Geometry Workshop Dec2014

Arthur Lee, Dec 5, 2014

This is a collection of tasks about transformation, symmetry and tessellation. This presentation provides some reference materials: https://dl.dropboxusercontent.com/u/4210789/me/mini/tiling/index.html More workshop materials are developed in a summer course for secondary students on these topics: 1. transformation and symmetry http://tube.geogebra.org/student/b320095 2. tessellation http://tube.geogebra.org/student/b320097

Table of Contents

  1. reflection in a square
    1. Reflection in a Square
    2. Reflection in regular polygon
  2. find triangles
    1. find two congruent triangles
    2. Find 2 similar triangles
  3. two rotations
    1. Two rotations
    2. Two reflections
    3. Two rotations investigate
    4. Two Reflections investigate
  4. squares around parallelogram
    1. squares around parallelogram
    2. more squares and parallelograms
    3. pentagonal tiling from squares and parallelograms
  5. pentagonal tiles
    1. Cairo Tiling 1
    2. Cairo Tiling 2
  6. others
    1. PMC 2006-07 Q16a
    2. PMC 2006-07 Q16b
    3. TMC 2008 Q1
    4. TMC 2006-07 Q13
    5. TMC 2008 Q2
    6. TMC 2007 Q14
    7. PMC 2007 Q6
    8. PMC 2007 Q7
    9. PMC 2012-13 Q8
    10. TMC 2007 Q12
    11. PMC 2008-09 Q4
    12. TMC 2008 Q17
    13. PMC 2010-11 Q2
  7. treasure island
    1. Treasure Island Puzzle
  8. squares around quadrilateral
    1. 4 squares on quadrilateral
  9. opposite sides and angles of hexagon
    1. opposite sides and angles of hexagon
    2. opposite angles of hexagon
    3. opposite side length of hexagon
    4. parallel opposite sides of hexagon

1.

reflection in a square

A point is reflected in the 4 sides of a square. The 4 images of reflection form a quadrilateral. Explore properties of this quadrilateral.

  • 1. Reflection in a Square
  • 2. Reflection in regular polygon

1.1.

Reflection in a Square

Point E is reflected in the sides of the square. Drag E to see the change in the images.[br]What properties do you notice in the quadrilateral formed by the red points?[br]By dragging E to different positions, what kind of quadrilaterals can be made.
If we start with other shapes (instead of a square) for generating the mirror images, what different results will we have?[br]In what other ways would you extend this problem?
– Arthur Lee

1.2.

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2.

find triangles

Here are investigations modified from textbook problems regarding identification of congruent or similar triangles.

  • 1. find two congruent triangles
  • 2. Find 2 similar triangles

2.1.

find two congruent triangles

Find two congruent triangles in this figure. You can freely rearrange the given equilateral triangles (orange and blue).
find two congruent triangles
– Arthur Lee

2.2.

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3.

two rotations

Are two successive rotations the same as a single rotation? How can you check whether it is the result of a single rotation? What special cases you would like to investigate? The last two applets provide hints for understanding effect of combined transformations.

  • 1. Two rotations
  • 2. Two reflections
  • 3. Two rotations investigate
  • 4. Two Reflections investigate

3.1.

Two rotations

Change the original objects or the centers and angles to explore the effect of two rotations.
Two rotations
– GeoGebra Institute of Hong Kong

3.2.

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3.3.

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3.4.

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4.

squares around parallelogram

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I

  • 1. squares around parallelogram
  • 2. more squares and parallelograms
  • 3. pentagonal tiling from squares and parallelograms

4.1.

squares around parallelogram

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I]https://www.evernote.com/l/AAPkK3E9enNHAKc6KqQPOUW77qJNNl5Hz7I[/url]
Thebault's Theorem[br][url=http://nrich.maths.org/1939]http://nrich.maths.org/1939[/url]
– Arthur Lee

4.2.

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4.3.

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5.

pentagonal tiles

[color=#c51414]Open the task sheet:[/color] [url]https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM[/url] More about tiling: [url]https://dl.dropboxusercontent.com/u/4210789/me/mini/tiling/index.html[/url] Reflection on students' work in this task: [url]https://www.evernote.com/shard/s3/sh/12da29a7-c078-4c9c-9a3c-763771e4d3e9/8731931f52d657877c8d72d51445af9e[/url] and [url]https://www.evernote.com/shard/s3/sh/7eb4f1d6-94b0-4bdb-bb3f-d0017f0d5ef9/92adfd7fe28dfb45fdcb88d99117a95a[/url]

  • 1. Cairo Tiling 1
  • 2. Cairo Tiling 2

5.1.

Cairo Tiling 1

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM]https://www.evernote.com/l/AAO-URTJbMVF5rsObg8eNmgNIEWaahXb5aM[/url]
– Arthur Lee

5.2.

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6.

others

These are questions from math contests for primary or junior secondary students. They require students to create symmetric shapes or rotate a figure.

  • 1. PMC 2006-07 Q16a
  • 2. PMC 2006-07 Q16b
  • 3. TMC 2008 Q1
  • 4. TMC 2006-07 Q13
  • 5. TMC 2008 Q2
  • 6. TMC 2007 Q14
  • 7. PMC 2007 Q6
  • 8. PMC 2007 Q7
  • 9. PMC 2012-13 Q8
  • 10. TMC 2007 Q12
  • 11. PMC 2008-09 Q4
  • 12. TMC 2008 Q17
  • 13. PMC 2010-11 Q2

6.1.

PMC 2006-07 Q16a

– Arthur Lee

6.2.

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6.3.

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6.4.

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6.5.

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6.6.

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6.7.

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6.8.

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6.9.

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6.10.

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6.11.

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6.12.

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6.13.

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7.

treasure island

[color=#c51414]Open the task sheet:[/color] https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I Find the treasure with a missing gallows. These are links related to this problem: https://www.ma.utexas.edu/users/gilbert/M333L/BinderDay2.pdf http://frink.machighway.com/~dynamicm/hunt.pdf http://jwilson.coe.uga.edu/EMAT6680Su07/McKee/6690/islandtreasure/it1.html http://books.google.com.hk/books?id=et6WMlkQlFcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false https://www.academia.edu/5600848/Back_to_Treasure_Island

  • 1. Treasure Island Puzzle

7.1.

Treasure Island Puzzle

[color=#c51414]Open the task sheet:[/color][br][url=https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I]https://www.evernote.com/l/AAPZAYsI93FDyqPqbOfRDHlw6FtHThNJY4I[/url]
Find the treasure with a missing gallows.[br]These are links related to this problem:[br][list][br][*]https://www.ma.utexas.edu/users/gilbert/M333L/BinderDay2.pdf[br][br][/*][*]http://frink.machighway.com/~dynamicm/hunt.pdf[br][br][/*][*]http://jwilson.coe.uga.edu/EMAT6680Su07/McKee/6690/islandtreasure/it1.html[br][br][/*][*]http://books.google.com.hk/books?id=et6WMlkQlFcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false[br][br][/*][*]https://www.academia.edu/5600848/Back_to_Treasure_Island[br][/*][/list]
– Arthur Lee

8.

squares around quadrilateral

4 squares are drawn on the sides of a quadrilateral. Join the centres of opposite squares to form 2 segments. Explore relation between these two segments.

  • 1. 4 squares on quadrilateral

8.1.

4 squares on quadrilateral

P, Q, R, S are centres of squares on sides of the quadrilateral ABCD.
4 squares on quadrilateral
– Arthur Lee

9.

opposite sides and angles of hexagon

[color=#c51414]Open the task sheet:[/color] [url]https://www.evernote.com/l/AANyIWZaqvBBaIUzGt7b_T6DZZjW3Og9zOA[/url] more info: [url]https://dl.dropboxusercontent.com/u/4210789/me/mini/hexagons/index.html[/url]

  • 1. opposite sides and angles of hexagon
  • 2. opposite angles of hexagon
  • 3. opposite side length of hexagon
  • 4. parallel opposite sides of hexagon

9.1.

opposite sides and angles of hexagon

[color=#c51414]Open the task sheet:[/color][br]https://www.evernote.com/l/AANyIWZaqvBBaIUzGt7b_T6DZZjW3Og9zOA
more info:[br]https://dl.dropboxusercontent.com/u/4210789/me/mini/hexagons/index.html
– Arthur Lee

9.2.

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9.3.

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9.4.

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