear Equation Generator
  • 4. Perimeter of a Rectangle: Solving Linear Equations
  • 5. Graphing Linear Inequalities with 2 Variables (Quiz)
  • 6. Quiz: Writing Exponential Growth & Decay Functions
  • 7. Writing Equivalent Polar Coordinates: Quiz

5.1.

Slope (Quiz) I

[b]Directions:[/b][br][br]1) In the applet below, [b]note the slope displayed.[/b][br]2) Drag one (or both) of the white points to move the line [b]so that this line has this slope.[br][/b][br]3) Click the [b]"Check Your Graph"[/b] checkbox to determine if you're correct. [br] If the slope of your line matches the slope displayed, you'll see a [color=#ff00ff][b][i]"CORRECT"[/i][/b][/color] sign appear. [br] If your line does not have the slope displayed, you'll see a [b]"Not Correct"[/b] sign appear. [br] If this happens, try again. If you need a refresher, [url=https://www.geogebra.org/m/Xutr9rPj]click here[/url]. [br]
– Tim Brzezinski

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Information