Ignite children’s prior knowledge by discussing home or local community gardens that they are familiar with. It may also be helpful to introduce this unit by reading a book about garden settings or viewing images of garden settings online. The overall aim of the unit is to create a classroom display of a garden using the activities as starting points. Be as creative as you can! 

Session 1: Up the garden path

In this session students will explore shapes that tessellate or repeat to cover the plane without gaps or overlaps. Although the students will only be covering a strip (path) any covering of a path can be used to tessellate the plane simply by putting paths together.

1. Explain to the students that they have the task of building a garden path. If possible, show them examples of garden paths in the local area. Use online images if no real-world examples are available. 
2. Ask students to build a path using shape blocks. All the shapes must fit together without any gaps. Students are to select 1 or 2 shapes to build their path. The path needs to have at least 3 or 4 rows of blocks. 
3. Students draw the paths they have created (or take digital photos) and present them to the class, describing the shapes that they have selected.
4. Create garden designs around the paths. 
   

Session 2: Bugs, Beetles and Butterflies

In this session students will be investigating line symmetry by making butterflies out of coloured paper.

1. Show pictures of native butterflies such as the Red Admiral (Kahukura) or Rauparaha's Copper. Look at the wings and discuss reflective symmetry.
2. Ask students to make their own butterflies by folding and cutting.
   
3. Encourage them to cut out pieces in the wings to add detail.
4. Ask students to share their work and talk about the reflective symmetry it contains.
5. Extend the activity to making other native bugs and beetles such as the Huhu Beetle or Puriri Moth by folding and cutting.
6. If adapting using the marae as the context, symmetrical tekoteko could be made in the same way. If using a skate park as the context, symmetrical people and dogs could be created.

Session 3: Butterfly Painting

In this session students will make symmetrical butterflies with paint. Refer to the pictures of native butterflies from the previous session as inspiration.

1. Fold a piece of paper in half. On one half draw the outline of half of a butterfly. Create designs on this half of the wings with paint. Carefully fold the other half of the paper onto the wet paint. Unfold it to get a symmetrical pattern.
2. Ask students to share their work and talk about the reflective symmetry it contains.
3. Students could then make other bugs and beetles for the garden using the same technique.

Session 4: The Flower Garden

In this session students will be introduced to making symmetrical patterns with shape blocks. The theme for this lesson is flowers for the garden, so showing the students images of flowers and reading or viewing a story about flowers would be beneficial. Sunflowers would be a great example of a flower to use in this session.

1.  Give students a piece of paper with a line drawn down the middle.
2. Students use shape blocks to make half of a flower pattern on one side of the line. They give this pattern to a partner who has to then repeat the pattern on the other side of the line making sure that it is symmetrical.
   
3. Ask students to trace around the shape blocks to make the petal shapes. Coloured paper could be used to cut out the petals. Glue the petals onto the paper to make symmetrical flowers.
4. This activity could be extended by encouraging students to create their own symmetrical designs. They could experiment with cutting the paper shapes in half to create other pieces for their designs.
5. These could then be displayed alongside the path designs from Session 1.

Session 5: The Garden Wall

In this session introduce students to the idea of translation. Students will be making tiles for the garden wall. Introduce the activity by showing them examples of some wall tiles from the local area.

1. Give each student a piece of square grid paper, for example a 4x4 grid. Students are to draw a design by colouring in the squares to make a pattern.
2. They make 3 or 4 copies of this pattern.
3. Stick these in a row to make a row of tiles with repeating patterns.
   
4. These could then be displayed above the flowers made in the activity from Session 4.
5. This session could be extended by encouraging students to use more grid squares or by creating more complex designs within each grid.
6. If adapting using the marae as the context, students could make tukutuku panels for a wharenui by showing them some examples and then having them copy and translate some of the patterns they have seen.

Session 6: Wind Catcher in the Garden

In this session students will make a wind catcher, which illustrates rotation, as an ornament for the garden .

1. Give each student a square piece of paper.
2. Fold the square along its diagonals.
3. Make cuts along the diagonals leaving about 1 cm uncut at the centre of the square.
4. Take one of the cut ends at each corner and fold into the centre.
5. Repeat this at each corner.
6. Pin the folded pieces together with a split pin.
7. Put a little piece of blue tack onto the back of the pin to hold the pieces in place.
8. Attach the pin to a stick or paper straw.
9. Blow to watch it rotate.
   
   Cut along lines in first image

Note: The wind catcher has rotational symmetry but not reflective symmetry. This is because it can be rotated around onto itself but it doesn't have a line of symmetry in the plane.

Other Ideas

• Make designs for a dinner set for a picnic in the garden. Students could design a pattern for the pieces in the dinner set. The Willow Pattern story and plates could be used as motivation for this. Patterns around the edges of the plates would need to be repeating patterns. This could also be adapted to include Māori or Pasifika desgins.
• Paint patterns around the rim of pots. These designs could include Māori and Pasifika aspects. Plants could be planted in these pots.
• Make a patchwork picnic tapa cloth with designs in each patch piece. This could be made out of paper or fabric. The patch pieces could show a tessellation or reflective symmetry of Māori or Pasifika designs.
• Have a touch table in your classroom of items from nature that show symmetry and transformation (for example, leaves, flowers, insects). These could be added to by the students in your class (see whānau link). Encourage students to bring fallen things rather than harming our environment. 
• Go to your school or community kāri/garden and notice any natural symmetry and transformation. Or use online images of kāri/gardens from around Aotearoa. Draw pictures of what you see and label any symmetry and/or transformation.
   

This activity gives students a simple method for creating an original tessellating shape. This method is sometimes called the “bite” or “nibble” method. The main rule to follow when changing shapes to create a tessellating tile is that the shape must retain the same area as the original, so each piece that is cut off must be rejoined. Students need to be careful to just translate pieces and not to inadvertently flip them
over.
When they have completed the tile, they translate the tile by sliding it along the plane – up, down, left or right – to create a tessellating pattern. The shape that the students start with must be a polygon in which opposite sides are parallel and congruent because an operation on one side will always affect the opposite side. The square is the easiest shape to start with, but encourage the students to experiment with other
shapes using this bite method. The more sides the polygon has, the more sides that can be altered.
Therefore, starting with a shape such as the regular hexagon can lead to an interesting design.
This activity gives students an opportunity to explore their own artistic creativity. They can add detail to the shapes to give the tessellation added appeal. However, it is important to use this activity not simply as an artistic activity, but as one that gives the students an opportunity to develop an understanding of translation and tessellation and to talk about their creations using the language of geometry.
Under the Sea, Figure It Out, Levels 2–3, page 11 and its accompanying teachers’ notes discuss adapting a triangle to make a whole shape that tessellates.

Answers to Activity

1. Practical activity
2. The shape you start with must be a polygon that has opposite sides parallel and congruent (for example, a rectangle or a regular hexagon).
Your new shape must have the same area as the original shape.

Activity One

This pattern is taken from a section of tapa cloth. The design is based on tessellations of triangles.
Students will need to count the triangles systematically for questions 1–4 and record their counting carefully. A knowledge of symmetry is very useful in solving these problems. For example, students can use the following method to count the number of small brown triangles:

Students can use the total number of small brown triangles to help find the number of small white
triangles. In the central section of the tapa cloth design, there is one more row of small white triangles
than there is of brown triangles, so the total number is: (1 + 2 + 3 + 4 + 5) + (5 + 5) = 25.

In question 3, students may realise that each row contains two more triangles than the previous row: 1 + 3 + 5 + 7 + 9 = 25.
In question 4, students will need to think how many triangles of different sizes there are in the pattern.

Question 5 asks students to generate the next section of the pattern. Students may extend the pattern
down or sideways. Two possible answers are given in the Answers section.
Students can find the next pattern using translation (shifting) and half-turn rotation.

Answers to Activities

Activity One
1. 20
2. 25
3. 25
4. 65
5. a. Answers will vary. They include:
the next section sideways:

i.

or the next section downwards:
ii.

b. Answers will vary, but the two examples
above will give i. 35 or ii. 15 new triangles.
Activity Two
Answers will vary.

This activity gets students to apply translational and rotational symmetry to create freeze-type patterns. The method suggested uses a printing block.
The patterns created by translation need to preserve the orientation of the figure. For example:

By contrast, the patterns created by rotation involve a change in orientation. For example:

Students may find that they can apply translation and rotation in combination. For example:

Another way of making patterns by reflection is to cut a tracing block from a square of card. Note that it is wise to mark the original corners of the square before cutting to help keep track of the original shape. For example:

The pattern can be duplicated across the whole page by repeated reflection.

Answers to Activity

Teacher to check

Problem One

Encourage the students to predict the number of fold lines as they follow the fold instructions. They may be able to draw where they think the fold lines will be after each step, and this can be confirmed by unfolding the square:

Visualisation is very difficult, so the students need experience with simpler examples, such as:

How many fold lines?
Students may also like to make up simple fold problems for someone else.

Problem Two

Students will need to identify mirror relationships between the digits.

No other mirror relationships exist. This limits the number of two-digit mirror numbers to:

This knowledge can be applied to finding the different three-digit mirror numbers.

Problem Three

A systematic approach will be needed to find all the possibilities. This might begin with all the possibilities involving any green rods:
Six green rods                                                     (6 x 3 = 18)
Three green rods, one yellow rod, one pink rod   (3 x 3) + 5 + 4 = 18
Two green rods, three pink rods                          (2 x 3) + (3 x 4) = 18
One green rod, three yellow rods                         3 + (3 x 5) = 18
Students can then find all the possibilities that can be made without using green rods:
Two yellow rods, two pink rods                          (2 x 5) + (2 x 4) = 18
Students might go on to investigate how many ways a length of 24 centimetres could be made with the three different-coloured rods.

Problem Four

If Henry’s pocket money is removed from the $12 total, that leaves $8 to be shared among the remaining younger children. There are several possible combinations, which can be found in an organised way (see the Answers section).
Note: If Henry is the elder by birth of twins, that opens up other possibilities, such as ages 4, 4, 3, 1 or 4, 4, 2, 2.
Students may enjoy making up their own problems about children’s ages.

Answers to Problems

1. 3
2. a.

b. 25, 11
3. 1 green and 3 yellow
2 green and 3 pink
3 green, 1 yellow, 1 pink
6 green
2 yellow and 2 pink
4. Some possibilities:
$4, $3.50, $3.50, $1
$4, $3.50, $3, $1.50
$4, $3.50, $2.50, $2
$4, $3, $3, $2
$4, $3, $2.50, $2.50
 

Getting Started

1. Fold a circle in half and cut pieces out of it in any way you like. Ask the students to anticipate the pattern that the opened-up shape might have. Write down any mathematical vocabulary that might arise, in particular, terms like symmetry, or reflection. Open the cut circle to confirm the students’ predictions. The shape has reflective or line symmetry along the fold line. 
   
   What pattern can you see in the shape?
   How did the way we made the shape affect the symmetry it has?
   You may want to show how the whole shape is visible if the mirror is located along the fold line. 
   Mirror symmetry and fold symmetry have the same meaning with 2-dimensional shapes.
2. Ask: Why is reflective symmetry sometimes called mirror symmetry, fold symmetry, and flip symmetry?
   It is important for students to see that one half of the shape maps onto the other half by folding and flipping actions. Each alternative term can be demonstrated. Holding a mirror along the fold line enables students to see an image of the whole shape even when one half is masked. Similarly, the amended (cut) circle can be refolded in half and traced around. Then flip the paper over the fold line and trace around it again. The traced figure will be that of the whole amended circle.
3. Ask students to make their own shape by folding a circle in half and cutting out pieces (they must not cut all the way along the fold line). This will enable them to conjecture that all such shapes have at least one line of reflective symmetry (some may have two depending on the cuts). The symmetric shapes may be displayed on a chart.
4. Move onto folding a circle into quarters (in half then in half again) and cutting pieces out. 
   
   Before opening up the circle ask:
   What pattern do you expect the shape to have?
5. Confirm that the two fold lines are lines of reflective symmetry.
   Is there another type of pattern in this shape?
   Students may not recognise that the cutout shape has rotational symmetry as well. Trace around the shape on a whiteboard. Turn the shape one half turn (180⁰) to show that the shape maps onto itself.
   The shape has half turn symmetry. How many times will it map onto itself in a full turn?
   Students should predict order 2 rotational symmetry. That means the shape maps onto itself twice, in a full turn.
6. Ask students to create their own pattern using quarter folds of a circle. Ask them to compare their shape with that of a partner. 
   How are the shapes the same? How are they different?
   Do both shapes have the same symmetry?
   Most students will realise that there are two lines of reflective symmetry (the fold lines) but the half turn rotational symmetry is harder to spot.
7. Ask your students to anticipate then investigate what symmetry the circle shape will have if folded in eighths or sixths before cutting. Sixths can be created by folding the circle in half and then looping the half into thirds (see diagram below). (The circle folded into eighths will have at least four lines of reflection symmetry and rotational symmetry of order 4; with sixths the symmetry is six lines and order 3). Ask the students if they can see a pattern in the "least number" of lines and orders.
   
    
8. Challenge the students to use paper circles to create a shape that has rotational symmetry of order 3 but no lines of reflective symmetry. Next ask them to produce a shape that has rotational symmetry of order 4 but with no lines of reflective symmetry. Then ask for a shape that has rotational symmetry of order 5 with no lines of reflective symmetry, etc.

Session Two: Car logos

1. Begin by showing a short film clip of a car from a popular brand.
   What make of car was that?
   How do you know?
2. Investigate the logos found on motor vehicles by looking at cars in the school car park, magazine advertisements or images from the internet. PowerPoint 1 provides some common car logos. Car advertisements in magazines can also be cut out and used. Most manufacturers use symmetry of some kind in designing their logos. For example, Audi uses four intersecting circles in a line. This pattern has one line of reflection symmetry. This logo is created by translating (shifting) one circle three times.
   
3. Discuss the symmetry of each logo and compile a list of car manufacturing companies for future reference. (You may need to omit the manufacturer’s name from some of the logos to get any symmetry. For instance, removing ‘Ford’ from its logo gives an elliptical shape that has two lines of symmetry.) It is just as important to identify logos that are non-examples of symmetry. For example, the logos for Volvo and Jaguar, and Peugeot have no symmetry, even when the company name is removed.
   
4. Provide the students with drawing instruments such as rulers, protractors, drawing compasses, or jar lids, and tell them to recreate the car logos they saw in the car park, online or in magazines. Have the images available for them to refer to, if needed. At times it may be necessary to bring the class together to discuss construction skills. For example, for a logo involving rotational symmetry of order 3, a protractor will be useful. Since there are 360° in a full turn, one third of that is 120°, which gives the angle measured at the centre for dividing a circle in thirds. Construction skills like drawing a right angle by using a protractor or compass construction may be modelled if necessary.

Session Three: Logos in the media

1. As homework (see Homelink) ask the students to find other examples of logos. Obviously not all logos have symmetry. Sporting goods manufacturers are good examples of this. Nike uses a “swish” that was designed to embody movement. Adidas uses three stripes etc. Examples will illustrate to the students that logos have to be both aesthetically pleasing (i.e. often symmetric) and suggestive of the nature of the company. Share the logos students bring along and group them by symmetry discussing what message is suggested by the logo image. This has strong links to visual language in the English curriculum. For example, the Canterbury Clothing Company has a logo of three translating, overlapping C’s with a kiwi inside them that give the impression of a single ball moving from left to right.
   
2. Set up a matrix for classifying the logos. Create a chart by pasting logos in the appropriate cell. The Canterbury logo belongs in the bottom right cell as it has no reflective or rotational symmetry. It does have translation symmetry.
   
   The Starbucks logo belongs in the bottom left cell as it has reflective symmetry but no rotational symmetry.

3. Set up the following scenario for the students: 
   You work for an advertising company as a logo designer. There are five new companies that need new logos. They have stipulated that the logo must have some symmetry but must also suggest what goods and services they provide. (If you wish, they may also be required to come up with a slogan that captures the message, e.g. "Just do it".)
   Here are the companies:
   • Sweeties - a company that makes sugar-free lollies that taste great and don’t ruin people’s teeth.
   • Gadgets - who make neat construction gadgets (gears, blocks, wheels, etc.), so people can create their own toys.
   • Mana - an after-school club that supports students in learning te reo Māori.
   • Duds - makers of cool clothes especially for primary school children.
   • Hapori māra - a community organisation that specialises in planting native trees
   • Brainbuilders - the people who provide one-on-one tutoring service for students. You get one-on-one help so you are in a class of your own!
4. Give the students sufficient time to design logos for one or more of the companies. They will need to present the logo in a short report to each company that shows what symmetries are involved and how the design suggests the goods or services the company provides. You may decide to set up a voting system for the class to decide on a winning logo for each company. 

Session Four

1. Provide your students with paper copies of logos (Copymaster 1). Display the logos and ask the students to collaborate (mahi tahi) with a partner and write down the symmetry that each design has (this is useful for assessment purposes). Get pairs to have a korero about what they have written with another pair group, then bring the class together for a collective discussion.
2. For each logo get students to demonstrate what symmetry the design has by using a mirror, or folding and flipping, and by tracing and rotating. Make a list of symmetries for each design.
3. Tell the students to look at their first list and add any information they may have missed. If they do this in a new colour you will have evidence of their initial independent understanding and their new shared understanding.
4. If time permits explore how a simple graphic programme, like PowerPoint, can be used to create simple design elements. By copying the element, reflecting or rotating it, then grouping elements together, complex symmetric design can be created.

This activity builds on the ideas of Shifting Shapes (page 13 in the students' book). The students create patterns using single transformations and combinations of transformations.
Kōwhaiwhai are often used as ornamentation on the uncarved heke (rafters) of a wharenui. Unlike carvings, kōwhaiwhai are not normally made by a master artist. However, their intricate, elegant curves require a designer's eye. Today, a cut-out stencil is sometimes used for the repeated design, and the painting is done
as a team project involving young as well as old. All kōwhaiwhai have meanings and are not just ornamental.
You could have your students study the geometry of kōwhaiwhai as a purely mathematical topic but, if possible, you should take them to a building or museum where they can see actual examples. You could ask someone knowledgeable to explain how they were made and what their significance is.
Both questions in this activity involve practical work, and the students may create very different patterns. Some may choose to modify the given designs, while others may try to create something quite different.

Question 2 says that the three basic transformations can be combined to produce more complex patterns. The complete list of transformations used for kōwhaiwhai is as follows:

Auckland Museum produced an excellent educational kit, kōwhaiwhai Tuturu Māori (PDF, 420KB), which gives a background to the history and significance of kōwhaiwhai, examples of kōwhaiwhai, and an illustrated list of the mathematical transformations used. 

•  reflection in a vertical axis
•  rotation of 180 degrees
•  translation
•  glide reflection (translation followed by reflection)
•  rotation of 180 degrees followed by reflection in a vertical axis
•  reflection in a horizontal axis followed by reflection in a vertical axis.
•  reflection in a horizontal axis

Answers to Activity

1.- 2. Practical activities. Results will vary.
 

The Problem

Grace’s kitchen floor is square and is fitted by 64 square tiles in a 8 x 8 array. Grace choses black and white tiles. She can have the tiles laid so that they look like a chessboard but she is hoping for something a bit unusual. The tile man sketches something that has reflective and rotational symmetry. What does he suggest?

Teaching Sequence

1. Introduce the pattern using a chess board (64 squares of alternating colour). Discuss the symmetries of the board.
2. Pose the problem.
3. Brainstorm for ways to solve the problem – (use equipment, draw, guess and check).
4. As the students work on the problem in pairs ask questions that focus their thinking on the symmetries of the pattern.
   How have you used reflection?
   How have you used rotation?
5. After the students have completed the floor pattern ask them to record the symmetries that it contains.
6. Share patterns – display on wall and discuss.

Extension

Grace decides to have the floor tiles laid like a chessboard after all. While redecorating her kitchen, Grace has some cupboards built. Two of these are placed in the opposite corners of the room and take up a whole tile each. (She needs to use 62 square tiles now.)

The tile man says there's a special on. He has a combined tile that consists of a black tile stuck to a white tile. Can Grace tile her floor with these combination tiles and save herself some money?

Solution

There are a large number of possible answers here. Each one can easily be checked to see that it has the right symmetries.

Solution to the Extension

For the extension, colour the squares like a chessboard. When you remove two opposite squares you remove two squares of the same colour. Thus, you have 30 squares left of one colour and 32 of the other. You can’t cover these with the combination tiles as each combination covers one square of each colour.