Book B TITLE Series Name: Book Name: Book Code: ISBN 13: Published:

About Geometry Book B: Symmetry and Transformations 602B 978-1-86968-482-2 2008

AUTHOR John Thompson

ACKNOWLEDGEMENTS The publishers wish to acknowledge the work of the following people in the various stages of publishing this resource. Design: Editor:

Glen Honeybone Murray Quartly

PUBLISHERS User Friendly Resources New Zealand PO Box 1820 Christchurch Tel: 0508-500-393 Fax: 0508-500-399

Australia PO Box 914 Mascot NSW 2020 Tel: 1800-553-890 Fax: 1800-553-891

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COPYING NOTICE This is a photocopiable book and permission is given to schools or teachers who buy this resource to make photocopies or transparencies of all pages. The copies must be for internal school use only, and may not be given or sold to other educational institutions or teachers from other institutions.

COPYRIGHT User Friendly Resources, 2008

User Friendly Resources specialises in publishing educational resources for teachers and students across a wide range of curriculum areas, at both primary and secondary levels. If you wish to know more about our resources, or if you think your resource ideas have publishing potential, please contact us at the above address.

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Book B

Contents SYMMETRY AND TRANSFORMATIONS

To the Teacher.............................................................................................................................. 5

Activity 1: Introducing Symmetry Activity Guidelines for Teachers ............................................................................................ 6 Activity 1A: What is Symmetry?............................................................................................. 7 Activity 1B: Putting it to Use ................................................................................................... 8 Activity 1C: Review Questions .............................................................................................10 Self-Assessment ........................................................................................................................15 Answers........................................................................................................................................15

Activity 2: Learning About Symmetry Activity Guidelines for Teachers ..........................................................................................16 Activity 2A: Making Tangrams .............................................................................................17 Activity 2B: Making up Shapes ...........................................................................................19 Activity 2C: Dissections ..........................................................................................................22 Answers .......................................................................................................................................24

Activity 3: Symmetry and Art Activity Guidelines for Teachers ..........................................................................................25 Activity 3A: Tesselations .........................................................................................................26 Activity 3B: Māori Art ..............................................................................................................29 Activity 3C: Review Questions .............................................................................................31 Self-Assessment ........................................................................................................................33 Answers........................................................................................................................................33

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Contents SYMMETRY AND TRANSFORMATIONS

Activity 4: Patterns and Designs Activity Guidelines for Teachers ........................................................................................... 6 Activity 4A: Wallpaper Designs ............................................................................................. 7 Activity 4B: Weaving Patterns ............................................................................................... 8 Activity 4C: Wrapping Paper ................................................................................................10 Activity 4D: Review Questions ............................................................................................10 Self-Assessment .......................................................................................................................15 Answers .......................................................................................................................................15

Activity 5: Similar Shapes Activity Guidelines for Teachers .........................................................................................16 Activity 5A: Making Things Bigger or Smaller ...............................................................17 Activity 5B: Making Exact Copies .......................................................................................19 Activity 5C: Similar Triangles ad Measurement ............................................................22 Activity 5D: Review Questions ............................................................................................22 Answers .......................................................................................................................................24

Activity 6: Symmetry and Art Activity Guidelines for Teachers .........................................................................................16 Activity 6A: Proofs, Problems and Puzzles ......................................................................17 Activity 6B: Turning Triangles into Polygons .................................................................19 Activity 6C: A Puzzle ...............................................................................................................22 Activity 6D: A Geometric Proof ...........................................................................................22 Activity 6E: Review Questions .............................................................................................22 Self-Assessment .......................................................................................................................24 Answers .......................................................................................................................................24 1cm Grid Template ..................................................................................................................24

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To the Teacher SYMMETRY AND TRANSFORMATIONS

Geometry is perhaps the oldest of mathematical disciplines and is certainly one that provides many applications to the ‘real world’. In About Geometry students are provided with a variety of activities which enable them to explore and apply geometric concepts. As with the other books in the About mathematics series the activities encourage problem-solving and co-operative group work. Within most of the activities there is a range of difficulty. Often the final sections of an activity can be used for extension or enrichment work with more capable students. There is plenty of scope for the use of educational technology in these activities. A photocopier can be used to make copies of designs which need to be repeated. If a scanner is available then designs can be scanned and put into a publishing or word processing program. Internet sites contain a wealth of lesson plans, software, and images for use in teaching geometry at all primary and secondary levels.

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1 F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation Describe the reflection or rotational symmetry of a figure or object Apply the symmetries of regular polygons

Introducing Symmetry

Book B

ACTIVITY GUIDELINES FOR TEACHERS

Introduction Have a brainstorming session on the word ‘symmetry’ – does anyone know what it means? Revise earlier work students have done on the three basic transformations: reflection, rotation, translation. Students can use informal methods for studying transformations. The work they do here on identifying symmetry patterns and creating their own symmetric designs can be used in later activities. Encourage students to use the language of geometry to describe the patterns they study. Ensure that students know the meaning of words like ‘axis’, ‘infinite’. Use the following items in the activity: rulers, protractors, Perspex mirrors, tracing paper.

Teaching Ideas • There are many examples available, both of social and cultural significance, that use symmetry to enhance design. Use newspapers and magazines to find and analyse designs which utilise symmetry. • The patterns used by many cultures, e.g. New Zealand Māori, use symmetry for artistic and spiritual purposes. Find examples and discuss the types of symmetry used. • Artists or architects may be willing to visit the classroom and discuss the use of symmetry in their work. • Get students to design a badge or logo for their school, sporting, or cultural group.

Reinforcement • The use of perspective and symmetry by Escher. • Identify symmetries used in commercial or artistic paper design.

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1A F O C U S Describe patterns in terms of symmetries

What is Symmetry? SYMMETRY AND TRANSFORMATIONS

What you need to know There are several types of symmetry in geometry. Here are two: Line symmetry

Design and make a pattern using rotation, reflection or translation

This is symmetry about a mirror line. You can test whether any pattern has a mirror line (line of symmetry) like this:

Describe the reflection or rotational symmetry of a figure or object

1. Trace or copy the pattern on to a piece of paper. Cut out the pattern with a pair of scissors. 2. Fold along where you think the mirror line is. 3. If the two halves now match up exactly, you have found the mirror line.

Apply the symmetries of regular polygons

You can also test for line symmetry using a mirror: 1. Place the mirror along where you think the mirror line is. 2. If the visible half and the image in the mirror look exactly the same as the original shape then you have found the mirror line. Hereâ€™s an example of a pattern with one mirror line or line of symmetry.

Mirror line In your own words, write a definition for line symmetry. Share your definition with other class members.

Rotational symmetry If any shape fits on top of itself when it is rotated, it has rotational symmetry. You can test whether a shape has rotational symmetry by doing this: 1. Copy the shape onto a piece of tracing paper. 2. Place the shape on the tracing paper on top of the original one. Put a pin where you think the centre of rotation is. You could use the point of your compass. 3. Turn the tracing paper. If the two shapes match up at any time before you have finished making a full turn then the shape has rotational symmetry. ÂŠ Copyright User Friendly Resources.

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1A F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation Describe the reflection or rotational symmetry of a figure or object Apply the symmetries of regular polygons

What is Symmetry? SYMMETRY AND TRANSFORMATIONS The number of times the shape matches up during one full turn of the tracing paper is called the order of rotational symmetry. (This means all shapes have order of rotational symmetry of at least 1 because a shape always fits on top of itself after one full turn.) Now try these questions: 1. Where is the centre of rotation for this shape?

2. What is the order of rotational symmetry for the shape?

In your own words, write a definition for rotational symmetry. Share your definition with other class members.

Translations Translations can be used to give symmetry. For a translation, a copy of a pattern is made. This copy is then placed some distance away from the original without being changed or rotated. Here is an example of a translation.

You can test for any translation by doing this: 1. Trace a copy of the pattern. 2. Move the tracing paper onto the translated shape without turning it. 3. If the copy exactly matches the shape you have moved to, then that shape is a translation of the one you copied onto the tracing paper. In your own words, write a definition for translation. Share your definition with other class members.

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1A F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation Describe the reflection or rotational symmetry of a figure or object Apply the symmetries of regular polygons

What is Symmetry? SYMMETRY AND TRANSFORMATIONS

What you need to do Sample Shapes 1. Draw in the lines of symmetry for the following patterns: a)

b)

c)

d)

2. For each of these shapes work out the order of rotational symmetry: a)

b)

Order of rotational symmetry: c)

Order of rotational symmetry: d)

Order of rotational symmetry:

Order of rotational symmetry:

3. Draw one shape of your own for each of the following. You will need to draw these shapes on your own sheet of paper. a) No lines of symmetry. b) Only one line of symmetry. c) Three lines of symmetry. d) An infinite number of lines of symmetry. e) Order of rotational symmetry of 2. f ) Order of rotational symmetry of 4. g) Draw a face that has two axes of symmetry and rotational symmetry of order two. ÂŠ Copyright User Friendly Resources.

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1B F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation Describe the reflection or rotational symmetry of a figure or object Apply the symmetries of regular polygons

Putting it to Use SYMMETRY AND TRANSFORMATIONS

What you need to do For these activities the you will need newspapers and/or magazines, scissors and glue. Many sporting or cultural groups have a logo or design that uses some type of symmetry. 1. Using newspapers and magazines, find three examples of such designs. Trace (or photocopy) them and put them in your book.

2. For each of the three designs you have copied, work out what kind of symmetry it has. Look for: •

lines of symmetry

•

rotational symmetry

•

translational symmetry

3. A group that you belong to has asked you to design a logo for them. Your design should fit onto half a page of A4 paper. The design must have some kind of symmetry and should represent what the group does. For example, if it is for a sports club then the design may have a piece of equipment in it, or if it is for a church group it may use a religious symbol.

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1C F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation Describe the reflection or rotational symmetry of a figure or object Apply the symmetries of regular polygons

Review Questions SYMMETRY AND TRANSFORMATIONS

What you need to do A. For each shape draw in the line(s) of symmetry. a)

b)

B. For each shape find the order of rotational symmetry. a)

b)

Order of rotational symmetry:

Order of rotational symmetry:

C. For this design:

1. How many mirror lines does it have?

2. What is its order of rotational symmetry?

3. Does it have translational symmetry?

D. Describe the two transformations that are needed to move this shape from its position on the left to its position on the right of the page.

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Self-Assessment

1

SYMMETRY AND TRANSFORMATIONS Use the confidence level scale on the right to assess how confident you feel about each of the activity questions. Tick on the scale for each concept.

CONCEPT

Confidence level scale 1 = I don’t understand it at all. 2 = I would find it difficult 3 = I could do it, but not easily. 4 = I know how to do this, but would like more practice. 5 = Too easy, give me a challenge!

EXAMPLE

CONFIDENCE LEVEL

I know how to describe patterns in terms of symmetries.

For each pattern say whether it has rotational symmetry or line symmetry.

I know how to design and make patterns that use reflection, rotation or translation.

Draw a design, which is based on a translation and reflection of this shape.

1

2

3

4

5

1

2

3

4

5

Answers SYMMETRY AND TRANSFORMATIONS

Activity 1A

Review Questions

Sample Shapes a)

A. a)

b)

B. c) 2 c)

C. 1. None

d)

b)

d) 2 2. 1

3. Yes

D. Reflection in a vertical mirror line and then translation downward in the direction of that line. (A glide reflection)

Rotational Symmetry a) 2

b) 4

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c) 3

d) 1

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2 F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation

Learning about Symmetry

Book B

SYMMETRY AND TRANSFORMATIONS

Introduction In the following activities students explore symmetry and do some problem solving at the same time. The tangram is a puzzle whose origins appear to be in ancient China, but in fact is a 19th century invention by the puzzle designer Sam Lloyd. The allegedly Chinese origins of the puzzle were part of an elaborate hoax perpetrated by Lloyd on the historians of his day. Students may be familiar with sets of tangrams that are ready made. By getting them to make their own tangrams they might appreciate that the puzzle arises in quite a natural way from folding a square of paper. Like many great puzzles its elegance lies in its simplicity. The making of shapes to pre-set patterns encourages students to use trial-and-error to arrive at a solution. As they become more adept they can challenge other students to see if they can duplicate some of their own designs. To do this the shapes can be traced and photocopied so that the way in which the individual shapes are used is not shown.

Teaching Ideas • Discuss the ‘ history’ of tangrams. • Ask the students to describe what symmetric patterns they can see in the tangram template. • Consider what type of symmetry each tangram piece has. • Discuss the symmetry of the completed shapes.

Reinforcement • Dissections of squares and triangles. • Greek cross problems. These problems involve cutting a shape in a ‘symmetric manner’ into three or four pieces and then putting them together to form a shape such as a square or triangle. Many books on recreational mathematics contact such examples.

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Book B

2A F O C U S Describe patterns in terms of symmetries Design and make a pattern using rotation, reflection or translation

Making Tangrams SYMMETRY AND TRANSFORMATIONS

What you need to know Tangram puzzles were invented over 100 years ago and have been popular ever since. In this activity you will find out how to make them and be given some challenges to make them into some well-known mathematical shapes.

What you need to do 1. Fold and cut your square along the lines as shown in the diagram. 2. Label the seven pieces in the way that is shown.

b c a

d e f

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2A

Making Tangrams SYMMETRY AND TRANSFORMATIONS 3. For each of the shapes complete this table, which describes the type of symmetry that each shape has.

F O C U S Describe patterns in terms of symmetries

Shape letter

Design and make a pattern using rotation, reflection or translation

a

Number of lines of symmetry

Order of rotational symmetry

b c d e f g Keep the pieces for Activity 2B.

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