Transforming shapes

Purpose

This unit applies geometry and measurement in the creation of bags, door stops and other fabric items.  

Achievement Objectives
GM5-4: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders.
GM5-6: Create accurate nets for simple polyhedra and connect three-dimensional solids with different two-dimensional representations.
GM5-9: Define and use transformations and describe the invariant properties of figures and objects under these transformations.
Specific Learning Outcomes
  • Use translations, reflections and rotations to create friezes and Escher type tessellations.
  • Create tivaevae and other symmetric patterns and describe the lines of symmetry and rotational symmetry.
  • Apply knowledge of circles (perimeters, area and surface area) to create nets for cylinders.
Description of Mathematics

The idea of this unit is for students to develop their understanding of the mathematical concepts that underlie the creation of different items out of fabric.  Students will make a geometric design to transfer onto a bag or similar item; they will develop the pattern for a cylindrical item, for example a sports bag, pencil case or pillow; and they will develop the net to create a doorstop.

 

Associated Achievement Objectives:

Technology:

Technological Knowledge

  • Understand how different forms of functional modelling are used to explore possibilities and to justify decision making and how prototyping can be used to justify refinement of technological outcomes.
Activity

Whilst this unit is presented as sequence of nine sessions, more sessions than this may be required. It is also expected that any session may extend beyond one teaching period. 

Session 1

This session introduces the topic and then focuses on exploring line, rotational and total symmetry. This knowledge is applied to the creation of tivaevae inspired designs. It is intended that it would take two to three lessons to complete.

SLOs:

  • Setting the scene for the unit and making the connection between the mathematics and how it will be used in fabric technology.
  • Learning about line symmetry, rotational symmetry and total symmetry of figures.
  • Able to recognise symmetries in everyday symbols and designs.
  • Using line symmetry and rotational symmetry to create designs based on Cook Island’s tivaevae. 

Activity 1

Introduction to the unit.

  • Students will be engaging in three projects throughout the unit.
    1. Creating geometric designs to transfer onto fabric as part of making a bag or for decorating a t-shirt or similar.
    2. Using their knowledge of circles, including surface area of cylinders to create a cylindrical fabric object e.g. sports bag, pencil case, pillow.
    3. Using their knowledge of the nets of pyramids to create a fabric door stop.
  • All projects require using both mathematics and technology knowledge and skills to create the final product.

Introduction to transformations.  The teacher may seek baseline knowledge from students to see what they know about transformations such as translation, rotation and reflection.  This knowledge is built on by exploring line, rotational and total symmetry of figures. 

Line symmetry

A figure has line symmetry if there is a mirror line such that each point in the figure can be reflected into the figure. In the picture below the drawing is symmetric with respect to the line of symmetry, m. 

 
There are a few ways to check if a figure has line symmetry.  For example, you can fold the paper along the assumed line of symmetry.  If each point of the figure on one side of the fold corresponds to the same point on the other side of the fold, then the figure is symmetric with respect to the fold line. Alternatively, you could place a mirror along the assumed line of symmetry.  If the figure and its refection are identical to the original figure, then the figure is symmetric with respect to the mirror line. 

Figures can have more than one line of symmetry.  For example, if exploring regular polygons, an equilateral triangle has three lines of symmetry, a square has four lines of symmetry and a regular n-sides polygon has n lines of symmetry.  For example, the regular hexagon below has six lines of symmetry.

Many trademarks, logos and designs have one or more lines of symmetry. 

Exploring line symmetry in logos: Familiar car brand logos can be explored


The Toyota logo has one line of symmetry as does the VW logo

Toyota logo Volkswagen logo


The Mitsubishi logo has three lines of symmetry. Using the logos provided in Copymaster 1, explore which other logos have line symmetry.  For some of the logos they might have symmetry in parts of the logo, and sometimes without the colour or shading the symmetry is present. 


 
Exploring line symmetry in capital letters: Explore which of the capital letters have vertical symmetry (vertical mirror line), and which have horizontal symmetry (horizontal mirror line).

Vertical symmetry: A, H, I, M, O, T, U, V, W, X, Y
Horizontal symmetry: B, C, D, E, H, I, O, X

At the Vivid Light Festival in Sydney in 2017 one of the light installations used these symmetrical letters to make a display using motivational words, see below. Get students to create their own words that could be used in a similar inspirational light and reflection show. Can they make words with only vertical symmetry letters or with only horizontal symmetry letters? 


   
Rotational symmetry

A figure has rotational symmetry if it can be rotated through less than 360o about a point to coincide with itself. For example, an equilateral triangle can rotate through 120o, 240o and 360o to coincide with itself. This means that an equilateral triangle has rotational symmetry of order three.
Point symmetry is a special case of rotational symmetry.  A figure has point symmetry if a half-turn makes the figure coincide with itself.
Exploring rotational symmetry in logos: A similar activity with the car logos can be done with rotational symmetry.  For example, the Mitsubishi logo has rotational symmetry of order three and the Suzuki logo has rotational symmetry of order two.

Mitsubishi logoSuzuki logo
Other figures and logos can be explored for rotational symmetry. 

Total order of symmetry

By combining the line symmetry and the rotational symmetry the total order of symmetry can be found.  For example:

Symbol Line symmetry Rotational symmetry Total Symmetry

Toyota logo

1 1 2
Volkswagen logo 1 1 2
Mitsubishi logo 3 3 6
Suzuki logo 0 2 2

Activity 2

The following activity is developed from the book The Cook Islands, Patterns of Polynesia by Ailsa Robertson – schools may have this in their library. Another good resource is The Art of Tivaevae by Lynnsay Rongokea.  Understanding about tivaevae and the place they have in the daily and ceremonial life of Cook Islands society is as important as the geometry that we seek as mathematics educators. 

     

This activity works well as a station and could be combined with other similar activities that use the idea of paper folding to create designs with line and rotational symmetry.

TIVAEVAE STATION

SNAZZY SNOWFLAKES

Have students complete the Snazzy Snowflakes Figure it Out activity (Level 4+, Geometry, Book Two, page 20)

AXES OF SYMMETRY 

Copymaster 2 provides another similar activity.

Regardless of the activity chosen to do with your students, it is important that as well as their design and creativity that the mathematics is identified within the design.  Once students have created their designs a copy of the design can be made, and the students can show the lines of symmetry and the order of rotational symmetry on the copy of their design. Below is an example of what students might do to show their understanding of line, rotational and total symmetry.

Session 2

This session explores translations, reflections and rotations within the context of the seven frieze patterns.  This knowledge is applied to the creation of kōwhaiwhai inspired designs. It is intended that it would take two to three lessons to complete. The activities explore translation, horizontal and vertical reflection, half-turn rotation and glide reflections. 

The process within each activity is the same:

  • Define the transformation
  • Discuss how to describe the transformation 
  • Students practice using the transformation(s) to generate a frieze
  • Students create a kōwhaiwhai rafter pattern for the frieze

SLOs:

  • Learning about translation, vertical and horizontal reflection, glide reflections and half turn rotations.
  • Using translation, vertical and horizontal reflection, glide reflections and half turn rotations to generate the seven frieze patterns. 
  • Using translation, vertical and horizontal reflection, glide reflections and half turn rotations to create friezes based on kōwhaiwhai rafter patterns. 

Activity 1

In this activity students are learning about translations.

Information about translations

Translations involve a linear shift or slide of a figure in a plane. 

In the figure above quadrilateral ABCD has been translated to a new position in the plane (A’B’C’D’).  Note that lines AA’, BB’, CC’ and DD’ are all parallel.

We would say quadrilateral ABCD maps to quadrilateral A’B’C’D’ under translation to the right of 6.5cm.

The properties of size, shape and orientation remain invariant (unchanged) under the operation of translation.

Translation allow us to repeat patterns. 

Teacher and student activity using translations

Using pre-prepared pattern blocks (Copymaster 3) teachers can show translation on their whiteboard or similar.  

  • Teacher pattern blocks can be printed onto card and then have magnets put on the back to use on their whiteboard.
  • Note that the teacher pattern block master has pre-reflected pattern blocks to use. 

ABCD maps to A’B’C’D’ under translation 11.5cm to the right.

Then show how a pattern piece (base pattern) can be repeated using translation in a single direction to create a frieze pattern (picture below).
 

  • Students can practice using Copymaster 4 part A using translations only.  
  • Students describe the distance and the direction of the translation on their sheet, marking the sheet to show the translation. 

Creating kōwhaiwhai rafter patterns

In this first activity students will create their kōwhaiwhai base pattern shape most likely based on the koru.  The station below can be used, this is based on ideas from Kōwhaiwhai Geometry of Aotearoa by D F McKenzie. As with the tivaevae understanding about the historical aspects of kōwhaiwhai and the significance they have for local iwi will enhance students’ engagement with the craft. See the Auckland Museum resource kōwhaiwhai Tuturu Māori for background to the history and significance of kōwhaiwhai. 

KŌWHAIWHAI STATION

Note: the stem part of the koru should be even.

Linked activity https://nzmaths.co.nz/resource/k-whaiwhai


In the third part of this activity students design a base template that they can use in subsequent activities to generate their kōwhaiwhai rafter patterns based on each of the seven friezes.  As each activity is completed students will create a kōwhaiwhai rafter pattern for the frieze.  At the end of the activities students pick one of their seven kōwhaiwhai friezes to complete fully with colour and then they notate a copy showing the different transformations involved. 

Students design a base template based on the koru. This should be made from robust card to support heavy use. 

  • Students cut out their base template.
  • Students use the base template to generate a frieze using translation only into their maths book (or similar).
  • Base template is retained for further use as the remaining six friezes are explored.

Activity 2

In this activity students are learning about horizontal and vertical reflections. 

Information about reflections

A reflection is the flipping of points of the plane about a line, called a mirror line.  The properties of size and shape remain invariant (unchanged) under the operation of reflection. In the figure below, the figure is reflected through line m. D’ is the reflected image of D, and D is the reflected image of D’.

Mirror lines are usually labelled with small italic letters. 

Horizontal and vertical reflections

A reflection is described by the direction of the mirror line.  In the figure above the mirror line is vertical, so this would be an example of a vertical reflection. That is D is reflected in the vertical mirror line m. 

Similarly, a horizontal reflection happens when a figure is reflected in a horizontal mirror line.  In the example below D is reflected in the horizontal mirror line. D maps to D’ under reflection in mirror line n.

Teacher and student activity using horizontal and vertical reflections

Using pre-prepared pattern blocks teachers can show the reflections on their whiteboard or similar.  

  • The teacher pattern block master has pre-reflected pattern blocks to use for reflections. 

Horizontal reflection

ABCD maps to A’D’C’B’ under horizontal reflection in mirror line m.

By making the horizontal reflection a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (picture below).

Vertical reflection

ABCD maps to A’D’C’B’ under reflection in the vertical mirror line n.

By making the vertical reflection a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (picture below).

Horizontal and vertical reflection

ABCD maps to A’D’C’B’ under horizontal reflection in mirror line m. This gives a new shape ABB’A’.
ABB’A’ maps to B’’’A’’’A’’B’’ under vertical reflection in mirror line n.

By making the horizontal and vertical reflection a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (picture below).

  • Students can practice using the student worksheet Copymaster 4 part B, C and D using horizontal reflections, using vertical reflections and using horizontal and vertical reflections.  
  • Students describe the reflections on their sheet, marking in the mirror lines and labelling them and showing the distance and direction of the translation on their sheet, marking the sheet to show the translation. 

Creating kōwhaiwhai rafter patterns

  • Students use the base template to generate the three friezes generated by reflections and translation into their maths book (or similar).

Activity 3

In this activity students are learning about half turn rotations. 

Information about rotations

A rotation involves turning a figure in a plane about a given point in the plane, called the centre of rotation.  The properties of size, shape and orientation remain invariant (unchanged) under the operation of rotation. In the picture below, the figure is rotated 90o clockwise about the point P. S has been rotated to a new position in the plane (object S’).  Point P is the centre of rotation.

The centre of rotation may or may not be a point on the figure itself.

Half turn rotations

Half turn rotations are when the amount of the rotation is 180o.  Only half turn rotations are used in frieze patterns.

Teacher and student activity using half turns and vertical reflections

Using pre-prepared pattern blocks teachers can show two further friezes on their whiteboard or similar.  

  • The teacher pattern block master has pre-reflected pattern blocks to use for reflections. 

Half-turn rotation

ABCD maps to A’B’C’D’ under half turn rotation about point P.

By making the half turn rotation a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (picture below).

Half turn rotation and vertical reflection

ABCD maps to A’B’C’D’ under half turn rotation about point P creating figure AD’A’D. AD’A’D maps to D’’A’’’D’’’A’’ under vertical reflection in mirror line n.

By making the half turn rotation and vertical reflection a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (picture below).

  • Students can practice using the student worksheet Copymaster 4 part E and F using half turn rotations and using half turn rotations and vertical reflections.
  • Students describe the rotations on their sheet marking the centre of rotation, describe the reflections on their sheet, marking in the mirror lines and labelling them and showing the distance and direction of the translation on their sheet, marking the sheet to show the translation. 

Creating kōwhaiwhai rafter patterns

Students use the base template to generate the two friezes generated by half turns and vertical reflections and translation into their maths book (or similar).

Activity 4

In this activity students are learning about glide reflections. 

Information about glide reflections

A glide reflection involves both a reflection and a translation along the mirror line in a single operation.  The properties of size and shape remain invariant (unchanged) under the operation of glide reflection. In the picture below, the figure is reflected in the horizontal mirror line m and then translated to the right. 

Teacher and student activity using glide reflections

Using pre-prepared pattern blocks teachers can show the final frieze on their whiteboard or similar.  

The teacher pattern block master has pre-reflected pattern blocks to use for reflections. 

Glide reflection

ABCD maps to A’D’C’B’ under horizontal reflection in mirror line m and translation to the right of 3.7cm.

ABCD maps to A’D’C’B’ under horizontal reflection in mirror line m and translation to the right of 7.5cm.

By making the glide reflection a new “base” pattern is generated. This base pattern is repeated using translation in a single direction to create a frieze pattern (two example pictures below).

  • Students can practice using Copymaster 4 part G using glide reflections.
  • Students describe the glide reflections on their sheet, marking in the mirror line, labelling the mirror line and describing the translation for the glide reflection, they then show the distance and direction of the translation on their sheet, marking the sheet to show the translation. 

Creating kōwhaiwhai rafter patterns

Students use the base template to generate the final frieze generated by glide reflections and translation into their maths book (or similar).

Activity 5

Students select one of the friezes to do a final design.  Students will get to do this one on paper and colour in.

  • The frieze can be drawn onto paper.  Using strips of A3 paper cut longways works well.  
  • Students can draw a border to “contain” their frieze and once completed they can use the traditional colours of red and black to finish the kōwhaiwhai rafter pattern. They need to check that the transformations still hold with the colour. 
  • As with the tivaevae, a copy of the student’s finished frieze can be made and then students can indicate the various transformations in the frieze. 

A follow up activity could be to find examples of kōwhaiwhai online, for example the national library (search national library + kōwhaiwhai), and to get the students to identify the different transformations in the kōwhaiwhai. They should show mirror lines, centres of rotation and the translation of the base pattern. 
They can then identify which of the seven frieze patterns using Dr Gordon Knight’s classification for friezes.  Knight used letters to show the differences between the seven patterns. Use the flow chart below.


 
P – no special properties (translation only)
B – horizontal mirror line
A – vertical mirror line
H – vertical and horizontal mirror lines
Z – half-turn rotation
V – half-turn rotation and vertical mirror line
L – glide reflection 


Session 3

This session explores translations and rotations within the context of Escher-type tessellations.  In the previous sessions students have developed the knowledge about translations, rotations, reflections and symmetry.  In this session this knowledge is applied to the creation of Escher-type tessellation inspired designs. It is intended that it would take one to two lessons to complete.  Given the previous work on transformations in this session students are focused on applying their knowledge.  Note, students could use reflections in their designs, this is a harder skill.  

SLOs:

  • Applying translation and rotations to create Escher-type tessellations.

Activity 1

Tessellations are like frieze patterns.  They both repeat a base pattern using transformations.  Frieze patterns are one dimensional, that is they repeat through translation in one direction. Tessellations repeat in two dimensions, through translation in two directions.

As with frieze patterns a base pattern is created that is then repeated.  The resulting tessellation has different transformations present as with the friezes.
To create a base pattern for an Escher-type tessellation there are some techniques that are used:

  • The “nibble” and slide
  • The “nibble and rotate (at a corner, at the midpoint of a side)
  • Combination of above

With tessellations the design covers the page.  There are no gaps and all the pattern pieces fit together.

Show examples of Escher and Escher-type tessellations. These can be found by searching Escher tessellations. See also Escher Envy and Tessellating Art and the Maths and Art thematic unit.

Activity 2

Teachers can choose to go as deep as they want to with this session.  The main aim is for students to create an Escher-type tessellation that they can add to their design pool of ideas for their fabric project.  The Escher-type tessellations station below would provide enough to create Escher-type tessellations. Alternatively, teachers could work with each of the transformations before the students make their design piece.  See Tessellating Art https://nzmaths.co.nz/resource/tessellating-art  for one possible approach to this. 

ESCHER TYPE TESSELLATIONS STATION

Students create their own tessellating template and use this to cover the page. If possible, they should look to see if an object or animal can be made from their template (see Escher examples). Encourage students to colour in their design, being aware of how colour might affect the transformations present.  Once they have finished make a copy and get students to identify and note the transformations present in their design on the copy. 

Activity 3

Collate the best of each of the three sessions together for each student.  This will provide good evidence for teachers to use for formative or summative assessment.  Students will select one of their designs to take forward into their fabric project. 

Session 4

Students print their completed design onto t-shirt transfer paper to be ironed onto their fabric or t-shirt. An inkjet printer is needed for this. Some students chose to make a bag using their geometric design as a feature strip, others used their design for a pencil case. Allow three to four lessons in the sewing room to complete the project.

SLOs:

  • Students create a fabric project using their selected geometric design.

Examples of student work
  
Bags


  
Hanging picture and cushion cover


  
Pencil cases

Session 5

In this session students are learning about circles and cylinders with a view to making a cylindrical bag, pillow or pencil case. This could take three to four lessons. 

SLOs:

  • Identifying features of circles
  • Finding pi through practical measuring activities and deriving the formula for the circumference.
  • Solving problems involving circumference.
  • Deriving the formula for the area of a circle through a practical activity.
  • Solving problems involving area of circles.
  • Connecting surface area of a cylinder to the net of a cylindrical bag and using this to design a cylindrical bag, pillow or pencil case. 

Activity 1

Introduction to circles and finding the circumference of a circle.  These activities could be done in the sewing room alongside students finishing up their first project.  

Check in with students to confirm the features of circles – circumference, radius and diameter.

Do the following activity to find pi, π, or select your own favourite activity or use one from nzmaths listed below. 

What is the ratio of diameter to circumference?

Place circular items such as jar lids, wastebaskets, hula hoops, cans, circular containers around the room. 
Explain to students that they will be going to each of the items and measuring the diameter and the circumference.  Each group will need string and a ruler or measuring tape. Model how to measure each attribute.
Students should record the measurements for at least 10 objects in a table. 

Object     Diameter of the circle (cm) Circumference of the circle (cm)
     
     
     

Students could make a plot of diameter (x) and circumference (y).  This could be done using CODAP https://codap.concord.org/ by putting the information into a table and then graphing diameter and circumference on the same graph.


 
Students should notice that the points are roughly in a line, which suggests that there is a relationship between the diameter and the circumference of a circle.  This can be explored further by adding a new column and calculating the ratio of circumference to diameter. 

Picture above shows the formula to put in (RHS window) and the column in yellow in the table shows the calculations made. 

nzmaths activities

From any of the activities the formula for finding the circumference of a circle can be developed.

The ratio found in the activity above is called pi and is represented by the symbol π. 

We have π = Circumference ÷ diameter.  If we rearrange this, we get Circumference (C) = π x diameter (d) or C=πd.

We also know that the diameter = 2 x radius so another formula for Circumference is 2 x π x radius (r) or C = 2πr.

C = πd    or    C = 2πr

Appropriate practice activities could follow including a mixture of diameters and radius problems with diameter/radius given or circumference given find diameter/radius.  Also, problems that require measuring diameter/radius to find the circumference. See On the right track https://nzmaths.co.nz/resource/right-track for a practical application of circumference.

Activity 2

Finding the area of a circle. Students should investigate the area formula for circles rather than just be given the formula.  

  • Students could cover a circle with square units and see how many cover a circle. The key to this approach is the students need to get a measure of the radius from the sides of the tiles. See the arithmetic approach in the area of a circle activity.
  • Another approach is to cut the circle into sectors and rearrange them to look more like a parallelogram.  The more sectors the closer the shape is to a parallelogram.  See the conceptual algebra approach in the area of a circle activity. You can use Copymaster 5 for students to cut and paste to make their own parallelogram. 

Students discover the rule for the area of a circle.

A = πr2

Activity 3

Connecting ideas about the net of a cylinder to surface area of a cylinder. Set the scene.  Students will be creating their own cylindrical project.  For example, they might be going to make a sports bag, a pillow or a pencil case.  In order to design the pattern and get the dimensions right what do we need to know?
Have a selection of cylinders to look at.
Describe the surface area, what shapes make up the surface area of a cylinder. If we “unwrapped” the cylinder what would we have?

A cylinder when “unwrapped” has a circle at the top and the bottom and the wrap undoes to become a rectangle. 

How do we know how big to make the rectangle if we have a fixed top/bottom (or end) circles?
H
ow do we know what size to make the end circles if we have a fixed rectangle?

Make the connection that the circumference of the end circle is the same length as the rectangle. 

Circle to rectangle – find the circumference, this is the length (base) of the rectangle.
Rectangle to circle – take the length of the rectangle, this equals the circumference. Circumference ÷ π = diameter of the circle.  Diameter ÷ 2 = radius.  The radius is the compass distance to draw the circle. 

As required students can do activities about surface area, but the main focus is to apply the ideas that connect the rectangle part of the cylinder net to the circle parts of the cylinder net to design their own cylindrical project.  

Students make their own pattern for their cylindrical project, remembering important sewing conventions such as allowing for seams, zips and other items. 
An extension could be to look at the volume of a cylinder.

Session 6

In this session students create their own cylindrical project from the pattern design through the construction of the project.  This will require three to four lessons.

SLOs:

  • Creating a cylindrical bag, pencil case or pillow.

Session 7

Exploring 3D shapes. An initial session that looks at 3D shapes, their properties and nets. This could be two to three lessons. 

SLOs:

  • Working with solids including regular and semi-regular polyhedra. 
  • Creating nets and constructing solids.

Activity 1

Students explore different polyhedra with attention to the common properties of nets for prisms and pyramids. Possible activities include:

Students should create nets and construct solids, so ensure activities chosen include these types of activities. 

Session 8

In this session students are designing nets for their last fabric project.  They will design door stops. This session builds on knowledge from the previous session where students identify which of the different solids they have looked at would be suitable as door stops. 

SLOs:

  • Designing nets for their door stop.
  • Designing nets for pyramids. 

Activity 1

Students to design a door stop that they will create from fabric.  They need to consider the net of their door stop and then how they will adapt it for making from fabric, considering things such as if any of the pieces will be connected and fold up, seam allowances, type of fabric, what they will fill their door stop with. 

Session 9

In this last session students are creating their door stops from fabric and tidying up any of their projects that are still outstanding.  Allow two to three lessons. 

SLOs:

  • Creating door stops that are based on 3D solids. 

Examples of student work
    
Cubes

Pyramids


   


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