CCSS High School: Geometry (Expressing GEO. Prop's w/Equns.)

Tim Brzezinski, Mr. Mundey, Nov 13, 2016

[b]This GeoGebra Book contains lots of discovery-based learning activities, investigations, and meaningful remediation worksheets that were designed to help enhance students' learning of geometry concepts both inside and outside of the mathematics classroom. [color=#1551b5]This GeoGebra Book also contains contributions from Samantha Cruz ([url]https://www.geogebra.org/scruz10[/url])[/color] [/b]

Table of Contents

  1. HSG.GPE.A.1
    1. Circle Equation: Center (0,0)
    2. Circle Equation: Center NOT (0,0)
    3. Circle Equation: Center (h,k)
    4. Equation & Graph of a Circle
  2. HSG.GPE.A.2
    1. Locus Problem (1)
    2. Locus Construction 1
    3. Animation 85
    4. Parabola as a Locus
    5. Animation 4
    6. Locus Constructions (via Paper Folding)
    7. Parabola (Locus Construction)
    8. Parabola (Graph & Equation Anatomy)
  3. HSG.GPE.A.3
    1. Locus Problem (2)
    2. Locus Construction 2
    3. Ellipse (Locus Construction)
    4. Animation 86
    5. Ellipse: Magic Sum = ?
    6. Ellipse: Magic Sum = ? (V2)
    7. Animation 10
    8. Locus Problem (3)
    9. Locus Construction 3
    10. Animation 87
    11. Hyperbola: Difference = ?
    12. Animation 30
    13. Locus Constructions (via Paper Folding)
    14. Hyperbola (Locus Construction)
  4. HSG.GPE.B.4
    1. Parallelogram Creation Exercises (I)
    2. Parallelogram Creation Exercise (II)
    3. Parallelogram Creation Exercise (III)
    4. Parallelogram: Coordinate Geometry Setup
    5. Parallelogram Diagonals
    6. Isosceles Triangle Medians
    7. Rectangle Diagonals
    8. Isosceles Trapezoid Diagonals
    9. Isosceles Trapezoid Midpoints
    10. Rhombus Diagonals
    11. Inscribed Angle Intercepts Semicircle
    12. 3 Special Points!
    13. Finsler-Hadwiger Exercise (VA)
    14. Finsler-Hadwiger Exercise (VB)
    15. Slope to Angle Measure Calculator (I)
    16. Slope to Angle Measure Calculator (II)
  5. HSG.GPE.B.5
    1. Slope Illustrator
    2. Parallel Consequence? (VA)
    3. Parallel Consequence? (VB)
    4. Perpendicular Lines Investigation (VA)
    5. Perpendicular Lines Investigation (VB)
    6. Slope Triangle Rotation (VA)
    7. Animation 103
    8. Parallel & Perpendicular Consequence
    9. Parallel Lines Definition
    10. Perpendicular Lines Definition (& Consequence)
    11. Writing Equations of Parallel and Perpendicular Lines
  6. HSG.GPE.B.6
    1. Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint
    2. Midpoint (x,y) of a Segment in the Coordinate Plane (Quiz)
    3. Partition Line Segments
  7. HSG.GPE.B.7
    1. Distance in the Coordinate Plane (With Hints)
    2. Distance in the Coordinate Plane (II)
    3. Distance in the Coordinate Plane (Quiz without Grid)
    4. Distance in the Coordinate Plane (Quiz With Grid)
  8. Links to Other CCSS High School: Geometry Standards
    1. Geometry Resources

1.

HSG.GPE.A.1

  • 1. Circle Equation: Center (0,0)
  • 2. Circle Equation: Center NOT (0,0)
  • 3. Circle Equation: Center (h,k)
  • 4. Equation & Graph of a Circle

1.1.

Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo
– Tim Brzezinski

1.2.

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1.3.

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1.4.

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2.

HSG.GPE.A.2

  • 1. Locus Problem (1)
  • 2. Locus Construction 1
  • 3. Animation 85
  • 4. Parabola as a Locus
  • 5. Animation 4
  • 6. Locus Constructions (via Paper Folding)
  • 7. Parabola (Locus Construction)
  • 8. Parabola (Graph & Equation Anatomy)

2.1.

Locus Problem (1)

In the applet below, [br][br][i][color=#1e84cc][b]F[/b][/color][/i][color=#1e84cc][b] is a point[/b] [/color]that is not on line [i]d[/i]. [br][b]Point [i]D[/i] is a point that lies ON [color=#666666]the line that passes through [i]A [/i]and [i]B[/i][/color]. [br][/b]The [color=#ff00ff][b]pink line[/b][/color] is the [color=#ff00ff][b]perpendicular bisector[/b][/color] of the segment with endpoints [i][color=#1e84cc][b]F[/b][/color][/i] and [i][b]D[/b][/i]. [br][br]Drag [b]point [i]D[/i][/b] along the line. What do you see? Describe in detail! [br] [br]Feel free to alter the locations of [color=#666666][i][b]A,[/b][/i][/color] [i][color=#666666][b]B[/b][/color], [/i]and/or [i][color=#1e84cc][b]F[/b][/color][/i][color=#666666]. [br][/color]Then [color=#38761d][b]clear the trace[/b][/color] and drag [b]point [i]D[/i][/b] along [color=#666666][b]the line[/b][/color] again. [br][br]Why does this occur?
Please go to the [url=https://www.geogebra.org/m/sduBXC6P]Locus Construction 1 Task[/url] and begin!
– Tim Brzezinski

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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2.7.

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2.8.

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3.

HSG.GPE.A.3

  • 1. Locus Problem (2)
  • 2. Locus Construction 2
  • 3. Ellipse (Locus Construction)
  • 4. Animation 86
  • 5. Ellipse: Magic Sum = ?
  • 6. Ellipse: Magic Sum = ? (V2)
  • 7. Animation 10
  • 8. Locus Problem (3)
  • 9. Locus Construction 3
  • 10. Animation 87
  • 11. Hyperbola: Difference = ?
  • 12. Animation 30
  • 13. Locus Constructions (via Paper Folding)
  • 14. Hyperbola (Locus Construction)

3.1.

Locus Problem (2)

In the applet below, [br][br][color=#1e84cc][b][i]O[/i] is the center [/b][/color]of the circle shown. [br][b]Point [i]D[/i] is a point that lies ON this circle. [/b][br][color=#1e84cc][b]Point [i]A[/i] is a point that ALWAYS LIES INSIDE[/b] [/color]the circle. (You can move it anywhere you'd like). [br]The [color=#ff00ff][b]pink line[/b][/color] is the [color=#ff00ff][b]perpendicular bisector[/b][/color] of the segment with endpoints [i]A[/i] and [i]D[/i]. [br][br]Drag [b]point [i]D[/i][/b] around the circle a few times. What do you see? Describe in detail! [br] [br]Feel free to alter the locations of [i][color=#1e84cc][b]A[/b][/color][/i] and [i][color=#1e84cc][b]R[/b][/color]. [/i]Then clear the trace and drag [b]point [i]D[/i][/b] around again. [br][br]Why does this occur?
Please go to the [url=https://www.geogebra.org/m/TZu6tRwE]Locus Construction 2 Task[/url] & begin!
– Tim Brzezinski

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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3.10.

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3.11.

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3.12.

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3.13.

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3.14.

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4.

HSG.GPE.B.4

  • 1. Parallelogram Creation Exercises (I)
  • 2. Parallelogram Creation Exercise (II)
  • 3. Parallelogram Creation Exercise (III)
  • 4. Parallelogram: Coordinate Geometry Setup
  • 5. Parallelogram Diagonals
  • 6. Isosceles Triangle Medians
  • 7. Rectangle Diagonals
  • 8. Isosceles Trapezoid Diagonals
  • 9. Isosceles Trapezoid Midpoints
  • 10. Rhombus Diagonals
  • 11. Inscribed Angle Intercepts Semicircle
  • 12. 3 Special Points!
  • 13. Finsler-Hadwiger Exercise (VA)
  • 14. Finsler-Hadwiger Exercise (VB)
  • 15. Slope to Angle Measure Calculator (I)
  • 16. Slope to Angle Measure Calculator (II)

4.1.

Parallelogram Creation Exercises (I)

[color=#000000]In the applets below, 3 vertices of a parallelogram are shown.[br]The coordinates (x,y) of these vertices are also displayed as well. [br][br]In each task below, determine the coordinates of each parallelogram's 4th vertex. [br]Plot this point in the coordinate plane. [br]Feel free to use any of the tools of the limited toolbar when doing so. [br]Then, use the Polygon tool to construct the parallelogram.[br][br]Afterwards, use any of the tools of each applet's limited toolbar to clearly show that the quadrilateral you've constructed is indeed a parallelogram. [/color]
– Tim Brzezinski

4.2.

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4.3.

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4.4.

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4.5.

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4.6.

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4.7.

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4.8.

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4.9.

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4.10.

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4.11.

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4.12.

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4.13.

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4.14.

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4.15.

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4.16.

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5.

HSG.GPE.B.5

  • 1. Slope Illustrator
  • 2. Parallel Consequence? (VA)
  • 3. Parallel Consequence? (VB)
  • 4. Perpendicular Lines Investigation (VA)
  • 5. Perpendicular Lines Investigation (VB)
  • 6. Slope Triangle Rotation (VA)
  • 7. Animation 103
  • 8. Parallel & Perpendicular Consequence
  • 9. Parallel Lines Definition
  • 10. Perpendicular Lines Definition (& Consequence)
  • 11. Writing Equations of Parallel and Perpendicular Lines

5.1.

Slope Illustrator

Move the two points around to change the slope of the line. Be sure to press the EXPLORE BUTTON when you see it appear. After doing so, move the new colored point (that will appear) around.
– Tim Brzezinski

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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5.9.

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5.10.

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5.11.

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6.

HSG.GPE.B.6

  • 1. Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint
  • 2. Midpoint (x,y) of a Segment in the Coordinate Plane (Quiz)
  • 3. Partition Line Segments

6.1.

Endpoint (x,y) of a Segment Given Midpoint & Other Endpoint

[b][color=#c51414]Directions:[/color][/b][br][br]1) In the applet below, note the [color=#0a971e]segment [/color]with [color=#1551b5]endpoint B[/color] and [color=#c51414]MIDPOINT M[/color].[br]2) Algebraically determine the coordinates of the [color=#1551b5]other endpoint (x, y)[/color]. [br]3) [color=#1551b5]Enter in these coordinates in the input boxes provided in the upper left hand corner.[/color][br]4) If you get the problem correct, the applet will indicate this to you. If not, keep trying.[br]5) Click the "New Problem" button to create a new segment with different endpoints.[br]6) Repeat steps (1) - (5).[br][b]7) Repeat steps (5) - (7) as many times as you need in order to master this concept![/b]
– Tim Brzezinski

6.2.

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6.3.

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7.

HSG.GPE.B.7

  • 1. Distance in the Coordinate Plane (With Hints)
  • 2. Distance in the Coordinate Plane (II)
  • 3. Distance in the Coordinate Plane (Quiz without Grid)
  • 4. Distance in the Coordinate Plane (Quiz With Grid)

7.1.

Distance in the Coordinate Plane (With Hints)

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

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8.

Links to Other CCSS High School: Geometry Standards

  • 1. Geometry Resources

8.1.

Geometry Resources

[list][*][b][url=https://www.geogebra.org/m/z8nvD94T]Congruence (Volume 1)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/munhXmzx]Congruence (Volume 2)[/url][/b][/*][*][b][url=https://www.geogebra.org/m/dPqv8ACE]Similarity, Right Triangles, Trigonometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/C7dutQHh]Circles[/url][/b][/*][*][b][url=https://www.geogebra.org/m/K2YbdFk8]Coordinate and Analytic Geometry[/url][/b][/*][*][b][url=https://www.geogebra.org/m/xDNjSjEK]Area, Surface Area, Volume, 3D, Cross Section[/url] [/b][/*][*][b][url=https://www.geogebra.org/m/NjmEPs3t]Proof Challenges[/url]  [/b][/*][/list]
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
What phenomenon is dynamically being illustrated here? (Vertices are moveable.)
– Tim Brzezinski

Information