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GM4-8: Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).

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Elaboration on this Achievement Objective

This means students will know invariant properties are those features of a figure that do not change as it is reflected, rotated, translated or enlarged.

 rotation stars.

Under rotation lengths, areas, angles do not change but orientation does.

 reflection signs.

Under reflection lengths, areas and angles do not change but orientation does.

translation cross.

Under translation lengths, areas, angles and orientation do not change.

enlargement.

Under positive enlargement angles and orientation do not change but lengths and areas do.

At Level Four students should be able to use the above invariant properties to create symmetrical patterns such as tessellations, logos and friezes, and to create enlarged copies of graphics.

 

Teaching resources for this Achievement Objective
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A Model World

Level Four
Integrated
Units of Work
The purpose of this unit is for students to apply and refine their measurement skills and to develop an understanding of enlargement and reduction, through making scale models to be used in a short, animated stop-motion film.
  • Understand and describe proportion using the language of mathematics.
  • Understand the principles of scale.
  • Create scale drawings.
  • Calculate scale dimensions.
  • Make accurate metric length measurements.
  • Understand that the value of the scale factor of a reduction is less than 1 and for an enlargement is300
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Bigger Triangles

Level Four
Geometry and Measurement
Rich learning activities
The purpose of this activity is to engage students in finding a pattern from multiplying dimensions and to generalise this pattern with a rule.
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Christchurch '74

Level Four
Geometry and Measurement
Rich learning activities
The purpose of this activity is to provide students with the opportunity to explore the invariant properties of a shape under transformations, in order to complete a given design and to communicate the transformations necessary to achieve this design.
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Design a Logo

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

design a logo using transformation elements

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Enlargement Explosion

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

enlarge a two dimensional shape using a scale factor

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Escher Envy

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore transformations through tessellations

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Fantastic Fractals

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore fractals

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Golden Spirals

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore fractals

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Growing Up and Down!

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore the properties of shapes under enlargement

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Hexagon Hunt

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

identify attributes of polygons

rotate an object

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Kōwhaiwhai

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

describe patterns using the language of transformation

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More no three in a line

Level Four
Integrated
Problem solving activities
This problem solving activity has a logic and reasoning focus.
  • Create geometric shapes that satisfy the no three in a line condition on 4 by 4 grids.
  • Devise a systematic approach to find possible outcomes.
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Noughts

Level Four
Geometry and Measurement
Problem solving activities
This problem solving activity has a geometry focus.
  • Use symmetry as a game strategy.
  • Apply problem solving strategies to a game context.
  • Devise and use problem solving strategies to explore situations mathematically (guess and check, make a drawing, use equipment).
  • Make conjectures in a mathematical context.
  • Critically follow a chain of reasoning.
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Reptiles

Level Four
Geometry and Measurement
Rich learning activities
The purpose of this activity is to engage students in applying transformations to create repeating tiling patterns.
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Re-Zoom

Level Four
Geometry and Measurement
Picture book activities
This is an activity based on the picture book Re-Zoom
  • Students will be able to express the enlargement relationship between two figures in multiplicative terms.
  • Students will be able to transform an existing image to a specified enlargement.
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Shifting Shapes

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

describe patterns using the language of transformation

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Snazzy Snowflakes

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore rotational symmetry

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Starry-eyed

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series. A PDF of the student activity is included.

explore symmetry in geometric patterns

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Stencil Style

Level Four
Geometry and Measurement
Figure It Out activities
This is a level 4 geometry strand activity from the Figure It Out series.

explore rotational symmetry

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Team Puzzles

Level Four
Geometry and Measurement
Units of Work
This unit requires students to enlarge and reduce two-dimensional mathematical shapes by a scale factor. As students work in co-operative teams of four to complete a puzzle, they use a ruler, compass and protractor.
  • Follow instructions, in diagram form, to construct two-dimensional mathematical shapes, e.g. triangles, quadrilaterals, pentagons and hexagons.
  • Enlarge and reduce two-dimensional mathematical shapes by a given scale factor.
  • Identify invariant properties when enlarging and reducing two300
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Tessellating Art

Level Four
Geometry and Measurement
Units of Work
In this unit we apply our understanding of why tessellations work to form our own unique tessellating shapes. We use these shapes to create interesting pieces of art in the style of M.C. Escher , a famous Dutch artist.
  • Alter polygons to create unique shapes that tessellate.
  • Describe the reflection or rotational symmetry of a shape or tessellation.
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Tessellation, Transformation and Art

Level Four
Geometry and Measurement
Units of Work
In this unit students learn about transformations (translations, rotations, reflections) and tessellations, then apply this knowledge to create a piece of art that creatively demonstrates one or more tessellated transformations.
  • Recognise invariant properties of tessellations.
  • Apply invariant properties to continue and complete given tessellating patterns.
  • Use translations, rotations and reflections to create Escher-type tessellations.
  • Apply knowledge of tessellations to the creation of a piece of art.
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The clumsy tiler A

Level Four
Geometry and Measurement
Problem solving activities
This problem solving activity has a geometry focus.
  • Recognise and invent patterns with reflection symmetry.
  • Devise and use problem solving strategies to explore situations mathematically (make a drawing, use equipment).
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Transformations

Level Four
Geometry and Measurement
Units of Work
This unit explores important ideas about reflective and rotational symmetry.
  • Recognise when one figure is a reflection of another using invariant properties such as perpendicular distance from the mirror line, equalities of lengths, areas and angles, and opposite orientation in relation to the mirror line.
  • Recognise the rotational symmetry of a figure, including identifying300